%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP280^3 : TPTP v9.2.1. Bugfixed v9.2.1.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n009.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Feb 28 09:16:58 AM UTC 2026
% Result : Theorem 52.96s 13.03s
% Output : Refutation 52.96s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 11
% Syntax : Number of formulae : 54 ( 45 unt; 0 typ; 0 def)
% Number of atoms : 122 ( 60 equ; 0 cnn)
% Maximal formula atoms : 3 ( 2 avg)
% Number of connectives : 720 ( 19 ~; 8 |; 0 &; 685 @)
% ( 0 <=>; 8 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 3 avg)
% Number of types : 60 ( 59 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 883 ( 880 usr; 57 con; 0-8 aty)
% Number of variables : 28 ( 0 ^; 28 !; 0 ?; 28 :)
% 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,
product_prod_int_int: $tType ).
thf(type_def_11,type,
product_prod_nat_nat: $tType ).
thf(type_def_12,type,
num: $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_fi4554929511873752355omplex: $tType ).
thf(type_def_20,type,
filter6041513312241820739omplex: $tType ).
thf(type_def_21,type,
set_fi7789364187291644575l_real: $tType ).
thf(type_def_22,type,
filter2146258269922977983l_real: $tType ).
thf(type_def_23,type,
set_real: $tType ).
thf(type_def_24,type,
set_set_nat: $tType ).
thf(type_def_25,type,
set_nat: $tType ).
thf(type_def_26,type,
set_int: $tType ).
thf(type_def_27,type,
filter_real: $tType ).
thf(type_def_28,type,
filter_int: $tType ).
thf(type_def_29,type,
filter_nat: $tType ).
thf(type_def_30,type,
set_Pr5085853215250843933omplex: $tType ).
thf(type_def_31,type,
set_Pr6218003697084177305l_real: $tType ).
thf(type_def_32,type,
set_o: $tType ).
thf(type_def_33,type,
set_complex: $tType ).
thf(type_def_34,type,
set_list_nat: $tType ).
thf(type_def_35,type,
set_char: $tType ).
thf(type_def_36,type,
list_nat: $tType ).
thf(type_def_37,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(type_def_38,type,
char: $tType ).
thf(type_def_39,type,
set_Pr8693737435421807431at_nat: $tType ).
thf(type_def_40,type,
extended_enat: $tType ).
thf(type_def_41,type,
list_o: $tType ).
thf(type_def_42,type,
list_int: $tType ).
thf(type_def_43,type,
set_Pr958786334691620121nt_int: $tType ).
thf(type_def_44,type,
set_Code_integer: $tType ).
thf(type_def_45,type,
set_rat: $tType ).
thf(type_def_46,type,
list_list_o: $tType ).
thf(type_def_47,type,
list_list_int: $tType ).
thf(type_def_48,type,
list_list_nat: $tType ).
thf(type_def_49,type,
list_list_real: $tType ).
thf(type_def_50,type,
list_real: $tType ).
thf(type_def_51,type,
list_complex: $tType ).
thf(type_def_52,type,
list_set_nat: $tType ).
thf(type_def_53,type,
vEBT_VEBT: $tType ).
thf(type_def_54,type,
list_VEBT_VEBT: $tType ).
thf(type_def_55,type,
set_list_o: $tType ).
thf(type_def_56,type,
set_list_int: $tType ).
thf(type_def_57,type,
set_list_real: $tType ).
thf(type_def_58,type,
set_VEBT_VEBT: $tType ).
thf(type_def_59,type,
option4927543243414619207at_nat: $tType ).
thf(type_def_60,type,
product_prod_nat_num: $tType ).
thf(type_def_61,type,
produc859450856879609959at_nat: $tType ).
thf(type_def_62,type,
produc9072475918466114483BT_nat: $tType ).
thf(type_def_63,type,
produc4411394909380815293omplex: $tType ).
thf(type_def_64,type,
produc2422161461964618553l_real: $tType ).
thf(func_def_0,type,
set_Pr8693737435421807431at_nat: $tType ).
thf(func_def_1,type,
produc859450856879609959at_nat: $tType ).
thf(func_def_2,type,
set_fi4554929511873752355omplex: $tType ).
thf(func_def_3,type,
set_fi7789364187291644575l_real: $tType ).
thf(func_def_4,type,
filter6041513312241820739omplex: $tType ).
thf(func_def_5,type,
set_Pr5085853215250843933omplex: $tType ).
thf(func_def_6,type,
produc8923325533196201883nteger: $tType ).
thf(func_def_7,type,
filter2146258269922977983l_real: $tType ).
thf(func_def_8,type,
option4927543243414619207at_nat: $tType ).
thf(func_def_9,type,
set_Pr6218003697084177305l_real: $tType ).
thf(func_def_10,type,
produc9072475918466114483BT_nat: $tType ).
thf(func_def_11,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(func_def_12,type,
set_Pr958786334691620121nt_int: $tType ).
thf(func_def_13,type,
produc4411394909380815293omplex: $tType ).
thf(func_def_14,type,
produc6271795597528267376eger_o: $tType ).
thf(func_def_15,type,
produc2422161461964618553l_real: $tType ).
thf(func_def_16,type,
product_prod_num_num: $tType ).
thf(func_def_17,type,
product_prod_nat_num: $tType ).
thf(func_def_18,type,
product_prod_nat_nat: $tType ).
thf(func_def_19,type,
product_prod_int_int: $tType ).
thf(func_def_20,type,
list_list_real: $tType ).
thf(func_def_21,type,
set_list_real: $tType ).
thf(func_def_22,type,
list_list_nat: $tType ).
thf(func_def_23,type,
list_list_int: $tType ).
thf(func_def_24,type,
list_VEBT_VEBT: $tType ).
thf(func_def_25,type,
set_list_nat: $tType ).
thf(func_def_26,type,
set_list_int: $tType ).
thf(func_def_27,type,
list_set_nat: $tType ).
thf(func_def_28,type,
set_VEBT_VEBT: $tType ).
thf(func_def_29,type,
set_set_nat: $tType ).
thf(func_def_30,type,
set_Code_integer: $tType ).
thf(func_def_31,type,
list_list_o: $tType ).
thf(func_def_32,type,
list_complex: $tType ).
thf(func_def_33,type,
set_list_o: $tType ).
thf(func_def_34,type,
set_complex: $tType ).
thf(func_def_35,type,
filter_real: $tType ).
thf(func_def_36,type,
option_num: $tType ).
thf(func_def_37,type,
filter_nat: $tType ).
thf(func_def_38,type,
filter_int: $tType ).
thf(func_def_39,type,
set_char: $tType ).
thf(func_def_40,type,
list_real: $tType ).
thf(func_def_41,type,
set_real: $tType ).
thf(func_def_42,type,
list_nat: $tType ).
thf(func_def_43,type,
list_int: $tType ).
thf(func_def_44,type,
vEBT_VEBT: $tType ).
thf(func_def_45,type,
set_rat: $tType ).
thf(func_def_46,type,
set_nat: $tType ).
thf(func_def_47,type,
set_int: $tType ).
thf(func_def_48,type,
code_integer: $tType ).
thf(func_def_49,type,
extended_enat: $tType ).
thf(func_def_50,type,
list_o: $tType ).
thf(func_def_51,type,
complex: $tType ).
thf(func_def_52,type,
set_o: $tType ).
thf(func_def_53,type,
char: $tType ).
thf(func_def_54,type,
real: $tType ).
thf(func_def_55,type,
rat: $tType ).
thf(func_def_56,type,
num: $tType ).
thf(func_def_57,type,
nat: $tType ).
thf(func_def_58,type,
int: $tType ).
thf(func_def_59,type,
archim2889992004027027881ng_rat: rat > int ).
thf(func_def_60,type,
archim7802044766580827645g_real: real > int ).
thf(func_def_61,type,
archim3151403230148437115or_rat: rat > int ).
thf(func_def_62,type,
archim6058952711729229775r_real: real > int ).
thf(func_def_63,type,
archimedean_frac_rat: rat > rat ).
thf(func_def_64,type,
archim2898591450579166408c_real: real > real ).
thf(func_def_65,type,
archim7778729529865785530nd_rat: rat > int ).
thf(func_def_66,type,
archim8280529875227126926d_real: real > int ).
thf(func_def_67,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_68,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_69,type,
bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).
thf(func_def_70,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_71,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_72,type,
bNF_re3023117138289059399t_real: ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ).
thf(func_def_73,type,
bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $o ).
thf(func_def_74,type,
bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).
thf(func_def_75,type,
bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).
thf(func_def_76,type,
bNF_re157797125943740599nt_int: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > product_prod_int_int ) > $o ).
thf(func_def_77,type,
bNF_re3461391660133120880nt_rat: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > rat ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ).
thf(func_def_78,type,
bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_79,type,
bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).
thf(func_def_80,type,
bNF_re6250860962936578807nt_int: ( int > int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ).
thf(func_def_81,type,
bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).
thf(func_def_82,type,
bNF_re4785983289428654063nt_int: ( nat > nat > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( nat > int > int ) > ( nat > int > int ) > $o ).
thf(func_def_83,type,
bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(func_def_84,type,
bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
thf(func_def_85,type,
bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_86,type,
bNF_re6650684261131312217nt_int: ( nat > nat > $o ) > ( int > int > $o ) > ( nat > int ) > ( nat > int ) > $o ).
thf(func_def_87,type,
bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).
thf(func_def_88,type,
bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).
thf(func_def_89,type,
bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).
thf(func_def_90,type,
bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).
thf(func_def_91,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_92,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_93,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_94,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_95,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_96,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_97,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_98,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_99,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_100,type,
bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).
thf(func_def_101,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_102,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_103,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_104,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_105,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_106,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_107,type,
binomial: nat > nat > nat ).
thf(func_def_108,type,
gbinomial_complex: complex > nat > complex ).
thf(func_def_109,type,
gbinomial_int: int > nat > int ).
thf(func_def_110,type,
gbinomial_nat: nat > nat > nat ).
thf(func_def_111,type,
gbinomial_rat: rat > nat > rat ).
thf(func_def_112,type,
gbinomial_real: real > nat > real ).
thf(func_def_113,type,
bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_114,type,
bit_and_not_num: num > num > option_num ).
thf(func_def_115,type,
bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_116,type,
bit_concat_bit: nat > int > int > int ).
thf(func_def_117,type,
bit_or_not_num_neg: num > num > num ).
thf(func_def_118,type,
bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_119,type,
bit_ri7632146776885996613nteger: code_integer > code_integer ).
thf(func_def_120,type,
bit_ri7919022796975470100ot_int: int > int ).
thf(func_def_121,type,
bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).
thf(func_def_122,type,
bit_ri631733984087533419it_int: nat > int > int ).
thf(func_def_123,type,
bit_se3949692690581998587nteger: code_integer > code_integer > code_integer ).
thf(func_def_124,type,
bit_se725231765392027082nd_int: int > int > int ).
thf(func_def_125,type,
bit_se727722235901077358nd_nat: nat > nat > nat ).
thf(func_def_126,type,
bit_se3928097537394005634nteger: nat > code_integer > code_integer ).
thf(func_def_127,type,
bit_se8568078237143864401it_int: nat > int > int ).
thf(func_def_128,type,
bit_se8570568707652914677it_nat: nat > nat > nat ).
thf(func_def_129,type,
bit_se1345352211410354436nteger: nat > code_integer > code_integer ).
thf(func_def_130,type,
bit_se2159334234014336723it_int: nat > int > int ).
thf(func_def_131,type,
bit_se2161824704523386999it_nat: nat > nat > nat ).
thf(func_def_132,type,
bit_se2119862282449309892nteger: nat > code_integer ).
thf(func_def_133,type,
bit_se2000444600071755411sk_int: nat > int ).
thf(func_def_134,type,
bit_se2002935070580805687sk_nat: nat > nat ).
thf(func_def_135,type,
bit_se1080825931792720795nteger: code_integer > code_integer > code_integer ).
thf(func_def_136,type,
bit_se1409905431419307370or_int: int > int > int ).
thf(func_def_137,type,
bit_se1412395901928357646or_nat: nat > nat > nat ).
thf(func_def_138,type,
bit_se7788150548672797655nteger: nat > code_integer > code_integer ).
thf(func_def_139,type,
bit_se545348938243370406it_int: nat > int > int ).
thf(func_def_140,type,
bit_se547839408752420682it_nat: nat > nat > nat ).
thf(func_def_141,type,
bit_se2793503036327961859nteger: nat > code_integer > code_integer ).
thf(func_def_142,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(func_def_143,type,
bit_se7882103937844011126it_nat: nat > nat > nat ).
thf(func_def_144,type,
bit_se1745604003318907178nteger: nat > code_integer > code_integer ).
thf(func_def_145,type,
bit_se2923211474154528505it_int: nat > int > int ).
thf(func_def_146,type,
bit_se2925701944663578781it_nat: nat > nat > nat ).
thf(func_def_147,type,
bit_se8260200283734997820nteger: nat > code_integer > code_integer ).
thf(func_def_148,type,
bit_se4203085406695923979it_int: nat > int > int ).
thf(func_def_149,type,
bit_se4205575877204974255it_nat: nat > nat > nat ).
thf(func_def_150,type,
bit_se3222712562003087583nteger: code_integer > code_integer > code_integer ).
thf(func_def_151,type,
bit_se6526347334894502574or_int: int > int > int ).
thf(func_def_152,type,
bit_se6528837805403552850or_nat: nat > nat > nat ).
thf(func_def_153,type,
bit_se9216721137139052372nteger: code_integer > nat > $o ).
thf(func_def_154,type,
bit_se1146084159140164899it_int: int > nat > $o ).
thf(func_def_155,type,
bit_se1148574629649215175it_nat: nat > nat > $o ).
thf(func_def_156,type,
bit_take_bit_num: nat > num > option_num ).
thf(func_def_157,type,
bit_un1837492267222099188nd_num: num > num > option_num ).
thf(func_def_158,type,
bit_un5425074673868309765um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_159,type,
bit_un6178654185764691216or_num: num > num > option_num ).
thf(func_def_160,type,
bit_un3595099601533988841um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_161,type,
bit_un7362597486090784418nd_num: num > num > option_num ).
thf(func_def_162,type,
bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_163,type,
bit_un2480387367778600638or_num: num > num > option_num ).
thf(func_def_164,type,
bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_165,type,
boolea2445317508997433345nteger: ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer ) > code_integer > code_integer > ( code_integer > code_integer > code_integer ) > $o ).
thf(func_def_166,type,
boolea8527374999097803216ff_int: ( int > int > int ) > ( int > int > int ) > ( int > int ) > int > int > ( int > int > int ) > $o ).
thf(func_def_167,type,
code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).
thf(func_def_168,type,
code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_169,type,
code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_170,type,
code_int_of_integer: code_integer > int ).
thf(func_def_171,type,
code_integer_of_int: int > code_integer ).
thf(func_def_172,type,
code_integer_of_num: num > code_integer ).
thf(func_def_173,type,
code_nat_of_integer: code_integer > nat ).
thf(func_def_174,type,
code_num_of_integer: code_integer > num ).
thf(func_def_175,type,
code_Target_negative: num > int ).
thf(func_def_176,type,
code_Target_positive: num > int ).
thf(func_def_177,type,
code_T6385005292777649522of_nat: nat > int ).
thf(func_def_178,type,
comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).
thf(func_def_179,type,
comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).
thf(func_def_180,type,
comple4887499456419720421f_real: set_real > real ).
thf(func_def_181,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(func_def_182,type,
complete_Sup_Sup_int: set_int > int ).
thf(func_def_183,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(func_def_184,type,
comple1385675409528146559p_real: set_real > real ).
thf(func_def_185,type,
arg: complex > real ).
thf(func_def_186,type,
cis: real > complex ).
thf(func_def_187,type,
cnj: complex > complex ).
thf(func_def_188,type,
complex2: real > real > complex ).
thf(func_def_189,type,
im: complex > real ).
thf(func_def_190,type,
re: complex > real ).
thf(func_def_191,type,
csqrt: complex > complex ).
thf(func_def_192,type,
imaginary_unit: complex ).
thf(func_def_193,type,
rcis: real > real > complex ).
thf(func_def_194,type,
differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).
thf(func_def_195,type,
has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).
thf(func_def_196,type,
has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).
thf(func_def_197,type,
adjust_div: product_prod_int_int > int ).
thf(func_def_198,type,
adjust_mod: int > int > int ).
thf(func_def_199,type,
divmod_nat: nat > nat > product_prod_nat_nat ).
thf(func_def_200,type,
eucl_rel_int: int > int > product_prod_int_int > $o ).
thf(func_def_201,type,
unique5052692396658037445od_int: num > num > product_prod_int_int ).
thf(func_def_202,type,
unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).
thf(func_def_203,type,
unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_204,type,
unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).
thf(func_def_205,type,
unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_206,type,
comm_s8582702949713902594nteger: code_integer > nat > code_integer ).
thf(func_def_207,type,
comm_s2602460028002588243omplex: complex > nat > complex ).
thf(func_def_208,type,
comm_s4660882817536571857er_int: int > nat > int ).
thf(func_def_209,type,
comm_s4663373288045622133er_nat: nat > nat > nat ).
thf(func_def_210,type,
comm_s4028243227959126397er_rat: rat > nat > rat ).
thf(func_def_211,type,
comm_s7457072308508201937r_real: real > nat > real ).
thf(func_def_212,type,
semiri3624122377584611663nteger: nat > code_integer ).
thf(func_def_213,type,
semiri5044797733671781792omplex: nat > complex ).
thf(func_def_214,type,
semiri1406184849735516958ct_int: nat > int ).
thf(func_def_215,type,
semiri1408675320244567234ct_nat: nat > nat ).
thf(func_def_216,type,
semiri773545260158071498ct_rat: nat > rat ).
thf(func_def_217,type,
semiri2265585572941072030t_real: nat > real ).
thf(func_def_218,type,
invers8013647133539491842omplex: complex > complex ).
thf(func_def_219,type,
inverse_inverse_rat: rat > rat ).
thf(func_def_220,type,
inverse_inverse_real: real > real ).
thf(func_def_221,type,
at_bot_real: filter_real ).
thf(func_def_222,type,
at_top_int: filter_int ).
thf(func_def_223,type,
at_top_nat: filter_nat ).
thf(func_def_224,type,
at_top_real: filter_real ).
thf(func_def_225,type,
eventually_nat: ( nat > $o ) > filter_nat > $o ).
thf(func_def_226,type,
eventually_real: ( real > $o ) > filter_real > $o ).
thf(func_def_227,type,
filterlim_nat_int: ( nat > int ) > filter_int > filter_nat > $o ).
thf(func_def_228,type,
filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
thf(func_def_229,type,
filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
thf(func_def_230,type,
filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
thf(func_def_231,type,
filtermap_real_real: ( real > real ) > filter_real > filter_real ).
thf(func_def_232,type,
princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).
thf(func_def_233,type,
princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).
thf(func_def_234,type,
finite_card_o: set_o > nat ).
thf(func_def_235,type,
finite_card_complex: set_complex > nat ).
thf(func_def_236,type,
finite_card_int: set_int > nat ).
thf(func_def_237,type,
finite_card_list_nat: set_list_nat > nat ).
thf(func_def_238,type,
finite_card_nat: set_nat > nat ).
thf(func_def_239,type,
finite_card_set_nat: set_set_nat > nat ).
thf(func_def_240,type,
finite_card_char: set_char > nat ).
thf(func_def_241,type,
finite3207457112153483333omplex: set_complex > $o ).
thf(func_def_242,type,
finite_finite_int: set_int > $o ).
thf(func_def_243,type,
finite_finite_nat: set_nat > $o ).
thf(func_def_244,type,
bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
thf(func_def_245,type,
bij_betw_int_nat: ( int > nat ) > set_int > set_nat > $o ).
thf(func_def_246,type,
bij_be8532844293280997160at_nat: ( list_nat > nat ) > set_list_nat > set_nat > $o ).
thf(func_def_247,type,
bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).
thf(func_def_248,type,
bij_betw_nat_int: ( nat > int ) > set_nat > set_int > $o ).
thf(func_def_249,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(func_def_250,type,
bij_be5333170631980326235at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > set_nat > $o ).
thf(func_def_251,type,
comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_252,type,
comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_253,type,
comp_int_int_num: ( int > int ) > ( num > int ) > num > int ).
thf(func_def_254,type,
comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).
thf(func_def_255,type,
comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(func_def_256,type,
comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).
thf(func_def_257,type,
id_o: $o > $o ).
thf(func_def_258,type,
id_nat: nat > nat ).
thf(func_def_259,type,
inj_on_int_nat: ( int > nat ) > set_int > $o ).
thf(func_def_260,type,
inj_on_list_nat_nat: ( list_nat > nat ) > set_list_nat > $o ).
thf(func_def_261,type,
inj_on_nat_int: ( nat > int ) > set_nat > $o ).
thf(func_def_262,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(func_def_263,type,
inj_on_nat_char: ( nat > char ) > set_nat > $o ).
thf(func_def_264,type,
inj_on2178005380612969504at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_265,type,
inj_on_real_real: ( real > real ) > set_real > $o ).
thf(func_def_266,type,
inj_on_set_nat_nat: ( set_nat > nat ) > set_set_nat > $o ).
thf(func_def_267,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_268,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_269,type,
map_fu4826362097070443709at_o_o: ( int > product_prod_nat_nat ) > ( $o > $o ) > ( product_prod_nat_nat > $o ) > int > $o ).
thf(func_def_270,type,
map_fu2345160673673942751at_nat: ( int > product_prod_nat_nat ) > ( nat > nat ) > ( product_prod_nat_nat > nat ) > int > nat ).
thf(func_def_271,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_272,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_273,type,
map_fu898904425404107465nt_o_o: ( rat > product_prod_int_int ) > ( $o > $o ) > ( product_prod_int_int > $o ) > rat > $o ).
thf(func_def_274,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_275,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_276,type,
map_fu7146612038024189824t_real: ( real > nat > rat ) > ( ( nat > rat ) > real ) > ( ( nat > rat ) > nat > rat ) > real > real ).
thf(func_def_277,type,
map_fu1856342031159181835at_o_o: ( real > nat > rat ) > ( $o > $o ) > ( ( nat > rat ) > $o ) > real > $o ).
thf(func_def_278,type,
the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).
thf(func_def_279,type,
fun_is_measure_int: ( int > nat ) > $o ).
thf(func_def_280,type,
fun_pair_leq: set_Pr8693737435421807431at_nat ).
thf(func_def_281,type,
fun_pair_less: set_Pr8693737435421807431at_nat ).
thf(func_def_282,type,
gcd_Gcd_int: set_int > int ).
thf(func_def_283,type,
gcd_Gcd_nat: set_nat > nat ).
thf(func_def_284,type,
bezw: nat > nat > product_prod_int_int ).
thf(func_def_285,type,
bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_286,type,
gcd_gcd_Code_integer: code_integer > code_integer > code_integer ).
thf(func_def_287,type,
gcd_gcd_int: int > int > int ).
thf(func_def_288,type,
gcd_gcd_nat: nat > nat > nat ).
thf(func_def_289,type,
gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_290,type,
abs_abs_Code_integer: code_integer > code_integer ).
thf(func_def_291,type,
abs_abs_complex: complex > complex ).
thf(func_def_292,type,
abs_abs_int: int > int ).
thf(func_def_293,type,
abs_abs_rat: rat > rat ).
thf(func_def_294,type,
abs_abs_real: real > real ).
thf(func_def_295,type,
minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).
thf(func_def_296,type,
minus_minus_complex: complex > complex > complex ).
thf(func_def_297,type,
minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_298,type,
minus_minus_int: int > int > int ).
thf(func_def_299,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_300,type,
minus_minus_rat: rat > rat > rat ).
thf(func_def_301,type,
minus_minus_real: real > real > real ).
thf(func_def_302,type,
minus_811609699411566653omplex: set_complex > set_complex > set_complex ).
thf(func_def_303,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(func_def_304,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_305,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(func_def_306,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(func_def_307,type,
one_one_Code_integer: code_integer ).
thf(func_def_308,type,
one_one_complex: complex ).
thf(func_def_309,type,
one_on7984719198319812577d_enat: extended_enat ).
thf(func_def_310,type,
one_one_int: int ).
thf(func_def_311,type,
one_one_nat: nat ).
thf(func_def_312,type,
one_one_rat: rat ).
thf(func_def_313,type,
one_one_real: real ).
thf(func_def_314,type,
plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).
thf(func_def_315,type,
plus_plus_complex: complex > complex > complex ).
thf(func_def_316,type,
plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_317,type,
plus_plus_int: int > int > int ).
thf(func_def_318,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_319,type,
plus_plus_num: num > num > num ).
thf(func_def_320,type,
plus_plus_rat: rat > rat > rat ).
thf(func_def_321,type,
plus_plus_real: real > real > real ).
thf(func_def_322,type,
sgn_sgn_Code_integer: code_integer > code_integer ).
thf(func_def_323,type,
sgn_sgn_complex: complex > complex ).
thf(func_def_324,type,
sgn_sgn_int: int > int ).
thf(func_def_325,type,
sgn_sgn_rat: rat > rat ).
thf(func_def_326,type,
sgn_sgn_real: real > real ).
thf(func_def_327,type,
times_3573771949741848930nteger: code_integer > code_integer > code_integer ).
thf(func_def_328,type,
times_times_complex: complex > complex > complex ).
thf(func_def_329,type,
times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_330,type,
times_times_int: int > int > int ).
thf(func_def_331,type,
times_times_nat: nat > nat > nat ).
thf(func_def_332,type,
times_times_num: num > num > num ).
thf(func_def_333,type,
times_times_rat: rat > rat > rat ).
thf(func_def_334,type,
times_times_real: real > real > real ).
thf(func_def_335,type,
uminus1351360451143612070nteger: code_integer > code_integer ).
thf(func_def_336,type,
uminus1482373934393186551omplex: complex > complex ).
thf(func_def_337,type,
uminus_uminus_int: int > int ).
thf(func_def_338,type,
uminus_uminus_rat: rat > rat ).
thf(func_def_339,type,
uminus_uminus_real: real > real ).
thf(func_def_340,type,
uminus8566677241136511917omplex: set_complex > set_complex ).
thf(func_def_341,type,
uminus1532241313380277803et_int: set_int > set_int ).
thf(func_def_342,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(func_def_343,type,
uminus612125837232591019t_real: set_real > set_real ).
thf(func_def_344,type,
uminus613421341184616069et_nat: set_set_nat > set_set_nat ).
thf(func_def_345,type,
zero_z3403309356797280102nteger: code_integer ).
thf(func_def_346,type,
zero_zero_complex: complex ).
thf(func_def_347,type,
zero_z5237406670263579293d_enat: extended_enat ).
thf(func_def_348,type,
zero_zero_int: int ).
thf(func_def_349,type,
zero_zero_nat: nat ).
thf(func_def_350,type,
zero_zero_rat: rat ).
thf(func_def_351,type,
zero_zero_real: real ).
thf(func_def_352,type,
groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).
thf(func_def_353,type,
groups4538972089207619220nt_int: ( int > int ) > set_int > int ).
thf(func_def_354,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_355,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(func_def_356,type,
groups1705073143266064639nt_int: ( int > int ) > set_int > int ).
thf(func_def_357,type,
groups705719431365010083at_int: ( nat > int ) > set_nat > int ).
thf(func_def_358,type,
groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_359,type,
groups3417619833198082522nteger: ( $o > code_integer ) > code_integer > list_o > code_integer ).
thf(func_def_360,type,
groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).
thf(func_def_361,type,
groups9119017779487936845_o_nat: ( $o > nat ) > nat > list_o > nat ).
thf(func_def_362,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(func_def_363,type,
the_int: ( int > $o ) > int ).
thf(func_def_364,type,
the_Pr4378521158711661632nt_int: ( product_prod_int_int > $o ) > product_prod_int_int ).
thf(func_def_365,type,
the_real: ( real > $o ) > real ).
thf(func_def_366,type,
if_int_int: $o > ( int > int ) > ( int > int ) > int > int ).
thf(func_def_367,type,
if_nat5617392847756311170nteger: $o > ( nat > code_integer > code_integer ) > ( nat > code_integer > code_integer ) > nat > code_integer > code_integer ).
thf(func_def_368,type,
if_nat_int_int: $o > ( nat > int > int ) > ( nat > int > int ) > nat > int > int ).
thf(func_def_369,type,
if_nat_nat_nat: $o > ( nat > nat > nat ) > ( nat > nat > nat ) > nat > nat > nat ).
thf(func_def_370,type,
if_nat_rat: $o > ( nat > rat ) > ( nat > rat ) > nat > rat ).
thf(func_def_371,type,
if_Code_integer: $o > code_integer > code_integer > code_integer ).
thf(func_def_372,type,
if_complex: $o > complex > complex > complex ).
thf(func_def_373,type,
if_int: $o > int > int > int ).
thf(func_def_374,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(func_def_375,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(func_def_376,type,
if_nat: $o > nat > nat > nat ).
thf(func_def_377,type,
if_num: $o > num > num > num ).
thf(func_def_378,type,
if_option_num: $o > option_num > option_num > option_num ).
thf(func_def_379,type,
if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).
thf(func_def_380,type,
if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_381,type,
if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).
thf(func_def_382,type,
if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_383,type,
if_rat: $o > rat > rat > rat ).
thf(func_def_384,type,
if_real: $o > real > real > real ).
thf(func_def_385,type,
infini8530281810654367211te_nat: set_nat > nat > nat ).
thf(func_def_386,type,
abs_Integ: product_prod_nat_nat > int ).
thf(func_def_387,type,
rep_Integ: int > product_prod_nat_nat ).
thf(func_def_388,type,
int_ge_less_than: int > set_Pr958786334691620121nt_int ).
thf(func_def_389,type,
int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
thf(func_def_390,type,
intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_391,type,
nat2: int > nat ).
thf(func_def_392,type,
pcr_int: product_prod_nat_nat > int > $o ).
thf(func_def_393,type,
power_int_real: real > int > real ).
thf(func_def_394,type,
ring_11222124179247155820nteger: set_Code_integer ).
thf(func_def_395,type,
ring_1_Ints_complex: set_complex ).
thf(func_def_396,type,
ring_1_Ints_int: set_int ).
thf(func_def_397,type,
ring_1_Ints_rat: set_rat ).
thf(func_def_398,type,
ring_1_Ints_real: set_real ).
thf(func_def_399,type,
ring_18347121197199848620nteger: int > code_integer ).
thf(func_def_400,type,
ring_17405671764205052669omplex: int > complex ).
thf(func_def_401,type,
ring_1_of_int_int: int > int ).
thf(func_def_402,type,
ring_1_of_int_rat: int > rat ).
thf(func_def_403,type,
ring_1_of_int_real: int > real ).
thf(func_def_404,type,
inf_in1870772243966228564d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_405,type,
inf_inf_int: int > int > int ).
thf(func_def_406,type,
inf_inf_nat: nat > nat > nat ).
thf(func_def_407,type,
semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(func_def_408,type,
sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_409,type,
sup_sup_int: int > int > int ).
thf(func_def_410,type,
sup_sup_nat: nat > nat > nat ).
thf(func_def_411,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_412,type,
sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_413,type,
lattic8263393255366662781ax_int: set_int > int ).
thf(func_def_414,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(func_def_415,type,
bfun_nat_real: ( nat > real ) > filter_nat > $o ).
thf(func_def_416,type,
at_infinity_real: filter_real ).
thf(func_def_417,type,
append_int: list_int > list_int > list_int ).
thf(func_def_418,type,
append_nat: list_nat > list_nat > list_nat ).
thf(func_def_419,type,
concat_o: list_list_o > list_o ).
thf(func_def_420,type,
concat_int: list_list_int > list_int ).
thf(func_def_421,type,
concat_nat: list_list_nat > list_nat ).
thf(func_def_422,type,
concat_real: list_list_real > list_real ).
thf(func_def_423,type,
foldr_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).
thf(func_def_424,type,
foldr_real_real: ( real > real > real ) > list_real > real > real ).
thf(func_def_425,type,
last_nat: list_nat > nat ).
thf(func_def_426,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(func_def_427,type,
cons_int: int > list_int > list_int ).
thf(func_def_428,type,
cons_nat: nat > list_nat > list_nat ).
thf(func_def_429,type,
nil_int: list_int ).
thf(func_def_430,type,
nil_nat: list_nat ).
thf(func_def_431,type,
map_complex_real: ( complex > real ) > list_complex > list_real ).
thf(func_def_432,type,
map_int_real: ( int > real ) > list_int > list_real ).
thf(func_def_433,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(func_def_434,type,
map_nat_real: ( nat > real ) > list_nat > list_real ).
thf(func_def_435,type,
map_real_real: ( real > real ) > list_real > list_real ).
thf(func_def_436,type,
map_set_nat_real: ( set_nat > real ) > list_set_nat > list_real ).
thf(func_def_437,type,
map_VEBT_VEBT_nat: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > list_nat ).
thf(func_def_438,type,
map_VEBT_VEBT_real: ( vEBT_VEBT > real ) > list_VEBT_VEBT > list_real ).
thf(func_def_439,type,
set_o2: list_o > set_o ).
thf(func_def_440,type,
set_complex2: list_complex > set_complex ).
thf(func_def_441,type,
set_int2: list_int > set_int ).
thf(func_def_442,type,
set_list_o2: list_list_o > set_list_o ).
thf(func_def_443,type,
set_list_int2: list_list_int > set_list_int ).
thf(func_def_444,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(func_def_445,type,
set_list_real2: list_list_real > set_list_real ).
thf(func_def_446,type,
set_nat2: list_nat > set_nat ).
thf(func_def_447,type,
set_real2: list_real > set_real ).
thf(func_def_448,type,
set_set_nat2: list_set_nat > set_set_nat ).
thf(func_def_449,type,
set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
thf(func_def_450,type,
size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
thf(func_def_451,type,
tl_nat: list_nat > list_nat ).
thf(func_def_452,type,
nth_o: list_o > nat > $o ).
thf(func_def_453,type,
nth_complex: list_complex > nat > complex ).
thf(func_def_454,type,
nth_int: list_int > nat > int ).
thf(func_def_455,type,
nth_nat: list_nat > nat > nat ).
thf(func_def_456,type,
nth_real: list_real > nat > real ).
thf(func_def_457,type,
nth_set_nat: list_set_nat > nat > set_nat ).
thf(func_def_458,type,
nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_459,type,
replicate_complex: nat > complex > list_complex ).
thf(func_def_460,type,
replicate_int: nat > int > list_int ).
thf(func_def_461,type,
replicate_nat: nat > nat > list_nat ).
thf(func_def_462,type,
replicate_real: nat > real > list_real ).
thf(func_def_463,type,
replicate_set_nat: nat > set_nat > list_set_nat ).
thf(func_def_464,type,
replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_465,type,
sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
thf(func_def_466,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(func_def_467,type,
subseqs_o: list_o > list_list_o ).
thf(func_def_468,type,
subseqs_int: list_int > list_list_int ).
thf(func_def_469,type,
subseqs_nat: list_nat > list_list_nat ).
thf(func_def_470,type,
subseqs_real: list_real > list_list_real ).
thf(func_def_471,type,
take_nat: nat > list_nat > list_nat ).
thf(func_def_472,type,
upt: nat > nat > list_nat ).
thf(func_def_473,type,
upto: int > int > list_int ).
thf(func_def_474,type,
upto_aux: int > int > list_int > list_int ).
thf(func_def_475,type,
upto_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_476,type,
suc: nat > nat ).
thf(func_def_477,type,
compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(func_def_478,type,
case_nat_o: $o > ( nat > $o ) > nat > $o ).
thf(func_def_479,type,
case_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(func_def_480,type,
case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
thf(func_def_481,type,
pred: nat > nat ).
thf(func_def_482,type,
semiri3842193898606819883omplex: set_complex ).
thf(func_def_483,type,
semiring_1_Nats_int: set_int ).
thf(func_def_484,type,
semiri4939895301339042750nteger: nat > code_integer ).
thf(func_def_485,type,
semiri8010041392384452111omplex: nat > complex ).
thf(func_def_486,type,
semiri4216267220026989637d_enat: nat > extended_enat ).
thf(func_def_487,type,
semiri1314217659103216013at_int: nat > int ).
thf(func_def_488,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(func_def_489,type,
semiri681578069525770553at_rat: nat > rat ).
thf(func_def_490,type,
semiri5074537144036343181t_real: nat > real ).
thf(func_def_491,type,
size_size_list_o: list_o > nat ).
thf(func_def_492,type,
size_s3451745648224563538omplex: list_complex > nat ).
thf(func_def_493,type,
size_size_list_int: list_int > nat ).
thf(func_def_494,type,
size_s2710708370519433104list_o: list_list_o > nat ).
thf(func_def_495,type,
size_s533118279054570080st_int: list_list_int > nat ).
thf(func_def_496,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(func_def_497,type,
size_s6660260683639930848t_real: list_list_real > nat ).
thf(func_def_498,type,
size_size_list_nat: list_nat > nat ).
thf(func_def_499,type,
size_size_list_real: list_real > nat ).
thf(func_def_500,type,
size_s3254054031482475050et_nat: list_set_nat > nat ).
thf(func_def_501,type,
size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
thf(func_def_502,type,
size_size_num: num > nat ).
thf(func_def_503,type,
size_size_char: char > nat ).
thf(func_def_504,type,
size_size_VEBT_VEBT: vEBT_VEBT > nat ).
thf(func_def_505,type,
nat_int_decode: nat > int ).
thf(func_def_506,type,
nat_int_encode: int > nat ).
thf(func_def_507,type,
nat_list_encode: list_nat > nat ).
thf(func_def_508,type,
nat_list_encode_rel: list_nat > list_nat > $o ).
thf(func_def_509,type,
nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
thf(func_def_510,type,
nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_511,type,
nat_prod_encode: product_prod_nat_nat > nat ).
thf(func_def_512,type,
nat_set_decode: nat > set_nat ).
thf(func_def_513,type,
nat_set_encode: set_nat > nat ).
thf(func_def_514,type,
nat_triangle: nat > nat ).
thf(func_def_515,type,
root: nat > real > real ).
thf(func_def_516,type,
sqrt: real > real ).
thf(func_def_517,type,
bitM: num > num ).
thf(func_def_518,type,
inc: num > num ).
thf(func_def_519,type,
neg_nu8804712462038260780nteger: code_integer > code_integer ).
thf(func_def_520,type,
neg_nu7009210354673126013omplex: complex > complex ).
thf(func_def_521,type,
neg_numeral_dbl_int: int > int ).
thf(func_def_522,type,
neg_numeral_dbl_rat: rat > rat ).
thf(func_def_523,type,
neg_numeral_dbl_real: real > real ).
thf(func_def_524,type,
neg_nu7757733837767384882nteger: code_integer > code_integer ).
thf(func_def_525,type,
neg_nu6511756317524482435omplex: complex > complex ).
thf(func_def_526,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(func_def_527,type,
neg_nu3179335615603231917ec_rat: rat > rat ).
thf(func_def_528,type,
neg_nu6075765906172075777c_real: real > real ).
thf(func_def_529,type,
neg_nu5831290666863070958nteger: code_integer > code_integer ).
thf(func_def_530,type,
neg_nu8557863876264182079omplex: complex > complex ).
thf(func_def_531,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(func_def_532,type,
neg_nu5219082963157363817nc_rat: rat > rat ).
thf(func_def_533,type,
neg_nu8295874005876285629c_real: real > real ).
thf(func_def_534,type,
neg_numeral_sub_int: num > num > int ).
thf(func_def_535,type,
bit0: num > num ).
thf(func_def_536,type,
bit1: num > num ).
thf(func_def_537,type,
one: num ).
thf(func_def_538,type,
case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
thf(func_def_539,type,
size_num: num > nat ).
thf(func_def_540,type,
num_of_nat: nat > num ).
thf(func_def_541,type,
numera6620942414471956472nteger: num > code_integer ).
thf(func_def_542,type,
numera6690914467698888265omplex: num > complex ).
thf(func_def_543,type,
numera1916890842035813515d_enat: num > extended_enat ).
thf(func_def_544,type,
numeral_numeral_int: num > int ).
thf(func_def_545,type,
numeral_numeral_nat: num > nat ).
thf(func_def_546,type,
numeral_numeral_rat: num > rat ).
thf(func_def_547,type,
numeral_numeral_real: num > real ).
thf(func_def_548,type,
pow: num > num > num ).
thf(func_def_549,type,
pred_numeral: num > nat ).
thf(func_def_550,type,
sqr: num > num ).
thf(func_def_551,type,
none_num: option_num ).
thf(func_def_552,type,
none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
thf(func_def_553,type,
some_num: num > option_num ).
thf(func_def_554,type,
some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
thf(func_def_555,type,
case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
thf(func_def_556,type,
case_option_int_num: int > ( num > int ) > option_num > int ).
thf(func_def_557,type,
case_option_num_num: num > ( num > num ) > option_num > num ).
thf(func_def_558,type,
case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).
thf(func_def_559,type,
map_option_num_num: ( num > num ) > option_num > option_num ).
thf(func_def_560,type,
bot_bo4199563552545308370d_enat: extended_enat ).
thf(func_def_561,type,
bot_bot_nat: nat ).
thf(func_def_562,type,
bot_bot_set_o: set_o ).
thf(func_def_563,type,
bot_bot_set_int: set_int ).
thf(func_def_564,type,
bot_bot_set_nat: set_nat ).
thf(func_def_565,type,
bot_bot_set_real: set_real ).
thf(func_def_566,type,
ord_Least_nat: ( nat > $o ) > nat ).
thf(func_def_567,type,
ord_Least_real: ( real > $o ) > real ).
thf(func_def_568,type,
ord_le6747313008572928689nteger: code_integer > code_integer > $o ).
thf(func_def_569,type,
ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
thf(func_def_570,type,
ord_less_int: int > int > $o ).
thf(func_def_571,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_572,type,
ord_less_num: num > num > $o ).
thf(func_def_573,type,
ord_less_rat: rat > rat > $o ).
thf(func_def_574,type,
ord_less_real: real > real > $o ).
thf(func_def_575,type,
ord_less_set_complex: set_complex > set_complex > $o ).
thf(func_def_576,type,
ord_less_set_int: set_int > set_int > $o ).
thf(func_def_577,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(func_def_578,type,
ord_less_set_real: set_real > set_real > $o ).
thf(func_def_579,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(func_def_580,type,
ord_le3102999989581377725nteger: code_integer > code_integer > $o ).
thf(func_def_581,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
thf(func_def_582,type,
ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).
thf(func_def_583,type,
ord_le4104064031414453916r_real: filter_real > filter_real > $o ).
thf(func_def_584,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_585,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_586,type,
ord_less_eq_num: num > num > $o ).
thf(func_def_587,type,
ord_less_eq_rat: rat > rat > $o ).
thf(func_def_588,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_589,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(func_def_590,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(func_def_591,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(func_def_592,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(func_def_593,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(func_def_594,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(func_def_595,type,
ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).
thf(func_def_596,type,
ord_max_Code_integer: code_integer > code_integer > code_integer ).
thf(func_def_597,type,
ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_598,type,
ord_max_int: int > int > int ).
thf(func_def_599,type,
ord_max_nat: nat > nat > nat ).
thf(func_def_600,type,
ord_max_num: num > num > num ).
thf(func_def_601,type,
ord_max_rat: rat > rat > rat ).
thf(func_def_602,type,
ord_max_real: real > real > real ).
thf(func_def_603,type,
ord_max_set_int: set_int > set_int > set_int ).
thf(func_def_604,type,
ord_mi8085742599997312461d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_605,type,
ord_min_int: int > int > int ).
thf(func_def_606,type,
ord_min_nat: nat > nat > nat ).
thf(func_def_607,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(func_def_608,type,
order_mono_nat_nat: ( nat > nat ) > $o ).
thf(func_def_609,type,
order_mono_nat_real: ( nat > real ) > $o ).
thf(func_def_610,type,
order_mono_real_real: ( real > real ) > $o ).
thf(func_def_611,type,
order_5726023648592871131at_nat: ( nat > nat ) > $o ).
thf(func_def_612,type,
order_7092887310737990675l_real: ( real > real ) > $o ).
thf(func_def_613,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(func_def_614,type,
top_top_set_o: set_o ).
thf(func_def_615,type,
top_top_set_int: set_int ).
thf(func_def_616,type,
top_top_set_list_nat: set_list_nat ).
thf(func_def_617,type,
top_top_set_nat: set_nat ).
thf(func_def_618,type,
top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).
thf(func_def_619,type,
top_top_set_real: set_real ).
thf(func_def_620,type,
top_top_set_char: set_char ).
thf(func_def_621,type,
power_8256067586552552935nteger: code_integer > nat > code_integer ).
thf(func_def_622,type,
power_power_complex: complex > nat > complex ).
thf(func_def_623,type,
power_power_int: int > nat > int ).
thf(func_def_624,type,
power_power_nat: nat > nat > nat ).
thf(func_def_625,type,
power_power_rat: rat > nat > rat ).
thf(func_def_626,type,
power_power_real: real > nat > real ).
thf(func_def_627,type,
produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).
thf(func_def_628,type,
produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_629,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(func_def_630,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(func_def_631,type,
product_Pair_nat_num: nat > num > product_prod_nat_num ).
thf(func_def_632,type,
product_Pair_num_num: num > num > product_prod_num_num ).
thf(func_def_633,type,
produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).
thf(func_def_634,type,
produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
thf(func_def_635,type,
produc457027306803732586at_nat: set_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).
thf(func_def_636,type,
produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_637,type,
produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
thf(func_def_638,type,
produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
thf(func_def_639,type,
produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).
thf(func_def_640,type,
produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
thf(func_def_641,type,
produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_642,type,
produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).
thf(func_def_643,type,
produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
thf(func_def_644,type,
produc8211389475949308722nt_int: ( int > int > int ) > product_prod_int_int > int ).
thf(func_def_645,type,
produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
thf(func_def_646,type,
produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_647,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_648,type,
produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
thf(func_def_649,type,
produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
thf(func_def_650,type,
produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_651,type,
produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
thf(func_def_652,type,
produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).
thf(func_def_653,type,
product_fst_int_int: product_prod_int_int > int ).
thf(func_def_654,type,
product_fst_nat_nat: product_prod_nat_nat > nat ).
thf(func_def_655,type,
product_snd_int_int: product_prod_int_int > int ).
thf(func_def_656,type,
product_snd_nat_nat: product_prod_nat_nat > nat ).
thf(func_def_657,type,
abs_Rat: product_prod_int_int > rat ).
thf(func_def_658,type,
fract: int > int > rat ).
thf(func_def_659,type,
frct: product_prod_int_int > rat ).
thf(func_def_660,type,
rep_Rat: rat > product_prod_int_int ).
thf(func_def_661,type,
field_5140801741446780682s_real: set_real ).
thf(func_def_662,type,
field_7254667332652039916t_real: rat > real ).
thf(func_def_663,type,
normalize: product_prod_int_int > product_prod_int_int ).
thf(func_def_664,type,
pcr_rat: product_prod_int_int > rat > $o ).
thf(func_def_665,type,
positive: rat > $o ).
thf(func_def_666,type,
quotient_of: rat > product_prod_int_int ).
thf(func_def_667,type,
ratrel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_668,type,
real2: ( nat > rat ) > real ).
thf(func_def_669,type,
cauchy: ( nat > rat ) > $o ).
thf(func_def_670,type,
cr_real: ( nat > rat ) > real > $o ).
thf(func_def_671,type,
pcr_real: ( nat > rat ) > real > $o ).
thf(func_def_672,type,
positive2: real > $o ).
thf(func_def_673,type,
realrel: ( nat > rat ) > ( nat > rat ) > $o ).
thf(func_def_674,type,
rep_real: real > nat > rat ).
thf(func_def_675,type,
vanishes: ( nat > rat ) > $o ).
thf(func_def_676,type,
real_V2521375963428798218omplex: set_complex ).
thf(func_def_677,type,
real_V470468836141973256s_real: set_real ).
thf(func_def_678,type,
real_V5970128139526366754l_real: ( real > real ) > $o ).
thf(func_def_679,type,
real_V3694042436643373181omplex: complex > complex > real ).
thf(func_def_680,type,
real_V975177566351809787t_real: real > real > real ).
thf(func_def_681,type,
real_V1022390504157884413omplex: complex > real ).
thf(func_def_682,type,
real_V7735802525324610683m_real: real > real ).
thf(func_def_683,type,
real_V4546457046886955230omplex: real > complex ).
thf(func_def_684,type,
real_V1803761363581548252l_real: real > real ).
thf(func_def_685,type,
real_V2046097035970521341omplex: real > complex > complex ).
thf(func_def_686,type,
real_V1485227260804924795R_real: real > real > real ).
thf(func_def_687,type,
algebr932160517623751201me_int: int > int > $o ).
thf(func_def_688,type,
algebr934650988132801477me_nat: nat > nat > $o ).
thf(func_def_689,type,
divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
thf(func_def_690,type,
divide1717551699836669952omplex: complex > complex > complex ).
thf(func_def_691,type,
divide_divide_int: int > int > int ).
thf(func_def_692,type,
divide_divide_nat: nat > nat > nat ).
thf(func_def_693,type,
divide_divide_rat: rat > rat > rat ).
thf(func_def_694,type,
divide_divide_real: real > real > real ).
thf(func_def_695,type,
dvd_dvd_Code_integer: code_integer > code_integer > $o ).
thf(func_def_696,type,
dvd_dvd_complex: complex > complex > $o ).
thf(func_def_697,type,
dvd_dvd_int: int > int > $o ).
thf(func_def_698,type,
dvd_dvd_nat: nat > nat > $o ).
thf(func_def_699,type,
dvd_dvd_rat: rat > rat > $o ).
thf(func_def_700,type,
dvd_dvd_real: real > real > $o ).
thf(func_def_701,type,
modulo364778990260209775nteger: code_integer > code_integer > code_integer ).
thf(func_def_702,type,
modulo_modulo_int: int > int > int ).
thf(func_def_703,type,
modulo_modulo_nat: nat > nat > nat ).
thf(func_def_704,type,
zero_n356916108424825756nteger: $o > code_integer ).
thf(func_def_705,type,
zero_n1201886186963655149omplex: $o > complex ).
thf(func_def_706,type,
zero_n2684676970156552555ol_int: $o > int ).
thf(func_def_707,type,
zero_n2687167440665602831ol_nat: $o > nat ).
thf(func_def_708,type,
zero_n2052037380579107095ol_rat: $o > rat ).
thf(func_def_709,type,
zero_n3304061248610475627l_real: $o > real ).
thf(func_def_710,type,
suminf_real: ( nat > real ) > real ).
thf(func_def_711,type,
summable_real: ( nat > real ) > $o ).
thf(func_def_712,type,
sums_real: ( nat > real ) > real > $o ).
thf(func_def_713,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(func_def_714,type,
collect_int: ( int > $o ) > set_int ).
thf(func_def_715,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(func_def_716,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_717,type,
collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).
thf(func_def_718,type,
collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
thf(func_def_719,type,
collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
thf(func_def_720,type,
collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).
thf(func_def_721,type,
collect_real: ( real > $o ) > set_real ).
thf(func_def_722,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(func_def_723,type,
pow_nat: set_nat > set_set_nat ).
thf(func_def_724,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(func_def_725,type,
image_int_nat: ( int > nat ) > set_int > set_nat ).
thf(func_def_726,type,
image_list_nat_nat: ( list_nat > nat ) > set_list_nat > set_nat ).
thf(func_def_727,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(func_def_728,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_729,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(func_def_730,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(func_def_731,type,
image_nat_char: ( nat > char ) > set_nat > set_char ).
thf(func_def_732,type,
image_2486076414777270412at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > set_nat ).
thf(func_def_733,type,
image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).
thf(func_def_734,type,
image_2178119161166701260l_real: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).
thf(func_def_735,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(func_def_736,type,
image_char_nat: ( char > nat ) > set_char > set_nat ).
thf(func_def_737,type,
insert_o: $o > set_o > set_o ).
thf(func_def_738,type,
insert_int: int > set_int > set_int ).
thf(func_def_739,type,
insert_nat: nat > set_nat > set_nat ).
thf(func_def_740,type,
insert_real: real > set_real > set_real ).
thf(func_def_741,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_742,type,
set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
thf(func_def_743,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(func_def_744,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(func_def_745,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(func_def_746,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(func_def_747,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(func_def_748,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(func_def_749,type,
set_ord_atLeast_real: real > set_real ).
thf(func_def_750,type,
set_ord_atMost_int: int > set_int ).
thf(func_def_751,type,
set_ord_atMost_nat: nat > set_nat ).
thf(func_def_752,type,
set_or6656581121297822940st_int: int > int > set_int ).
thf(func_def_753,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(func_def_754,type,
set_or5832277885323065728an_int: int > int > set_int ).
thf(func_def_755,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(func_def_756,type,
set_or1633881224788618240n_real: real > real > set_real ).
thf(func_def_757,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(func_def_758,type,
set_or5849166863359141190n_real: real > set_real ).
thf(func_def_759,type,
set_ord_lessThan_int: int > set_int ).
thf(func_def_760,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(func_def_761,type,
set_or5984915006950818249n_real: real > set_real ).
thf(func_def_762,type,
ascii_of: char > char ).
thf(func_def_763,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(func_def_764,type,
size_char: char > nat ).
thf(func_def_765,type,
comm_s629917340098488124ar_nat: char > nat ).
thf(func_def_766,type,
integer_of_char: char > code_integer ).
thf(func_def_767,type,
unique3096191561947761185of_nat: nat > char ).
thf(func_def_768,type,
topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
thf(func_def_769,type,
topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
thf(func_def_770,type,
topolo6980174941875973593q_real: ( nat > real ) > $o ).
thf(func_def_771,type,
topolo2177554685111907308n_real: real > set_real > filter_real ).
thf(func_def_772,type,
topolo7531315842566124627t_real: ( nat > real ) > $o ).
thf(func_def_773,type,
topolo2815343760600316023s_real: real > filter_real ).
thf(func_def_774,type,
topolo6517432010174082258omplex: ( nat > complex ) > $o ).
thf(func_def_775,type,
topolo4055970368930404560y_real: ( nat > real ) > $o ).
thf(func_def_776,type,
topolo896644834953643431omplex: filter6041513312241820739omplex ).
thf(func_def_777,type,
topolo1511823702728130853y_real: filter2146258269922977983l_real ).
thf(func_def_778,type,
arccos: real > real ).
thf(func_def_779,type,
arcosh_real: real > real ).
thf(func_def_780,type,
arcsin: real > real ).
thf(func_def_781,type,
arctan: real > real ).
thf(func_def_782,type,
arsinh_real: real > real ).
thf(func_def_783,type,
artanh_real: real > real ).
thf(func_def_784,type,
cos_complex: complex > complex ).
thf(func_def_785,type,
cos_real: real > real ).
thf(func_def_786,type,
cos_coeff: nat > real ).
thf(func_def_787,type,
cosh_complex: complex > complex ).
thf(func_def_788,type,
cosh_real: real > real ).
thf(func_def_789,type,
cot_complex: complex > complex ).
thf(func_def_790,type,
cot_real: real > real ).
thf(func_def_791,type,
exp_complex: complex > complex ).
thf(func_def_792,type,
exp_real: real > real ).
thf(func_def_793,type,
ln_ln_real: real > real ).
thf(func_def_794,type,
log: real > real > real ).
thf(func_def_795,type,
pi: real ).
thf(func_def_796,type,
powr_real: real > real > real ).
thf(func_def_797,type,
sin_complex: complex > complex ).
thf(func_def_798,type,
sin_real: real > real ).
thf(func_def_799,type,
sin_coeff: nat > real ).
thf(func_def_800,type,
sinh_complex: complex > complex ).
thf(func_def_801,type,
sinh_real: real > real ).
thf(func_def_802,type,
tan_complex: complex > complex ).
thf(func_def_803,type,
tan_real: real > real ).
thf(func_def_804,type,
tanh_complex: complex > complex ).
thf(func_def_805,type,
tanh_real: real > real ).
thf(func_def_806,type,
bi_tot896582865486249351at_int: ( product_prod_nat_nat > int > $o ) > $o ).
thf(func_def_807,type,
transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_808,type,
vEBT_Leaf: $o > $o > vEBT_VEBT ).
thf(func_def_809,type,
vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_810,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(func_def_811,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(func_def_812,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(func_def_813,type,
vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
thf(func_def_814,type,
vEBT_VEBT_low: nat > nat > nat ).
thf(func_def_815,type,
vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
thf(func_def_816,type,
vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_817,type,
vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
thf(func_def_818,type,
vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_819,type,
vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
thf(func_def_820,type,
vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_821,type,
vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
thf(func_def_822,type,
vEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(func_def_823,type,
vEBT_vebt_buildup: nat > vEBT_VEBT ).
thf(func_def_824,type,
vEBT_v4011308405150292612up_rel: nat > nat > $o ).
thf(func_def_825,type,
vEBT_V8646137997579335489_i_l_d: nat > nat ).
thf(func_def_826,type,
vEBT_V8346862874174094_d_u_p: nat > nat ).
thf(func_def_827,type,
vEBT_V1247956027447740395_p_rel: nat > nat > $o ).
thf(func_def_828,type,
vEBT_V5144397997797733112_d_rel: nat > nat > $o ).
thf(func_def_829,type,
vEBT_VEBT_cnt: vEBT_VEBT > real ).
thf(func_def_830,type,
vEBT_VEBT_cnt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_831,type,
vEBT_VEBT_space: vEBT_VEBT > nat ).
thf(func_def_832,type,
vEBT_VEBT_space2: vEBT_VEBT > nat ).
thf(func_def_833,type,
vEBT_VEBT_space_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_834,type,
vEBT_VEBT_space_rel2: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_835,type,
accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
thf(func_def_836,type,
accp_nat: ( nat > nat > $o ) > nat > $o ).
thf(func_def_837,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(func_def_838,type,
accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(func_def_839,type,
accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
thf(func_def_840,type,
accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
thf(func_def_841,type,
accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
thf(func_def_842,type,
pred_nat: set_Pr1261947904930325089at_nat ).
thf(func_def_843,type,
fChoice_real: ( real > $o ) > real ).
thf(func_def_844,type,
member_o: $o > set_o > $o ).
thf(func_def_845,type,
member_Code_integer: code_integer > set_Code_integer > $o ).
thf(func_def_846,type,
member_complex: complex > set_complex > $o ).
thf(func_def_847,type,
member_int: int > set_int > $o ).
thf(func_def_848,type,
member_list_o: list_o > set_list_o > $o ).
thf(func_def_849,type,
member_list_int: list_int > set_list_int > $o ).
thf(func_def_850,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(func_def_851,type,
member_list_real: list_real > set_list_real > $o ).
thf(func_def_852,type,
member_nat: nat > set_nat > $o ).
thf(func_def_853,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_854,type,
member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(func_def_855,type,
member_rat: rat > set_rat > $o ).
thf(func_def_856,type,
member_real: real > set_real > $o ).
thf(func_def_857,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(func_def_858,type,
member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
thf(func_def_859,type,
va: nat ).
thf(func_def_878,type,
sP0: nat > vEBT_VEBT > $o ).
thf(func_def_879,type,
sP1: real > real > real > $o ).
thf(func_def_880,type,
sP2: real > real > real > $o ).
thf(func_def_881,type,
sP3: real > real > real > $o ).
thf(func_def_882,type,
sP4: nat > nat > $o ).
thf(func_def_883,type,
sP5: nat > nat > $o ).
thf(func_def_884,type,
sP6: nat > nat > $o ).
thf(func_def_885,type,
sP7: nat > vEBT_VEBT > $o ).
thf(func_def_886,type,
sP8: nat > vEBT_VEBT > $o ).
thf(func_def_887,type,
sP9: vEBT_VEBT > real > $o ).
thf(func_def_888,type,
sK10: ( nat > nat > $o ) > nat ).
thf(func_def_889,type,
sK11: ( nat > nat > $o ) > nat ).
thf(func_def_890,type,
sK12: ( nat > nat > $o ) > nat ).
thf(func_def_891,type,
sK13: ( nat > nat > $o ) > nat ).
thf(func_def_892,type,
sK14: real > real > nat ).
thf(func_def_893,type,
sK15: real > nat ).
thf(func_def_894,type,
sK16: ( nat > real ) > nat ).
thf(func_def_895,type,
sK17: real > real > nat ).
thf(func_def_896,type,
sK18: num > num ).
thf(func_def_897,type,
sK19: num > num ).
thf(func_def_898,type,
sK20: nat > nat ).
thf(func_def_899,type,
sK21: nat > nat ).
thf(func_def_900,type,
sK22: ( nat > $o ) > nat ).
thf(func_def_901,type,
sK23: nat > nat ).
thf(func_def_902,type,
sK24: nat > nat > nat ).
thf(func_def_903,type,
sK25: nat > vEBT_VEBT > nat ).
thf(func_def_904,type,
sK26: ( nat > real ) > nat ).
thf(func_def_905,type,
sK27: nat > nat ).
thf(func_def_906,type,
sK28: real > real > nat ).
thf(func_def_907,type,
sK29: nat > nat > nat ).
thf(func_def_908,type,
sK30: nat > nat > ( nat > $o ) > nat ).
thf(func_def_909,type,
sK31: nat > nat ).
thf(func_def_910,type,
sK32: nat > nat > ( nat > $o ) > nat ).
thf(func_def_911,type,
sK33: ( nat > real ) > nat ).
thf(func_def_912,type,
sK34: ( nat > nat ) > nat ).
thf(func_def_913,type,
sK35: real > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_914,type,
sK36: real > vEBT_VEBT > nat ).
thf(func_def_915,type,
sK37: real > vEBT_VEBT > option4927543243414619207at_nat ).
thf(func_def_916,type,
sK38: real > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_917,type,
sK39: vEBT_VEBT > $o ).
thf(func_def_918,type,
sK40: vEBT_VEBT > $o ).
thf(func_def_919,type,
sK41: ( nat > nat > $o ) > nat ).
thf(func_def_920,type,
sK42: ( nat > nat > $o ) > nat ).
thf(func_def_921,type,
sK43: ( nat > nat > $o ) > nat ).
thf(func_def_922,type,
sK44: ( nat > nat > $o ) > nat ).
thf(func_def_923,type,
sK45: ( nat > $o ) > nat > nat ).
thf(func_def_924,type,
sK46: nat > nat > nat ).
thf(func_def_925,type,
sK47: ( nat > $o ) > nat > nat ).
thf(func_def_926,type,
sK48: ( nat > $o ) > nat > nat ).
thf(func_def_927,type,
sK49: nat > nat ).
thf(func_def_928,type,
sK50: ( nat > nat ) > nat ).
thf(func_def_929,type,
sK51: ( nat > $o ) > nat ).
thf(func_def_930,type,
sK52: ( nat > nat ) > nat ).
thf(func_def_931,type,
sK53: ( nat > $o ) > nat > nat ).
thf(func_def_932,type,
sK54: ( nat > nat ) > nat ).
thf(func_def_933,type,
sK55: ( nat > $o ) > nat ).
thf(func_def_934,type,
sK56: ( nat > nat > $o ) > nat ).
thf(func_def_935,type,
sK57: ( nat > nat > $o ) > nat ).
thf(func_def_936,type,
sK58: ( nat > nat > $o ) > nat ).
thf(func_def_937,type,
sK59: ( nat > nat > $o ) > nat ).
thf(func_def_938,type,
sK60: ( nat > nat > $o ) > nat ).
thf(func_def_939,type,
sK61: nat > nat > nat ).
thf(func_def_940,type,
sK62: nat > nat > nat ).
thf(func_def_941,type,
sK63: nat > vEBT_VEBT > nat ).
thf(func_def_942,type,
sK64: real > nat ).
thf(func_def_943,type,
sK65: real > nat ).
thf(func_def_944,type,
sK66: ( nat > $o ) > nat ).
thf(func_def_945,type,
sK67: vEBT_VEBT > real > vEBT_VEBT ).
thf(func_def_946,type,
sK68: vEBT_VEBT > real > nat ).
thf(func_def_947,type,
sK69: vEBT_VEBT > real > option4927543243414619207at_nat ).
thf(func_def_948,type,
sK70: vEBT_VEBT > real > list_VEBT_VEBT ).
thf(func_def_949,type,
sK71: real > vEBT_VEBT > $o ).
thf(func_def_950,type,
sK72: real > vEBT_VEBT > $o ).
thf(func_def_951,type,
sK73: ( real > real > $o ) > real ).
thf(func_def_952,type,
sK74: ( real > real > $o ) > real ).
thf(func_def_953,type,
sK75: ( real > real > $o ) > real ).
thf(func_def_954,type,
sK76: ( real > real > $o ) > real ).
thf(func_def_955,type,
sK77: ( nat > real ) > nat ).
thf(func_def_957,type,
ph79:
!>[X0: $tType] : X0 ).
thf(f26624,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f26623,f24630]) ).
thf(f24630,plain,
( $true
!= ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( suc @ ( numeral_numeral_nat @ one ) ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( suc @ ( numeral_numeral_nat @ one ) ) ) ) ) ) ) ) ) ),
inference(forward_demodulation,[status(thm)],[f23258,f24379]) ).
thf(f24379,plain,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( numeral_numeral_nat @ one ) ) ),
inference(forward_demodulation,[status(thm)],[f23785,f23657]) ).
thf(f23657,plain,
( ( numeral_numeral_nat @ one )
= ( suc @ ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f23179,f23338]) ).
thf(f23338,plain,
( zero_zero_nat
= ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ),
inference(cnf_transformation,[status(thm)],[f1305]) ).
thf(f1305,axiom,
( zero_zero_nat
= ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_1304_bits__1__div__2) ).
thf(f23179,plain,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ),
inference(cnf_transformation,[status(thm)],[f866]) ).
thf(f866,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ),
file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_865_numeral__1__eq__Suc__0) ).
thf(f23785,plain,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f23400,f23338]) ).
thf(f23400,plain,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ),
inference(cnf_transformation,[status(thm)],[f872]) ).
thf(f872,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_871_numeral__2__eq__2) ).
thf(f23258,plain,
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f22547]) ).
thf(f22547,plain,
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
!= $true ),
inference(flattening,[status(thm)],[f11353]) ).
thf(f11353,plain,
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f11352]) ).
thf(f11352,plain,
~ ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f10258]) ).
thf(f10258,negated_conjecture,
~ ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f10257]) ).
thf(f10257,conjecture,
ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',conj_0) ).
thf(f26623,plain,
( $true
= ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( suc @ ( numeral_numeral_nat @ one ) ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( suc @ ( numeral_numeral_nat @ one ) ) ) ) ) ) ) ) ) ),
inference(backward_demodulation,[status(thm)],[f25106,f26621]) ).
thf(f26621,plain,
! [X0: real,X1: real] :
( ( times_times_real @ X0 @ X1 )
= ( times_times_real @ X1 @ X0 ) ),
inference(forward_demodulation,[status(thm)],[f26597,f24728]) ).
thf(f24728,plain,
! [X0: real] :
( ( times_times_real @ X0 @ ( semiri5074537144036343181t_real @ one_one_nat ) )
= X0 ),
inference(backward_demodulation,[status(thm)],[f23381,f23753]) ).
thf(f23753,plain,
( ( numeral_numeral_real @ one )
= ( semiri5074537144036343181t_real @ one_one_nat ) ),
inference(definition_unfolding,[status(thm)],[f23350,f23173]) ).
thf(f23173,plain,
( ( numeral_numeral_real @ one )
= one_one_real ),
inference(cnf_transformation,[status(thm)],[f2101]) ).
thf(f2101,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ),
file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_2100_numeral__One) ).
thf(f23350,plain,
( one_one_real
= ( semiri5074537144036343181t_real @ one_one_nat ) ),
inference(cnf_transformation,[status(thm)],[f1235]) ).
thf(f1235,axiom,
( one_one_real
= ( semiri5074537144036343181t_real @ one_one_nat ) ),
file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_1234_of__nat__1) ).
thf(f23381,plain,
! [X0: real] :
( ( times_times_real @ X0 @ ( numeral_numeral_real @ one ) )
= X0 ),
inference(cnf_transformation,[status(thm)],[f22630]) ).
thf(f22630,plain,
! [X0: real] :
( ( times_times_real @ X0 @ ( numeral_numeral_real @ one ) )
= X0 ),
inference(rectify,[status(thm)],[f151]) ).
thf(f151,axiom,
! [X5: real] :
( ( times_times_real @ X5 @ ( numeral_numeral_real @ one ) )
= X5 ),
file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_150_mult__numeral__1__right) ).
thf(f26597,plain,
! [X0: real,X1: real] :
( ( times_times_real @ X1 @ X0 )
= ( times_times_real @ X0 @ ( times_times_real @ X1 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ),
inference(superposition,[status(thm)],[f23425,f24728]) ).
thf(f23425,plain,
! [X2: real,X0: real,X1: real] :
( ( times_times_real @ X1 @ ( times_times_real @ X0 @ X2 ) )
= ( times_times_real @ X0 @ ( times_times_real @ X1 @ X2 ) ) ),
inference(cnf_transformation,[status(thm)],[f22555]) ).
thf(f22555,plain,
! [X0: real,X1: real,X2: real] :
( ( times_times_real @ X1 @ ( times_times_real @ X0 @ X2 ) )
= ( times_times_real @ X0 @ ( times_times_real @ X1 @ X2 ) ) ),
inference(rectify,[status(thm)],[f1042]) ).
thf(f1042,axiom,
! [X5: real,X3: real,X36: real] :
( ( times_times_real @ X3 @ ( times_times_real @ X5 @ X36 ) )
= ( times_times_real @ X5 @ ( times_times_real @ X3 @ X36 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_1041_mult_Oleft__commute) ).
thf(f25106,plain,
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( suc @ ( numeral_numeral_nat @ one ) ) ) ) ) ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( suc @ ( numeral_numeral_nat @ one ) ) ) ) ) ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) )
= $true ),
inference(forward_demodulation,[status(thm)],[f25105,f24453]) ).
thf(f24453,plain,
! [X0: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ X0 ) ) @ ( suc @ ( numeral_numeral_nat @ one ) ) )
= ( suc @ ( divide_divide_nat @ X0 @ ( suc @ ( numeral_numeral_nat @ one ) ) ) ) ),
inference(forward_demodulation,[status(thm)],[f23593,f24379]) ).
thf(f23593,plain,
! [X0: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ X0 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ X0 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f22662]) ).
thf(f22662,plain,
! [X0: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ X0 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ X0 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,axiom,
! [X2: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ X2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_5_div2__Suc__Suc) ).
thf(f25105,plain,
( $true
= ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( suc @ ( numeral_numeral_nat @ one ) ) ) ) ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( suc @ ( numeral_numeral_nat @ one ) ) ) ) ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ),
inference(forward_demodulation,[status(thm)],[f25104,f24379]) ).
thf(f25104,plain,
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) )
= $true ),
inference(subsumption_resolution,[status(thm)],[f25103,f24488]) ).
thf(f24488,plain,
( ( dvd_dvd_nat @ ( suc @ ( numeral_numeral_nat @ one ) ) @ ( suc @ ( suc @ va ) ) )
!= $true ),
inference(forward_demodulation,[status(thm)],[f23614,f24379]) ).
thf(f23614,plain,
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f22696]) ).
thf(f22696,plain,
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
!= $true ),
inference(flattening,[status(thm)],[f16640]) ).
thf(f16640,plain,
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f16639]) ).
thf(f16639,plain,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_0_False) ).
thf(f25103,plain,
( ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) )
= $true )
| ( ( dvd_dvd_nat @ ( suc @ ( numeral_numeral_nat @ one ) ) @ ( suc @ ( suc @ va ) ) )
= $true ) ),
inference(forward_demodulation,[status(thm)],[f23941,f24379]) ).
thf(f23941,plain,
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
= $true )
| ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) )
= $true ) ),
inference(equality_resolution,[status(thm)],[f23349]) ).
thf(f23349,plain,
! [X0: nat] :
( ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= X0 )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
= $true )
| ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ X0 ) ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ X0 ) ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f22949]) ).
thf(f22949,plain,
! [X0: nat] :
( ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= X0 )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
= $true )
| ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ X0 ) ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ X0 ) ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) )
= $true ) ),
inference(flattening,[status(thm)],[f22948]) ).
thf(f22948,plain,
! [X0: nat] :
( ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ X0 ) ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ X0 ) ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) )
= $true )
| ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= X0 )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f22688]) ).
thf(f22688,plain,
! [X0: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
!= $true )
=> ( ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X0 )
=> ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ X0 ) ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ X0 ) ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) )
= $true ) ) ),
inference(flattening,[status(thm)],[f13406]) ).
thf(f13406,plain,
! [X0: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
!= $true )
=> ( ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X0 )
=> ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ X0 ) ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ X0 ) ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f13405]) ).
thf(f13405,plain,
! [X0: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
=> ( ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X0 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ X0 ) ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ X0 ) ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
! [X0: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
=> ( ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X0 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ X0 ) ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ X0 ) ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_1__C3_OIH_C_I3_J) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : ITP280^3 : 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 : n009.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
% 0.11/0.29 % CPULimit : 300
% 0.11/0.29 % WCLimit : 300
% 0.11/0.29 % DateTime : Fri Feb 27 14:50:10 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
% 0.16/0.39 ---- Original DTF file ---
% 0.16/0.39 thf(spec,logic,$$dhol).
% 0.16/0.39 %------------------------------------------------------------------------------
% 0.16/0.39 % File : ITP280^3 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.16/0.39 % Domain : Interactive Theorem Proving
% 0.16/0.39 % Problem : Sledgehammer problem VEBT_Space 00398_018442
% 0.16/0.39 % Version : [Des22] axioms.
% 0.16/0.39 % English :
% 0.16/0.39
% 0.16/0.39 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.16/0.39 % : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% 0.16/0.39 % Source : [Des22]
% 0.16/0.39 % Names : 0076_VEBT_Space_00398_018442 [Des22]
% 0.16/0.39
% 0.16/0.39 % Status : Theorem
% 0.16/0.39 % Rating : ? v9.2.1
% 0.16/0.39 % Syntax : Number of formulae : 11117 (6321 unt; 860 typ; 0 def)
% 0.16/0.39 % Number of atoms : 25845 (11910 equ; 0 cnn)
% 0.16/0.39 % Maximal formula atoms : 71 ( 2 avg)
% 0.16/0.39 % Number of connectives : 99808 (2428 ~; 483 |;1427 &;87236 @)
% 0.16/0.39 % ( 0 <=>;8234 =>; 0 <=; 0 <~>)
% 0.16/0.39 % Maximal formula depth : 39 ( 5 avg)
% 0.16/0.39 % Number of types : 60 ( 59 usr)
% 0.16/0.39 % Number of type conns : 2776 (2776 >; 0 *; 0 +; 0 <<)
% 0.16/0.39 % Number of symbols : 804 ( 801 usr; 57 con; 0-8 aty)
% 0.16/0.39 % Number of variables : 22270 (1711 ^;20086 !; 473 ?;22270 :)
% 0.16/0.39 % SPC : TH0_THM_EQU_NAR
% 0.16/0.39
% 0.16/0.39 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.16/0.39 % from the van Emde Boas Trees session in the Archive of Formal
% 0.16/0.39 % proofs -
% 0.16/0.39 % www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 0.16/0.39 % 2022-02-18 17:11:08.768
% 0.16/0.39 % Bugfixes : Added ()s around negated sides of equations.
% 0.16/0.39 %------------------------------------------------------------------------------
% 0.16/0.39 % Could-be-implicit typings (59)
% 0.16/0.39 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,
% 0.16/0.39 set_Pr8693737435421807431at_nat: $tType ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 produc859450856879609959at_nat: $tType ).
% 0.16/0.39
% 0.16/0.39 thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J_J,type,
% 0.16/0.39 set_fi4554929511873752355omplex: $tType ).
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% 0.16/0.39 thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J_J,type,
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% 0.16/0.39
% 0.16/0.39 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
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% 0.16/0.39 produc8923325533196201883nteger: $tType ).
% 0.16/0.39
% 0.16/0.39 thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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% 0.16/0.39 produc9072475918466114483BT_nat: $tType ).
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% 0.16/0.39 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.16/0.39 set_Pr1261947904930325089at_nat: $tType ).
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% 0.16/0.39 produc4411394909380815293omplex: $tType ).
% 0.16/0.39
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% 0.16/0.39
% 0.16/0.39 thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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% 0.16/0.39
% 0.16/0.39 thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
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% 0.16/0.39
% 0.16/0.39 thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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% 0.16/0.39
% 0.16/0.39 thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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% 0.16/0.39 thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
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% 0.16/0.39
% 0.16/0.39 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
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% 0.16/0.39
% 0.16/0.39 thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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% 0.16/0.39
% 0.16/0.39 thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
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% 0.16/0.39
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% 0.16/0.39
% 0.16/0.39 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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% 0.16/0.39 thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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% 0.16/0.39 thf(ty_n_t__Rat__Orat,type,
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% 0.16/0.39 thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
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% 0.16/0.39 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
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% 0.16/0.39 thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
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% 0.16/0.39 thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
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% 0.16/0.39 thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
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% 0.16/0.39 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,
% 0.16/0.39 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 ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 bNF_re895249473297799549at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 bNF_re4695409256820837752l_real: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( real > real > real ) > $o ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 bNF_re4521903465945308077real_o: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > $o ) > ( real > $o ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( real > real > $o ) > $o ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 bNF_re3023117138289059399t_real: ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_Eo_001_Eo,type,
% 0.16/0.39 bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $o ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.16/0.39 bNF_re157797125943740599nt_int: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > product_prod_int_int ) > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Rat__Orat_J,type,
% 0.16/0.39 bNF_re3461391660133120880nt_rat: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > rat ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
% 0.16/0.39 bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
% 0.16/0.39 bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).
% 0.16/0.39
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% 0.16/0.39 bNF_re6250860962936578807nt_int: ( int > int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ).
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% 0.16/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 0.16/0.39 bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.16/0.39 bNF_re4785983289428654063nt_int: ( nat > nat > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( nat > int > int ) > ( nat > int > int ) > $o ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
% 0.16/0.39 bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint_001t__Int__Oint,type,
% 0.16/0.39 bNF_re6650684261131312217nt_int: ( nat > nat > $o ) > ( int > int > $o ) > ( nat > int ) > ( nat > int ) > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.16/0.39 bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 0.16/0.39 bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Int__Oint,type,
% 0.16/0.39 bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 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 ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 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 ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 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 ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 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 ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 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 ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 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 ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 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 ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 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 ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 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 ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 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 ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 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 ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 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 ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 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 ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Binomial_Obinomial,type,
% 0.16/0.39 binomial: nat > nat > nat ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
% 0.16/0.39 gbinomial_complex: complex > nat > complex ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
% 0.16/0.39 gbinomial_int: int > nat > int ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
% 0.16/0.39 gbinomial_nat: nat > nat > nat ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
% 0.16/0.39 gbinomial_rat: rat > nat > rat ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
% 0.16/0.39 gbinomial_real: real > nat > real ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Oand__int__rel,type,
% 0.16/0.39 bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Oand__not__num,type,
% 0.16/0.39 bit_and_not_num: num > num > option_num ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
% 0.16/0.39 bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Oconcat__bit,type,
% 0.16/0.39 bit_concat_bit: nat > int > int > int ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
% 0.16/0.39 bit_or_not_num_neg: num > num > num ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
% 0.16/0.39 bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Code____Numeral__Ointeger,type,
% 0.16/0.39 bit_ri7632146776885996613nteger: code_integer > code_integer ).
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% 0.16/0.39 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
% 0.16/0.39 bit_ri7919022796975470100ot_int: int > int ).
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% 0.16/0.39 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
% 0.16/0.39 bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).
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% 0.16/0.39 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
% 0.16/0.39 bit_se727722235901077358nd_nat: nat > nat > nat ).
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Code____Numeral__Ointeger,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Code____Numeral__Ointeger,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Code____Numeral__Ointeger,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
% 0.16/0.39 bit_se1412395901928357646or_nat: nat > nat > nat ).
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Code____Numeral__Ointeger,type,
% 0.16/0.39 bit_se7788150548672797655nteger: nat > code_integer > code_integer ).
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
% 0.16/0.39 bit_se7882103937844011126it_nat: nat > nat > nat ).
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Code____Numeral__Ointeger,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Code____Numeral__Ointeger,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Code____Numeral__Ointeger,type,
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% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
% 0.16/0.39 bit_se1146084159140164899it_int: int > nat > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
% 0.16/0.39 bit_se1148574629649215175it_nat: nat > nat > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Otake__bit__num,type,
% 0.16/0.39 bit_take_bit_num: nat > num > option_num ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num__rel,type,
% 0.16/0.39 bit_un5425074673868309765um_rel: product_prod_num_num > product_prod_num_num > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num,type,
% 0.16/0.39 bit_un6178654185764691216or_num: num > num > option_num ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num__rel,type,
% 0.16/0.39 bit_un3595099601533988841um_rel: product_prod_num_num > product_prod_num_num > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
% 0.16/0.39 bit_un7362597486090784418nd_num: num > num > option_num ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
% 0.16/0.39 bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
% 0.16/0.39 bit_un2480387367778600638or_num: num > num > option_num ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
% 0.16/0.39 bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o ).
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% 0.16/0.39 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff_001t__Code____Numeral__Ointeger,type,
% 0.16/0.39 boolea2445317508997433345nteger: ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer ) > code_integer > code_integer > ( code_integer > code_integer > code_integer ) > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff_001t__Int__Oint,type,
% 0.16/0.39 boolea8527374999097803216ff_int: ( int > int > int ) > ( int > int > int ) > ( int > int ) > int > int > ( int > int > int ) > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Code__Numeral_Obit__cut__integer,type,
% 0.16/0.39 code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Code__Numeral_Odivmod__abs,type,
% 0.16/0.39 code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Code__Numeral_Odivmod__integer,type,
% 0.16/0.39 code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
% 0.16/0.39 code_int_of_integer: code_integer > int ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
% 0.16/0.39 code_integer_of_int: int > code_integer ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Code__Numeral_Ointeger__of__num,type,
% 0.16/0.39 code_integer_of_num: num > code_integer ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Code__Numeral_Onat__of__integer,type,
% 0.16/0.39 code_nat_of_integer: code_integer > nat ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Code__Numeral_Onum__of__integer,type,
% 0.16/0.39 code_num_of_integer: code_integer > num ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Code__Target__Int_Onegative,type,
% 0.16/0.39 code_Target_negative: num > int ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Code__Target__Int_Opositive,type,
% 0.16/0.39 code_Target_positive: num > int ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Code__Target__Nat_Oint__of__nat,type,
% 0.16/0.39 code_T6385005292777649522of_nat: nat > int ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 0.16/0.39 comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 0.16/0.39 comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Real__Oreal,type,
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% 0.16/0.39 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Complex_OArg,type,
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% 0.16/0.39 thf(sy_c_Complex_Ocnj,type,
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% 0.16/0.39 thf(sy_c_Complex_Ocomplex_OComplex,type,
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% 0.16/0.39 thf(sy_c_Complex_Ocomplex_OIm,type,
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% 0.16/0.39 thf(sy_c_Complex_Oimaginary__unit,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Complex_Orcis,type,
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% 0.16/0.39 thf(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
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% 0.16/0.39 thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
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% 0.16/0.39 thf(sy_c_Divides_Oadjust__div,type,
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% 0.16/0.39 thf(sy_c_Divides_Oadjust__mod,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Divides_Odivmod__nat,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Divides_Oeucl__rel__int,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
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% 0.16/0.39 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
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% 0.16/0.39 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
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% 0.16/0.39 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Nat__Onat,type,
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% 0.16/0.39 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Rat__Orat,type,
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% 0.16/0.39 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
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% 0.16/0.39
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
% 0.16/0.39 semiri2265585572941072030t_real: nat > real ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
% 0.16/0.39 invers8013647133539491842omplex: complex > complex ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
% 0.16/0.39 inverse_inverse_rat: rat > rat ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
% 0.16/0.39 inverse_inverse_real: real > real ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
% 0.16/0.39 at_bot_real: filter_real ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Filter_Oat__top_001t__Int__Oint,type,
% 0.16/0.39 at_top_int: filter_int ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
% 0.16/0.39 at_top_nat: filter_nat ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
% 0.16/0.39 at_top_real: filter_real ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
% 0.16/0.39 eventually_nat: ( nat > $o ) > filter_nat > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
% 0.16/0.39 eventually_real: ( real > $o ) > filter_real > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Int__Oint,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.16/0.39 filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.16/0.39 filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Filter_Ofiltermap_001t__Real__Oreal_001t__Real__Oreal,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 0.16/0.39 princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).
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% 0.16/0.39 thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.16/0.39 princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Finite__Set_Ocard_001_Eo,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
% 0.16/0.39 finite_card_int: set_int > nat ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
% 0.16/0.39 finite_card_list_nat: set_list_nat > nat ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Finite__Set_Ocard_001t__String__Ochar,type,
% 0.16/0.39 finite_card_char: set_char > nat ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
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% 0.16/0.39
% 0.16/0.39 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 0.16/0.39 finite_finite_nat: set_nat > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 0.16/0.39 bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Fun_Obij__betw_001t__Int__Oint_001t__Nat__Onat,type,
% 0.16/0.39 bij_betw_int_nat: ( int > nat ) > set_int > set_nat > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Fun_Obij__betw_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
% 0.16/0.39 bij_be8532844293280997160at_nat: ( list_nat > nat ) > set_list_nat > set_nat > $o ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
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% 0.16/0.39 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Int__Oint,type,
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% 0.16/0.39 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.16/0.39 comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.16/0.39
% 0.16/0.39 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,
% 0.16/0.39 comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.16/0.39
% 0.16/0.39 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Num__Onum,type,
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% 0.16/0.39 thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
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% 0.16/0.39 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Int__Oint,type,
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% 0.16/0.39 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.16/0.39 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
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% 0.16/0.39 thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
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% 0.16/0.39 thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
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% 0.16/0.39 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 ).
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% 0.16/0.40 thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Real__Oreal_J,type,
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% 0.16/0.40 thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
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% 0.16/0.40 thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
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% 0.16/0.40 thf(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
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% 0.16/0.40 thf(sy_c_List_Osubseqs_001t__Real__Oreal,type,
% 0.16/0.40 subseqs_real: list_real > list_list_real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_List_Otake_001t__Nat__Onat,type,
% 0.16/0.40 take_nat: nat > list_nat > list_nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_List_Oupt,type,
% 0.16/0.40 upt: nat > nat > list_nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_List_Oupto,type,
% 0.16/0.40 upto: int > int > list_int ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_List_Oupto__aux,type,
% 0.16/0.40 upto_aux: int > int > list_int > list_int ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_List_Oupto__rel,type,
% 0.16/0.40 upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_OSuc,type,
% 0.16/0.40 suc: nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.16/0.40 compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 0.16/0.40 case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 0.16/0.40 case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.16/0.40 case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Onat_Opred,type,
% 0.16/0.40 pred: nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Complex__Ocomplex,type,
% 0.16/0.40 semiri3842193898606819883omplex: set_complex ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
% 0.16/0.40 semiring_1_Nats_int: set_int ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
% 0.16/0.40 semiri4939895301339042750nteger: nat > code_integer ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
% 0.16/0.40 semiri8010041392384452111omplex: nat > complex ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
% 0.16/0.40 semiri4216267220026989637d_enat: nat > extended_enat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 0.16/0.40 semiri1314217659103216013at_int: nat > int ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 0.16/0.40 semiri1316708129612266289at_nat: nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
% 0.16/0.40 semiri681578069525770553at_rat: nat > rat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 0.16/0.40 semiri5074537144036343181t_real: nat > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
% 0.16/0.40 size_size_list_o: list_o > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 0.16/0.40 size_s3451745648224563538omplex: list_complex > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
% 0.16/0.40 size_size_list_int: list_int > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
% 0.16/0.40 size_s2710708370519433104list_o: list_list_o > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
% 0.16/0.40 size_s533118279054570080st_int: list_list_int > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
% 0.16/0.40 size_s3023201423986296836st_nat: list_list_nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Real__Oreal_J_J,type,
% 0.16/0.40 size_s6660260683639930848t_real: list_list_real > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 0.16/0.40 size_size_list_nat: list_nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
% 0.16/0.40 size_size_list_real: list_real > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.16/0.40 size_s3254054031482475050et_nat: list_set_nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 0.16/0.40 size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
% 0.16/0.40 size_size_num: num > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
% 0.16/0.40 size_size_char: char > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
% 0.16/0.40 size_size_VEBT_VEBT: vEBT_VEBT > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat__Bijection_Oint__decode,type,
% 0.16/0.40 nat_int_decode: nat > int ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat__Bijection_Oint__encode,type,
% 0.16/0.40 nat_int_encode: int > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat__Bijection_Olist__encode,type,
% 0.16/0.40 nat_list_encode: list_nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
% 0.16/0.40 nat_list_encode_rel: list_nat > list_nat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 0.16/0.40 nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 0.16/0.40 nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 0.16/0.40 nat_prod_encode: product_prod_nat_nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat__Bijection_Oset__decode,type,
% 0.16/0.40 nat_set_decode: nat > set_nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat__Bijection_Oset__encode,type,
% 0.16/0.40 nat_set_encode: set_nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Nat__Bijection_Otriangle,type,
% 0.16/0.40 nat_triangle: nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_NthRoot_Oroot,type,
% 0.16/0.40 root: nat > real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_NthRoot_Osqrt,type,
% 0.16/0.40 sqrt: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_OBitM,type,
% 0.16/0.40 bitM: num > num ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Oinc,type,
% 0.16/0.40 inc: num > num ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
% 0.16/0.40 neg_nu8804712462038260780nteger: code_integer > code_integer ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
% 0.16/0.40 neg_nu7009210354673126013omplex: complex > complex ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
% 0.16/0.40 neg_numeral_dbl_int: int > int ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
% 0.16/0.40 neg_numeral_dbl_rat: rat > rat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
% 0.16/0.40 neg_numeral_dbl_real: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
% 0.16/0.40 neg_nu7757733837767384882nteger: code_integer > code_integer ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
% 0.16/0.40 neg_nu6511756317524482435omplex: complex > complex ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
% 0.16/0.40 neg_nu3811975205180677377ec_int: int > int ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
% 0.16/0.40 neg_nu3179335615603231917ec_rat: rat > rat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
% 0.16/0.40 neg_nu6075765906172075777c_real: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
% 0.16/0.40 neg_nu5831290666863070958nteger: code_integer > code_integer ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
% 0.16/0.40 neg_nu8557863876264182079omplex: complex > complex ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
% 0.16/0.40 neg_nu5851722552734809277nc_int: int > int ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
% 0.16/0.40 neg_nu5219082963157363817nc_rat: rat > rat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
% 0.16/0.40 neg_nu8295874005876285629c_real: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
% 0.16/0.40 neg_numeral_sub_int: num > num > int ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Onum_OBit0,type,
% 0.16/0.40 bit0: num > num ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Onum_OBit1,type,
% 0.16/0.40 bit1: num > num ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Onum_OOne,type,
% 0.16/0.40 one: num ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.16/0.40 case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Onum_Osize__num,type,
% 0.16/0.40 size_num: num > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Onum__of__nat,type,
% 0.16/0.40 num_of_nat: nat > num ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
% 0.16/0.40 numera6620942414471956472nteger: num > code_integer ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
% 0.16/0.40 numera6690914467698888265omplex: num > complex ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
% 0.16/0.40 numera1916890842035813515d_enat: num > extended_enat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
% 0.16/0.40 numeral_numeral_int: num > int ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
% 0.16/0.40 numeral_numeral_nat: num > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
% 0.16/0.40 numeral_numeral_rat: num > rat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
% 0.16/0.40 numeral_numeral_real: num > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Opow,type,
% 0.16/0.40 pow: num > num > num ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Opred__numeral,type,
% 0.16/0.40 pred_numeral: num > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Num_Osqr,type,
% 0.16/0.40 sqr: num > num ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
% 0.16/0.40 none_num: option_num ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.16/0.40 none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
% 0.16/0.40 some_num: num > option_num ).
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% 0.16/0.40 thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.16/0.40 some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.16/0.40 case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
% 0.16/0.40 case_option_int_num: int > ( num > int ) > option_num > int ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
% 0.16/0.40 case_option_num_num: num > ( num > num ) > option_num > num ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
% 0.16/0.40 case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
% 0.16/0.40 map_option_num_num: ( num > num ) > option_num > option_num ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
% 0.16/0.40 bot_bo4199563552545308370d_enat: extended_enat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
% 0.16/0.40 bot_bot_nat: nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
% 0.16/0.40 bot_bot_set_o: set_o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
% 0.16/0.40 bot_bot_set_int: set_int ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.16/0.40 bot_bot_set_nat: set_nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.16/0.40 bot_bot_set_real: set_real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
% 0.16/0.40 ord_Least_nat: ( nat > $o ) > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Oord__class_OLeast_001t__Real__Oreal,type,
% 0.16/0.40 ord_Least_real: ( real > $o ) > real ).
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% 0.16/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
% 0.16/0.40 ord_le6747313008572928689nteger: code_integer > code_integer > $o ).
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% 0.16/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
% 0.16/0.40 ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 0.16/0.40 ord_less_int: int > int > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.16/0.40 ord_less_nat: nat > nat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
% 0.16/0.40 ord_less_num: num > num > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
% 0.16/0.40 ord_less_rat: rat > rat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 0.16/0.40 ord_less_real: real > real > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 0.16/0.40 ord_less_set_complex: set_complex > set_complex > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
% 0.16/0.40 ord_less_set_int: set_int > set_int > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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% 0.16/0.40
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% 0.16/0.40 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
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% 0.16/0.40 thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
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% 0.16/0.40 thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
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% 0.16/0.40 thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
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% 0.16/0.40 thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
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% 0.16/0.40 thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.16/0.40 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
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% 0.16/0.40 thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
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% 0.16/0.40 thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.16/0.40
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Rat_Oratrel,type,
% 0.16/0.40 ratrel: product_prod_int_int > product_prod_int_int > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Real_OReal,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Real_Ocauchy,type,
% 0.16/0.40 cauchy: ( nat > rat ) > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Real_Ocr__real,type,
% 0.16/0.40 cr_real: ( nat > rat ) > real > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Real_Opcr__real,type,
% 0.16/0.40 pcr_real: ( nat > rat ) > real > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Real_Opositive,type,
% 0.16/0.40 positive2: real > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Real_Orealrel,type,
% 0.16/0.40 realrel: ( nat > rat ) > ( nat > rat ) > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Real_Orep__real,type,
% 0.16/0.40 rep_real: real > nat > rat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Real_Ovanishes,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Real__Vector__Spaces_OReals_001t__Real__Oreal,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
% 0.16/0.40 algebr932160517623751201me_int: int > int > $o ).
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% 0.16/0.40 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
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% 0.16/0.40 thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
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% 0.16/0.40 thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
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% 0.16/0.40 thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
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% 0.16/0.40 thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
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% 0.16/0.40 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
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% 0.16/0.40 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
% 0.16/0.40 zero_n356916108424825756nteger: $o > code_integer ).
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
% 0.16/0.40 zero_n2684676970156552555ol_int: $o > int ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
% 0.16/0.40 zero_n2687167440665602831ol_nat: $o > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
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% 0.16/0.40 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Series_Osums_001t__Real__Oreal,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Set_OPow_001t__Nat__Onat,type,
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% 0.16/0.40 thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
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% 0.16/0.40 thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
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% 0.16/0.40
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.16/0.40 topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
% 0.16/0.40 topolo6980174941875973593q_real: ( nat > real ) > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
% 0.16/0.40 topolo2177554685111907308n_real: real > set_real > filter_real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent_001t__Real__Oreal,type,
% 0.16/0.40 topolo7531315842566124627t_real: ( nat > real ) > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
% 0.16/0.40 topolo2815343760600316023s_real: real > filter_real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
% 0.16/0.40 topolo6517432010174082258omplex: ( nat > complex ) > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
% 0.16/0.40 topolo4055970368930404560y_real: ( nat > real ) > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
% 0.16/0.40 topolo896644834953643431omplex: filter6041513312241820739omplex ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
% 0.16/0.40 topolo1511823702728130853y_real: filter2146258269922977983l_real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Oarccos,type,
% 0.16/0.40 arccos: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
% 0.16/0.40 arcosh_real: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Oarcsin,type,
% 0.16/0.40 arcsin: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Oarctan,type,
% 0.16/0.40 arctan: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
% 0.16/0.40 arsinh_real: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
% 0.16/0.40 artanh_real: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
% 0.16/0.40 cos_complex: complex > complex ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
% 0.16/0.40 cos_real: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Ocos__coeff,type,
% 0.16/0.40 cos_coeff: nat > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Ocosh_001t__Complex__Ocomplex,type,
% 0.16/0.40 cosh_complex: complex > complex ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
% 0.16/0.40 cosh_real: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Ocot_001t__Complex__Ocomplex,type,
% 0.16/0.40 cot_complex: complex > complex ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
% 0.16/0.40 cot_real: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
% 0.16/0.40 exp_complex: complex > complex ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
% 0.16/0.40 exp_real: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
% 0.16/0.40 ln_ln_real: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Olog,type,
% 0.16/0.40 log: real > real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Opi,type,
% 0.16/0.40 pi: real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 0.16/0.40 powr_real: real > real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
% 0.16/0.40 sin_complex: complex > complex ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
% 0.16/0.40 sin_real: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Osin__coeff,type,
% 0.16/0.40 sin_coeff: nat > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Osinh_001t__Complex__Ocomplex,type,
% 0.16/0.40 sinh_complex: complex > complex ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
% 0.16/0.40 sinh_real: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
% 0.16/0.40 tan_complex: complex > complex ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
% 0.16/0.40 tan_real: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
% 0.16/0.40 tanh_complex: complex > complex ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
% 0.16/0.40 tanh_real: real > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transfer_Obi__total_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 0.16/0.40 bi_tot896582865486249351at_int: ( product_prod_nat_nat > int > $o ) > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
% 0.16/0.40 transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
% 0.16/0.40 vEBT_Leaf: $o > $o > vEBT_VEBT ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
% 0.16/0.40 vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
% 0.16/0.40 vEBT_size_VEBT: vEBT_VEBT > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
% 0.16/0.40 vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
% 0.16/0.40 vEBT_VEBT_high: nat > nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
% 0.16/0.40 vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
% 0.16/0.40 vEBT_VEBT_low: nat > nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
% 0.16/0.40 vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
% 0.16/0.40 vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
% 0.16/0.40 vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
% 0.16/0.40 vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
% 0.16/0.40 vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
% 0.16/0.40 vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
% 0.16/0.40 vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Definitions_Oset__vebt,type,
% 0.16/0.40 vEBT_set_vebt: vEBT_VEBT > set_nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
% 0.16/0.40 vEBT_vebt_buildup: nat > vEBT_VEBT ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
% 0.16/0.40 vEBT_v4011308405150292612up_rel: nat > nat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
% 0.16/0.40 vEBT_V8646137997579335489_i_l_d: nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
% 0.16/0.40 vEBT_V8346862874174094_d_u_p: nat > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
% 0.16/0.40 vEBT_V1247956027447740395_p_rel: nat > nat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
% 0.16/0.40 vEBT_V5144397997797733112_d_rel: nat > nat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
% 0.16/0.40 vEBT_VEBT_cnt: vEBT_VEBT > real ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
% 0.16/0.40 vEBT_VEBT_cnt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
% 0.16/0.40 vEBT_VEBT_space: vEBT_VEBT > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
% 0.16/0.40 vEBT_VEBT_space2: vEBT_VEBT > nat ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
% 0.16/0.40 vEBT_VEBT_space_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
% 0.16/0.40 vEBT_VEBT_space_rel2: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
% 0.16/0.40 accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
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% 0.16/0.40 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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% 0.16/0.40
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
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% 0.16/0.40
% 0.16/0.40 thf(sy_c_Wellfounded_Opred__nat,type,
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% 0.16/0.40 thf(sy_c_member_001t__Int__Oint,type,
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% 0.16/0.40 thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
% 0.16/0.40 member_list_int: list_int > set_list_int > $o ).
% 0.16/0.40
% 0.16/0.40 thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
% 0.16/0.40 member_list_nat: list_nat > set_list_nat > $o ).
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% 0.16/0.40 thf(sy_c_member_001t__List__Olist_It__Real__Oreal_J,type,
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% 0.16/0.40 thf(sy_c_member_001t__Nat__Onat,type,
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% 0.16/0.40 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
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% 0.16/0.40 thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
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% 0.16/0.40 thf(sy_v_va____,type,
% 0.16/0.40 va: nat ).
% 0.16/0.40
% 0.16/0.40 % Relevant facts (10216)
% 0.16/0.40 thf(fact_0_False,axiom,
% 0.16/0.40 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % False
% 0.16/0.40 thf(fact_1__C3_OIH_C_I3_J,axiom,
% 0.16/0.40 ! [X: nat] :
% 0.16/0.40 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
% 0.16/0.40 => ( ( X
% 0.16/0.40 = ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.40 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ X ) ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ X ) ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % "3.IH"(3)
% 0.16/0.40 thf(fact_2__C3_OIH_C_I4_J,axiom,
% 0.16/0.40 ! [X: nat] :
% 0.16/0.40 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
% 0.16/0.40 => ( ( X
% 0.16/0.40 = ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.40 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ X ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ X ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % "3.IH"(4)
% 0.16/0.40 thf(fact_3__C3_OIH_C_I1_J,axiom,
% 0.16/0.40 ! [X: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
% 0.16/0.40 => ( ( X
% 0.16/0.40 = ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.40 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ X ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ X ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % "3.IH"(1)
% 0.16/0.40 thf(fact_4_space__cnt,axiom,
% 0.16/0.40 ! [T: vEBT_VEBT] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_space2 @ T ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_cnt @ T ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % space_cnt
% 0.16/0.40 thf(fact_5_div2__Suc__Suc,axiom,
% 0.16/0.40 ! [M: nat] :
% 0.16/0.40 ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.40 = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div2_Suc_Suc
% 0.16/0.40 thf(fact_6_divide__le__eq__numeral1_I1_J,axiom,
% 0.16/0.40 ! [B: real,W: num,A: real] :
% 0.16/0.40 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 0.16/0.40 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % divide_le_eq_numeral1(1)
% 0.16/0.40 thf(fact_7_divide__le__eq__numeral1_I1_J,axiom,
% 0.16/0.40 ! [B: rat,W: num,A: rat] :
% 0.16/0.40 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 0.16/0.40 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % divide_le_eq_numeral1(1)
% 0.16/0.40 thf(fact_8_le__divide__eq__numeral1_I1_J,axiom,
% 0.16/0.40 ! [A: real,B: real,W: num] :
% 0.16/0.40 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 0.16/0.40 = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 0.16/0.40
% 0.16/0.40 % le_divide_eq_numeral1(1)
% 0.16/0.40 thf(fact_9_le__divide__eq__numeral1_I1_J,axiom,
% 0.16/0.40 ! [A: rat,B: rat,W: num] :
% 0.16/0.40 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 0.16/0.40 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 0.16/0.40
% 0.16/0.40 % le_divide_eq_numeral1(1)
% 0.16/0.40 thf(fact_10_of__nat__numeral,axiom,
% 0.16/0.40 ! [N: num] :
% 0.16/0.40 ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
% 0.16/0.40 = ( numera6690914467698888265omplex @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_numeral
% 0.16/0.40 thf(fact_11_of__nat__numeral,axiom,
% 0.16/0.40 ! [N: num] :
% 0.16/0.40 ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
% 0.16/0.40 = ( numeral_numeral_real @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_numeral
% 0.16/0.40 thf(fact_12_of__nat__numeral,axiom,
% 0.16/0.40 ! [N: num] :
% 0.16/0.40 ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
% 0.16/0.40 = ( numeral_numeral_rat @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_numeral
% 0.16/0.40 thf(fact_13_of__nat__numeral,axiom,
% 0.16/0.40 ! [N: num] :
% 0.16/0.40 ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
% 0.16/0.40 = ( numeral_numeral_nat @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_numeral
% 0.16/0.40 thf(fact_14_of__nat__numeral,axiom,
% 0.16/0.40 ! [N: num] :
% 0.16/0.40 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 0.16/0.40 = ( numeral_numeral_int @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_numeral
% 0.16/0.40 thf(fact_15_semiring__norm_I86_J,axiom,
% 0.16/0.40 ! [M: num] :
% 0.16/0.40 ( ( bit1 @ M )
% 0.16/0.40 != one ) ).
% 0.16/0.40
% 0.16/0.40 % semiring_norm(86)
% 0.16/0.40 thf(fact_16_semiring__norm_I84_J,axiom,
% 0.16/0.40 ! [N: num] :
% 0.16/0.40 ( one
% 0.16/0.40 != ( bit1 @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % semiring_norm(84)
% 0.16/0.40 thf(fact_17_semiring__norm_I89_J,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( bit1 @ M )
% 0.16/0.40 != ( bit0 @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % semiring_norm(89)
% 0.16/0.40 thf(fact_18_semiring__norm_I88_J,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( bit0 @ M )
% 0.16/0.40 != ( bit1 @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % semiring_norm(88)
% 0.16/0.40 thf(fact_19_of__nat__mult,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
% 0.16/0.40 = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_mult
% 0.16/0.40 thf(fact_20_of__nat__mult,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
% 0.16/0.40 = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_mult
% 0.16/0.40 thf(fact_21_of__nat__mult,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
% 0.16/0.40 = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_mult
% 0.16/0.40 thf(fact_22_of__nat__mult,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
% 0.16/0.40 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_mult
% 0.16/0.40 thf(fact_23_of__nat__mult,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
% 0.16/0.40 = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_mult
% 0.16/0.40 thf(fact_24_numeral__Bit1__div__2,axiom,
% 0.16/0.40 ! [N: num] :
% 0.16/0.40 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.40 = ( numeral_numeral_nat @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % numeral_Bit1_div_2
% 0.16/0.40 thf(fact_25_numeral__Bit1__div__2,axiom,
% 0.16/0.40 ! [N: num] :
% 0.16/0.40 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.16/0.40 = ( numeral_numeral_int @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % numeral_Bit1_div_2
% 0.16/0.40 thf(fact_26_of__nat__le__iff,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.16/0.40 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_le_iff
% 0.16/0.40 thf(fact_27_of__nat__le__iff,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.16/0.40 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_le_iff
% 0.16/0.40 thf(fact_28_of__nat__le__iff,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.16/0.40 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_le_iff
% 0.16/0.40 thf(fact_29_of__nat__le__iff,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.16/0.40 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_le_iff
% 0.16/0.40 thf(fact_30_numeral__eq__iff,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( ( numera6690914467698888265omplex @ M )
% 0.16/0.40 = ( numera6690914467698888265omplex @ N ) )
% 0.16/0.40 = ( M = N ) ) ).
% 0.16/0.40
% 0.16/0.40 % numeral_eq_iff
% 0.16/0.40 thf(fact_31_numeral__eq__iff,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( ( numeral_numeral_real @ M )
% 0.16/0.40 = ( numeral_numeral_real @ N ) )
% 0.16/0.40 = ( M = N ) ) ).
% 0.16/0.40
% 0.16/0.40 % numeral_eq_iff
% 0.16/0.40 thf(fact_32_numeral__eq__iff,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( ( numeral_numeral_rat @ M )
% 0.16/0.40 = ( numeral_numeral_rat @ N ) )
% 0.16/0.40 = ( M = N ) ) ).
% 0.16/0.40
% 0.16/0.40 % numeral_eq_iff
% 0.16/0.40 thf(fact_33_numeral__eq__iff,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( ( numeral_numeral_nat @ M )
% 0.16/0.40 = ( numeral_numeral_nat @ N ) )
% 0.16/0.40 = ( M = N ) ) ).
% 0.16/0.40
% 0.16/0.40 % numeral_eq_iff
% 0.16/0.40 thf(fact_34_numeral__eq__iff,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( ( numeral_numeral_int @ M )
% 0.16/0.40 = ( numeral_numeral_int @ N ) )
% 0.16/0.40 = ( M = N ) ) ).
% 0.16/0.40
% 0.16/0.40 % numeral_eq_iff
% 0.16/0.40 thf(fact_35_semiring__norm_I13_J,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.16/0.40 = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % semiring_norm(13)
% 0.16/0.40 thf(fact_36_semiring__norm_I71_J,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.16/0.40 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % semiring_norm(71)
% 0.16/0.40 thf(fact_37_semiring__norm_I87_J,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( ( bit0 @ M )
% 0.16/0.40 = ( bit0 @ N ) )
% 0.16/0.40 = ( M = N ) ) ).
% 0.16/0.40
% 0.16/0.40 % semiring_norm(87)
% 0.16/0.40 thf(fact_38_semiring__norm_I11_J,axiom,
% 0.16/0.40 ! [M: num] :
% 0.16/0.40 ( ( times_times_num @ M @ one )
% 0.16/0.40 = M ) ).
% 0.16/0.40
% 0.16/0.40 % semiring_norm(11)
% 0.16/0.40 thf(fact_39_semiring__norm_I12_J,axiom,
% 0.16/0.40 ! [N: num] :
% 0.16/0.40 ( ( times_times_num @ one @ N )
% 0.16/0.40 = N ) ).
% 0.16/0.40
% 0.16/0.40 % semiring_norm(12)
% 0.16/0.40 thf(fact_40_semiring__norm_I68_J,axiom,
% 0.16/0.40 ! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% 0.16/0.40
% 0.16/0.40 % semiring_norm(68)
% 0.16/0.40 thf(fact_41_old_Onat_Oinject,axiom,
% 0.16/0.40 ! [Nat: nat,Nat2: nat] :
% 0.16/0.40 ( ( ( suc @ Nat )
% 0.16/0.40 = ( suc @ Nat2 ) )
% 0.16/0.40 = ( Nat = Nat2 ) ) ).
% 0.16/0.40
% 0.16/0.40 % old.nat.inject
% 0.16/0.40 thf(fact_42_nat_Oinject,axiom,
% 0.16/0.40 ! [X2: nat,Y2: nat] :
% 0.16/0.40 ( ( ( suc @ X2 )
% 0.16/0.40 = ( suc @ Y2 ) )
% 0.16/0.40 = ( X2 = Y2 ) ) ).
% 0.16/0.40
% 0.16/0.40 % nat.inject
% 0.16/0.40 thf(fact_43_of__nat__eq__iff,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( ( semiri8010041392384452111omplex @ M )
% 0.16/0.40 = ( semiri8010041392384452111omplex @ N ) )
% 0.16/0.40 = ( M = N ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_eq_iff
% 0.16/0.40 thf(fact_44_of__nat__eq__iff,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( ( semiri5074537144036343181t_real @ M )
% 0.16/0.40 = ( semiri5074537144036343181t_real @ N ) )
% 0.16/0.40 = ( M = N ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_eq_iff
% 0.16/0.40 thf(fact_45_of__nat__eq__iff,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( ( semiri681578069525770553at_rat @ M )
% 0.16/0.40 = ( semiri681578069525770553at_rat @ N ) )
% 0.16/0.40 = ( M = N ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_eq_iff
% 0.16/0.40 thf(fact_46_of__nat__eq__iff,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( ( semiri1316708129612266289at_nat @ M )
% 0.16/0.40 = ( semiri1316708129612266289at_nat @ N ) )
% 0.16/0.40 = ( M = N ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_eq_iff
% 0.16/0.40 thf(fact_47_of__nat__eq__iff,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( ( semiri1314217659103216013at_int @ M )
% 0.16/0.40 = ( semiri1314217659103216013at_int @ N ) )
% 0.16/0.40 = ( M = N ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_eq_iff
% 0.16/0.40 thf(fact_48_semiring__norm_I73_J,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.16/0.40 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % semiring_norm(73)
% 0.16/0.40 thf(fact_49_semiring__norm_I90_J,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( ( bit1 @ M )
% 0.16/0.40 = ( bit1 @ N ) )
% 0.16/0.40 = ( M = N ) ) ).
% 0.16/0.40
% 0.16/0.40 % semiring_norm(90)
% 0.16/0.40 thf(fact_50_numeral__le__iff,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.16/0.40 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % numeral_le_iff
% 0.16/0.40 thf(fact_51_numeral__le__iff,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.16/0.40 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % numeral_le_iff
% 0.16/0.40 thf(fact_52_numeral__le__iff,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.16/0.40 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % numeral_le_iff
% 0.16/0.40 thf(fact_53_numeral__le__iff,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.16/0.40 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % numeral_le_iff
% 0.16/0.40 thf(fact_54_mult__numeral__left__semiring__numeral,axiom,
% 0.16/0.40 ! [V: num,W: num,Z: complex] :
% 0.16/0.40 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 0.16/0.40 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.16/0.40
% 0.16/0.40 % mult_numeral_left_semiring_numeral
% 0.16/0.40 thf(fact_55_mult__numeral__left__semiring__numeral,axiom,
% 0.16/0.40 ! [V: num,W: num,Z: real] :
% 0.16/0.40 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 0.16/0.40 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.16/0.40
% 0.16/0.40 % mult_numeral_left_semiring_numeral
% 0.16/0.40 thf(fact_56_mult__numeral__left__semiring__numeral,axiom,
% 0.16/0.40 ! [V: num,W: num,Z: rat] :
% 0.16/0.40 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 0.16/0.40 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.16/0.40
% 0.16/0.40 % mult_numeral_left_semiring_numeral
% 0.16/0.40 thf(fact_57_mult__numeral__left__semiring__numeral,axiom,
% 0.16/0.40 ! [V: num,W: num,Z: nat] :
% 0.16/0.40 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 0.16/0.40 = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.16/0.40
% 0.16/0.40 % mult_numeral_left_semiring_numeral
% 0.16/0.40 thf(fact_58_mult__numeral__left__semiring__numeral,axiom,
% 0.16/0.40 ! [V: num,W: num,Z: int] :
% 0.16/0.40 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 0.16/0.40 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.16/0.40
% 0.16/0.40 % mult_numeral_left_semiring_numeral
% 0.16/0.40 thf(fact_59_numeral__times__numeral,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 0.16/0.40 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % numeral_times_numeral
% 0.16/0.40 thf(fact_60_numeral__times__numeral,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.16/0.40 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % numeral_times_numeral
% 0.16/0.40 thf(fact_61_numeral__times__numeral,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.16/0.40 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % numeral_times_numeral
% 0.16/0.40 thf(fact_62_numeral__times__numeral,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.16/0.40 = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % numeral_times_numeral
% 0.16/0.40 thf(fact_63_numeral__times__numeral,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.16/0.40 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % numeral_times_numeral
% 0.16/0.40 thf(fact_64_num__double,axiom,
% 0.16/0.40 ! [N: num] :
% 0.16/0.40 ( ( times_times_num @ ( bit0 @ one ) @ N )
% 0.16/0.40 = ( bit0 @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % num_double
% 0.16/0.40 thf(fact_65_semiring__norm_I69_J,axiom,
% 0.16/0.40 ! [M: num] :
% 0.16/0.40 ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% 0.16/0.40
% 0.16/0.40 % semiring_norm(69)
% 0.16/0.40 thf(fact_66_semiring__norm_I83_J,axiom,
% 0.16/0.40 ! [N: num] :
% 0.16/0.40 ( one
% 0.16/0.40 != ( bit0 @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % semiring_norm(83)
% 0.16/0.40 thf(fact_67_semiring__norm_I85_J,axiom,
% 0.16/0.40 ! [M: num] :
% 0.16/0.40 ( ( bit0 @ M )
% 0.16/0.40 != one ) ).
% 0.16/0.40
% 0.16/0.40 % semiring_norm(85)
% 0.16/0.40 thf(fact_68_semiring__norm_I14_J,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.16/0.40 = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % semiring_norm(14)
% 0.16/0.40 thf(fact_69_semiring__norm_I15_J,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.16/0.40 = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % semiring_norm(15)
% 0.16/0.40 thf(fact_70_semiring__norm_I72_J,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.16/0.40 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % semiring_norm(72)
% 0.16/0.40 thf(fact_71_semiring__norm_I70_J,axiom,
% 0.16/0.40 ! [M: num] :
% 0.16/0.40 ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% 0.16/0.40
% 0.16/0.40 % semiring_norm(70)
% 0.16/0.40 thf(fact_72_Suc__le__mono,axiom,
% 0.16/0.40 ! [N: nat,M: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.16/0.40 = ( ord_less_eq_nat @ N @ M ) ) ).
% 0.16/0.40
% 0.16/0.40 % Suc_le_mono
% 0.16/0.40 thf(fact_73_le__cube,axiom,
% 0.16/0.40 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % le_cube
% 0.16/0.40 thf(fact_74_le__refl,axiom,
% 0.16/0.40 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.16/0.40
% 0.16/0.40 % le_refl
% 0.16/0.40 thf(fact_75_le__trans,axiom,
% 0.16/0.40 ! [I: nat,J: nat,K: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ I @ J )
% 0.16/0.40 => ( ( ord_less_eq_nat @ J @ K )
% 0.16/0.40 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % le_trans
% 0.16/0.40 thf(fact_76_mem__Collect__eq,axiom,
% 0.16/0.40 ! [A: complex,P: complex > $o] :
% 0.16/0.40 ( ( member_complex @ A @ ( collect_complex @ P ) )
% 0.16/0.40 = ( P @ A ) ) ).
% 0.16/0.40
% 0.16/0.40 % mem_Collect_eq
% 0.16/0.40 thf(fact_77_mem__Collect__eq,axiom,
% 0.16/0.40 ! [A: real,P: real > $o] :
% 0.16/0.40 ( ( member_real @ A @ ( collect_real @ P ) )
% 0.16/0.40 = ( P @ A ) ) ).
% 0.16/0.40
% 0.16/0.40 % mem_Collect_eq
% 0.16/0.40 thf(fact_78_mem__Collect__eq,axiom,
% 0.16/0.40 ! [A: list_nat,P: list_nat > $o] :
% 0.16/0.40 ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
% 0.16/0.40 = ( P @ A ) ) ).
% 0.16/0.40
% 0.16/0.40 % mem_Collect_eq
% 0.16/0.40 thf(fact_79_mem__Collect__eq,axiom,
% 0.16/0.40 ! [A: set_nat,P: set_nat > $o] :
% 0.16/0.40 ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
% 0.16/0.40 = ( P @ A ) ) ).
% 0.16/0.40
% 0.16/0.40 % mem_Collect_eq
% 0.16/0.40 thf(fact_80_mem__Collect__eq,axiom,
% 0.16/0.40 ! [A: nat,P: nat > $o] :
% 0.16/0.40 ( ( member_nat @ A @ ( collect_nat @ P ) )
% 0.16/0.40 = ( P @ A ) ) ).
% 0.16/0.40
% 0.16/0.40 % mem_Collect_eq
% 0.16/0.40 thf(fact_81_mem__Collect__eq,axiom,
% 0.16/0.40 ! [A: int,P: int > $o] :
% 0.16/0.40 ( ( member_int @ A @ ( collect_int @ P ) )
% 0.16/0.40 = ( P @ A ) ) ).
% 0.16/0.40
% 0.16/0.40 % mem_Collect_eq
% 0.16/0.40 thf(fact_82_Collect__mem__eq,axiom,
% 0.16/0.40 ! [A2: set_complex] :
% 0.16/0.40 ( ( collect_complex
% 0.16/0.40 @ ^ [X3: complex] : ( member_complex @ X3 @ A2 ) )
% 0.16/0.40 = A2 ) ).
% 0.16/0.40
% 0.16/0.40 % Collect_mem_eq
% 0.16/0.40 thf(fact_83_Collect__mem__eq,axiom,
% 0.16/0.40 ! [A2: set_real] :
% 0.16/0.40 ( ( collect_real
% 0.16/0.40 @ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
% 0.16/0.40 = A2 ) ).
% 0.16/0.40
% 0.16/0.40 % Collect_mem_eq
% 0.16/0.40 thf(fact_84_Collect__mem__eq,axiom,
% 0.16/0.40 ! [A2: set_list_nat] :
% 0.16/0.40 ( ( collect_list_nat
% 0.16/0.40 @ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A2 ) )
% 0.16/0.40 = A2 ) ).
% 0.16/0.40
% 0.16/0.40 % Collect_mem_eq
% 0.16/0.40 thf(fact_85_Collect__mem__eq,axiom,
% 0.16/0.40 ! [A2: set_set_nat] :
% 0.16/0.40 ( ( collect_set_nat
% 0.16/0.40 @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
% 0.16/0.40 = A2 ) ).
% 0.16/0.40
% 0.16/0.40 % Collect_mem_eq
% 0.16/0.40 thf(fact_86_Collect__mem__eq,axiom,
% 0.16/0.40 ! [A2: set_nat] :
% 0.16/0.40 ( ( collect_nat
% 0.16/0.40 @ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
% 0.16/0.40 = A2 ) ).
% 0.16/0.40
% 0.16/0.40 % Collect_mem_eq
% 0.16/0.40 thf(fact_87_Collect__mem__eq,axiom,
% 0.16/0.40 ! [A2: set_int] :
% 0.16/0.40 ( ( collect_int
% 0.16/0.40 @ ^ [X3: int] : ( member_int @ X3 @ A2 ) )
% 0.16/0.40 = A2 ) ).
% 0.16/0.40
% 0.16/0.40 % Collect_mem_eq
% 0.16/0.40 thf(fact_88_Collect__cong,axiom,
% 0.16/0.40 ! [P: real > $o,Q: real > $o] :
% 0.16/0.40 ( ! [X4: real] :
% 0.16/0.40 ( ( P @ X4 )
% 0.16/0.40 = ( Q @ X4 ) )
% 0.16/0.40 => ( ( collect_real @ P )
% 0.16/0.40 = ( collect_real @ Q ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % Collect_cong
% 0.16/0.40 thf(fact_89_Collect__cong,axiom,
% 0.16/0.40 ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.16/0.40 ( ! [X4: list_nat] :
% 0.16/0.40 ( ( P @ X4 )
% 0.16/0.40 = ( Q @ X4 ) )
% 0.16/0.40 => ( ( collect_list_nat @ P )
% 0.16/0.40 = ( collect_list_nat @ Q ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % Collect_cong
% 0.16/0.40 thf(fact_90_Collect__cong,axiom,
% 0.16/0.40 ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.16/0.40 ( ! [X4: set_nat] :
% 0.16/0.40 ( ( P @ X4 )
% 0.16/0.40 = ( Q @ X4 ) )
% 0.16/0.40 => ( ( collect_set_nat @ P )
% 0.16/0.40 = ( collect_set_nat @ Q ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % Collect_cong
% 0.16/0.40 thf(fact_91_Collect__cong,axiom,
% 0.16/0.40 ! [P: nat > $o,Q: nat > $o] :
% 0.16/0.40 ( ! [X4: nat] :
% 0.16/0.40 ( ( P @ X4 )
% 0.16/0.40 = ( Q @ X4 ) )
% 0.16/0.40 => ( ( collect_nat @ P )
% 0.16/0.40 = ( collect_nat @ Q ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % Collect_cong
% 0.16/0.40 thf(fact_92_Collect__cong,axiom,
% 0.16/0.40 ! [P: int > $o,Q: int > $o] :
% 0.16/0.40 ( ! [X4: int] :
% 0.16/0.40 ( ( P @ X4 )
% 0.16/0.40 = ( Q @ X4 ) )
% 0.16/0.40 => ( ( collect_int @ P )
% 0.16/0.40 = ( collect_int @ Q ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % Collect_cong
% 0.16/0.40 thf(fact_93_eq__imp__le,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( M = N )
% 0.16/0.40 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % eq_imp_le
% 0.16/0.40 thf(fact_94_le__square,axiom,
% 0.16/0.40 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 0.16/0.40
% 0.16/0.40 % le_square
% 0.16/0.40 thf(fact_95_le__antisym,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.40 => ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.40 => ( M = N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % le_antisym
% 0.16/0.40 thf(fact_96_dvd__antisym,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ M @ N )
% 0.16/0.40 => ( ( dvd_dvd_nat @ N @ M )
% 0.16/0.40 => ( M = N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_antisym
% 0.16/0.40 thf(fact_97_mult__le__mono,axiom,
% 0.16/0.40 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ I @ J )
% 0.16/0.40 => ( ( ord_less_eq_nat @ K @ L )
% 0.16/0.40 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % mult_le_mono
% 0.16/0.40 thf(fact_98_mult__le__mono1,axiom,
% 0.16/0.40 ! [I: nat,J: nat,K: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ I @ J )
% 0.16/0.40 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % mult_le_mono1
% 0.16/0.40 thf(fact_99_mult__le__mono2,axiom,
% 0.16/0.40 ! [I: nat,J: nat,K: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ I @ J )
% 0.16/0.40 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % mult_le_mono2
% 0.16/0.40 thf(fact_100_nat__le__linear,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.40 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.16/0.40
% 0.16/0.40 % nat_le_linear
% 0.16/0.40 thf(fact_101_Suc__mult__le__cancel1,axiom,
% 0.16/0.40 ! [K: nat,M: nat,N: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.16/0.40 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % Suc_mult_le_cancel1
% 0.16/0.40 thf(fact_102_Nat_Oex__has__greatest__nat,axiom,
% 0.16/0.40 ! [P: nat > $o,K: nat,B: nat] :
% 0.16/0.40 ( ( P @ K )
% 0.16/0.40 => ( ! [Y: nat] :
% 0.16/0.40 ( ( P @ Y )
% 0.16/0.40 => ( ord_less_eq_nat @ Y @ B ) )
% 0.16/0.40 => ? [X4: nat] :
% 0.16/0.40 ( ( P @ X4 )
% 0.16/0.40 & ! [Y3: nat] :
% 0.16/0.40 ( ( P @ Y3 )
% 0.16/0.40 => ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % Nat.ex_has_greatest_nat
% 0.16/0.40 thf(fact_103_div__times__less__eq__dividend,axiom,
% 0.16/0.40 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% 0.16/0.40
% 0.16/0.40 % div_times_less_eq_dividend
% 0.16/0.40 thf(fact_104_times__div__less__eq__dividend,axiom,
% 0.16/0.40 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% 0.16/0.40
% 0.16/0.40 % times_div_less_eq_dividend
% 0.16/0.40 thf(fact_105_le__num__One__iff,axiom,
% 0.16/0.40 ! [X: num] :
% 0.16/0.40 ( ( ord_less_eq_num @ X @ one )
% 0.16/0.40 = ( X = one ) ) ).
% 0.16/0.40
% 0.16/0.40 % le_num_One_iff
% 0.16/0.40 thf(fact_106_Suc__mult__cancel1,axiom,
% 0.16/0.40 ! [K: nat,M: nat,N: nat] :
% 0.16/0.40 ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 0.16/0.40 = ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.16/0.40 = ( M = N ) ) ).
% 0.16/0.40
% 0.16/0.40 % Suc_mult_cancel1
% 0.16/0.40 thf(fact_107_transitive__stepwise__le,axiom,
% 0.16/0.40 ! [M: nat,N: nat,R: nat > nat > $o] :
% 0.16/0.40 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.40 => ( ! [X4: nat] : ( R @ X4 @ X4 )
% 0.16/0.40 => ( ! [X4: nat,Y: nat,Z2: nat] :
% 0.16/0.40 ( ( R @ X4 @ Y )
% 0.16/0.40 => ( ( R @ Y @ Z2 )
% 0.16/0.40 => ( R @ X4 @ Z2 ) ) )
% 0.16/0.40 => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
% 0.16/0.40 => ( R @ M @ N ) ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % transitive_stepwise_le
% 0.16/0.40 thf(fact_108_nat__induct__at__least,axiom,
% 0.16/0.40 ! [M: nat,N: nat,P: nat > $o] :
% 0.16/0.40 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.40 => ( ( P @ M )
% 0.16/0.40 => ( ! [N2: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ M @ N2 )
% 0.16/0.40 => ( ( P @ N2 )
% 0.16/0.40 => ( P @ ( suc @ N2 ) ) ) )
% 0.16/0.40 => ( P @ N ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % nat_induct_at_least
% 0.16/0.40 thf(fact_109_full__nat__induct,axiom,
% 0.16/0.40 ! [P: nat > $o,N: nat] :
% 0.16/0.40 ( ! [N2: nat] :
% 0.16/0.40 ( ! [M2: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
% 0.16/0.40 => ( P @ M2 ) )
% 0.16/0.40 => ( P @ N2 ) )
% 0.16/0.40 => ( P @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % full_nat_induct
% 0.16/0.40 thf(fact_110_not__less__eq__eq,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 0.16/0.40 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 0.16/0.40
% 0.16/0.40 % not_less_eq_eq
% 0.16/0.40 thf(fact_111_Suc__n__not__le__n,axiom,
% 0.16/0.40 ! [N: nat] :
% 0.16/0.40 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 0.16/0.40
% 0.16/0.40 % Suc_n_not_le_n
% 0.16/0.40 thf(fact_112_le__Suc__eq,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.16/0.40 = ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.40 | ( M
% 0.16/0.40 = ( suc @ N ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % le_Suc_eq
% 0.16/0.40 thf(fact_113_Suc__le__D,axiom,
% 0.16/0.40 ! [N: nat,M3: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ ( suc @ N ) @ M3 )
% 0.16/0.40 => ? [M4: nat] :
% 0.16/0.40 ( M3
% 0.16/0.40 = ( suc @ M4 ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % Suc_le_D
% 0.16/0.40 thf(fact_114_le__SucI,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.40 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % le_SucI
% 0.16/0.40 thf(fact_115_le__SucE,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.16/0.40 => ( ~ ( ord_less_eq_nat @ M @ N )
% 0.16/0.40 => ( M
% 0.16/0.40 = ( suc @ N ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % le_SucE
% 0.16/0.40 thf(fact_116_Suc__leD,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.16/0.40 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % Suc_leD
% 0.16/0.40 thf(fact_117_div__mult2__eq,axiom,
% 0.16/0.40 ! [M: nat,N: nat,Q2: nat] :
% 0.16/0.40 ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 0.16/0.40 = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_mult2_eq
% 0.16/0.40 thf(fact_118_div__le__dividend,axiom,
% 0.16/0.40 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 0.16/0.40
% 0.16/0.40 % div_le_dividend
% 0.16/0.40 thf(fact_119_div__le__mono,axiom,
% 0.16/0.40 ! [M: nat,N: nat,K: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.40 => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_le_mono
% 0.16/0.40 thf(fact_120_of__nat__dvd__iff,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
% 0.16/0.40 = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_dvd_iff
% 0.16/0.40 thf(fact_121_of__nat__dvd__iff,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.16/0.40 = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_dvd_iff
% 0.16/0.40 thf(fact_122_of__nat__dvd__iff,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.16/0.40 = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_dvd_iff
% 0.16/0.40 thf(fact_123_div__mult2__numeral__eq,axiom,
% 0.16/0.40 ! [A: nat,K: num,L: num] :
% 0.16/0.40 ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
% 0.16/0.40 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_mult2_numeral_eq
% 0.16/0.40 thf(fact_124_div__mult2__numeral__eq,axiom,
% 0.16/0.40 ! [A: int,K: num,L: num] :
% 0.16/0.40 ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
% 0.16/0.40 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_mult2_numeral_eq
% 0.16/0.40 thf(fact_125_Suc__div__le__mono,axiom,
% 0.16/0.40 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % Suc_div_le_mono
% 0.16/0.40 thf(fact_126_n__not__Suc__n,axiom,
% 0.16/0.40 ! [N: nat] :
% 0.16/0.40 ( N
% 0.16/0.40 != ( suc @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % n_not_Suc_n
% 0.16/0.40 thf(fact_127_Suc__inject,axiom,
% 0.16/0.40 ! [X: nat,Y4: nat] :
% 0.16/0.40 ( ( ( suc @ X )
% 0.16/0.40 = ( suc @ Y4 ) )
% 0.16/0.40 => ( X = Y4 ) ) ).
% 0.16/0.40
% 0.16/0.40 % Suc_inject
% 0.16/0.40 thf(fact_128_lift__Suc__antimono__le,axiom,
% 0.16/0.40 ! [F: nat > set_int,N: nat,N3: nat] :
% 0.16/0.40 ( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.16/0.40 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.16/0.40 => ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % lift_Suc_antimono_le
% 0.16/0.40 thf(fact_129_lift__Suc__antimono__le,axiom,
% 0.16/0.40 ! [F: nat > rat,N: nat,N3: nat] :
% 0.16/0.40 ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.16/0.40 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.16/0.40 => ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % lift_Suc_antimono_le
% 0.16/0.40 thf(fact_130_lift__Suc__antimono__le,axiom,
% 0.16/0.40 ! [F: nat > num,N: nat,N3: nat] :
% 0.16/0.40 ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.16/0.40 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.16/0.40 => ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % lift_Suc_antimono_le
% 0.16/0.40 thf(fact_131_lift__Suc__antimono__le,axiom,
% 0.16/0.40 ! [F: nat > nat,N: nat,N3: nat] :
% 0.16/0.40 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.16/0.40 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.16/0.40 => ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % lift_Suc_antimono_le
% 0.16/0.40 thf(fact_132_lift__Suc__antimono__le,axiom,
% 0.16/0.40 ! [F: nat > int,N: nat,N3: nat] :
% 0.16/0.40 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.16/0.40 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.16/0.40 => ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % lift_Suc_antimono_le
% 0.16/0.40 thf(fact_133_lift__Suc__antimono__le,axiom,
% 0.16/0.40 ! [F: nat > real,N: nat,N3: nat] :
% 0.16/0.40 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.16/0.40 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.16/0.40 => ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % lift_Suc_antimono_le
% 0.16/0.40 thf(fact_134_lift__Suc__mono__le,axiom,
% 0.16/0.40 ! [F: nat > set_int,N: nat,N3: nat] :
% 0.16/0.40 ( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.16/0.40 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.16/0.40 => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % lift_Suc_mono_le
% 0.16/0.40 thf(fact_135_lift__Suc__mono__le,axiom,
% 0.16/0.40 ! [F: nat > rat,N: nat,N3: nat] :
% 0.16/0.40 ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.16/0.40 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.16/0.40 => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % lift_Suc_mono_le
% 0.16/0.40 thf(fact_136_lift__Suc__mono__le,axiom,
% 0.16/0.40 ! [F: nat > num,N: nat,N3: nat] :
% 0.16/0.40 ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.16/0.40 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.16/0.40 => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % lift_Suc_mono_le
% 0.16/0.40 thf(fact_137_lift__Suc__mono__le,axiom,
% 0.16/0.40 ! [F: nat > nat,N: nat,N3: nat] :
% 0.16/0.40 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.16/0.40 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.16/0.40 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % lift_Suc_mono_le
% 0.16/0.40 thf(fact_138_lift__Suc__mono__le,axiom,
% 0.16/0.40 ! [F: nat > int,N: nat,N3: nat] :
% 0.16/0.40 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.16/0.40 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.16/0.40 => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % lift_Suc_mono_le
% 0.16/0.40 thf(fact_139_lift__Suc__mono__le,axiom,
% 0.16/0.40 ! [F: nat > real,N: nat,N3: nat] :
% 0.16/0.40 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.16/0.40 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.16/0.40 => ( ord_less_eq_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % lift_Suc_mono_le
% 0.16/0.40 thf(fact_140_of__nat__mono,axiom,
% 0.16/0.40 ! [I: nat,J: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ I @ J )
% 0.16/0.40 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_mono
% 0.16/0.40 thf(fact_141_of__nat__mono,axiom,
% 0.16/0.40 ! [I: nat,J: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ I @ J )
% 0.16/0.40 => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_mono
% 0.16/0.40 thf(fact_142_of__nat__mono,axiom,
% 0.16/0.40 ! [I: nat,J: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ I @ J )
% 0.16/0.40 => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_mono
% 0.16/0.40 thf(fact_143_of__nat__mono,axiom,
% 0.16/0.40 ! [I: nat,J: nat] :
% 0.16/0.40 ( ( ord_less_eq_nat @ I @ J )
% 0.16/0.40 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % of_nat_mono
% 0.16/0.40 thf(fact_144_mult__of__nat__commute,axiom,
% 0.16/0.40 ! [X: nat,Y4: complex] :
% 0.16/0.40 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y4 )
% 0.16/0.40 = ( times_times_complex @ Y4 @ ( semiri8010041392384452111omplex @ X ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % mult_of_nat_commute
% 0.16/0.40 thf(fact_145_mult__of__nat__commute,axiom,
% 0.16/0.40 ! [X: nat,Y4: real] :
% 0.16/0.40 ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y4 )
% 0.16/0.40 = ( times_times_real @ Y4 @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % mult_of_nat_commute
% 0.16/0.40 thf(fact_146_mult__of__nat__commute,axiom,
% 0.16/0.40 ! [X: nat,Y4: rat] :
% 0.16/0.40 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y4 )
% 0.16/0.40 = ( times_times_rat @ Y4 @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % mult_of_nat_commute
% 0.16/0.40 thf(fact_147_mult__of__nat__commute,axiom,
% 0.16/0.40 ! [X: nat,Y4: nat] :
% 0.16/0.40 ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y4 )
% 0.16/0.40 = ( times_times_nat @ Y4 @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % mult_of_nat_commute
% 0.16/0.40 thf(fact_148_mult__of__nat__commute,axiom,
% 0.16/0.40 ! [X: nat,Y4: int] :
% 0.16/0.40 ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y4 )
% 0.16/0.40 = ( times_times_int @ Y4 @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % mult_of_nat_commute
% 0.16/0.40 thf(fact_149_mult__numeral__1__right,axiom,
% 0.16/0.40 ! [A: complex] :
% 0.16/0.40 ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
% 0.16/0.40 = A ) ).
% 0.16/0.40
% 0.16/0.40 % mult_numeral_1_right
% 0.16/0.40 thf(fact_150_mult__numeral__1__right,axiom,
% 0.16/0.40 ! [A: real] :
% 0.16/0.40 ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
% 0.16/0.40 = A ) ).
% 0.16/0.40
% 0.16/0.40 % mult_numeral_1_right
% 0.16/0.40 thf(fact_151_mult__numeral__1__right,axiom,
% 0.16/0.40 ! [A: rat] :
% 0.16/0.40 ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
% 0.16/0.40 = A ) ).
% 0.16/0.40
% 0.16/0.40 % mult_numeral_1_right
% 0.16/0.40 thf(fact_152_mult__numeral__1__right,axiom,
% 0.16/0.40 ! [A: nat] :
% 0.16/0.40 ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
% 0.16/0.40 = A ) ).
% 0.16/0.40
% 0.16/0.40 % mult_numeral_1_right
% 0.16/0.40 thf(fact_153_mult__numeral__1__right,axiom,
% 0.16/0.40 ! [A: int] :
% 0.16/0.40 ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
% 0.16/0.40 = A ) ).
% 0.16/0.40
% 0.16/0.40 % mult_numeral_1_right
% 0.16/0.40 thf(fact_154_mult__numeral__1,axiom,
% 0.16/0.40 ! [A: complex] :
% 0.16/0.40 ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
% 0.16/0.40 = A ) ).
% 0.16/0.40
% 0.16/0.40 % mult_numeral_1
% 0.16/0.40 thf(fact_155_mult__numeral__1,axiom,
% 0.16/0.40 ! [A: real] :
% 0.16/0.40 ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
% 0.16/0.40 = A ) ).
% 0.16/0.40
% 0.16/0.40 % mult_numeral_1
% 0.16/0.40 thf(fact_156_mult__numeral__1,axiom,
% 0.16/0.40 ! [A: rat] :
% 0.16/0.40 ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
% 0.16/0.40 = A ) ).
% 0.16/0.40
% 0.16/0.40 % mult_numeral_1
% 0.16/0.40 thf(fact_157_mult__numeral__1,axiom,
% 0.16/0.40 ! [A: nat] :
% 0.16/0.40 ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
% 0.16/0.40 = A ) ).
% 0.16/0.40
% 0.16/0.40 % mult_numeral_1
% 0.16/0.40 thf(fact_158_mult__numeral__1,axiom,
% 0.16/0.40 ! [A: int] :
% 0.16/0.40 ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
% 0.16/0.40 = A ) ).
% 0.16/0.40
% 0.16/0.40 % mult_numeral_1
% 0.16/0.40 thf(fact_159_divide__numeral__1,axiom,
% 0.16/0.40 ! [A: complex] :
% 0.16/0.40 ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
% 0.16/0.40 = A ) ).
% 0.16/0.40
% 0.16/0.40 % divide_numeral_1
% 0.16/0.40 thf(fact_160_divide__numeral__1,axiom,
% 0.16/0.40 ! [A: real] :
% 0.16/0.40 ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
% 0.16/0.40 = A ) ).
% 0.16/0.40
% 0.16/0.40 % divide_numeral_1
% 0.16/0.40 thf(fact_161_divide__numeral__1,axiom,
% 0.16/0.40 ! [A: rat] :
% 0.16/0.40 ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
% 0.16/0.40 = A ) ).
% 0.16/0.40
% 0.16/0.40 % divide_numeral_1
% 0.16/0.40 thf(fact_162_num_Oexhaust,axiom,
% 0.16/0.40 ! [Y4: num] :
% 0.16/0.40 ( ( Y4 != one )
% 0.16/0.40 => ( ! [X22: num] :
% 0.16/0.40 ( Y4
% 0.16/0.40 != ( bit0 @ X22 ) )
% 0.16/0.40 => ~ ! [X32: num] :
% 0.16/0.40 ( Y4
% 0.16/0.40 != ( bit1 @ X32 ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % num.exhaust
% 0.16/0.40 thf(fact_163_div__mult2__eq_H,axiom,
% 0.16/0.40 ! [A: nat,M: nat,N: nat] :
% 0.16/0.40 ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 0.16/0.40 = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_mult2_eq'
% 0.16/0.40 thf(fact_164_div__mult2__eq_H,axiom,
% 0.16/0.40 ! [A: int,M: nat,N: nat] :
% 0.16/0.40 ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.16/0.40 = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_mult2_eq'
% 0.16/0.40 thf(fact_165_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
% 0.16/0.40 = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % unique_euclidean_semiring_with_nat_class.of_nat_div
% 0.16/0.40 thf(fact_166_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
% 0.16/0.40 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % unique_euclidean_semiring_with_nat_class.of_nat_div
% 0.16/0.40 thf(fact_167_numeral__Bit0__div__2,axiom,
% 0.16/0.40 ! [N: num] :
% 0.16/0.40 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.40 = ( numeral_numeral_nat @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % numeral_Bit0_div_2
% 0.16/0.40 thf(fact_168_numeral__Bit0__div__2,axiom,
% 0.16/0.40 ! [N: num] :
% 0.16/0.40 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.16/0.40 = ( numeral_numeral_int @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % numeral_Bit0_div_2
% 0.16/0.40 thf(fact_169_eval__nat__numeral_I3_J,axiom,
% 0.16/0.40 ! [N: num] :
% 0.16/0.40 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 0.16/0.40 = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % eval_nat_numeral(3)
% 0.16/0.40 thf(fact_170_even__of__nat,axiom,
% 0.16/0.40 ! [N: nat] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
% 0.16/0.40 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % even_of_nat
% 0.16/0.40 thf(fact_171_even__of__nat,axiom,
% 0.16/0.40 ! [N: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.16/0.40 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % even_of_nat
% 0.16/0.40 thf(fact_172_even__of__nat,axiom,
% 0.16/0.40 ! [N: nat] :
% 0.16/0.40 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.16/0.40 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % even_of_nat
% 0.16/0.40 thf(fact_173_even__Suc__div__two,axiom,
% 0.16/0.40 ! [N: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.40 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.40 = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % even_Suc_div_two
% 0.16/0.40 thf(fact_174_odd__Suc__div__two,axiom,
% 0.16/0.40 ! [N: nat] :
% 0.16/0.40 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.40 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.40 = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % odd_Suc_div_two
% 0.16/0.40 thf(fact_175_numeral__le__real__of__nat__iff,axiom,
% 0.16/0.40 ! [N: num,M: nat] :
% 0.16/0.40 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
% 0.16/0.40 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 0.16/0.40
% 0.16/0.40 % numeral_le_real_of_nat_iff
% 0.16/0.40 thf(fact_176_even__Suc__Suc__iff,axiom,
% 0.16/0.40 ! [N: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
% 0.16/0.40 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % even_Suc_Suc_iff
% 0.16/0.40 thf(fact_177_even__Suc,axiom,
% 0.16/0.40 ! [N: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
% 0.16/0.40 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % even_Suc
% 0.16/0.40 thf(fact_178_even__mult__iff,axiom,
% 0.16/0.40 ! [A: code_integer,B: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.16/0.40 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.16/0.40 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % even_mult_iff
% 0.16/0.40 thf(fact_179_even__mult__iff,axiom,
% 0.16/0.40 ! [A: nat,B: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
% 0.16/0.40 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.16/0.40 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % even_mult_iff
% 0.16/0.40 thf(fact_180_even__mult__iff,axiom,
% 0.16/0.40 ! [A: int,B: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
% 0.16/0.40 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.16/0.40 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % even_mult_iff
% 0.16/0.40 thf(fact_181_dvd__mult__div__cancel,axiom,
% 0.16/0.40 ! [A: code_integer,B: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.16/0.40 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
% 0.16/0.40 = B ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult_div_cancel
% 0.16/0.40 thf(fact_182_dvd__mult__div__cancel,axiom,
% 0.16/0.40 ! [A: nat,B: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.40 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
% 0.16/0.40 = B ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult_div_cancel
% 0.16/0.40 thf(fact_183_dvd__mult__div__cancel,axiom,
% 0.16/0.40 ! [A: int,B: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ A @ B )
% 0.16/0.40 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
% 0.16/0.40 = B ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult_div_cancel
% 0.16/0.40 thf(fact_184_dvd__div__mult__self,axiom,
% 0.16/0.40 ! [A: code_integer,B: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.16/0.40 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 0.16/0.40 = B ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_mult_self
% 0.16/0.40 thf(fact_185_dvd__div__mult__self,axiom,
% 0.16/0.40 ! [A: nat,B: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.40 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 0.16/0.40 = B ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_mult_self
% 0.16/0.40 thf(fact_186_dvd__div__mult__self,axiom,
% 0.16/0.40 ! [A: int,B: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ A @ B )
% 0.16/0.40 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 0.16/0.40 = B ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_mult_self
% 0.16/0.40 thf(fact_187_even__two__times__div__two,axiom,
% 0.16/0.40 ! [A: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.16/0.40 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.16/0.40 = A ) ) ).
% 0.16/0.40
% 0.16/0.40 % even_two_times_div_two
% 0.16/0.40 thf(fact_188_even__two__times__div__two,axiom,
% 0.16/0.40 ! [A: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.16/0.40 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.40 = A ) ) ).
% 0.16/0.40
% 0.16/0.40 % even_two_times_div_two
% 0.16/0.40 thf(fact_189_even__two__times__div__two,axiom,
% 0.16/0.40 ! [A: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.16/0.40 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.16/0.40 = A ) ) ).
% 0.16/0.40
% 0.16/0.40 % even_two_times_div_two
% 0.16/0.40 thf(fact_190_div2__even__ext__nat,axiom,
% 0.16/0.40 ! [X: nat,Y4: nat] :
% 0.16/0.40 ( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.40 = ( divide_divide_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.40 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
% 0.16/0.40 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y4 ) )
% 0.16/0.40 => ( X = Y4 ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div2_even_ext_nat
% 0.16/0.40 thf(fact_191_enat__ord__number_I1_J,axiom,
% 0.16/0.40 ! [M: num,N: num] :
% 0.16/0.40 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.16/0.40 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % enat_ord_number(1)
% 0.16/0.40 thf(fact_192_div__dvd__div,axiom,
% 0.16/0.40 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.16/0.40 => ( ( dvd_dvd_Code_integer @ A @ C )
% 0.16/0.40 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
% 0.16/0.40 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_dvd_div
% 0.16/0.40 thf(fact_193_div__dvd__div,axiom,
% 0.16/0.40 ! [A: nat,B: nat,C: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.40 => ( ( dvd_dvd_nat @ A @ C )
% 0.16/0.40 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
% 0.16/0.40 = ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_dvd_div
% 0.16/0.40 thf(fact_194_div__dvd__div,axiom,
% 0.16/0.40 ! [A: int,B: int,C: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ A @ B )
% 0.16/0.40 => ( ( dvd_dvd_int @ A @ C )
% 0.16/0.40 => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
% 0.16/0.40 = ( dvd_dvd_int @ B @ C ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_dvd_div
% 0.16/0.40 thf(fact_195_dvd__trans,axiom,
% 0.16/0.40 ! [A: nat,B: nat,C: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.40 => ( ( dvd_dvd_nat @ B @ C )
% 0.16/0.40 => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_trans
% 0.16/0.40 thf(fact_196_dvd__trans,axiom,
% 0.16/0.40 ! [A: int,B: int,C: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ A @ B )
% 0.16/0.40 => ( ( dvd_dvd_int @ B @ C )
% 0.16/0.40 => ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_trans
% 0.16/0.40 thf(fact_197_dvd__trans,axiom,
% 0.16/0.40 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.16/0.40 => ( ( dvd_dvd_Code_integer @ B @ C )
% 0.16/0.40 => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_trans
% 0.16/0.40 thf(fact_198_dvd__refl,axiom,
% 0.16/0.40 ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_refl
% 0.16/0.40 thf(fact_199_dvd__refl,axiom,
% 0.16/0.40 ! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_refl
% 0.16/0.40 thf(fact_200_dvd__refl,axiom,
% 0.16/0.40 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_refl
% 0.16/0.40 thf(fact_201_complete__real,axiom,
% 0.16/0.40 ! [S: set_real] :
% 0.16/0.40 ( ? [X5: real] : ( member_real @ X5 @ S )
% 0.16/0.40 => ( ? [Z3: real] :
% 0.16/0.40 ! [X4: real] :
% 0.16/0.40 ( ( member_real @ X4 @ S )
% 0.16/0.40 => ( ord_less_eq_real @ X4 @ Z3 ) )
% 0.16/0.40 => ? [Y: real] :
% 0.16/0.40 ( ! [X5: real] :
% 0.16/0.40 ( ( member_real @ X5 @ S )
% 0.16/0.40 => ( ord_less_eq_real @ X5 @ Y ) )
% 0.16/0.40 & ! [Z3: real] :
% 0.16/0.40 ( ! [X4: real] :
% 0.16/0.40 ( ( member_real @ X4 @ S )
% 0.16/0.40 => ( ord_less_eq_real @ X4 @ Z3 ) )
% 0.16/0.40 => ( ord_less_eq_real @ Y @ Z3 ) ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % complete_real
% 0.16/0.40 thf(fact_202_dvdE,axiom,
% 0.16/0.40 ! [B: code_integer,A: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.16/0.40 => ~ ! [K2: code_integer] :
% 0.16/0.40 ( A
% 0.16/0.40 != ( times_3573771949741848930nteger @ B @ K2 ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvdE
% 0.16/0.40 thf(fact_203_dvdE,axiom,
% 0.16/0.40 ! [B: real,A: real] :
% 0.16/0.40 ( ( dvd_dvd_real @ B @ A )
% 0.16/0.40 => ~ ! [K2: real] :
% 0.16/0.40 ( A
% 0.16/0.40 != ( times_times_real @ B @ K2 ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvdE
% 0.16/0.40 thf(fact_204_dvdE,axiom,
% 0.16/0.40 ! [B: rat,A: rat] :
% 0.16/0.40 ( ( dvd_dvd_rat @ B @ A )
% 0.16/0.40 => ~ ! [K2: rat] :
% 0.16/0.40 ( A
% 0.16/0.40 != ( times_times_rat @ B @ K2 ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvdE
% 0.16/0.40 thf(fact_205_dvdE,axiom,
% 0.16/0.40 ! [B: nat,A: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ B @ A )
% 0.16/0.40 => ~ ! [K2: nat] :
% 0.16/0.40 ( A
% 0.16/0.40 != ( times_times_nat @ B @ K2 ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvdE
% 0.16/0.40 thf(fact_206_dvdE,axiom,
% 0.16/0.40 ! [B: int,A: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ B @ A )
% 0.16/0.40 => ~ ! [K2: int] :
% 0.16/0.40 ( A
% 0.16/0.40 != ( times_times_int @ B @ K2 ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvdE
% 0.16/0.40 thf(fact_207_dvdI,axiom,
% 0.16/0.40 ! [A: code_integer,B: code_integer,K: code_integer] :
% 0.16/0.40 ( ( A
% 0.16/0.40 = ( times_3573771949741848930nteger @ B @ K ) )
% 0.16/0.40 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvdI
% 0.16/0.40 thf(fact_208_dvdI,axiom,
% 0.16/0.40 ! [A: real,B: real,K: real] :
% 0.16/0.40 ( ( A
% 0.16/0.40 = ( times_times_real @ B @ K ) )
% 0.16/0.40 => ( dvd_dvd_real @ B @ A ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvdI
% 0.16/0.40 thf(fact_209_dvdI,axiom,
% 0.16/0.40 ! [A: rat,B: rat,K: rat] :
% 0.16/0.40 ( ( A
% 0.16/0.40 = ( times_times_rat @ B @ K ) )
% 0.16/0.40 => ( dvd_dvd_rat @ B @ A ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvdI
% 0.16/0.40 thf(fact_210_dvdI,axiom,
% 0.16/0.40 ! [A: nat,B: nat,K: nat] :
% 0.16/0.40 ( ( A
% 0.16/0.40 = ( times_times_nat @ B @ K ) )
% 0.16/0.40 => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvdI
% 0.16/0.40 thf(fact_211_dvdI,axiom,
% 0.16/0.40 ! [A: int,B: int,K: int] :
% 0.16/0.40 ( ( A
% 0.16/0.40 = ( times_times_int @ B @ K ) )
% 0.16/0.40 => ( dvd_dvd_int @ B @ A ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvdI
% 0.16/0.40 thf(fact_212_dvd__def,axiom,
% 0.16/0.40 ( dvd_dvd_Code_integer
% 0.16/0.40 = ( ^ [B2: code_integer,A3: code_integer] :
% 0.16/0.40 ? [K3: code_integer] :
% 0.16/0.40 ( A3
% 0.16/0.40 = ( times_3573771949741848930nteger @ B2 @ K3 ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_def
% 0.16/0.40 thf(fact_213_dvd__def,axiom,
% 0.16/0.40 ( dvd_dvd_real
% 0.16/0.40 = ( ^ [B2: real,A3: real] :
% 0.16/0.40 ? [K3: real] :
% 0.16/0.40 ( A3
% 0.16/0.40 = ( times_times_real @ B2 @ K3 ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_def
% 0.16/0.40 thf(fact_214_dvd__def,axiom,
% 0.16/0.40 ( dvd_dvd_rat
% 0.16/0.40 = ( ^ [B2: rat,A3: rat] :
% 0.16/0.40 ? [K3: rat] :
% 0.16/0.40 ( A3
% 0.16/0.40 = ( times_times_rat @ B2 @ K3 ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_def
% 0.16/0.40 thf(fact_215_dvd__def,axiom,
% 0.16/0.40 ( dvd_dvd_nat
% 0.16/0.40 = ( ^ [B2: nat,A3: nat] :
% 0.16/0.40 ? [K3: nat] :
% 0.16/0.40 ( A3
% 0.16/0.40 = ( times_times_nat @ B2 @ K3 ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_def
% 0.16/0.40 thf(fact_216_dvd__def,axiom,
% 0.16/0.40 ( dvd_dvd_int
% 0.16/0.40 = ( ^ [B2: int,A3: int] :
% 0.16/0.40 ? [K3: int] :
% 0.16/0.40 ( A3
% 0.16/0.40 = ( times_times_int @ B2 @ K3 ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_def
% 0.16/0.40 thf(fact_217_dvd__mult,axiom,
% 0.16/0.40 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ A @ C )
% 0.16/0.40 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult
% 0.16/0.40 thf(fact_218_dvd__mult,axiom,
% 0.16/0.40 ! [A: real,C: real,B: real] :
% 0.16/0.40 ( ( dvd_dvd_real @ A @ C )
% 0.16/0.40 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult
% 0.16/0.40 thf(fact_219_dvd__mult,axiom,
% 0.16/0.40 ! [A: rat,C: rat,B: rat] :
% 0.16/0.40 ( ( dvd_dvd_rat @ A @ C )
% 0.16/0.40 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult
% 0.16/0.40 thf(fact_220_dvd__mult,axiom,
% 0.16/0.40 ! [A: nat,C: nat,B: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ A @ C )
% 0.16/0.40 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult
% 0.16/0.40 thf(fact_221_dvd__mult,axiom,
% 0.16/0.40 ! [A: int,C: int,B: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ A @ C )
% 0.16/0.40 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult
% 0.16/0.40 thf(fact_222_dvd__mult2,axiom,
% 0.16/0.40 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.16/0.40 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult2
% 0.16/0.40 thf(fact_223_dvd__mult2,axiom,
% 0.16/0.40 ! [A: real,B: real,C: real] :
% 0.16/0.40 ( ( dvd_dvd_real @ A @ B )
% 0.16/0.40 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult2
% 0.16/0.40 thf(fact_224_dvd__mult2,axiom,
% 0.16/0.40 ! [A: rat,B: rat,C: rat] :
% 0.16/0.40 ( ( dvd_dvd_rat @ A @ B )
% 0.16/0.40 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult2
% 0.16/0.40 thf(fact_225_dvd__mult2,axiom,
% 0.16/0.40 ! [A: nat,B: nat,C: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.40 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult2
% 0.16/0.40 thf(fact_226_dvd__mult2,axiom,
% 0.16/0.40 ! [A: int,B: int,C: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ A @ B )
% 0.16/0.40 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult2
% 0.16/0.40 thf(fact_227_dvd__mult__left,axiom,
% 0.16/0.40 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.16/0.40 => ( dvd_dvd_Code_integer @ A @ C ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult_left
% 0.16/0.40 thf(fact_228_dvd__mult__left,axiom,
% 0.16/0.40 ! [A: real,B: real,C: real] :
% 0.16/0.40 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 0.16/0.40 => ( dvd_dvd_real @ A @ C ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult_left
% 0.16/0.40 thf(fact_229_dvd__mult__left,axiom,
% 0.16/0.40 ! [A: rat,B: rat,C: rat] :
% 0.16/0.40 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.16/0.40 => ( dvd_dvd_rat @ A @ C ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult_left
% 0.16/0.40 thf(fact_230_dvd__mult__left,axiom,
% 0.16/0.40 ! [A: nat,B: nat,C: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.16/0.40 => ( dvd_dvd_nat @ A @ C ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult_left
% 0.16/0.40 thf(fact_231_dvd__mult__left,axiom,
% 0.16/0.40 ! [A: int,B: int,C: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.16/0.40 => ( dvd_dvd_int @ A @ C ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult_left
% 0.16/0.40 thf(fact_232_dvd__triv__left,axiom,
% 0.16/0.40 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_triv_left
% 0.16/0.40 thf(fact_233_dvd__triv__left,axiom,
% 0.16/0.40 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_triv_left
% 0.16/0.40 thf(fact_234_dvd__triv__left,axiom,
% 0.16/0.40 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_triv_left
% 0.16/0.40 thf(fact_235_dvd__triv__left,axiom,
% 0.16/0.40 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_triv_left
% 0.16/0.40 thf(fact_236_dvd__triv__left,axiom,
% 0.16/0.40 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_triv_left
% 0.16/0.40 thf(fact_237_mult__dvd__mono,axiom,
% 0.16/0.40 ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.16/0.40 => ( ( dvd_dvd_Code_integer @ C @ D )
% 0.16/0.40 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % mult_dvd_mono
% 0.16/0.40 thf(fact_238_mult__dvd__mono,axiom,
% 0.16/0.40 ! [A: real,B: real,C: real,D: real] :
% 0.16/0.40 ( ( dvd_dvd_real @ A @ B )
% 0.16/0.40 => ( ( dvd_dvd_real @ C @ D )
% 0.16/0.40 => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % mult_dvd_mono
% 0.16/0.40 thf(fact_239_mult__dvd__mono,axiom,
% 0.16/0.40 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.16/0.40 ( ( dvd_dvd_rat @ A @ B )
% 0.16/0.40 => ( ( dvd_dvd_rat @ C @ D )
% 0.16/0.40 => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % mult_dvd_mono
% 0.16/0.40 thf(fact_240_mult__dvd__mono,axiom,
% 0.16/0.40 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.40 => ( ( dvd_dvd_nat @ C @ D )
% 0.16/0.40 => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % mult_dvd_mono
% 0.16/0.40 thf(fact_241_mult__dvd__mono,axiom,
% 0.16/0.40 ! [A: int,B: int,C: int,D: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ A @ B )
% 0.16/0.40 => ( ( dvd_dvd_int @ C @ D )
% 0.16/0.40 => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % mult_dvd_mono
% 0.16/0.40 thf(fact_242_dvd__mult__right,axiom,
% 0.16/0.40 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.16/0.40 => ( dvd_dvd_Code_integer @ B @ C ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult_right
% 0.16/0.40 thf(fact_243_dvd__mult__right,axiom,
% 0.16/0.40 ! [A: real,B: real,C: real] :
% 0.16/0.40 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 0.16/0.40 => ( dvd_dvd_real @ B @ C ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult_right
% 0.16/0.40 thf(fact_244_dvd__mult__right,axiom,
% 0.16/0.40 ! [A: rat,B: rat,C: rat] :
% 0.16/0.40 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.16/0.40 => ( dvd_dvd_rat @ B @ C ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult_right
% 0.16/0.40 thf(fact_245_dvd__mult__right,axiom,
% 0.16/0.40 ! [A: nat,B: nat,C: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.16/0.40 => ( dvd_dvd_nat @ B @ C ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult_right
% 0.16/0.40 thf(fact_246_dvd__mult__right,axiom,
% 0.16/0.40 ! [A: int,B: int,C: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.16/0.40 => ( dvd_dvd_int @ B @ C ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult_right
% 0.16/0.40 thf(fact_247_dvd__triv__right,axiom,
% 0.16/0.40 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_triv_right
% 0.16/0.40 thf(fact_248_dvd__triv__right,axiom,
% 0.16/0.40 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_triv_right
% 0.16/0.40 thf(fact_249_dvd__triv__right,axiom,
% 0.16/0.40 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_triv_right
% 0.16/0.40 thf(fact_250_dvd__triv__right,axiom,
% 0.16/0.40 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_triv_right
% 0.16/0.40 thf(fact_251_dvd__triv__right,axiom,
% 0.16/0.40 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_triv_right
% 0.16/0.40 thf(fact_252_dvd__div__eq__iff,axiom,
% 0.16/0.40 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.16/0.40 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.16/0.40 => ( ( ( divide6298287555418463151nteger @ A @ C )
% 0.16/0.40 = ( divide6298287555418463151nteger @ B @ C ) )
% 0.16/0.40 = ( A = B ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_eq_iff
% 0.16/0.40 thf(fact_253_dvd__div__eq__iff,axiom,
% 0.16/0.40 ! [C: complex,A: complex,B: complex] :
% 0.16/0.40 ( ( dvd_dvd_complex @ C @ A )
% 0.16/0.40 => ( ( dvd_dvd_complex @ C @ B )
% 0.16/0.40 => ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.16/0.40 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.16/0.40 = ( A = B ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_eq_iff
% 0.16/0.40 thf(fact_254_dvd__div__eq__iff,axiom,
% 0.16/0.40 ! [C: real,A: real,B: real] :
% 0.16/0.40 ( ( dvd_dvd_real @ C @ A )
% 0.16/0.40 => ( ( dvd_dvd_real @ C @ B )
% 0.16/0.40 => ( ( ( divide_divide_real @ A @ C )
% 0.16/0.40 = ( divide_divide_real @ B @ C ) )
% 0.16/0.40 = ( A = B ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_eq_iff
% 0.16/0.40 thf(fact_255_dvd__div__eq__iff,axiom,
% 0.16/0.40 ! [C: rat,A: rat,B: rat] :
% 0.16/0.40 ( ( dvd_dvd_rat @ C @ A )
% 0.16/0.40 => ( ( dvd_dvd_rat @ C @ B )
% 0.16/0.40 => ( ( ( divide_divide_rat @ A @ C )
% 0.16/0.40 = ( divide_divide_rat @ B @ C ) )
% 0.16/0.40 = ( A = B ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_eq_iff
% 0.16/0.40 thf(fact_256_dvd__div__eq__iff,axiom,
% 0.16/0.40 ! [C: nat,A: nat,B: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ C @ A )
% 0.16/0.40 => ( ( dvd_dvd_nat @ C @ B )
% 0.16/0.40 => ( ( ( divide_divide_nat @ A @ C )
% 0.16/0.40 = ( divide_divide_nat @ B @ C ) )
% 0.16/0.40 = ( A = B ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_eq_iff
% 0.16/0.40 thf(fact_257_dvd__div__eq__iff,axiom,
% 0.16/0.40 ! [C: int,A: int,B: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ C @ A )
% 0.16/0.40 => ( ( dvd_dvd_int @ C @ B )
% 0.16/0.40 => ( ( ( divide_divide_int @ A @ C )
% 0.16/0.40 = ( divide_divide_int @ B @ C ) )
% 0.16/0.40 = ( A = B ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_eq_iff
% 0.16/0.40 thf(fact_258_dvd__div__eq__cancel,axiom,
% 0.16/0.40 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.16/0.40 ( ( ( divide6298287555418463151nteger @ A @ C )
% 0.16/0.40 = ( divide6298287555418463151nteger @ B @ C ) )
% 0.16/0.40 => ( ( dvd_dvd_Code_integer @ C @ A )
% 0.16/0.40 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.16/0.40 => ( A = B ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_eq_cancel
% 0.16/0.40 thf(fact_259_dvd__div__eq__cancel,axiom,
% 0.16/0.40 ! [A: complex,C: complex,B: complex] :
% 0.16/0.40 ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.16/0.40 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.16/0.40 => ( ( dvd_dvd_complex @ C @ A )
% 0.16/0.40 => ( ( dvd_dvd_complex @ C @ B )
% 0.16/0.40 => ( A = B ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_eq_cancel
% 0.16/0.40 thf(fact_260_dvd__div__eq__cancel,axiom,
% 0.16/0.40 ! [A: real,C: real,B: real] :
% 0.16/0.40 ( ( ( divide_divide_real @ A @ C )
% 0.16/0.40 = ( divide_divide_real @ B @ C ) )
% 0.16/0.40 => ( ( dvd_dvd_real @ C @ A )
% 0.16/0.40 => ( ( dvd_dvd_real @ C @ B )
% 0.16/0.40 => ( A = B ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_eq_cancel
% 0.16/0.40 thf(fact_261_dvd__div__eq__cancel,axiom,
% 0.16/0.40 ! [A: rat,C: rat,B: rat] :
% 0.16/0.40 ( ( ( divide_divide_rat @ A @ C )
% 0.16/0.40 = ( divide_divide_rat @ B @ C ) )
% 0.16/0.40 => ( ( dvd_dvd_rat @ C @ A )
% 0.16/0.40 => ( ( dvd_dvd_rat @ C @ B )
% 0.16/0.40 => ( A = B ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_eq_cancel
% 0.16/0.40 thf(fact_262_dvd__div__eq__cancel,axiom,
% 0.16/0.40 ! [A: nat,C: nat,B: nat] :
% 0.16/0.40 ( ( ( divide_divide_nat @ A @ C )
% 0.16/0.40 = ( divide_divide_nat @ B @ C ) )
% 0.16/0.40 => ( ( dvd_dvd_nat @ C @ A )
% 0.16/0.40 => ( ( dvd_dvd_nat @ C @ B )
% 0.16/0.40 => ( A = B ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_eq_cancel
% 0.16/0.40 thf(fact_263_dvd__div__eq__cancel,axiom,
% 0.16/0.40 ! [A: int,C: int,B: int] :
% 0.16/0.40 ( ( ( divide_divide_int @ A @ C )
% 0.16/0.40 = ( divide_divide_int @ B @ C ) )
% 0.16/0.40 => ( ( dvd_dvd_int @ C @ A )
% 0.16/0.40 => ( ( dvd_dvd_int @ C @ B )
% 0.16/0.40 => ( A = B ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_eq_cancel
% 0.16/0.40 thf(fact_264_div__div__div__same,axiom,
% 0.16/0.40 ! [D: code_integer,B: code_integer,A: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ D @ B )
% 0.16/0.40 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.16/0.40 => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
% 0.16/0.40 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_div_div_same
% 0.16/0.40 thf(fact_265_div__div__div__same,axiom,
% 0.16/0.40 ! [D: nat,B: nat,A: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ D @ B )
% 0.16/0.40 => ( ( dvd_dvd_nat @ B @ A )
% 0.16/0.40 => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
% 0.16/0.40 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_div_div_same
% 0.16/0.40 thf(fact_266_div__div__div__same,axiom,
% 0.16/0.40 ! [D: int,B: int,A: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ D @ B )
% 0.16/0.40 => ( ( dvd_dvd_int @ B @ A )
% 0.16/0.40 => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
% 0.16/0.40 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_div_div_same
% 0.16/0.40 thf(fact_267_dvd__div__mult,axiom,
% 0.16/0.40 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.16/0.40 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
% 0.16/0.40 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_mult
% 0.16/0.40 thf(fact_268_dvd__div__mult,axiom,
% 0.16/0.40 ! [C: nat,B: nat,A: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ C @ B )
% 0.16/0.40 => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
% 0.16/0.40 = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_mult
% 0.16/0.40 thf(fact_269_dvd__div__mult,axiom,
% 0.16/0.40 ! [C: int,B: int,A: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ C @ B )
% 0.16/0.40 => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
% 0.16/0.40 = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_mult
% 0.16/0.40 thf(fact_270_div__mult__swap,axiom,
% 0.16/0.40 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.16/0.40 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.16/0.40 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_mult_swap
% 0.16/0.40 thf(fact_271_div__mult__swap,axiom,
% 0.16/0.40 ! [C: nat,B: nat,A: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ C @ B )
% 0.16/0.40 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.16/0.40 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_mult_swap
% 0.16/0.40 thf(fact_272_div__mult__swap,axiom,
% 0.16/0.40 ! [C: int,B: int,A: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ C @ B )
% 0.16/0.40 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.16/0.40 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_mult_swap
% 0.16/0.40 thf(fact_273_div__div__eq__right,axiom,
% 0.16/0.40 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.16/0.40 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.16/0.40 => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.16/0.40 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_div_eq_right
% 0.16/0.40 thf(fact_274_div__div__eq__right,axiom,
% 0.16/0.40 ! [C: nat,B: nat,A: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ C @ B )
% 0.16/0.40 => ( ( dvd_dvd_nat @ B @ A )
% 0.16/0.40 => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.16/0.40 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_div_eq_right
% 0.16/0.40 thf(fact_275_div__div__eq__right,axiom,
% 0.16/0.40 ! [C: int,B: int,A: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ C @ B )
% 0.16/0.40 => ( ( dvd_dvd_int @ B @ A )
% 0.16/0.40 => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.16/0.40 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_div_eq_right
% 0.16/0.40 thf(fact_276_dvd__div__mult2__eq,axiom,
% 0.16/0.40 ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
% 0.16/0.40 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.16/0.40 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_mult2_eq
% 0.16/0.40 thf(fact_277_dvd__div__mult2__eq,axiom,
% 0.16/0.40 ! [B: nat,C: nat,A: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
% 0.16/0.40 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.16/0.40 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_mult2_eq
% 0.16/0.40 thf(fact_278_dvd__div__mult2__eq,axiom,
% 0.16/0.40 ! [B: int,C: int,A: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
% 0.16/0.40 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.16/0.40 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_div_mult2_eq
% 0.16/0.40 thf(fact_279_dvd__mult__imp__div,axiom,
% 0.16/0.40 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
% 0.16/0.40 => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult_imp_div
% 0.16/0.40 thf(fact_280_dvd__mult__imp__div,axiom,
% 0.16/0.40 ! [A: nat,C: nat,B: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
% 0.16/0.40 => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult_imp_div
% 0.16/0.40 thf(fact_281_dvd__mult__imp__div,axiom,
% 0.16/0.40 ! [A: int,C: int,B: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
% 0.16/0.40 => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % dvd_mult_imp_div
% 0.16/0.40 thf(fact_282_div__mult__div__if__dvd,axiom,
% 0.16/0.40 ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.16/0.40 => ( ( dvd_dvd_Code_integer @ D @ C )
% 0.16/0.40 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
% 0.16/0.40 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_mult_div_if_dvd
% 0.16/0.40 thf(fact_283_div__mult__div__if__dvd,axiom,
% 0.16/0.40 ! [B: nat,A: nat,D: nat,C: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ B @ A )
% 0.16/0.40 => ( ( dvd_dvd_nat @ D @ C )
% 0.16/0.40 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
% 0.16/0.40 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_mult_div_if_dvd
% 0.16/0.40 thf(fact_284_div__mult__div__if__dvd,axiom,
% 0.16/0.40 ! [B: int,A: int,D: int,C: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ B @ A )
% 0.16/0.40 => ( ( dvd_dvd_int @ D @ C )
% 0.16/0.40 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
% 0.16/0.40 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % div_mult_div_if_dvd
% 0.16/0.40 thf(fact_285_real__of__nat__div4,axiom,
% 0.16/0.40 ! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % real_of_nat_div4
% 0.16/0.40 thf(fact_286_real__of__nat__div,axiom,
% 0.16/0.40 ! [D: nat,N: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ D @ N )
% 0.16/0.40 => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
% 0.16/0.40 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % real_of_nat_div
% 0.16/0.40 thf(fact_287_even__numeral,axiom,
% 0.16/0.40 ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % even_numeral
% 0.16/0.40 thf(fact_288_even__numeral,axiom,
% 0.16/0.40 ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % even_numeral
% 0.16/0.40 thf(fact_289_even__numeral,axiom,
% 0.16/0.40 ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % even_numeral
% 0.16/0.40 thf(fact_290_evenE,axiom,
% 0.16/0.40 ! [A: code_integer] :
% 0.16/0.40 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.16/0.40 => ~ ! [B3: code_integer] :
% 0.16/0.40 ( A
% 0.16/0.40 != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % evenE
% 0.16/0.40 thf(fact_291_evenE,axiom,
% 0.16/0.40 ! [A: nat] :
% 0.16/0.40 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.16/0.40 => ~ ! [B3: nat] :
% 0.16/0.40 ( A
% 0.16/0.40 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % evenE
% 0.16/0.40 thf(fact_292_evenE,axiom,
% 0.16/0.40 ! [A: int] :
% 0.16/0.40 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.16/0.40 => ~ ! [B3: int] :
% 0.16/0.40 ( A
% 0.16/0.40 != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % evenE
% 0.16/0.40 thf(fact_293_odd__numeral,axiom,
% 0.16/0.40 ! [N: num] :
% 0.16/0.40 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % odd_numeral
% 0.16/0.40 thf(fact_294_odd__numeral,axiom,
% 0.16/0.40 ! [N: num] :
% 0.16/0.40 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % odd_numeral
% 0.16/0.40 thf(fact_295_odd__numeral,axiom,
% 0.16/0.40 ! [N: num] :
% 0.16/0.40 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % odd_numeral
% 0.16/0.40 thf(fact_296_Suc__double__not__eq__double,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.16/0.40 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % Suc_double_not_eq_double
% 0.16/0.40 thf(fact_297_double__not__eq__Suc__double,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.16/0.40 != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % double_not_eq_Suc_double
% 0.16/0.40 thf(fact_298_zdiv__numeral__Bit1,axiom,
% 0.16/0.40 ! [V: num,W: num] :
% 0.16/0.40 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.16/0.40 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % zdiv_numeral_Bit1
% 0.16/0.40 thf(fact_299_bit__eq__rec,axiom,
% 0.16/0.40 ( ( ^ [Y5: code_integer,Z4: code_integer] : ( Y5 = Z4 ) )
% 0.16/0.40 = ( ^ [A3: code_integer,B2: code_integer] :
% 0.16/0.40 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
% 0.16/0.40 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) )
% 0.16/0.40 & ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.16/0.40 = ( divide6298287555418463151nteger @ B2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % bit_eq_rec
% 0.16/0.40 thf(fact_300_bit__eq__rec,axiom,
% 0.16/0.40 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.16/0.40 = ( ^ [A3: nat,B2: nat] :
% 0.16/0.40 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
% 0.16/0.40 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
% 0.16/0.40 & ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.40 = ( divide_divide_nat @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % bit_eq_rec
% 0.16/0.40 thf(fact_301_bit__eq__rec,axiom,
% 0.16/0.40 ( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.16/0.40 = ( ^ [A3: int,B2: int] :
% 0.16/0.40 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
% 0.16/0.40 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
% 0.16/0.40 & ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.16/0.40 = ( divide_divide_int @ B2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % bit_eq_rec
% 0.16/0.40 thf(fact_302_times__divide__eq__left,axiom,
% 0.16/0.40 ! [B: complex,C: complex,A: complex] :
% 0.16/0.40 ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
% 0.16/0.40 = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).
% 0.16/0.40
% 0.16/0.40 % times_divide_eq_left
% 0.16/0.40 thf(fact_303_times__divide__eq__left,axiom,
% 0.16/0.40 ! [B: real,C: real,A: real] :
% 0.16/0.40 ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.16/0.40 = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% 0.16/0.40
% 0.16/0.40 % times_divide_eq_left
% 0.16/0.40 thf(fact_304_times__divide__eq__left,axiom,
% 0.16/0.40 ! [B: rat,C: rat,A: rat] :
% 0.16/0.40 ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.16/0.40 = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).
% 0.16/0.40
% 0.16/0.40 % times_divide_eq_left
% 0.16/0.40 thf(fact_305_divide__divide__eq__left,axiom,
% 0.16/0.40 ! [A: complex,B: complex,C: complex] :
% 0.16/0.40 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 0.16/0.40 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % divide_divide_eq_left
% 0.16/0.40 thf(fact_306_divide__divide__eq__left,axiom,
% 0.16/0.40 ! [A: real,B: real,C: real] :
% 0.16/0.40 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 0.16/0.40 = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % divide_divide_eq_left
% 0.16/0.40 thf(fact_307_divide__divide__eq__left,axiom,
% 0.16/0.40 ! [A: rat,B: rat,C: rat] :
% 0.16/0.40 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 0.16/0.40 = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % divide_divide_eq_left
% 0.16/0.40 thf(fact_308_divide__divide__eq__right,axiom,
% 0.16/0.40 ! [A: complex,B: complex,C: complex] :
% 0.16/0.40 ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.16/0.40 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).
% 0.16/0.40
% 0.16/0.40 % divide_divide_eq_right
% 0.16/0.40 thf(fact_309_divide__divide__eq__right,axiom,
% 0.16/0.40 ! [A: real,B: real,C: real] :
% 0.16/0.40 ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.16/0.40 = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% 0.16/0.40
% 0.16/0.40 % divide_divide_eq_right
% 0.16/0.40 thf(fact_310_divide__divide__eq__right,axiom,
% 0.16/0.40 ! [A: rat,B: rat,C: rat] :
% 0.16/0.40 ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.16/0.40 = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).
% 0.16/0.40
% 0.16/0.40 % divide_divide_eq_right
% 0.16/0.40 thf(fact_311_times__divide__eq__right,axiom,
% 0.16/0.40 ! [A: complex,B: complex,C: complex] :
% 0.16/0.40 ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.16/0.40 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).
% 0.16/0.40
% 0.16/0.40 % times_divide_eq_right
% 0.16/0.40 thf(fact_312_times__divide__eq__right,axiom,
% 0.16/0.40 ! [A: real,B: real,C: real] :
% 0.16/0.40 ( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.16/0.40 = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% 0.16/0.40
% 0.16/0.40 % times_divide_eq_right
% 0.16/0.40 thf(fact_313_times__divide__eq__right,axiom,
% 0.16/0.40 ! [A: rat,B: rat,C: rat] :
% 0.16/0.40 ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.16/0.40 = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).
% 0.16/0.40
% 0.16/0.40 % times_divide_eq_right
% 0.16/0.40 thf(fact_314_buildup__nothing__in__leaf,axiom,
% 0.16/0.40 ! [N: nat,X: nat] :
% 0.16/0.40 ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% 0.16/0.40
% 0.16/0.40 % buildup_nothing_in_leaf
% 0.16/0.40 thf(fact_315_triangle__def,axiom,
% 0.16/0.40 ( nat_triangle
% 0.16/0.40 = ( ^ [N4: nat] : ( divide_divide_nat @ ( times_times_nat @ N4 @ ( suc @ N4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % triangle_def
% 0.16/0.40 thf(fact_316_set__decode__Suc,axiom,
% 0.16/0.40 ! [N: nat,X: nat] :
% 0.16/0.40 ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
% 0.16/0.40 = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % set_decode_Suc
% 0.16/0.40 thf(fact_317_int__dvd__int__iff,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.16/0.40 = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % int_dvd_int_iff
% 0.16/0.40 thf(fact_318_real__divide__square__eq,axiom,
% 0.16/0.40 ! [R2: real,A: real] :
% 0.16/0.40 ( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
% 0.16/0.40 = ( divide_divide_real @ A @ R2 ) ) ).
% 0.16/0.40
% 0.16/0.40 % real_divide_square_eq
% 0.16/0.40 thf(fact_319_int__eq__iff__numeral,axiom,
% 0.16/0.40 ! [M: nat,V: num] :
% 0.16/0.40 ( ( ( semiri1314217659103216013at_int @ M )
% 0.16/0.40 = ( numeral_numeral_int @ V ) )
% 0.16/0.40 = ( M
% 0.16/0.40 = ( numeral_numeral_nat @ V ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % int_eq_iff_numeral
% 0.16/0.40 thf(fact_320_zdiv__numeral__Bit0,axiom,
% 0.16/0.40 ! [V: num,W: num] :
% 0.16/0.40 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.16/0.40 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % zdiv_numeral_Bit0
% 0.16/0.40 thf(fact_321_int__int__eq,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( ( semiri1314217659103216013at_int @ M )
% 0.16/0.40 = ( semiri1314217659103216013at_int @ N ) )
% 0.16/0.40 = ( M = N ) ) ).
% 0.16/0.40
% 0.16/0.40 % int_int_eq
% 0.16/0.40 thf(fact_322_subset__decode__imp__le,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 0.16/0.40 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % subset_decode_imp_le
% 0.16/0.40 thf(fact_323_zle__int,axiom,
% 0.16/0.40 ! [M: nat,N: nat] :
% 0.16/0.40 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.16/0.40 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % zle_int
% 0.16/0.40 thf(fact_324_zdiv__int,axiom,
% 0.16/0.40 ! [A: nat,B: nat] :
% 0.16/0.40 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
% 0.16/0.40 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % zdiv_int
% 0.16/0.40 thf(fact_325_divide__divide__eq__left_H,axiom,
% 0.16/0.40 ! [A: complex,B: complex,C: complex] :
% 0.16/0.40 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 0.16/0.40 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % divide_divide_eq_left'
% 0.16/0.40 thf(fact_326_divide__divide__eq__left_H,axiom,
% 0.16/0.40 ! [A: real,B: real,C: real] :
% 0.16/0.40 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 0.16/0.40 = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % divide_divide_eq_left'
% 0.16/0.40 thf(fact_327_divide__divide__eq__left_H,axiom,
% 0.16/0.40 ! [A: rat,B: rat,C: rat] :
% 0.16/0.40 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 0.16/0.40 = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % divide_divide_eq_left'
% 0.16/0.40 thf(fact_328_divide__divide__times__eq,axiom,
% 0.16/0.40 ! [X: complex,Y4: complex,Z: complex,W: complex] :
% 0.16/0.40 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y4 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 0.16/0.40 = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y4 @ Z ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % divide_divide_times_eq
% 0.16/0.40 thf(fact_329_divide__divide__times__eq,axiom,
% 0.16/0.40 ! [X: real,Y4: real,Z: real,W: real] :
% 0.16/0.40 ( ( divide_divide_real @ ( divide_divide_real @ X @ Y4 ) @ ( divide_divide_real @ Z @ W ) )
% 0.16/0.40 = ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y4 @ Z ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % divide_divide_times_eq
% 0.16/0.40 thf(fact_330_divide__divide__times__eq,axiom,
% 0.16/0.40 ! [X: rat,Y4: rat,Z: rat,W: rat] :
% 0.16/0.40 ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y4 ) @ ( divide_divide_rat @ Z @ W ) )
% 0.16/0.40 = ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y4 @ Z ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % divide_divide_times_eq
% 0.16/0.40 thf(fact_331_times__divide__times__eq,axiom,
% 0.16/0.40 ! [X: complex,Y4: complex,Z: complex,W: complex] :
% 0.16/0.40 ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y4 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 0.16/0.40 = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y4 @ W ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % times_divide_times_eq
% 0.16/0.40 thf(fact_332_times__divide__times__eq,axiom,
% 0.16/0.40 ! [X: real,Y4: real,Z: real,W: real] :
% 0.16/0.40 ( ( times_times_real @ ( divide_divide_real @ X @ Y4 ) @ ( divide_divide_real @ Z @ W ) )
% 0.16/0.40 = ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y4 @ W ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % times_divide_times_eq
% 0.16/0.40 thf(fact_333_times__divide__times__eq,axiom,
% 0.16/0.40 ! [X: rat,Y4: rat,Z: rat,W: rat] :
% 0.16/0.40 ( ( times_times_rat @ ( divide_divide_rat @ X @ Y4 ) @ ( divide_divide_rat @ Z @ W ) )
% 0.16/0.40 = ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y4 @ W ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % times_divide_times_eq
% 0.16/0.40 thf(fact_334_int__ops_I3_J,axiom,
% 0.16/0.40 ! [N: num] :
% 0.16/0.40 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 0.16/0.40 = ( numeral_numeral_int @ N ) ) ).
% 0.16/0.40
% 0.16/0.40 % int_ops(3)
% 0.16/0.40 thf(fact_335_verit__eq__simplify_I9_J,axiom,
% 0.16/0.40 ! [X33: num,Y32: num] :
% 0.16/0.40 ( ( ( bit1 @ X33 )
% 0.16/0.40 = ( bit1 @ Y32 ) )
% 0.16/0.40 = ( X33 = Y32 ) ) ).
% 0.16/0.40
% 0.16/0.40 % verit_eq_simplify(9)
% 0.16/0.40 thf(fact_336_verit__eq__simplify_I8_J,axiom,
% 0.16/0.40 ! [X2: num,Y2: num] :
% 0.16/0.40 ( ( ( bit0 @ X2 )
% 0.16/0.40 = ( bit0 @ Y2 ) )
% 0.16/0.40 = ( X2 = Y2 ) ) ).
% 0.16/0.40
% 0.16/0.40 % verit_eq_simplify(8)
% 0.16/0.40 thf(fact_337_dual__order_Orefl,axiom,
% 0.16/0.40 ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 0.16/0.40
% 0.16/0.40 % dual_order.refl
% 0.16/0.40 thf(fact_338_dual__order_Orefl,axiom,
% 0.16/0.40 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 0.16/0.40
% 0.16/0.40 % dual_order.refl
% 0.16/0.40 thf(fact_339_dual__order_Orefl,axiom,
% 0.16/0.40 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 0.16/0.40
% 0.16/0.40 % dual_order.refl
% 0.16/0.40 thf(fact_340_dual__order_Orefl,axiom,
% 0.16/0.40 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.16/0.40
% 0.16/0.40 % dual_order.refl
% 0.16/0.40 thf(fact_341_dual__order_Orefl,axiom,
% 0.16/0.40 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.16/0.40
% 0.16/0.40 % dual_order.refl
% 0.16/0.40 thf(fact_342_dual__order_Orefl,axiom,
% 0.16/0.40 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 0.16/0.40
% 0.16/0.40 % dual_order.refl
% 0.16/0.40 thf(fact_343_order__refl,axiom,
% 0.16/0.40 ! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).
% 0.16/0.40
% 0.16/0.40 % order_refl
% 0.16/0.40 thf(fact_344_order__refl,axiom,
% 0.16/0.40 ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).
% 0.16/0.40
% 0.16/0.40 % order_refl
% 0.16/0.40 thf(fact_345_order__refl,axiom,
% 0.16/0.40 ! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% 0.16/0.40
% 0.16/0.40 % order_refl
% 0.16/0.40 thf(fact_346_order__refl,axiom,
% 0.16/0.40 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.16/0.40
% 0.16/0.40 % order_refl
% 0.16/0.40 thf(fact_347_order__refl,axiom,
% 0.16/0.40 ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 0.16/0.40
% 0.16/0.40 % order_refl
% 0.16/0.40 thf(fact_348_order__refl,axiom,
% 0.16/0.40 ! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% 0.16/0.40
% 0.16/0.40 % order_refl
% 0.16/0.40 thf(fact_349_cnt__non__neg,axiom,
% 0.16/0.40 ! [T: vEBT_VEBT] : ( ord_less_eq_real @ zero_zero_real @ ( vEBT_VEBT_cnt @ T ) ) ).
% 0.16/0.40
% 0.16/0.40 % cnt_non_neg
% 0.16/0.40 thf(fact_350_buildup__nothing__in__min__max,axiom,
% 0.16/0.40 ! [N: nat,X: nat] :
% 0.16/0.40 ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% 0.16/0.40
% 0.16/0.40 % buildup_nothing_in_min_max
% 0.16/0.40 thf(fact_351_set__decode__0,axiom,
% 0.16/0.40 ! [X: nat] :
% 0.16/0.40 ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
% 0.16/0.40 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % set_decode_0
% 0.16/0.40 thf(fact_352_int__ops_I7_J,axiom,
% 0.16/0.40 ! [A: nat,B: nat] :
% 0.16/0.40 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
% 0.16/0.40 = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.16/0.40
% 0.16/0.40 % int_ops(7)
% 0.16/0.40 thf(fact_353_nat__int__comparison_I3_J,axiom,
% 0.16/0.40 ( ord_less_eq_nat
% 0.16/0.40 = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nat_int_comparison(3)
% 0.16/0.41 thf(fact_354_verit__eq__simplify_I12_J,axiom,
% 0.16/0.41 ! [X33: num] :
% 0.16/0.41 ( one
% 0.16/0.41 != ( bit1 @ X33 ) ) ).
% 0.16/0.41
% 0.16/0.41 % verit_eq_simplify(12)
% 0.16/0.41 thf(fact_355_verit__eq__simplify_I14_J,axiom,
% 0.16/0.41 ! [X2: num,X33: num] :
% 0.16/0.41 ( ( bit0 @ X2 )
% 0.16/0.41 != ( bit1 @ X33 ) ) ).
% 0.16/0.41
% 0.16/0.41 % verit_eq_simplify(14)
% 0.16/0.41 thf(fact_356_mult__zero__left,axiom,
% 0.16/0.41 ! [A: complex] :
% 0.16/0.41 ( ( times_times_complex @ zero_zero_complex @ A )
% 0.16/0.41 = zero_zero_complex ) ).
% 0.16/0.41
% 0.16/0.41 % mult_zero_left
% 0.16/0.41 thf(fact_357_mult__zero__left,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( times_times_real @ zero_zero_real @ A )
% 0.16/0.41 = zero_zero_real ) ).
% 0.16/0.41
% 0.16/0.41 % mult_zero_left
% 0.16/0.41 thf(fact_358_mult__zero__left,axiom,
% 0.16/0.41 ! [A: rat] :
% 0.16/0.41 ( ( times_times_rat @ zero_zero_rat @ A )
% 0.16/0.41 = zero_zero_rat ) ).
% 0.16/0.41
% 0.16/0.41 % mult_zero_left
% 0.16/0.41 thf(fact_359_mult__zero__left,axiom,
% 0.16/0.41 ! [A: nat] :
% 0.16/0.41 ( ( times_times_nat @ zero_zero_nat @ A )
% 0.16/0.41 = zero_zero_nat ) ).
% 0.16/0.41
% 0.16/0.41 % mult_zero_left
% 0.16/0.41 thf(fact_360_mult__zero__left,axiom,
% 0.16/0.41 ! [A: int] :
% 0.16/0.41 ( ( times_times_int @ zero_zero_int @ A )
% 0.16/0.41 = zero_zero_int ) ).
% 0.16/0.41
% 0.16/0.41 % mult_zero_left
% 0.16/0.41 thf(fact_361_mult__zero__right,axiom,
% 0.16/0.41 ! [A: complex] :
% 0.16/0.41 ( ( times_times_complex @ A @ zero_zero_complex )
% 0.16/0.41 = zero_zero_complex ) ).
% 0.16/0.41
% 0.16/0.41 % mult_zero_right
% 0.16/0.41 thf(fact_362_mult__zero__right,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( times_times_real @ A @ zero_zero_real )
% 0.16/0.41 = zero_zero_real ) ).
% 0.16/0.41
% 0.16/0.41 % mult_zero_right
% 0.16/0.41 thf(fact_363_mult__zero__right,axiom,
% 0.16/0.41 ! [A: rat] :
% 0.16/0.41 ( ( times_times_rat @ A @ zero_zero_rat )
% 0.16/0.41 = zero_zero_rat ) ).
% 0.16/0.41
% 0.16/0.41 % mult_zero_right
% 0.16/0.41 thf(fact_364_mult__zero__right,axiom,
% 0.16/0.41 ! [A: nat] :
% 0.16/0.41 ( ( times_times_nat @ A @ zero_zero_nat )
% 0.16/0.41 = zero_zero_nat ) ).
% 0.16/0.41
% 0.16/0.41 % mult_zero_right
% 0.16/0.41 thf(fact_365_mult__zero__right,axiom,
% 0.16/0.41 ! [A: int] :
% 0.16/0.41 ( ( times_times_int @ A @ zero_zero_int )
% 0.16/0.41 = zero_zero_int ) ).
% 0.16/0.41
% 0.16/0.41 % mult_zero_right
% 0.16/0.41 thf(fact_366_mult__eq__0__iff,axiom,
% 0.16/0.41 ! [A: complex,B: complex] :
% 0.16/0.41 ( ( ( times_times_complex @ A @ B )
% 0.16/0.41 = zero_zero_complex )
% 0.16/0.41 = ( ( A = zero_zero_complex )
% 0.16/0.41 | ( B = zero_zero_complex ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_eq_0_iff
% 0.16/0.41 thf(fact_367_mult__eq__0__iff,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ( times_times_real @ A @ B )
% 0.16/0.41 = zero_zero_real )
% 0.16/0.41 = ( ( A = zero_zero_real )
% 0.16/0.41 | ( B = zero_zero_real ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_eq_0_iff
% 0.16/0.41 thf(fact_368_mult__eq__0__iff,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( ( times_times_rat @ A @ B )
% 0.16/0.41 = zero_zero_rat )
% 0.16/0.41 = ( ( A = zero_zero_rat )
% 0.16/0.41 | ( B = zero_zero_rat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_eq_0_iff
% 0.16/0.41 thf(fact_369_mult__eq__0__iff,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ( times_times_nat @ A @ B )
% 0.16/0.41 = zero_zero_nat )
% 0.16/0.41 = ( ( A = zero_zero_nat )
% 0.16/0.41 | ( B = zero_zero_nat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_eq_0_iff
% 0.16/0.41 thf(fact_370_mult__eq__0__iff,axiom,
% 0.16/0.41 ! [A: int,B: int] :
% 0.16/0.41 ( ( ( times_times_int @ A @ B )
% 0.16/0.41 = zero_zero_int )
% 0.16/0.41 = ( ( A = zero_zero_int )
% 0.16/0.41 | ( B = zero_zero_int ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_eq_0_iff
% 0.16/0.41 thf(fact_371_mult__cancel__left,axiom,
% 0.16/0.41 ! [C: complex,A: complex,B: complex] :
% 0.16/0.41 ( ( ( times_times_complex @ C @ A )
% 0.16/0.41 = ( times_times_complex @ C @ B ) )
% 0.16/0.41 = ( ( C = zero_zero_complex )
% 0.16/0.41 | ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_cancel_left
% 0.16/0.41 thf(fact_372_mult__cancel__left,axiom,
% 0.16/0.41 ! [C: real,A: real,B: real] :
% 0.16/0.41 ( ( ( times_times_real @ C @ A )
% 0.16/0.41 = ( times_times_real @ C @ B ) )
% 0.16/0.41 = ( ( C = zero_zero_real )
% 0.16/0.41 | ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_cancel_left
% 0.16/0.41 thf(fact_373_mult__cancel__left,axiom,
% 0.16/0.41 ! [C: rat,A: rat,B: rat] :
% 0.16/0.41 ( ( ( times_times_rat @ C @ A )
% 0.16/0.41 = ( times_times_rat @ C @ B ) )
% 0.16/0.41 = ( ( C = zero_zero_rat )
% 0.16/0.41 | ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_cancel_left
% 0.16/0.41 thf(fact_374_mult__cancel__left,axiom,
% 0.16/0.41 ! [C: nat,A: nat,B: nat] :
% 0.16/0.41 ( ( ( times_times_nat @ C @ A )
% 0.16/0.41 = ( times_times_nat @ C @ B ) )
% 0.16/0.41 = ( ( C = zero_zero_nat )
% 0.16/0.41 | ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_cancel_left
% 0.16/0.41 thf(fact_375_mult__cancel__left,axiom,
% 0.16/0.41 ! [C: int,A: int,B: int] :
% 0.16/0.41 ( ( ( times_times_int @ C @ A )
% 0.16/0.41 = ( times_times_int @ C @ B ) )
% 0.16/0.41 = ( ( C = zero_zero_int )
% 0.16/0.41 | ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_cancel_left
% 0.16/0.41 thf(fact_376_mult__cancel__right,axiom,
% 0.16/0.41 ! [A: complex,C: complex,B: complex] :
% 0.16/0.41 ( ( ( times_times_complex @ A @ C )
% 0.16/0.41 = ( times_times_complex @ B @ C ) )
% 0.16/0.41 = ( ( C = zero_zero_complex )
% 0.16/0.41 | ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_cancel_right
% 0.16/0.41 thf(fact_377_mult__cancel__right,axiom,
% 0.16/0.41 ! [A: real,C: real,B: real] :
% 0.16/0.41 ( ( ( times_times_real @ A @ C )
% 0.16/0.41 = ( times_times_real @ B @ C ) )
% 0.16/0.41 = ( ( C = zero_zero_real )
% 0.16/0.41 | ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_cancel_right
% 0.16/0.41 thf(fact_378_mult__cancel__right,axiom,
% 0.16/0.41 ! [A: rat,C: rat,B: rat] :
% 0.16/0.41 ( ( ( times_times_rat @ A @ C )
% 0.16/0.41 = ( times_times_rat @ B @ C ) )
% 0.16/0.41 = ( ( C = zero_zero_rat )
% 0.16/0.41 | ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_cancel_right
% 0.16/0.41 thf(fact_379_mult__cancel__right,axiom,
% 0.16/0.41 ! [A: nat,C: nat,B: nat] :
% 0.16/0.41 ( ( ( times_times_nat @ A @ C )
% 0.16/0.41 = ( times_times_nat @ B @ C ) )
% 0.16/0.41 = ( ( C = zero_zero_nat )
% 0.16/0.41 | ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_cancel_right
% 0.16/0.41 thf(fact_380_mult__cancel__right,axiom,
% 0.16/0.41 ! [A: int,C: int,B: int] :
% 0.16/0.41 ( ( ( times_times_int @ A @ C )
% 0.16/0.41 = ( times_times_int @ B @ C ) )
% 0.16/0.41 = ( ( C = zero_zero_int )
% 0.16/0.41 | ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_cancel_right
% 0.16/0.41 thf(fact_381_div__0,axiom,
% 0.16/0.41 ! [A: complex] :
% 0.16/0.41 ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 0.16/0.41 = zero_zero_complex ) ).
% 0.16/0.41
% 0.16/0.41 % div_0
% 0.16/0.41 thf(fact_382_div__0,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( divide_divide_real @ zero_zero_real @ A )
% 0.16/0.41 = zero_zero_real ) ).
% 0.16/0.41
% 0.16/0.41 % div_0
% 0.16/0.41 thf(fact_383_div__0,axiom,
% 0.16/0.41 ! [A: rat] :
% 0.16/0.41 ( ( divide_divide_rat @ zero_zero_rat @ A )
% 0.16/0.41 = zero_zero_rat ) ).
% 0.16/0.41
% 0.16/0.41 % div_0
% 0.16/0.41 thf(fact_384_div__0,axiom,
% 0.16/0.41 ! [A: nat] :
% 0.16/0.41 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.16/0.41 = zero_zero_nat ) ).
% 0.16/0.41
% 0.16/0.41 % div_0
% 0.16/0.41 thf(fact_385_div__0,axiom,
% 0.16/0.41 ! [A: int] :
% 0.16/0.41 ( ( divide_divide_int @ zero_zero_int @ A )
% 0.16/0.41 = zero_zero_int ) ).
% 0.16/0.41
% 0.16/0.41 % div_0
% 0.16/0.41 thf(fact_386_div__by__0,axiom,
% 0.16/0.41 ! [A: complex] :
% 0.16/0.41 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.16/0.41 = zero_zero_complex ) ).
% 0.16/0.41
% 0.16/0.41 % div_by_0
% 0.16/0.41 thf(fact_387_div__by__0,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( divide_divide_real @ A @ zero_zero_real )
% 0.16/0.41 = zero_zero_real ) ).
% 0.16/0.41
% 0.16/0.41 % div_by_0
% 0.16/0.41 thf(fact_388_div__by__0,axiom,
% 0.16/0.41 ! [A: rat] :
% 0.16/0.41 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.16/0.41 = zero_zero_rat ) ).
% 0.16/0.41
% 0.16/0.41 % div_by_0
% 0.16/0.41 thf(fact_389_div__by__0,axiom,
% 0.16/0.41 ! [A: nat] :
% 0.16/0.41 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.16/0.41 = zero_zero_nat ) ).
% 0.16/0.41
% 0.16/0.41 % div_by_0
% 0.16/0.41 thf(fact_390_div__by__0,axiom,
% 0.16/0.41 ! [A: int] :
% 0.16/0.41 ( ( divide_divide_int @ A @ zero_zero_int )
% 0.16/0.41 = zero_zero_int ) ).
% 0.16/0.41
% 0.16/0.41 % div_by_0
% 0.16/0.41 thf(fact_391_divide__eq__0__iff,axiom,
% 0.16/0.41 ! [A: complex,B: complex] :
% 0.16/0.41 ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.16/0.41 = zero_zero_complex )
% 0.16/0.41 = ( ( A = zero_zero_complex )
% 0.16/0.41 | ( B = zero_zero_complex ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_eq_0_iff
% 0.16/0.41 thf(fact_392_divide__eq__0__iff,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ( divide_divide_real @ A @ B )
% 0.16/0.41 = zero_zero_real )
% 0.16/0.41 = ( ( A = zero_zero_real )
% 0.16/0.41 | ( B = zero_zero_real ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_eq_0_iff
% 0.16/0.41 thf(fact_393_divide__eq__0__iff,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( ( divide_divide_rat @ A @ B )
% 0.16/0.41 = zero_zero_rat )
% 0.16/0.41 = ( ( A = zero_zero_rat )
% 0.16/0.41 | ( B = zero_zero_rat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_eq_0_iff
% 0.16/0.41 thf(fact_394_divide__cancel__left,axiom,
% 0.16/0.41 ! [C: complex,A: complex,B: complex] :
% 0.16/0.41 ( ( ( divide1717551699836669952omplex @ C @ A )
% 0.16/0.41 = ( divide1717551699836669952omplex @ C @ B ) )
% 0.16/0.41 = ( ( C = zero_zero_complex )
% 0.16/0.41 | ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_cancel_left
% 0.16/0.41 thf(fact_395_divide__cancel__left,axiom,
% 0.16/0.41 ! [C: real,A: real,B: real] :
% 0.16/0.41 ( ( ( divide_divide_real @ C @ A )
% 0.16/0.41 = ( divide_divide_real @ C @ B ) )
% 0.16/0.41 = ( ( C = zero_zero_real )
% 0.16/0.41 | ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_cancel_left
% 0.16/0.41 thf(fact_396_divide__cancel__left,axiom,
% 0.16/0.41 ! [C: rat,A: rat,B: rat] :
% 0.16/0.41 ( ( ( divide_divide_rat @ C @ A )
% 0.16/0.41 = ( divide_divide_rat @ C @ B ) )
% 0.16/0.41 = ( ( C = zero_zero_rat )
% 0.16/0.41 | ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_cancel_left
% 0.16/0.41 thf(fact_397_divide__cancel__right,axiom,
% 0.16/0.41 ! [A: complex,C: complex,B: complex] :
% 0.16/0.41 ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.16/0.41 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.16/0.41 = ( ( C = zero_zero_complex )
% 0.16/0.41 | ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_cancel_right
% 0.16/0.41 thf(fact_398_divide__cancel__right,axiom,
% 0.16/0.41 ! [A: real,C: real,B: real] :
% 0.16/0.41 ( ( ( divide_divide_real @ A @ C )
% 0.16/0.41 = ( divide_divide_real @ B @ C ) )
% 0.16/0.41 = ( ( C = zero_zero_real )
% 0.16/0.41 | ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_cancel_right
% 0.16/0.41 thf(fact_399_divide__cancel__right,axiom,
% 0.16/0.41 ! [A: rat,C: rat,B: rat] :
% 0.16/0.41 ( ( ( divide_divide_rat @ A @ C )
% 0.16/0.41 = ( divide_divide_rat @ B @ C ) )
% 0.16/0.41 = ( ( C = zero_zero_rat )
% 0.16/0.41 | ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_cancel_right
% 0.16/0.41 thf(fact_400_bits__div__0,axiom,
% 0.16/0.41 ! [A: nat] :
% 0.16/0.41 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.16/0.41 = zero_zero_nat ) ).
% 0.16/0.41
% 0.16/0.41 % bits_div_0
% 0.16/0.41 thf(fact_401_bits__div__0,axiom,
% 0.16/0.41 ! [A: int] :
% 0.16/0.41 ( ( divide_divide_int @ zero_zero_int @ A )
% 0.16/0.41 = zero_zero_int ) ).
% 0.16/0.41
% 0.16/0.41 % bits_div_0
% 0.16/0.41 thf(fact_402_bits__div__by__0,axiom,
% 0.16/0.41 ! [A: nat] :
% 0.16/0.41 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.16/0.41 = zero_zero_nat ) ).
% 0.16/0.41
% 0.16/0.41 % bits_div_by_0
% 0.16/0.41 thf(fact_403_bits__div__by__0,axiom,
% 0.16/0.41 ! [A: int] :
% 0.16/0.41 ( ( divide_divide_int @ A @ zero_zero_int )
% 0.16/0.41 = zero_zero_int ) ).
% 0.16/0.41
% 0.16/0.41 % bits_div_by_0
% 0.16/0.41 thf(fact_404_division__ring__divide__zero,axiom,
% 0.16/0.41 ! [A: complex] :
% 0.16/0.41 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.16/0.41 = zero_zero_complex ) ).
% 0.16/0.41
% 0.16/0.41 % division_ring_divide_zero
% 0.16/0.41 thf(fact_405_division__ring__divide__zero,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( divide_divide_real @ A @ zero_zero_real )
% 0.16/0.41 = zero_zero_real ) ).
% 0.16/0.41
% 0.16/0.41 % division_ring_divide_zero
% 0.16/0.41 thf(fact_406_division__ring__divide__zero,axiom,
% 0.16/0.41 ! [A: rat] :
% 0.16/0.41 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.16/0.41 = zero_zero_rat ) ).
% 0.16/0.41
% 0.16/0.41 % division_ring_divide_zero
% 0.16/0.41 thf(fact_407_dvd__0__left__iff,axiom,
% 0.16/0.41 ! [A: code_integer] :
% 0.16/0.41 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 0.16/0.41 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_0_left_iff
% 0.16/0.41 thf(fact_408_dvd__0__left__iff,axiom,
% 0.16/0.41 ! [A: complex] :
% 0.16/0.41 ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 0.16/0.41 = ( A = zero_zero_complex ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_0_left_iff
% 0.16/0.41 thf(fact_409_dvd__0__left__iff,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 0.16/0.41 = ( A = zero_zero_real ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_0_left_iff
% 0.16/0.41 thf(fact_410_dvd__0__left__iff,axiom,
% 0.16/0.41 ! [A: rat] :
% 0.16/0.41 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 0.16/0.41 = ( A = zero_zero_rat ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_0_left_iff
% 0.16/0.41 thf(fact_411_dvd__0__left__iff,axiom,
% 0.16/0.41 ! [A: nat] :
% 0.16/0.41 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.16/0.41 = ( A = zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_0_left_iff
% 0.16/0.41 thf(fact_412_dvd__0__left__iff,axiom,
% 0.16/0.41 ! [A: int] :
% 0.16/0.41 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 0.16/0.41 = ( A = zero_zero_int ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_0_left_iff
% 0.16/0.41 thf(fact_413_dvd__0__right,axiom,
% 0.16/0.41 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_0_right
% 0.16/0.41 thf(fact_414_dvd__0__right,axiom,
% 0.16/0.41 ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_0_right
% 0.16/0.41 thf(fact_415_dvd__0__right,axiom,
% 0.16/0.41 ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_0_right
% 0.16/0.41 thf(fact_416_dvd__0__right,axiom,
% 0.16/0.41 ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_0_right
% 0.16/0.41 thf(fact_417_dvd__0__right,axiom,
% 0.16/0.41 ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_0_right
% 0.16/0.41 thf(fact_418_dvd__0__right,axiom,
% 0.16/0.41 ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_0_right
% 0.16/0.41 thf(fact_419_le0,axiom,
% 0.16/0.41 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.16/0.41
% 0.16/0.41 % le0
% 0.16/0.41 thf(fact_420_bot__nat__0_Oextremum,axiom,
% 0.16/0.41 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.16/0.41
% 0.16/0.41 % bot_nat_0.extremum
% 0.16/0.41 thf(fact_421_mult__cancel2,axiom,
% 0.16/0.41 ! [M: nat,K: nat,N: nat] :
% 0.16/0.41 ( ( ( times_times_nat @ M @ K )
% 0.16/0.41 = ( times_times_nat @ N @ K ) )
% 0.16/0.41 = ( ( M = N )
% 0.16/0.41 | ( K = zero_zero_nat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_cancel2
% 0.16/0.41 thf(fact_422_mult__cancel1,axiom,
% 0.16/0.41 ! [K: nat,M: nat,N: nat] :
% 0.16/0.41 ( ( ( times_times_nat @ K @ M )
% 0.16/0.41 = ( times_times_nat @ K @ N ) )
% 0.16/0.41 = ( ( M = N )
% 0.16/0.41 | ( K = zero_zero_nat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_cancel1
% 0.16/0.41 thf(fact_423_mult__0__right,axiom,
% 0.16/0.41 ! [M: nat] :
% 0.16/0.41 ( ( times_times_nat @ M @ zero_zero_nat )
% 0.16/0.41 = zero_zero_nat ) ).
% 0.16/0.41
% 0.16/0.41 % mult_0_right
% 0.16/0.41 thf(fact_424_mult__is__0,axiom,
% 0.16/0.41 ! [M: nat,N: nat] :
% 0.16/0.41 ( ( ( times_times_nat @ M @ N )
% 0.16/0.41 = zero_zero_nat )
% 0.16/0.41 = ( ( M = zero_zero_nat )
% 0.16/0.41 | ( N = zero_zero_nat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_is_0
% 0.16/0.41 thf(fact_425_triangle__0,axiom,
% 0.16/0.41 ( ( nat_triangle @ zero_zero_nat )
% 0.16/0.41 = zero_zero_nat ) ).
% 0.16/0.41
% 0.16/0.41 % triangle_0
% 0.16/0.41 thf(fact_426_nonzero__mult__div__cancel__left,axiom,
% 0.16/0.41 ! [A: complex,B: complex] :
% 0.16/0.41 ( ( A != zero_zero_complex )
% 0.16/0.41 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
% 0.16/0.41 = B ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_div_cancel_left
% 0.16/0.41 thf(fact_427_nonzero__mult__div__cancel__left,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( A != zero_zero_real )
% 0.16/0.41 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
% 0.16/0.41 = B ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_div_cancel_left
% 0.16/0.41 thf(fact_428_nonzero__mult__div__cancel__left,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( A != zero_zero_rat )
% 0.16/0.41 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
% 0.16/0.41 = B ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_div_cancel_left
% 0.16/0.41 thf(fact_429_nonzero__mult__div__cancel__left,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( A != zero_zero_nat )
% 0.16/0.41 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
% 0.16/0.41 = B ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_div_cancel_left
% 0.16/0.41 thf(fact_430_nonzero__mult__div__cancel__left,axiom,
% 0.16/0.41 ! [A: int,B: int] :
% 0.16/0.41 ( ( A != zero_zero_int )
% 0.16/0.41 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
% 0.16/0.41 = B ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_div_cancel_left
% 0.16/0.41 thf(fact_431_nonzero__mult__div__cancel__right,axiom,
% 0.16/0.41 ! [B: complex,A: complex] :
% 0.16/0.41 ( ( B != zero_zero_complex )
% 0.16/0.41 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
% 0.16/0.41 = A ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_div_cancel_right
% 0.16/0.41 thf(fact_432_nonzero__mult__div__cancel__right,axiom,
% 0.16/0.41 ! [B: real,A: real] :
% 0.16/0.41 ( ( B != zero_zero_real )
% 0.16/0.41 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
% 0.16/0.41 = A ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_div_cancel_right
% 0.16/0.41 thf(fact_433_nonzero__mult__div__cancel__right,axiom,
% 0.16/0.41 ! [B: rat,A: rat] :
% 0.16/0.41 ( ( B != zero_zero_rat )
% 0.16/0.41 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
% 0.16/0.41 = A ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_div_cancel_right
% 0.16/0.41 thf(fact_434_nonzero__mult__div__cancel__right,axiom,
% 0.16/0.41 ! [B: nat,A: nat] :
% 0.16/0.41 ( ( B != zero_zero_nat )
% 0.16/0.41 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
% 0.16/0.41 = A ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_div_cancel_right
% 0.16/0.41 thf(fact_435_nonzero__mult__div__cancel__right,axiom,
% 0.16/0.41 ! [B: int,A: int] :
% 0.16/0.41 ( ( B != zero_zero_int )
% 0.16/0.41 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
% 0.16/0.41 = A ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_div_cancel_right
% 0.16/0.41 thf(fact_436_div__mult__mult1,axiom,
% 0.16/0.41 ! [C: nat,A: nat,B: nat] :
% 0.16/0.41 ( ( C != zero_zero_nat )
% 0.16/0.41 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.16/0.41 = ( divide_divide_nat @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % div_mult_mult1
% 0.16/0.41 thf(fact_437_div__mult__mult1,axiom,
% 0.16/0.41 ! [C: int,A: int,B: int] :
% 0.16/0.41 ( ( C != zero_zero_int )
% 0.16/0.41 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.16/0.41 = ( divide_divide_int @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % div_mult_mult1
% 0.16/0.41 thf(fact_438_div__mult__mult2,axiom,
% 0.16/0.41 ! [C: nat,A: nat,B: nat] :
% 0.16/0.41 ( ( C != zero_zero_nat )
% 0.16/0.41 => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.16/0.41 = ( divide_divide_nat @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % div_mult_mult2
% 0.16/0.41 thf(fact_439_div__mult__mult2,axiom,
% 0.16/0.41 ! [C: int,A: int,B: int] :
% 0.16/0.41 ( ( C != zero_zero_int )
% 0.16/0.41 => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.16/0.41 = ( divide_divide_int @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % div_mult_mult2
% 0.16/0.41 thf(fact_440_div__mult__mult1__if,axiom,
% 0.16/0.41 ! [C: nat,A: nat,B: nat] :
% 0.16/0.41 ( ( ( C = zero_zero_nat )
% 0.16/0.41 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.16/0.41 = zero_zero_nat ) )
% 0.16/0.41 & ( ( C != zero_zero_nat )
% 0.16/0.41 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.16/0.41 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % div_mult_mult1_if
% 0.16/0.41 thf(fact_441_div__mult__mult1__if,axiom,
% 0.16/0.41 ! [C: int,A: int,B: int] :
% 0.16/0.41 ( ( ( C = zero_zero_int )
% 0.16/0.41 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.16/0.41 = zero_zero_int ) )
% 0.16/0.41 & ( ( C != zero_zero_int )
% 0.16/0.41 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.16/0.41 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % div_mult_mult1_if
% 0.16/0.41 thf(fact_442_mult__divide__mult__cancel__left__if,axiom,
% 0.16/0.41 ! [C: complex,A: complex,B: complex] :
% 0.16/0.41 ( ( ( C = zero_zero_complex )
% 0.16/0.41 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.16/0.41 = zero_zero_complex ) )
% 0.16/0.41 & ( ( C != zero_zero_complex )
% 0.16/0.41 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.16/0.41 = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_divide_mult_cancel_left_if
% 0.16/0.41 thf(fact_443_mult__divide__mult__cancel__left__if,axiom,
% 0.16/0.41 ! [C: real,A: real,B: real] :
% 0.16/0.41 ( ( ( C = zero_zero_real )
% 0.16/0.41 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.16/0.41 = zero_zero_real ) )
% 0.16/0.41 & ( ( C != zero_zero_real )
% 0.16/0.41 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.16/0.41 = ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_divide_mult_cancel_left_if
% 0.16/0.41 thf(fact_444_mult__divide__mult__cancel__left__if,axiom,
% 0.16/0.41 ! [C: rat,A: rat,B: rat] :
% 0.16/0.41 ( ( ( C = zero_zero_rat )
% 0.16/0.41 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.16/0.41 = zero_zero_rat ) )
% 0.16/0.41 & ( ( C != zero_zero_rat )
% 0.16/0.41 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.16/0.41 = ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_divide_mult_cancel_left_if
% 0.16/0.41 thf(fact_445_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.16/0.41 ! [C: complex,A: complex,B: complex] :
% 0.16/0.41 ( ( C != zero_zero_complex )
% 0.16/0.41 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.16/0.41 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_divide_mult_cancel_left
% 0.16/0.41 thf(fact_446_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.16/0.41 ! [C: real,A: real,B: real] :
% 0.16/0.41 ( ( C != zero_zero_real )
% 0.16/0.41 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.16/0.41 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_divide_mult_cancel_left
% 0.16/0.41 thf(fact_447_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.16/0.41 ! [C: rat,A: rat,B: rat] :
% 0.16/0.41 ( ( C != zero_zero_rat )
% 0.16/0.41 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.16/0.41 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_divide_mult_cancel_left
% 0.16/0.41 thf(fact_448_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.16/0.41 ! [C: complex,A: complex,B: complex] :
% 0.16/0.41 ( ( C != zero_zero_complex )
% 0.16/0.41 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
% 0.16/0.41 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_divide_mult_cancel_left2
% 0.16/0.41 thf(fact_449_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.16/0.41 ! [C: real,A: real,B: real] :
% 0.16/0.41 ( ( C != zero_zero_real )
% 0.16/0.41 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
% 0.16/0.41 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_divide_mult_cancel_left2
% 0.16/0.41 thf(fact_450_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.16/0.41 ! [C: rat,A: rat,B: rat] :
% 0.16/0.41 ( ( C != zero_zero_rat )
% 0.16/0.41 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
% 0.16/0.41 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_divide_mult_cancel_left2
% 0.16/0.41 thf(fact_451_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.16/0.41 ! [C: complex,A: complex,B: complex] :
% 0.16/0.41 ( ( C != zero_zero_complex )
% 0.16/0.41 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 0.16/0.41 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_divide_mult_cancel_right
% 0.16/0.41 thf(fact_452_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.16/0.41 ! [C: real,A: real,B: real] :
% 0.16/0.41 ( ( C != zero_zero_real )
% 0.16/0.41 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.16/0.41 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_divide_mult_cancel_right
% 0.16/0.41 thf(fact_453_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.16/0.41 ! [C: rat,A: rat,B: rat] :
% 0.16/0.41 ( ( C != zero_zero_rat )
% 0.16/0.41 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.16/0.41 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_divide_mult_cancel_right
% 0.16/0.41 thf(fact_454_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.16/0.41 ! [C: complex,A: complex,B: complex] :
% 0.16/0.41 ( ( C != zero_zero_complex )
% 0.16/0.41 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
% 0.16/0.41 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_divide_mult_cancel_right2
% 0.16/0.41 thf(fact_455_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.16/0.41 ! [C: real,A: real,B: real] :
% 0.16/0.41 ( ( C != zero_zero_real )
% 0.16/0.41 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
% 0.16/0.41 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_divide_mult_cancel_right2
% 0.16/0.41 thf(fact_456_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.16/0.41 ! [C: rat,A: rat,B: rat] :
% 0.16/0.41 ( ( C != zero_zero_rat )
% 0.16/0.41 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
% 0.16/0.41 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_mult_divide_mult_cancel_right2
% 0.16/0.41 thf(fact_457_dvd__mult__cancel__left,axiom,
% 0.16/0.41 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.16/0.41 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.16/0.41 = ( ( C = zero_z3403309356797280102nteger )
% 0.16/0.41 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_mult_cancel_left
% 0.16/0.41 thf(fact_458_dvd__mult__cancel__left,axiom,
% 0.16/0.41 ! [C: complex,A: complex,B: complex] :
% 0.16/0.41 ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.16/0.41 = ( ( C = zero_zero_complex )
% 0.16/0.41 | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_mult_cancel_left
% 0.16/0.41 thf(fact_459_dvd__mult__cancel__left,axiom,
% 0.16/0.41 ! [C: real,A: real,B: real] :
% 0.16/0.41 ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.16/0.41 = ( ( C = zero_zero_real )
% 0.16/0.41 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_mult_cancel_left
% 0.16/0.41 thf(fact_460_dvd__mult__cancel__left,axiom,
% 0.16/0.41 ! [C: rat,A: rat,B: rat] :
% 0.16/0.41 ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.16/0.41 = ( ( C = zero_zero_rat )
% 0.16/0.41 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_mult_cancel_left
% 0.16/0.41 thf(fact_461_dvd__mult__cancel__left,axiom,
% 0.16/0.41 ! [C: int,A: int,B: int] :
% 0.16/0.41 ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.16/0.41 = ( ( C = zero_zero_int )
% 0.16/0.41 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_mult_cancel_left
% 0.16/0.41 thf(fact_462_dvd__mult__cancel__right,axiom,
% 0.16/0.41 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.16/0.41 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.16/0.41 = ( ( C = zero_z3403309356797280102nteger )
% 0.16/0.41 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_mult_cancel_right
% 0.16/0.41 thf(fact_463_dvd__mult__cancel__right,axiom,
% 0.16/0.41 ! [A: complex,C: complex,B: complex] :
% 0.16/0.41 ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 0.16/0.41 = ( ( C = zero_zero_complex )
% 0.16/0.41 | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_mult_cancel_right
% 0.16/0.41 thf(fact_464_dvd__mult__cancel__right,axiom,
% 0.16/0.41 ! [A: real,C: real,B: real] :
% 0.16/0.41 ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.16/0.41 = ( ( C = zero_zero_real )
% 0.16/0.41 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_mult_cancel_right
% 0.16/0.41 thf(fact_465_dvd__mult__cancel__right,axiom,
% 0.16/0.41 ! [A: rat,C: rat,B: rat] :
% 0.16/0.41 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.16/0.41 = ( ( C = zero_zero_rat )
% 0.16/0.41 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_mult_cancel_right
% 0.16/0.41 thf(fact_466_dvd__mult__cancel__right,axiom,
% 0.16/0.41 ! [A: int,C: int,B: int] :
% 0.16/0.41 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.16/0.41 = ( ( C = zero_zero_int )
% 0.16/0.41 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_mult_cancel_right
% 0.16/0.41 thf(fact_467_dvd__times__left__cancel__iff,axiom,
% 0.16/0.41 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.16/0.41 ( ( A != zero_z3403309356797280102nteger )
% 0.16/0.41 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 0.16/0.41 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_times_left_cancel_iff
% 0.16/0.41 thf(fact_468_dvd__times__left__cancel__iff,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( A != zero_zero_nat )
% 0.16/0.41 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
% 0.16/0.41 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_times_left_cancel_iff
% 0.16/0.41 thf(fact_469_dvd__times__left__cancel__iff,axiom,
% 0.16/0.41 ! [A: int,B: int,C: int] :
% 0.16/0.41 ( ( A != zero_zero_int )
% 0.16/0.41 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
% 0.16/0.41 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_times_left_cancel_iff
% 0.16/0.41 thf(fact_470_dvd__times__right__cancel__iff,axiom,
% 0.16/0.41 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.16/0.41 ( ( A != zero_z3403309356797280102nteger )
% 0.16/0.41 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 0.16/0.41 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_times_right_cancel_iff
% 0.16/0.41 thf(fact_471_dvd__times__right__cancel__iff,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( A != zero_zero_nat )
% 0.16/0.41 => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
% 0.16/0.41 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_times_right_cancel_iff
% 0.16/0.41 thf(fact_472_dvd__times__right__cancel__iff,axiom,
% 0.16/0.41 ! [A: int,B: int,C: int] :
% 0.16/0.41 ( ( A != zero_zero_int )
% 0.16/0.41 => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
% 0.16/0.41 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_times_right_cancel_iff
% 0.16/0.41 thf(fact_473_of__nat__eq__0__iff,axiom,
% 0.16/0.41 ! [M: nat] :
% 0.16/0.41 ( ( ( semiri8010041392384452111omplex @ M )
% 0.16/0.41 = zero_zero_complex )
% 0.16/0.41 = ( M = zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_eq_0_iff
% 0.16/0.41 thf(fact_474_of__nat__eq__0__iff,axiom,
% 0.16/0.41 ! [M: nat] :
% 0.16/0.41 ( ( ( semiri5074537144036343181t_real @ M )
% 0.16/0.41 = zero_zero_real )
% 0.16/0.41 = ( M = zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_eq_0_iff
% 0.16/0.41 thf(fact_475_of__nat__eq__0__iff,axiom,
% 0.16/0.41 ! [M: nat] :
% 0.16/0.41 ( ( ( semiri681578069525770553at_rat @ M )
% 0.16/0.41 = zero_zero_rat )
% 0.16/0.41 = ( M = zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_eq_0_iff
% 0.16/0.41 thf(fact_476_of__nat__eq__0__iff,axiom,
% 0.16/0.41 ! [M: nat] :
% 0.16/0.41 ( ( ( semiri1316708129612266289at_nat @ M )
% 0.16/0.41 = zero_zero_nat )
% 0.16/0.41 = ( M = zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_eq_0_iff
% 0.16/0.41 thf(fact_477_of__nat__eq__0__iff,axiom,
% 0.16/0.41 ! [M: nat] :
% 0.16/0.41 ( ( ( semiri1314217659103216013at_int @ M )
% 0.16/0.41 = zero_zero_int )
% 0.16/0.41 = ( M = zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_eq_0_iff
% 0.16/0.41 thf(fact_478_of__nat__0__eq__iff,axiom,
% 0.16/0.41 ! [N: nat] :
% 0.16/0.41 ( ( zero_zero_complex
% 0.16/0.41 = ( semiri8010041392384452111omplex @ N ) )
% 0.16/0.41 = ( zero_zero_nat = N ) ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_0_eq_iff
% 0.16/0.41 thf(fact_479_of__nat__0__eq__iff,axiom,
% 0.16/0.41 ! [N: nat] :
% 0.16/0.41 ( ( zero_zero_real
% 0.16/0.41 = ( semiri5074537144036343181t_real @ N ) )
% 0.16/0.41 = ( zero_zero_nat = N ) ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_0_eq_iff
% 0.16/0.41 thf(fact_480_of__nat__0__eq__iff,axiom,
% 0.16/0.41 ! [N: nat] :
% 0.16/0.41 ( ( zero_zero_rat
% 0.16/0.41 = ( semiri681578069525770553at_rat @ N ) )
% 0.16/0.41 = ( zero_zero_nat = N ) ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_0_eq_iff
% 0.16/0.41 thf(fact_481_of__nat__0__eq__iff,axiom,
% 0.16/0.41 ! [N: nat] :
% 0.16/0.41 ( ( zero_zero_nat
% 0.16/0.41 = ( semiri1316708129612266289at_nat @ N ) )
% 0.16/0.41 = ( zero_zero_nat = N ) ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_0_eq_iff
% 0.16/0.41 thf(fact_482_of__nat__0__eq__iff,axiom,
% 0.16/0.41 ! [N: nat] :
% 0.16/0.41 ( ( zero_zero_int
% 0.16/0.41 = ( semiri1314217659103216013at_int @ N ) )
% 0.16/0.41 = ( zero_zero_nat = N ) ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_0_eq_iff
% 0.16/0.41 thf(fact_483_of__nat__0,axiom,
% 0.16/0.41 ( ( semiri8010041392384452111omplex @ zero_zero_nat )
% 0.16/0.41 = zero_zero_complex ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_0
% 0.16/0.41 thf(fact_484_of__nat__0,axiom,
% 0.16/0.41 ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 0.16/0.41 = zero_zero_real ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_0
% 0.16/0.41 thf(fact_485_of__nat__0,axiom,
% 0.16/0.41 ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 0.16/0.41 = zero_zero_rat ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_0
% 0.16/0.41 thf(fact_486_of__nat__0,axiom,
% 0.16/0.41 ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 0.16/0.41 = zero_zero_nat ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_0
% 0.16/0.41 thf(fact_487_of__nat__0,axiom,
% 0.16/0.41 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 0.16/0.41 = zero_zero_int ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_0
% 0.16/0.41 thf(fact_488_one__eq__mult__iff,axiom,
% 0.16/0.41 ! [M: nat,N: nat] :
% 0.16/0.41 ( ( ( suc @ zero_zero_nat )
% 0.16/0.41 = ( times_times_nat @ M @ N ) )
% 0.16/0.41 = ( ( M
% 0.16/0.41 = ( suc @ zero_zero_nat ) )
% 0.16/0.41 & ( N
% 0.16/0.41 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % one_eq_mult_iff
% 0.16/0.41 thf(fact_489_mult__eq__1__iff,axiom,
% 0.16/0.41 ! [M: nat,N: nat] :
% 0.16/0.41 ( ( ( times_times_nat @ M @ N )
% 0.16/0.41 = ( suc @ zero_zero_nat ) )
% 0.16/0.41 = ( ( M
% 0.16/0.41 = ( suc @ zero_zero_nat ) )
% 0.16/0.41 & ( N
% 0.16/0.41 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_eq_1_iff
% 0.16/0.41 thf(fact_490_dvd__1__left,axiom,
% 0.16/0.41 ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_1_left
% 0.16/0.41 thf(fact_491_dvd__1__iff__1,axiom,
% 0.16/0.41 ! [M: nat] :
% 0.16/0.41 ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.16/0.41 = ( M
% 0.16/0.41 = ( suc @ zero_zero_nat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_1_iff_1
% 0.16/0.41 thf(fact_492_div__by__Suc__0,axiom,
% 0.16/0.41 ! [M: nat] :
% 0.16/0.41 ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.16/0.41 = M ) ).
% 0.16/0.41
% 0.16/0.41 % div_by_Suc_0
% 0.16/0.41 thf(fact_493_nat__mult__dvd__cancel__disj,axiom,
% 0.16/0.41 ! [K: nat,M: nat,N: nat] :
% 0.16/0.41 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.16/0.41 = ( ( K = zero_zero_nat )
% 0.16/0.41 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nat_mult_dvd_cancel_disj
% 0.16/0.41 thf(fact_494_nat__mult__div__cancel__disj,axiom,
% 0.16/0.41 ! [K: nat,M: nat,N: nat] :
% 0.16/0.41 ( ( ( K = zero_zero_nat )
% 0.16/0.41 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.16/0.41 = zero_zero_nat ) )
% 0.16/0.41 & ( ( K != zero_zero_nat )
% 0.16/0.41 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.16/0.41 = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nat_mult_div_cancel_disj
% 0.16/0.41 thf(fact_495_divide__eq__eq__numeral1_I1_J,axiom,
% 0.16/0.41 ! [B: complex,W: num,A: complex] :
% 0.16/0.41 ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
% 0.16/0.41 = A )
% 0.16/0.41 = ( ( ( ( numera6690914467698888265omplex @ W )
% 0.16/0.41 != zero_zero_complex )
% 0.16/0.41 => ( B
% 0.16/0.41 = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
% 0.16/0.41 & ( ( ( numera6690914467698888265omplex @ W )
% 0.16/0.41 = zero_zero_complex )
% 0.16/0.41 => ( A = zero_zero_complex ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_eq_eq_numeral1(1)
% 0.16/0.41 thf(fact_496_divide__eq__eq__numeral1_I1_J,axiom,
% 0.16/0.41 ! [B: real,W: num,A: real] :
% 0.16/0.41 ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
% 0.16/0.41 = A )
% 0.16/0.41 = ( ( ( ( numeral_numeral_real @ W )
% 0.16/0.41 != zero_zero_real )
% 0.16/0.41 => ( B
% 0.16/0.41 = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
% 0.16/0.41 & ( ( ( numeral_numeral_real @ W )
% 0.16/0.41 = zero_zero_real )
% 0.16/0.41 => ( A = zero_zero_real ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_eq_eq_numeral1(1)
% 0.16/0.41 thf(fact_497_divide__eq__eq__numeral1_I1_J,axiom,
% 0.16/0.41 ! [B: rat,W: num,A: rat] :
% 0.16/0.41 ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
% 0.16/0.41 = A )
% 0.16/0.41 = ( ( ( ( numeral_numeral_rat @ W )
% 0.16/0.41 != zero_zero_rat )
% 0.16/0.41 => ( B
% 0.16/0.41 = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
% 0.16/0.41 & ( ( ( numeral_numeral_rat @ W )
% 0.16/0.41 = zero_zero_rat )
% 0.16/0.41 => ( A = zero_zero_rat ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_eq_eq_numeral1(1)
% 0.16/0.41 thf(fact_498_eq__divide__eq__numeral1_I1_J,axiom,
% 0.16/0.41 ! [A: complex,B: complex,W: num] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
% 0.16/0.41 = ( ( ( ( numera6690914467698888265omplex @ W )
% 0.16/0.41 != zero_zero_complex )
% 0.16/0.41 => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
% 0.16/0.41 = B ) )
% 0.16/0.41 & ( ( ( numera6690914467698888265omplex @ W )
% 0.16/0.41 = zero_zero_complex )
% 0.16/0.41 => ( A = zero_zero_complex ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_divide_eq_numeral1(1)
% 0.16/0.41 thf(fact_499_eq__divide__eq__numeral1_I1_J,axiom,
% 0.16/0.41 ! [A: real,B: real,W: num] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 0.16/0.41 = ( ( ( ( numeral_numeral_real @ W )
% 0.16/0.41 != zero_zero_real )
% 0.16/0.41 => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
% 0.16/0.41 = B ) )
% 0.16/0.41 & ( ( ( numeral_numeral_real @ W )
% 0.16/0.41 = zero_zero_real )
% 0.16/0.41 => ( A = zero_zero_real ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_divide_eq_numeral1(1)
% 0.16/0.41 thf(fact_500_eq__divide__eq__numeral1_I1_J,axiom,
% 0.16/0.41 ! [A: rat,B: rat,W: num] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 0.16/0.41 = ( ( ( ( numeral_numeral_rat @ W )
% 0.16/0.41 != zero_zero_rat )
% 0.16/0.41 => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
% 0.16/0.41 = B ) )
% 0.16/0.41 & ( ( ( numeral_numeral_rat @ W )
% 0.16/0.41 = zero_zero_rat )
% 0.16/0.41 => ( A = zero_zero_rat ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_divide_eq_numeral1(1)
% 0.16/0.41 thf(fact_501_of__nat__le__0__iff,axiom,
% 0.16/0.41 ! [M: nat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 0.16/0.41 = ( M = zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_le_0_iff
% 0.16/0.41 thf(fact_502_of__nat__le__0__iff,axiom,
% 0.16/0.41 ! [M: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 0.16/0.41 = ( M = zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_le_0_iff
% 0.16/0.41 thf(fact_503_of__nat__le__0__iff,axiom,
% 0.16/0.41 ! [M: nat] :
% 0.16/0.41 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 0.16/0.41 = ( M = zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_le_0_iff
% 0.16/0.41 thf(fact_504_of__nat__le__0__iff,axiom,
% 0.16/0.41 ! [M: nat] :
% 0.16/0.41 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 0.16/0.41 = ( M = zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_le_0_iff
% 0.16/0.41 thf(fact_505_one__le__mult__iff,axiom,
% 0.16/0.41 ! [M: nat,N: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 0.16/0.41 = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.16/0.41 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % one_le_mult_iff
% 0.16/0.41 thf(fact_506_verit__la__generic,axiom,
% 0.16/0.41 ! [A: int,X: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ A @ X )
% 0.16/0.41 | ( A = X )
% 0.16/0.41 | ( ord_less_eq_int @ X @ A ) ) ).
% 0.16/0.41
% 0.16/0.41 % verit_la_generic
% 0.16/0.41 thf(fact_507_le__numeral__extra_I3_J,axiom,
% 0.16/0.41 ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 0.16/0.41
% 0.16/0.41 % le_numeral_extra(3)
% 0.16/0.41 thf(fact_508_le__numeral__extra_I3_J,axiom,
% 0.16/0.41 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.16/0.41
% 0.16/0.41 % le_numeral_extra(3)
% 0.16/0.41 thf(fact_509_le__numeral__extra_I3_J,axiom,
% 0.16/0.41 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.16/0.41
% 0.16/0.41 % le_numeral_extra(3)
% 0.16/0.41 thf(fact_510_le__numeral__extra_I3_J,axiom,
% 0.16/0.41 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.16/0.41
% 0.16/0.41 % le_numeral_extra(3)
% 0.16/0.41 thf(fact_511_zero__neq__numeral,axiom,
% 0.16/0.41 ! [N: num] :
% 0.16/0.41 ( zero_zero_complex
% 0.16/0.41 != ( numera6690914467698888265omplex @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_neq_numeral
% 0.16/0.41 thf(fact_512_zero__neq__numeral,axiom,
% 0.16/0.41 ! [N: num] :
% 0.16/0.41 ( zero_zero_real
% 0.16/0.41 != ( numeral_numeral_real @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_neq_numeral
% 0.16/0.41 thf(fact_513_zero__neq__numeral,axiom,
% 0.16/0.41 ! [N: num] :
% 0.16/0.41 ( zero_zero_rat
% 0.16/0.41 != ( numeral_numeral_rat @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_neq_numeral
% 0.16/0.41 thf(fact_514_zero__neq__numeral,axiom,
% 0.16/0.41 ! [N: num] :
% 0.16/0.41 ( zero_zero_nat
% 0.16/0.41 != ( numeral_numeral_nat @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_neq_numeral
% 0.16/0.41 thf(fact_515_zero__neq__numeral,axiom,
% 0.16/0.41 ! [N: num] :
% 0.16/0.41 ( zero_zero_int
% 0.16/0.41 != ( numeral_numeral_int @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_neq_numeral
% 0.16/0.41 thf(fact_516_mult__not__zero,axiom,
% 0.16/0.41 ! [A: complex,B: complex] :
% 0.16/0.41 ( ( ( times_times_complex @ A @ B )
% 0.16/0.41 != zero_zero_complex )
% 0.16/0.41 => ( ( A != zero_zero_complex )
% 0.16/0.41 & ( B != zero_zero_complex ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_not_zero
% 0.16/0.41 thf(fact_517_mult__not__zero,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ( times_times_real @ A @ B )
% 0.16/0.41 != zero_zero_real )
% 0.16/0.41 => ( ( A != zero_zero_real )
% 0.16/0.41 & ( B != zero_zero_real ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_not_zero
% 0.16/0.41 thf(fact_518_mult__not__zero,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( ( times_times_rat @ A @ B )
% 0.16/0.41 != zero_zero_rat )
% 0.16/0.41 => ( ( A != zero_zero_rat )
% 0.16/0.41 & ( B != zero_zero_rat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_not_zero
% 0.16/0.41 thf(fact_519_mult__not__zero,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ( times_times_nat @ A @ B )
% 0.16/0.41 != zero_zero_nat )
% 0.16/0.41 => ( ( A != zero_zero_nat )
% 0.16/0.41 & ( B != zero_zero_nat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_not_zero
% 0.16/0.41 thf(fact_520_mult__not__zero,axiom,
% 0.16/0.41 ! [A: int,B: int] :
% 0.16/0.41 ( ( ( times_times_int @ A @ B )
% 0.16/0.41 != zero_zero_int )
% 0.16/0.41 => ( ( A != zero_zero_int )
% 0.16/0.41 & ( B != zero_zero_int ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_not_zero
% 0.16/0.41 thf(fact_521_divisors__zero,axiom,
% 0.16/0.41 ! [A: complex,B: complex] :
% 0.16/0.41 ( ( ( times_times_complex @ A @ B )
% 0.16/0.41 = zero_zero_complex )
% 0.16/0.41 => ( ( A = zero_zero_complex )
% 0.16/0.41 | ( B = zero_zero_complex ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divisors_zero
% 0.16/0.41 thf(fact_522_divisors__zero,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ( times_times_real @ A @ B )
% 0.16/0.41 = zero_zero_real )
% 0.16/0.41 => ( ( A = zero_zero_real )
% 0.16/0.41 | ( B = zero_zero_real ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divisors_zero
% 0.16/0.41 thf(fact_523_divisors__zero,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( ( times_times_rat @ A @ B )
% 0.16/0.41 = zero_zero_rat )
% 0.16/0.41 => ( ( A = zero_zero_rat )
% 0.16/0.41 | ( B = zero_zero_rat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divisors_zero
% 0.16/0.41 thf(fact_524_divisors__zero,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ( times_times_nat @ A @ B )
% 0.16/0.41 = zero_zero_nat )
% 0.16/0.41 => ( ( A = zero_zero_nat )
% 0.16/0.41 | ( B = zero_zero_nat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divisors_zero
% 0.16/0.41 thf(fact_525_divisors__zero,axiom,
% 0.16/0.41 ! [A: int,B: int] :
% 0.16/0.41 ( ( ( times_times_int @ A @ B )
% 0.16/0.41 = zero_zero_int )
% 0.16/0.41 => ( ( A = zero_zero_int )
% 0.16/0.41 | ( B = zero_zero_int ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divisors_zero
% 0.16/0.41 thf(fact_526_no__zero__divisors,axiom,
% 0.16/0.41 ! [A: complex,B: complex] :
% 0.16/0.41 ( ( A != zero_zero_complex )
% 0.16/0.41 => ( ( B != zero_zero_complex )
% 0.16/0.41 => ( ( times_times_complex @ A @ B )
% 0.16/0.41 != zero_zero_complex ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % no_zero_divisors
% 0.16/0.41 thf(fact_527_no__zero__divisors,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( A != zero_zero_real )
% 0.16/0.41 => ( ( B != zero_zero_real )
% 0.16/0.41 => ( ( times_times_real @ A @ B )
% 0.16/0.41 != zero_zero_real ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % no_zero_divisors
% 0.16/0.41 thf(fact_528_no__zero__divisors,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( A != zero_zero_rat )
% 0.16/0.41 => ( ( B != zero_zero_rat )
% 0.16/0.41 => ( ( times_times_rat @ A @ B )
% 0.16/0.41 != zero_zero_rat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % no_zero_divisors
% 0.16/0.41 thf(fact_529_no__zero__divisors,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( A != zero_zero_nat )
% 0.16/0.41 => ( ( B != zero_zero_nat )
% 0.16/0.41 => ( ( times_times_nat @ A @ B )
% 0.16/0.41 != zero_zero_nat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % no_zero_divisors
% 0.16/0.41 thf(fact_530_no__zero__divisors,axiom,
% 0.16/0.41 ! [A: int,B: int] :
% 0.16/0.41 ( ( A != zero_zero_int )
% 0.16/0.41 => ( ( B != zero_zero_int )
% 0.16/0.41 => ( ( times_times_int @ A @ B )
% 0.16/0.41 != zero_zero_int ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % no_zero_divisors
% 0.16/0.41 thf(fact_531_mult__left__cancel,axiom,
% 0.16/0.41 ! [C: complex,A: complex,B: complex] :
% 0.16/0.41 ( ( C != zero_zero_complex )
% 0.16/0.41 => ( ( ( times_times_complex @ C @ A )
% 0.16/0.41 = ( times_times_complex @ C @ B ) )
% 0.16/0.41 = ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_left_cancel
% 0.16/0.41 thf(fact_532_mult__left__cancel,axiom,
% 0.16/0.41 ! [C: real,A: real,B: real] :
% 0.16/0.41 ( ( C != zero_zero_real )
% 0.16/0.41 => ( ( ( times_times_real @ C @ A )
% 0.16/0.41 = ( times_times_real @ C @ B ) )
% 0.16/0.41 = ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_left_cancel
% 0.16/0.41 thf(fact_533_mult__left__cancel,axiom,
% 0.16/0.41 ! [C: rat,A: rat,B: rat] :
% 0.16/0.41 ( ( C != zero_zero_rat )
% 0.16/0.41 => ( ( ( times_times_rat @ C @ A )
% 0.16/0.41 = ( times_times_rat @ C @ B ) )
% 0.16/0.41 = ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_left_cancel
% 0.16/0.41 thf(fact_534_mult__left__cancel,axiom,
% 0.16/0.41 ! [C: nat,A: nat,B: nat] :
% 0.16/0.41 ( ( C != zero_zero_nat )
% 0.16/0.41 => ( ( ( times_times_nat @ C @ A )
% 0.16/0.41 = ( times_times_nat @ C @ B ) )
% 0.16/0.41 = ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_left_cancel
% 0.16/0.41 thf(fact_535_mult__left__cancel,axiom,
% 0.16/0.41 ! [C: int,A: int,B: int] :
% 0.16/0.41 ( ( C != zero_zero_int )
% 0.16/0.41 => ( ( ( times_times_int @ C @ A )
% 0.16/0.41 = ( times_times_int @ C @ B ) )
% 0.16/0.41 = ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_left_cancel
% 0.16/0.41 thf(fact_536_mult__right__cancel,axiom,
% 0.16/0.41 ! [C: complex,A: complex,B: complex] :
% 0.16/0.41 ( ( C != zero_zero_complex )
% 0.16/0.41 => ( ( ( times_times_complex @ A @ C )
% 0.16/0.41 = ( times_times_complex @ B @ C ) )
% 0.16/0.41 = ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_right_cancel
% 0.16/0.41 thf(fact_537_mult__right__cancel,axiom,
% 0.16/0.41 ! [C: real,A: real,B: real] :
% 0.16/0.41 ( ( C != zero_zero_real )
% 0.16/0.41 => ( ( ( times_times_real @ A @ C )
% 0.16/0.41 = ( times_times_real @ B @ C ) )
% 0.16/0.41 = ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_right_cancel
% 0.16/0.41 thf(fact_538_mult__right__cancel,axiom,
% 0.16/0.41 ! [C: rat,A: rat,B: rat] :
% 0.16/0.41 ( ( C != zero_zero_rat )
% 0.16/0.41 => ( ( ( times_times_rat @ A @ C )
% 0.16/0.41 = ( times_times_rat @ B @ C ) )
% 0.16/0.41 = ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_right_cancel
% 0.16/0.41 thf(fact_539_mult__right__cancel,axiom,
% 0.16/0.41 ! [C: nat,A: nat,B: nat] :
% 0.16/0.41 ( ( C != zero_zero_nat )
% 0.16/0.41 => ( ( ( times_times_nat @ A @ C )
% 0.16/0.41 = ( times_times_nat @ B @ C ) )
% 0.16/0.41 = ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_right_cancel
% 0.16/0.41 thf(fact_540_mult__right__cancel,axiom,
% 0.16/0.41 ! [C: int,A: int,B: int] :
% 0.16/0.41 ( ( C != zero_zero_int )
% 0.16/0.41 => ( ( ( times_times_int @ A @ C )
% 0.16/0.41 = ( times_times_int @ B @ C ) )
% 0.16/0.41 = ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_right_cancel
% 0.16/0.41 thf(fact_541_dvd__0__left,axiom,
% 0.16/0.41 ! [A: code_integer] :
% 0.16/0.41 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 0.16/0.41 => ( A = zero_z3403309356797280102nteger ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_0_left
% 0.16/0.41 thf(fact_542_dvd__0__left,axiom,
% 0.16/0.41 ! [A: complex] :
% 0.16/0.41 ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 0.16/0.41 => ( A = zero_zero_complex ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_0_left
% 0.16/0.41 thf(fact_543_dvd__0__left,axiom,
% 0.16/0.41 ! [A: real] :
% 0.16/0.41 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 0.16/0.41 => ( A = zero_zero_real ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_0_left
% 0.16/0.41 thf(fact_544_dvd__0__left,axiom,
% 0.16/0.41 ! [A: rat] :
% 0.16/0.41 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 0.16/0.41 => ( A = zero_zero_rat ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_0_left
% 0.16/0.41 thf(fact_545_dvd__0__left,axiom,
% 0.16/0.41 ! [A: nat] :
% 0.16/0.41 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.16/0.41 => ( A = zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_0_left
% 0.16/0.41 thf(fact_546_dvd__0__left,axiom,
% 0.16/0.41 ! [A: int] :
% 0.16/0.41 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 0.16/0.41 => ( A = zero_zero_int ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_0_left
% 0.16/0.41 thf(fact_547_dvd__field__iff,axiom,
% 0.16/0.41 ( dvd_dvd_complex
% 0.16/0.41 = ( ^ [A3: complex,B2: complex] :
% 0.16/0.41 ( ( A3 = zero_zero_complex )
% 0.16/0.41 => ( B2 = zero_zero_complex ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_field_iff
% 0.16/0.41 thf(fact_548_dvd__field__iff,axiom,
% 0.16/0.41 ( dvd_dvd_real
% 0.16/0.41 = ( ^ [A3: real,B2: real] :
% 0.16/0.41 ( ( A3 = zero_zero_real )
% 0.16/0.41 => ( B2 = zero_zero_real ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_field_iff
% 0.16/0.41 thf(fact_549_dvd__field__iff,axiom,
% 0.16/0.41 ( dvd_dvd_rat
% 0.16/0.41 = ( ^ [A3: rat,B2: rat] :
% 0.16/0.41 ( ( A3 = zero_zero_rat )
% 0.16/0.41 => ( B2 = zero_zero_rat ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_field_iff
% 0.16/0.41 thf(fact_550_list__decode_Ocases,axiom,
% 0.16/0.41 ! [X: nat] :
% 0.16/0.41 ( ( X != zero_zero_nat )
% 0.16/0.41 => ~ ! [N2: nat] :
% 0.16/0.41 ( X
% 0.16/0.41 != ( suc @ N2 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % list_decode.cases
% 0.16/0.41 thf(fact_551_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
% 0.16/0.41 ! [X: nat] :
% 0.16/0.41 ( ( X != zero_zero_nat )
% 0.16/0.41 => ( ( X
% 0.16/0.41 != ( suc @ zero_zero_nat ) )
% 0.16/0.41 => ~ ! [Va: nat] :
% 0.16/0.41 ( X
% 0.16/0.41 != ( suc @ ( suc @ Va ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
% 0.16/0.41 thf(fact_552_nat_Odistinct_I1_J,axiom,
% 0.16/0.41 ! [X2: nat] :
% 0.16/0.41 ( zero_zero_nat
% 0.16/0.41 != ( suc @ X2 ) ) ).
% 0.16/0.41
% 0.16/0.41 % nat.distinct(1)
% 0.16/0.41 thf(fact_553_old_Onat_Odistinct_I2_J,axiom,
% 0.16/0.41 ! [Nat2: nat] :
% 0.16/0.41 ( ( suc @ Nat2 )
% 0.16/0.41 != zero_zero_nat ) ).
% 0.16/0.41
% 0.16/0.41 % old.nat.distinct(2)
% 0.16/0.41 thf(fact_554_old_Onat_Odistinct_I1_J,axiom,
% 0.16/0.41 ! [Nat2: nat] :
% 0.16/0.41 ( zero_zero_nat
% 0.16/0.41 != ( suc @ Nat2 ) ) ).
% 0.16/0.41
% 0.16/0.41 % old.nat.distinct(1)
% 0.16/0.41 thf(fact_555_nat_OdiscI,axiom,
% 0.16/0.41 ! [Nat: nat,X2: nat] :
% 0.16/0.41 ( ( Nat
% 0.16/0.41 = ( suc @ X2 ) )
% 0.16/0.41 => ( Nat != zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % nat.discI
% 0.16/0.41 thf(fact_556_old_Onat_Oexhaust,axiom,
% 0.16/0.41 ! [Y4: nat] :
% 0.16/0.41 ( ( Y4 != zero_zero_nat )
% 0.16/0.41 => ~ ! [Nat3: nat] :
% 0.16/0.41 ( Y4
% 0.16/0.41 != ( suc @ Nat3 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % old.nat.exhaust
% 0.16/0.41 thf(fact_557_nat__induct,axiom,
% 0.16/0.41 ! [P: nat > $o,N: nat] :
% 0.16/0.41 ( ( P @ zero_zero_nat )
% 0.16/0.41 => ( ! [N2: nat] :
% 0.16/0.41 ( ( P @ N2 )
% 0.16/0.41 => ( P @ ( suc @ N2 ) ) )
% 0.16/0.41 => ( P @ N ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nat_induct
% 0.16/0.41 thf(fact_558_diff__induct,axiom,
% 0.16/0.41 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.16/0.41 ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
% 0.16/0.41 => ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
% 0.16/0.41 => ( ! [X4: nat,Y: nat] :
% 0.16/0.41 ( ( P @ X4 @ Y )
% 0.16/0.41 => ( P @ ( suc @ X4 ) @ ( suc @ Y ) ) )
% 0.16/0.41 => ( P @ M @ N ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % diff_induct
% 0.16/0.41 thf(fact_559_zero__induct,axiom,
% 0.16/0.41 ! [P: nat > $o,K: nat] :
% 0.16/0.41 ( ( P @ K )
% 0.16/0.41 => ( ! [N2: nat] :
% 0.16/0.41 ( ( P @ ( suc @ N2 ) )
% 0.16/0.41 => ( P @ N2 ) )
% 0.16/0.41 => ( P @ zero_zero_nat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_induct
% 0.16/0.41 thf(fact_560_Suc__neq__Zero,axiom,
% 0.16/0.41 ! [M: nat] :
% 0.16/0.41 ( ( suc @ M )
% 0.16/0.41 != zero_zero_nat ) ).
% 0.16/0.41
% 0.16/0.41 % Suc_neq_Zero
% 0.16/0.41 thf(fact_561_Zero__neq__Suc,axiom,
% 0.16/0.41 ! [M: nat] :
% 0.16/0.41 ( zero_zero_nat
% 0.16/0.41 != ( suc @ M ) ) ).
% 0.16/0.41
% 0.16/0.41 % Zero_neq_Suc
% 0.16/0.41 thf(fact_562_Zero__not__Suc,axiom,
% 0.16/0.41 ! [M: nat] :
% 0.16/0.41 ( zero_zero_nat
% 0.16/0.41 != ( suc @ M ) ) ).
% 0.16/0.41
% 0.16/0.41 % Zero_not_Suc
% 0.16/0.41 thf(fact_563_not0__implies__Suc,axiom,
% 0.16/0.41 ! [N: nat] :
% 0.16/0.41 ( ( N != zero_zero_nat )
% 0.16/0.41 => ? [M4: nat] :
% 0.16/0.41 ( N
% 0.16/0.41 = ( suc @ M4 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % not0_implies_Suc
% 0.16/0.41 thf(fact_564_le__0__eq,axiom,
% 0.16/0.41 ! [N: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.16/0.41 = ( N = zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_0_eq
% 0.16/0.41 thf(fact_565_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.16/0.41 ! [A: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.16/0.41 => ( A = zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % bot_nat_0.extremum_uniqueI
% 0.16/0.41 thf(fact_566_bot__nat__0_Oextremum__unique,axiom,
% 0.16/0.41 ! [A: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.16/0.41 = ( A = zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % bot_nat_0.extremum_unique
% 0.16/0.41 thf(fact_567_less__eq__nat_Osimps_I1_J,axiom,
% 0.16/0.41 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.16/0.41
% 0.16/0.41 % less_eq_nat.simps(1)
% 0.16/0.41 thf(fact_568_nat__mult__eq__cancel__disj,axiom,
% 0.16/0.41 ! [K: nat,M: nat,N: nat] :
% 0.16/0.41 ( ( ( times_times_nat @ K @ M )
% 0.16/0.41 = ( times_times_nat @ K @ N ) )
% 0.16/0.41 = ( ( K = zero_zero_nat )
% 0.16/0.41 | ( M = N ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nat_mult_eq_cancel_disj
% 0.16/0.41 thf(fact_569_mult__0,axiom,
% 0.16/0.41 ! [N: nat] :
% 0.16/0.41 ( ( times_times_nat @ zero_zero_nat @ N )
% 0.16/0.41 = zero_zero_nat ) ).
% 0.16/0.41
% 0.16/0.41 % mult_0
% 0.16/0.41 thf(fact_570_zero__le__numeral,axiom,
% 0.16/0.41 ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_le_numeral
% 0.16/0.41 thf(fact_571_zero__le__numeral,axiom,
% 0.16/0.41 ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_le_numeral
% 0.16/0.41 thf(fact_572_zero__le__numeral,axiom,
% 0.16/0.41 ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_le_numeral
% 0.16/0.41 thf(fact_573_zero__le__numeral,axiom,
% 0.16/0.41 ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_le_numeral
% 0.16/0.41 thf(fact_574_not__numeral__le__zero,axiom,
% 0.16/0.41 ! [N: num] :
% 0.16/0.41 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 0.16/0.41
% 0.16/0.41 % not_numeral_le_zero
% 0.16/0.41 thf(fact_575_not__numeral__le__zero,axiom,
% 0.16/0.41 ! [N: num] :
% 0.16/0.41 ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 0.16/0.41
% 0.16/0.41 % not_numeral_le_zero
% 0.16/0.41 thf(fact_576_not__numeral__le__zero,axiom,
% 0.16/0.41 ! [N: num] :
% 0.16/0.41 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 0.16/0.41
% 0.16/0.41 % not_numeral_le_zero
% 0.16/0.41 thf(fact_577_not__numeral__le__zero,axiom,
% 0.16/0.41 ! [N: num] :
% 0.16/0.41 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 0.16/0.41
% 0.16/0.41 % not_numeral_le_zero
% 0.16/0.41 thf(fact_578_mult__mono,axiom,
% 0.16/0.41 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_rat @ C @ D )
% 0.16/0.41 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.16/0.41 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.16/0.41 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_mono
% 0.16/0.41 thf(fact_579_mult__mono,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_nat @ C @ D )
% 0.16/0.41 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.16/0.41 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.16/0.41 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_mono
% 0.16/0.41 thf(fact_580_mult__mono,axiom,
% 0.16/0.41 ! [A: int,B: int,C: int,D: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_int @ C @ D )
% 0.16/0.41 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.16/0.41 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.16/0.41 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_mono
% 0.16/0.41 thf(fact_581_mult__mono,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real,D: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_real @ C @ D )
% 0.16/0.41 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.16/0.41 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.41 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_mono
% 0.16/0.41 thf(fact_582_mult__mono_H,axiom,
% 0.16/0.41 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_rat @ C @ D )
% 0.16/0.41 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.41 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.16/0.41 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_mono'
% 0.16/0.41 thf(fact_583_mult__mono_H,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_nat @ C @ D )
% 0.16/0.41 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.41 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.16/0.41 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_mono'
% 0.16/0.41 thf(fact_584_mult__mono_H,axiom,
% 0.16/0.41 ! [A: int,B: int,C: int,D: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_int @ C @ D )
% 0.16/0.41 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.41 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.16/0.41 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_mono'
% 0.16/0.41 thf(fact_585_mult__mono_H,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real,D: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_real @ C @ D )
% 0.16/0.41 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.41 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.41 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_mono'
% 0.16/0.41 thf(fact_586_zero__le__square,axiom,
% 0.16/0.41 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_le_square
% 0.16/0.41 thf(fact_587_zero__le__square,axiom,
% 0.16/0.41 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_le_square
% 0.16/0.41 thf(fact_588_zero__le__square,axiom,
% 0.16/0.41 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_le_square
% 0.16/0.41 thf(fact_589_split__mult__pos__le,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.41 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.16/0.41 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.16/0.41 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 0.16/0.41 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % split_mult_pos_le
% 0.16/0.41 thf(fact_590_split__mult__pos__le,axiom,
% 0.16/0.41 ! [A: int,B: int] :
% 0.16/0.41 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.41 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 0.16/0.41 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.16/0.41 & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 0.16/0.41 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % split_mult_pos_le
% 0.16/0.41 thf(fact_591_split__mult__pos__le,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.41 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.16/0.41 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.16/0.41 & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % split_mult_pos_le
% 0.16/0.41 thf(fact_592_mult__left__mono__neg,axiom,
% 0.16/0.41 ! [B: rat,A: rat,C: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.16/0.41 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_left_mono_neg
% 0.16/0.41 thf(fact_593_mult__left__mono__neg,axiom,
% 0.16/0.41 ! [B: int,A: int,C: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.16/0.41 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_left_mono_neg
% 0.16/0.41 thf(fact_594_mult__left__mono__neg,axiom,
% 0.16/0.41 ! [B: real,A: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.16/0.41 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_left_mono_neg
% 0.16/0.41 thf(fact_595_mult__nonpos__nonpos,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.16/0.41 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.16/0.41 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_nonpos_nonpos
% 0.16/0.41 thf(fact_596_mult__nonpos__nonpos,axiom,
% 0.16/0.41 ! [A: int,B: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.16/0.41 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.16/0.41 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_nonpos_nonpos
% 0.16/0.41 thf(fact_597_mult__nonpos__nonpos,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.16/0.41 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.16/0.41 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_nonpos_nonpos
% 0.16/0.41 thf(fact_598_mult__left__mono,axiom,
% 0.16/0.41 ! [A: rat,B: rat,C: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.16/0.41 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_left_mono
% 0.16/0.41 thf(fact_599_mult__left__mono,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.16/0.41 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_left_mono
% 0.16/0.41 thf(fact_600_mult__left__mono,axiom,
% 0.16/0.41 ! [A: int,B: int,C: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.16/0.41 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_left_mono
% 0.16/0.41 thf(fact_601_mult__left__mono,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.41 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_left_mono
% 0.16/0.41 thf(fact_602_mult__right__mono__neg,axiom,
% 0.16/0.41 ! [B: rat,A: rat,C: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.16/0.41 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_right_mono_neg
% 0.16/0.41 thf(fact_603_mult__right__mono__neg,axiom,
% 0.16/0.41 ! [B: int,A: int,C: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.16/0.41 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_right_mono_neg
% 0.16/0.41 thf(fact_604_mult__right__mono__neg,axiom,
% 0.16/0.41 ! [B: real,A: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.16/0.41 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_right_mono_neg
% 0.16/0.41 thf(fact_605_mult__right__mono,axiom,
% 0.16/0.41 ! [A: rat,B: rat,C: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.16/0.41 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_right_mono
% 0.16/0.41 thf(fact_606_mult__right__mono,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.16/0.41 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_right_mono
% 0.16/0.41 thf(fact_607_mult__right__mono,axiom,
% 0.16/0.41 ! [A: int,B: int,C: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.16/0.41 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_right_mono
% 0.16/0.41 thf(fact_608_mult__right__mono,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.41 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_right_mono
% 0.16/0.41 thf(fact_609_mult__le__0__iff,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.16/0.41 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.41 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.16/0.41 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.16/0.41 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_le_0_iff
% 0.16/0.41 thf(fact_610_mult__le__0__iff,axiom,
% 0.16/0.41 ! [A: int,B: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 0.16/0.41 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.41 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 0.16/0.41 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.16/0.41 & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_le_0_iff
% 0.16/0.41 thf(fact_611_mult__le__0__iff,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.16/0.41 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.41 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.16/0.41 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.16/0.41 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_le_0_iff
% 0.16/0.41 thf(fact_612_split__mult__neg__le,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.41 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.16/0.41 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.16/0.41 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
% 0.16/0.41 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% 0.16/0.41
% 0.16/0.41 % split_mult_neg_le
% 0.16/0.41 thf(fact_613_split__mult__neg__le,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.41 & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 0.16/0.41 | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.16/0.41 & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 0.16/0.41 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % split_mult_neg_le
% 0.16/0.41 thf(fact_614_split__mult__neg__le,axiom,
% 0.16/0.41 ! [A: int,B: int] :
% 0.16/0.41 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.41 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 0.16/0.41 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.16/0.41 & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
% 0.16/0.41 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% 0.16/0.41
% 0.16/0.41 % split_mult_neg_le
% 0.16/0.41 thf(fact_615_split__mult__neg__le,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.41 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.16/0.41 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.16/0.41 & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% 0.16/0.41
% 0.16/0.41 % split_mult_neg_le
% 0.16/0.41 thf(fact_616_mult__nonneg__nonneg,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.41 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.16/0.41 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_nonneg_nonneg
% 0.16/0.41 thf(fact_617_mult__nonneg__nonneg,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.41 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.16/0.41 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_nonneg_nonneg
% 0.16/0.41 thf(fact_618_mult__nonneg__nonneg,axiom,
% 0.16/0.41 ! [A: int,B: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.41 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.16/0.41 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_nonneg_nonneg
% 0.16/0.41 thf(fact_619_mult__nonneg__nonneg,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.41 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.16/0.41 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_nonneg_nonneg
% 0.16/0.41 thf(fact_620_mult__nonneg__nonpos,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.41 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.16/0.41 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_nonneg_nonpos
% 0.16/0.41 thf(fact_621_mult__nonneg__nonpos,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.41 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.16/0.41 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_nonneg_nonpos
% 0.16/0.41 thf(fact_622_mult__nonneg__nonpos,axiom,
% 0.16/0.41 ! [A: int,B: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.41 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.16/0.41 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_nonneg_nonpos
% 0.16/0.41 thf(fact_623_mult__nonneg__nonpos,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.41 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.16/0.41 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_nonneg_nonpos
% 0.16/0.41 thf(fact_624_mult__nonpos__nonneg,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.16/0.41 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.16/0.41 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_nonpos_nonneg
% 0.16/0.41 thf(fact_625_mult__nonpos__nonneg,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.16/0.41 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.16/0.41 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_nonpos_nonneg
% 0.16/0.41 thf(fact_626_mult__nonpos__nonneg,axiom,
% 0.16/0.41 ! [A: int,B: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.16/0.41 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.16/0.41 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_nonpos_nonneg
% 0.16/0.41 thf(fact_627_mult__nonpos__nonneg,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.16/0.41 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.16/0.41 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_nonpos_nonneg
% 0.16/0.41 thf(fact_628_mult__nonneg__nonpos2,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.41 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.16/0.41 => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_nonneg_nonpos2
% 0.16/0.41 thf(fact_629_mult__nonneg__nonpos2,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.41 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.16/0.41 => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_nonneg_nonpos2
% 0.16/0.41 thf(fact_630_mult__nonneg__nonpos2,axiom,
% 0.16/0.41 ! [A: int,B: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.41 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.16/0.41 => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_nonneg_nonpos2
% 0.16/0.41 thf(fact_631_mult__nonneg__nonpos2,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.41 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.16/0.41 => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % mult_nonneg_nonpos2
% 0.16/0.41 thf(fact_632_zero__le__mult__iff,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.16/0.41 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.41 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.16/0.41 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.16/0.41 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_le_mult_iff
% 0.16/0.41 thf(fact_633_zero__le__mult__iff,axiom,
% 0.16/0.41 ! [A: int,B: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.16/0.41 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.41 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 0.16/0.41 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.16/0.41 & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_le_mult_iff
% 0.16/0.41 thf(fact_634_zero__le__mult__iff,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.16/0.41 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.41 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.16/0.41 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.16/0.41 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_le_mult_iff
% 0.16/0.41 thf(fact_635_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.16/0.41 ! [A: rat,B: rat,C: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.16/0.41 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.16/0.41 thf(fact_636_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.16/0.41 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.16/0.41 thf(fact_637_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.16/0.41 ! [A: int,B: int,C: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.16/0.41 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.16/0.41 thf(fact_638_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.41 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.16/0.41 thf(fact_639_divide__le__0__iff,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 0.16/0.41 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.41 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.16/0.41 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.16/0.41 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_le_0_iff
% 0.16/0.41 thf(fact_640_divide__le__0__iff,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.16/0.41 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.41 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.16/0.41 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.16/0.41 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_le_0_iff
% 0.16/0.41 thf(fact_641_divide__right__mono,axiom,
% 0.16/0.41 ! [A: rat,B: rat,C: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.16/0.41 => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_right_mono
% 0.16/0.41 thf(fact_642_divide__right__mono,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.41 => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_right_mono
% 0.16/0.41 thf(fact_643_zero__le__divide__iff,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 0.16/0.41 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.41 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.16/0.41 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.16/0.41 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_le_divide_iff
% 0.16/0.41 thf(fact_644_zero__le__divide__iff,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 0.16/0.41 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.41 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.16/0.41 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.16/0.41 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % zero_le_divide_iff
% 0.16/0.41 thf(fact_645_divide__nonneg__nonneg,axiom,
% 0.16/0.41 ! [X: rat,Y4: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.16/0.41 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 0.16/0.41 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_nonneg_nonneg
% 0.16/0.41 thf(fact_646_divide__nonneg__nonneg,axiom,
% 0.16/0.41 ! [X: real,Y4: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.16/0.41 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.16/0.41 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_nonneg_nonneg
% 0.16/0.41 thf(fact_647_divide__nonneg__nonpos,axiom,
% 0.16/0.41 ! [X: rat,Y4: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.16/0.41 => ( ( ord_less_eq_rat @ Y4 @ zero_zero_rat )
% 0.16/0.41 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y4 ) @ zero_zero_rat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_nonneg_nonpos
% 0.16/0.41 thf(fact_648_divide__nonneg__nonpos,axiom,
% 0.16/0.41 ! [X: real,Y4: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.16/0.41 => ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
% 0.16/0.41 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_nonneg_nonpos
% 0.16/0.41 thf(fact_649_divide__nonpos__nonneg,axiom,
% 0.16/0.41 ! [X: rat,Y4: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.16/0.41 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 0.16/0.41 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y4 ) @ zero_zero_rat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_nonpos_nonneg
% 0.16/0.41 thf(fact_650_divide__nonpos__nonneg,axiom,
% 0.16/0.41 ! [X: real,Y4: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.16/0.41 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.16/0.41 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_nonpos_nonneg
% 0.16/0.41 thf(fact_651_divide__nonpos__nonpos,axiom,
% 0.16/0.41 ! [X: rat,Y4: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.16/0.41 => ( ( ord_less_eq_rat @ Y4 @ zero_zero_rat )
% 0.16/0.41 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_nonpos_nonpos
% 0.16/0.41 thf(fact_652_divide__nonpos__nonpos,axiom,
% 0.16/0.41 ! [X: real,Y4: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.16/0.41 => ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
% 0.16/0.41 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_nonpos_nonpos
% 0.16/0.41 thf(fact_653_divide__right__mono__neg,axiom,
% 0.16/0.41 ! [A: rat,B: rat,C: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.16/0.41 => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_right_mono_neg
% 0.16/0.41 thf(fact_654_divide__right__mono__neg,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.16/0.41 => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_right_mono_neg
% 0.16/0.41 thf(fact_655_frac__eq__eq,axiom,
% 0.16/0.41 ! [Y4: complex,Z: complex,X: complex,W: complex] :
% 0.16/0.41 ( ( Y4 != zero_zero_complex )
% 0.16/0.41 => ( ( Z != zero_zero_complex )
% 0.16/0.41 => ( ( ( divide1717551699836669952omplex @ X @ Y4 )
% 0.16/0.41 = ( divide1717551699836669952omplex @ W @ Z ) )
% 0.16/0.41 = ( ( times_times_complex @ X @ Z )
% 0.16/0.41 = ( times_times_complex @ W @ Y4 ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % frac_eq_eq
% 0.16/0.41 thf(fact_656_frac__eq__eq,axiom,
% 0.16/0.41 ! [Y4: real,Z: real,X: real,W: real] :
% 0.16/0.41 ( ( Y4 != zero_zero_real )
% 0.16/0.41 => ( ( Z != zero_zero_real )
% 0.16/0.41 => ( ( ( divide_divide_real @ X @ Y4 )
% 0.16/0.41 = ( divide_divide_real @ W @ Z ) )
% 0.16/0.41 = ( ( times_times_real @ X @ Z )
% 0.16/0.41 = ( times_times_real @ W @ Y4 ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % frac_eq_eq
% 0.16/0.41 thf(fact_657_frac__eq__eq,axiom,
% 0.16/0.41 ! [Y4: rat,Z: rat,X: rat,W: rat] :
% 0.16/0.41 ( ( Y4 != zero_zero_rat )
% 0.16/0.41 => ( ( Z != zero_zero_rat )
% 0.16/0.41 => ( ( ( divide_divide_rat @ X @ Y4 )
% 0.16/0.41 = ( divide_divide_rat @ W @ Z ) )
% 0.16/0.41 = ( ( times_times_rat @ X @ Z )
% 0.16/0.41 = ( times_times_rat @ W @ Y4 ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % frac_eq_eq
% 0.16/0.41 thf(fact_658_divide__eq__eq,axiom,
% 0.16/0.41 ! [B: complex,C: complex,A: complex] :
% 0.16/0.41 ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.16/0.41 = A )
% 0.16/0.41 = ( ( ( C != zero_zero_complex )
% 0.16/0.41 => ( B
% 0.16/0.41 = ( times_times_complex @ A @ C ) ) )
% 0.16/0.41 & ( ( C = zero_zero_complex )
% 0.16/0.41 => ( A = zero_zero_complex ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_eq_eq
% 0.16/0.41 thf(fact_659_divide__eq__eq,axiom,
% 0.16/0.41 ! [B: real,C: real,A: real] :
% 0.16/0.41 ( ( ( divide_divide_real @ B @ C )
% 0.16/0.41 = A )
% 0.16/0.41 = ( ( ( C != zero_zero_real )
% 0.16/0.41 => ( B
% 0.16/0.41 = ( times_times_real @ A @ C ) ) )
% 0.16/0.41 & ( ( C = zero_zero_real )
% 0.16/0.41 => ( A = zero_zero_real ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_eq_eq
% 0.16/0.41 thf(fact_660_divide__eq__eq,axiom,
% 0.16/0.41 ! [B: rat,C: rat,A: rat] :
% 0.16/0.41 ( ( ( divide_divide_rat @ B @ C )
% 0.16/0.41 = A )
% 0.16/0.41 = ( ( ( C != zero_zero_rat )
% 0.16/0.41 => ( B
% 0.16/0.41 = ( times_times_rat @ A @ C ) ) )
% 0.16/0.41 & ( ( C = zero_zero_rat )
% 0.16/0.41 => ( A = zero_zero_rat ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_eq_eq
% 0.16/0.41 thf(fact_661_eq__divide__eq,axiom,
% 0.16/0.41 ! [A: complex,B: complex,C: complex] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.16/0.41 = ( ( ( C != zero_zero_complex )
% 0.16/0.41 => ( ( times_times_complex @ A @ C )
% 0.16/0.41 = B ) )
% 0.16/0.41 & ( ( C = zero_zero_complex )
% 0.16/0.41 => ( A = zero_zero_complex ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_divide_eq
% 0.16/0.41 thf(fact_662_eq__divide__eq,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( divide_divide_real @ B @ C ) )
% 0.16/0.41 = ( ( ( C != zero_zero_real )
% 0.16/0.41 => ( ( times_times_real @ A @ C )
% 0.16/0.41 = B ) )
% 0.16/0.41 & ( ( C = zero_zero_real )
% 0.16/0.41 => ( A = zero_zero_real ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_divide_eq
% 0.16/0.41 thf(fact_663_eq__divide__eq,axiom,
% 0.16/0.41 ! [A: rat,B: rat,C: rat] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( divide_divide_rat @ B @ C ) )
% 0.16/0.41 = ( ( ( C != zero_zero_rat )
% 0.16/0.41 => ( ( times_times_rat @ A @ C )
% 0.16/0.41 = B ) )
% 0.16/0.41 & ( ( C = zero_zero_rat )
% 0.16/0.41 => ( A = zero_zero_rat ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_divide_eq
% 0.16/0.41 thf(fact_664_divide__eq__imp,axiom,
% 0.16/0.41 ! [C: complex,B: complex,A: complex] :
% 0.16/0.41 ( ( C != zero_zero_complex )
% 0.16/0.41 => ( ( B
% 0.16/0.41 = ( times_times_complex @ A @ C ) )
% 0.16/0.41 => ( ( divide1717551699836669952omplex @ B @ C )
% 0.16/0.41 = A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_eq_imp
% 0.16/0.41 thf(fact_665_divide__eq__imp,axiom,
% 0.16/0.41 ! [C: real,B: real,A: real] :
% 0.16/0.41 ( ( C != zero_zero_real )
% 0.16/0.41 => ( ( B
% 0.16/0.41 = ( times_times_real @ A @ C ) )
% 0.16/0.41 => ( ( divide_divide_real @ B @ C )
% 0.16/0.41 = A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_eq_imp
% 0.16/0.41 thf(fact_666_divide__eq__imp,axiom,
% 0.16/0.41 ! [C: rat,B: rat,A: rat] :
% 0.16/0.41 ( ( C != zero_zero_rat )
% 0.16/0.41 => ( ( B
% 0.16/0.41 = ( times_times_rat @ A @ C ) )
% 0.16/0.41 => ( ( divide_divide_rat @ B @ C )
% 0.16/0.41 = A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_eq_imp
% 0.16/0.41 thf(fact_667_eq__divide__imp,axiom,
% 0.16/0.41 ! [C: complex,A: complex,B: complex] :
% 0.16/0.41 ( ( C != zero_zero_complex )
% 0.16/0.41 => ( ( ( times_times_complex @ A @ C )
% 0.16/0.41 = B )
% 0.16/0.41 => ( A
% 0.16/0.41 = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_divide_imp
% 0.16/0.41 thf(fact_668_eq__divide__imp,axiom,
% 0.16/0.41 ! [C: real,A: real,B: real] :
% 0.16/0.41 ( ( C != zero_zero_real )
% 0.16/0.41 => ( ( ( times_times_real @ A @ C )
% 0.16/0.41 = B )
% 0.16/0.41 => ( A
% 0.16/0.41 = ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_divide_imp
% 0.16/0.41 thf(fact_669_eq__divide__imp,axiom,
% 0.16/0.41 ! [C: rat,A: rat,B: rat] :
% 0.16/0.41 ( ( C != zero_zero_rat )
% 0.16/0.41 => ( ( ( times_times_rat @ A @ C )
% 0.16/0.41 = B )
% 0.16/0.41 => ( A
% 0.16/0.41 = ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_divide_imp
% 0.16/0.41 thf(fact_670_nonzero__divide__eq__eq,axiom,
% 0.16/0.41 ! [C: complex,B: complex,A: complex] :
% 0.16/0.41 ( ( C != zero_zero_complex )
% 0.16/0.41 => ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.16/0.41 = A )
% 0.16/0.41 = ( B
% 0.16/0.41 = ( times_times_complex @ A @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_divide_eq_eq
% 0.16/0.41 thf(fact_671_nonzero__divide__eq__eq,axiom,
% 0.16/0.41 ! [C: real,B: real,A: real] :
% 0.16/0.41 ( ( C != zero_zero_real )
% 0.16/0.41 => ( ( ( divide_divide_real @ B @ C )
% 0.16/0.41 = A )
% 0.16/0.41 = ( B
% 0.16/0.41 = ( times_times_real @ A @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_divide_eq_eq
% 0.16/0.41 thf(fact_672_nonzero__divide__eq__eq,axiom,
% 0.16/0.41 ! [C: rat,B: rat,A: rat] :
% 0.16/0.41 ( ( C != zero_zero_rat )
% 0.16/0.41 => ( ( ( divide_divide_rat @ B @ C )
% 0.16/0.41 = A )
% 0.16/0.41 = ( B
% 0.16/0.41 = ( times_times_rat @ A @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_divide_eq_eq
% 0.16/0.41 thf(fact_673_nonzero__eq__divide__eq,axiom,
% 0.16/0.41 ! [C: complex,A: complex,B: complex] :
% 0.16/0.41 ( ( C != zero_zero_complex )
% 0.16/0.41 => ( ( A
% 0.16/0.41 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.16/0.41 = ( ( times_times_complex @ A @ C )
% 0.16/0.41 = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_eq_divide_eq
% 0.16/0.41 thf(fact_674_nonzero__eq__divide__eq,axiom,
% 0.16/0.41 ! [C: real,A: real,B: real] :
% 0.16/0.41 ( ( C != zero_zero_real )
% 0.16/0.41 => ( ( A
% 0.16/0.41 = ( divide_divide_real @ B @ C ) )
% 0.16/0.41 = ( ( times_times_real @ A @ C )
% 0.16/0.41 = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_eq_divide_eq
% 0.16/0.41 thf(fact_675_nonzero__eq__divide__eq,axiom,
% 0.16/0.41 ! [C: rat,A: rat,B: rat] :
% 0.16/0.41 ( ( C != zero_zero_rat )
% 0.16/0.41 => ( ( A
% 0.16/0.41 = ( divide_divide_rat @ B @ C ) )
% 0.16/0.41 = ( ( times_times_rat @ A @ C )
% 0.16/0.41 = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nonzero_eq_divide_eq
% 0.16/0.41 thf(fact_676_of__nat__0__le__iff,axiom,
% 0.16/0.41 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_0_le_iff
% 0.16/0.41 thf(fact_677_of__nat__0__le__iff,axiom,
% 0.16/0.41 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_0_le_iff
% 0.16/0.41 thf(fact_678_of__nat__0__le__iff,axiom,
% 0.16/0.41 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_0_le_iff
% 0.16/0.41 thf(fact_679_of__nat__0__le__iff,axiom,
% 0.16/0.41 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_0_le_iff
% 0.16/0.41 thf(fact_680_dvd__div__eq__0__iff,axiom,
% 0.16/0.41 ! [B: code_integer,A: code_integer] :
% 0.16/0.41 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.16/0.41 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.16/0.41 = zero_z3403309356797280102nteger )
% 0.16/0.41 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_div_eq_0_iff
% 0.16/0.41 thf(fact_681_dvd__div__eq__0__iff,axiom,
% 0.16/0.41 ! [B: complex,A: complex] :
% 0.16/0.41 ( ( dvd_dvd_complex @ B @ A )
% 0.16/0.41 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.16/0.41 = zero_zero_complex )
% 0.16/0.41 = ( A = zero_zero_complex ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_div_eq_0_iff
% 0.16/0.41 thf(fact_682_dvd__div__eq__0__iff,axiom,
% 0.16/0.41 ! [B: real,A: real] :
% 0.16/0.41 ( ( dvd_dvd_real @ B @ A )
% 0.16/0.41 => ( ( ( divide_divide_real @ A @ B )
% 0.16/0.41 = zero_zero_real )
% 0.16/0.41 = ( A = zero_zero_real ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_div_eq_0_iff
% 0.16/0.41 thf(fact_683_dvd__div__eq__0__iff,axiom,
% 0.16/0.41 ! [B: rat,A: rat] :
% 0.16/0.41 ( ( dvd_dvd_rat @ B @ A )
% 0.16/0.41 => ( ( ( divide_divide_rat @ A @ B )
% 0.16/0.41 = zero_zero_rat )
% 0.16/0.41 = ( A = zero_zero_rat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_div_eq_0_iff
% 0.16/0.41 thf(fact_684_dvd__div__eq__0__iff,axiom,
% 0.16/0.41 ! [B: nat,A: nat] :
% 0.16/0.41 ( ( dvd_dvd_nat @ B @ A )
% 0.16/0.41 => ( ( ( divide_divide_nat @ A @ B )
% 0.16/0.41 = zero_zero_nat )
% 0.16/0.41 = ( A = zero_zero_nat ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_div_eq_0_iff
% 0.16/0.41 thf(fact_685_dvd__div__eq__0__iff,axiom,
% 0.16/0.41 ! [B: int,A: int] :
% 0.16/0.41 ( ( dvd_dvd_int @ B @ A )
% 0.16/0.41 => ( ( ( divide_divide_int @ A @ B )
% 0.16/0.41 = zero_zero_int )
% 0.16/0.41 = ( A = zero_zero_int ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_div_eq_0_iff
% 0.16/0.41 thf(fact_686_of__nat__neq__0,axiom,
% 0.16/0.41 ! [N: nat] :
% 0.16/0.41 ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
% 0.16/0.41 != zero_zero_complex ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_neq_0
% 0.16/0.41 thf(fact_687_of__nat__neq__0,axiom,
% 0.16/0.41 ! [N: nat] :
% 0.16/0.41 ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
% 0.16/0.41 != zero_zero_real ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_neq_0
% 0.16/0.41 thf(fact_688_of__nat__neq__0,axiom,
% 0.16/0.41 ! [N: nat] :
% 0.16/0.41 ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
% 0.16/0.41 != zero_zero_rat ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_neq_0
% 0.16/0.41 thf(fact_689_of__nat__neq__0,axiom,
% 0.16/0.41 ! [N: nat] :
% 0.16/0.41 ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
% 0.16/0.41 != zero_zero_nat ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_neq_0
% 0.16/0.41 thf(fact_690_of__nat__neq__0,axiom,
% 0.16/0.41 ! [N: nat] :
% 0.16/0.41 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 0.16/0.41 != zero_zero_int ) ).
% 0.16/0.41
% 0.16/0.41 % of_nat_neq_0
% 0.16/0.41 thf(fact_691_verit__comp__simplify1_I2_J,axiom,
% 0.16/0.41 ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 0.16/0.41
% 0.16/0.41 % verit_comp_simplify1(2)
% 0.16/0.41 thf(fact_692_verit__comp__simplify1_I2_J,axiom,
% 0.16/0.41 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 0.16/0.41
% 0.16/0.41 % verit_comp_simplify1(2)
% 0.16/0.41 thf(fact_693_verit__comp__simplify1_I2_J,axiom,
% 0.16/0.41 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 0.16/0.41
% 0.16/0.41 % verit_comp_simplify1(2)
% 0.16/0.41 thf(fact_694_verit__comp__simplify1_I2_J,axiom,
% 0.16/0.41 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.16/0.41
% 0.16/0.41 % verit_comp_simplify1(2)
% 0.16/0.41 thf(fact_695_verit__comp__simplify1_I2_J,axiom,
% 0.16/0.41 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.16/0.41
% 0.16/0.41 % verit_comp_simplify1(2)
% 0.16/0.41 thf(fact_696_verit__comp__simplify1_I2_J,axiom,
% 0.16/0.41 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 0.16/0.41
% 0.16/0.41 % verit_comp_simplify1(2)
% 0.16/0.41 thf(fact_697_nle__le,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( ~ ( ord_less_eq_rat @ A @ B ) )
% 0.16/0.41 = ( ( ord_less_eq_rat @ B @ A )
% 0.16/0.41 & ( B != A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nle_le
% 0.16/0.41 thf(fact_698_nle__le,axiom,
% 0.16/0.41 ! [A: num,B: num] :
% 0.16/0.41 ( ( ~ ( ord_less_eq_num @ A @ B ) )
% 0.16/0.41 = ( ( ord_less_eq_num @ B @ A )
% 0.16/0.41 & ( B != A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nle_le
% 0.16/0.41 thf(fact_699_nle__le,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ~ ( ord_less_eq_nat @ A @ B ) )
% 0.16/0.41 = ( ( ord_less_eq_nat @ B @ A )
% 0.16/0.41 & ( B != A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nle_le
% 0.16/0.41 thf(fact_700_nle__le,axiom,
% 0.16/0.41 ! [A: int,B: int] :
% 0.16/0.41 ( ( ~ ( ord_less_eq_int @ A @ B ) )
% 0.16/0.41 = ( ( ord_less_eq_int @ B @ A )
% 0.16/0.41 & ( B != A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nle_le
% 0.16/0.41 thf(fact_701_nle__le,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ~ ( ord_less_eq_real @ A @ B ) )
% 0.16/0.41 = ( ( ord_less_eq_real @ B @ A )
% 0.16/0.41 & ( B != A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nle_le
% 0.16/0.41 thf(fact_702_le__cases3,axiom,
% 0.16/0.41 ! [X: rat,Y4: rat,Z: rat] :
% 0.16/0.41 ( ( ( ord_less_eq_rat @ X @ Y4 )
% 0.16/0.41 => ~ ( ord_less_eq_rat @ Y4 @ Z ) )
% 0.16/0.41 => ( ( ( ord_less_eq_rat @ Y4 @ X )
% 0.16/0.41 => ~ ( ord_less_eq_rat @ X @ Z ) )
% 0.16/0.41 => ( ( ( ord_less_eq_rat @ X @ Z )
% 0.16/0.41 => ~ ( ord_less_eq_rat @ Z @ Y4 ) )
% 0.16/0.41 => ( ( ( ord_less_eq_rat @ Z @ Y4 )
% 0.16/0.41 => ~ ( ord_less_eq_rat @ Y4 @ X ) )
% 0.16/0.41 => ( ( ( ord_less_eq_rat @ Y4 @ Z )
% 0.16/0.41 => ~ ( ord_less_eq_rat @ Z @ X ) )
% 0.16/0.41 => ~ ( ( ord_less_eq_rat @ Z @ X )
% 0.16/0.41 => ~ ( ord_less_eq_rat @ X @ Y4 ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_cases3
% 0.16/0.41 thf(fact_703_le__cases3,axiom,
% 0.16/0.41 ! [X: num,Y4: num,Z: num] :
% 0.16/0.41 ( ( ( ord_less_eq_num @ X @ Y4 )
% 0.16/0.41 => ~ ( ord_less_eq_num @ Y4 @ Z ) )
% 0.16/0.41 => ( ( ( ord_less_eq_num @ Y4 @ X )
% 0.16/0.41 => ~ ( ord_less_eq_num @ X @ Z ) )
% 0.16/0.41 => ( ( ( ord_less_eq_num @ X @ Z )
% 0.16/0.41 => ~ ( ord_less_eq_num @ Z @ Y4 ) )
% 0.16/0.41 => ( ( ( ord_less_eq_num @ Z @ Y4 )
% 0.16/0.41 => ~ ( ord_less_eq_num @ Y4 @ X ) )
% 0.16/0.41 => ( ( ( ord_less_eq_num @ Y4 @ Z )
% 0.16/0.41 => ~ ( ord_less_eq_num @ Z @ X ) )
% 0.16/0.41 => ~ ( ( ord_less_eq_num @ Z @ X )
% 0.16/0.41 => ~ ( ord_less_eq_num @ X @ Y4 ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_cases3
% 0.16/0.41 thf(fact_704_le__cases3,axiom,
% 0.16/0.41 ! [X: nat,Y4: nat,Z: nat] :
% 0.16/0.41 ( ( ( ord_less_eq_nat @ X @ Y4 )
% 0.16/0.41 => ~ ( ord_less_eq_nat @ Y4 @ Z ) )
% 0.16/0.41 => ( ( ( ord_less_eq_nat @ Y4 @ X )
% 0.16/0.41 => ~ ( ord_less_eq_nat @ X @ Z ) )
% 0.16/0.41 => ( ( ( ord_less_eq_nat @ X @ Z )
% 0.16/0.41 => ~ ( ord_less_eq_nat @ Z @ Y4 ) )
% 0.16/0.41 => ( ( ( ord_less_eq_nat @ Z @ Y4 )
% 0.16/0.41 => ~ ( ord_less_eq_nat @ Y4 @ X ) )
% 0.16/0.41 => ( ( ( ord_less_eq_nat @ Y4 @ Z )
% 0.16/0.41 => ~ ( ord_less_eq_nat @ Z @ X ) )
% 0.16/0.41 => ~ ( ( ord_less_eq_nat @ Z @ X )
% 0.16/0.41 => ~ ( ord_less_eq_nat @ X @ Y4 ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_cases3
% 0.16/0.41 thf(fact_705_le__cases3,axiom,
% 0.16/0.41 ! [X: int,Y4: int,Z: int] :
% 0.16/0.41 ( ( ( ord_less_eq_int @ X @ Y4 )
% 0.16/0.41 => ~ ( ord_less_eq_int @ Y4 @ Z ) )
% 0.16/0.41 => ( ( ( ord_less_eq_int @ Y4 @ X )
% 0.16/0.41 => ~ ( ord_less_eq_int @ X @ Z ) )
% 0.16/0.41 => ( ( ( ord_less_eq_int @ X @ Z )
% 0.16/0.41 => ~ ( ord_less_eq_int @ Z @ Y4 ) )
% 0.16/0.41 => ( ( ( ord_less_eq_int @ Z @ Y4 )
% 0.16/0.41 => ~ ( ord_less_eq_int @ Y4 @ X ) )
% 0.16/0.41 => ( ( ( ord_less_eq_int @ Y4 @ Z )
% 0.16/0.41 => ~ ( ord_less_eq_int @ Z @ X ) )
% 0.16/0.41 => ~ ( ( ord_less_eq_int @ Z @ X )
% 0.16/0.41 => ~ ( ord_less_eq_int @ X @ Y4 ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_cases3
% 0.16/0.41 thf(fact_706_le__cases3,axiom,
% 0.16/0.41 ! [X: real,Y4: real,Z: real] :
% 0.16/0.41 ( ( ( ord_less_eq_real @ X @ Y4 )
% 0.16/0.41 => ~ ( ord_less_eq_real @ Y4 @ Z ) )
% 0.16/0.41 => ( ( ( ord_less_eq_real @ Y4 @ X )
% 0.16/0.41 => ~ ( ord_less_eq_real @ X @ Z ) )
% 0.16/0.41 => ( ( ( ord_less_eq_real @ X @ Z )
% 0.16/0.41 => ~ ( ord_less_eq_real @ Z @ Y4 ) )
% 0.16/0.41 => ( ( ( ord_less_eq_real @ Z @ Y4 )
% 0.16/0.41 => ~ ( ord_less_eq_real @ Y4 @ X ) )
% 0.16/0.41 => ( ( ( ord_less_eq_real @ Y4 @ Z )
% 0.16/0.41 => ~ ( ord_less_eq_real @ Z @ X ) )
% 0.16/0.41 => ~ ( ( ord_less_eq_real @ Z @ X )
% 0.16/0.41 => ~ ( ord_less_eq_real @ X @ Y4 ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % le_cases3
% 0.16/0.41 thf(fact_707_order__class_Oorder__eq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y5: set_int,Z4: set_int] : ( Y5 = Z4 ) )
% 0.16/0.41 = ( ^ [X3: set_int,Y6: set_int] :
% 0.16/0.41 ( ( ord_less_eq_set_int @ X3 @ Y6 )
% 0.16/0.41 & ( ord_less_eq_set_int @ Y6 @ X3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_class.order_eq_iff
% 0.16/0.41 thf(fact_708_order__class_Oorder__eq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 ) )
% 0.16/0.41 = ( ^ [X3: rat,Y6: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X3 @ Y6 )
% 0.16/0.41 & ( ord_less_eq_rat @ Y6 @ X3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_class.order_eq_iff
% 0.16/0.41 thf(fact_709_order__class_Oorder__eq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
% 0.16/0.41 = ( ^ [X3: num,Y6: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X3 @ Y6 )
% 0.16/0.41 & ( ord_less_eq_num @ Y6 @ X3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_class.order_eq_iff
% 0.16/0.41 thf(fact_710_order__class_Oorder__eq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.16/0.41 = ( ^ [X3: nat,Y6: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ X3 @ Y6 )
% 0.16/0.41 & ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_class.order_eq_iff
% 0.16/0.41 thf(fact_711_order__class_Oorder__eq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.16/0.41 = ( ^ [X3: int,Y6: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ X3 @ Y6 )
% 0.16/0.41 & ( ord_less_eq_int @ Y6 @ X3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_class.order_eq_iff
% 0.16/0.41 thf(fact_712_order__class_Oorder__eq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
% 0.16/0.41 = ( ^ [X3: real,Y6: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X3 @ Y6 )
% 0.16/0.41 & ( ord_less_eq_real @ Y6 @ X3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_class.order_eq_iff
% 0.16/0.41 thf(fact_713_ord__eq__le__trans,axiom,
% 0.16/0.41 ! [A: set_int,B: set_int,C: set_int] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ( ( ord_less_eq_set_int @ B @ C )
% 0.16/0.41 => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_trans
% 0.16/0.41 thf(fact_714_ord__eq__le__trans,axiom,
% 0.16/0.41 ! [A: rat,B: rat,C: rat] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ( ( ord_less_eq_rat @ B @ C )
% 0.16/0.41 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_trans
% 0.16/0.41 thf(fact_715_ord__eq__le__trans,axiom,
% 0.16/0.41 ! [A: num,B: num,C: num] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ( ( ord_less_eq_num @ B @ C )
% 0.16/0.41 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_trans
% 0.16/0.41 thf(fact_716_ord__eq__le__trans,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ( ( ord_less_eq_nat @ B @ C )
% 0.16/0.41 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_trans
% 0.16/0.41 thf(fact_717_ord__eq__le__trans,axiom,
% 0.16/0.41 ! [A: int,B: int,C: int] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ( ( ord_less_eq_int @ B @ C )
% 0.16/0.41 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_trans
% 0.16/0.41 thf(fact_718_ord__eq__le__trans,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ( ( ord_less_eq_real @ B @ C )
% 0.16/0.41 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_trans
% 0.16/0.41 thf(fact_719_ord__le__eq__trans,axiom,
% 0.16/0.41 ! [A: set_int,B: set_int,C: set_int] :
% 0.16/0.41 ( ( ord_less_eq_set_int @ A @ B )
% 0.16/0.41 => ( ( B = C )
% 0.16/0.41 => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_trans
% 0.16/0.41 thf(fact_720_ord__le__eq__trans,axiom,
% 0.16/0.41 ! [A: rat,B: rat,C: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( B = C )
% 0.16/0.41 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_trans
% 0.16/0.41 thf(fact_721_ord__le__eq__trans,axiom,
% 0.16/0.41 ! [A: num,B: num,C: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.41 => ( ( B = C )
% 0.16/0.41 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_trans
% 0.16/0.41 thf(fact_722_ord__le__eq__trans,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( B = C )
% 0.16/0.41 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_trans
% 0.16/0.41 thf(fact_723_ord__le__eq__trans,axiom,
% 0.16/0.41 ! [A: int,B: int,C: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ A @ B )
% 0.16/0.41 => ( ( B = C )
% 0.16/0.41 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_trans
% 0.16/0.41 thf(fact_724_ord__le__eq__trans,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( B = C )
% 0.16/0.41 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_trans
% 0.16/0.41 thf(fact_725_order__antisym,axiom,
% 0.16/0.41 ! [X: set_int,Y4: set_int] :
% 0.16/0.41 ( ( ord_less_eq_set_int @ X @ Y4 )
% 0.16/0.41 => ( ( ord_less_eq_set_int @ Y4 @ X )
% 0.16/0.41 => ( X = Y4 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_antisym
% 0.16/0.41 thf(fact_726_order__antisym,axiom,
% 0.16/0.41 ! [X: rat,Y4: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X @ Y4 )
% 0.16/0.41 => ( ( ord_less_eq_rat @ Y4 @ X )
% 0.16/0.41 => ( X = Y4 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_antisym
% 0.16/0.41 thf(fact_727_order__antisym,axiom,
% 0.16/0.41 ! [X: num,Y4: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X @ Y4 )
% 0.16/0.41 => ( ( ord_less_eq_num @ Y4 @ X )
% 0.16/0.41 => ( X = Y4 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_antisym
% 0.16/0.41 thf(fact_728_order__antisym,axiom,
% 0.16/0.41 ! [X: nat,Y4: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.16/0.41 => ( ( ord_less_eq_nat @ Y4 @ X )
% 0.16/0.41 => ( X = Y4 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_antisym
% 0.16/0.41 thf(fact_729_order__antisym,axiom,
% 0.16/0.41 ! [X: int,Y4: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ X @ Y4 )
% 0.16/0.41 => ( ( ord_less_eq_int @ Y4 @ X )
% 0.16/0.41 => ( X = Y4 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_antisym
% 0.16/0.41 thf(fact_730_order__antisym,axiom,
% 0.16/0.41 ! [X: real,Y4: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X @ Y4 )
% 0.16/0.41 => ( ( ord_less_eq_real @ Y4 @ X )
% 0.16/0.41 => ( X = Y4 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_antisym
% 0.16/0.41 thf(fact_731_order_Otrans,axiom,
% 0.16/0.41 ! [A: set_int,B: set_int,C: set_int] :
% 0.16/0.41 ( ( ord_less_eq_set_int @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_set_int @ B @ C )
% 0.16/0.41 => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order.trans
% 0.16/0.41 thf(fact_732_order_Otrans,axiom,
% 0.16/0.41 ! [A: rat,B: rat,C: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_rat @ B @ C )
% 0.16/0.41 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order.trans
% 0.16/0.41 thf(fact_733_order_Otrans,axiom,
% 0.16/0.41 ! [A: num,B: num,C: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_num @ B @ C )
% 0.16/0.41 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order.trans
% 0.16/0.41 thf(fact_734_order_Otrans,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_nat @ B @ C )
% 0.16/0.41 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order.trans
% 0.16/0.41 thf(fact_735_order_Otrans,axiom,
% 0.16/0.41 ! [A: int,B: int,C: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_int @ B @ C )
% 0.16/0.41 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order.trans
% 0.16/0.41 thf(fact_736_order_Otrans,axiom,
% 0.16/0.41 ! [A: real,B: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_real @ B @ C )
% 0.16/0.41 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order.trans
% 0.16/0.41 thf(fact_737_order__trans,axiom,
% 0.16/0.41 ! [X: set_int,Y4: set_int,Z: set_int] :
% 0.16/0.41 ( ( ord_less_eq_set_int @ X @ Y4 )
% 0.16/0.41 => ( ( ord_less_eq_set_int @ Y4 @ Z )
% 0.16/0.41 => ( ord_less_eq_set_int @ X @ Z ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_trans
% 0.16/0.41 thf(fact_738_order__trans,axiom,
% 0.16/0.41 ! [X: rat,Y4: rat,Z: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X @ Y4 )
% 0.16/0.41 => ( ( ord_less_eq_rat @ Y4 @ Z )
% 0.16/0.41 => ( ord_less_eq_rat @ X @ Z ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_trans
% 0.16/0.41 thf(fact_739_order__trans,axiom,
% 0.16/0.41 ! [X: num,Y4: num,Z: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X @ Y4 )
% 0.16/0.41 => ( ( ord_less_eq_num @ Y4 @ Z )
% 0.16/0.41 => ( ord_less_eq_num @ X @ Z ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_trans
% 0.16/0.41 thf(fact_740_order__trans,axiom,
% 0.16/0.41 ! [X: nat,Y4: nat,Z: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.16/0.41 => ( ( ord_less_eq_nat @ Y4 @ Z )
% 0.16/0.41 => ( ord_less_eq_nat @ X @ Z ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_trans
% 0.16/0.41 thf(fact_741_order__trans,axiom,
% 0.16/0.41 ! [X: int,Y4: int,Z: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ X @ Y4 )
% 0.16/0.41 => ( ( ord_less_eq_int @ Y4 @ Z )
% 0.16/0.41 => ( ord_less_eq_int @ X @ Z ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_trans
% 0.16/0.41 thf(fact_742_order__trans,axiom,
% 0.16/0.41 ! [X: real,Y4: real,Z: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X @ Y4 )
% 0.16/0.41 => ( ( ord_less_eq_real @ Y4 @ Z )
% 0.16/0.41 => ( ord_less_eq_real @ X @ Z ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_trans
% 0.16/0.41 thf(fact_743_linorder__wlog,axiom,
% 0.16/0.41 ! [P: rat > rat > $o,A: rat,B: rat] :
% 0.16/0.41 ( ! [A4: rat,B3: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A4 @ B3 )
% 0.16/0.41 => ( P @ A4 @ B3 ) )
% 0.16/0.41 => ( ! [A4: rat,B3: rat] :
% 0.16/0.41 ( ( P @ B3 @ A4 )
% 0.16/0.41 => ( P @ A4 @ B3 ) )
% 0.16/0.41 => ( P @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % linorder_wlog
% 0.16/0.41 thf(fact_744_linorder__wlog,axiom,
% 0.16/0.41 ! [P: num > num > $o,A: num,B: num] :
% 0.16/0.41 ( ! [A4: num,B3: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ A4 @ B3 )
% 0.16/0.41 => ( P @ A4 @ B3 ) )
% 0.16/0.41 => ( ! [A4: num,B3: num] :
% 0.16/0.41 ( ( P @ B3 @ A4 )
% 0.16/0.41 => ( P @ A4 @ B3 ) )
% 0.16/0.41 => ( P @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % linorder_wlog
% 0.16/0.41 thf(fact_745_linorder__wlog,axiom,
% 0.16/0.41 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.16/0.41 ( ! [A4: nat,B3: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A4 @ B3 )
% 0.16/0.41 => ( P @ A4 @ B3 ) )
% 0.16/0.41 => ( ! [A4: nat,B3: nat] :
% 0.16/0.41 ( ( P @ B3 @ A4 )
% 0.16/0.41 => ( P @ A4 @ B3 ) )
% 0.16/0.41 => ( P @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % linorder_wlog
% 0.16/0.41 thf(fact_746_linorder__wlog,axiom,
% 0.16/0.41 ! [P: int > int > $o,A: int,B: int] :
% 0.16/0.41 ( ! [A4: int,B3: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ A4 @ B3 )
% 0.16/0.41 => ( P @ A4 @ B3 ) )
% 0.16/0.41 => ( ! [A4: int,B3: int] :
% 0.16/0.41 ( ( P @ B3 @ A4 )
% 0.16/0.41 => ( P @ A4 @ B3 ) )
% 0.16/0.41 => ( P @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % linorder_wlog
% 0.16/0.41 thf(fact_747_linorder__wlog,axiom,
% 0.16/0.41 ! [P: real > real > $o,A: real,B: real] :
% 0.16/0.41 ( ! [A4: real,B3: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A4 @ B3 )
% 0.16/0.41 => ( P @ A4 @ B3 ) )
% 0.16/0.41 => ( ! [A4: real,B3: real] :
% 0.16/0.41 ( ( P @ B3 @ A4 )
% 0.16/0.41 => ( P @ A4 @ B3 ) )
% 0.16/0.41 => ( P @ A @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % linorder_wlog
% 0.16/0.41 thf(fact_748_dual__order_Oeq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y5: set_int,Z4: set_int] : ( Y5 = Z4 ) )
% 0.16/0.41 = ( ^ [A3: set_int,B2: set_int] :
% 0.16/0.41 ( ( ord_less_eq_set_int @ B2 @ A3 )
% 0.16/0.41 & ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.eq_iff
% 0.16/0.41 thf(fact_749_dual__order_Oeq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 ) )
% 0.16/0.41 = ( ^ [A3: rat,B2: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ B2 @ A3 )
% 0.16/0.41 & ( ord_less_eq_rat @ A3 @ B2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.eq_iff
% 0.16/0.41 thf(fact_750_dual__order_Oeq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
% 0.16/0.41 = ( ^ [A3: num,B2: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ B2 @ A3 )
% 0.16/0.41 & ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.eq_iff
% 0.16/0.41 thf(fact_751_dual__order_Oeq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.16/0.41 = ( ^ [A3: nat,B2: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ B2 @ A3 )
% 0.16/0.41 & ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.eq_iff
% 0.16/0.41 thf(fact_752_dual__order_Oeq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.16/0.41 = ( ^ [A3: int,B2: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ B2 @ A3 )
% 0.16/0.41 & ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.eq_iff
% 0.16/0.41 thf(fact_753_dual__order_Oeq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
% 0.16/0.41 = ( ^ [A3: real,B2: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ B2 @ A3 )
% 0.16/0.41 & ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.eq_iff
% 0.16/0.41 thf(fact_754_dual__order_Oantisym,axiom,
% 0.16/0.41 ! [B: set_int,A: set_int] :
% 0.16/0.41 ( ( ord_less_eq_set_int @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_set_int @ A @ B )
% 0.16/0.41 => ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.antisym
% 0.16/0.41 thf(fact_755_dual__order_Oantisym,axiom,
% 0.16/0.41 ! [B: rat,A: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.antisym
% 0.16/0.41 thf(fact_756_dual__order_Oantisym,axiom,
% 0.16/0.41 ! [B: num,A: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_num @ A @ B )
% 0.16/0.41 => ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.antisym
% 0.16/0.41 thf(fact_757_dual__order_Oantisym,axiom,
% 0.16/0.41 ! [B: nat,A: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.antisym
% 0.16/0.41 thf(fact_758_dual__order_Oantisym,axiom,
% 0.16/0.41 ! [B: int,A: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_int @ A @ B )
% 0.16/0.41 => ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.antisym
% 0.16/0.41 thf(fact_759_dual__order_Oantisym,axiom,
% 0.16/0.41 ! [B: real,A: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.antisym
% 0.16/0.41 thf(fact_760_dual__order_Otrans,axiom,
% 0.16/0.41 ! [B: set_int,A: set_int,C: set_int] :
% 0.16/0.41 ( ( ord_less_eq_set_int @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_set_int @ C @ B )
% 0.16/0.41 => ( ord_less_eq_set_int @ C @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.trans
% 0.16/0.41 thf(fact_761_dual__order_Otrans,axiom,
% 0.16/0.41 ! [B: rat,A: rat,C: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_rat @ C @ B )
% 0.16/0.41 => ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.trans
% 0.16/0.41 thf(fact_762_dual__order_Otrans,axiom,
% 0.16/0.41 ! [B: num,A: num,C: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_num @ C @ B )
% 0.16/0.41 => ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.trans
% 0.16/0.41 thf(fact_763_dual__order_Otrans,axiom,
% 0.16/0.41 ! [B: nat,A: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_nat @ C @ B )
% 0.16/0.41 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.trans
% 0.16/0.41 thf(fact_764_dual__order_Otrans,axiom,
% 0.16/0.41 ! [B: int,A: int,C: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_int @ C @ B )
% 0.16/0.41 => ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.trans
% 0.16/0.41 thf(fact_765_dual__order_Otrans,axiom,
% 0.16/0.41 ! [B: real,A: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ B @ A )
% 0.16/0.41 => ( ( ord_less_eq_real @ C @ B )
% 0.16/0.41 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dual_order.trans
% 0.16/0.41 thf(fact_766_antisym,axiom,
% 0.16/0.41 ! [A: set_int,B: set_int] :
% 0.16/0.41 ( ( ord_less_eq_set_int @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_set_int @ B @ A )
% 0.16/0.41 => ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % antisym
% 0.16/0.41 thf(fact_767_antisym,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_rat @ B @ A )
% 0.16/0.41 => ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % antisym
% 0.16/0.41 thf(fact_768_antisym,axiom,
% 0.16/0.41 ! [A: num,B: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_num @ B @ A )
% 0.16/0.41 => ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % antisym
% 0.16/0.41 thf(fact_769_antisym,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_nat @ B @ A )
% 0.16/0.41 => ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % antisym
% 0.16/0.41 thf(fact_770_antisym,axiom,
% 0.16/0.41 ! [A: int,B: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_int @ B @ A )
% 0.16/0.41 => ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % antisym
% 0.16/0.41 thf(fact_771_antisym,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_real @ B @ A )
% 0.16/0.41 => ( A = B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % antisym
% 0.16/0.41 thf(fact_772_Orderings_Oorder__eq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y5: set_int,Z4: set_int] : ( Y5 = Z4 ) )
% 0.16/0.41 = ( ^ [A3: set_int,B2: set_int] :
% 0.16/0.41 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.16/0.41 & ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % Orderings.order_eq_iff
% 0.16/0.41 thf(fact_773_Orderings_Oorder__eq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 ) )
% 0.16/0.41 = ( ^ [A3: rat,B2: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A3 @ B2 )
% 0.16/0.41 & ( ord_less_eq_rat @ B2 @ A3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % Orderings.order_eq_iff
% 0.16/0.41 thf(fact_774_Orderings_Oorder__eq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
% 0.16/0.41 = ( ^ [A3: num,B2: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ A3 @ B2 )
% 0.16/0.41 & ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % Orderings.order_eq_iff
% 0.16/0.41 thf(fact_775_Orderings_Oorder__eq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.16/0.41 = ( ^ [A3: nat,B2: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ A3 @ B2 )
% 0.16/0.41 & ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % Orderings.order_eq_iff
% 0.16/0.41 thf(fact_776_Orderings_Oorder__eq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.16/0.41 = ( ^ [A3: int,B2: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ A3 @ B2 )
% 0.16/0.41 & ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % Orderings.order_eq_iff
% 0.16/0.41 thf(fact_777_Orderings_Oorder__eq__iff,axiom,
% 0.16/0.41 ( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
% 0.16/0.41 = ( ^ [A3: real,B2: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ A3 @ B2 )
% 0.16/0.41 & ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % Orderings.order_eq_iff
% 0.16/0.41 thf(fact_778_order__subst1,axiom,
% 0.16/0.41 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_rat @ B @ C )
% 0.16/0.41 => ( ! [X4: rat,Y: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst1
% 0.16/0.41 thf(fact_779_order__subst1,axiom,
% 0.16/0.41 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_num @ B @ C )
% 0.16/0.41 => ( ! [X4: num,Y: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst1
% 0.16/0.41 thf(fact_780_order__subst1,axiom,
% 0.16/0.41 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_nat @ B @ C )
% 0.16/0.41 => ( ! [X4: nat,Y: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst1
% 0.16/0.41 thf(fact_781_order__subst1,axiom,
% 0.16/0.41 ! [A: rat,F: int > rat,B: int,C: int] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_int @ B @ C )
% 0.16/0.41 => ( ! [X4: int,Y: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst1
% 0.16/0.41 thf(fact_782_order__subst1,axiom,
% 0.16/0.41 ! [A: rat,F: real > rat,B: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_real @ B @ C )
% 0.16/0.41 => ( ! [X4: real,Y: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst1
% 0.16/0.41 thf(fact_783_order__subst1,axiom,
% 0.16/0.41 ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.16/0.41 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_rat @ B @ C )
% 0.16/0.41 => ( ! [X4: rat,Y: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst1
% 0.16/0.41 thf(fact_784_order__subst1,axiom,
% 0.16/0.41 ! [A: num,F: num > num,B: num,C: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_num @ B @ C )
% 0.16/0.41 => ( ! [X4: num,Y: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst1
% 0.16/0.41 thf(fact_785_order__subst1,axiom,
% 0.16/0.41 ! [A: num,F: nat > num,B: nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_nat @ B @ C )
% 0.16/0.41 => ( ! [X4: nat,Y: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst1
% 0.16/0.41 thf(fact_786_order__subst1,axiom,
% 0.16/0.41 ! [A: num,F: int > num,B: int,C: int] :
% 0.16/0.41 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_int @ B @ C )
% 0.16/0.41 => ( ! [X4: int,Y: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst1
% 0.16/0.41 thf(fact_787_order__subst1,axiom,
% 0.16/0.41 ! [A: num,F: real > num,B: real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_real @ B @ C )
% 0.16/0.41 => ( ! [X4: real,Y: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst1
% 0.16/0.41 thf(fact_788_order__subst2,axiom,
% 0.16/0.41 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.16/0.41 => ( ! [X4: rat,Y: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst2
% 0.16/0.41 thf(fact_789_order__subst2,axiom,
% 0.16/0.41 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.16/0.41 => ( ! [X4: rat,Y: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst2
% 0.16/0.41 thf(fact_790_order__subst2,axiom,
% 0.16/0.41 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.16/0.41 => ( ! [X4: rat,Y: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst2
% 0.16/0.41 thf(fact_791_order__subst2,axiom,
% 0.16/0.41 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.16/0.41 => ( ! [X4: rat,Y: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst2
% 0.16/0.41 thf(fact_792_order__subst2,axiom,
% 0.16/0.41 ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.16/0.41 => ( ! [X4: rat,Y: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst2
% 0.16/0.41 thf(fact_793_order__subst2,axiom,
% 0.16/0.41 ! [A: num,B: num,F: num > rat,C: rat] :
% 0.16/0.41 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.16/0.41 => ( ! [X4: num,Y: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst2
% 0.16/0.41 thf(fact_794_order__subst2,axiom,
% 0.16/0.41 ! [A: num,B: num,F: num > num,C: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.16/0.41 => ( ! [X4: num,Y: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst2
% 0.16/0.41 thf(fact_795_order__subst2,axiom,
% 0.16/0.41 ! [A: num,B: num,F: num > nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.16/0.41 => ( ! [X4: num,Y: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst2
% 0.16/0.41 thf(fact_796_order__subst2,axiom,
% 0.16/0.41 ! [A: num,B: num,F: num > int,C: int] :
% 0.16/0.41 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.16/0.41 => ( ! [X4: num,Y: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst2
% 0.16/0.41 thf(fact_797_order__subst2,axiom,
% 0.16/0.41 ! [A: num,B: num,F: num > real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.41 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.16/0.41 => ( ! [X4: num,Y: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_subst2
% 0.16/0.41 thf(fact_798_order__eq__refl,axiom,
% 0.16/0.41 ! [X: set_int,Y4: set_int] :
% 0.16/0.41 ( ( X = Y4 )
% 0.16/0.41 => ( ord_less_eq_set_int @ X @ Y4 ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_eq_refl
% 0.16/0.41 thf(fact_799_order__eq__refl,axiom,
% 0.16/0.41 ! [X: rat,Y4: rat] :
% 0.16/0.41 ( ( X = Y4 )
% 0.16/0.41 => ( ord_less_eq_rat @ X @ Y4 ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_eq_refl
% 0.16/0.41 thf(fact_800_order__eq__refl,axiom,
% 0.16/0.41 ! [X: num,Y4: num] :
% 0.16/0.41 ( ( X = Y4 )
% 0.16/0.41 => ( ord_less_eq_num @ X @ Y4 ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_eq_refl
% 0.16/0.41 thf(fact_801_order__eq__refl,axiom,
% 0.16/0.41 ! [X: nat,Y4: nat] :
% 0.16/0.41 ( ( X = Y4 )
% 0.16/0.41 => ( ord_less_eq_nat @ X @ Y4 ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_eq_refl
% 0.16/0.41 thf(fact_802_order__eq__refl,axiom,
% 0.16/0.41 ! [X: int,Y4: int] :
% 0.16/0.41 ( ( X = Y4 )
% 0.16/0.41 => ( ord_less_eq_int @ X @ Y4 ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_eq_refl
% 0.16/0.41 thf(fact_803_order__eq__refl,axiom,
% 0.16/0.41 ! [X: real,Y4: real] :
% 0.16/0.41 ( ( X = Y4 )
% 0.16/0.41 => ( ord_less_eq_real @ X @ Y4 ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_eq_refl
% 0.16/0.41 thf(fact_804_verit__la__disequality,axiom,
% 0.16/0.41 ! [A: rat,B: rat] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 | ~ ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 | ~ ( ord_less_eq_rat @ B @ A ) ) ).
% 0.16/0.41
% 0.16/0.41 % verit_la_disequality
% 0.16/0.41 thf(fact_805_verit__la__disequality,axiom,
% 0.16/0.41 ! [A: num,B: num] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 | ~ ( ord_less_eq_num @ A @ B )
% 0.16/0.41 | ~ ( ord_less_eq_num @ B @ A ) ) ).
% 0.16/0.41
% 0.16/0.41 % verit_la_disequality
% 0.16/0.41 thf(fact_806_verit__la__disequality,axiom,
% 0.16/0.41 ! [A: nat,B: nat] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 | ~ ( ord_less_eq_nat @ A @ B )
% 0.16/0.41 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 0.16/0.41
% 0.16/0.41 % verit_la_disequality
% 0.16/0.41 thf(fact_807_verit__la__disequality,axiom,
% 0.16/0.41 ! [A: int,B: int] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 | ~ ( ord_less_eq_int @ A @ B )
% 0.16/0.41 | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 0.16/0.41
% 0.16/0.41 % verit_la_disequality
% 0.16/0.41 thf(fact_808_verit__la__disequality,axiom,
% 0.16/0.41 ! [A: real,B: real] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 | ~ ( ord_less_eq_real @ A @ B )
% 0.16/0.41 | ~ ( ord_less_eq_real @ B @ A ) ) ).
% 0.16/0.41
% 0.16/0.41 % verit_la_disequality
% 0.16/0.41 thf(fact_809_linorder__linear,axiom,
% 0.16/0.41 ! [X: rat,Y4: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X @ Y4 )
% 0.16/0.41 | ( ord_less_eq_rat @ Y4 @ X ) ) ).
% 0.16/0.41
% 0.16/0.41 % linorder_linear
% 0.16/0.41 thf(fact_810_linorder__linear,axiom,
% 0.16/0.41 ! [X: num,Y4: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X @ Y4 )
% 0.16/0.41 | ( ord_less_eq_num @ Y4 @ X ) ) ).
% 0.16/0.41
% 0.16/0.41 % linorder_linear
% 0.16/0.41 thf(fact_811_linorder__linear,axiom,
% 0.16/0.41 ! [X: nat,Y4: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.16/0.41 | ( ord_less_eq_nat @ Y4 @ X ) ) ).
% 0.16/0.41
% 0.16/0.41 % linorder_linear
% 0.16/0.41 thf(fact_812_linorder__linear,axiom,
% 0.16/0.41 ! [X: int,Y4: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ X @ Y4 )
% 0.16/0.41 | ( ord_less_eq_int @ Y4 @ X ) ) ).
% 0.16/0.41
% 0.16/0.41 % linorder_linear
% 0.16/0.41 thf(fact_813_linorder__linear,axiom,
% 0.16/0.41 ! [X: real,Y4: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ X @ Y4 )
% 0.16/0.41 | ( ord_less_eq_real @ Y4 @ X ) ) ).
% 0.16/0.41
% 0.16/0.41 % linorder_linear
% 0.16/0.41 thf(fact_814_ord__eq__le__subst,axiom,
% 0.16/0.41 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_rat @ B @ C )
% 0.16/0.41 => ( ! [X4: rat,Y: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_subst
% 0.16/0.41 thf(fact_815_ord__eq__le__subst,axiom,
% 0.16/0.41 ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_rat @ B @ C )
% 0.16/0.41 => ( ! [X4: rat,Y: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_subst
% 0.16/0.41 thf(fact_816_ord__eq__le__subst,axiom,
% 0.16/0.41 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_rat @ B @ C )
% 0.16/0.41 => ( ! [X4: rat,Y: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_subst
% 0.16/0.41 thf(fact_817_ord__eq__le__subst,axiom,
% 0.16/0.41 ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_rat @ B @ C )
% 0.16/0.41 => ( ! [X4: rat,Y: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_subst
% 0.16/0.41 thf(fact_818_ord__eq__le__subst,axiom,
% 0.16/0.41 ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_rat @ B @ C )
% 0.16/0.41 => ( ! [X4: rat,Y: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_subst
% 0.16/0.41 thf(fact_819_ord__eq__le__subst,axiom,
% 0.16/0.41 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_num @ B @ C )
% 0.16/0.41 => ( ! [X4: num,Y: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_subst
% 0.16/0.41 thf(fact_820_ord__eq__le__subst,axiom,
% 0.16/0.41 ! [A: num,F: num > num,B: num,C: num] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_num @ B @ C )
% 0.16/0.41 => ( ! [X4: num,Y: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_subst
% 0.16/0.41 thf(fact_821_ord__eq__le__subst,axiom,
% 0.16/0.41 ! [A: nat,F: num > nat,B: num,C: num] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_num @ B @ C )
% 0.16/0.41 => ( ! [X4: num,Y: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_subst
% 0.16/0.41 thf(fact_822_ord__eq__le__subst,axiom,
% 0.16/0.41 ! [A: int,F: num > int,B: num,C: num] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_num @ B @ C )
% 0.16/0.41 => ( ! [X4: num,Y: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_subst
% 0.16/0.41 thf(fact_823_ord__eq__le__subst,axiom,
% 0.16/0.41 ! [A: real,F: num > real,B: num,C: num] :
% 0.16/0.41 ( ( A
% 0.16/0.41 = ( F @ B ) )
% 0.16/0.41 => ( ( ord_less_eq_num @ B @ C )
% 0.16/0.41 => ( ! [X4: num,Y: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_eq_le_subst
% 0.16/0.41 thf(fact_824_ord__le__eq__subst,axiom,
% 0.16/0.41 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( ( F @ B )
% 0.16/0.41 = C )
% 0.16/0.41 => ( ! [X4: rat,Y: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_subst
% 0.16/0.41 thf(fact_825_ord__le__eq__subst,axiom,
% 0.16/0.41 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( ( F @ B )
% 0.16/0.41 = C )
% 0.16/0.41 => ( ! [X4: rat,Y: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_subst
% 0.16/0.41 thf(fact_826_ord__le__eq__subst,axiom,
% 0.16/0.41 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( ( F @ B )
% 0.16/0.41 = C )
% 0.16/0.41 => ( ! [X4: rat,Y: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_subst
% 0.16/0.41 thf(fact_827_ord__le__eq__subst,axiom,
% 0.16/0.41 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( ( F @ B )
% 0.16/0.41 = C )
% 0.16/0.41 => ( ! [X4: rat,Y: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_subst
% 0.16/0.41 thf(fact_828_ord__le__eq__subst,axiom,
% 0.16/0.41 ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.41 => ( ( ( F @ B )
% 0.16/0.41 = C )
% 0.16/0.41 => ( ! [X4: rat,Y: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_subst
% 0.16/0.41 thf(fact_829_ord__le__eq__subst,axiom,
% 0.16/0.41 ! [A: num,B: num,F: num > rat,C: rat] :
% 0.16/0.41 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.41 => ( ( ( F @ B )
% 0.16/0.41 = C )
% 0.16/0.41 => ( ! [X4: num,Y: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_subst
% 0.16/0.41 thf(fact_830_ord__le__eq__subst,axiom,
% 0.16/0.41 ! [A: num,B: num,F: num > num,C: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.41 => ( ( ( F @ B )
% 0.16/0.41 = C )
% 0.16/0.41 => ( ! [X4: num,Y: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_subst
% 0.16/0.41 thf(fact_831_ord__le__eq__subst,axiom,
% 0.16/0.41 ! [A: num,B: num,F: num > nat,C: nat] :
% 0.16/0.41 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.41 => ( ( ( F @ B )
% 0.16/0.41 = C )
% 0.16/0.41 => ( ! [X4: num,Y: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_subst
% 0.16/0.41 thf(fact_832_ord__le__eq__subst,axiom,
% 0.16/0.41 ! [A: num,B: num,F: num > int,C: int] :
% 0.16/0.41 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.41 => ( ( ( F @ B )
% 0.16/0.41 = C )
% 0.16/0.41 => ( ! [X4: num,Y: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_subst
% 0.16/0.41 thf(fact_833_ord__le__eq__subst,axiom,
% 0.16/0.41 ! [A: num,B: num,F: num > real,C: real] :
% 0.16/0.41 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.41 => ( ( ( F @ B )
% 0.16/0.41 = C )
% 0.16/0.41 => ( ! [X4: num,Y: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.41 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % ord_le_eq_subst
% 0.16/0.41 thf(fact_834_linorder__le__cases,axiom,
% 0.16/0.41 ! [X: rat,Y4: rat] :
% 0.16/0.41 ( ~ ( ord_less_eq_rat @ X @ Y4 )
% 0.16/0.41 => ( ord_less_eq_rat @ Y4 @ X ) ) ).
% 0.16/0.41
% 0.16/0.41 % linorder_le_cases
% 0.16/0.41 thf(fact_835_linorder__le__cases,axiom,
% 0.16/0.41 ! [X: num,Y4: num] :
% 0.16/0.41 ( ~ ( ord_less_eq_num @ X @ Y4 )
% 0.16/0.41 => ( ord_less_eq_num @ Y4 @ X ) ) ).
% 0.16/0.41
% 0.16/0.41 % linorder_le_cases
% 0.16/0.41 thf(fact_836_linorder__le__cases,axiom,
% 0.16/0.41 ! [X: nat,Y4: nat] :
% 0.16/0.41 ( ~ ( ord_less_eq_nat @ X @ Y4 )
% 0.16/0.41 => ( ord_less_eq_nat @ Y4 @ X ) ) ).
% 0.16/0.41
% 0.16/0.41 % linorder_le_cases
% 0.16/0.41 thf(fact_837_linorder__le__cases,axiom,
% 0.16/0.41 ! [X: int,Y4: int] :
% 0.16/0.41 ( ~ ( ord_less_eq_int @ X @ Y4 )
% 0.16/0.41 => ( ord_less_eq_int @ Y4 @ X ) ) ).
% 0.16/0.41
% 0.16/0.41 % linorder_le_cases
% 0.16/0.41 thf(fact_838_linorder__le__cases,axiom,
% 0.16/0.41 ! [X: real,Y4: real] :
% 0.16/0.41 ( ~ ( ord_less_eq_real @ X @ Y4 )
% 0.16/0.41 => ( ord_less_eq_real @ Y4 @ X ) ) ).
% 0.16/0.41
% 0.16/0.41 % linorder_le_cases
% 0.16/0.41 thf(fact_839_order__antisym__conv,axiom,
% 0.16/0.41 ! [Y4: set_int,X: set_int] :
% 0.16/0.41 ( ( ord_less_eq_set_int @ Y4 @ X )
% 0.16/0.41 => ( ( ord_less_eq_set_int @ X @ Y4 )
% 0.16/0.41 = ( X = Y4 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_antisym_conv
% 0.16/0.41 thf(fact_840_order__antisym__conv,axiom,
% 0.16/0.41 ! [Y4: rat,X: rat] :
% 0.16/0.41 ( ( ord_less_eq_rat @ Y4 @ X )
% 0.16/0.41 => ( ( ord_less_eq_rat @ X @ Y4 )
% 0.16/0.41 = ( X = Y4 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_antisym_conv
% 0.16/0.41 thf(fact_841_order__antisym__conv,axiom,
% 0.16/0.41 ! [Y4: num,X: num] :
% 0.16/0.41 ( ( ord_less_eq_num @ Y4 @ X )
% 0.16/0.41 => ( ( ord_less_eq_num @ X @ Y4 )
% 0.16/0.41 = ( X = Y4 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_antisym_conv
% 0.16/0.41 thf(fact_842_order__antisym__conv,axiom,
% 0.16/0.41 ! [Y4: nat,X: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ Y4 @ X )
% 0.16/0.41 => ( ( ord_less_eq_nat @ X @ Y4 )
% 0.16/0.41 = ( X = Y4 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_antisym_conv
% 0.16/0.41 thf(fact_843_order__antisym__conv,axiom,
% 0.16/0.41 ! [Y4: int,X: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ Y4 @ X )
% 0.16/0.41 => ( ( ord_less_eq_int @ X @ Y4 )
% 0.16/0.41 = ( X = Y4 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_antisym_conv
% 0.16/0.41 thf(fact_844_order__antisym__conv,axiom,
% 0.16/0.41 ! [Y4: real,X: real] :
% 0.16/0.41 ( ( ord_less_eq_real @ Y4 @ X )
% 0.16/0.41 => ( ( ord_less_eq_real @ X @ Y4 )
% 0.16/0.41 = ( X = Y4 ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % order_antisym_conv
% 0.16/0.41 thf(fact_845_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.16/0.41 ! [C: nat,A: nat,B: nat] :
% 0.16/0.41 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.16/0.41 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.16/0.41 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.16/0.41 thf(fact_846_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.16/0.41 ! [C: int,A: int,B: int] :
% 0.16/0.41 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.16/0.41 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.16/0.41 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.16/0.41 thf(fact_847_divide__eq__eq__numeral_I1_J,axiom,
% 0.16/0.41 ! [B: complex,C: complex,W: num] :
% 0.16/0.41 ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.16/0.41 = ( numera6690914467698888265omplex @ W ) )
% 0.16/0.41 = ( ( ( C != zero_zero_complex )
% 0.16/0.41 => ( B
% 0.16/0.41 = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
% 0.16/0.41 & ( ( C = zero_zero_complex )
% 0.16/0.41 => ( ( numera6690914467698888265omplex @ W )
% 0.16/0.41 = zero_zero_complex ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_eq_eq_numeral(1)
% 0.16/0.41 thf(fact_848_divide__eq__eq__numeral_I1_J,axiom,
% 0.16/0.41 ! [B: real,C: real,W: num] :
% 0.16/0.41 ( ( ( divide_divide_real @ B @ C )
% 0.16/0.41 = ( numeral_numeral_real @ W ) )
% 0.16/0.41 = ( ( ( C != zero_zero_real )
% 0.16/0.41 => ( B
% 0.16/0.41 = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.16/0.41 & ( ( C = zero_zero_real )
% 0.16/0.41 => ( ( numeral_numeral_real @ W )
% 0.16/0.41 = zero_zero_real ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_eq_eq_numeral(1)
% 0.16/0.41 thf(fact_849_divide__eq__eq__numeral_I1_J,axiom,
% 0.16/0.41 ! [B: rat,C: rat,W: num] :
% 0.16/0.41 ( ( ( divide_divide_rat @ B @ C )
% 0.16/0.41 = ( numeral_numeral_rat @ W ) )
% 0.16/0.41 = ( ( ( C != zero_zero_rat )
% 0.16/0.41 => ( B
% 0.16/0.41 = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.16/0.41 & ( ( C = zero_zero_rat )
% 0.16/0.41 => ( ( numeral_numeral_rat @ W )
% 0.16/0.41 = zero_zero_rat ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % divide_eq_eq_numeral(1)
% 0.16/0.41 thf(fact_850_eq__divide__eq__numeral_I1_J,axiom,
% 0.16/0.41 ! [W: num,B: complex,C: complex] :
% 0.16/0.41 ( ( ( numera6690914467698888265omplex @ W )
% 0.16/0.41 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.16/0.41 = ( ( ( C != zero_zero_complex )
% 0.16/0.41 => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
% 0.16/0.41 = B ) )
% 0.16/0.41 & ( ( C = zero_zero_complex )
% 0.16/0.41 => ( ( numera6690914467698888265omplex @ W )
% 0.16/0.41 = zero_zero_complex ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_divide_eq_numeral(1)
% 0.16/0.41 thf(fact_851_eq__divide__eq__numeral_I1_J,axiom,
% 0.16/0.41 ! [W: num,B: real,C: real] :
% 0.16/0.41 ( ( ( numeral_numeral_real @ W )
% 0.16/0.41 = ( divide_divide_real @ B @ C ) )
% 0.16/0.41 = ( ( ( C != zero_zero_real )
% 0.16/0.41 => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
% 0.16/0.41 = B ) )
% 0.16/0.41 & ( ( C = zero_zero_real )
% 0.16/0.41 => ( ( numeral_numeral_real @ W )
% 0.16/0.41 = zero_zero_real ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_divide_eq_numeral(1)
% 0.16/0.41 thf(fact_852_eq__divide__eq__numeral_I1_J,axiom,
% 0.16/0.41 ! [W: num,B: rat,C: rat] :
% 0.16/0.41 ( ( ( numeral_numeral_rat @ W )
% 0.16/0.41 = ( divide_divide_rat @ B @ C ) )
% 0.16/0.41 = ( ( ( C != zero_zero_rat )
% 0.16/0.41 => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
% 0.16/0.41 = B ) )
% 0.16/0.41 & ( ( C = zero_zero_rat )
% 0.16/0.41 => ( ( numeral_numeral_rat @ W )
% 0.16/0.41 = zero_zero_rat ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % eq_divide_eq_numeral(1)
% 0.16/0.41 thf(fact_853_dvd__div__eq__mult,axiom,
% 0.16/0.41 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.16/0.41 ( ( A != zero_z3403309356797280102nteger )
% 0.16/0.41 => ( ( dvd_dvd_Code_integer @ A @ B )
% 0.16/0.41 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.16/0.41 = C )
% 0.16/0.41 = ( B
% 0.16/0.41 = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_div_eq_mult
% 0.16/0.41 thf(fact_854_dvd__div__eq__mult,axiom,
% 0.16/0.41 ! [A: nat,B: nat,C: nat] :
% 0.16/0.41 ( ( A != zero_zero_nat )
% 0.16/0.41 => ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.41 => ( ( ( divide_divide_nat @ B @ A )
% 0.16/0.41 = C )
% 0.16/0.41 = ( B
% 0.16/0.41 = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_div_eq_mult
% 0.16/0.41 thf(fact_855_dvd__div__eq__mult,axiom,
% 0.16/0.41 ! [A: int,B: int,C: int] :
% 0.16/0.41 ( ( A != zero_zero_int )
% 0.16/0.41 => ( ( dvd_dvd_int @ A @ B )
% 0.16/0.41 => ( ( ( divide_divide_int @ B @ A )
% 0.16/0.41 = C )
% 0.16/0.41 = ( B
% 0.16/0.41 = ( times_times_int @ C @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_div_eq_mult
% 0.16/0.41 thf(fact_856_div__dvd__iff__mult,axiom,
% 0.16/0.41 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.16/0.41 ( ( B != zero_z3403309356797280102nteger )
% 0.16/0.41 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.16/0.41 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.16/0.41 = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % div_dvd_iff_mult
% 0.16/0.41 thf(fact_857_div__dvd__iff__mult,axiom,
% 0.16/0.41 ! [B: nat,A: nat,C: nat] :
% 0.16/0.41 ( ( B != zero_zero_nat )
% 0.16/0.41 => ( ( dvd_dvd_nat @ B @ A )
% 0.16/0.41 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.16/0.41 = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % div_dvd_iff_mult
% 0.16/0.41 thf(fact_858_div__dvd__iff__mult,axiom,
% 0.16/0.41 ! [B: int,A: int,C: int] :
% 0.16/0.41 ( ( B != zero_zero_int )
% 0.16/0.41 => ( ( dvd_dvd_int @ B @ A )
% 0.16/0.41 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.16/0.41 = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % div_dvd_iff_mult
% 0.16/0.41 thf(fact_859_dvd__div__iff__mult,axiom,
% 0.16/0.41 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.16/0.41 ( ( C != zero_z3403309356797280102nteger )
% 0.16/0.41 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.16/0.41 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.16/0.41 = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_div_iff_mult
% 0.16/0.41 thf(fact_860_dvd__div__iff__mult,axiom,
% 0.16/0.41 ! [C: nat,B: nat,A: nat] :
% 0.16/0.41 ( ( C != zero_zero_nat )
% 0.16/0.41 => ( ( dvd_dvd_nat @ C @ B )
% 0.16/0.41 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.16/0.41 = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_div_iff_mult
% 0.16/0.41 thf(fact_861_dvd__div__iff__mult,axiom,
% 0.16/0.41 ! [C: int,B: int,A: int] :
% 0.16/0.41 ( ( C != zero_zero_int )
% 0.16/0.41 => ( ( dvd_dvd_int @ C @ B )
% 0.16/0.41 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.16/0.41 = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_div_iff_mult
% 0.16/0.41 thf(fact_862_dvd__div__div__eq__mult,axiom,
% 0.16/0.41 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 0.16/0.41 ( ( A != zero_z3403309356797280102nteger )
% 0.16/0.41 => ( ( C != zero_z3403309356797280102nteger )
% 0.16/0.41 => ( ( dvd_dvd_Code_integer @ A @ B )
% 0.16/0.41 => ( ( dvd_dvd_Code_integer @ C @ D )
% 0.16/0.41 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.16/0.41 = ( divide6298287555418463151nteger @ D @ C ) )
% 0.16/0.41 = ( ( times_3573771949741848930nteger @ B @ C )
% 0.16/0.41 = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_div_div_eq_mult
% 0.16/0.41 thf(fact_863_dvd__div__div__eq__mult,axiom,
% 0.16/0.41 ! [A: nat,C: nat,B: nat,D: nat] :
% 0.16/0.41 ( ( A != zero_zero_nat )
% 0.16/0.41 => ( ( C != zero_zero_nat )
% 0.16/0.41 => ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.41 => ( ( dvd_dvd_nat @ C @ D )
% 0.16/0.41 => ( ( ( divide_divide_nat @ B @ A )
% 0.16/0.41 = ( divide_divide_nat @ D @ C ) )
% 0.16/0.41 = ( ( times_times_nat @ B @ C )
% 0.16/0.41 = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_div_div_eq_mult
% 0.16/0.41 thf(fact_864_dvd__div__div__eq__mult,axiom,
% 0.16/0.41 ! [A: int,C: int,B: int,D: int] :
% 0.16/0.41 ( ( A != zero_zero_int )
% 0.16/0.41 => ( ( C != zero_zero_int )
% 0.16/0.41 => ( ( dvd_dvd_int @ A @ B )
% 0.16/0.41 => ( ( dvd_dvd_int @ C @ D )
% 0.16/0.41 => ( ( ( divide_divide_int @ B @ A )
% 0.16/0.41 = ( divide_divide_int @ D @ C ) )
% 0.16/0.41 = ( ( times_times_int @ B @ C )
% 0.16/0.41 = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % dvd_div_div_eq_mult
% 0.16/0.41 thf(fact_865_numeral__1__eq__Suc__0,axiom,
% 0.16/0.41 ( ( numeral_numeral_nat @ one )
% 0.16/0.41 = ( suc @ zero_zero_nat ) ) ).
% 0.16/0.41
% 0.16/0.41 % numeral_1_eq_Suc_0
% 0.16/0.41 thf(fact_866_nat__int__comparison_I1_J,axiom,
% 0.16/0.41 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.16/0.41 = ( ^ [A3: nat,B2: nat] :
% 0.16/0.41 ( ( semiri1314217659103216013at_int @ A3 )
% 0.16/0.41 = ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % nat_int_comparison(1)
% 0.16/0.41 thf(fact_867_int__if,axiom,
% 0.16/0.41 ! [P: $o,A: nat,B: nat] :
% 0.16/0.41 ( ( P
% 0.16/0.41 => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
% 0.16/0.41 = ( semiri1314217659103216013at_int @ A ) ) )
% 0.16/0.41 & ( ~ P
% 0.16/0.41 => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
% 0.16/0.41 = ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 % int_if
% 0.16/0.41 thf(fact_868_even__zero,axiom,
% 0.16/0.41 dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 0.16/0.41
% 0.16/0.41 % even_zero
% 0.16/0.41 thf(fact_869_even__zero,axiom,
% 0.16/0.41 dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 0.16/0.42
% 0.16/0.42 % even_zero
% 0.16/0.42 thf(fact_870_even__zero,axiom,
% 0.16/0.42 dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 0.16/0.42
% 0.16/0.42 % even_zero
% 0.16/0.42 thf(fact_871_numeral__2__eq__2,axiom,
% 0.16/0.42 ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 0.16/0.42 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % numeral_2_eq_2
% 0.16/0.42 thf(fact_872_numeral__3__eq__3,axiom,
% 0.16/0.42 ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 0.16/0.42 = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % numeral_3_eq_3
% 0.16/0.42 thf(fact_873_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
% 0.16/0.42 ( ( vEBT_V8646137997579335489_i_l_d @ zero_zero_nat )
% 0.16/0.42 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
% 0.16/0.42 thf(fact_874_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
% 0.16/0.42 ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ zero_zero_nat ) )
% 0.16/0.42 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
% 0.16/0.42 thf(fact_875_verit__eq__simplify_I10_J,axiom,
% 0.16/0.42 ! [X2: num] :
% 0.16/0.42 ( one
% 0.16/0.42 != ( bit0 @ X2 ) ) ).
% 0.16/0.42
% 0.16/0.42 % verit_eq_simplify(10)
% 0.16/0.42 thf(fact_876_le__zero__eq,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.16/0.42 = ( N = zero_zero_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % le_zero_eq
% 0.16/0.42 thf(fact_877_unset__bit__0,axiom,
% 0.16/0.42 ! [A: int] :
% 0.16/0.42 ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
% 0.16/0.42 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % unset_bit_0
% 0.16/0.42 thf(fact_878_unset__bit__0,axiom,
% 0.16/0.42 ! [A: nat] :
% 0.16/0.42 ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
% 0.16/0.42 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % unset_bit_0
% 0.16/0.42 thf(fact_879_unset__bit__0,axiom,
% 0.16/0.42 ! [A: code_integer] :
% 0.16/0.42 ( ( bit_se8260200283734997820nteger @ zero_zero_nat @ A )
% 0.16/0.42 = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % unset_bit_0
% 0.16/0.42 thf(fact_880_even__set__bit__iff,axiom,
% 0.16/0.42 ! [M: nat,A: int] :
% 0.16/0.42 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.16/0.42 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.16/0.42 & ( M != zero_zero_nat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % even_set_bit_iff
% 0.16/0.42 thf(fact_881_even__set__bit__iff,axiom,
% 0.16/0.42 ! [M: nat,A: nat] :
% 0.16/0.42 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.16/0.42 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.16/0.42 & ( M != zero_zero_nat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % even_set_bit_iff
% 0.16/0.42 thf(fact_882_even__set__bit__iff,axiom,
% 0.16/0.42 ! [M: nat,A: code_integer] :
% 0.16/0.42 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 0.16/0.42 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.16/0.42 & ( M != zero_zero_nat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % even_set_bit_iff
% 0.16/0.42 thf(fact_883_even__flip__bit__iff,axiom,
% 0.16/0.42 ! [M: nat,A: int] :
% 0.16/0.42 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.16/0.42 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.16/0.42 != ( M = zero_zero_nat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % even_flip_bit_iff
% 0.16/0.42 thf(fact_884_even__flip__bit__iff,axiom,
% 0.16/0.42 ! [M: nat,A: nat] :
% 0.16/0.42 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.16/0.42 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.16/0.42 != ( M = zero_zero_nat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % even_flip_bit_iff
% 0.16/0.42 thf(fact_885_even__flip__bit__iff,axiom,
% 0.16/0.42 ! [M: nat,A: code_integer] :
% 0.16/0.42 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 0.16/0.42 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.16/0.42 != ( M = zero_zero_nat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % even_flip_bit_iff
% 0.16/0.42 thf(fact_886_even__unset__bit__iff,axiom,
% 0.16/0.42 ! [M: nat,A: int] :
% 0.16/0.42 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.16/0.42 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.16/0.42 | ( M = zero_zero_nat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % even_unset_bit_iff
% 0.16/0.42 thf(fact_887_even__unset__bit__iff,axiom,
% 0.16/0.42 ! [M: nat,A: nat] :
% 0.16/0.42 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.16/0.42 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.16/0.42 | ( M = zero_zero_nat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % even_unset_bit_iff
% 0.16/0.42 thf(fact_888_even__unset__bit__iff,axiom,
% 0.16/0.42 ! [M: nat,A: code_integer] :
% 0.16/0.42 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 0.16/0.42 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.16/0.42 | ( M = zero_zero_nat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % even_unset_bit_iff
% 0.16/0.42 thf(fact_889_subset__antisym,axiom,
% 0.16/0.42 ! [A2: set_int,B4: set_int] :
% 0.16/0.42 ( ( ord_less_eq_set_int @ A2 @ B4 )
% 0.16/0.42 => ( ( ord_less_eq_set_int @ B4 @ A2 )
% 0.16/0.42 => ( A2 = B4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % subset_antisym
% 0.16/0.42 thf(fact_890_subsetI,axiom,
% 0.16/0.42 ! [A2: set_complex,B4: set_complex] :
% 0.16/0.42 ( ! [X4: complex] :
% 0.16/0.42 ( ( member_complex @ X4 @ A2 )
% 0.16/0.42 => ( member_complex @ X4 @ B4 ) )
% 0.16/0.42 => ( ord_le211207098394363844omplex @ A2 @ B4 ) ) ).
% 0.16/0.42
% 0.16/0.42 % subsetI
% 0.16/0.42 thf(fact_891_subsetI,axiom,
% 0.16/0.42 ! [A2: set_real,B4: set_real] :
% 0.16/0.42 ( ! [X4: real] :
% 0.16/0.42 ( ( member_real @ X4 @ A2 )
% 0.16/0.42 => ( member_real @ X4 @ B4 ) )
% 0.16/0.42 => ( ord_less_eq_set_real @ A2 @ B4 ) ) ).
% 0.16/0.42
% 0.16/0.42 % subsetI
% 0.16/0.42 thf(fact_892_subsetI,axiom,
% 0.16/0.42 ! [A2: set_set_nat,B4: set_set_nat] :
% 0.16/0.42 ( ! [X4: set_nat] :
% 0.16/0.42 ( ( member_set_nat @ X4 @ A2 )
% 0.16/0.42 => ( member_set_nat @ X4 @ B4 ) )
% 0.16/0.42 => ( ord_le6893508408891458716et_nat @ A2 @ B4 ) ) ).
% 0.16/0.42
% 0.16/0.42 % subsetI
% 0.16/0.42 thf(fact_893_subsetI,axiom,
% 0.16/0.42 ! [A2: set_nat,B4: set_nat] :
% 0.16/0.42 ( ! [X4: nat] :
% 0.16/0.42 ( ( member_nat @ X4 @ A2 )
% 0.16/0.42 => ( member_nat @ X4 @ B4 ) )
% 0.16/0.42 => ( ord_less_eq_set_nat @ A2 @ B4 ) ) ).
% 0.16/0.42
% 0.16/0.42 % subsetI
% 0.16/0.42 thf(fact_894_subsetI,axiom,
% 0.16/0.42 ! [A2: set_int,B4: set_int] :
% 0.16/0.42 ( ! [X4: int] :
% 0.16/0.42 ( ( member_int @ X4 @ A2 )
% 0.16/0.42 => ( member_int @ X4 @ B4 ) )
% 0.16/0.42 => ( ord_less_eq_set_int @ A2 @ B4 ) ) ).
% 0.16/0.42
% 0.16/0.42 % subsetI
% 0.16/0.42 thf(fact_895_odd__Suc__minus__one,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.42 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.16/0.42 = N ) ) ).
% 0.16/0.42
% 0.16/0.42 % odd_Suc_minus_one
% 0.16/0.42 thf(fact_896_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
% 0.16/0.42 ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ zero_zero_nat ) )
% 0.16/0.42 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
% 0.16/0.42 thf(fact_897_unset__bit__nonnegative__int__iff,axiom,
% 0.16/0.42 ! [N: nat,K: int] :
% 0.16/0.42 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
% 0.16/0.42 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.16/0.42
% 0.16/0.42 % unset_bit_nonnegative_int_iff
% 0.16/0.42 thf(fact_898_set__bit__nonnegative__int__iff,axiom,
% 0.16/0.42 ! [N: nat,K: int] :
% 0.16/0.42 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
% 0.16/0.42 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.16/0.42
% 0.16/0.42 % set_bit_nonnegative_int_iff
% 0.16/0.42 thf(fact_899_flip__bit__nonnegative__int__iff,axiom,
% 0.16/0.42 ! [N: nat,K: int] :
% 0.16/0.42 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
% 0.16/0.42 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.16/0.42
% 0.16/0.42 % flip_bit_nonnegative_int_iff
% 0.16/0.42 thf(fact_900_diff__self,axiom,
% 0.16/0.42 ! [A: complex] :
% 0.16/0.42 ( ( minus_minus_complex @ A @ A )
% 0.16/0.42 = zero_zero_complex ) ).
% 0.16/0.42
% 0.16/0.42 % diff_self
% 0.16/0.42 thf(fact_901_diff__self,axiom,
% 0.16/0.42 ! [A: real] :
% 0.16/0.42 ( ( minus_minus_real @ A @ A )
% 0.16/0.42 = zero_zero_real ) ).
% 0.16/0.42
% 0.16/0.42 % diff_self
% 0.16/0.42 thf(fact_902_diff__self,axiom,
% 0.16/0.42 ! [A: rat] :
% 0.16/0.42 ( ( minus_minus_rat @ A @ A )
% 0.16/0.42 = zero_zero_rat ) ).
% 0.16/0.42
% 0.16/0.42 % diff_self
% 0.16/0.42 thf(fact_903_diff__self,axiom,
% 0.16/0.42 ! [A: int] :
% 0.16/0.42 ( ( minus_minus_int @ A @ A )
% 0.16/0.42 = zero_zero_int ) ).
% 0.16/0.42
% 0.16/0.42 % diff_self
% 0.16/0.42 thf(fact_904_diff__0__right,axiom,
% 0.16/0.42 ! [A: complex] :
% 0.16/0.42 ( ( minus_minus_complex @ A @ zero_zero_complex )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % diff_0_right
% 0.16/0.42 thf(fact_905_diff__0__right,axiom,
% 0.16/0.42 ! [A: real] :
% 0.16/0.42 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % diff_0_right
% 0.16/0.42 thf(fact_906_diff__0__right,axiom,
% 0.16/0.42 ! [A: rat] :
% 0.16/0.42 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % diff_0_right
% 0.16/0.42 thf(fact_907_diff__0__right,axiom,
% 0.16/0.42 ! [A: int] :
% 0.16/0.42 ( ( minus_minus_int @ A @ zero_zero_int )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % diff_0_right
% 0.16/0.42 thf(fact_908_zero__diff,axiom,
% 0.16/0.42 ! [A: nat] :
% 0.16/0.42 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 0.16/0.42 = zero_zero_nat ) ).
% 0.16/0.42
% 0.16/0.42 % zero_diff
% 0.16/0.42 thf(fact_909_diff__zero,axiom,
% 0.16/0.42 ! [A: complex] :
% 0.16/0.42 ( ( minus_minus_complex @ A @ zero_zero_complex )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % diff_zero
% 0.16/0.42 thf(fact_910_diff__zero,axiom,
% 0.16/0.42 ! [A: real] :
% 0.16/0.42 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % diff_zero
% 0.16/0.42 thf(fact_911_diff__zero,axiom,
% 0.16/0.42 ! [A: rat] :
% 0.16/0.42 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % diff_zero
% 0.16/0.42 thf(fact_912_diff__zero,axiom,
% 0.16/0.42 ! [A: nat] :
% 0.16/0.42 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % diff_zero
% 0.16/0.42 thf(fact_913_diff__zero,axiom,
% 0.16/0.42 ! [A: int] :
% 0.16/0.42 ( ( minus_minus_int @ A @ zero_zero_int )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % diff_zero
% 0.16/0.42 thf(fact_914_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.16/0.42 ! [A: complex] :
% 0.16/0.42 ( ( minus_minus_complex @ A @ A )
% 0.16/0.42 = zero_zero_complex ) ).
% 0.16/0.42
% 0.16/0.42 % cancel_comm_monoid_add_class.diff_cancel
% 0.16/0.42 thf(fact_915_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.16/0.42 ! [A: real] :
% 0.16/0.42 ( ( minus_minus_real @ A @ A )
% 0.16/0.42 = zero_zero_real ) ).
% 0.16/0.42
% 0.16/0.42 % cancel_comm_monoid_add_class.diff_cancel
% 0.16/0.42 thf(fact_916_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.16/0.42 ! [A: rat] :
% 0.16/0.42 ( ( minus_minus_rat @ A @ A )
% 0.16/0.42 = zero_zero_rat ) ).
% 0.16/0.42
% 0.16/0.42 % cancel_comm_monoid_add_class.diff_cancel
% 0.16/0.42 thf(fact_917_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.16/0.42 ! [A: nat] :
% 0.16/0.42 ( ( minus_minus_nat @ A @ A )
% 0.16/0.42 = zero_zero_nat ) ).
% 0.16/0.42
% 0.16/0.42 % cancel_comm_monoid_add_class.diff_cancel
% 0.16/0.42 thf(fact_918_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.16/0.42 ! [A: int] :
% 0.16/0.42 ( ( minus_minus_int @ A @ A )
% 0.16/0.42 = zero_zero_int ) ).
% 0.16/0.42
% 0.16/0.42 % cancel_comm_monoid_add_class.diff_cancel
% 0.16/0.42 thf(fact_919_Suc__diff__diff,axiom,
% 0.16/0.42 ! [M: nat,N: nat,K: nat] :
% 0.16/0.42 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 0.16/0.42 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 0.16/0.42
% 0.16/0.42 % Suc_diff_diff
% 0.16/0.42 thf(fact_920_diff__Suc__Suc,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.16/0.42 = ( minus_minus_nat @ M @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_Suc_Suc
% 0.16/0.42 thf(fact_921_diff__self__eq__0,axiom,
% 0.16/0.42 ! [M: nat] :
% 0.16/0.42 ( ( minus_minus_nat @ M @ M )
% 0.16/0.42 = zero_zero_nat ) ).
% 0.16/0.42
% 0.16/0.42 % diff_self_eq_0
% 0.16/0.42 thf(fact_922_diff__0__eq__0,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 0.16/0.42 = zero_zero_nat ) ).
% 0.16/0.42
% 0.16/0.42 % diff_0_eq_0
% 0.16/0.42 thf(fact_923_diff__diff__cancel,axiom,
% 0.16/0.42 ! [I: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_eq_nat @ I @ N )
% 0.16/0.42 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 0.16/0.42 = I ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_diff_cancel
% 0.16/0.42 thf(fact_924_diff__ge__0__iff__ge,axiom,
% 0.16/0.42 ! [A: rat,B: rat] :
% 0.16/0.42 ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 0.16/0.42 = ( ord_less_eq_rat @ B @ A ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_ge_0_iff_ge
% 0.16/0.42 thf(fact_925_diff__ge__0__iff__ge,axiom,
% 0.16/0.42 ! [A: int,B: int] :
% 0.16/0.42 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.16/0.42 = ( ord_less_eq_int @ B @ A ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_ge_0_iff_ge
% 0.16/0.42 thf(fact_926_diff__ge__0__iff__ge,axiom,
% 0.16/0.42 ! [A: real,B: real] :
% 0.16/0.42 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.16/0.42 = ( ord_less_eq_real @ B @ A ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_ge_0_iff_ge
% 0.16/0.42 thf(fact_927_right__diff__distrib__numeral,axiom,
% 0.16/0.42 ! [V: num,B: complex,C: complex] :
% 0.16/0.42 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B @ C ) )
% 0.16/0.42 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % right_diff_distrib_numeral
% 0.16/0.42 thf(fact_928_right__diff__distrib__numeral,axiom,
% 0.16/0.42 ! [V: num,B: real,C: real] :
% 0.16/0.42 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
% 0.16/0.42 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % right_diff_distrib_numeral
% 0.16/0.42 thf(fact_929_right__diff__distrib__numeral,axiom,
% 0.16/0.42 ! [V: num,B: rat,C: rat] :
% 0.16/0.42 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
% 0.16/0.42 = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % right_diff_distrib_numeral
% 0.16/0.42 thf(fact_930_right__diff__distrib__numeral,axiom,
% 0.16/0.42 ! [V: num,B: int,C: int] :
% 0.16/0.42 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
% 0.16/0.42 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % right_diff_distrib_numeral
% 0.16/0.42 thf(fact_931_left__diff__distrib__numeral,axiom,
% 0.16/0.42 ! [A: complex,B: complex,V: num] :
% 0.16/0.42 ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 0.16/0.42 = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % left_diff_distrib_numeral
% 0.16/0.42 thf(fact_932_left__diff__distrib__numeral,axiom,
% 0.16/0.42 ! [A: real,B: real,V: num] :
% 0.16/0.42 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 0.16/0.42 = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % left_diff_distrib_numeral
% 0.16/0.42 thf(fact_933_left__diff__distrib__numeral,axiom,
% 0.16/0.42 ! [A: rat,B: rat,V: num] :
% 0.16/0.42 ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 0.16/0.42 = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % left_diff_distrib_numeral
% 0.16/0.42 thf(fact_934_left__diff__distrib__numeral,axiom,
% 0.16/0.42 ! [A: int,B: int,V: num] :
% 0.16/0.42 ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 0.16/0.42 = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % left_diff_distrib_numeral
% 0.16/0.42 thf(fact_935_div__diff,axiom,
% 0.16/0.42 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.16/0.42 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.16/0.42 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.16/0.42 => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 0.16/0.42 = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % div_diff
% 0.16/0.42 thf(fact_936_div__diff,axiom,
% 0.16/0.42 ! [C: int,A: int,B: int] :
% 0.16/0.42 ( ( dvd_dvd_int @ C @ A )
% 0.16/0.42 => ( ( dvd_dvd_int @ C @ B )
% 0.16/0.42 => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.16/0.42 = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % div_diff
% 0.16/0.42 thf(fact_937_diff__is__0__eq,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( ( minus_minus_nat @ M @ N )
% 0.16/0.42 = zero_zero_nat )
% 0.16/0.42 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_is_0_eq
% 0.16/0.42 thf(fact_938_diff__is__0__eq_H,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.42 => ( ( minus_minus_nat @ M @ N )
% 0.16/0.42 = zero_zero_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_is_0_eq'
% 0.16/0.42 thf(fact_939_half__nonnegative__int__iff,axiom,
% 0.16/0.42 ! [K: int] :
% 0.16/0.42 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.16/0.42 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.16/0.42
% 0.16/0.42 % half_nonnegative_int_iff
% 0.16/0.42 thf(fact_940_diff__eq__diff__eq,axiom,
% 0.16/0.42 ! [A: real,B: real,C: real,D: real] :
% 0.16/0.42 ( ( ( minus_minus_real @ A @ B )
% 0.16/0.42 = ( minus_minus_real @ C @ D ) )
% 0.16/0.42 => ( ( A = B )
% 0.16/0.42 = ( C = D ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_eq_diff_eq
% 0.16/0.42 thf(fact_941_diff__eq__diff__eq,axiom,
% 0.16/0.42 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.16/0.42 ( ( ( minus_minus_rat @ A @ B )
% 0.16/0.42 = ( minus_minus_rat @ C @ D ) )
% 0.16/0.42 => ( ( A = B )
% 0.16/0.42 = ( C = D ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_eq_diff_eq
% 0.16/0.42 thf(fact_942_diff__eq__diff__eq,axiom,
% 0.16/0.42 ! [A: int,B: int,C: int,D: int] :
% 0.16/0.42 ( ( ( minus_minus_int @ A @ B )
% 0.16/0.42 = ( minus_minus_int @ C @ D ) )
% 0.16/0.42 => ( ( A = B )
% 0.16/0.42 = ( C = D ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_eq_diff_eq
% 0.16/0.42 thf(fact_943_diff__right__commute,axiom,
% 0.16/0.42 ! [A: real,C: real,B: real] :
% 0.16/0.42 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 0.16/0.42 = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_right_commute
% 0.16/0.42 thf(fact_944_diff__right__commute,axiom,
% 0.16/0.42 ! [A: rat,C: rat,B: rat] :
% 0.16/0.42 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
% 0.16/0.42 = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_right_commute
% 0.16/0.42 thf(fact_945_diff__right__commute,axiom,
% 0.16/0.42 ! [A: nat,C: nat,B: nat] :
% 0.16/0.42 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.16/0.42 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_right_commute
% 0.16/0.42 thf(fact_946_diff__right__commute,axiom,
% 0.16/0.42 ! [A: int,C: int,B: int] :
% 0.16/0.42 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 0.16/0.42 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_right_commute
% 0.16/0.42 thf(fact_947_diff__commute,axiom,
% 0.16/0.42 ! [I: nat,J: nat,K: nat] :
% 0.16/0.42 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.16/0.42 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_commute
% 0.16/0.42 thf(fact_948_diff__eq__diff__less__eq,axiom,
% 0.16/0.42 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.16/0.42 ( ( ( minus_minus_rat @ A @ B )
% 0.16/0.42 = ( minus_minus_rat @ C @ D ) )
% 0.16/0.42 => ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.42 = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_eq_diff_less_eq
% 0.16/0.42 thf(fact_949_diff__eq__diff__less__eq,axiom,
% 0.16/0.42 ! [A: int,B: int,C: int,D: int] :
% 0.16/0.42 ( ( ( minus_minus_int @ A @ B )
% 0.16/0.42 = ( minus_minus_int @ C @ D ) )
% 0.16/0.42 => ( ( ord_less_eq_int @ A @ B )
% 0.16/0.42 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_eq_diff_less_eq
% 0.16/0.42 thf(fact_950_diff__eq__diff__less__eq,axiom,
% 0.16/0.42 ! [A: real,B: real,C: real,D: real] :
% 0.16/0.42 ( ( ( minus_minus_real @ A @ B )
% 0.16/0.42 = ( minus_minus_real @ C @ D ) )
% 0.16/0.42 => ( ( ord_less_eq_real @ A @ B )
% 0.16/0.42 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_eq_diff_less_eq
% 0.16/0.42 thf(fact_951_diff__right__mono,axiom,
% 0.16/0.42 ! [A: rat,B: rat,C: rat] :
% 0.16/0.42 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.42 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_right_mono
% 0.16/0.42 thf(fact_952_diff__right__mono,axiom,
% 0.16/0.42 ! [A: int,B: int,C: int] :
% 0.16/0.42 ( ( ord_less_eq_int @ A @ B )
% 0.16/0.42 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_right_mono
% 0.16/0.42 thf(fact_953_diff__right__mono,axiom,
% 0.16/0.42 ! [A: real,B: real,C: real] :
% 0.16/0.42 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.42 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_right_mono
% 0.16/0.42 thf(fact_954_diff__left__mono,axiom,
% 0.16/0.42 ! [B: rat,A: rat,C: rat] :
% 0.16/0.42 ( ( ord_less_eq_rat @ B @ A )
% 0.16/0.42 => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_left_mono
% 0.16/0.42 thf(fact_955_diff__left__mono,axiom,
% 0.16/0.42 ! [B: int,A: int,C: int] :
% 0.16/0.42 ( ( ord_less_eq_int @ B @ A )
% 0.16/0.42 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_left_mono
% 0.16/0.42 thf(fact_956_diff__left__mono,axiom,
% 0.16/0.42 ! [B: real,A: real,C: real] :
% 0.16/0.42 ( ( ord_less_eq_real @ B @ A )
% 0.16/0.42 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_left_mono
% 0.16/0.42 thf(fact_957_diff__mono,axiom,
% 0.16/0.42 ! [A: rat,B: rat,D: rat,C: rat] :
% 0.16/0.42 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.42 => ( ( ord_less_eq_rat @ D @ C )
% 0.16/0.42 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_mono
% 0.16/0.42 thf(fact_958_diff__mono,axiom,
% 0.16/0.42 ! [A: int,B: int,D: int,C: int] :
% 0.16/0.42 ( ( ord_less_eq_int @ A @ B )
% 0.16/0.42 => ( ( ord_less_eq_int @ D @ C )
% 0.16/0.42 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_mono
% 0.16/0.42 thf(fact_959_diff__mono,axiom,
% 0.16/0.42 ! [A: real,B: real,D: real,C: real] :
% 0.16/0.42 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.42 => ( ( ord_less_eq_real @ D @ C )
% 0.16/0.42 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_mono
% 0.16/0.42 thf(fact_960_eq__iff__diff__eq__0,axiom,
% 0.16/0.42 ( ( ^ [Y5: complex,Z4: complex] : ( Y5 = Z4 ) )
% 0.16/0.42 = ( ^ [A3: complex,B2: complex] :
% 0.16/0.42 ( ( minus_minus_complex @ A3 @ B2 )
% 0.16/0.42 = zero_zero_complex ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % eq_iff_diff_eq_0
% 0.16/0.42 thf(fact_961_eq__iff__diff__eq__0,axiom,
% 0.16/0.42 ( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
% 0.16/0.42 = ( ^ [A3: real,B2: real] :
% 0.16/0.42 ( ( minus_minus_real @ A3 @ B2 )
% 0.16/0.42 = zero_zero_real ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % eq_iff_diff_eq_0
% 0.16/0.42 thf(fact_962_eq__iff__diff__eq__0,axiom,
% 0.16/0.42 ( ( ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 ) )
% 0.16/0.42 = ( ^ [A3: rat,B2: rat] :
% 0.16/0.42 ( ( minus_minus_rat @ A3 @ B2 )
% 0.16/0.42 = zero_zero_rat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % eq_iff_diff_eq_0
% 0.16/0.42 thf(fact_963_eq__iff__diff__eq__0,axiom,
% 0.16/0.42 ( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.16/0.42 = ( ^ [A3: int,B2: int] :
% 0.16/0.42 ( ( minus_minus_int @ A3 @ B2 )
% 0.16/0.42 = zero_zero_int ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % eq_iff_diff_eq_0
% 0.16/0.42 thf(fact_964_of__nat__diff,axiom,
% 0.16/0.42 ! [N: nat,M: nat] :
% 0.16/0.42 ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.42 => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N ) )
% 0.16/0.42 = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_diff
% 0.16/0.42 thf(fact_965_of__nat__diff,axiom,
% 0.16/0.42 ! [N: nat,M: nat] :
% 0.16/0.42 ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.42 => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
% 0.16/0.42 = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_diff
% 0.16/0.42 thf(fact_966_of__nat__diff,axiom,
% 0.16/0.42 ! [N: nat,M: nat] :
% 0.16/0.42 ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.42 => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
% 0.16/0.42 = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_diff
% 0.16/0.42 thf(fact_967_of__nat__diff,axiom,
% 0.16/0.42 ! [N: nat,M: nat] :
% 0.16/0.42 ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.42 => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
% 0.16/0.42 = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_diff
% 0.16/0.42 thf(fact_968_of__nat__diff,axiom,
% 0.16/0.42 ! [N: nat,M: nat] :
% 0.16/0.42 ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.42 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
% 0.16/0.42 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_diff
% 0.16/0.42 thf(fact_969_right__diff__distrib_H,axiom,
% 0.16/0.42 ! [A: real,B: real,C: real] :
% 0.16/0.42 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.16/0.42 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % right_diff_distrib'
% 0.16/0.42 thf(fact_970_right__diff__distrib_H,axiom,
% 0.16/0.42 ! [A: rat,B: rat,C: rat] :
% 0.16/0.42 ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.16/0.42 = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % right_diff_distrib'
% 0.16/0.42 thf(fact_971_right__diff__distrib_H,axiom,
% 0.16/0.42 ! [A: nat,B: nat,C: nat] :
% 0.16/0.42 ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
% 0.16/0.42 = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % right_diff_distrib'
% 0.16/0.42 thf(fact_972_right__diff__distrib_H,axiom,
% 0.16/0.42 ! [A: int,B: int,C: int] :
% 0.16/0.42 ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.16/0.42 = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % right_diff_distrib'
% 0.16/0.42 thf(fact_973_left__diff__distrib_H,axiom,
% 0.16/0.42 ! [B: real,C: real,A: real] :
% 0.16/0.42 ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
% 0.16/0.42 = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % left_diff_distrib'
% 0.16/0.42 thf(fact_974_left__diff__distrib_H,axiom,
% 0.16/0.42 ! [B: rat,C: rat,A: rat] :
% 0.16/0.42 ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
% 0.16/0.42 = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % left_diff_distrib'
% 0.16/0.42 thf(fact_975_left__diff__distrib_H,axiom,
% 0.16/0.42 ! [B: nat,C: nat,A: nat] :
% 0.16/0.42 ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
% 0.16/0.42 = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % left_diff_distrib'
% 0.16/0.42 thf(fact_976_left__diff__distrib_H,axiom,
% 0.16/0.42 ! [B: int,C: int,A: int] :
% 0.16/0.42 ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
% 0.16/0.42 = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % left_diff_distrib'
% 0.16/0.42 thf(fact_977_right__diff__distrib,axiom,
% 0.16/0.42 ! [A: real,B: real,C: real] :
% 0.16/0.42 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.16/0.42 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % right_diff_distrib
% 0.16/0.42 thf(fact_978_right__diff__distrib,axiom,
% 0.16/0.42 ! [A: rat,B: rat,C: rat] :
% 0.16/0.42 ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.16/0.42 = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % right_diff_distrib
% 0.16/0.42 thf(fact_979_right__diff__distrib,axiom,
% 0.16/0.42 ! [A: int,B: int,C: int] :
% 0.16/0.42 ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.16/0.42 = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % right_diff_distrib
% 0.16/0.42 thf(fact_980_left__diff__distrib,axiom,
% 0.16/0.42 ! [A: real,B: real,C: real] :
% 0.16/0.42 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.16/0.42 = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % left_diff_distrib
% 0.16/0.42 thf(fact_981_left__diff__distrib,axiom,
% 0.16/0.42 ! [A: rat,B: rat,C: rat] :
% 0.16/0.42 ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.16/0.42 = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % left_diff_distrib
% 0.16/0.42 thf(fact_982_left__diff__distrib,axiom,
% 0.16/0.42 ! [A: int,B: int,C: int] :
% 0.16/0.42 ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.16/0.42 = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % left_diff_distrib
% 0.16/0.42 thf(fact_983_diff__divide__distrib,axiom,
% 0.16/0.42 ! [A: complex,B: complex,C: complex] :
% 0.16/0.42 ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
% 0.16/0.42 = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_divide_distrib
% 0.16/0.42 thf(fact_984_diff__divide__distrib,axiom,
% 0.16/0.42 ! [A: real,B: real,C: real] :
% 0.16/0.42 ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.16/0.42 = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_divide_distrib
% 0.16/0.42 thf(fact_985_diff__divide__distrib,axiom,
% 0.16/0.42 ! [A: rat,B: rat,C: rat] :
% 0.16/0.42 ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.16/0.42 = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_divide_distrib
% 0.16/0.42 thf(fact_986_dvd__diff__commute,axiom,
% 0.16/0.42 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.16/0.42 ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
% 0.16/0.42 = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dvd_diff_commute
% 0.16/0.42 thf(fact_987_dvd__diff__commute,axiom,
% 0.16/0.42 ! [A: int,C: int,B: int] :
% 0.16/0.42 ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.16/0.42 = ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dvd_diff_commute
% 0.16/0.42 thf(fact_988_dvd__diff,axiom,
% 0.16/0.42 ! [X: code_integer,Y4: code_integer,Z: code_integer] :
% 0.16/0.42 ( ( dvd_dvd_Code_integer @ X @ Y4 )
% 0.16/0.42 => ( ( dvd_dvd_Code_integer @ X @ Z )
% 0.16/0.42 => ( dvd_dvd_Code_integer @ X @ ( minus_8373710615458151222nteger @ Y4 @ Z ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dvd_diff
% 0.16/0.42 thf(fact_989_dvd__diff,axiom,
% 0.16/0.42 ! [X: real,Y4: real,Z: real] :
% 0.16/0.42 ( ( dvd_dvd_real @ X @ Y4 )
% 0.16/0.42 => ( ( dvd_dvd_real @ X @ Z )
% 0.16/0.42 => ( dvd_dvd_real @ X @ ( minus_minus_real @ Y4 @ Z ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dvd_diff
% 0.16/0.42 thf(fact_990_dvd__diff,axiom,
% 0.16/0.42 ! [X: rat,Y4: rat,Z: rat] :
% 0.16/0.42 ( ( dvd_dvd_rat @ X @ Y4 )
% 0.16/0.42 => ( ( dvd_dvd_rat @ X @ Z )
% 0.16/0.42 => ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y4 @ Z ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dvd_diff
% 0.16/0.42 thf(fact_991_dvd__diff,axiom,
% 0.16/0.42 ! [X: int,Y4: int,Z: int] :
% 0.16/0.42 ( ( dvd_dvd_int @ X @ Y4 )
% 0.16/0.42 => ( ( dvd_dvd_int @ X @ Z )
% 0.16/0.42 => ( dvd_dvd_int @ X @ ( minus_minus_int @ Y4 @ Z ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dvd_diff
% 0.16/0.42 thf(fact_992_zero__induct__lemma,axiom,
% 0.16/0.42 ! [P: nat > $o,K: nat,I: nat] :
% 0.16/0.42 ( ( P @ K )
% 0.16/0.42 => ( ! [N2: nat] :
% 0.16/0.42 ( ( P @ ( suc @ N2 ) )
% 0.16/0.42 => ( P @ N2 ) )
% 0.16/0.42 => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_induct_lemma
% 0.16/0.42 thf(fact_993_diffs0__imp__equal,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( ( minus_minus_nat @ M @ N )
% 0.16/0.42 = zero_zero_nat )
% 0.16/0.42 => ( ( ( minus_minus_nat @ N @ M )
% 0.16/0.42 = zero_zero_nat )
% 0.16/0.42 => ( M = N ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diffs0_imp_equal
% 0.16/0.42 thf(fact_994_minus__nat_Odiff__0,axiom,
% 0.16/0.42 ! [M: nat] :
% 0.16/0.42 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 0.16/0.42 = M ) ).
% 0.16/0.42
% 0.16/0.42 % minus_nat.diff_0
% 0.16/0.42 thf(fact_995_eq__diff__iff,axiom,
% 0.16/0.42 ! [K: nat,M: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_eq_nat @ K @ M )
% 0.16/0.42 => ( ( ord_less_eq_nat @ K @ N )
% 0.16/0.42 => ( ( ( minus_minus_nat @ M @ K )
% 0.16/0.42 = ( minus_minus_nat @ N @ K ) )
% 0.16/0.42 = ( M = N ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % eq_diff_iff
% 0.16/0.42 thf(fact_996_le__diff__iff,axiom,
% 0.16/0.42 ! [K: nat,M: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_eq_nat @ K @ M )
% 0.16/0.42 => ( ( ord_less_eq_nat @ K @ N )
% 0.16/0.42 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.16/0.42 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % le_diff_iff
% 0.16/0.42 thf(fact_997_Nat_Odiff__diff__eq,axiom,
% 0.16/0.42 ! [K: nat,M: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_eq_nat @ K @ M )
% 0.16/0.42 => ( ( ord_less_eq_nat @ K @ N )
% 0.16/0.42 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.16/0.42 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % Nat.diff_diff_eq
% 0.16/0.42 thf(fact_998_diff__le__mono,axiom,
% 0.16/0.42 ! [M: nat,N: nat,L: nat] :
% 0.16/0.42 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.42 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_le_mono
% 0.16/0.42 thf(fact_999_diff__le__self,axiom,
% 0.16/0.42 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.16/0.42
% 0.16/0.42 % diff_le_self
% 0.16/0.42 thf(fact_1000_le__diff__iff_H,axiom,
% 0.16/0.42 ! [A: nat,C: nat,B: nat] :
% 0.16/0.42 ( ( ord_less_eq_nat @ A @ C )
% 0.16/0.42 => ( ( ord_less_eq_nat @ B @ C )
% 0.16/0.42 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 0.16/0.42 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % le_diff_iff'
% 0.16/0.42 thf(fact_1001_diff__le__mono2,axiom,
% 0.16/0.42 ! [M: nat,N: nat,L: nat] :
% 0.16/0.42 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.42 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_le_mono2
% 0.16/0.42 thf(fact_1002_diff__mult__distrib,axiom,
% 0.16/0.42 ! [M: nat,N: nat,K: nat] :
% 0.16/0.42 ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 0.16/0.42 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_mult_distrib
% 0.16/0.42 thf(fact_1003_diff__mult__distrib2,axiom,
% 0.16/0.42 ! [K: nat,M: nat,N: nat] :
% 0.16/0.42 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.16/0.42 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_mult_distrib2
% 0.16/0.42 thf(fact_1004_dvd__diff__nat,axiom,
% 0.16/0.42 ! [K: nat,M: nat,N: nat] :
% 0.16/0.42 ( ( dvd_dvd_nat @ K @ M )
% 0.16/0.42 => ( ( dvd_dvd_nat @ K @ N )
% 0.16/0.42 => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dvd_diff_nat
% 0.16/0.42 thf(fact_1005_le__iff__diff__le__0,axiom,
% 0.16/0.42 ( ord_less_eq_rat
% 0.16/0.42 = ( ^ [A3: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % le_iff_diff_le_0
% 0.16/0.42 thf(fact_1006_le__iff__diff__le__0,axiom,
% 0.16/0.42 ( ord_less_eq_int
% 0.16/0.42 = ( ^ [A3: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % le_iff_diff_le_0
% 0.16/0.42 thf(fact_1007_le__iff__diff__le__0,axiom,
% 0.16/0.42 ( ord_less_eq_real
% 0.16/0.42 = ( ^ [A3: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % le_iff_diff_le_0
% 0.16/0.42 thf(fact_1008_less__eq__int__code_I1_J,axiom,
% 0.16/0.42 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.16/0.42
% 0.16/0.42 % less_eq_int_code(1)
% 0.16/0.42 thf(fact_1009_times__int__code_I1_J,axiom,
% 0.16/0.42 ! [K: int] :
% 0.16/0.42 ( ( times_times_int @ K @ zero_zero_int )
% 0.16/0.42 = zero_zero_int ) ).
% 0.16/0.42
% 0.16/0.42 % times_int_code(1)
% 0.16/0.42 thf(fact_1010_times__int__code_I2_J,axiom,
% 0.16/0.42 ! [L: int] :
% 0.16/0.42 ( ( times_times_int @ zero_zero_int @ L )
% 0.16/0.42 = zero_zero_int ) ).
% 0.16/0.42
% 0.16/0.42 % times_int_code(2)
% 0.16/0.42 thf(fact_1011_ile0__eq,axiom,
% 0.16/0.42 ! [N: extended_enat] :
% 0.16/0.42 ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
% 0.16/0.42 = ( N = zero_z5237406670263579293d_enat ) ) ).
% 0.16/0.42
% 0.16/0.42 % ile0_eq
% 0.16/0.42 thf(fact_1012_i0__lb,axiom,
% 0.16/0.42 ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% 0.16/0.42
% 0.16/0.42 % i0_lb
% 0.16/0.42 thf(fact_1013_unset__bit__less__eq,axiom,
% 0.16/0.42 ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% 0.16/0.42
% 0.16/0.42 % unset_bit_less_eq
% 0.16/0.42 thf(fact_1014_set__bit__greater__eq,axiom,
% 0.16/0.42 ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% 0.16/0.42
% 0.16/0.42 % set_bit_greater_eq
% 0.16/0.42 thf(fact_1015_Suc__diff__le,axiom,
% 0.16/0.42 ! [N: nat,M: nat] :
% 0.16/0.42 ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.42 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.16/0.42 = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % Suc_diff_le
% 0.16/0.42 thf(fact_1016_dvd__diffD,axiom,
% 0.16/0.42 ! [K: nat,M: nat,N: nat] :
% 0.16/0.42 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.16/0.42 => ( ( dvd_dvd_nat @ K @ N )
% 0.16/0.42 => ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.42 => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dvd_diffD
% 0.16/0.42 thf(fact_1017_dvd__diffD1,axiom,
% 0.16/0.42 ! [K: nat,M: nat,N: nat] :
% 0.16/0.42 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.16/0.42 => ( ( dvd_dvd_nat @ K @ M )
% 0.16/0.42 => ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.42 => ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dvd_diffD1
% 0.16/0.42 thf(fact_1018_less__eq__dvd__minus,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.42 => ( ( dvd_dvd_nat @ M @ N )
% 0.16/0.42 = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_eq_dvd_minus
% 0.16/0.42 thf(fact_1019_int__ops_I1_J,axiom,
% 0.16/0.42 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 0.16/0.42 = zero_zero_int ) ).
% 0.16/0.42
% 0.16/0.42 % int_ops(1)
% 0.16/0.42 thf(fact_1020_nonneg__int__cases,axiom,
% 0.16/0.42 ! [K: int] :
% 0.16/0.42 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.16/0.42 => ~ ! [N2: nat] :
% 0.16/0.42 ( K
% 0.16/0.42 != ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % nonneg_int_cases
% 0.16/0.42 thf(fact_1021_zero__le__imp__eq__int,axiom,
% 0.16/0.42 ! [K: int] :
% 0.16/0.42 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.16/0.42 => ? [N2: nat] :
% 0.16/0.42 ( K
% 0.16/0.42 = ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_le_imp_eq_int
% 0.16/0.42 thf(fact_1022_zdvd__antisym__nonneg,axiom,
% 0.16/0.42 ! [M: int,N: int] :
% 0.16/0.42 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 0.16/0.42 => ( ( ord_less_eq_int @ zero_zero_int @ N )
% 0.16/0.42 => ( ( dvd_dvd_int @ M @ N )
% 0.16/0.42 => ( ( dvd_dvd_int @ N @ M )
% 0.16/0.42 => ( M = N ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % zdvd_antisym_nonneg
% 0.16/0.42 thf(fact_1023_zdvd__mult__cancel,axiom,
% 0.16/0.42 ! [K: int,M: int,N: int] :
% 0.16/0.42 ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
% 0.16/0.42 => ( ( K != zero_zero_int )
% 0.16/0.42 => ( dvd_dvd_int @ M @ N ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % zdvd_mult_cancel
% 0.16/0.42 thf(fact_1024_divide__diff__eq__iff,axiom,
% 0.16/0.42 ! [Z: complex,X: complex,Y4: complex] :
% 0.16/0.42 ( ( Z != zero_zero_complex )
% 0.16/0.42 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y4 )
% 0.16/0.42 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y4 @ Z ) ) @ Z ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_diff_eq_iff
% 0.16/0.42 thf(fact_1025_divide__diff__eq__iff,axiom,
% 0.16/0.42 ! [Z: real,X: real,Y4: real] :
% 0.16/0.42 ( ( Z != zero_zero_real )
% 0.16/0.42 => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y4 )
% 0.16/0.42 = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y4 @ Z ) ) @ Z ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_diff_eq_iff
% 0.16/0.42 thf(fact_1026_divide__diff__eq__iff,axiom,
% 0.16/0.42 ! [Z: rat,X: rat,Y4: rat] :
% 0.16/0.42 ( ( Z != zero_zero_rat )
% 0.16/0.42 => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y4 )
% 0.16/0.42 = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y4 @ Z ) ) @ Z ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_diff_eq_iff
% 0.16/0.42 thf(fact_1027_diff__divide__eq__iff,axiom,
% 0.16/0.42 ! [Z: complex,X: complex,Y4: complex] :
% 0.16/0.42 ( ( Z != zero_zero_complex )
% 0.16/0.42 => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y4 @ Z ) )
% 0.16/0.42 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ Y4 ) @ Z ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_divide_eq_iff
% 0.16/0.42 thf(fact_1028_diff__divide__eq__iff,axiom,
% 0.16/0.42 ! [Z: real,X: real,Y4: real] :
% 0.16/0.42 ( ( Z != zero_zero_real )
% 0.16/0.42 => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y4 @ Z ) )
% 0.16/0.42 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y4 ) @ Z ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_divide_eq_iff
% 0.16/0.42 thf(fact_1029_diff__divide__eq__iff,axiom,
% 0.16/0.42 ! [Z: rat,X: rat,Y4: rat] :
% 0.16/0.42 ( ( Z != zero_zero_rat )
% 0.16/0.42 => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y4 @ Z ) )
% 0.16/0.42 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y4 ) @ Z ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_divide_eq_iff
% 0.16/0.42 thf(fact_1030_diff__frac__eq,axiom,
% 0.16/0.42 ! [Y4: complex,Z: complex,X: complex,W: complex] :
% 0.16/0.42 ( ( Y4 != zero_zero_complex )
% 0.16/0.42 => ( ( Z != zero_zero_complex )
% 0.16/0.42 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y4 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 0.16/0.42 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y4 ) ) @ ( times_times_complex @ Y4 @ Z ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_frac_eq
% 0.16/0.42 thf(fact_1031_diff__frac__eq,axiom,
% 0.16/0.42 ! [Y4: real,Z: real,X: real,W: real] :
% 0.16/0.42 ( ( Y4 != zero_zero_real )
% 0.16/0.42 => ( ( Z != zero_zero_real )
% 0.16/0.42 => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y4 ) @ ( divide_divide_real @ W @ Z ) )
% 0.16/0.42 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y4 ) ) @ ( times_times_real @ Y4 @ Z ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_frac_eq
% 0.16/0.42 thf(fact_1032_diff__frac__eq,axiom,
% 0.16/0.42 ! [Y4: rat,Z: rat,X: rat,W: rat] :
% 0.16/0.42 ( ( Y4 != zero_zero_rat )
% 0.16/0.42 => ( ( Z != zero_zero_rat )
% 0.16/0.42 => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y4 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.16/0.42 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y4 ) ) @ ( times_times_rat @ Y4 @ Z ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_frac_eq
% 0.16/0.42 thf(fact_1033_add__divide__eq__if__simps_I4_J,axiom,
% 0.16/0.42 ! [Z: complex,A: complex,B: complex] :
% 0.16/0.42 ( ( ( Z = zero_zero_complex )
% 0.16/0.42 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.16/0.42 = A ) )
% 0.16/0.42 & ( ( Z != zero_zero_complex )
% 0.16/0.42 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.16/0.42 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % add_divide_eq_if_simps(4)
% 0.16/0.42 thf(fact_1034_add__divide__eq__if__simps_I4_J,axiom,
% 0.16/0.42 ! [Z: real,A: real,B: real] :
% 0.16/0.42 ( ( ( Z = zero_zero_real )
% 0.16/0.42 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.16/0.42 = A ) )
% 0.16/0.42 & ( ( Z != zero_zero_real )
% 0.16/0.42 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.16/0.42 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % add_divide_eq_if_simps(4)
% 0.16/0.42 thf(fact_1035_add__divide__eq__if__simps_I4_J,axiom,
% 0.16/0.42 ! [Z: rat,A: rat,B: rat] :
% 0.16/0.42 ( ( ( Z = zero_zero_rat )
% 0.16/0.42 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.16/0.42 = A ) )
% 0.16/0.42 & ( ( Z != zero_zero_rat )
% 0.16/0.42 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.16/0.42 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % add_divide_eq_if_simps(4)
% 0.16/0.42 thf(fact_1036_zero__reorient,axiom,
% 0.16/0.42 ! [X: complex] :
% 0.16/0.42 ( ( zero_zero_complex = X )
% 0.16/0.42 = ( X = zero_zero_complex ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_reorient
% 0.16/0.42 thf(fact_1037_zero__reorient,axiom,
% 0.16/0.42 ! [X: real] :
% 0.16/0.42 ( ( zero_zero_real = X )
% 0.16/0.42 = ( X = zero_zero_real ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_reorient
% 0.16/0.42 thf(fact_1038_zero__reorient,axiom,
% 0.16/0.42 ! [X: rat] :
% 0.16/0.42 ( ( zero_zero_rat = X )
% 0.16/0.42 = ( X = zero_zero_rat ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_reorient
% 0.16/0.42 thf(fact_1039_zero__reorient,axiom,
% 0.16/0.42 ! [X: nat] :
% 0.16/0.42 ( ( zero_zero_nat = X )
% 0.16/0.42 = ( X = zero_zero_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_reorient
% 0.16/0.42 thf(fact_1040_zero__reorient,axiom,
% 0.16/0.42 ! [X: int] :
% 0.16/0.42 ( ( zero_zero_int = X )
% 0.16/0.42 = ( X = zero_zero_int ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_reorient
% 0.16/0.42 thf(fact_1041_mult_Oleft__commute,axiom,
% 0.16/0.42 ! [B: real,A: real,C: real] :
% 0.16/0.42 ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 0.16/0.42 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult.left_commute
% 0.16/0.42 thf(fact_1042_mult_Oleft__commute,axiom,
% 0.16/0.42 ! [B: rat,A: rat,C: rat] :
% 0.16/0.42 ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
% 0.16/0.42 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult.left_commute
% 0.16/0.42 thf(fact_1043_mult_Oleft__commute,axiom,
% 0.16/0.42 ! [B: nat,A: nat,C: nat] :
% 0.16/0.42 ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 0.16/0.42 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult.left_commute
% 0.16/0.42 thf(fact_1044_mult_Oleft__commute,axiom,
% 0.16/0.42 ! [B: int,A: int,C: int] :
% 0.16/0.42 ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
% 0.16/0.42 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult.left_commute
% 0.16/0.42 thf(fact_1045_mult_Ocommute,axiom,
% 0.16/0.42 ( times_times_real
% 0.16/0.42 = ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult.commute
% 0.16/0.42 thf(fact_1046_mult_Ocommute,axiom,
% 0.16/0.42 ( times_times_rat
% 0.16/0.42 = ( ^ [A3: rat,B2: rat] : ( times_times_rat @ B2 @ A3 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult.commute
% 0.16/0.42 thf(fact_1047_mult_Ocommute,axiom,
% 0.16/0.42 ( times_times_nat
% 0.16/0.42 = ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult.commute
% 0.16/0.42 thf(fact_1048_mult_Ocommute,axiom,
% 0.16/0.42 ( times_times_int
% 0.16/0.42 = ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult.commute
% 0.16/0.42 thf(fact_1049_mult_Oassoc,axiom,
% 0.16/0.42 ! [A: real,B: real,C: real] :
% 0.16/0.42 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.16/0.42 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult.assoc
% 0.16/0.42 thf(fact_1050_mult_Oassoc,axiom,
% 0.16/0.42 ! [A: rat,B: rat,C: rat] :
% 0.16/0.42 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.16/0.42 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult.assoc
% 0.16/0.42 thf(fact_1051_mult_Oassoc,axiom,
% 0.16/0.42 ! [A: nat,B: nat,C: nat] :
% 0.16/0.42 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.16/0.42 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult.assoc
% 0.16/0.42 thf(fact_1052_mult_Oassoc,axiom,
% 0.16/0.42 ! [A: int,B: int,C: int] :
% 0.16/0.42 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 0.16/0.42 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult.assoc
% 0.16/0.42 thf(fact_1053_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.16/0.42 ! [A: real,B: real,C: real] :
% 0.16/0.42 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.16/0.42 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % ab_semigroup_mult_class.mult_ac(1)
% 0.16/0.42 thf(fact_1054_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.16/0.42 ! [A: rat,B: rat,C: rat] :
% 0.16/0.42 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.16/0.42 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % ab_semigroup_mult_class.mult_ac(1)
% 0.16/0.42 thf(fact_1055_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.16/0.42 ! [A: nat,B: nat,C: nat] :
% 0.16/0.42 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.16/0.42 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % ab_semigroup_mult_class.mult_ac(1)
% 0.16/0.42 thf(fact_1056_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.16/0.42 ! [A: int,B: int,C: int] :
% 0.16/0.42 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 0.16/0.42 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % ab_semigroup_mult_class.mult_ac(1)
% 0.16/0.42 thf(fact_1057_in__mono,axiom,
% 0.16/0.42 ! [A2: set_complex,B4: set_complex,X: complex] :
% 0.16/0.42 ( ( ord_le211207098394363844omplex @ A2 @ B4 )
% 0.16/0.42 => ( ( member_complex @ X @ A2 )
% 0.16/0.42 => ( member_complex @ X @ B4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % in_mono
% 0.16/0.42 thf(fact_1058_in__mono,axiom,
% 0.16/0.42 ! [A2: set_real,B4: set_real,X: real] :
% 0.16/0.42 ( ( ord_less_eq_set_real @ A2 @ B4 )
% 0.16/0.42 => ( ( member_real @ X @ A2 )
% 0.16/0.42 => ( member_real @ X @ B4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % in_mono
% 0.16/0.42 thf(fact_1059_in__mono,axiom,
% 0.16/0.42 ! [A2: set_set_nat,B4: set_set_nat,X: set_nat] :
% 0.16/0.42 ( ( ord_le6893508408891458716et_nat @ A2 @ B4 )
% 0.16/0.42 => ( ( member_set_nat @ X @ A2 )
% 0.16/0.42 => ( member_set_nat @ X @ B4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % in_mono
% 0.16/0.42 thf(fact_1060_in__mono,axiom,
% 0.16/0.42 ! [A2: set_nat,B4: set_nat,X: nat] :
% 0.16/0.42 ( ( ord_less_eq_set_nat @ A2 @ B4 )
% 0.16/0.42 => ( ( member_nat @ X @ A2 )
% 0.16/0.42 => ( member_nat @ X @ B4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % in_mono
% 0.16/0.42 thf(fact_1061_in__mono,axiom,
% 0.16/0.42 ! [A2: set_int,B4: set_int,X: int] :
% 0.16/0.42 ( ( ord_less_eq_set_int @ A2 @ B4 )
% 0.16/0.42 => ( ( member_int @ X @ A2 )
% 0.16/0.42 => ( member_int @ X @ B4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % in_mono
% 0.16/0.42 thf(fact_1062_subsetD,axiom,
% 0.16/0.42 ! [A2: set_complex,B4: set_complex,C: complex] :
% 0.16/0.42 ( ( ord_le211207098394363844omplex @ A2 @ B4 )
% 0.16/0.42 => ( ( member_complex @ C @ A2 )
% 0.16/0.42 => ( member_complex @ C @ B4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % subsetD
% 0.16/0.42 thf(fact_1063_subsetD,axiom,
% 0.16/0.42 ! [A2: set_real,B4: set_real,C: real] :
% 0.16/0.42 ( ( ord_less_eq_set_real @ A2 @ B4 )
% 0.16/0.42 => ( ( member_real @ C @ A2 )
% 0.16/0.42 => ( member_real @ C @ B4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % subsetD
% 0.16/0.42 thf(fact_1064_subsetD,axiom,
% 0.16/0.42 ! [A2: set_set_nat,B4: set_set_nat,C: set_nat] :
% 0.16/0.42 ( ( ord_le6893508408891458716et_nat @ A2 @ B4 )
% 0.16/0.42 => ( ( member_set_nat @ C @ A2 )
% 0.16/0.42 => ( member_set_nat @ C @ B4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % subsetD
% 0.16/0.42 thf(fact_1065_subsetD,axiom,
% 0.16/0.42 ! [A2: set_nat,B4: set_nat,C: nat] :
% 0.16/0.42 ( ( ord_less_eq_set_nat @ A2 @ B4 )
% 0.16/0.42 => ( ( member_nat @ C @ A2 )
% 0.16/0.42 => ( member_nat @ C @ B4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % subsetD
% 0.16/0.42 thf(fact_1066_subsetD,axiom,
% 0.16/0.42 ! [A2: set_int,B4: set_int,C: int] :
% 0.16/0.42 ( ( ord_less_eq_set_int @ A2 @ B4 )
% 0.16/0.42 => ( ( member_int @ C @ A2 )
% 0.16/0.42 => ( member_int @ C @ B4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % subsetD
% 0.16/0.42 thf(fact_1067_equalityE,axiom,
% 0.16/0.42 ! [A2: set_int,B4: set_int] :
% 0.16/0.42 ( ( A2 = B4 )
% 0.16/0.42 => ~ ( ( ord_less_eq_set_int @ A2 @ B4 )
% 0.16/0.42 => ~ ( ord_less_eq_set_int @ B4 @ A2 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % equalityE
% 0.16/0.42 thf(fact_1068_subset__eq,axiom,
% 0.16/0.42 ( ord_le211207098394363844omplex
% 0.16/0.42 = ( ^ [A5: set_complex,B5: set_complex] :
% 0.16/0.42 ! [X3: complex] :
% 0.16/0.42 ( ( member_complex @ X3 @ A5 )
% 0.16/0.42 => ( member_complex @ X3 @ B5 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % subset_eq
% 0.16/0.42 thf(fact_1069_subset__eq,axiom,
% 0.16/0.42 ( ord_less_eq_set_real
% 0.16/0.42 = ( ^ [A5: set_real,B5: set_real] :
% 0.16/0.42 ! [X3: real] :
% 0.16/0.42 ( ( member_real @ X3 @ A5 )
% 0.16/0.42 => ( member_real @ X3 @ B5 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % subset_eq
% 0.16/0.42 thf(fact_1070_subset__eq,axiom,
% 0.16/0.42 ( ord_le6893508408891458716et_nat
% 0.16/0.42 = ( ^ [A5: set_set_nat,B5: set_set_nat] :
% 0.16/0.42 ! [X3: set_nat] :
% 0.16/0.42 ( ( member_set_nat @ X3 @ A5 )
% 0.16/0.42 => ( member_set_nat @ X3 @ B5 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % subset_eq
% 0.16/0.42 thf(fact_1071_subset__eq,axiom,
% 0.16/0.42 ( ord_less_eq_set_nat
% 0.16/0.42 = ( ^ [A5: set_nat,B5: set_nat] :
% 0.16/0.42 ! [X3: nat] :
% 0.16/0.42 ( ( member_nat @ X3 @ A5 )
% 0.16/0.42 => ( member_nat @ X3 @ B5 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % subset_eq
% 0.16/0.42 thf(fact_1072_subset__eq,axiom,
% 0.16/0.42 ( ord_less_eq_set_int
% 0.16/0.42 = ( ^ [A5: set_int,B5: set_int] :
% 0.16/0.42 ! [X3: int] :
% 0.16/0.42 ( ( member_int @ X3 @ A5 )
% 0.16/0.42 => ( member_int @ X3 @ B5 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % subset_eq
% 0.16/0.42 thf(fact_1073_equalityD1,axiom,
% 0.16/0.42 ! [A2: set_int,B4: set_int] :
% 0.16/0.42 ( ( A2 = B4 )
% 0.16/0.42 => ( ord_less_eq_set_int @ A2 @ B4 ) ) ).
% 0.16/0.42
% 0.16/0.42 % equalityD1
% 0.16/0.42 thf(fact_1074_equalityD2,axiom,
% 0.16/0.42 ! [A2: set_int,B4: set_int] :
% 0.16/0.42 ( ( A2 = B4 )
% 0.16/0.42 => ( ord_less_eq_set_int @ B4 @ A2 ) ) ).
% 0.16/0.42
% 0.16/0.42 % equalityD2
% 0.16/0.42 thf(fact_1075_subset__iff,axiom,
% 0.16/0.42 ( ord_le211207098394363844omplex
% 0.16/0.42 = ( ^ [A5: set_complex,B5: set_complex] :
% 0.16/0.42 ! [T2: complex] :
% 0.16/0.42 ( ( member_complex @ T2 @ A5 )
% 0.16/0.42 => ( member_complex @ T2 @ B5 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % subset_iff
% 0.16/0.42 thf(fact_1076_subset__iff,axiom,
% 0.16/0.42 ( ord_less_eq_set_real
% 0.16/0.42 = ( ^ [A5: set_real,B5: set_real] :
% 0.16/0.42 ! [T2: real] :
% 0.16/0.42 ( ( member_real @ T2 @ A5 )
% 0.16/0.42 => ( member_real @ T2 @ B5 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % subset_iff
% 0.16/0.42 thf(fact_1077_subset__iff,axiom,
% 0.16/0.42 ( ord_le6893508408891458716et_nat
% 0.16/0.42 = ( ^ [A5: set_set_nat,B5: set_set_nat] :
% 0.16/0.42 ! [T2: set_nat] :
% 0.16/0.42 ( ( member_set_nat @ T2 @ A5 )
% 0.16/0.42 => ( member_set_nat @ T2 @ B5 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % subset_iff
% 0.16/0.42 thf(fact_1078_subset__iff,axiom,
% 0.16/0.42 ( ord_less_eq_set_nat
% 0.16/0.42 = ( ^ [A5: set_nat,B5: set_nat] :
% 0.16/0.42 ! [T2: nat] :
% 0.16/0.42 ( ( member_nat @ T2 @ A5 )
% 0.16/0.42 => ( member_nat @ T2 @ B5 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % subset_iff
% 0.16/0.42 thf(fact_1079_subset__iff,axiom,
% 0.16/0.42 ( ord_less_eq_set_int
% 0.16/0.42 = ( ^ [A5: set_int,B5: set_int] :
% 0.16/0.42 ! [T2: int] :
% 0.16/0.42 ( ( member_int @ T2 @ A5 )
% 0.16/0.42 => ( member_int @ T2 @ B5 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % subset_iff
% 0.16/0.42 thf(fact_1080_subset__refl,axiom,
% 0.16/0.42 ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% 0.16/0.42
% 0.16/0.42 % subset_refl
% 0.16/0.42 thf(fact_1081_Collect__mono,axiom,
% 0.16/0.42 ! [P: real > $o,Q: real > $o] :
% 0.16/0.42 ( ! [X4: real] :
% 0.16/0.42 ( ( P @ X4 )
% 0.16/0.42 => ( Q @ X4 ) )
% 0.16/0.42 => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % Collect_mono
% 0.16/0.42 thf(fact_1082_Collect__mono,axiom,
% 0.16/0.42 ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.16/0.42 ( ! [X4: list_nat] :
% 0.16/0.42 ( ( P @ X4 )
% 0.16/0.42 => ( Q @ X4 ) )
% 0.16/0.42 => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % Collect_mono
% 0.16/0.42 thf(fact_1083_Collect__mono,axiom,
% 0.16/0.42 ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.16/0.42 ( ! [X4: set_nat] :
% 0.16/0.42 ( ( P @ X4 )
% 0.16/0.42 => ( Q @ X4 ) )
% 0.16/0.42 => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % Collect_mono
% 0.16/0.42 thf(fact_1084_Collect__mono,axiom,
% 0.16/0.42 ! [P: nat > $o,Q: nat > $o] :
% 0.16/0.42 ( ! [X4: nat] :
% 0.16/0.42 ( ( P @ X4 )
% 0.16/0.42 => ( Q @ X4 ) )
% 0.16/0.42 => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % Collect_mono
% 0.16/0.42 thf(fact_1085_Collect__mono,axiom,
% 0.16/0.42 ! [P: int > $o,Q: int > $o] :
% 0.16/0.42 ( ! [X4: int] :
% 0.16/0.42 ( ( P @ X4 )
% 0.16/0.42 => ( Q @ X4 ) )
% 0.16/0.42 => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % Collect_mono
% 0.16/0.42 thf(fact_1086_subset__trans,axiom,
% 0.16/0.42 ! [A2: set_int,B4: set_int,C2: set_int] :
% 0.16/0.42 ( ( ord_less_eq_set_int @ A2 @ B4 )
% 0.16/0.42 => ( ( ord_less_eq_set_int @ B4 @ C2 )
% 0.16/0.42 => ( ord_less_eq_set_int @ A2 @ C2 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % subset_trans
% 0.16/0.42 thf(fact_1087_set__eq__subset,axiom,
% 0.16/0.42 ( ( ^ [Y5: set_int,Z4: set_int] : ( Y5 = Z4 ) )
% 0.16/0.42 = ( ^ [A5: set_int,B5: set_int] :
% 0.16/0.42 ( ( ord_less_eq_set_int @ A5 @ B5 )
% 0.16/0.42 & ( ord_less_eq_set_int @ B5 @ A5 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % set_eq_subset
% 0.16/0.42 thf(fact_1088_Collect__mono__iff,axiom,
% 0.16/0.42 ! [P: real > $o,Q: real > $o] :
% 0.16/0.42 ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
% 0.16/0.42 = ( ! [X3: real] :
% 0.16/0.42 ( ( P @ X3 )
% 0.16/0.42 => ( Q @ X3 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % Collect_mono_iff
% 0.16/0.42 thf(fact_1089_Collect__mono__iff,axiom,
% 0.16/0.42 ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.16/0.42 ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
% 0.16/0.42 = ( ! [X3: list_nat] :
% 0.16/0.42 ( ( P @ X3 )
% 0.16/0.42 => ( Q @ X3 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % Collect_mono_iff
% 0.16/0.42 thf(fact_1090_Collect__mono__iff,axiom,
% 0.16/0.42 ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.16/0.42 ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
% 0.16/0.42 = ( ! [X3: set_nat] :
% 0.16/0.42 ( ( P @ X3 )
% 0.16/0.42 => ( Q @ X3 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % Collect_mono_iff
% 0.16/0.42 thf(fact_1091_Collect__mono__iff,axiom,
% 0.16/0.42 ! [P: nat > $o,Q: nat > $o] :
% 0.16/0.42 ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 0.16/0.42 = ( ! [X3: nat] :
% 0.16/0.42 ( ( P @ X3 )
% 0.16/0.42 => ( Q @ X3 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % Collect_mono_iff
% 0.16/0.42 thf(fact_1092_Collect__mono__iff,axiom,
% 0.16/0.42 ! [P: int > $o,Q: int > $o] :
% 0.16/0.42 ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
% 0.16/0.42 = ( ! [X3: int] :
% 0.16/0.42 ( ( P @ X3 )
% 0.16/0.42 => ( Q @ X3 ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % Collect_mono_iff
% 0.16/0.42 thf(fact_1093_zdiv__zmult2__eq,axiom,
% 0.16/0.42 ! [C: int,A: int,B: int] :
% 0.16/0.42 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.16/0.42 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.16/0.42 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % zdiv_zmult2_eq
% 0.16/0.42 thf(fact_1094_frac__le__eq,axiom,
% 0.16/0.42 ! [Y4: rat,Z: rat,X: rat,W: rat] :
% 0.16/0.42 ( ( Y4 != zero_zero_rat )
% 0.16/0.42 => ( ( Z != zero_zero_rat )
% 0.16/0.42 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y4 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.16/0.42 = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y4 ) ) @ ( times_times_rat @ Y4 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % frac_le_eq
% 0.16/0.42 thf(fact_1095_frac__le__eq,axiom,
% 0.16/0.42 ! [Y4: real,Z: real,X: real,W: real] :
% 0.16/0.42 ( ( Y4 != zero_zero_real )
% 0.16/0.42 => ( ( Z != zero_zero_real )
% 0.16/0.42 => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ ( divide_divide_real @ W @ Z ) )
% 0.16/0.42 = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y4 ) ) @ ( times_times_real @ Y4 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % frac_le_eq
% 0.16/0.42 thf(fact_1096_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
% 0.16/0.42 ( ( vEBT_V8346862874174094_d_u_p @ zero_zero_nat )
% 0.16/0.42 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
% 0.16/0.42 thf(fact_1097_zero__le,axiom,
% 0.16/0.42 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.16/0.42
% 0.16/0.42 % zero_le
% 0.16/0.42 thf(fact_1098_bezout1__nat,axiom,
% 0.16/0.42 ! [A: nat,B: nat] :
% 0.16/0.42 ? [D2: nat,X4: nat,Y: nat] :
% 0.16/0.42 ( ( dvd_dvd_nat @ D2 @ A )
% 0.16/0.42 & ( dvd_dvd_nat @ D2 @ B )
% 0.16/0.42 & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y ) )
% 0.16/0.42 = D2 )
% 0.16/0.42 | ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y ) )
% 0.16/0.42 = D2 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % bezout1_nat
% 0.16/0.42 thf(fact_1099_odd__two__times__div__two__nat,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.42 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.42 = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % odd_two_times_div_two_nat
% 0.16/0.42 thf(fact_1100_buildup__build__time,axiom,
% 0.16/0.42 ! [N: nat] : ( ord_less_nat @ ( vEBT_V8346862874174094_d_u_p @ N ) @ ( vEBT_V8646137997579335489_i_l_d @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % buildup_build_time
% 0.16/0.42 thf(fact_1101_zdvd__mono,axiom,
% 0.16/0.42 ! [K: int,M: int,T: int] :
% 0.16/0.42 ( ( K != zero_zero_int )
% 0.16/0.42 => ( ( dvd_dvd_int @ M @ T )
% 0.16/0.42 = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % zdvd_mono
% 0.16/0.42 thf(fact_1102_zero__le__power__eq__numeral,axiom,
% 0.16/0.42 ! [A: rat,W: num] :
% 0.16/0.42 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.16/0.42 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_le_power_eq_numeral
% 0.16/0.42 thf(fact_1103_zero__le__power__eq__numeral,axiom,
% 0.16/0.42 ! [A: int,W: num] :
% 0.16/0.42 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.16/0.42 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_le_power_eq_numeral
% 0.16/0.42 thf(fact_1104_zero__le__power__eq__numeral,axiom,
% 0.16/0.42 ! [A: real,W: num] :
% 0.16/0.42 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.16/0.42 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_le_power_eq_numeral
% 0.16/0.42 thf(fact_1105_gcd__nat_Oextremum,axiom,
% 0.16/0.42 ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 0.16/0.42
% 0.16/0.42 % gcd_nat.extremum
% 0.16/0.42 thf(fact_1106_gcd__nat_Oextremum__strict,axiom,
% 0.16/0.42 ! [A: nat] :
% 0.16/0.42 ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.16/0.42 & ( zero_zero_nat != A ) ) ).
% 0.16/0.42
% 0.16/0.42 % gcd_nat.extremum_strict
% 0.16/0.42 thf(fact_1107_gcd__nat_Oextremum__unique,axiom,
% 0.16/0.42 ! [A: nat] :
% 0.16/0.42 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.16/0.42 = ( A = zero_zero_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % gcd_nat.extremum_unique
% 0.16/0.42 thf(fact_1108_gcd__nat_Onot__eq__extremum,axiom,
% 0.16/0.42 ! [A: nat] :
% 0.16/0.42 ( ( A != zero_zero_nat )
% 0.16/0.42 = ( ( dvd_dvd_nat @ A @ zero_zero_nat )
% 0.16/0.42 & ( A != zero_zero_nat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % gcd_nat.not_eq_extremum
% 0.16/0.42 thf(fact_1109_gcd__nat_Oextremum__uniqueI,axiom,
% 0.16/0.42 ! [A: nat] :
% 0.16/0.42 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.16/0.42 => ( A = zero_zero_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % gcd_nat.extremum_uniqueI
% 0.16/0.42 thf(fact_1110_idiff__0,axiom,
% 0.16/0.42 ! [N: extended_enat] :
% 0.16/0.42 ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.16/0.42 = zero_z5237406670263579293d_enat ) ).
% 0.16/0.42
% 0.16/0.42 % idiff_0
% 0.16/0.42 thf(fact_1111_idiff__0__right,axiom,
% 0.16/0.42 ! [N: extended_enat] :
% 0.16/0.42 ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
% 0.16/0.42 = N ) ).
% 0.16/0.42
% 0.16/0.42 % idiff_0_right
% 0.16/0.42 thf(fact_1112_not__gr__zero,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.16/0.42 = ( N = zero_zero_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % not_gr_zero
% 0.16/0.42 thf(fact_1113_numeral__less__iff,axiom,
% 0.16/0.42 ! [M: num,N: num] :
% 0.16/0.42 ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.16/0.42 = ( ord_less_num @ M @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % numeral_less_iff
% 0.16/0.42 thf(fact_1114_numeral__less__iff,axiom,
% 0.16/0.42 ! [M: num,N: num] :
% 0.16/0.42 ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.16/0.42 = ( ord_less_num @ M @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % numeral_less_iff
% 0.16/0.42 thf(fact_1115_numeral__less__iff,axiom,
% 0.16/0.42 ! [M: num,N: num] :
% 0.16/0.42 ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.16/0.42 = ( ord_less_num @ M @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % numeral_less_iff
% 0.16/0.42 thf(fact_1116_numeral__less__iff,axiom,
% 0.16/0.42 ! [M: num,N: num] :
% 0.16/0.42 ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.16/0.42 = ( ord_less_num @ M @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % numeral_less_iff
% 0.16/0.42 thf(fact_1117_mult_Oright__neutral,axiom,
% 0.16/0.42 ! [A: complex] :
% 0.16/0.42 ( ( times_times_complex @ A @ one_one_complex )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % mult.right_neutral
% 0.16/0.42 thf(fact_1118_mult_Oright__neutral,axiom,
% 0.16/0.42 ! [A: real] :
% 0.16/0.42 ( ( times_times_real @ A @ one_one_real )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % mult.right_neutral
% 0.16/0.42 thf(fact_1119_mult_Oright__neutral,axiom,
% 0.16/0.42 ! [A: rat] :
% 0.16/0.42 ( ( times_times_rat @ A @ one_one_rat )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % mult.right_neutral
% 0.16/0.42 thf(fact_1120_mult_Oright__neutral,axiom,
% 0.16/0.42 ! [A: nat] :
% 0.16/0.42 ( ( times_times_nat @ A @ one_one_nat )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % mult.right_neutral
% 0.16/0.42 thf(fact_1121_mult_Oright__neutral,axiom,
% 0.16/0.42 ! [A: int] :
% 0.16/0.42 ( ( times_times_int @ A @ one_one_int )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % mult.right_neutral
% 0.16/0.42 thf(fact_1122_mult__1,axiom,
% 0.16/0.42 ! [A: complex] :
% 0.16/0.42 ( ( times_times_complex @ one_one_complex @ A )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % mult_1
% 0.16/0.42 thf(fact_1123_mult__1,axiom,
% 0.16/0.42 ! [A: real] :
% 0.16/0.42 ( ( times_times_real @ one_one_real @ A )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % mult_1
% 0.16/0.42 thf(fact_1124_mult__1,axiom,
% 0.16/0.42 ! [A: rat] :
% 0.16/0.42 ( ( times_times_rat @ one_one_rat @ A )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % mult_1
% 0.16/0.42 thf(fact_1125_mult__1,axiom,
% 0.16/0.42 ! [A: nat] :
% 0.16/0.42 ( ( times_times_nat @ one_one_nat @ A )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % mult_1
% 0.16/0.42 thf(fact_1126_mult__1,axiom,
% 0.16/0.42 ! [A: int] :
% 0.16/0.42 ( ( times_times_int @ one_one_int @ A )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % mult_1
% 0.16/0.42 thf(fact_1127_div__by__1,axiom,
% 0.16/0.42 ! [A: complex] :
% 0.16/0.42 ( ( divide1717551699836669952omplex @ A @ one_one_complex )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % div_by_1
% 0.16/0.42 thf(fact_1128_div__by__1,axiom,
% 0.16/0.42 ! [A: real] :
% 0.16/0.42 ( ( divide_divide_real @ A @ one_one_real )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % div_by_1
% 0.16/0.42 thf(fact_1129_div__by__1,axiom,
% 0.16/0.42 ! [A: rat] :
% 0.16/0.42 ( ( divide_divide_rat @ A @ one_one_rat )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % div_by_1
% 0.16/0.42 thf(fact_1130_div__by__1,axiom,
% 0.16/0.42 ! [A: nat] :
% 0.16/0.42 ( ( divide_divide_nat @ A @ one_one_nat )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % div_by_1
% 0.16/0.42 thf(fact_1131_div__by__1,axiom,
% 0.16/0.42 ! [A: int] :
% 0.16/0.42 ( ( divide_divide_int @ A @ one_one_int )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % div_by_1
% 0.16/0.42 thf(fact_1132_bits__div__by__1,axiom,
% 0.16/0.42 ! [A: nat] :
% 0.16/0.42 ( ( divide_divide_nat @ A @ one_one_nat )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % bits_div_by_1
% 0.16/0.42 thf(fact_1133_bits__div__by__1,axiom,
% 0.16/0.42 ! [A: int] :
% 0.16/0.42 ( ( divide_divide_int @ A @ one_one_int )
% 0.16/0.42 = A ) ).
% 0.16/0.42
% 0.16/0.42 % bits_div_by_1
% 0.16/0.42 thf(fact_1134_lessI,axiom,
% 0.16/0.42 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % lessI
% 0.16/0.42 thf(fact_1135_Suc__mono,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_nat @ M @ N )
% 0.16/0.42 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % Suc_mono
% 0.16/0.42 thf(fact_1136_Suc__less__eq,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.16/0.42 = ( ord_less_nat @ M @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % Suc_less_eq
% 0.16/0.42 thf(fact_1137_bot__nat__0_Onot__eq__extremum,axiom,
% 0.16/0.42 ! [A: nat] :
% 0.16/0.42 ( ( A != zero_zero_nat )
% 0.16/0.42 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.16/0.42
% 0.16/0.42 % bot_nat_0.not_eq_extremum
% 0.16/0.42 thf(fact_1138_neq0__conv,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( N != zero_zero_nat )
% 0.16/0.42 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % neq0_conv
% 0.16/0.42 thf(fact_1139_less__nat__zero__code,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.16/0.42
% 0.16/0.42 % less_nat_zero_code
% 0.16/0.42 thf(fact_1140_nat__mult__eq__1__iff,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( ( times_times_nat @ M @ N )
% 0.16/0.42 = one_one_nat )
% 0.16/0.42 = ( ( M = one_one_nat )
% 0.16/0.42 & ( N = one_one_nat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % nat_mult_eq_1_iff
% 0.16/0.42 thf(fact_1141_nat__1__eq__mult__iff,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( one_one_nat
% 0.16/0.42 = ( times_times_nat @ M @ N ) )
% 0.16/0.42 = ( ( M = one_one_nat )
% 0.16/0.42 & ( N = one_one_nat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % nat_1_eq_mult_iff
% 0.16/0.42 thf(fact_1142_nat__dvd__1__iff__1,axiom,
% 0.16/0.42 ! [M: nat] :
% 0.16/0.42 ( ( dvd_dvd_nat @ M @ one_one_nat )
% 0.16/0.42 = ( M = one_one_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % nat_dvd_1_iff_1
% 0.16/0.42 thf(fact_1143_diff__gt__0__iff__gt,axiom,
% 0.16/0.42 ! [A: real,B: real] :
% 0.16/0.42 ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.16/0.42 = ( ord_less_real @ B @ A ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_gt_0_iff_gt
% 0.16/0.42 thf(fact_1144_diff__gt__0__iff__gt,axiom,
% 0.16/0.42 ! [A: rat,B: rat] :
% 0.16/0.42 ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 0.16/0.42 = ( ord_less_rat @ B @ A ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_gt_0_iff_gt
% 0.16/0.42 thf(fact_1145_diff__gt__0__iff__gt,axiom,
% 0.16/0.42 ! [A: int,B: int] :
% 0.16/0.42 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.16/0.42 = ( ord_less_int @ B @ A ) ) ).
% 0.16/0.42
% 0.16/0.42 % diff_gt_0_iff_gt
% 0.16/0.42 thf(fact_1146_mult__cancel__left1,axiom,
% 0.16/0.42 ! [C: complex,B: complex] :
% 0.16/0.42 ( ( C
% 0.16/0.42 = ( times_times_complex @ C @ B ) )
% 0.16/0.42 = ( ( C = zero_zero_complex )
% 0.16/0.42 | ( B = one_one_complex ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult_cancel_left1
% 0.16/0.42 thf(fact_1147_mult__cancel__left1,axiom,
% 0.16/0.42 ! [C: real,B: real] :
% 0.16/0.42 ( ( C
% 0.16/0.42 = ( times_times_real @ C @ B ) )
% 0.16/0.42 = ( ( C = zero_zero_real )
% 0.16/0.42 | ( B = one_one_real ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult_cancel_left1
% 0.16/0.42 thf(fact_1148_mult__cancel__left1,axiom,
% 0.16/0.42 ! [C: rat,B: rat] :
% 0.16/0.42 ( ( C
% 0.16/0.42 = ( times_times_rat @ C @ B ) )
% 0.16/0.42 = ( ( C = zero_zero_rat )
% 0.16/0.42 | ( B = one_one_rat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult_cancel_left1
% 0.16/0.42 thf(fact_1149_mult__cancel__left1,axiom,
% 0.16/0.42 ! [C: int,B: int] :
% 0.16/0.42 ( ( C
% 0.16/0.42 = ( times_times_int @ C @ B ) )
% 0.16/0.42 = ( ( C = zero_zero_int )
% 0.16/0.42 | ( B = one_one_int ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult_cancel_left1
% 0.16/0.42 thf(fact_1150_mult__cancel__left2,axiom,
% 0.16/0.42 ! [C: complex,A: complex] :
% 0.16/0.42 ( ( ( times_times_complex @ C @ A )
% 0.16/0.42 = C )
% 0.16/0.42 = ( ( C = zero_zero_complex )
% 0.16/0.42 | ( A = one_one_complex ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult_cancel_left2
% 0.16/0.42 thf(fact_1151_mult__cancel__left2,axiom,
% 0.16/0.42 ! [C: real,A: real] :
% 0.16/0.42 ( ( ( times_times_real @ C @ A )
% 0.16/0.42 = C )
% 0.16/0.42 = ( ( C = zero_zero_real )
% 0.16/0.42 | ( A = one_one_real ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult_cancel_left2
% 0.16/0.42 thf(fact_1152_mult__cancel__left2,axiom,
% 0.16/0.42 ! [C: rat,A: rat] :
% 0.16/0.42 ( ( ( times_times_rat @ C @ A )
% 0.16/0.42 = C )
% 0.16/0.42 = ( ( C = zero_zero_rat )
% 0.16/0.42 | ( A = one_one_rat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult_cancel_left2
% 0.16/0.42 thf(fact_1153_mult__cancel__left2,axiom,
% 0.16/0.42 ! [C: int,A: int] :
% 0.16/0.42 ( ( ( times_times_int @ C @ A )
% 0.16/0.42 = C )
% 0.16/0.42 = ( ( C = zero_zero_int )
% 0.16/0.42 | ( A = one_one_int ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult_cancel_left2
% 0.16/0.42 thf(fact_1154_mult__cancel__right1,axiom,
% 0.16/0.42 ! [C: complex,B: complex] :
% 0.16/0.42 ( ( C
% 0.16/0.42 = ( times_times_complex @ B @ C ) )
% 0.16/0.42 = ( ( C = zero_zero_complex )
% 0.16/0.42 | ( B = one_one_complex ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult_cancel_right1
% 0.16/0.42 thf(fact_1155_mult__cancel__right1,axiom,
% 0.16/0.42 ! [C: real,B: real] :
% 0.16/0.42 ( ( C
% 0.16/0.42 = ( times_times_real @ B @ C ) )
% 0.16/0.42 = ( ( C = zero_zero_real )
% 0.16/0.42 | ( B = one_one_real ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult_cancel_right1
% 0.16/0.42 thf(fact_1156_mult__cancel__right1,axiom,
% 0.16/0.42 ! [C: rat,B: rat] :
% 0.16/0.42 ( ( C
% 0.16/0.42 = ( times_times_rat @ B @ C ) )
% 0.16/0.42 = ( ( C = zero_zero_rat )
% 0.16/0.42 | ( B = one_one_rat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult_cancel_right1
% 0.16/0.42 thf(fact_1157_mult__cancel__right1,axiom,
% 0.16/0.42 ! [C: int,B: int] :
% 0.16/0.42 ( ( C
% 0.16/0.42 = ( times_times_int @ B @ C ) )
% 0.16/0.42 = ( ( C = zero_zero_int )
% 0.16/0.42 | ( B = one_one_int ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult_cancel_right1
% 0.16/0.42 thf(fact_1158_mult__cancel__right2,axiom,
% 0.16/0.42 ! [A: complex,C: complex] :
% 0.16/0.42 ( ( ( times_times_complex @ A @ C )
% 0.16/0.42 = C )
% 0.16/0.42 = ( ( C = zero_zero_complex )
% 0.16/0.42 | ( A = one_one_complex ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult_cancel_right2
% 0.16/0.42 thf(fact_1159_mult__cancel__right2,axiom,
% 0.16/0.42 ! [A: real,C: real] :
% 0.16/0.42 ( ( ( times_times_real @ A @ C )
% 0.16/0.42 = C )
% 0.16/0.42 = ( ( C = zero_zero_real )
% 0.16/0.42 | ( A = one_one_real ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult_cancel_right2
% 0.16/0.42 thf(fact_1160_mult__cancel__right2,axiom,
% 0.16/0.42 ! [A: rat,C: rat] :
% 0.16/0.42 ( ( ( times_times_rat @ A @ C )
% 0.16/0.42 = C )
% 0.16/0.42 = ( ( C = zero_zero_rat )
% 0.16/0.42 | ( A = one_one_rat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult_cancel_right2
% 0.16/0.42 thf(fact_1161_mult__cancel__right2,axiom,
% 0.16/0.42 ! [A: int,C: int] :
% 0.16/0.42 ( ( ( times_times_int @ A @ C )
% 0.16/0.42 = C )
% 0.16/0.42 = ( ( C = zero_zero_int )
% 0.16/0.42 | ( A = one_one_int ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult_cancel_right2
% 0.16/0.42 thf(fact_1162_diff__numeral__special_I9_J,axiom,
% 0.16/0.42 ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 0.16/0.42 = zero_zero_complex ) ).
% 0.16/0.42
% 0.16/0.42 % diff_numeral_special(9)
% 0.16/0.42 thf(fact_1163_diff__numeral__special_I9_J,axiom,
% 0.16/0.42 ( ( minus_minus_real @ one_one_real @ one_one_real )
% 0.16/0.42 = zero_zero_real ) ).
% 0.16/0.42
% 0.16/0.42 % diff_numeral_special(9)
% 0.16/0.42 thf(fact_1164_diff__numeral__special_I9_J,axiom,
% 0.16/0.42 ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 0.16/0.42 = zero_zero_rat ) ).
% 0.16/0.42
% 0.16/0.42 % diff_numeral_special(9)
% 0.16/0.42 thf(fact_1165_diff__numeral__special_I9_J,axiom,
% 0.16/0.42 ( ( minus_minus_int @ one_one_int @ one_one_int )
% 0.16/0.42 = zero_zero_int ) ).
% 0.16/0.42
% 0.16/0.42 % diff_numeral_special(9)
% 0.16/0.42 thf(fact_1166_div__self,axiom,
% 0.16/0.42 ! [A: complex] :
% 0.16/0.42 ( ( A != zero_zero_complex )
% 0.16/0.42 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.16/0.42 = one_one_complex ) ) ).
% 0.16/0.42
% 0.16/0.42 % div_self
% 0.16/0.42 thf(fact_1167_div__self,axiom,
% 0.16/0.42 ! [A: real] :
% 0.16/0.42 ( ( A != zero_zero_real )
% 0.16/0.42 => ( ( divide_divide_real @ A @ A )
% 0.16/0.42 = one_one_real ) ) ).
% 0.16/0.42
% 0.16/0.42 % div_self
% 0.16/0.42 thf(fact_1168_div__self,axiom,
% 0.16/0.42 ! [A: rat] :
% 0.16/0.42 ( ( A != zero_zero_rat )
% 0.16/0.42 => ( ( divide_divide_rat @ A @ A )
% 0.16/0.42 = one_one_rat ) ) ).
% 0.16/0.42
% 0.16/0.42 % div_self
% 0.16/0.42 thf(fact_1169_div__self,axiom,
% 0.16/0.42 ! [A: nat] :
% 0.16/0.42 ( ( A != zero_zero_nat )
% 0.16/0.42 => ( ( divide_divide_nat @ A @ A )
% 0.16/0.42 = one_one_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % div_self
% 0.16/0.42 thf(fact_1170_div__self,axiom,
% 0.16/0.42 ! [A: int] :
% 0.16/0.42 ( ( A != zero_zero_int )
% 0.16/0.42 => ( ( divide_divide_int @ A @ A )
% 0.16/0.42 = one_one_int ) ) ).
% 0.16/0.42
% 0.16/0.42 % div_self
% 0.16/0.42 thf(fact_1171_divide__eq__1__iff,axiom,
% 0.16/0.42 ! [A: complex,B: complex] :
% 0.16/0.42 ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.16/0.42 = one_one_complex )
% 0.16/0.42 = ( ( B != zero_zero_complex )
% 0.16/0.42 & ( A = B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_eq_1_iff
% 0.16/0.42 thf(fact_1172_divide__eq__1__iff,axiom,
% 0.16/0.42 ! [A: real,B: real] :
% 0.16/0.42 ( ( ( divide_divide_real @ A @ B )
% 0.16/0.42 = one_one_real )
% 0.16/0.42 = ( ( B != zero_zero_real )
% 0.16/0.42 & ( A = B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_eq_1_iff
% 0.16/0.42 thf(fact_1173_divide__eq__1__iff,axiom,
% 0.16/0.42 ! [A: rat,B: rat] :
% 0.16/0.42 ( ( ( divide_divide_rat @ A @ B )
% 0.16/0.42 = one_one_rat )
% 0.16/0.42 = ( ( B != zero_zero_rat )
% 0.16/0.42 & ( A = B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_eq_1_iff
% 0.16/0.42 thf(fact_1174_one__eq__divide__iff,axiom,
% 0.16/0.42 ! [A: complex,B: complex] :
% 0.16/0.42 ( ( one_one_complex
% 0.16/0.42 = ( divide1717551699836669952omplex @ A @ B ) )
% 0.16/0.42 = ( ( B != zero_zero_complex )
% 0.16/0.42 & ( A = B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % one_eq_divide_iff
% 0.16/0.42 thf(fact_1175_one__eq__divide__iff,axiom,
% 0.16/0.42 ! [A: real,B: real] :
% 0.16/0.42 ( ( one_one_real
% 0.16/0.42 = ( divide_divide_real @ A @ B ) )
% 0.16/0.42 = ( ( B != zero_zero_real )
% 0.16/0.42 & ( A = B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % one_eq_divide_iff
% 0.16/0.42 thf(fact_1176_one__eq__divide__iff,axiom,
% 0.16/0.42 ! [A: rat,B: rat] :
% 0.16/0.42 ( ( one_one_rat
% 0.16/0.42 = ( divide_divide_rat @ A @ B ) )
% 0.16/0.42 = ( ( B != zero_zero_rat )
% 0.16/0.42 & ( A = B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % one_eq_divide_iff
% 0.16/0.42 thf(fact_1177_divide__self,axiom,
% 0.16/0.42 ! [A: complex] :
% 0.16/0.42 ( ( A != zero_zero_complex )
% 0.16/0.42 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.16/0.42 = one_one_complex ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_self
% 0.16/0.42 thf(fact_1178_divide__self,axiom,
% 0.16/0.42 ! [A: real] :
% 0.16/0.42 ( ( A != zero_zero_real )
% 0.16/0.42 => ( ( divide_divide_real @ A @ A )
% 0.16/0.42 = one_one_real ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_self
% 0.16/0.42 thf(fact_1179_divide__self,axiom,
% 0.16/0.42 ! [A: rat] :
% 0.16/0.42 ( ( A != zero_zero_rat )
% 0.16/0.42 => ( ( divide_divide_rat @ A @ A )
% 0.16/0.42 = one_one_rat ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_self
% 0.16/0.42 thf(fact_1180_divide__self__if,axiom,
% 0.16/0.42 ! [A: complex] :
% 0.16/0.42 ( ( ( A = zero_zero_complex )
% 0.16/0.42 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.16/0.42 = zero_zero_complex ) )
% 0.16/0.42 & ( ( A != zero_zero_complex )
% 0.16/0.42 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.16/0.42 = one_one_complex ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_self_if
% 0.16/0.42 thf(fact_1181_divide__self__if,axiom,
% 0.16/0.42 ! [A: real] :
% 0.16/0.42 ( ( ( A = zero_zero_real )
% 0.16/0.42 => ( ( divide_divide_real @ A @ A )
% 0.16/0.42 = zero_zero_real ) )
% 0.16/0.42 & ( ( A != zero_zero_real )
% 0.16/0.42 => ( ( divide_divide_real @ A @ A )
% 0.16/0.42 = one_one_real ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_self_if
% 0.16/0.42 thf(fact_1182_divide__self__if,axiom,
% 0.16/0.42 ! [A: rat] :
% 0.16/0.42 ( ( ( A = zero_zero_rat )
% 0.16/0.42 => ( ( divide_divide_rat @ A @ A )
% 0.16/0.42 = zero_zero_rat ) )
% 0.16/0.42 & ( ( A != zero_zero_rat )
% 0.16/0.42 => ( ( divide_divide_rat @ A @ A )
% 0.16/0.42 = one_one_rat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_self_if
% 0.16/0.42 thf(fact_1183_divide__eq__eq__1,axiom,
% 0.16/0.42 ! [B: real,A: real] :
% 0.16/0.42 ( ( ( divide_divide_real @ B @ A )
% 0.16/0.42 = one_one_real )
% 0.16/0.42 = ( ( A != zero_zero_real )
% 0.16/0.42 & ( A = B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_eq_eq_1
% 0.16/0.42 thf(fact_1184_divide__eq__eq__1,axiom,
% 0.16/0.42 ! [B: rat,A: rat] :
% 0.16/0.42 ( ( ( divide_divide_rat @ B @ A )
% 0.16/0.42 = one_one_rat )
% 0.16/0.42 = ( ( A != zero_zero_rat )
% 0.16/0.42 & ( A = B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_eq_eq_1
% 0.16/0.42 thf(fact_1185_eq__divide__eq__1,axiom,
% 0.16/0.42 ! [B: real,A: real] :
% 0.16/0.42 ( ( one_one_real
% 0.16/0.42 = ( divide_divide_real @ B @ A ) )
% 0.16/0.42 = ( ( A != zero_zero_real )
% 0.16/0.42 & ( A = B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % eq_divide_eq_1
% 0.16/0.42 thf(fact_1186_eq__divide__eq__1,axiom,
% 0.16/0.42 ! [B: rat,A: rat] :
% 0.16/0.42 ( ( one_one_rat
% 0.16/0.42 = ( divide_divide_rat @ B @ A ) )
% 0.16/0.42 = ( ( A != zero_zero_rat )
% 0.16/0.42 & ( A = B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % eq_divide_eq_1
% 0.16/0.42 thf(fact_1187_one__divide__eq__0__iff,axiom,
% 0.16/0.42 ! [A: real] :
% 0.16/0.42 ( ( ( divide_divide_real @ one_one_real @ A )
% 0.16/0.42 = zero_zero_real )
% 0.16/0.42 = ( A = zero_zero_real ) ) ).
% 0.16/0.42
% 0.16/0.42 % one_divide_eq_0_iff
% 0.16/0.42 thf(fact_1188_one__divide__eq__0__iff,axiom,
% 0.16/0.42 ! [A: rat] :
% 0.16/0.42 ( ( ( divide_divide_rat @ one_one_rat @ A )
% 0.16/0.42 = zero_zero_rat )
% 0.16/0.42 = ( A = zero_zero_rat ) ) ).
% 0.16/0.42
% 0.16/0.42 % one_divide_eq_0_iff
% 0.16/0.42 thf(fact_1189_zero__eq__1__divide__iff,axiom,
% 0.16/0.42 ! [A: real] :
% 0.16/0.42 ( ( zero_zero_real
% 0.16/0.42 = ( divide_divide_real @ one_one_real @ A ) )
% 0.16/0.42 = ( A = zero_zero_real ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_eq_1_divide_iff
% 0.16/0.42 thf(fact_1190_zero__eq__1__divide__iff,axiom,
% 0.16/0.42 ! [A: rat] :
% 0.16/0.42 ( ( zero_zero_rat
% 0.16/0.42 = ( divide_divide_rat @ one_one_rat @ A ) )
% 0.16/0.42 = ( A = zero_zero_rat ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_eq_1_divide_iff
% 0.16/0.42 thf(fact_1191_numeral__eq__one__iff,axiom,
% 0.16/0.42 ! [N: num] :
% 0.16/0.42 ( ( ( numera6690914467698888265omplex @ N )
% 0.16/0.42 = one_one_complex )
% 0.16/0.42 = ( N = one ) ) ).
% 0.16/0.42
% 0.16/0.42 % numeral_eq_one_iff
% 0.16/0.42 thf(fact_1192_numeral__eq__one__iff,axiom,
% 0.16/0.42 ! [N: num] :
% 0.16/0.42 ( ( ( numeral_numeral_real @ N )
% 0.16/0.42 = one_one_real )
% 0.16/0.42 = ( N = one ) ) ).
% 0.16/0.42
% 0.16/0.42 % numeral_eq_one_iff
% 0.16/0.42 thf(fact_1193_numeral__eq__one__iff,axiom,
% 0.16/0.42 ! [N: num] :
% 0.16/0.42 ( ( ( numeral_numeral_rat @ N )
% 0.16/0.42 = one_one_rat )
% 0.16/0.42 = ( N = one ) ) ).
% 0.16/0.42
% 0.16/0.42 % numeral_eq_one_iff
% 0.16/0.42 thf(fact_1194_numeral__eq__one__iff,axiom,
% 0.16/0.42 ! [N: num] :
% 0.16/0.42 ( ( ( numeral_numeral_nat @ N )
% 0.16/0.42 = one_one_nat )
% 0.16/0.42 = ( N = one ) ) ).
% 0.16/0.42
% 0.16/0.42 % numeral_eq_one_iff
% 0.16/0.42 thf(fact_1195_numeral__eq__one__iff,axiom,
% 0.16/0.42 ! [N: num] :
% 0.16/0.42 ( ( ( numeral_numeral_int @ N )
% 0.16/0.42 = one_one_int )
% 0.16/0.42 = ( N = one ) ) ).
% 0.16/0.42
% 0.16/0.42 % numeral_eq_one_iff
% 0.16/0.42 thf(fact_1196_one__eq__numeral__iff,axiom,
% 0.16/0.42 ! [N: num] :
% 0.16/0.42 ( ( one_one_complex
% 0.16/0.42 = ( numera6690914467698888265omplex @ N ) )
% 0.16/0.42 = ( one = N ) ) ).
% 0.16/0.42
% 0.16/0.42 % one_eq_numeral_iff
% 0.16/0.42 thf(fact_1197_one__eq__numeral__iff,axiom,
% 0.16/0.42 ! [N: num] :
% 0.16/0.42 ( ( one_one_real
% 0.16/0.42 = ( numeral_numeral_real @ N ) )
% 0.16/0.42 = ( one = N ) ) ).
% 0.16/0.42
% 0.16/0.42 % one_eq_numeral_iff
% 0.16/0.42 thf(fact_1198_one__eq__numeral__iff,axiom,
% 0.16/0.42 ! [N: num] :
% 0.16/0.42 ( ( one_one_rat
% 0.16/0.42 = ( numeral_numeral_rat @ N ) )
% 0.16/0.42 = ( one = N ) ) ).
% 0.16/0.42
% 0.16/0.42 % one_eq_numeral_iff
% 0.16/0.42 thf(fact_1199_one__eq__numeral__iff,axiom,
% 0.16/0.42 ! [N: num] :
% 0.16/0.42 ( ( one_one_nat
% 0.16/0.42 = ( numeral_numeral_nat @ N ) )
% 0.16/0.42 = ( one = N ) ) ).
% 0.16/0.42
% 0.16/0.42 % one_eq_numeral_iff
% 0.16/0.42 thf(fact_1200_one__eq__numeral__iff,axiom,
% 0.16/0.42 ! [N: num] :
% 0.16/0.42 ( ( one_one_int
% 0.16/0.42 = ( numeral_numeral_int @ N ) )
% 0.16/0.42 = ( one = N ) ) ).
% 0.16/0.42
% 0.16/0.42 % one_eq_numeral_iff
% 0.16/0.42 thf(fact_1201_unit__prod,axiom,
% 0.16/0.42 ! [A: code_integer,B: code_integer] :
% 0.16/0.42 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.16/0.42 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.16/0.42 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % unit_prod
% 0.16/0.42 thf(fact_1202_unit__prod,axiom,
% 0.16/0.42 ! [A: nat,B: nat] :
% 0.16/0.42 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.16/0.42 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.16/0.42 => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % unit_prod
% 0.16/0.42 thf(fact_1203_unit__prod,axiom,
% 0.16/0.42 ! [A: int,B: int] :
% 0.16/0.42 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.16/0.42 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.16/0.42 => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % unit_prod
% 0.16/0.42 thf(fact_1204_unit__div,axiom,
% 0.16/0.42 ! [A: code_integer,B: code_integer] :
% 0.16/0.42 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.16/0.42 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.16/0.42 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % unit_div
% 0.16/0.42 thf(fact_1205_unit__div,axiom,
% 0.16/0.42 ! [A: nat,B: nat] :
% 0.16/0.42 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.16/0.42 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.16/0.42 => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % unit_div
% 0.16/0.42 thf(fact_1206_unit__div,axiom,
% 0.16/0.42 ! [A: int,B: int] :
% 0.16/0.42 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.16/0.42 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.16/0.42 => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % unit_div
% 0.16/0.42 thf(fact_1207_unit__div__1__unit,axiom,
% 0.16/0.42 ! [A: code_integer] :
% 0.16/0.42 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.16/0.42 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% 0.16/0.42
% 0.16/0.42 % unit_div_1_unit
% 0.16/0.42 thf(fact_1208_unit__div__1__unit,axiom,
% 0.16/0.42 ! [A: nat] :
% 0.16/0.42 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.16/0.42 => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % unit_div_1_unit
% 0.16/0.42 thf(fact_1209_unit__div__1__unit,axiom,
% 0.16/0.42 ! [A: int] :
% 0.16/0.42 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.16/0.42 => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% 0.16/0.42
% 0.16/0.42 % unit_div_1_unit
% 0.16/0.42 thf(fact_1210_unit__div__1__div__1,axiom,
% 0.16/0.42 ! [A: code_integer] :
% 0.16/0.42 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.16/0.42 => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 0.16/0.42 = A ) ) ).
% 0.16/0.42
% 0.16/0.42 % unit_div_1_div_1
% 0.16/0.42 thf(fact_1211_unit__div__1__div__1,axiom,
% 0.16/0.42 ! [A: nat] :
% 0.16/0.42 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.16/0.42 => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
% 0.16/0.42 = A ) ) ).
% 0.16/0.42
% 0.16/0.42 % unit_div_1_div_1
% 0.16/0.42 thf(fact_1212_unit__div__1__div__1,axiom,
% 0.16/0.42 ! [A: int] :
% 0.16/0.42 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.16/0.42 => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
% 0.16/0.42 = A ) ) ).
% 0.16/0.42
% 0.16/0.42 % unit_div_1_div_1
% 0.16/0.42 thf(fact_1213_of__nat__less__iff,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.16/0.42 = ( ord_less_nat @ M @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_less_iff
% 0.16/0.42 thf(fact_1214_of__nat__less__iff,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.16/0.42 = ( ord_less_nat @ M @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_less_iff
% 0.16/0.42 thf(fact_1215_of__nat__less__iff,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.16/0.42 = ( ord_less_nat @ M @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_less_iff
% 0.16/0.42 thf(fact_1216_of__nat__less__iff,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.16/0.42 = ( ord_less_nat @ M @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_less_iff
% 0.16/0.42 thf(fact_1217_zero__less__Suc,axiom,
% 0.16/0.42 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_less_Suc
% 0.16/0.42 thf(fact_1218_less__Suc0,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.16/0.42 = ( N = zero_zero_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_Suc0
% 0.16/0.42 thf(fact_1219_zero__less__diff,axiom,
% 0.16/0.42 ! [N: nat,M: nat] :
% 0.16/0.42 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 0.16/0.42 = ( ord_less_nat @ M @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_less_diff
% 0.16/0.42 thf(fact_1220_nat__mult__less__cancel__disj,axiom,
% 0.16/0.42 ! [K: nat,M: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.16/0.42 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.16/0.42 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % nat_mult_less_cancel_disj
% 0.16/0.42 thf(fact_1221_nat__0__less__mult__iff,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 0.16/0.42 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.16/0.42 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % nat_0_less_mult_iff
% 0.16/0.42 thf(fact_1222_mult__less__cancel2,axiom,
% 0.16/0.42 ! [M: nat,K: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.16/0.42 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.16/0.42 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult_less_cancel2
% 0.16/0.42 thf(fact_1223_of__nat__eq__1__iff,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( ( semiri8010041392384452111omplex @ N )
% 0.16/0.42 = one_one_complex )
% 0.16/0.42 = ( N = one_one_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_eq_1_iff
% 0.16/0.42 thf(fact_1224_of__nat__eq__1__iff,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( ( semiri5074537144036343181t_real @ N )
% 0.16/0.42 = one_one_real )
% 0.16/0.42 = ( N = one_one_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_eq_1_iff
% 0.16/0.42 thf(fact_1225_of__nat__eq__1__iff,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( ( semiri681578069525770553at_rat @ N )
% 0.16/0.42 = one_one_rat )
% 0.16/0.42 = ( N = one_one_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_eq_1_iff
% 0.16/0.42 thf(fact_1226_of__nat__eq__1__iff,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( ( semiri1316708129612266289at_nat @ N )
% 0.16/0.42 = one_one_nat )
% 0.16/0.42 = ( N = one_one_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_eq_1_iff
% 0.16/0.42 thf(fact_1227_of__nat__eq__1__iff,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( ( semiri1314217659103216013at_int @ N )
% 0.16/0.42 = one_one_int )
% 0.16/0.42 = ( N = one_one_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_eq_1_iff
% 0.16/0.42 thf(fact_1228_of__nat__1__eq__iff,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( one_one_complex
% 0.16/0.42 = ( semiri8010041392384452111omplex @ N ) )
% 0.16/0.42 = ( N = one_one_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_1_eq_iff
% 0.16/0.42 thf(fact_1229_of__nat__1__eq__iff,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( one_one_real
% 0.16/0.42 = ( semiri5074537144036343181t_real @ N ) )
% 0.16/0.42 = ( N = one_one_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_1_eq_iff
% 0.16/0.42 thf(fact_1230_of__nat__1__eq__iff,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( one_one_rat
% 0.16/0.42 = ( semiri681578069525770553at_rat @ N ) )
% 0.16/0.42 = ( N = one_one_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_1_eq_iff
% 0.16/0.42 thf(fact_1231_of__nat__1__eq__iff,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( one_one_nat
% 0.16/0.42 = ( semiri1316708129612266289at_nat @ N ) )
% 0.16/0.42 = ( N = one_one_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_1_eq_iff
% 0.16/0.42 thf(fact_1232_of__nat__1__eq__iff,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( one_one_int
% 0.16/0.42 = ( semiri1314217659103216013at_int @ N ) )
% 0.16/0.42 = ( N = one_one_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_1_eq_iff
% 0.16/0.42 thf(fact_1233_of__nat__1,axiom,
% 0.16/0.42 ( ( semiri8010041392384452111omplex @ one_one_nat )
% 0.16/0.42 = one_one_complex ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_1
% 0.16/0.42 thf(fact_1234_of__nat__1,axiom,
% 0.16/0.42 ( ( semiri5074537144036343181t_real @ one_one_nat )
% 0.16/0.42 = one_one_real ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_1
% 0.16/0.42 thf(fact_1235_of__nat__1,axiom,
% 0.16/0.42 ( ( semiri681578069525770553at_rat @ one_one_nat )
% 0.16/0.42 = one_one_rat ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_1
% 0.16/0.42 thf(fact_1236_of__nat__1,axiom,
% 0.16/0.42 ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 0.16/0.42 = one_one_nat ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_1
% 0.16/0.42 thf(fact_1237_of__nat__1,axiom,
% 0.16/0.42 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 0.16/0.42 = one_one_int ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_1
% 0.16/0.42 thf(fact_1238_less__one,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( ord_less_nat @ N @ one_one_nat )
% 0.16/0.42 = ( N = zero_zero_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_one
% 0.16/0.42 thf(fact_1239_div__less,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_nat @ M @ N )
% 0.16/0.42 => ( ( divide_divide_nat @ M @ N )
% 0.16/0.42 = zero_zero_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % div_less
% 0.16/0.42 thf(fact_1240_diff__Suc__1,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 0.16/0.42 = N ) ).
% 0.16/0.42
% 0.16/0.42 % diff_Suc_1
% 0.16/0.42 thf(fact_1241_divide__le__0__1__iff,axiom,
% 0.16/0.42 ! [A: rat] :
% 0.16/0.42 ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.16/0.42 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_le_0_1_iff
% 0.16/0.42 thf(fact_1242_divide__le__0__1__iff,axiom,
% 0.16/0.42 ! [A: real] :
% 0.16/0.42 ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.16/0.42 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_le_0_1_iff
% 0.16/0.42 thf(fact_1243_zero__le__divide__1__iff,axiom,
% 0.16/0.42 ! [A: rat] :
% 0.16/0.42 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.16/0.42 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_le_divide_1_iff
% 0.16/0.42 thf(fact_1244_zero__le__divide__1__iff,axiom,
% 0.16/0.42 ! [A: real] :
% 0.16/0.42 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.16/0.42 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_le_divide_1_iff
% 0.16/0.42 thf(fact_1245_zero__less__divide__1__iff,axiom,
% 0.16/0.42 ! [A: real] :
% 0.16/0.42 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.16/0.42 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_less_divide_1_iff
% 0.16/0.42 thf(fact_1246_zero__less__divide__1__iff,axiom,
% 0.16/0.42 ! [A: rat] :
% 0.16/0.42 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.16/0.42 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_less_divide_1_iff
% 0.16/0.42 thf(fact_1247_less__divide__eq__1__pos,axiom,
% 0.16/0.42 ! [A: real,B: real] :
% 0.16/0.42 ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.42 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.16/0.42 = ( ord_less_real @ A @ B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_divide_eq_1_pos
% 0.16/0.42 thf(fact_1248_less__divide__eq__1__pos,axiom,
% 0.16/0.42 ! [A: rat,B: rat] :
% 0.16/0.42 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.42 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.16/0.42 = ( ord_less_rat @ A @ B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_divide_eq_1_pos
% 0.16/0.42 thf(fact_1249_less__divide__eq__1__neg,axiom,
% 0.16/0.42 ! [A: real,B: real] :
% 0.16/0.42 ( ( ord_less_real @ A @ zero_zero_real )
% 0.16/0.42 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.16/0.42 = ( ord_less_real @ B @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_divide_eq_1_neg
% 0.16/0.42 thf(fact_1250_less__divide__eq__1__neg,axiom,
% 0.16/0.42 ! [A: rat,B: rat] :
% 0.16/0.42 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.16/0.42 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.16/0.42 = ( ord_less_rat @ B @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_divide_eq_1_neg
% 0.16/0.42 thf(fact_1251_divide__less__eq__1__pos,axiom,
% 0.16/0.42 ! [A: real,B: real] :
% 0.16/0.42 ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.42 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.16/0.42 = ( ord_less_real @ B @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_less_eq_1_pos
% 0.16/0.42 thf(fact_1252_divide__less__eq__1__pos,axiom,
% 0.16/0.42 ! [A: rat,B: rat] :
% 0.16/0.42 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.42 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.16/0.42 = ( ord_less_rat @ B @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_less_eq_1_pos
% 0.16/0.42 thf(fact_1253_divide__less__eq__1__neg,axiom,
% 0.16/0.42 ! [A: real,B: real] :
% 0.16/0.42 ( ( ord_less_real @ A @ zero_zero_real )
% 0.16/0.42 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.16/0.42 = ( ord_less_real @ A @ B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_less_eq_1_neg
% 0.16/0.42 thf(fact_1254_divide__less__eq__1__neg,axiom,
% 0.16/0.42 ! [A: rat,B: rat] :
% 0.16/0.42 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.16/0.42 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.16/0.42 = ( ord_less_rat @ A @ B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_less_eq_1_neg
% 0.16/0.42 thf(fact_1255_divide__less__0__1__iff,axiom,
% 0.16/0.42 ! [A: real] :
% 0.16/0.42 ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.16/0.42 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_less_0_1_iff
% 0.16/0.42 thf(fact_1256_divide__less__0__1__iff,axiom,
% 0.16/0.42 ! [A: rat] :
% 0.16/0.42 ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.16/0.42 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_less_0_1_iff
% 0.16/0.42 thf(fact_1257_one__less__numeral__iff,axiom,
% 0.16/0.42 ! [N: num] :
% 0.16/0.42 ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 0.16/0.42 = ( ord_less_num @ one @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % one_less_numeral_iff
% 0.16/0.42 thf(fact_1258_one__less__numeral__iff,axiom,
% 0.16/0.42 ! [N: num] :
% 0.16/0.42 ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 0.16/0.42 = ( ord_less_num @ one @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % one_less_numeral_iff
% 0.16/0.42 thf(fact_1259_one__less__numeral__iff,axiom,
% 0.16/0.42 ! [N: num] :
% 0.16/0.42 ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 0.16/0.42 = ( ord_less_num @ one @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % one_less_numeral_iff
% 0.16/0.42 thf(fact_1260_one__less__numeral__iff,axiom,
% 0.16/0.42 ! [N: num] :
% 0.16/0.42 ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 0.16/0.42 = ( ord_less_num @ one @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % one_less_numeral_iff
% 0.16/0.42 thf(fact_1261_divide__less__eq__numeral1_I1_J,axiom,
% 0.16/0.42 ! [B: real,W: num,A: real] :
% 0.16/0.42 ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 0.16/0.42 = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_less_eq_numeral1(1)
% 0.16/0.42 thf(fact_1262_divide__less__eq__numeral1_I1_J,axiom,
% 0.16/0.42 ! [B: rat,W: num,A: rat] :
% 0.16/0.42 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 0.16/0.42 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_less_eq_numeral1(1)
% 0.16/0.42 thf(fact_1263_less__divide__eq__numeral1_I1_J,axiom,
% 0.16/0.42 ! [A: real,B: real,W: num] :
% 0.16/0.42 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 0.16/0.42 = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_divide_eq_numeral1(1)
% 0.16/0.42 thf(fact_1264_less__divide__eq__numeral1_I1_J,axiom,
% 0.16/0.42 ! [A: rat,B: rat,W: num] :
% 0.16/0.42 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 0.16/0.42 = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_divide_eq_numeral1(1)
% 0.16/0.42 thf(fact_1265_nonzero__divide__mult__cancel__left,axiom,
% 0.16/0.42 ! [A: complex,B: complex] :
% 0.16/0.42 ( ( A != zero_zero_complex )
% 0.16/0.42 => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
% 0.16/0.42 = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % nonzero_divide_mult_cancel_left
% 0.16/0.42 thf(fact_1266_nonzero__divide__mult__cancel__left,axiom,
% 0.16/0.42 ! [A: real,B: real] :
% 0.16/0.42 ( ( A != zero_zero_real )
% 0.16/0.42 => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
% 0.16/0.42 = ( divide_divide_real @ one_one_real @ B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % nonzero_divide_mult_cancel_left
% 0.16/0.42 thf(fact_1267_nonzero__divide__mult__cancel__left,axiom,
% 0.16/0.42 ! [A: rat,B: rat] :
% 0.16/0.42 ( ( A != zero_zero_rat )
% 0.16/0.42 => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
% 0.16/0.42 = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % nonzero_divide_mult_cancel_left
% 0.16/0.42 thf(fact_1268_nonzero__divide__mult__cancel__right,axiom,
% 0.16/0.42 ! [B: complex,A: complex] :
% 0.16/0.42 ( ( B != zero_zero_complex )
% 0.16/0.42 => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
% 0.16/0.42 = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % nonzero_divide_mult_cancel_right
% 0.16/0.42 thf(fact_1269_nonzero__divide__mult__cancel__right,axiom,
% 0.16/0.42 ! [B: real,A: real] :
% 0.16/0.42 ( ( B != zero_zero_real )
% 0.16/0.42 => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
% 0.16/0.42 = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % nonzero_divide_mult_cancel_right
% 0.16/0.42 thf(fact_1270_nonzero__divide__mult__cancel__right,axiom,
% 0.16/0.42 ! [B: rat,A: rat] :
% 0.16/0.42 ( ( B != zero_zero_rat )
% 0.16/0.42 => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
% 0.16/0.42 = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % nonzero_divide_mult_cancel_right
% 0.16/0.42 thf(fact_1271_unit__mult__div__div,axiom,
% 0.16/0.42 ! [A: code_integer,B: code_integer] :
% 0.16/0.42 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.16/0.42 => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 0.16/0.42 = ( divide6298287555418463151nteger @ B @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % unit_mult_div_div
% 0.16/0.42 thf(fact_1272_unit__mult__div__div,axiom,
% 0.16/0.42 ! [A: nat,B: nat] :
% 0.16/0.42 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.16/0.42 => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
% 0.16/0.42 = ( divide_divide_nat @ B @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % unit_mult_div_div
% 0.16/0.42 thf(fact_1273_unit__mult__div__div,axiom,
% 0.16/0.42 ! [A: int,B: int] :
% 0.16/0.42 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.16/0.42 => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
% 0.16/0.42 = ( divide_divide_int @ B @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % unit_mult_div_div
% 0.16/0.42 thf(fact_1274_unit__div__mult__self,axiom,
% 0.16/0.42 ! [A: code_integer,B: code_integer] :
% 0.16/0.42 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.16/0.42 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 0.16/0.42 = B ) ) ).
% 0.16/0.42
% 0.16/0.42 % unit_div_mult_self
% 0.16/0.42 thf(fact_1275_unit__div__mult__self,axiom,
% 0.16/0.42 ! [A: nat,B: nat] :
% 0.16/0.42 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.16/0.42 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 0.16/0.42 = B ) ) ).
% 0.16/0.42
% 0.16/0.42 % unit_div_mult_self
% 0.16/0.42 thf(fact_1276_unit__div__mult__self,axiom,
% 0.16/0.42 ! [A: int,B: int] :
% 0.16/0.42 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.16/0.42 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 0.16/0.42 = B ) ) ).
% 0.16/0.42
% 0.16/0.42 % unit_div_mult_self
% 0.16/0.42 thf(fact_1277_pow__divides__pow__iff,axiom,
% 0.16/0.42 ! [N: nat,A: int,B: int] :
% 0.16/0.42 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.42 => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.16/0.42 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % pow_divides_pow_iff
% 0.16/0.42 thf(fact_1278_pow__divides__pow__iff,axiom,
% 0.16/0.42 ! [N: nat,A: nat,B: nat] :
% 0.16/0.42 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.42 => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.16/0.42 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % pow_divides_pow_iff
% 0.16/0.42 thf(fact_1279_Suc__pred,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.42 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.16/0.42 = N ) ) ).
% 0.16/0.42
% 0.16/0.42 % Suc_pred
% 0.16/0.42 thf(fact_1280_nat__mult__le__cancel__disj,axiom,
% 0.16/0.42 ! [K: nat,M: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.16/0.42 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.16/0.42 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % nat_mult_le_cancel_disj
% 0.16/0.42 thf(fact_1281_mult__le__cancel2,axiom,
% 0.16/0.42 ! [M: nat,K: nat,N: nat] :
% 0.16/0.42 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.16/0.42 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.16/0.42 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % mult_le_cancel2
% 0.16/0.42 thf(fact_1282_div__mult__self1__is__m,axiom,
% 0.16/0.42 ! [N: nat,M: nat] :
% 0.16/0.42 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.42 => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 0.16/0.42 = M ) ) ).
% 0.16/0.42
% 0.16/0.42 % div_mult_self1_is_m
% 0.16/0.42 thf(fact_1283_div__mult__self__is__m,axiom,
% 0.16/0.42 ! [N: nat,M: nat] :
% 0.16/0.42 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.42 => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 0.16/0.42 = M ) ) ).
% 0.16/0.42
% 0.16/0.42 % div_mult_self_is_m
% 0.16/0.42 thf(fact_1284_divide__le__eq__1__neg,axiom,
% 0.16/0.42 ! [A: rat,B: rat] :
% 0.16/0.42 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.16/0.42 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.16/0.42 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_le_eq_1_neg
% 0.16/0.42 thf(fact_1285_divide__le__eq__1__neg,axiom,
% 0.16/0.42 ! [A: real,B: real] :
% 0.16/0.42 ( ( ord_less_real @ A @ zero_zero_real )
% 0.16/0.42 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.16/0.42 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_le_eq_1_neg
% 0.16/0.42 thf(fact_1286_divide__le__eq__1__pos,axiom,
% 0.16/0.42 ! [A: rat,B: rat] :
% 0.16/0.42 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.42 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.16/0.42 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_le_eq_1_pos
% 0.16/0.42 thf(fact_1287_divide__le__eq__1__pos,axiom,
% 0.16/0.42 ! [A: real,B: real] :
% 0.16/0.42 ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.42 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.16/0.42 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % divide_le_eq_1_pos
% 0.16/0.42 thf(fact_1288_le__divide__eq__1__neg,axiom,
% 0.16/0.42 ! [A: rat,B: rat] :
% 0.16/0.42 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.16/0.42 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.16/0.42 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % le_divide_eq_1_neg
% 0.16/0.42 thf(fact_1289_le__divide__eq__1__neg,axiom,
% 0.16/0.42 ! [A: real,B: real] :
% 0.16/0.42 ( ( ord_less_real @ A @ zero_zero_real )
% 0.16/0.42 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.16/0.42 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % le_divide_eq_1_neg
% 0.16/0.42 thf(fact_1290_le__divide__eq__1__pos,axiom,
% 0.16/0.42 ! [A: rat,B: rat] :
% 0.16/0.42 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.42 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.16/0.42 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % le_divide_eq_1_pos
% 0.16/0.42 thf(fact_1291_le__divide__eq__1__pos,axiom,
% 0.16/0.42 ! [A: real,B: real] :
% 0.16/0.42 ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.42 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.16/0.42 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % le_divide_eq_1_pos
% 0.16/0.42 thf(fact_1292_of__nat__0__less__iff,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.16/0.42 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_0_less_iff
% 0.16/0.42 thf(fact_1293_of__nat__0__less__iff,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.16/0.42 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_0_less_iff
% 0.16/0.42 thf(fact_1294_of__nat__0__less__iff,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
% 0.16/0.42 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_0_less_iff
% 0.16/0.42 thf(fact_1295_of__nat__0__less__iff,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.16/0.42 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % of_nat_0_less_iff
% 0.16/0.42 thf(fact_1296_Suc__1,axiom,
% 0.16/0.42 ( ( suc @ one_one_nat )
% 0.16/0.42 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % Suc_1
% 0.16/0.42 thf(fact_1297_Suc__diff__1,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.42 => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.16/0.42 = N ) ) ).
% 0.16/0.42
% 0.16/0.42 % Suc_diff_1
% 0.16/0.42 thf(fact_1298_numeral__le__one__iff,axiom,
% 0.16/0.42 ! [N: num] :
% 0.16/0.42 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 0.16/0.42 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.16/0.42
% 0.16/0.42 % numeral_le_one_iff
% 0.16/0.42 thf(fact_1299_numeral__le__one__iff,axiom,
% 0.16/0.42 ! [N: num] :
% 0.16/0.42 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 0.16/0.42 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.16/0.42
% 0.16/0.42 % numeral_le_one_iff
% 0.16/0.42 thf(fact_1300_numeral__le__one__iff,axiom,
% 0.16/0.42 ! [N: num] :
% 0.16/0.42 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 0.16/0.42 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.16/0.42
% 0.16/0.42 % numeral_le_one_iff
% 0.16/0.42 thf(fact_1301_numeral__le__one__iff,axiom,
% 0.16/0.42 ! [N: num] :
% 0.16/0.42 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 0.16/0.42 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.16/0.42
% 0.16/0.42 % numeral_le_one_iff
% 0.16/0.42 thf(fact_1302_one__div__two__eq__zero,axiom,
% 0.16/0.42 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.42 = zero_zero_nat ) ).
% 0.16/0.42
% 0.16/0.42 % one_div_two_eq_zero
% 0.16/0.42 thf(fact_1303_one__div__two__eq__zero,axiom,
% 0.16/0.42 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.16/0.42 = zero_zero_int ) ).
% 0.16/0.42
% 0.16/0.42 % one_div_two_eq_zero
% 0.16/0.42 thf(fact_1304_bits__1__div__2,axiom,
% 0.16/0.42 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.42 = zero_zero_nat ) ).
% 0.16/0.42
% 0.16/0.42 % bits_1_div_2
% 0.16/0.42 thf(fact_1305_bits__1__div__2,axiom,
% 0.16/0.42 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.16/0.42 = zero_zero_int ) ).
% 0.16/0.42
% 0.16/0.42 % bits_1_div_2
% 0.16/0.42 thf(fact_1306_even__power,axiom,
% 0.16/0.42 ! [A: code_integer,N: nat] :
% 0.16/0.42 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 0.16/0.42 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.16/0.42 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % even_power
% 0.16/0.42 thf(fact_1307_even__power,axiom,
% 0.16/0.42 ! [A: nat,N: nat] :
% 0.16/0.42 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
% 0.16/0.42 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.16/0.42 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % even_power
% 0.16/0.42 thf(fact_1308_even__power,axiom,
% 0.16/0.42 ! [A: int,N: nat] :
% 0.16/0.42 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
% 0.16/0.42 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.16/0.42 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % even_power
% 0.16/0.42 thf(fact_1309_power__less__zero__eq,axiom,
% 0.16/0.42 ! [A: real,N: nat] :
% 0.16/0.42 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 0.16/0.42 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.42 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % power_less_zero_eq
% 0.16/0.42 thf(fact_1310_power__less__zero__eq,axiom,
% 0.16/0.42 ! [A: rat,N: nat] :
% 0.16/0.42 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 0.16/0.42 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.42 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % power_less_zero_eq
% 0.16/0.42 thf(fact_1311_power__less__zero__eq,axiom,
% 0.16/0.42 ! [A: int,N: nat] :
% 0.16/0.42 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 0.16/0.42 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.42 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % power_less_zero_eq
% 0.16/0.42 thf(fact_1312_power__less__zero__eq__numeral,axiom,
% 0.16/0.42 ! [A: real,W: num] :
% 0.16/0.42 ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 0.16/0.42 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % power_less_zero_eq_numeral
% 0.16/0.42 thf(fact_1313_power__less__zero__eq__numeral,axiom,
% 0.16/0.42 ! [A: rat,W: num] :
% 0.16/0.42 ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 0.16/0.42 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % power_less_zero_eq_numeral
% 0.16/0.42 thf(fact_1314_power__less__zero__eq__numeral,axiom,
% 0.16/0.42 ! [A: int,W: num] :
% 0.16/0.42 ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 0.16/0.42 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % power_less_zero_eq_numeral
% 0.16/0.42 thf(fact_1315_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
% 0.16/0.42 = ( N = zero_zero_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % semiring_parity_class.even_mask_iff
% 0.16/0.42 thf(fact_1316_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
% 0.16/0.42 = ( N = zero_zero_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % semiring_parity_class.even_mask_iff
% 0.16/0.42 thf(fact_1317_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 0.16/0.42 = ( N = zero_zero_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % semiring_parity_class.even_mask_iff
% 0.16/0.42 thf(fact_1318_zero__less__power__eq__numeral,axiom,
% 0.16/0.42 ! [A: real,W: num] :
% 0.16/0.42 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.16/0.42 = ( ( ( numeral_numeral_nat @ W )
% 0.16/0.42 = zero_zero_nat )
% 0.16/0.42 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( A != zero_zero_real ) )
% 0.16/0.42 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_less_power_eq_numeral
% 0.16/0.42 thf(fact_1319_zero__less__power__eq__numeral,axiom,
% 0.16/0.42 ! [A: rat,W: num] :
% 0.16/0.42 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.16/0.42 = ( ( ( numeral_numeral_nat @ W )
% 0.16/0.42 = zero_zero_nat )
% 0.16/0.42 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( A != zero_zero_rat ) )
% 0.16/0.42 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_less_power_eq_numeral
% 0.16/0.42 thf(fact_1320_zero__less__power__eq__numeral,axiom,
% 0.16/0.42 ! [A: int,W: num] :
% 0.16/0.42 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.16/0.42 = ( ( ( numeral_numeral_nat @ W )
% 0.16/0.42 = zero_zero_nat )
% 0.16/0.42 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( A != zero_zero_int ) )
% 0.16/0.42 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % zero_less_power_eq_numeral
% 0.16/0.42 thf(fact_1321_power__le__zero__eq__numeral,axiom,
% 0.16/0.42 ! [A: rat,W: num] :
% 0.16/0.42 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 0.16/0.42 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.16/0.42 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % power_le_zero_eq_numeral
% 0.16/0.42 thf(fact_1322_power__le__zero__eq__numeral,axiom,
% 0.16/0.42 ! [A: int,W: num] :
% 0.16/0.42 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 0.16/0.42 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.16/0.42 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( A = zero_zero_int ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % power_le_zero_eq_numeral
% 0.16/0.42 thf(fact_1323_power__le__zero__eq__numeral,axiom,
% 0.16/0.42 ! [A: real,W: num] :
% 0.16/0.42 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 0.16/0.42 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.16/0.42 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.16/0.42 & ( A = zero_zero_real ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % power_le_zero_eq_numeral
% 0.16/0.42 thf(fact_1324_minus__int__code_I1_J,axiom,
% 0.16/0.42 ! [K: int] :
% 0.16/0.42 ( ( minus_minus_int @ K @ zero_zero_int )
% 0.16/0.42 = K ) ).
% 0.16/0.42
% 0.16/0.42 % minus_int_code(1)
% 0.16/0.42 thf(fact_1325_imult__is__0,axiom,
% 0.16/0.42 ! [M: extended_enat,N: extended_enat] :
% 0.16/0.42 ( ( ( times_7803423173614009249d_enat @ M @ N )
% 0.16/0.42 = zero_z5237406670263579293d_enat )
% 0.16/0.42 = ( ( M = zero_z5237406670263579293d_enat )
% 0.16/0.42 | ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % imult_is_0
% 0.16/0.42 thf(fact_1326_one__reorient,axiom,
% 0.16/0.42 ! [X: complex] :
% 0.16/0.42 ( ( one_one_complex = X )
% 0.16/0.42 = ( X = one_one_complex ) ) ).
% 0.16/0.42
% 0.16/0.42 % one_reorient
% 0.16/0.42 thf(fact_1327_one__reorient,axiom,
% 0.16/0.42 ! [X: real] :
% 0.16/0.42 ( ( one_one_real = X )
% 0.16/0.42 = ( X = one_one_real ) ) ).
% 0.16/0.42
% 0.16/0.42 % one_reorient
% 0.16/0.42 thf(fact_1328_one__reorient,axiom,
% 0.16/0.42 ! [X: rat] :
% 0.16/0.42 ( ( one_one_rat = X )
% 0.16/0.42 = ( X = one_one_rat ) ) ).
% 0.16/0.42
% 0.16/0.42 % one_reorient
% 0.16/0.42 thf(fact_1329_one__reorient,axiom,
% 0.16/0.42 ! [X: nat] :
% 0.16/0.42 ( ( one_one_nat = X )
% 0.16/0.42 = ( X = one_one_nat ) ) ).
% 0.16/0.42
% 0.16/0.42 % one_reorient
% 0.16/0.42 thf(fact_1330_one__reorient,axiom,
% 0.16/0.42 ! [X: int] :
% 0.16/0.42 ( ( one_one_int = X )
% 0.16/0.42 = ( X = one_one_int ) ) ).
% 0.16/0.42
% 0.16/0.42 % one_reorient
% 0.16/0.42 thf(fact_1331_linordered__field__no__lb,axiom,
% 0.16/0.42 ! [X5: real] :
% 0.16/0.42 ? [Y: real] : ( ord_less_real @ Y @ X5 ) ).
% 0.16/0.42
% 0.16/0.42 % linordered_field_no_lb
% 0.16/0.42 thf(fact_1332_linordered__field__no__lb,axiom,
% 0.16/0.42 ! [X5: rat] :
% 0.16/0.42 ? [Y: rat] : ( ord_less_rat @ Y @ X5 ) ).
% 0.16/0.42
% 0.16/0.42 % linordered_field_no_lb
% 0.16/0.42 thf(fact_1333_linordered__field__no__ub,axiom,
% 0.16/0.42 ! [X5: real] :
% 0.16/0.42 ? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% 0.16/0.42
% 0.16/0.42 % linordered_field_no_ub
% 0.16/0.42 thf(fact_1334_linordered__field__no__ub,axiom,
% 0.16/0.42 ! [X5: rat] :
% 0.16/0.42 ? [X_1: rat] : ( ord_less_rat @ X5 @ X_1 ) ).
% 0.16/0.42
% 0.16/0.42 % linordered_field_no_ub
% 0.16/0.42 thf(fact_1335_less__numeral__extra_I4_J,axiom,
% 0.16/0.42 ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 0.16/0.42
% 0.16/0.42 % less_numeral_extra(4)
% 0.16/0.42 thf(fact_1336_less__numeral__extra_I4_J,axiom,
% 0.16/0.42 ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 0.16/0.42
% 0.16/0.42 % less_numeral_extra(4)
% 0.16/0.42 thf(fact_1337_less__numeral__extra_I4_J,axiom,
% 0.16/0.42 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 0.16/0.42
% 0.16/0.42 % less_numeral_extra(4)
% 0.16/0.42 thf(fact_1338_less__numeral__extra_I4_J,axiom,
% 0.16/0.42 ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 0.16/0.42
% 0.16/0.42 % less_numeral_extra(4)
% 0.16/0.42 thf(fact_1339_nat__neq__iff,axiom,
% 0.16/0.42 ! [M: nat,N: nat] :
% 0.16/0.42 ( ( M != N )
% 0.16/0.42 = ( ( ord_less_nat @ M @ N )
% 0.16/0.42 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % nat_neq_iff
% 0.16/0.42 thf(fact_1340_less__not__refl,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ~ ( ord_less_nat @ N @ N ) ).
% 0.16/0.42
% 0.16/0.42 % less_not_refl
% 0.16/0.42 thf(fact_1341_less__not__refl2,axiom,
% 0.16/0.42 ! [N: nat,M: nat] :
% 0.16/0.42 ( ( ord_less_nat @ N @ M )
% 0.16/0.42 => ( M != N ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_not_refl2
% 0.16/0.42 thf(fact_1342_less__not__refl3,axiom,
% 0.16/0.42 ! [S2: nat,T: nat] :
% 0.16/0.42 ( ( ord_less_nat @ S2 @ T )
% 0.16/0.42 => ( S2 != T ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_not_refl3
% 0.16/0.42 thf(fact_1343_less__irrefl__nat,axiom,
% 0.16/0.42 ! [N: nat] :
% 0.16/0.42 ~ ( ord_less_nat @ N @ N ) ).
% 0.16/0.42
% 0.16/0.42 % less_irrefl_nat
% 0.16/0.42 thf(fact_1344_nat__less__induct,axiom,
% 0.16/0.42 ! [P: nat > $o,N: nat] :
% 0.16/0.42 ( ! [N2: nat] :
% 0.16/0.42 ( ! [M2: nat] :
% 0.16/0.42 ( ( ord_less_nat @ M2 @ N2 )
% 0.16/0.42 => ( P @ M2 ) )
% 0.16/0.42 => ( P @ N2 ) )
% 0.16/0.42 => ( P @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % nat_less_induct
% 0.16/0.42 thf(fact_1345_infinite__descent,axiom,
% 0.16/0.42 ! [P: nat > $o,N: nat] :
% 0.16/0.42 ( ! [N2: nat] :
% 0.16/0.42 ( ~ ( P @ N2 )
% 0.16/0.42 => ? [M2: nat] :
% 0.16/0.42 ( ( ord_less_nat @ M2 @ N2 )
% 0.16/0.42 & ~ ( P @ M2 ) ) )
% 0.16/0.42 => ( P @ N ) ) ).
% 0.16/0.42
% 0.16/0.42 % infinite_descent
% 0.16/0.42 thf(fact_1346_linorder__neqE__nat,axiom,
% 0.16/0.42 ! [X: nat,Y4: nat] :
% 0.16/0.42 ( ( X != Y4 )
% 0.16/0.42 => ( ~ ( ord_less_nat @ X @ Y4 )
% 0.16/0.42 => ( ord_less_nat @ Y4 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_neqE_nat
% 0.16/0.42 thf(fact_1347_linorder__neqE__linordered__idom,axiom,
% 0.16/0.42 ! [X: real,Y4: real] :
% 0.16/0.42 ( ( X != Y4 )
% 0.16/0.42 => ( ~ ( ord_less_real @ X @ Y4 )
% 0.16/0.42 => ( ord_less_real @ Y4 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_neqE_linordered_idom
% 0.16/0.42 thf(fact_1348_linorder__neqE__linordered__idom,axiom,
% 0.16/0.42 ! [X: rat,Y4: rat] :
% 0.16/0.42 ( ( X != Y4 )
% 0.16/0.42 => ( ~ ( ord_less_rat @ X @ Y4 )
% 0.16/0.42 => ( ord_less_rat @ Y4 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_neqE_linordered_idom
% 0.16/0.42 thf(fact_1349_linorder__neqE__linordered__idom,axiom,
% 0.16/0.42 ! [X: int,Y4: int] :
% 0.16/0.42 ( ( X != Y4 )
% 0.16/0.42 => ( ~ ( ord_less_int @ X @ Y4 )
% 0.16/0.42 => ( ord_less_int @ Y4 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_neqE_linordered_idom
% 0.16/0.42 thf(fact_1350_verit__comp__simplify1_I1_J,axiom,
% 0.16/0.42 ! [A: real] :
% 0.16/0.42 ~ ( ord_less_real @ A @ A ) ).
% 0.16/0.42
% 0.16/0.42 % verit_comp_simplify1(1)
% 0.16/0.42 thf(fact_1351_verit__comp__simplify1_I1_J,axiom,
% 0.16/0.42 ! [A: rat] :
% 0.16/0.42 ~ ( ord_less_rat @ A @ A ) ).
% 0.16/0.42
% 0.16/0.42 % verit_comp_simplify1(1)
% 0.16/0.42 thf(fact_1352_verit__comp__simplify1_I1_J,axiom,
% 0.16/0.42 ! [A: num] :
% 0.16/0.42 ~ ( ord_less_num @ A @ A ) ).
% 0.16/0.42
% 0.16/0.42 % verit_comp_simplify1(1)
% 0.16/0.42 thf(fact_1353_verit__comp__simplify1_I1_J,axiom,
% 0.16/0.42 ! [A: nat] :
% 0.16/0.42 ~ ( ord_less_nat @ A @ A ) ).
% 0.16/0.42
% 0.16/0.42 % verit_comp_simplify1(1)
% 0.16/0.42 thf(fact_1354_verit__comp__simplify1_I1_J,axiom,
% 0.16/0.42 ! [A: int] :
% 0.16/0.42 ~ ( ord_less_int @ A @ A ) ).
% 0.16/0.42
% 0.16/0.42 % verit_comp_simplify1(1)
% 0.16/0.42 thf(fact_1355_lt__ex,axiom,
% 0.16/0.42 ! [X: real] :
% 0.16/0.42 ? [Y: real] : ( ord_less_real @ Y @ X ) ).
% 0.16/0.42
% 0.16/0.42 % lt_ex
% 0.16/0.42 thf(fact_1356_lt__ex,axiom,
% 0.16/0.42 ! [X: rat] :
% 0.16/0.42 ? [Y: rat] : ( ord_less_rat @ Y @ X ) ).
% 0.16/0.42
% 0.16/0.42 % lt_ex
% 0.16/0.42 thf(fact_1357_lt__ex,axiom,
% 0.16/0.42 ! [X: int] :
% 0.16/0.42 ? [Y: int] : ( ord_less_int @ Y @ X ) ).
% 0.16/0.42
% 0.16/0.42 % lt_ex
% 0.16/0.42 thf(fact_1358_gt__ex,axiom,
% 0.16/0.42 ! [X: real] :
% 0.16/0.42 ? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% 0.16/0.42
% 0.16/0.42 % gt_ex
% 0.16/0.42 thf(fact_1359_gt__ex,axiom,
% 0.16/0.42 ! [X: rat] :
% 0.16/0.42 ? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).
% 0.16/0.42
% 0.16/0.42 % gt_ex
% 0.16/0.42 thf(fact_1360_gt__ex,axiom,
% 0.16/0.42 ! [X: nat] :
% 0.16/0.42 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 0.16/0.42
% 0.16/0.42 % gt_ex
% 0.16/0.42 thf(fact_1361_gt__ex,axiom,
% 0.16/0.42 ! [X: int] :
% 0.16/0.42 ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% 0.16/0.42
% 0.16/0.42 % gt_ex
% 0.16/0.42 thf(fact_1362_dense,axiom,
% 0.16/0.42 ! [X: real,Y4: real] :
% 0.16/0.42 ( ( ord_less_real @ X @ Y4 )
% 0.16/0.42 => ? [Z2: real] :
% 0.16/0.42 ( ( ord_less_real @ X @ Z2 )
% 0.16/0.42 & ( ord_less_real @ Z2 @ Y4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dense
% 0.16/0.42 thf(fact_1363_dense,axiom,
% 0.16/0.42 ! [X: rat,Y4: rat] :
% 0.16/0.42 ( ( ord_less_rat @ X @ Y4 )
% 0.16/0.42 => ? [Z2: rat] :
% 0.16/0.42 ( ( ord_less_rat @ X @ Z2 )
% 0.16/0.42 & ( ord_less_rat @ Z2 @ Y4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dense
% 0.16/0.42 thf(fact_1364_less__imp__neq,axiom,
% 0.16/0.42 ! [X: real,Y4: real] :
% 0.16/0.42 ( ( ord_less_real @ X @ Y4 )
% 0.16/0.42 => ( X != Y4 ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_imp_neq
% 0.16/0.42 thf(fact_1365_less__imp__neq,axiom,
% 0.16/0.42 ! [X: rat,Y4: rat] :
% 0.16/0.42 ( ( ord_less_rat @ X @ Y4 )
% 0.16/0.42 => ( X != Y4 ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_imp_neq
% 0.16/0.42 thf(fact_1366_less__imp__neq,axiom,
% 0.16/0.42 ! [X: num,Y4: num] :
% 0.16/0.42 ( ( ord_less_num @ X @ Y4 )
% 0.16/0.42 => ( X != Y4 ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_imp_neq
% 0.16/0.42 thf(fact_1367_less__imp__neq,axiom,
% 0.16/0.42 ! [X: nat,Y4: nat] :
% 0.16/0.42 ( ( ord_less_nat @ X @ Y4 )
% 0.16/0.42 => ( X != Y4 ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_imp_neq
% 0.16/0.42 thf(fact_1368_less__imp__neq,axiom,
% 0.16/0.42 ! [X: int,Y4: int] :
% 0.16/0.42 ( ( ord_less_int @ X @ Y4 )
% 0.16/0.42 => ( X != Y4 ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_imp_neq
% 0.16/0.42 thf(fact_1369_order_Oasym,axiom,
% 0.16/0.42 ! [A: real,B: real] :
% 0.16/0.42 ( ( ord_less_real @ A @ B )
% 0.16/0.42 => ~ ( ord_less_real @ B @ A ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.asym
% 0.16/0.42 thf(fact_1370_order_Oasym,axiom,
% 0.16/0.42 ! [A: rat,B: rat] :
% 0.16/0.42 ( ( ord_less_rat @ A @ B )
% 0.16/0.42 => ~ ( ord_less_rat @ B @ A ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.asym
% 0.16/0.42 thf(fact_1371_order_Oasym,axiom,
% 0.16/0.42 ! [A: num,B: num] :
% 0.16/0.42 ( ( ord_less_num @ A @ B )
% 0.16/0.42 => ~ ( ord_less_num @ B @ A ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.asym
% 0.16/0.42 thf(fact_1372_order_Oasym,axiom,
% 0.16/0.42 ! [A: nat,B: nat] :
% 0.16/0.42 ( ( ord_less_nat @ A @ B )
% 0.16/0.42 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.asym
% 0.16/0.42 thf(fact_1373_order_Oasym,axiom,
% 0.16/0.42 ! [A: int,B: int] :
% 0.16/0.42 ( ( ord_less_int @ A @ B )
% 0.16/0.42 => ~ ( ord_less_int @ B @ A ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.asym
% 0.16/0.42 thf(fact_1374_ord__eq__less__trans,axiom,
% 0.16/0.42 ! [A: real,B: real,C: real] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ( ( ord_less_real @ B @ C )
% 0.16/0.42 => ( ord_less_real @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % ord_eq_less_trans
% 0.16/0.42 thf(fact_1375_ord__eq__less__trans,axiom,
% 0.16/0.42 ! [A: rat,B: rat,C: rat] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ( ( ord_less_rat @ B @ C )
% 0.16/0.42 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % ord_eq_less_trans
% 0.16/0.42 thf(fact_1376_ord__eq__less__trans,axiom,
% 0.16/0.42 ! [A: num,B: num,C: num] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ( ( ord_less_num @ B @ C )
% 0.16/0.42 => ( ord_less_num @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % ord_eq_less_trans
% 0.16/0.42 thf(fact_1377_ord__eq__less__trans,axiom,
% 0.16/0.42 ! [A: nat,B: nat,C: nat] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ( ( ord_less_nat @ B @ C )
% 0.16/0.42 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % ord_eq_less_trans
% 0.16/0.42 thf(fact_1378_ord__eq__less__trans,axiom,
% 0.16/0.42 ! [A: int,B: int,C: int] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ( ( ord_less_int @ B @ C )
% 0.16/0.42 => ( ord_less_int @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % ord_eq_less_trans
% 0.16/0.42 thf(fact_1379_ord__less__eq__trans,axiom,
% 0.16/0.42 ! [A: real,B: real,C: real] :
% 0.16/0.42 ( ( ord_less_real @ A @ B )
% 0.16/0.42 => ( ( B = C )
% 0.16/0.42 => ( ord_less_real @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % ord_less_eq_trans
% 0.16/0.42 thf(fact_1380_ord__less__eq__trans,axiom,
% 0.16/0.42 ! [A: rat,B: rat,C: rat] :
% 0.16/0.42 ( ( ord_less_rat @ A @ B )
% 0.16/0.42 => ( ( B = C )
% 0.16/0.42 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % ord_less_eq_trans
% 0.16/0.42 thf(fact_1381_ord__less__eq__trans,axiom,
% 0.16/0.42 ! [A: num,B: num,C: num] :
% 0.16/0.42 ( ( ord_less_num @ A @ B )
% 0.16/0.42 => ( ( B = C )
% 0.16/0.42 => ( ord_less_num @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % ord_less_eq_trans
% 0.16/0.42 thf(fact_1382_ord__less__eq__trans,axiom,
% 0.16/0.42 ! [A: nat,B: nat,C: nat] :
% 0.16/0.42 ( ( ord_less_nat @ A @ B )
% 0.16/0.42 => ( ( B = C )
% 0.16/0.42 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % ord_less_eq_trans
% 0.16/0.42 thf(fact_1383_ord__less__eq__trans,axiom,
% 0.16/0.42 ! [A: int,B: int,C: int] :
% 0.16/0.42 ( ( ord_less_int @ A @ B )
% 0.16/0.42 => ( ( B = C )
% 0.16/0.42 => ( ord_less_int @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % ord_less_eq_trans
% 0.16/0.42 thf(fact_1384_less__induct,axiom,
% 0.16/0.42 ! [P: nat > $o,A: nat] :
% 0.16/0.42 ( ! [X4: nat] :
% 0.16/0.42 ( ! [Y3: nat] :
% 0.16/0.42 ( ( ord_less_nat @ Y3 @ X4 )
% 0.16/0.42 => ( P @ Y3 ) )
% 0.16/0.42 => ( P @ X4 ) )
% 0.16/0.42 => ( P @ A ) ) ).
% 0.16/0.42
% 0.16/0.42 % less_induct
% 0.16/0.42 thf(fact_1385_antisym__conv3,axiom,
% 0.16/0.42 ! [Y4: real,X: real] :
% 0.16/0.42 ( ~ ( ord_less_real @ Y4 @ X )
% 0.16/0.42 => ( ( ~ ( ord_less_real @ X @ Y4 ) )
% 0.16/0.42 = ( X = Y4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % antisym_conv3
% 0.16/0.42 thf(fact_1386_antisym__conv3,axiom,
% 0.16/0.42 ! [Y4: rat,X: rat] :
% 0.16/0.42 ( ~ ( ord_less_rat @ Y4 @ X )
% 0.16/0.42 => ( ( ~ ( ord_less_rat @ X @ Y4 ) )
% 0.16/0.42 = ( X = Y4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % antisym_conv3
% 0.16/0.42 thf(fact_1387_antisym__conv3,axiom,
% 0.16/0.42 ! [Y4: num,X: num] :
% 0.16/0.42 ( ~ ( ord_less_num @ Y4 @ X )
% 0.16/0.42 => ( ( ~ ( ord_less_num @ X @ Y4 ) )
% 0.16/0.42 = ( X = Y4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % antisym_conv3
% 0.16/0.42 thf(fact_1388_antisym__conv3,axiom,
% 0.16/0.42 ! [Y4: nat,X: nat] :
% 0.16/0.42 ( ~ ( ord_less_nat @ Y4 @ X )
% 0.16/0.42 => ( ( ~ ( ord_less_nat @ X @ Y4 ) )
% 0.16/0.42 = ( X = Y4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % antisym_conv3
% 0.16/0.42 thf(fact_1389_antisym__conv3,axiom,
% 0.16/0.42 ! [Y4: int,X: int] :
% 0.16/0.42 ( ~ ( ord_less_int @ Y4 @ X )
% 0.16/0.42 => ( ( ~ ( ord_less_int @ X @ Y4 ) )
% 0.16/0.42 = ( X = Y4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % antisym_conv3
% 0.16/0.42 thf(fact_1390_linorder__cases,axiom,
% 0.16/0.42 ! [X: real,Y4: real] :
% 0.16/0.42 ( ~ ( ord_less_real @ X @ Y4 )
% 0.16/0.42 => ( ( X != Y4 )
% 0.16/0.42 => ( ord_less_real @ Y4 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_cases
% 0.16/0.42 thf(fact_1391_linorder__cases,axiom,
% 0.16/0.42 ! [X: rat,Y4: rat] :
% 0.16/0.42 ( ~ ( ord_less_rat @ X @ Y4 )
% 0.16/0.42 => ( ( X != Y4 )
% 0.16/0.42 => ( ord_less_rat @ Y4 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_cases
% 0.16/0.42 thf(fact_1392_linorder__cases,axiom,
% 0.16/0.42 ! [X: num,Y4: num] :
% 0.16/0.42 ( ~ ( ord_less_num @ X @ Y4 )
% 0.16/0.42 => ( ( X != Y4 )
% 0.16/0.42 => ( ord_less_num @ Y4 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_cases
% 0.16/0.42 thf(fact_1393_linorder__cases,axiom,
% 0.16/0.42 ! [X: nat,Y4: nat] :
% 0.16/0.42 ( ~ ( ord_less_nat @ X @ Y4 )
% 0.16/0.42 => ( ( X != Y4 )
% 0.16/0.42 => ( ord_less_nat @ Y4 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_cases
% 0.16/0.42 thf(fact_1394_linorder__cases,axiom,
% 0.16/0.42 ! [X: int,Y4: int] :
% 0.16/0.42 ( ~ ( ord_less_int @ X @ Y4 )
% 0.16/0.42 => ( ( X != Y4 )
% 0.16/0.42 => ( ord_less_int @ Y4 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_cases
% 0.16/0.42 thf(fact_1395_dual__order_Oasym,axiom,
% 0.16/0.42 ! [B: real,A: real] :
% 0.16/0.42 ( ( ord_less_real @ B @ A )
% 0.16/0.42 => ~ ( ord_less_real @ A @ B ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.asym
% 0.16/0.42 thf(fact_1396_dual__order_Oasym,axiom,
% 0.16/0.42 ! [B: rat,A: rat] :
% 0.16/0.42 ( ( ord_less_rat @ B @ A )
% 0.16/0.42 => ~ ( ord_less_rat @ A @ B ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.asym
% 0.16/0.42 thf(fact_1397_dual__order_Oasym,axiom,
% 0.16/0.42 ! [B: num,A: num] :
% 0.16/0.42 ( ( ord_less_num @ B @ A )
% 0.16/0.42 => ~ ( ord_less_num @ A @ B ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.asym
% 0.16/0.42 thf(fact_1398_dual__order_Oasym,axiom,
% 0.16/0.42 ! [B: nat,A: nat] :
% 0.16/0.42 ( ( ord_less_nat @ B @ A )
% 0.16/0.42 => ~ ( ord_less_nat @ A @ B ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.asym
% 0.16/0.42 thf(fact_1399_dual__order_Oasym,axiom,
% 0.16/0.42 ! [B: int,A: int] :
% 0.16/0.42 ( ( ord_less_int @ B @ A )
% 0.16/0.42 => ~ ( ord_less_int @ A @ B ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.asym
% 0.16/0.42 thf(fact_1400_dual__order_Oirrefl,axiom,
% 0.16/0.42 ! [A: real] :
% 0.16/0.42 ~ ( ord_less_real @ A @ A ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.irrefl
% 0.16/0.42 thf(fact_1401_dual__order_Oirrefl,axiom,
% 0.16/0.42 ! [A: rat] :
% 0.16/0.42 ~ ( ord_less_rat @ A @ A ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.irrefl
% 0.16/0.42 thf(fact_1402_dual__order_Oirrefl,axiom,
% 0.16/0.42 ! [A: num] :
% 0.16/0.42 ~ ( ord_less_num @ A @ A ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.irrefl
% 0.16/0.42 thf(fact_1403_dual__order_Oirrefl,axiom,
% 0.16/0.42 ! [A: nat] :
% 0.16/0.42 ~ ( ord_less_nat @ A @ A ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.irrefl
% 0.16/0.42 thf(fact_1404_dual__order_Oirrefl,axiom,
% 0.16/0.42 ! [A: int] :
% 0.16/0.42 ~ ( ord_less_int @ A @ A ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.irrefl
% 0.16/0.42 thf(fact_1405_exists__least__iff,axiom,
% 0.16/0.42 ( ( ^ [P2: nat > $o] :
% 0.16/0.42 ? [X6: nat] : ( P2 @ X6 ) )
% 0.16/0.42 = ( ^ [P3: nat > $o] :
% 0.16/0.42 ? [N4: nat] :
% 0.16/0.42 ( ( P3 @ N4 )
% 0.16/0.42 & ! [M5: nat] :
% 0.16/0.42 ( ( ord_less_nat @ M5 @ N4 )
% 0.16/0.42 => ~ ( P3 @ M5 ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % exists_least_iff
% 0.16/0.42 thf(fact_1406_linorder__less__wlog,axiom,
% 0.16/0.42 ! [P: real > real > $o,A: real,B: real] :
% 0.16/0.42 ( ! [A4: real,B3: real] :
% 0.16/0.42 ( ( ord_less_real @ A4 @ B3 )
% 0.16/0.42 => ( P @ A4 @ B3 ) )
% 0.16/0.42 => ( ! [A4: real] : ( P @ A4 @ A4 )
% 0.16/0.42 => ( ! [A4: real,B3: real] :
% 0.16/0.42 ( ( P @ B3 @ A4 )
% 0.16/0.42 => ( P @ A4 @ B3 ) )
% 0.16/0.42 => ( P @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_less_wlog
% 0.16/0.42 thf(fact_1407_linorder__less__wlog,axiom,
% 0.16/0.42 ! [P: rat > rat > $o,A: rat,B: rat] :
% 0.16/0.42 ( ! [A4: rat,B3: rat] :
% 0.16/0.42 ( ( ord_less_rat @ A4 @ B3 )
% 0.16/0.42 => ( P @ A4 @ B3 ) )
% 0.16/0.42 => ( ! [A4: rat] : ( P @ A4 @ A4 )
% 0.16/0.42 => ( ! [A4: rat,B3: rat] :
% 0.16/0.42 ( ( P @ B3 @ A4 )
% 0.16/0.42 => ( P @ A4 @ B3 ) )
% 0.16/0.42 => ( P @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_less_wlog
% 0.16/0.42 thf(fact_1408_linorder__less__wlog,axiom,
% 0.16/0.42 ! [P: num > num > $o,A: num,B: num] :
% 0.16/0.42 ( ! [A4: num,B3: num] :
% 0.16/0.42 ( ( ord_less_num @ A4 @ B3 )
% 0.16/0.42 => ( P @ A4 @ B3 ) )
% 0.16/0.42 => ( ! [A4: num] : ( P @ A4 @ A4 )
% 0.16/0.42 => ( ! [A4: num,B3: num] :
% 0.16/0.42 ( ( P @ B3 @ A4 )
% 0.16/0.42 => ( P @ A4 @ B3 ) )
% 0.16/0.42 => ( P @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_less_wlog
% 0.16/0.42 thf(fact_1409_linorder__less__wlog,axiom,
% 0.16/0.42 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.16/0.42 ( ! [A4: nat,B3: nat] :
% 0.16/0.42 ( ( ord_less_nat @ A4 @ B3 )
% 0.16/0.42 => ( P @ A4 @ B3 ) )
% 0.16/0.42 => ( ! [A4: nat] : ( P @ A4 @ A4 )
% 0.16/0.42 => ( ! [A4: nat,B3: nat] :
% 0.16/0.42 ( ( P @ B3 @ A4 )
% 0.16/0.42 => ( P @ A4 @ B3 ) )
% 0.16/0.42 => ( P @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_less_wlog
% 0.16/0.42 thf(fact_1410_linorder__less__wlog,axiom,
% 0.16/0.42 ! [P: int > int > $o,A: int,B: int] :
% 0.16/0.42 ( ! [A4: int,B3: int] :
% 0.16/0.42 ( ( ord_less_int @ A4 @ B3 )
% 0.16/0.42 => ( P @ A4 @ B3 ) )
% 0.16/0.42 => ( ! [A4: int] : ( P @ A4 @ A4 )
% 0.16/0.42 => ( ! [A4: int,B3: int] :
% 0.16/0.42 ( ( P @ B3 @ A4 )
% 0.16/0.42 => ( P @ A4 @ B3 ) )
% 0.16/0.42 => ( P @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_less_wlog
% 0.16/0.42 thf(fact_1411_order_Ostrict__trans,axiom,
% 0.16/0.42 ! [A: real,B: real,C: real] :
% 0.16/0.42 ( ( ord_less_real @ A @ B )
% 0.16/0.42 => ( ( ord_less_real @ B @ C )
% 0.16/0.42 => ( ord_less_real @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.strict_trans
% 0.16/0.42 thf(fact_1412_order_Ostrict__trans,axiom,
% 0.16/0.42 ! [A: rat,B: rat,C: rat] :
% 0.16/0.42 ( ( ord_less_rat @ A @ B )
% 0.16/0.42 => ( ( ord_less_rat @ B @ C )
% 0.16/0.42 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.strict_trans
% 0.16/0.42 thf(fact_1413_order_Ostrict__trans,axiom,
% 0.16/0.42 ! [A: num,B: num,C: num] :
% 0.16/0.42 ( ( ord_less_num @ A @ B )
% 0.16/0.42 => ( ( ord_less_num @ B @ C )
% 0.16/0.42 => ( ord_less_num @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.strict_trans
% 0.16/0.42 thf(fact_1414_order_Ostrict__trans,axiom,
% 0.16/0.42 ! [A: nat,B: nat,C: nat] :
% 0.16/0.42 ( ( ord_less_nat @ A @ B )
% 0.16/0.42 => ( ( ord_less_nat @ B @ C )
% 0.16/0.42 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.strict_trans
% 0.16/0.42 thf(fact_1415_order_Ostrict__trans,axiom,
% 0.16/0.42 ! [A: int,B: int,C: int] :
% 0.16/0.42 ( ( ord_less_int @ A @ B )
% 0.16/0.42 => ( ( ord_less_int @ B @ C )
% 0.16/0.42 => ( ord_less_int @ A @ C ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.strict_trans
% 0.16/0.42 thf(fact_1416_not__less__iff__gr__or__eq,axiom,
% 0.16/0.42 ! [X: real,Y4: real] :
% 0.16/0.42 ( ( ~ ( ord_less_real @ X @ Y4 ) )
% 0.16/0.42 = ( ( ord_less_real @ Y4 @ X )
% 0.16/0.42 | ( X = Y4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % not_less_iff_gr_or_eq
% 0.16/0.42 thf(fact_1417_not__less__iff__gr__or__eq,axiom,
% 0.16/0.42 ! [X: rat,Y4: rat] :
% 0.16/0.42 ( ( ~ ( ord_less_rat @ X @ Y4 ) )
% 0.16/0.42 = ( ( ord_less_rat @ Y4 @ X )
% 0.16/0.42 | ( X = Y4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % not_less_iff_gr_or_eq
% 0.16/0.42 thf(fact_1418_not__less__iff__gr__or__eq,axiom,
% 0.16/0.42 ! [X: num,Y4: num] :
% 0.16/0.42 ( ( ~ ( ord_less_num @ X @ Y4 ) )
% 0.16/0.42 = ( ( ord_less_num @ Y4 @ X )
% 0.16/0.42 | ( X = Y4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % not_less_iff_gr_or_eq
% 0.16/0.42 thf(fact_1419_not__less__iff__gr__or__eq,axiom,
% 0.16/0.42 ! [X: nat,Y4: nat] :
% 0.16/0.42 ( ( ~ ( ord_less_nat @ X @ Y4 ) )
% 0.16/0.42 = ( ( ord_less_nat @ Y4 @ X )
% 0.16/0.42 | ( X = Y4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % not_less_iff_gr_or_eq
% 0.16/0.42 thf(fact_1420_not__less__iff__gr__or__eq,axiom,
% 0.16/0.42 ! [X: int,Y4: int] :
% 0.16/0.42 ( ( ~ ( ord_less_int @ X @ Y4 ) )
% 0.16/0.42 = ( ( ord_less_int @ Y4 @ X )
% 0.16/0.42 | ( X = Y4 ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % not_less_iff_gr_or_eq
% 0.16/0.42 thf(fact_1421_dual__order_Ostrict__trans,axiom,
% 0.16/0.42 ! [B: real,A: real,C: real] :
% 0.16/0.42 ( ( ord_less_real @ B @ A )
% 0.16/0.42 => ( ( ord_less_real @ C @ B )
% 0.16/0.42 => ( ord_less_real @ C @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.strict_trans
% 0.16/0.42 thf(fact_1422_dual__order_Ostrict__trans,axiom,
% 0.16/0.42 ! [B: rat,A: rat,C: rat] :
% 0.16/0.42 ( ( ord_less_rat @ B @ A )
% 0.16/0.42 => ( ( ord_less_rat @ C @ B )
% 0.16/0.42 => ( ord_less_rat @ C @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.strict_trans
% 0.16/0.42 thf(fact_1423_dual__order_Ostrict__trans,axiom,
% 0.16/0.42 ! [B: num,A: num,C: num] :
% 0.16/0.42 ( ( ord_less_num @ B @ A )
% 0.16/0.42 => ( ( ord_less_num @ C @ B )
% 0.16/0.42 => ( ord_less_num @ C @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.strict_trans
% 0.16/0.42 thf(fact_1424_dual__order_Ostrict__trans,axiom,
% 0.16/0.42 ! [B: nat,A: nat,C: nat] :
% 0.16/0.42 ( ( ord_less_nat @ B @ A )
% 0.16/0.42 => ( ( ord_less_nat @ C @ B )
% 0.16/0.42 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.strict_trans
% 0.16/0.42 thf(fact_1425_dual__order_Ostrict__trans,axiom,
% 0.16/0.42 ! [B: int,A: int,C: int] :
% 0.16/0.42 ( ( ord_less_int @ B @ A )
% 0.16/0.42 => ( ( ord_less_int @ C @ B )
% 0.16/0.42 => ( ord_less_int @ C @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.strict_trans
% 0.16/0.42 thf(fact_1426_order_Ostrict__implies__not__eq,axiom,
% 0.16/0.42 ! [A: real,B: real] :
% 0.16/0.42 ( ( ord_less_real @ A @ B )
% 0.16/0.42 => ( A != B ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.strict_implies_not_eq
% 0.16/0.42 thf(fact_1427_order_Ostrict__implies__not__eq,axiom,
% 0.16/0.42 ! [A: rat,B: rat] :
% 0.16/0.42 ( ( ord_less_rat @ A @ B )
% 0.16/0.42 => ( A != B ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.strict_implies_not_eq
% 0.16/0.42 thf(fact_1428_order_Ostrict__implies__not__eq,axiom,
% 0.16/0.42 ! [A: num,B: num] :
% 0.16/0.42 ( ( ord_less_num @ A @ B )
% 0.16/0.42 => ( A != B ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.strict_implies_not_eq
% 0.16/0.42 thf(fact_1429_order_Ostrict__implies__not__eq,axiom,
% 0.16/0.42 ! [A: nat,B: nat] :
% 0.16/0.42 ( ( ord_less_nat @ A @ B )
% 0.16/0.42 => ( A != B ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.strict_implies_not_eq
% 0.16/0.42 thf(fact_1430_order_Ostrict__implies__not__eq,axiom,
% 0.16/0.42 ! [A: int,B: int] :
% 0.16/0.42 ( ( ord_less_int @ A @ B )
% 0.16/0.42 => ( A != B ) ) ).
% 0.16/0.42
% 0.16/0.42 % order.strict_implies_not_eq
% 0.16/0.42 thf(fact_1431_dual__order_Ostrict__implies__not__eq,axiom,
% 0.16/0.42 ! [B: real,A: real] :
% 0.16/0.42 ( ( ord_less_real @ B @ A )
% 0.16/0.42 => ( A != B ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.strict_implies_not_eq
% 0.16/0.42 thf(fact_1432_dual__order_Ostrict__implies__not__eq,axiom,
% 0.16/0.42 ! [B: rat,A: rat] :
% 0.16/0.42 ( ( ord_less_rat @ B @ A )
% 0.16/0.42 => ( A != B ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.strict_implies_not_eq
% 0.16/0.42 thf(fact_1433_dual__order_Ostrict__implies__not__eq,axiom,
% 0.16/0.42 ! [B: num,A: num] :
% 0.16/0.42 ( ( ord_less_num @ B @ A )
% 0.16/0.42 => ( A != B ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.strict_implies_not_eq
% 0.16/0.42 thf(fact_1434_dual__order_Ostrict__implies__not__eq,axiom,
% 0.16/0.42 ! [B: nat,A: nat] :
% 0.16/0.42 ( ( ord_less_nat @ B @ A )
% 0.16/0.42 => ( A != B ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.strict_implies_not_eq
% 0.16/0.42 thf(fact_1435_dual__order_Ostrict__implies__not__eq,axiom,
% 0.16/0.42 ! [B: int,A: int] :
% 0.16/0.42 ( ( ord_less_int @ B @ A )
% 0.16/0.42 => ( A != B ) ) ).
% 0.16/0.42
% 0.16/0.42 % dual_order.strict_implies_not_eq
% 0.16/0.42 thf(fact_1436_linorder__neqE,axiom,
% 0.16/0.42 ! [X: real,Y4: real] :
% 0.16/0.42 ( ( X != Y4 )
% 0.16/0.42 => ( ~ ( ord_less_real @ X @ Y4 )
% 0.16/0.42 => ( ord_less_real @ Y4 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_neqE
% 0.16/0.42 thf(fact_1437_linorder__neqE,axiom,
% 0.16/0.42 ! [X: rat,Y4: rat] :
% 0.16/0.42 ( ( X != Y4 )
% 0.16/0.42 => ( ~ ( ord_less_rat @ X @ Y4 )
% 0.16/0.42 => ( ord_less_rat @ Y4 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_neqE
% 0.16/0.42 thf(fact_1438_linorder__neqE,axiom,
% 0.16/0.42 ! [X: num,Y4: num] :
% 0.16/0.42 ( ( X != Y4 )
% 0.16/0.42 => ( ~ ( ord_less_num @ X @ Y4 )
% 0.16/0.42 => ( ord_less_num @ Y4 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_neqE
% 0.16/0.42 thf(fact_1439_linorder__neqE,axiom,
% 0.16/0.42 ! [X: nat,Y4: nat] :
% 0.16/0.42 ( ( X != Y4 )
% 0.16/0.42 => ( ~ ( ord_less_nat @ X @ Y4 )
% 0.16/0.42 => ( ord_less_nat @ Y4 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_neqE
% 0.16/0.42 thf(fact_1440_linorder__neqE,axiom,
% 0.16/0.42 ! [X: int,Y4: int] :
% 0.16/0.42 ( ( X != Y4 )
% 0.16/0.42 => ( ~ ( ord_less_int @ X @ Y4 )
% 0.16/0.42 => ( ord_less_int @ Y4 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_neqE
% 0.16/0.42 thf(fact_1441_order__less__asym,axiom,
% 0.16/0.42 ! [X: real,Y4: real] :
% 0.16/0.42 ( ( ord_less_real @ X @ Y4 )
% 0.16/0.42 => ~ ( ord_less_real @ Y4 @ X ) ) ).
% 0.16/0.42
% 0.16/0.42 % order_less_asym
% 0.16/0.42 thf(fact_1442_order__less__asym,axiom,
% 0.16/0.42 ! [X: rat,Y4: rat] :
% 0.16/0.42 ( ( ord_less_rat @ X @ Y4 )
% 0.16/0.42 => ~ ( ord_less_rat @ Y4 @ X ) ) ).
% 0.16/0.42
% 0.16/0.42 % order_less_asym
% 0.16/0.42 thf(fact_1443_order__less__asym,axiom,
% 0.16/0.42 ! [X: num,Y4: num] :
% 0.16/0.42 ( ( ord_less_num @ X @ Y4 )
% 0.16/0.42 => ~ ( ord_less_num @ Y4 @ X ) ) ).
% 0.16/0.42
% 0.16/0.42 % order_less_asym
% 0.16/0.42 thf(fact_1444_order__less__asym,axiom,
% 0.16/0.42 ! [X: nat,Y4: nat] :
% 0.16/0.42 ( ( ord_less_nat @ X @ Y4 )
% 0.16/0.42 => ~ ( ord_less_nat @ Y4 @ X ) ) ).
% 0.16/0.42
% 0.16/0.42 % order_less_asym
% 0.16/0.42 thf(fact_1445_order__less__asym,axiom,
% 0.16/0.42 ! [X: int,Y4: int] :
% 0.16/0.42 ( ( ord_less_int @ X @ Y4 )
% 0.16/0.42 => ~ ( ord_less_int @ Y4 @ X ) ) ).
% 0.16/0.42
% 0.16/0.42 % order_less_asym
% 0.16/0.42 thf(fact_1446_linorder__neq__iff,axiom,
% 0.16/0.42 ! [X: real,Y4: real] :
% 0.16/0.42 ( ( X != Y4 )
% 0.16/0.42 = ( ( ord_less_real @ X @ Y4 )
% 0.16/0.42 | ( ord_less_real @ Y4 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_neq_iff
% 0.16/0.42 thf(fact_1447_linorder__neq__iff,axiom,
% 0.16/0.42 ! [X: rat,Y4: rat] :
% 0.16/0.42 ( ( X != Y4 )
% 0.16/0.42 = ( ( ord_less_rat @ X @ Y4 )
% 0.16/0.42 | ( ord_less_rat @ Y4 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_neq_iff
% 0.16/0.42 thf(fact_1448_linorder__neq__iff,axiom,
% 0.16/0.42 ! [X: num,Y4: num] :
% 0.16/0.42 ( ( X != Y4 )
% 0.16/0.42 = ( ( ord_less_num @ X @ Y4 )
% 0.16/0.42 | ( ord_less_num @ Y4 @ X ) ) ) ).
% 0.16/0.42
% 0.16/0.42 % linorder_neq_iff
% 0.16/0.42 thf(fact_1449_linorder__neq__iff,axiom,
% 0.16/0.42 ! [X: nat,Y4: nat] :
% 0.16/0.42 ( ( X != Y4 )
% 0.16/0.42 = ( ( ord_less_nat @ X @ Y4 )
% 0.16/0.42 | ( ord_less_nat @ Y4 @ X ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_neq_iff
% 0.16/0.43 thf(fact_1450_linorder__neq__iff,axiom,
% 0.16/0.43 ! [X: int,Y4: int] :
% 0.16/0.43 ( ( X != Y4 )
% 0.16/0.43 = ( ( ord_less_int @ X @ Y4 )
% 0.16/0.43 | ( ord_less_int @ Y4 @ X ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_neq_iff
% 0.16/0.43 thf(fact_1451_order__less__asym_H,axiom,
% 0.16/0.43 ! [A: real,B: real] :
% 0.16/0.43 ( ( ord_less_real @ A @ B )
% 0.16/0.43 => ~ ( ord_less_real @ B @ A ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_asym'
% 0.16/0.43 thf(fact_1452_order__less__asym_H,axiom,
% 0.16/0.43 ! [A: rat,B: rat] :
% 0.16/0.43 ( ( ord_less_rat @ A @ B )
% 0.16/0.43 => ~ ( ord_less_rat @ B @ A ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_asym'
% 0.16/0.43 thf(fact_1453_order__less__asym_H,axiom,
% 0.16/0.43 ! [A: num,B: num] :
% 0.16/0.43 ( ( ord_less_num @ A @ B )
% 0.16/0.43 => ~ ( ord_less_num @ B @ A ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_asym'
% 0.16/0.43 thf(fact_1454_order__less__asym_H,axiom,
% 0.16/0.43 ! [A: nat,B: nat] :
% 0.16/0.43 ( ( ord_less_nat @ A @ B )
% 0.16/0.43 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_asym'
% 0.16/0.43 thf(fact_1455_order__less__asym_H,axiom,
% 0.16/0.43 ! [A: int,B: int] :
% 0.16/0.43 ( ( ord_less_int @ A @ B )
% 0.16/0.43 => ~ ( ord_less_int @ B @ A ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_asym'
% 0.16/0.43 thf(fact_1456_order__less__trans,axiom,
% 0.16/0.43 ! [X: real,Y4: real,Z: real] :
% 0.16/0.43 ( ( ord_less_real @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_real @ Y4 @ Z )
% 0.16/0.43 => ( ord_less_real @ X @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_trans
% 0.16/0.43 thf(fact_1457_order__less__trans,axiom,
% 0.16/0.43 ! [X: rat,Y4: rat,Z: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_rat @ Y4 @ Z )
% 0.16/0.43 => ( ord_less_rat @ X @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_trans
% 0.16/0.43 thf(fact_1458_order__less__trans,axiom,
% 0.16/0.43 ! [X: num,Y4: num,Z: num] :
% 0.16/0.43 ( ( ord_less_num @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_num @ Y4 @ Z )
% 0.16/0.43 => ( ord_less_num @ X @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_trans
% 0.16/0.43 thf(fact_1459_order__less__trans,axiom,
% 0.16/0.43 ! [X: nat,Y4: nat,Z: nat] :
% 0.16/0.43 ( ( ord_less_nat @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_nat @ Y4 @ Z )
% 0.16/0.43 => ( ord_less_nat @ X @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_trans
% 0.16/0.43 thf(fact_1460_order__less__trans,axiom,
% 0.16/0.43 ! [X: int,Y4: int,Z: int] :
% 0.16/0.43 ( ( ord_less_int @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_int @ Y4 @ Z )
% 0.16/0.43 => ( ord_less_int @ X @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_trans
% 0.16/0.43 thf(fact_1461_ord__eq__less__subst,axiom,
% 0.16/0.43 ! [A: real,F: real > real,B: real,C: real] :
% 0.16/0.43 ( ( A
% 0.16/0.43 = ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_real @ B @ C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_eq_less_subst
% 0.16/0.43 thf(fact_1462_ord__eq__less__subst,axiom,
% 0.16/0.43 ! [A: rat,F: real > rat,B: real,C: real] :
% 0.16/0.43 ( ( A
% 0.16/0.43 = ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_real @ B @ C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_eq_less_subst
% 0.16/0.43 thf(fact_1463_ord__eq__less__subst,axiom,
% 0.16/0.43 ! [A: num,F: real > num,B: real,C: real] :
% 0.16/0.43 ( ( A
% 0.16/0.43 = ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_real @ B @ C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_eq_less_subst
% 0.16/0.43 thf(fact_1464_ord__eq__less__subst,axiom,
% 0.16/0.43 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.16/0.43 ( ( A
% 0.16/0.43 = ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_real @ B @ C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_eq_less_subst
% 0.16/0.43 thf(fact_1465_ord__eq__less__subst,axiom,
% 0.16/0.43 ! [A: int,F: real > int,B: real,C: real] :
% 0.16/0.43 ( ( A
% 0.16/0.43 = ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_real @ B @ C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_eq_less_subst
% 0.16/0.43 thf(fact_1466_ord__eq__less__subst,axiom,
% 0.16/0.43 ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.16/0.43 ( ( A
% 0.16/0.43 = ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_rat @ B @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_eq_less_subst
% 0.16/0.43 thf(fact_1467_ord__eq__less__subst,axiom,
% 0.16/0.43 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.16/0.43 ( ( A
% 0.16/0.43 = ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_rat @ B @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_eq_less_subst
% 0.16/0.43 thf(fact_1468_ord__eq__less__subst,axiom,
% 0.16/0.43 ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.16/0.43 ( ( A
% 0.16/0.43 = ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_rat @ B @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_eq_less_subst
% 0.16/0.43 thf(fact_1469_ord__eq__less__subst,axiom,
% 0.16/0.43 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.16/0.43 ( ( A
% 0.16/0.43 = ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_rat @ B @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_eq_less_subst
% 0.16/0.43 thf(fact_1470_ord__eq__less__subst,axiom,
% 0.16/0.43 ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.16/0.43 ( ( A
% 0.16/0.43 = ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_rat @ B @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_eq_less_subst
% 0.16/0.43 thf(fact_1471_ord__less__eq__subst,axiom,
% 0.16/0.43 ! [A: real,B: real,F: real > real,C: real] :
% 0.16/0.43 ( ( ord_less_real @ A @ B )
% 0.16/0.43 => ( ( ( F @ B )
% 0.16/0.43 = C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_less_eq_subst
% 0.16/0.43 thf(fact_1472_ord__less__eq__subst,axiom,
% 0.16/0.43 ! [A: real,B: real,F: real > rat,C: rat] :
% 0.16/0.43 ( ( ord_less_real @ A @ B )
% 0.16/0.43 => ( ( ( F @ B )
% 0.16/0.43 = C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_less_eq_subst
% 0.16/0.43 thf(fact_1473_ord__less__eq__subst,axiom,
% 0.16/0.43 ! [A: real,B: real,F: real > num,C: num] :
% 0.16/0.43 ( ( ord_less_real @ A @ B )
% 0.16/0.43 => ( ( ( F @ B )
% 0.16/0.43 = C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_less_eq_subst
% 0.16/0.43 thf(fact_1474_ord__less__eq__subst,axiom,
% 0.16/0.43 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.16/0.43 ( ( ord_less_real @ A @ B )
% 0.16/0.43 => ( ( ( F @ B )
% 0.16/0.43 = C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_less_eq_subst
% 0.16/0.43 thf(fact_1475_ord__less__eq__subst,axiom,
% 0.16/0.43 ! [A: real,B: real,F: real > int,C: int] :
% 0.16/0.43 ( ( ord_less_real @ A @ B )
% 0.16/0.43 => ( ( ( F @ B )
% 0.16/0.43 = C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_less_eq_subst
% 0.16/0.43 thf(fact_1476_ord__less__eq__subst,axiom,
% 0.16/0.43 ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.16/0.43 ( ( ord_less_rat @ A @ B )
% 0.16/0.43 => ( ( ( F @ B )
% 0.16/0.43 = C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_less_eq_subst
% 0.16/0.43 thf(fact_1477_ord__less__eq__subst,axiom,
% 0.16/0.43 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.16/0.43 ( ( ord_less_rat @ A @ B )
% 0.16/0.43 => ( ( ( F @ B )
% 0.16/0.43 = C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_less_eq_subst
% 0.16/0.43 thf(fact_1478_ord__less__eq__subst,axiom,
% 0.16/0.43 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.16/0.43 ( ( ord_less_rat @ A @ B )
% 0.16/0.43 => ( ( ( F @ B )
% 0.16/0.43 = C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_less_eq_subst
% 0.16/0.43 thf(fact_1479_ord__less__eq__subst,axiom,
% 0.16/0.43 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.16/0.43 ( ( ord_less_rat @ A @ B )
% 0.16/0.43 => ( ( ( F @ B )
% 0.16/0.43 = C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_less_eq_subst
% 0.16/0.43 thf(fact_1480_ord__less__eq__subst,axiom,
% 0.16/0.43 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.16/0.43 ( ( ord_less_rat @ A @ B )
% 0.16/0.43 => ( ( ( F @ B )
% 0.16/0.43 = C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ord_less_eq_subst
% 0.16/0.43 thf(fact_1481_order__less__irrefl,axiom,
% 0.16/0.43 ! [X: real] :
% 0.16/0.43 ~ ( ord_less_real @ X @ X ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_irrefl
% 0.16/0.43 thf(fact_1482_order__less__irrefl,axiom,
% 0.16/0.43 ! [X: rat] :
% 0.16/0.43 ~ ( ord_less_rat @ X @ X ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_irrefl
% 0.16/0.43 thf(fact_1483_order__less__irrefl,axiom,
% 0.16/0.43 ! [X: num] :
% 0.16/0.43 ~ ( ord_less_num @ X @ X ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_irrefl
% 0.16/0.43 thf(fact_1484_order__less__irrefl,axiom,
% 0.16/0.43 ! [X: nat] :
% 0.16/0.43 ~ ( ord_less_nat @ X @ X ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_irrefl
% 0.16/0.43 thf(fact_1485_order__less__irrefl,axiom,
% 0.16/0.43 ! [X: int] :
% 0.16/0.43 ~ ( ord_less_int @ X @ X ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_irrefl
% 0.16/0.43 thf(fact_1486_order__less__subst1,axiom,
% 0.16/0.43 ! [A: real,F: real > real,B: real,C: real] :
% 0.16/0.43 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_real @ B @ C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst1
% 0.16/0.43 thf(fact_1487_order__less__subst1,axiom,
% 0.16/0.43 ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.16/0.43 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_rat @ B @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst1
% 0.16/0.43 thf(fact_1488_order__less__subst1,axiom,
% 0.16/0.43 ! [A: real,F: num > real,B: num,C: num] :
% 0.16/0.43 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_num @ B @ C )
% 0.16/0.43 => ( ! [X4: num,Y: num] :
% 0.16/0.43 ( ( ord_less_num @ X4 @ Y )
% 0.16/0.43 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst1
% 0.16/0.43 thf(fact_1489_order__less__subst1,axiom,
% 0.16/0.43 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.16/0.43 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_nat @ B @ C )
% 0.16/0.43 => ( ! [X4: nat,Y: nat] :
% 0.16/0.43 ( ( ord_less_nat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst1
% 0.16/0.43 thf(fact_1490_order__less__subst1,axiom,
% 0.16/0.43 ! [A: real,F: int > real,B: int,C: int] :
% 0.16/0.43 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_int @ B @ C )
% 0.16/0.43 => ( ! [X4: int,Y: int] :
% 0.16/0.43 ( ( ord_less_int @ X4 @ Y )
% 0.16/0.43 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst1
% 0.16/0.43 thf(fact_1491_order__less__subst1,axiom,
% 0.16/0.43 ! [A: rat,F: real > rat,B: real,C: real] :
% 0.16/0.43 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_real @ B @ C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst1
% 0.16/0.43 thf(fact_1492_order__less__subst1,axiom,
% 0.16/0.43 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.16/0.43 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_rat @ B @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst1
% 0.16/0.43 thf(fact_1493_order__less__subst1,axiom,
% 0.16/0.43 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.16/0.43 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_num @ B @ C )
% 0.16/0.43 => ( ! [X4: num,Y: num] :
% 0.16/0.43 ( ( ord_less_num @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst1
% 0.16/0.43 thf(fact_1494_order__less__subst1,axiom,
% 0.16/0.43 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.16/0.43 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_nat @ B @ C )
% 0.16/0.43 => ( ! [X4: nat,Y: nat] :
% 0.16/0.43 ( ( ord_less_nat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst1
% 0.16/0.43 thf(fact_1495_order__less__subst1,axiom,
% 0.16/0.43 ! [A: rat,F: int > rat,B: int,C: int] :
% 0.16/0.43 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_int @ B @ C )
% 0.16/0.43 => ( ! [X4: int,Y: int] :
% 0.16/0.43 ( ( ord_less_int @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst1
% 0.16/0.43 thf(fact_1496_order__less__subst2,axiom,
% 0.16/0.43 ! [A: real,B: real,F: real > real,C: real] :
% 0.16/0.43 ( ( ord_less_real @ A @ B )
% 0.16/0.43 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst2
% 0.16/0.43 thf(fact_1497_order__less__subst2,axiom,
% 0.16/0.43 ! [A: real,B: real,F: real > rat,C: rat] :
% 0.16/0.43 ( ( ord_less_real @ A @ B )
% 0.16/0.43 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst2
% 0.16/0.43 thf(fact_1498_order__less__subst2,axiom,
% 0.16/0.43 ! [A: real,B: real,F: real > num,C: num] :
% 0.16/0.43 ( ( ord_less_real @ A @ B )
% 0.16/0.43 => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst2
% 0.16/0.43 thf(fact_1499_order__less__subst2,axiom,
% 0.16/0.43 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.16/0.43 ( ( ord_less_real @ A @ B )
% 0.16/0.43 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst2
% 0.16/0.43 thf(fact_1500_order__less__subst2,axiom,
% 0.16/0.43 ! [A: real,B: real,F: real > int,C: int] :
% 0.16/0.43 ( ( ord_less_real @ A @ B )
% 0.16/0.43 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst2
% 0.16/0.43 thf(fact_1501_order__less__subst2,axiom,
% 0.16/0.43 ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.16/0.43 ( ( ord_less_rat @ A @ B )
% 0.16/0.43 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst2
% 0.16/0.43 thf(fact_1502_order__less__subst2,axiom,
% 0.16/0.43 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.16/0.43 ( ( ord_less_rat @ A @ B )
% 0.16/0.43 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst2
% 0.16/0.43 thf(fact_1503_order__less__subst2,axiom,
% 0.16/0.43 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.16/0.43 ( ( ord_less_rat @ A @ B )
% 0.16/0.43 => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst2
% 0.16/0.43 thf(fact_1504_order__less__subst2,axiom,
% 0.16/0.43 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.16/0.43 ( ( ord_less_rat @ A @ B )
% 0.16/0.43 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst2
% 0.16/0.43 thf(fact_1505_order__less__subst2,axiom,
% 0.16/0.43 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.16/0.43 ( ( ord_less_rat @ A @ B )
% 0.16/0.43 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_subst2
% 0.16/0.43 thf(fact_1506_order__less__not__sym,axiom,
% 0.16/0.43 ! [X: real,Y4: real] :
% 0.16/0.43 ( ( ord_less_real @ X @ Y4 )
% 0.16/0.43 => ~ ( ord_less_real @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_not_sym
% 0.16/0.43 thf(fact_1507_order__less__not__sym,axiom,
% 0.16/0.43 ! [X: rat,Y4: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X @ Y4 )
% 0.16/0.43 => ~ ( ord_less_rat @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_not_sym
% 0.16/0.43 thf(fact_1508_order__less__not__sym,axiom,
% 0.16/0.43 ! [X: num,Y4: num] :
% 0.16/0.43 ( ( ord_less_num @ X @ Y4 )
% 0.16/0.43 => ~ ( ord_less_num @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_not_sym
% 0.16/0.43 thf(fact_1509_order__less__not__sym,axiom,
% 0.16/0.43 ! [X: nat,Y4: nat] :
% 0.16/0.43 ( ( ord_less_nat @ X @ Y4 )
% 0.16/0.43 => ~ ( ord_less_nat @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_not_sym
% 0.16/0.43 thf(fact_1510_order__less__not__sym,axiom,
% 0.16/0.43 ! [X: int,Y4: int] :
% 0.16/0.43 ( ( ord_less_int @ X @ Y4 )
% 0.16/0.43 => ~ ( ord_less_int @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_not_sym
% 0.16/0.43 thf(fact_1511_order__less__imp__triv,axiom,
% 0.16/0.43 ! [X: real,Y4: real,P: $o] :
% 0.16/0.43 ( ( ord_less_real @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_real @ Y4 @ X )
% 0.16/0.43 => P ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_triv
% 0.16/0.43 thf(fact_1512_order__less__imp__triv,axiom,
% 0.16/0.43 ! [X: rat,Y4: rat,P: $o] :
% 0.16/0.43 ( ( ord_less_rat @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_rat @ Y4 @ X )
% 0.16/0.43 => P ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_triv
% 0.16/0.43 thf(fact_1513_order__less__imp__triv,axiom,
% 0.16/0.43 ! [X: num,Y4: num,P: $o] :
% 0.16/0.43 ( ( ord_less_num @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_num @ Y4 @ X )
% 0.16/0.43 => P ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_triv
% 0.16/0.43 thf(fact_1514_order__less__imp__triv,axiom,
% 0.16/0.43 ! [X: nat,Y4: nat,P: $o] :
% 0.16/0.43 ( ( ord_less_nat @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_nat @ Y4 @ X )
% 0.16/0.43 => P ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_triv
% 0.16/0.43 thf(fact_1515_order__less__imp__triv,axiom,
% 0.16/0.43 ! [X: int,Y4: int,P: $o] :
% 0.16/0.43 ( ( ord_less_int @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_int @ Y4 @ X )
% 0.16/0.43 => P ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_triv
% 0.16/0.43 thf(fact_1516_linorder__less__linear,axiom,
% 0.16/0.43 ! [X: real,Y4: real] :
% 0.16/0.43 ( ( ord_less_real @ X @ Y4 )
% 0.16/0.43 | ( X = Y4 )
% 0.16/0.43 | ( ord_less_real @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_less_linear
% 0.16/0.43 thf(fact_1517_linorder__less__linear,axiom,
% 0.16/0.43 ! [X: rat,Y4: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X @ Y4 )
% 0.16/0.43 | ( X = Y4 )
% 0.16/0.43 | ( ord_less_rat @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_less_linear
% 0.16/0.43 thf(fact_1518_linorder__less__linear,axiom,
% 0.16/0.43 ! [X: num,Y4: num] :
% 0.16/0.43 ( ( ord_less_num @ X @ Y4 )
% 0.16/0.43 | ( X = Y4 )
% 0.16/0.43 | ( ord_less_num @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_less_linear
% 0.16/0.43 thf(fact_1519_linorder__less__linear,axiom,
% 0.16/0.43 ! [X: nat,Y4: nat] :
% 0.16/0.43 ( ( ord_less_nat @ X @ Y4 )
% 0.16/0.43 | ( X = Y4 )
% 0.16/0.43 | ( ord_less_nat @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_less_linear
% 0.16/0.43 thf(fact_1520_linorder__less__linear,axiom,
% 0.16/0.43 ! [X: int,Y4: int] :
% 0.16/0.43 ( ( ord_less_int @ X @ Y4 )
% 0.16/0.43 | ( X = Y4 )
% 0.16/0.43 | ( ord_less_int @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_less_linear
% 0.16/0.43 thf(fact_1521_order__less__imp__not__eq,axiom,
% 0.16/0.43 ! [X: real,Y4: real] :
% 0.16/0.43 ( ( ord_less_real @ X @ Y4 )
% 0.16/0.43 => ( X != Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_not_eq
% 0.16/0.43 thf(fact_1522_order__less__imp__not__eq,axiom,
% 0.16/0.43 ! [X: rat,Y4: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X @ Y4 )
% 0.16/0.43 => ( X != Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_not_eq
% 0.16/0.43 thf(fact_1523_order__less__imp__not__eq,axiom,
% 0.16/0.43 ! [X: num,Y4: num] :
% 0.16/0.43 ( ( ord_less_num @ X @ Y4 )
% 0.16/0.43 => ( X != Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_not_eq
% 0.16/0.43 thf(fact_1524_order__less__imp__not__eq,axiom,
% 0.16/0.43 ! [X: nat,Y4: nat] :
% 0.16/0.43 ( ( ord_less_nat @ X @ Y4 )
% 0.16/0.43 => ( X != Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_not_eq
% 0.16/0.43 thf(fact_1525_order__less__imp__not__eq,axiom,
% 0.16/0.43 ! [X: int,Y4: int] :
% 0.16/0.43 ( ( ord_less_int @ X @ Y4 )
% 0.16/0.43 => ( X != Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_not_eq
% 0.16/0.43 thf(fact_1526_order__less__imp__not__eq2,axiom,
% 0.16/0.43 ! [X: real,Y4: real] :
% 0.16/0.43 ( ( ord_less_real @ X @ Y4 )
% 0.16/0.43 => ( Y4 != X ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_not_eq2
% 0.16/0.43 thf(fact_1527_order__less__imp__not__eq2,axiom,
% 0.16/0.43 ! [X: rat,Y4: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X @ Y4 )
% 0.16/0.43 => ( Y4 != X ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_not_eq2
% 0.16/0.43 thf(fact_1528_order__less__imp__not__eq2,axiom,
% 0.16/0.43 ! [X: num,Y4: num] :
% 0.16/0.43 ( ( ord_less_num @ X @ Y4 )
% 0.16/0.43 => ( Y4 != X ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_not_eq2
% 0.16/0.43 thf(fact_1529_order__less__imp__not__eq2,axiom,
% 0.16/0.43 ! [X: nat,Y4: nat] :
% 0.16/0.43 ( ( ord_less_nat @ X @ Y4 )
% 0.16/0.43 => ( Y4 != X ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_not_eq2
% 0.16/0.43 thf(fact_1530_order__less__imp__not__eq2,axiom,
% 0.16/0.43 ! [X: int,Y4: int] :
% 0.16/0.43 ( ( ord_less_int @ X @ Y4 )
% 0.16/0.43 => ( Y4 != X ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_not_eq2
% 0.16/0.43 thf(fact_1531_order__less__imp__not__less,axiom,
% 0.16/0.43 ! [X: real,Y4: real] :
% 0.16/0.43 ( ( ord_less_real @ X @ Y4 )
% 0.16/0.43 => ~ ( ord_less_real @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_not_less
% 0.16/0.43 thf(fact_1532_order__less__imp__not__less,axiom,
% 0.16/0.43 ! [X: rat,Y4: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X @ Y4 )
% 0.16/0.43 => ~ ( ord_less_rat @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_not_less
% 0.16/0.43 thf(fact_1533_order__less__imp__not__less,axiom,
% 0.16/0.43 ! [X: num,Y4: num] :
% 0.16/0.43 ( ( ord_less_num @ X @ Y4 )
% 0.16/0.43 => ~ ( ord_less_num @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_not_less
% 0.16/0.43 thf(fact_1534_order__less__imp__not__less,axiom,
% 0.16/0.43 ! [X: nat,Y4: nat] :
% 0.16/0.43 ( ( ord_less_nat @ X @ Y4 )
% 0.16/0.43 => ~ ( ord_less_nat @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_not_less
% 0.16/0.43 thf(fact_1535_order__less__imp__not__less,axiom,
% 0.16/0.43 ! [X: int,Y4: int] :
% 0.16/0.43 ( ( ord_less_int @ X @ Y4 )
% 0.16/0.43 => ~ ( ord_less_int @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_not_less
% 0.16/0.43 thf(fact_1536_less__numeral__extra_I1_J,axiom,
% 0.16/0.43 ord_less_real @ zero_zero_real @ one_one_real ).
% 0.16/0.43
% 0.16/0.43 % less_numeral_extra(1)
% 0.16/0.43 thf(fact_1537_less__numeral__extra_I1_J,axiom,
% 0.16/0.43 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.16/0.43
% 0.16/0.43 % less_numeral_extra(1)
% 0.16/0.43 thf(fact_1538_less__numeral__extra_I1_J,axiom,
% 0.16/0.43 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.16/0.43
% 0.16/0.43 % less_numeral_extra(1)
% 0.16/0.43 thf(fact_1539_less__numeral__extra_I1_J,axiom,
% 0.16/0.43 ord_less_int @ zero_zero_int @ one_one_int ).
% 0.16/0.43
% 0.16/0.43 % less_numeral_extra(1)
% 0.16/0.43 thf(fact_1540_zero__less__one,axiom,
% 0.16/0.43 ord_less_real @ zero_zero_real @ one_one_real ).
% 0.16/0.43
% 0.16/0.43 % zero_less_one
% 0.16/0.43 thf(fact_1541_zero__less__one,axiom,
% 0.16/0.43 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.16/0.43
% 0.16/0.43 % zero_less_one
% 0.16/0.43 thf(fact_1542_zero__less__one,axiom,
% 0.16/0.43 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.16/0.43
% 0.16/0.43 % zero_less_one
% 0.16/0.43 thf(fact_1543_zero__less__one,axiom,
% 0.16/0.43 ord_less_int @ zero_zero_int @ one_one_int ).
% 0.16/0.43
% 0.16/0.43 % zero_less_one
% 0.16/0.43 thf(fact_1544_not__one__less__zero,axiom,
% 0.16/0.43 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 0.16/0.43
% 0.16/0.43 % not_one_less_zero
% 0.16/0.43 thf(fact_1545_not__one__less__zero,axiom,
% 0.16/0.43 ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.16/0.43
% 0.16/0.43 % not_one_less_zero
% 0.16/0.43 thf(fact_1546_not__one__less__zero,axiom,
% 0.16/0.43 ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.16/0.43
% 0.16/0.43 % not_one_less_zero
% 0.16/0.43 thf(fact_1547_not__one__less__zero,axiom,
% 0.16/0.43 ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 0.16/0.43
% 0.16/0.43 % not_one_less_zero
% 0.16/0.43 thf(fact_1548_not__numeral__less__one,axiom,
% 0.16/0.43 ! [N: num] :
% 0.16/0.43 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% 0.16/0.43
% 0.16/0.43 % not_numeral_less_one
% 0.16/0.43 thf(fact_1549_not__numeral__less__one,axiom,
% 0.16/0.43 ! [N: num] :
% 0.16/0.43 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% 0.16/0.43
% 0.16/0.43 % not_numeral_less_one
% 0.16/0.43 thf(fact_1550_not__numeral__less__one,axiom,
% 0.16/0.43 ! [N: num] :
% 0.16/0.43 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% 0.16/0.43
% 0.16/0.43 % not_numeral_less_one
% 0.16/0.43 thf(fact_1551_not__numeral__less__one,axiom,
% 0.16/0.43 ! [N: num] :
% 0.16/0.43 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% 0.16/0.43
% 0.16/0.43 % not_numeral_less_one
% 0.16/0.43 thf(fact_1552_less__1__mult,axiom,
% 0.16/0.43 ! [M: real,N: real] :
% 0.16/0.43 ( ( ord_less_real @ one_one_real @ M )
% 0.16/0.43 => ( ( ord_less_real @ one_one_real @ N )
% 0.16/0.43 => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_1_mult
% 0.16/0.43 thf(fact_1553_less__1__mult,axiom,
% 0.16/0.43 ! [M: rat,N: rat] :
% 0.16/0.43 ( ( ord_less_rat @ one_one_rat @ M )
% 0.16/0.43 => ( ( ord_less_rat @ one_one_rat @ N )
% 0.16/0.43 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_1_mult
% 0.16/0.43 thf(fact_1554_less__1__mult,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ one_one_nat @ M )
% 0.16/0.43 => ( ( ord_less_nat @ one_one_nat @ N )
% 0.16/0.43 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_1_mult
% 0.16/0.43 thf(fact_1555_less__1__mult,axiom,
% 0.16/0.43 ! [M: int,N: int] :
% 0.16/0.43 ( ( ord_less_int @ one_one_int @ M )
% 0.16/0.43 => ( ( ord_less_int @ one_one_int @ N )
% 0.16/0.43 => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_1_mult
% 0.16/0.43 thf(fact_1556_lift__Suc__mono__less__iff,axiom,
% 0.16/0.43 ! [F: nat > real,N: nat,M: nat] :
% 0.16/0.43 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.16/0.43 => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 0.16/0.43 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % lift_Suc_mono_less_iff
% 0.16/0.43 thf(fact_1557_lift__Suc__mono__less__iff,axiom,
% 0.16/0.43 ! [F: nat > rat,N: nat,M: nat] :
% 0.16/0.43 ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.16/0.43 => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
% 0.16/0.43 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % lift_Suc_mono_less_iff
% 0.16/0.43 thf(fact_1558_lift__Suc__mono__less__iff,axiom,
% 0.16/0.43 ! [F: nat > num,N: nat,M: nat] :
% 0.16/0.43 ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.16/0.43 => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
% 0.16/0.43 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % lift_Suc_mono_less_iff
% 0.16/0.43 thf(fact_1559_lift__Suc__mono__less__iff,axiom,
% 0.16/0.43 ! [F: nat > nat,N: nat,M: nat] :
% 0.16/0.43 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.16/0.43 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 0.16/0.43 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % lift_Suc_mono_less_iff
% 0.16/0.43 thf(fact_1560_lift__Suc__mono__less__iff,axiom,
% 0.16/0.43 ! [F: nat > int,N: nat,M: nat] :
% 0.16/0.43 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.16/0.43 => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 0.16/0.43 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % lift_Suc_mono_less_iff
% 0.16/0.43 thf(fact_1561_lift__Suc__mono__less,axiom,
% 0.16/0.43 ! [F: nat > real,N: nat,N3: nat] :
% 0.16/0.43 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.16/0.43 => ( ( ord_less_nat @ N @ N3 )
% 0.16/0.43 => ( ord_less_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % lift_Suc_mono_less
% 0.16/0.43 thf(fact_1562_lift__Suc__mono__less,axiom,
% 0.16/0.43 ! [F: nat > rat,N: nat,N3: nat] :
% 0.16/0.43 ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.16/0.43 => ( ( ord_less_nat @ N @ N3 )
% 0.16/0.43 => ( ord_less_rat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % lift_Suc_mono_less
% 0.16/0.43 thf(fact_1563_lift__Suc__mono__less,axiom,
% 0.16/0.43 ! [F: nat > num,N: nat,N3: nat] :
% 0.16/0.43 ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.16/0.43 => ( ( ord_less_nat @ N @ N3 )
% 0.16/0.43 => ( ord_less_num @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % lift_Suc_mono_less
% 0.16/0.43 thf(fact_1564_lift__Suc__mono__less,axiom,
% 0.16/0.43 ! [F: nat > nat,N: nat,N3: nat] :
% 0.16/0.43 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.16/0.43 => ( ( ord_less_nat @ N @ N3 )
% 0.16/0.43 => ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % lift_Suc_mono_less
% 0.16/0.43 thf(fact_1565_lift__Suc__mono__less,axiom,
% 0.16/0.43 ! [F: nat > int,N: nat,N3: nat] :
% 0.16/0.43 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.16/0.43 => ( ( ord_less_nat @ N @ N3 )
% 0.16/0.43 => ( ord_less_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % lift_Suc_mono_less
% 0.16/0.43 thf(fact_1566_of__nat__less__imp__less,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.16/0.43 => ( ord_less_nat @ M @ N ) ) ).
% 0.16/0.43
% 0.16/0.43 % of_nat_less_imp_less
% 0.16/0.43 thf(fact_1567_of__nat__less__imp__less,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.16/0.43 => ( ord_less_nat @ M @ N ) ) ).
% 0.16/0.43
% 0.16/0.43 % of_nat_less_imp_less
% 0.16/0.43 thf(fact_1568_of__nat__less__imp__less,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.16/0.43 => ( ord_less_nat @ M @ N ) ) ).
% 0.16/0.43
% 0.16/0.43 % of_nat_less_imp_less
% 0.16/0.43 thf(fact_1569_of__nat__less__imp__less,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.16/0.43 => ( ord_less_nat @ M @ N ) ) ).
% 0.16/0.43
% 0.16/0.43 % of_nat_less_imp_less
% 0.16/0.43 thf(fact_1570_less__imp__of__nat__less,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ M @ N )
% 0.16/0.43 => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_imp_of_nat_less
% 0.16/0.43 thf(fact_1571_less__imp__of__nat__less,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ M @ N )
% 0.16/0.43 => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_imp_of_nat_less
% 0.16/0.43 thf(fact_1572_less__imp__of__nat__less,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ M @ N )
% 0.16/0.43 => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_imp_of_nat_less
% 0.16/0.43 thf(fact_1573_less__imp__of__nat__less,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ M @ N )
% 0.16/0.43 => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_imp_of_nat_less
% 0.16/0.43 thf(fact_1574_minf_I8_J,axiom,
% 0.16/0.43 ! [T: rat] :
% 0.16/0.43 ? [Z2: rat] :
% 0.16/0.43 ! [X5: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X5 @ Z2 )
% 0.16/0.43 => ~ ( ord_less_eq_rat @ T @ X5 ) ) ).
% 0.16/0.43
% 0.16/0.43 % minf(8)
% 0.16/0.43 thf(fact_1575_minf_I8_J,axiom,
% 0.16/0.43 ! [T: num] :
% 0.16/0.43 ? [Z2: num] :
% 0.16/0.43 ! [X5: num] :
% 0.16/0.43 ( ( ord_less_num @ X5 @ Z2 )
% 0.16/0.43 => ~ ( ord_less_eq_num @ T @ X5 ) ) ).
% 0.16/0.43
% 0.16/0.43 % minf(8)
% 0.16/0.43 thf(fact_1576_minf_I8_J,axiom,
% 0.16/0.43 ! [T: nat] :
% 0.16/0.43 ? [Z2: nat] :
% 0.16/0.43 ! [X5: nat] :
% 0.16/0.43 ( ( ord_less_nat @ X5 @ Z2 )
% 0.16/0.43 => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% 0.16/0.43
% 0.16/0.43 % minf(8)
% 0.16/0.43 thf(fact_1577_minf_I8_J,axiom,
% 0.16/0.43 ! [T: int] :
% 0.16/0.43 ? [Z2: int] :
% 0.16/0.43 ! [X5: int] :
% 0.16/0.43 ( ( ord_less_int @ X5 @ Z2 )
% 0.16/0.43 => ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% 0.16/0.43
% 0.16/0.43 % minf(8)
% 0.16/0.43 thf(fact_1578_minf_I8_J,axiom,
% 0.16/0.43 ! [T: real] :
% 0.16/0.43 ? [Z2: real] :
% 0.16/0.43 ! [X5: real] :
% 0.16/0.43 ( ( ord_less_real @ X5 @ Z2 )
% 0.16/0.43 => ~ ( ord_less_eq_real @ T @ X5 ) ) ).
% 0.16/0.43
% 0.16/0.43 % minf(8)
% 0.16/0.43 thf(fact_1579_minf_I6_J,axiom,
% 0.16/0.43 ! [T: rat] :
% 0.16/0.43 ? [Z2: rat] :
% 0.16/0.43 ! [X5: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X5 @ Z2 )
% 0.16/0.43 => ( ord_less_eq_rat @ X5 @ T ) ) ).
% 0.16/0.43
% 0.16/0.43 % minf(6)
% 0.16/0.43 thf(fact_1580_minf_I6_J,axiom,
% 0.16/0.43 ! [T: num] :
% 0.16/0.43 ? [Z2: num] :
% 0.16/0.43 ! [X5: num] :
% 0.16/0.43 ( ( ord_less_num @ X5 @ Z2 )
% 0.16/0.43 => ( ord_less_eq_num @ X5 @ T ) ) ).
% 0.16/0.43
% 0.16/0.43 % minf(6)
% 0.16/0.43 thf(fact_1581_minf_I6_J,axiom,
% 0.16/0.43 ! [T: nat] :
% 0.16/0.43 ? [Z2: nat] :
% 0.16/0.43 ! [X5: nat] :
% 0.16/0.43 ( ( ord_less_nat @ X5 @ Z2 )
% 0.16/0.43 => ( ord_less_eq_nat @ X5 @ T ) ) ).
% 0.16/0.43
% 0.16/0.43 % minf(6)
% 0.16/0.43 thf(fact_1582_minf_I6_J,axiom,
% 0.16/0.43 ! [T: int] :
% 0.16/0.43 ? [Z2: int] :
% 0.16/0.43 ! [X5: int] :
% 0.16/0.43 ( ( ord_less_int @ X5 @ Z2 )
% 0.16/0.43 => ( ord_less_eq_int @ X5 @ T ) ) ).
% 0.16/0.43
% 0.16/0.43 % minf(6)
% 0.16/0.43 thf(fact_1583_minf_I6_J,axiom,
% 0.16/0.43 ! [T: real] :
% 0.16/0.43 ? [Z2: real] :
% 0.16/0.43 ! [X5: real] :
% 0.16/0.43 ( ( ord_less_real @ X5 @ Z2 )
% 0.16/0.43 => ( ord_less_eq_real @ X5 @ T ) ) ).
% 0.16/0.43
% 0.16/0.43 % minf(6)
% 0.16/0.43 thf(fact_1584_pinf_I8_J,axiom,
% 0.16/0.43 ! [T: rat] :
% 0.16/0.43 ? [Z2: rat] :
% 0.16/0.43 ! [X5: rat] :
% 0.16/0.43 ( ( ord_less_rat @ Z2 @ X5 )
% 0.16/0.43 => ( ord_less_eq_rat @ T @ X5 ) ) ).
% 0.16/0.43
% 0.16/0.43 % pinf(8)
% 0.16/0.43 thf(fact_1585_pinf_I8_J,axiom,
% 0.16/0.43 ! [T: num] :
% 0.16/0.43 ? [Z2: num] :
% 0.16/0.43 ! [X5: num] :
% 0.16/0.43 ( ( ord_less_num @ Z2 @ X5 )
% 0.16/0.43 => ( ord_less_eq_num @ T @ X5 ) ) ).
% 0.16/0.43
% 0.16/0.43 % pinf(8)
% 0.16/0.43 thf(fact_1586_pinf_I8_J,axiom,
% 0.16/0.43 ! [T: nat] :
% 0.16/0.43 ? [Z2: nat] :
% 0.16/0.43 ! [X5: nat] :
% 0.16/0.43 ( ( ord_less_nat @ Z2 @ X5 )
% 0.16/0.43 => ( ord_less_eq_nat @ T @ X5 ) ) ).
% 0.16/0.43
% 0.16/0.43 % pinf(8)
% 0.16/0.43 thf(fact_1587_pinf_I8_J,axiom,
% 0.16/0.43 ! [T: int] :
% 0.16/0.43 ? [Z2: int] :
% 0.16/0.43 ! [X5: int] :
% 0.16/0.43 ( ( ord_less_int @ Z2 @ X5 )
% 0.16/0.43 => ( ord_less_eq_int @ T @ X5 ) ) ).
% 0.16/0.43
% 0.16/0.43 % pinf(8)
% 0.16/0.43 thf(fact_1588_pinf_I8_J,axiom,
% 0.16/0.43 ! [T: real] :
% 0.16/0.43 ? [Z2: real] :
% 0.16/0.43 ! [X5: real] :
% 0.16/0.43 ( ( ord_less_real @ Z2 @ X5 )
% 0.16/0.43 => ( ord_less_eq_real @ T @ X5 ) ) ).
% 0.16/0.43
% 0.16/0.43 % pinf(8)
% 0.16/0.43 thf(fact_1589_pinf_I6_J,axiom,
% 0.16/0.43 ! [T: rat] :
% 0.16/0.43 ? [Z2: rat] :
% 0.16/0.43 ! [X5: rat] :
% 0.16/0.43 ( ( ord_less_rat @ Z2 @ X5 )
% 0.16/0.43 => ~ ( ord_less_eq_rat @ X5 @ T ) ) ).
% 0.16/0.43
% 0.16/0.43 % pinf(6)
% 0.16/0.43 thf(fact_1590_pinf_I6_J,axiom,
% 0.16/0.43 ! [T: num] :
% 0.16/0.43 ? [Z2: num] :
% 0.16/0.43 ! [X5: num] :
% 0.16/0.43 ( ( ord_less_num @ Z2 @ X5 )
% 0.16/0.43 => ~ ( ord_less_eq_num @ X5 @ T ) ) ).
% 0.16/0.43
% 0.16/0.43 % pinf(6)
% 0.16/0.43 thf(fact_1591_pinf_I6_J,axiom,
% 0.16/0.43 ! [T: nat] :
% 0.16/0.43 ? [Z2: nat] :
% 0.16/0.43 ! [X5: nat] :
% 0.16/0.43 ( ( ord_less_nat @ Z2 @ X5 )
% 0.16/0.43 => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% 0.16/0.43
% 0.16/0.43 % pinf(6)
% 0.16/0.43 thf(fact_1592_pinf_I6_J,axiom,
% 0.16/0.43 ! [T: int] :
% 0.16/0.43 ? [Z2: int] :
% 0.16/0.43 ! [X5: int] :
% 0.16/0.43 ( ( ord_less_int @ Z2 @ X5 )
% 0.16/0.43 => ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% 0.16/0.43
% 0.16/0.43 % pinf(6)
% 0.16/0.43 thf(fact_1593_pinf_I6_J,axiom,
% 0.16/0.43 ! [T: real] :
% 0.16/0.43 ? [Z2: real] :
% 0.16/0.43 ! [X5: real] :
% 0.16/0.43 ( ( ord_less_real @ Z2 @ X5 )
% 0.16/0.43 => ~ ( ord_less_eq_real @ X5 @ T ) ) ).
% 0.16/0.43
% 0.16/0.43 % pinf(6)
% 0.16/0.43 thf(fact_1594_less__divide__eq__1,axiom,
% 0.16/0.43 ! [B: real,A: real] :
% 0.16/0.43 ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.16/0.43 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.43 & ( ord_less_real @ A @ B ) )
% 0.16/0.43 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.16/0.43 & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_divide_eq_1
% 0.16/0.43 thf(fact_1595_less__divide__eq__1,axiom,
% 0.16/0.43 ! [B: rat,A: rat] :
% 0.16/0.43 ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.16/0.43 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.43 & ( ord_less_rat @ A @ B ) )
% 0.16/0.43 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.16/0.43 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_divide_eq_1
% 0.16/0.43 thf(fact_1596_divide__less__eq__1,axiom,
% 0.16/0.43 ! [B: real,A: real] :
% 0.16/0.43 ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.16/0.43 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.43 & ( ord_less_real @ B @ A ) )
% 0.16/0.43 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.16/0.43 & ( ord_less_real @ A @ B ) )
% 0.16/0.43 | ( A = zero_zero_real ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % divide_less_eq_1
% 0.16/0.43 thf(fact_1597_divide__less__eq__1,axiom,
% 0.16/0.43 ! [B: rat,A: rat] :
% 0.16/0.43 ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.16/0.43 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.43 & ( ord_less_rat @ B @ A ) )
% 0.16/0.43 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.16/0.43 & ( ord_less_rat @ A @ B ) )
% 0.16/0.43 | ( A = zero_zero_rat ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % divide_less_eq_1
% 0.16/0.43 thf(fact_1598_nat__induct__non__zero,axiom,
% 0.16/0.43 ! [N: nat,P: nat > $o] :
% 0.16/0.43 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.43 => ( ( P @ one_one_nat )
% 0.16/0.43 => ( ! [N2: nat] :
% 0.16/0.43 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.16/0.43 => ( ( P @ N2 )
% 0.16/0.43 => ( P @ ( suc @ N2 ) ) ) )
% 0.16/0.43 => ( P @ N ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % nat_induct_non_zero
% 0.16/0.43 thf(fact_1599_div__eq__dividend__iff,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.16/0.43 => ( ( ( divide_divide_nat @ M @ N )
% 0.16/0.43 = M )
% 0.16/0.43 = ( N = one_one_nat ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % div_eq_dividend_iff
% 0.16/0.43 thf(fact_1600_div__less__dividend,axiom,
% 0.16/0.43 ! [N: nat,M: nat] :
% 0.16/0.43 ( ( ord_less_nat @ one_one_nat @ N )
% 0.16/0.43 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.16/0.43 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % div_less_dividend
% 0.16/0.43 thf(fact_1601_verit__comp__simplify1_I3_J,axiom,
% 0.16/0.43 ! [B6: rat,A6: rat] :
% 0.16/0.43 ( ( ~ ( ord_less_eq_rat @ B6 @ A6 ) )
% 0.16/0.43 = ( ord_less_rat @ A6 @ B6 ) ) ).
% 0.16/0.43
% 0.16/0.43 % verit_comp_simplify1(3)
% 0.16/0.43 thf(fact_1602_verit__comp__simplify1_I3_J,axiom,
% 0.16/0.43 ! [B6: num,A6: num] :
% 0.16/0.43 ( ( ~ ( ord_less_eq_num @ B6 @ A6 ) )
% 0.16/0.43 = ( ord_less_num @ A6 @ B6 ) ) ).
% 0.16/0.43
% 0.16/0.43 % verit_comp_simplify1(3)
% 0.16/0.43 thf(fact_1603_verit__comp__simplify1_I3_J,axiom,
% 0.16/0.43 ! [B6: nat,A6: nat] :
% 0.16/0.43 ( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
% 0.16/0.43 = ( ord_less_nat @ A6 @ B6 ) ) ).
% 0.16/0.43
% 0.16/0.43 % verit_comp_simplify1(3)
% 0.16/0.43 thf(fact_1604_verit__comp__simplify1_I3_J,axiom,
% 0.16/0.43 ! [B6: int,A6: int] :
% 0.16/0.43 ( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
% 0.16/0.43 = ( ord_less_int @ A6 @ B6 ) ) ).
% 0.16/0.43
% 0.16/0.43 % verit_comp_simplify1(3)
% 0.16/0.43 thf(fact_1605_verit__comp__simplify1_I3_J,axiom,
% 0.16/0.43 ! [B6: real,A6: real] :
% 0.16/0.43 ( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
% 0.16/0.43 = ( ord_less_real @ A6 @ B6 ) ) ).
% 0.16/0.43
% 0.16/0.43 % verit_comp_simplify1(3)
% 0.16/0.43 thf(fact_1606_leD,axiom,
% 0.16/0.43 ! [Y4: set_int,X: set_int] :
% 0.16/0.43 ( ( ord_less_eq_set_int @ Y4 @ X )
% 0.16/0.43 => ~ ( ord_less_set_int @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % leD
% 0.16/0.43 thf(fact_1607_leD,axiom,
% 0.16/0.43 ! [Y4: rat,X: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ Y4 @ X )
% 0.16/0.43 => ~ ( ord_less_rat @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % leD
% 0.16/0.43 thf(fact_1608_leD,axiom,
% 0.16/0.43 ! [Y4: num,X: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ Y4 @ X )
% 0.16/0.43 => ~ ( ord_less_num @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % leD
% 0.16/0.43 thf(fact_1609_leD,axiom,
% 0.16/0.43 ! [Y4: nat,X: nat] :
% 0.16/0.43 ( ( ord_less_eq_nat @ Y4 @ X )
% 0.16/0.43 => ~ ( ord_less_nat @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % leD
% 0.16/0.43 thf(fact_1610_leD,axiom,
% 0.16/0.43 ! [Y4: int,X: int] :
% 0.16/0.43 ( ( ord_less_eq_int @ Y4 @ X )
% 0.16/0.43 => ~ ( ord_less_int @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % leD
% 0.16/0.43 thf(fact_1611_leD,axiom,
% 0.16/0.43 ! [Y4: real,X: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ Y4 @ X )
% 0.16/0.43 => ~ ( ord_less_real @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % leD
% 0.16/0.43 thf(fact_1612_leI,axiom,
% 0.16/0.43 ! [X: rat,Y4: rat] :
% 0.16/0.43 ( ~ ( ord_less_rat @ X @ Y4 )
% 0.16/0.43 => ( ord_less_eq_rat @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % leI
% 0.16/0.43 thf(fact_1613_leI,axiom,
% 0.16/0.43 ! [X: num,Y4: num] :
% 0.16/0.43 ( ~ ( ord_less_num @ X @ Y4 )
% 0.16/0.43 => ( ord_less_eq_num @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % leI
% 0.16/0.43 thf(fact_1614_leI,axiom,
% 0.16/0.43 ! [X: nat,Y4: nat] :
% 0.16/0.43 ( ~ ( ord_less_nat @ X @ Y4 )
% 0.16/0.43 => ( ord_less_eq_nat @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % leI
% 0.16/0.43 thf(fact_1615_leI,axiom,
% 0.16/0.43 ! [X: int,Y4: int] :
% 0.16/0.43 ( ~ ( ord_less_int @ X @ Y4 )
% 0.16/0.43 => ( ord_less_eq_int @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % leI
% 0.16/0.43 thf(fact_1616_leI,axiom,
% 0.16/0.43 ! [X: real,Y4: real] :
% 0.16/0.43 ( ~ ( ord_less_real @ X @ Y4 )
% 0.16/0.43 => ( ord_less_eq_real @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % leI
% 0.16/0.43 thf(fact_1617_nless__le,axiom,
% 0.16/0.43 ! [A: set_int,B: set_int] :
% 0.16/0.43 ( ( ~ ( ord_less_set_int @ A @ B ) )
% 0.16/0.43 = ( ~ ( ord_less_eq_set_int @ A @ B )
% 0.16/0.43 | ( A = B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % nless_le
% 0.16/0.43 thf(fact_1618_nless__le,axiom,
% 0.16/0.43 ! [A: rat,B: rat] :
% 0.16/0.43 ( ( ~ ( ord_less_rat @ A @ B ) )
% 0.16/0.43 = ( ~ ( ord_less_eq_rat @ A @ B )
% 0.16/0.43 | ( A = B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % nless_le
% 0.16/0.43 thf(fact_1619_nless__le,axiom,
% 0.16/0.43 ! [A: num,B: num] :
% 0.16/0.43 ( ( ~ ( ord_less_num @ A @ B ) )
% 0.16/0.43 = ( ~ ( ord_less_eq_num @ A @ B )
% 0.16/0.43 | ( A = B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % nless_le
% 0.16/0.43 thf(fact_1620_nless__le,axiom,
% 0.16/0.43 ! [A: nat,B: nat] :
% 0.16/0.43 ( ( ~ ( ord_less_nat @ A @ B ) )
% 0.16/0.43 = ( ~ ( ord_less_eq_nat @ A @ B )
% 0.16/0.43 | ( A = B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % nless_le
% 0.16/0.43 thf(fact_1621_nless__le,axiom,
% 0.16/0.43 ! [A: int,B: int] :
% 0.16/0.43 ( ( ~ ( ord_less_int @ A @ B ) )
% 0.16/0.43 = ( ~ ( ord_less_eq_int @ A @ B )
% 0.16/0.43 | ( A = B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % nless_le
% 0.16/0.43 thf(fact_1622_nless__le,axiom,
% 0.16/0.43 ! [A: real,B: real] :
% 0.16/0.43 ( ( ~ ( ord_less_real @ A @ B ) )
% 0.16/0.43 = ( ~ ( ord_less_eq_real @ A @ B )
% 0.16/0.43 | ( A = B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % nless_le
% 0.16/0.43 thf(fact_1623_antisym__conv1,axiom,
% 0.16/0.43 ! [X: set_int,Y4: set_int] :
% 0.16/0.43 ( ~ ( ord_less_set_int @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_eq_set_int @ X @ Y4 )
% 0.16/0.43 = ( X = Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % antisym_conv1
% 0.16/0.43 thf(fact_1624_antisym__conv1,axiom,
% 0.16/0.43 ! [X: rat,Y4: rat] :
% 0.16/0.43 ( ~ ( ord_less_rat @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_eq_rat @ X @ Y4 )
% 0.16/0.43 = ( X = Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % antisym_conv1
% 0.16/0.43 thf(fact_1625_antisym__conv1,axiom,
% 0.16/0.43 ! [X: num,Y4: num] :
% 0.16/0.43 ( ~ ( ord_less_num @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_eq_num @ X @ Y4 )
% 0.16/0.43 = ( X = Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % antisym_conv1
% 0.16/0.43 thf(fact_1626_antisym__conv1,axiom,
% 0.16/0.43 ! [X: nat,Y4: nat] :
% 0.16/0.43 ( ~ ( ord_less_nat @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_eq_nat @ X @ Y4 )
% 0.16/0.43 = ( X = Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % antisym_conv1
% 0.16/0.43 thf(fact_1627_antisym__conv1,axiom,
% 0.16/0.43 ! [X: int,Y4: int] :
% 0.16/0.43 ( ~ ( ord_less_int @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_eq_int @ X @ Y4 )
% 0.16/0.43 = ( X = Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % antisym_conv1
% 0.16/0.43 thf(fact_1628_antisym__conv1,axiom,
% 0.16/0.43 ! [X: real,Y4: real] :
% 0.16/0.43 ( ~ ( ord_less_real @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_eq_real @ X @ Y4 )
% 0.16/0.43 = ( X = Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % antisym_conv1
% 0.16/0.43 thf(fact_1629_antisym__conv2,axiom,
% 0.16/0.43 ! [X: set_int,Y4: set_int] :
% 0.16/0.43 ( ( ord_less_eq_set_int @ X @ Y4 )
% 0.16/0.43 => ( ( ~ ( ord_less_set_int @ X @ Y4 ) )
% 0.16/0.43 = ( X = Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % antisym_conv2
% 0.16/0.43 thf(fact_1630_antisym__conv2,axiom,
% 0.16/0.43 ! [X: rat,Y4: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ X @ Y4 )
% 0.16/0.43 => ( ( ~ ( ord_less_rat @ X @ Y4 ) )
% 0.16/0.43 = ( X = Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % antisym_conv2
% 0.16/0.43 thf(fact_1631_antisym__conv2,axiom,
% 0.16/0.43 ! [X: num,Y4: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ X @ Y4 )
% 0.16/0.43 => ( ( ~ ( ord_less_num @ X @ Y4 ) )
% 0.16/0.43 = ( X = Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % antisym_conv2
% 0.16/0.43 thf(fact_1632_antisym__conv2,axiom,
% 0.16/0.43 ! [X: nat,Y4: nat] :
% 0.16/0.43 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.16/0.43 => ( ( ~ ( ord_less_nat @ X @ Y4 ) )
% 0.16/0.43 = ( X = Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % antisym_conv2
% 0.16/0.43 thf(fact_1633_antisym__conv2,axiom,
% 0.16/0.43 ! [X: int,Y4: int] :
% 0.16/0.43 ( ( ord_less_eq_int @ X @ Y4 )
% 0.16/0.43 => ( ( ~ ( ord_less_int @ X @ Y4 ) )
% 0.16/0.43 = ( X = Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % antisym_conv2
% 0.16/0.43 thf(fact_1634_antisym__conv2,axiom,
% 0.16/0.43 ! [X: real,Y4: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ X @ Y4 )
% 0.16/0.43 => ( ( ~ ( ord_less_real @ X @ Y4 ) )
% 0.16/0.43 = ( X = Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % antisym_conv2
% 0.16/0.43 thf(fact_1635_dense__ge,axiom,
% 0.16/0.43 ! [Z: rat,Y4: rat] :
% 0.16/0.43 ( ! [X4: rat] :
% 0.16/0.43 ( ( ord_less_rat @ Z @ X4 )
% 0.16/0.43 => ( ord_less_eq_rat @ Y4 @ X4 ) )
% 0.16/0.43 => ( ord_less_eq_rat @ Y4 @ Z ) ) ).
% 0.16/0.43
% 0.16/0.43 % dense_ge
% 0.16/0.43 thf(fact_1636_dense__ge,axiom,
% 0.16/0.43 ! [Z: real,Y4: real] :
% 0.16/0.43 ( ! [X4: real] :
% 0.16/0.43 ( ( ord_less_real @ Z @ X4 )
% 0.16/0.43 => ( ord_less_eq_real @ Y4 @ X4 ) )
% 0.16/0.43 => ( ord_less_eq_real @ Y4 @ Z ) ) ).
% 0.16/0.43
% 0.16/0.43 % dense_ge
% 0.16/0.43 thf(fact_1637_dense__le,axiom,
% 0.16/0.43 ! [Y4: rat,Z: rat] :
% 0.16/0.43 ( ! [X4: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y4 )
% 0.16/0.43 => ( ord_less_eq_rat @ X4 @ Z ) )
% 0.16/0.43 => ( ord_less_eq_rat @ Y4 @ Z ) ) ).
% 0.16/0.43
% 0.16/0.43 % dense_le
% 0.16/0.43 thf(fact_1638_dense__le,axiom,
% 0.16/0.43 ! [Y4: real,Z: real] :
% 0.16/0.43 ( ! [X4: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y4 )
% 0.16/0.43 => ( ord_less_eq_real @ X4 @ Z ) )
% 0.16/0.43 => ( ord_less_eq_real @ Y4 @ Z ) ) ).
% 0.16/0.43
% 0.16/0.43 % dense_le
% 0.16/0.43 thf(fact_1639_less__le__not__le,axiom,
% 0.16/0.43 ( ord_less_set_int
% 0.16/0.43 = ( ^ [X3: set_int,Y6: set_int] :
% 0.16/0.43 ( ( ord_less_eq_set_int @ X3 @ Y6 )
% 0.16/0.43 & ~ ( ord_less_eq_set_int @ Y6 @ X3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_le_not_le
% 0.16/0.43 thf(fact_1640_less__le__not__le,axiom,
% 0.16/0.43 ( ord_less_rat
% 0.16/0.43 = ( ^ [X3: rat,Y6: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ X3 @ Y6 )
% 0.16/0.43 & ~ ( ord_less_eq_rat @ Y6 @ X3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_le_not_le
% 0.16/0.43 thf(fact_1641_less__le__not__le,axiom,
% 0.16/0.43 ( ord_less_num
% 0.16/0.43 = ( ^ [X3: num,Y6: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ X3 @ Y6 )
% 0.16/0.43 & ~ ( ord_less_eq_num @ Y6 @ X3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_le_not_le
% 0.16/0.43 thf(fact_1642_less__le__not__le,axiom,
% 0.16/0.43 ( ord_less_nat
% 0.16/0.43 = ( ^ [X3: nat,Y6: nat] :
% 0.16/0.43 ( ( ord_less_eq_nat @ X3 @ Y6 )
% 0.16/0.43 & ~ ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_le_not_le
% 0.16/0.43 thf(fact_1643_less__le__not__le,axiom,
% 0.16/0.43 ( ord_less_int
% 0.16/0.43 = ( ^ [X3: int,Y6: int] :
% 0.16/0.43 ( ( ord_less_eq_int @ X3 @ Y6 )
% 0.16/0.43 & ~ ( ord_less_eq_int @ Y6 @ X3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_le_not_le
% 0.16/0.43 thf(fact_1644_less__le__not__le,axiom,
% 0.16/0.43 ( ord_less_real
% 0.16/0.43 = ( ^ [X3: real,Y6: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ X3 @ Y6 )
% 0.16/0.43 & ~ ( ord_less_eq_real @ Y6 @ X3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_le_not_le
% 0.16/0.43 thf(fact_1645_not__le__imp__less,axiom,
% 0.16/0.43 ! [Y4: rat,X: rat] :
% 0.16/0.43 ( ~ ( ord_less_eq_rat @ Y4 @ X )
% 0.16/0.43 => ( ord_less_rat @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % not_le_imp_less
% 0.16/0.43 thf(fact_1646_not__le__imp__less,axiom,
% 0.16/0.43 ! [Y4: num,X: num] :
% 0.16/0.43 ( ~ ( ord_less_eq_num @ Y4 @ X )
% 0.16/0.43 => ( ord_less_num @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % not_le_imp_less
% 0.16/0.43 thf(fact_1647_not__le__imp__less,axiom,
% 0.16/0.43 ! [Y4: nat,X: nat] :
% 0.16/0.43 ( ~ ( ord_less_eq_nat @ Y4 @ X )
% 0.16/0.43 => ( ord_less_nat @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % not_le_imp_less
% 0.16/0.43 thf(fact_1648_not__le__imp__less,axiom,
% 0.16/0.43 ! [Y4: int,X: int] :
% 0.16/0.43 ( ~ ( ord_less_eq_int @ Y4 @ X )
% 0.16/0.43 => ( ord_less_int @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % not_le_imp_less
% 0.16/0.43 thf(fact_1649_not__le__imp__less,axiom,
% 0.16/0.43 ! [Y4: real,X: real] :
% 0.16/0.43 ( ~ ( ord_less_eq_real @ Y4 @ X )
% 0.16/0.43 => ( ord_less_real @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % not_le_imp_less
% 0.16/0.43 thf(fact_1650_order_Oorder__iff__strict,axiom,
% 0.16/0.43 ( ord_less_eq_set_int
% 0.16/0.43 = ( ^ [A3: set_int,B2: set_int] :
% 0.16/0.43 ( ( ord_less_set_int @ A3 @ B2 )
% 0.16/0.43 | ( A3 = B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.order_iff_strict
% 0.16/0.43 thf(fact_1651_order_Oorder__iff__strict,axiom,
% 0.16/0.43 ( ord_less_eq_rat
% 0.16/0.43 = ( ^ [A3: rat,B2: rat] :
% 0.16/0.43 ( ( ord_less_rat @ A3 @ B2 )
% 0.16/0.43 | ( A3 = B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.order_iff_strict
% 0.16/0.43 thf(fact_1652_order_Oorder__iff__strict,axiom,
% 0.16/0.43 ( ord_less_eq_num
% 0.16/0.43 = ( ^ [A3: num,B2: num] :
% 0.16/0.43 ( ( ord_less_num @ A3 @ B2 )
% 0.16/0.43 | ( A3 = B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.order_iff_strict
% 0.16/0.43 thf(fact_1653_order_Oorder__iff__strict,axiom,
% 0.16/0.43 ( ord_less_eq_nat
% 0.16/0.43 = ( ^ [A3: nat,B2: nat] :
% 0.16/0.43 ( ( ord_less_nat @ A3 @ B2 )
% 0.16/0.43 | ( A3 = B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.order_iff_strict
% 0.16/0.43 thf(fact_1654_order_Oorder__iff__strict,axiom,
% 0.16/0.43 ( ord_less_eq_int
% 0.16/0.43 = ( ^ [A3: int,B2: int] :
% 0.16/0.43 ( ( ord_less_int @ A3 @ B2 )
% 0.16/0.43 | ( A3 = B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.order_iff_strict
% 0.16/0.43 thf(fact_1655_order_Oorder__iff__strict,axiom,
% 0.16/0.43 ( ord_less_eq_real
% 0.16/0.43 = ( ^ [A3: real,B2: real] :
% 0.16/0.43 ( ( ord_less_real @ A3 @ B2 )
% 0.16/0.43 | ( A3 = B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.order_iff_strict
% 0.16/0.43 thf(fact_1656_order_Ostrict__iff__order,axiom,
% 0.16/0.43 ( ord_less_set_int
% 0.16/0.43 = ( ^ [A3: set_int,B2: set_int] :
% 0.16/0.43 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.16/0.43 & ( A3 != B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_iff_order
% 0.16/0.43 thf(fact_1657_order_Ostrict__iff__order,axiom,
% 0.16/0.43 ( ord_less_rat
% 0.16/0.43 = ( ^ [A3: rat,B2: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ A3 @ B2 )
% 0.16/0.43 & ( A3 != B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_iff_order
% 0.16/0.43 thf(fact_1658_order_Ostrict__iff__order,axiom,
% 0.16/0.43 ( ord_less_num
% 0.16/0.43 = ( ^ [A3: num,B2: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ A3 @ B2 )
% 0.16/0.43 & ( A3 != B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_iff_order
% 0.16/0.43 thf(fact_1659_order_Ostrict__iff__order,axiom,
% 0.16/0.43 ( ord_less_nat
% 0.16/0.43 = ( ^ [A3: nat,B2: nat] :
% 0.16/0.43 ( ( ord_less_eq_nat @ A3 @ B2 )
% 0.16/0.43 & ( A3 != B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_iff_order
% 0.16/0.43 thf(fact_1660_order_Ostrict__iff__order,axiom,
% 0.16/0.43 ( ord_less_int
% 0.16/0.43 = ( ^ [A3: int,B2: int] :
% 0.16/0.43 ( ( ord_less_eq_int @ A3 @ B2 )
% 0.16/0.43 & ( A3 != B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_iff_order
% 0.16/0.43 thf(fact_1661_order_Ostrict__iff__order,axiom,
% 0.16/0.43 ( ord_less_real
% 0.16/0.43 = ( ^ [A3: real,B2: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ A3 @ B2 )
% 0.16/0.43 & ( A3 != B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_iff_order
% 0.16/0.43 thf(fact_1662_order_Ostrict__trans1,axiom,
% 0.16/0.43 ! [A: set_int,B: set_int,C: set_int] :
% 0.16/0.43 ( ( ord_less_eq_set_int @ A @ B )
% 0.16/0.43 => ( ( ord_less_set_int @ B @ C )
% 0.16/0.43 => ( ord_less_set_int @ A @ C ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_trans1
% 0.16/0.43 thf(fact_1663_order_Ostrict__trans1,axiom,
% 0.16/0.43 ! [A: rat,B: rat,C: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.43 => ( ( ord_less_rat @ B @ C )
% 0.16/0.43 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_trans1
% 0.16/0.43 thf(fact_1664_order_Ostrict__trans1,axiom,
% 0.16/0.43 ! [A: num,B: num,C: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.43 => ( ( ord_less_num @ B @ C )
% 0.16/0.43 => ( ord_less_num @ A @ C ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_trans1
% 0.16/0.43 thf(fact_1665_order_Ostrict__trans1,axiom,
% 0.16/0.43 ! [A: nat,B: nat,C: nat] :
% 0.16/0.43 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.43 => ( ( ord_less_nat @ B @ C )
% 0.16/0.43 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_trans1
% 0.16/0.43 thf(fact_1666_order_Ostrict__trans1,axiom,
% 0.16/0.43 ! [A: int,B: int,C: int] :
% 0.16/0.43 ( ( ord_less_eq_int @ A @ B )
% 0.16/0.43 => ( ( ord_less_int @ B @ C )
% 0.16/0.43 => ( ord_less_int @ A @ C ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_trans1
% 0.16/0.43 thf(fact_1667_order_Ostrict__trans1,axiom,
% 0.16/0.43 ! [A: real,B: real,C: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.43 => ( ( ord_less_real @ B @ C )
% 0.16/0.43 => ( ord_less_real @ A @ C ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_trans1
% 0.16/0.43 thf(fact_1668_order_Ostrict__trans2,axiom,
% 0.16/0.43 ! [A: set_int,B: set_int,C: set_int] :
% 0.16/0.43 ( ( ord_less_set_int @ A @ B )
% 0.16/0.43 => ( ( ord_less_eq_set_int @ B @ C )
% 0.16/0.43 => ( ord_less_set_int @ A @ C ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_trans2
% 0.16/0.43 thf(fact_1669_order_Ostrict__trans2,axiom,
% 0.16/0.43 ! [A: rat,B: rat,C: rat] :
% 0.16/0.43 ( ( ord_less_rat @ A @ B )
% 0.16/0.43 => ( ( ord_less_eq_rat @ B @ C )
% 0.16/0.43 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_trans2
% 0.16/0.43 thf(fact_1670_order_Ostrict__trans2,axiom,
% 0.16/0.43 ! [A: num,B: num,C: num] :
% 0.16/0.43 ( ( ord_less_num @ A @ B )
% 0.16/0.43 => ( ( ord_less_eq_num @ B @ C )
% 0.16/0.43 => ( ord_less_num @ A @ C ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_trans2
% 0.16/0.43 thf(fact_1671_order_Ostrict__trans2,axiom,
% 0.16/0.43 ! [A: nat,B: nat,C: nat] :
% 0.16/0.43 ( ( ord_less_nat @ A @ B )
% 0.16/0.43 => ( ( ord_less_eq_nat @ B @ C )
% 0.16/0.43 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_trans2
% 0.16/0.43 thf(fact_1672_order_Ostrict__trans2,axiom,
% 0.16/0.43 ! [A: int,B: int,C: int] :
% 0.16/0.43 ( ( ord_less_int @ A @ B )
% 0.16/0.43 => ( ( ord_less_eq_int @ B @ C )
% 0.16/0.43 => ( ord_less_int @ A @ C ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_trans2
% 0.16/0.43 thf(fact_1673_order_Ostrict__trans2,axiom,
% 0.16/0.43 ! [A: real,B: real,C: real] :
% 0.16/0.43 ( ( ord_less_real @ A @ B )
% 0.16/0.43 => ( ( ord_less_eq_real @ B @ C )
% 0.16/0.43 => ( ord_less_real @ A @ C ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_trans2
% 0.16/0.43 thf(fact_1674_order_Ostrict__iff__not,axiom,
% 0.16/0.43 ( ord_less_set_int
% 0.16/0.43 = ( ^ [A3: set_int,B2: set_int] :
% 0.16/0.43 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.16/0.43 & ~ ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_iff_not
% 0.16/0.43 thf(fact_1675_order_Ostrict__iff__not,axiom,
% 0.16/0.43 ( ord_less_rat
% 0.16/0.43 = ( ^ [A3: rat,B2: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ A3 @ B2 )
% 0.16/0.43 & ~ ( ord_less_eq_rat @ B2 @ A3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_iff_not
% 0.16/0.43 thf(fact_1676_order_Ostrict__iff__not,axiom,
% 0.16/0.43 ( ord_less_num
% 0.16/0.43 = ( ^ [A3: num,B2: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ A3 @ B2 )
% 0.16/0.43 & ~ ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_iff_not
% 0.16/0.43 thf(fact_1677_order_Ostrict__iff__not,axiom,
% 0.16/0.43 ( ord_less_nat
% 0.16/0.43 = ( ^ [A3: nat,B2: nat] :
% 0.16/0.43 ( ( ord_less_eq_nat @ A3 @ B2 )
% 0.16/0.43 & ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_iff_not
% 0.16/0.43 thf(fact_1678_order_Ostrict__iff__not,axiom,
% 0.16/0.43 ( ord_less_int
% 0.16/0.43 = ( ^ [A3: int,B2: int] :
% 0.16/0.43 ( ( ord_less_eq_int @ A3 @ B2 )
% 0.16/0.43 & ~ ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_iff_not
% 0.16/0.43 thf(fact_1679_order_Ostrict__iff__not,axiom,
% 0.16/0.43 ( ord_less_real
% 0.16/0.43 = ( ^ [A3: real,B2: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ A3 @ B2 )
% 0.16/0.43 & ~ ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_iff_not
% 0.16/0.43 thf(fact_1680_dense__ge__bounded,axiom,
% 0.16/0.43 ! [Z: rat,X: rat,Y4: rat] :
% 0.16/0.43 ( ( ord_less_rat @ Z @ X )
% 0.16/0.43 => ( ! [W2: rat] :
% 0.16/0.43 ( ( ord_less_rat @ Z @ W2 )
% 0.16/0.43 => ( ( ord_less_rat @ W2 @ X )
% 0.16/0.43 => ( ord_less_eq_rat @ Y4 @ W2 ) ) )
% 0.16/0.43 => ( ord_less_eq_rat @ Y4 @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dense_ge_bounded
% 0.16/0.43 thf(fact_1681_dense__ge__bounded,axiom,
% 0.16/0.43 ! [Z: real,X: real,Y4: real] :
% 0.16/0.43 ( ( ord_less_real @ Z @ X )
% 0.16/0.43 => ( ! [W2: real] :
% 0.16/0.43 ( ( ord_less_real @ Z @ W2 )
% 0.16/0.43 => ( ( ord_less_real @ W2 @ X )
% 0.16/0.43 => ( ord_less_eq_real @ Y4 @ W2 ) ) )
% 0.16/0.43 => ( ord_less_eq_real @ Y4 @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dense_ge_bounded
% 0.16/0.43 thf(fact_1682_dense__le__bounded,axiom,
% 0.16/0.43 ! [X: rat,Y4: rat,Z: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X @ Y4 )
% 0.16/0.43 => ( ! [W2: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X @ W2 )
% 0.16/0.43 => ( ( ord_less_rat @ W2 @ Y4 )
% 0.16/0.43 => ( ord_less_eq_rat @ W2 @ Z ) ) )
% 0.16/0.43 => ( ord_less_eq_rat @ Y4 @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dense_le_bounded
% 0.16/0.43 thf(fact_1683_dense__le__bounded,axiom,
% 0.16/0.43 ! [X: real,Y4: real,Z: real] :
% 0.16/0.43 ( ( ord_less_real @ X @ Y4 )
% 0.16/0.43 => ( ! [W2: real] :
% 0.16/0.43 ( ( ord_less_real @ X @ W2 )
% 0.16/0.43 => ( ( ord_less_real @ W2 @ Y4 )
% 0.16/0.43 => ( ord_less_eq_real @ W2 @ Z ) ) )
% 0.16/0.43 => ( ord_less_eq_real @ Y4 @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dense_le_bounded
% 0.16/0.43 thf(fact_1684_dual__order_Oorder__iff__strict,axiom,
% 0.16/0.43 ( ord_less_eq_set_int
% 0.16/0.43 = ( ^ [B2: set_int,A3: set_int] :
% 0.16/0.43 ( ( ord_less_set_int @ B2 @ A3 )
% 0.16/0.43 | ( A3 = B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.order_iff_strict
% 0.16/0.43 thf(fact_1685_dual__order_Oorder__iff__strict,axiom,
% 0.16/0.43 ( ord_less_eq_rat
% 0.16/0.43 = ( ^ [B2: rat,A3: rat] :
% 0.16/0.43 ( ( ord_less_rat @ B2 @ A3 )
% 0.16/0.43 | ( A3 = B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.order_iff_strict
% 0.16/0.43 thf(fact_1686_dual__order_Oorder__iff__strict,axiom,
% 0.16/0.43 ( ord_less_eq_num
% 0.16/0.43 = ( ^ [B2: num,A3: num] :
% 0.16/0.43 ( ( ord_less_num @ B2 @ A3 )
% 0.16/0.43 | ( A3 = B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.order_iff_strict
% 0.16/0.43 thf(fact_1687_dual__order_Oorder__iff__strict,axiom,
% 0.16/0.43 ( ord_less_eq_nat
% 0.16/0.43 = ( ^ [B2: nat,A3: nat] :
% 0.16/0.43 ( ( ord_less_nat @ B2 @ A3 )
% 0.16/0.43 | ( A3 = B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.order_iff_strict
% 0.16/0.43 thf(fact_1688_dual__order_Oorder__iff__strict,axiom,
% 0.16/0.43 ( ord_less_eq_int
% 0.16/0.43 = ( ^ [B2: int,A3: int] :
% 0.16/0.43 ( ( ord_less_int @ B2 @ A3 )
% 0.16/0.43 | ( A3 = B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.order_iff_strict
% 0.16/0.43 thf(fact_1689_dual__order_Oorder__iff__strict,axiom,
% 0.16/0.43 ( ord_less_eq_real
% 0.16/0.43 = ( ^ [B2: real,A3: real] :
% 0.16/0.43 ( ( ord_less_real @ B2 @ A3 )
% 0.16/0.43 | ( A3 = B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.order_iff_strict
% 0.16/0.43 thf(fact_1690_dual__order_Ostrict__iff__order,axiom,
% 0.16/0.43 ( ord_less_set_int
% 0.16/0.43 = ( ^ [B2: set_int,A3: set_int] :
% 0.16/0.43 ( ( ord_less_eq_set_int @ B2 @ A3 )
% 0.16/0.43 & ( A3 != B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_iff_order
% 0.16/0.43 thf(fact_1691_dual__order_Ostrict__iff__order,axiom,
% 0.16/0.43 ( ord_less_rat
% 0.16/0.43 = ( ^ [B2: rat,A3: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ B2 @ A3 )
% 0.16/0.43 & ( A3 != B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_iff_order
% 0.16/0.43 thf(fact_1692_dual__order_Ostrict__iff__order,axiom,
% 0.16/0.43 ( ord_less_num
% 0.16/0.43 = ( ^ [B2: num,A3: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ B2 @ A3 )
% 0.16/0.43 & ( A3 != B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_iff_order
% 0.16/0.43 thf(fact_1693_dual__order_Ostrict__iff__order,axiom,
% 0.16/0.43 ( ord_less_nat
% 0.16/0.43 = ( ^ [B2: nat,A3: nat] :
% 0.16/0.43 ( ( ord_less_eq_nat @ B2 @ A3 )
% 0.16/0.43 & ( A3 != B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_iff_order
% 0.16/0.43 thf(fact_1694_dual__order_Ostrict__iff__order,axiom,
% 0.16/0.43 ( ord_less_int
% 0.16/0.43 = ( ^ [B2: int,A3: int] :
% 0.16/0.43 ( ( ord_less_eq_int @ B2 @ A3 )
% 0.16/0.43 & ( A3 != B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_iff_order
% 0.16/0.43 thf(fact_1695_dual__order_Ostrict__iff__order,axiom,
% 0.16/0.43 ( ord_less_real
% 0.16/0.43 = ( ^ [B2: real,A3: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ B2 @ A3 )
% 0.16/0.43 & ( A3 != B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_iff_order
% 0.16/0.43 thf(fact_1696_dual__order_Ostrict__trans1,axiom,
% 0.16/0.43 ! [B: set_int,A: set_int,C: set_int] :
% 0.16/0.43 ( ( ord_less_eq_set_int @ B @ A )
% 0.16/0.43 => ( ( ord_less_set_int @ C @ B )
% 0.16/0.43 => ( ord_less_set_int @ C @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_trans1
% 0.16/0.43 thf(fact_1697_dual__order_Ostrict__trans1,axiom,
% 0.16/0.43 ! [B: rat,A: rat,C: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ B @ A )
% 0.16/0.43 => ( ( ord_less_rat @ C @ B )
% 0.16/0.43 => ( ord_less_rat @ C @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_trans1
% 0.16/0.43 thf(fact_1698_dual__order_Ostrict__trans1,axiom,
% 0.16/0.43 ! [B: num,A: num,C: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ B @ A )
% 0.16/0.43 => ( ( ord_less_num @ C @ B )
% 0.16/0.43 => ( ord_less_num @ C @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_trans1
% 0.16/0.43 thf(fact_1699_dual__order_Ostrict__trans1,axiom,
% 0.16/0.43 ! [B: nat,A: nat,C: nat] :
% 0.16/0.43 ( ( ord_less_eq_nat @ B @ A )
% 0.16/0.43 => ( ( ord_less_nat @ C @ B )
% 0.16/0.43 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_trans1
% 0.16/0.43 thf(fact_1700_dual__order_Ostrict__trans1,axiom,
% 0.16/0.43 ! [B: int,A: int,C: int] :
% 0.16/0.43 ( ( ord_less_eq_int @ B @ A )
% 0.16/0.43 => ( ( ord_less_int @ C @ B )
% 0.16/0.43 => ( ord_less_int @ C @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_trans1
% 0.16/0.43 thf(fact_1701_dual__order_Ostrict__trans1,axiom,
% 0.16/0.43 ! [B: real,A: real,C: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ B @ A )
% 0.16/0.43 => ( ( ord_less_real @ C @ B )
% 0.16/0.43 => ( ord_less_real @ C @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_trans1
% 0.16/0.43 thf(fact_1702_dual__order_Ostrict__trans2,axiom,
% 0.16/0.43 ! [B: set_int,A: set_int,C: set_int] :
% 0.16/0.43 ( ( ord_less_set_int @ B @ A )
% 0.16/0.43 => ( ( ord_less_eq_set_int @ C @ B )
% 0.16/0.43 => ( ord_less_set_int @ C @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_trans2
% 0.16/0.43 thf(fact_1703_dual__order_Ostrict__trans2,axiom,
% 0.16/0.43 ! [B: rat,A: rat,C: rat] :
% 0.16/0.43 ( ( ord_less_rat @ B @ A )
% 0.16/0.43 => ( ( ord_less_eq_rat @ C @ B )
% 0.16/0.43 => ( ord_less_rat @ C @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_trans2
% 0.16/0.43 thf(fact_1704_dual__order_Ostrict__trans2,axiom,
% 0.16/0.43 ! [B: num,A: num,C: num] :
% 0.16/0.43 ( ( ord_less_num @ B @ A )
% 0.16/0.43 => ( ( ord_less_eq_num @ C @ B )
% 0.16/0.43 => ( ord_less_num @ C @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_trans2
% 0.16/0.43 thf(fact_1705_dual__order_Ostrict__trans2,axiom,
% 0.16/0.43 ! [B: nat,A: nat,C: nat] :
% 0.16/0.43 ( ( ord_less_nat @ B @ A )
% 0.16/0.43 => ( ( ord_less_eq_nat @ C @ B )
% 0.16/0.43 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_trans2
% 0.16/0.43 thf(fact_1706_dual__order_Ostrict__trans2,axiom,
% 0.16/0.43 ! [B: int,A: int,C: int] :
% 0.16/0.43 ( ( ord_less_int @ B @ A )
% 0.16/0.43 => ( ( ord_less_eq_int @ C @ B )
% 0.16/0.43 => ( ord_less_int @ C @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_trans2
% 0.16/0.43 thf(fact_1707_dual__order_Ostrict__trans2,axiom,
% 0.16/0.43 ! [B: real,A: real,C: real] :
% 0.16/0.43 ( ( ord_less_real @ B @ A )
% 0.16/0.43 => ( ( ord_less_eq_real @ C @ B )
% 0.16/0.43 => ( ord_less_real @ C @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_trans2
% 0.16/0.43 thf(fact_1708_dual__order_Ostrict__iff__not,axiom,
% 0.16/0.43 ( ord_less_set_int
% 0.16/0.43 = ( ^ [B2: set_int,A3: set_int] :
% 0.16/0.43 ( ( ord_less_eq_set_int @ B2 @ A3 )
% 0.16/0.43 & ~ ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_iff_not
% 0.16/0.43 thf(fact_1709_dual__order_Ostrict__iff__not,axiom,
% 0.16/0.43 ( ord_less_rat
% 0.16/0.43 = ( ^ [B2: rat,A3: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ B2 @ A3 )
% 0.16/0.43 & ~ ( ord_less_eq_rat @ A3 @ B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_iff_not
% 0.16/0.43 thf(fact_1710_dual__order_Ostrict__iff__not,axiom,
% 0.16/0.43 ( ord_less_num
% 0.16/0.43 = ( ^ [B2: num,A3: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ B2 @ A3 )
% 0.16/0.43 & ~ ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_iff_not
% 0.16/0.43 thf(fact_1711_dual__order_Ostrict__iff__not,axiom,
% 0.16/0.43 ( ord_less_nat
% 0.16/0.43 = ( ^ [B2: nat,A3: nat] :
% 0.16/0.43 ( ( ord_less_eq_nat @ B2 @ A3 )
% 0.16/0.43 & ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_iff_not
% 0.16/0.43 thf(fact_1712_dual__order_Ostrict__iff__not,axiom,
% 0.16/0.43 ( ord_less_int
% 0.16/0.43 = ( ^ [B2: int,A3: int] :
% 0.16/0.43 ( ( ord_less_eq_int @ B2 @ A3 )
% 0.16/0.43 & ~ ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_iff_not
% 0.16/0.43 thf(fact_1713_dual__order_Ostrict__iff__not,axiom,
% 0.16/0.43 ( ord_less_real
% 0.16/0.43 = ( ^ [B2: real,A3: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ B2 @ A3 )
% 0.16/0.43 & ~ ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_iff_not
% 0.16/0.43 thf(fact_1714_order_Ostrict__implies__order,axiom,
% 0.16/0.43 ! [A: set_int,B: set_int] :
% 0.16/0.43 ( ( ord_less_set_int @ A @ B )
% 0.16/0.43 => ( ord_less_eq_set_int @ A @ B ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_implies_order
% 0.16/0.43 thf(fact_1715_order_Ostrict__implies__order,axiom,
% 0.16/0.43 ! [A: rat,B: rat] :
% 0.16/0.43 ( ( ord_less_rat @ A @ B )
% 0.16/0.43 => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_implies_order
% 0.16/0.43 thf(fact_1716_order_Ostrict__implies__order,axiom,
% 0.16/0.43 ! [A: num,B: num] :
% 0.16/0.43 ( ( ord_less_num @ A @ B )
% 0.16/0.43 => ( ord_less_eq_num @ A @ B ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_implies_order
% 0.16/0.43 thf(fact_1717_order_Ostrict__implies__order,axiom,
% 0.16/0.43 ! [A: nat,B: nat] :
% 0.16/0.43 ( ( ord_less_nat @ A @ B )
% 0.16/0.43 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_implies_order
% 0.16/0.43 thf(fact_1718_order_Ostrict__implies__order,axiom,
% 0.16/0.43 ! [A: int,B: int] :
% 0.16/0.43 ( ( ord_less_int @ A @ B )
% 0.16/0.43 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_implies_order
% 0.16/0.43 thf(fact_1719_order_Ostrict__implies__order,axiom,
% 0.16/0.43 ! [A: real,B: real] :
% 0.16/0.43 ( ( ord_less_real @ A @ B )
% 0.16/0.43 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.16/0.43
% 0.16/0.43 % order.strict_implies_order
% 0.16/0.43 thf(fact_1720_dual__order_Ostrict__implies__order,axiom,
% 0.16/0.43 ! [B: set_int,A: set_int] :
% 0.16/0.43 ( ( ord_less_set_int @ B @ A )
% 0.16/0.43 => ( ord_less_eq_set_int @ B @ A ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_implies_order
% 0.16/0.43 thf(fact_1721_dual__order_Ostrict__implies__order,axiom,
% 0.16/0.43 ! [B: rat,A: rat] :
% 0.16/0.43 ( ( ord_less_rat @ B @ A )
% 0.16/0.43 => ( ord_less_eq_rat @ B @ A ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_implies_order
% 0.16/0.43 thf(fact_1722_dual__order_Ostrict__implies__order,axiom,
% 0.16/0.43 ! [B: num,A: num] :
% 0.16/0.43 ( ( ord_less_num @ B @ A )
% 0.16/0.43 => ( ord_less_eq_num @ B @ A ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_implies_order
% 0.16/0.43 thf(fact_1723_dual__order_Ostrict__implies__order,axiom,
% 0.16/0.43 ! [B: nat,A: nat] :
% 0.16/0.43 ( ( ord_less_nat @ B @ A )
% 0.16/0.43 => ( ord_less_eq_nat @ B @ A ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_implies_order
% 0.16/0.43 thf(fact_1724_dual__order_Ostrict__implies__order,axiom,
% 0.16/0.43 ! [B: int,A: int] :
% 0.16/0.43 ( ( ord_less_int @ B @ A )
% 0.16/0.43 => ( ord_less_eq_int @ B @ A ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_implies_order
% 0.16/0.43 thf(fact_1725_dual__order_Ostrict__implies__order,axiom,
% 0.16/0.43 ! [B: real,A: real] :
% 0.16/0.43 ( ( ord_less_real @ B @ A )
% 0.16/0.43 => ( ord_less_eq_real @ B @ A ) ) ).
% 0.16/0.43
% 0.16/0.43 % dual_order.strict_implies_order
% 0.16/0.43 thf(fact_1726_order__le__less,axiom,
% 0.16/0.43 ( ord_less_eq_set_int
% 0.16/0.43 = ( ^ [X3: set_int,Y6: set_int] :
% 0.16/0.43 ( ( ord_less_set_int @ X3 @ Y6 )
% 0.16/0.43 | ( X3 = Y6 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less
% 0.16/0.43 thf(fact_1727_order__le__less,axiom,
% 0.16/0.43 ( ord_less_eq_rat
% 0.16/0.43 = ( ^ [X3: rat,Y6: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X3 @ Y6 )
% 0.16/0.43 | ( X3 = Y6 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less
% 0.16/0.43 thf(fact_1728_order__le__less,axiom,
% 0.16/0.43 ( ord_less_eq_num
% 0.16/0.43 = ( ^ [X3: num,Y6: num] :
% 0.16/0.43 ( ( ord_less_num @ X3 @ Y6 )
% 0.16/0.43 | ( X3 = Y6 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less
% 0.16/0.43 thf(fact_1729_order__le__less,axiom,
% 0.16/0.43 ( ord_less_eq_nat
% 0.16/0.43 = ( ^ [X3: nat,Y6: nat] :
% 0.16/0.43 ( ( ord_less_nat @ X3 @ Y6 )
% 0.16/0.43 | ( X3 = Y6 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less
% 0.16/0.43 thf(fact_1730_order__le__less,axiom,
% 0.16/0.43 ( ord_less_eq_int
% 0.16/0.43 = ( ^ [X3: int,Y6: int] :
% 0.16/0.43 ( ( ord_less_int @ X3 @ Y6 )
% 0.16/0.43 | ( X3 = Y6 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less
% 0.16/0.43 thf(fact_1731_order__le__less,axiom,
% 0.16/0.43 ( ord_less_eq_real
% 0.16/0.43 = ( ^ [X3: real,Y6: real] :
% 0.16/0.43 ( ( ord_less_real @ X3 @ Y6 )
% 0.16/0.43 | ( X3 = Y6 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less
% 0.16/0.43 thf(fact_1732_order__less__le,axiom,
% 0.16/0.43 ( ord_less_set_int
% 0.16/0.43 = ( ^ [X3: set_int,Y6: set_int] :
% 0.16/0.43 ( ( ord_less_eq_set_int @ X3 @ Y6 )
% 0.16/0.43 & ( X3 != Y6 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le
% 0.16/0.43 thf(fact_1733_order__less__le,axiom,
% 0.16/0.43 ( ord_less_rat
% 0.16/0.43 = ( ^ [X3: rat,Y6: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ X3 @ Y6 )
% 0.16/0.43 & ( X3 != Y6 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le
% 0.16/0.43 thf(fact_1734_order__less__le,axiom,
% 0.16/0.43 ( ord_less_num
% 0.16/0.43 = ( ^ [X3: num,Y6: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ X3 @ Y6 )
% 0.16/0.43 & ( X3 != Y6 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le
% 0.16/0.43 thf(fact_1735_order__less__le,axiom,
% 0.16/0.43 ( ord_less_nat
% 0.16/0.43 = ( ^ [X3: nat,Y6: nat] :
% 0.16/0.43 ( ( ord_less_eq_nat @ X3 @ Y6 )
% 0.16/0.43 & ( X3 != Y6 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le
% 0.16/0.43 thf(fact_1736_order__less__le,axiom,
% 0.16/0.43 ( ord_less_int
% 0.16/0.43 = ( ^ [X3: int,Y6: int] :
% 0.16/0.43 ( ( ord_less_eq_int @ X3 @ Y6 )
% 0.16/0.43 & ( X3 != Y6 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le
% 0.16/0.43 thf(fact_1737_order__less__le,axiom,
% 0.16/0.43 ( ord_less_real
% 0.16/0.43 = ( ^ [X3: real,Y6: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ X3 @ Y6 )
% 0.16/0.43 & ( X3 != Y6 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le
% 0.16/0.43 thf(fact_1738_linorder__not__le,axiom,
% 0.16/0.43 ! [X: rat,Y4: rat] :
% 0.16/0.43 ( ( ~ ( ord_less_eq_rat @ X @ Y4 ) )
% 0.16/0.43 = ( ord_less_rat @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_not_le
% 0.16/0.43 thf(fact_1739_linorder__not__le,axiom,
% 0.16/0.43 ! [X: num,Y4: num] :
% 0.16/0.43 ( ( ~ ( ord_less_eq_num @ X @ Y4 ) )
% 0.16/0.43 = ( ord_less_num @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_not_le
% 0.16/0.43 thf(fact_1740_linorder__not__le,axiom,
% 0.16/0.43 ! [X: nat,Y4: nat] :
% 0.16/0.43 ( ( ~ ( ord_less_eq_nat @ X @ Y4 ) )
% 0.16/0.43 = ( ord_less_nat @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_not_le
% 0.16/0.43 thf(fact_1741_linorder__not__le,axiom,
% 0.16/0.43 ! [X: int,Y4: int] :
% 0.16/0.43 ( ( ~ ( ord_less_eq_int @ X @ Y4 ) )
% 0.16/0.43 = ( ord_less_int @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_not_le
% 0.16/0.43 thf(fact_1742_linorder__not__le,axiom,
% 0.16/0.43 ! [X: real,Y4: real] :
% 0.16/0.43 ( ( ~ ( ord_less_eq_real @ X @ Y4 ) )
% 0.16/0.43 = ( ord_less_real @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_not_le
% 0.16/0.43 thf(fact_1743_linorder__not__less,axiom,
% 0.16/0.43 ! [X: rat,Y4: rat] :
% 0.16/0.43 ( ( ~ ( ord_less_rat @ X @ Y4 ) )
% 0.16/0.43 = ( ord_less_eq_rat @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_not_less
% 0.16/0.43 thf(fact_1744_linorder__not__less,axiom,
% 0.16/0.43 ! [X: num,Y4: num] :
% 0.16/0.43 ( ( ~ ( ord_less_num @ X @ Y4 ) )
% 0.16/0.43 = ( ord_less_eq_num @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_not_less
% 0.16/0.43 thf(fact_1745_linorder__not__less,axiom,
% 0.16/0.43 ! [X: nat,Y4: nat] :
% 0.16/0.43 ( ( ~ ( ord_less_nat @ X @ Y4 ) )
% 0.16/0.43 = ( ord_less_eq_nat @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_not_less
% 0.16/0.43 thf(fact_1746_linorder__not__less,axiom,
% 0.16/0.43 ! [X: int,Y4: int] :
% 0.16/0.43 ( ( ~ ( ord_less_int @ X @ Y4 ) )
% 0.16/0.43 = ( ord_less_eq_int @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_not_less
% 0.16/0.43 thf(fact_1747_linorder__not__less,axiom,
% 0.16/0.43 ! [X: real,Y4: real] :
% 0.16/0.43 ( ( ~ ( ord_less_real @ X @ Y4 ) )
% 0.16/0.43 = ( ord_less_eq_real @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_not_less
% 0.16/0.43 thf(fact_1748_order__less__imp__le,axiom,
% 0.16/0.43 ! [X: set_int,Y4: set_int] :
% 0.16/0.43 ( ( ord_less_set_int @ X @ Y4 )
% 0.16/0.43 => ( ord_less_eq_set_int @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_le
% 0.16/0.43 thf(fact_1749_order__less__imp__le,axiom,
% 0.16/0.43 ! [X: rat,Y4: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X @ Y4 )
% 0.16/0.43 => ( ord_less_eq_rat @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_le
% 0.16/0.43 thf(fact_1750_order__less__imp__le,axiom,
% 0.16/0.43 ! [X: num,Y4: num] :
% 0.16/0.43 ( ( ord_less_num @ X @ Y4 )
% 0.16/0.43 => ( ord_less_eq_num @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_le
% 0.16/0.43 thf(fact_1751_order__less__imp__le,axiom,
% 0.16/0.43 ! [X: nat,Y4: nat] :
% 0.16/0.43 ( ( ord_less_nat @ X @ Y4 )
% 0.16/0.43 => ( ord_less_eq_nat @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_le
% 0.16/0.43 thf(fact_1752_order__less__imp__le,axiom,
% 0.16/0.43 ! [X: int,Y4: int] :
% 0.16/0.43 ( ( ord_less_int @ X @ Y4 )
% 0.16/0.43 => ( ord_less_eq_int @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_le
% 0.16/0.43 thf(fact_1753_order__less__imp__le,axiom,
% 0.16/0.43 ! [X: real,Y4: real] :
% 0.16/0.43 ( ( ord_less_real @ X @ Y4 )
% 0.16/0.43 => ( ord_less_eq_real @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_imp_le
% 0.16/0.43 thf(fact_1754_order__le__neq__trans,axiom,
% 0.16/0.43 ! [A: set_int,B: set_int] :
% 0.16/0.43 ( ( ord_less_eq_set_int @ A @ B )
% 0.16/0.43 => ( ( A != B )
% 0.16/0.43 => ( ord_less_set_int @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_neq_trans
% 0.16/0.43 thf(fact_1755_order__le__neq__trans,axiom,
% 0.16/0.43 ! [A: rat,B: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.43 => ( ( A != B )
% 0.16/0.43 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_neq_trans
% 0.16/0.43 thf(fact_1756_order__le__neq__trans,axiom,
% 0.16/0.43 ! [A: num,B: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.43 => ( ( A != B )
% 0.16/0.43 => ( ord_less_num @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_neq_trans
% 0.16/0.43 thf(fact_1757_order__le__neq__trans,axiom,
% 0.16/0.43 ! [A: nat,B: nat] :
% 0.16/0.43 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.43 => ( ( A != B )
% 0.16/0.43 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_neq_trans
% 0.16/0.43 thf(fact_1758_order__le__neq__trans,axiom,
% 0.16/0.43 ! [A: int,B: int] :
% 0.16/0.43 ( ( ord_less_eq_int @ A @ B )
% 0.16/0.43 => ( ( A != B )
% 0.16/0.43 => ( ord_less_int @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_neq_trans
% 0.16/0.43 thf(fact_1759_order__le__neq__trans,axiom,
% 0.16/0.43 ! [A: real,B: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.43 => ( ( A != B )
% 0.16/0.43 => ( ord_less_real @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_neq_trans
% 0.16/0.43 thf(fact_1760_order__neq__le__trans,axiom,
% 0.16/0.43 ! [A: set_int,B: set_int] :
% 0.16/0.43 ( ( A != B )
% 0.16/0.43 => ( ( ord_less_eq_set_int @ A @ B )
% 0.16/0.43 => ( ord_less_set_int @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_neq_le_trans
% 0.16/0.43 thf(fact_1761_order__neq__le__trans,axiom,
% 0.16/0.43 ! [A: rat,B: rat] :
% 0.16/0.43 ( ( A != B )
% 0.16/0.43 => ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.43 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_neq_le_trans
% 0.16/0.43 thf(fact_1762_order__neq__le__trans,axiom,
% 0.16/0.43 ! [A: num,B: num] :
% 0.16/0.43 ( ( A != B )
% 0.16/0.43 => ( ( ord_less_eq_num @ A @ B )
% 0.16/0.43 => ( ord_less_num @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_neq_le_trans
% 0.16/0.43 thf(fact_1763_order__neq__le__trans,axiom,
% 0.16/0.43 ! [A: nat,B: nat] :
% 0.16/0.43 ( ( A != B )
% 0.16/0.43 => ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.43 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_neq_le_trans
% 0.16/0.43 thf(fact_1764_order__neq__le__trans,axiom,
% 0.16/0.43 ! [A: int,B: int] :
% 0.16/0.43 ( ( A != B )
% 0.16/0.43 => ( ( ord_less_eq_int @ A @ B )
% 0.16/0.43 => ( ord_less_int @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_neq_le_trans
% 0.16/0.43 thf(fact_1765_order__neq__le__trans,axiom,
% 0.16/0.43 ! [A: real,B: real] :
% 0.16/0.43 ( ( A != B )
% 0.16/0.43 => ( ( ord_less_eq_real @ A @ B )
% 0.16/0.43 => ( ord_less_real @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_neq_le_trans
% 0.16/0.43 thf(fact_1766_order__le__less__trans,axiom,
% 0.16/0.43 ! [X: set_int,Y4: set_int,Z: set_int] :
% 0.16/0.43 ( ( ord_less_eq_set_int @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_set_int @ Y4 @ Z )
% 0.16/0.43 => ( ord_less_set_int @ X @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_trans
% 0.16/0.43 thf(fact_1767_order__le__less__trans,axiom,
% 0.16/0.43 ! [X: rat,Y4: rat,Z: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_rat @ Y4 @ Z )
% 0.16/0.43 => ( ord_less_rat @ X @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_trans
% 0.16/0.43 thf(fact_1768_order__le__less__trans,axiom,
% 0.16/0.43 ! [X: num,Y4: num,Z: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_num @ Y4 @ Z )
% 0.16/0.43 => ( ord_less_num @ X @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_trans
% 0.16/0.43 thf(fact_1769_order__le__less__trans,axiom,
% 0.16/0.43 ! [X: nat,Y4: nat,Z: nat] :
% 0.16/0.43 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_nat @ Y4 @ Z )
% 0.16/0.43 => ( ord_less_nat @ X @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_trans
% 0.16/0.43 thf(fact_1770_order__le__less__trans,axiom,
% 0.16/0.43 ! [X: int,Y4: int,Z: int] :
% 0.16/0.43 ( ( ord_less_eq_int @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_int @ Y4 @ Z )
% 0.16/0.43 => ( ord_less_int @ X @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_trans
% 0.16/0.43 thf(fact_1771_order__le__less__trans,axiom,
% 0.16/0.43 ! [X: real,Y4: real,Z: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_real @ Y4 @ Z )
% 0.16/0.43 => ( ord_less_real @ X @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_trans
% 0.16/0.43 thf(fact_1772_order__less__le__trans,axiom,
% 0.16/0.43 ! [X: set_int,Y4: set_int,Z: set_int] :
% 0.16/0.43 ( ( ord_less_set_int @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_eq_set_int @ Y4 @ Z )
% 0.16/0.43 => ( ord_less_set_int @ X @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_trans
% 0.16/0.43 thf(fact_1773_order__less__le__trans,axiom,
% 0.16/0.43 ! [X: rat,Y4: rat,Z: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_eq_rat @ Y4 @ Z )
% 0.16/0.43 => ( ord_less_rat @ X @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_trans
% 0.16/0.43 thf(fact_1774_order__less__le__trans,axiom,
% 0.16/0.43 ! [X: num,Y4: num,Z: num] :
% 0.16/0.43 ( ( ord_less_num @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_eq_num @ Y4 @ Z )
% 0.16/0.43 => ( ord_less_num @ X @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_trans
% 0.16/0.43 thf(fact_1775_order__less__le__trans,axiom,
% 0.16/0.43 ! [X: nat,Y4: nat,Z: nat] :
% 0.16/0.43 ( ( ord_less_nat @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_eq_nat @ Y4 @ Z )
% 0.16/0.43 => ( ord_less_nat @ X @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_trans
% 0.16/0.43 thf(fact_1776_order__less__le__trans,axiom,
% 0.16/0.43 ! [X: int,Y4: int,Z: int] :
% 0.16/0.43 ( ( ord_less_int @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_eq_int @ Y4 @ Z )
% 0.16/0.43 => ( ord_less_int @ X @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_trans
% 0.16/0.43 thf(fact_1777_order__less__le__trans,axiom,
% 0.16/0.43 ! [X: real,Y4: real,Z: real] :
% 0.16/0.43 ( ( ord_less_real @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_eq_real @ Y4 @ Z )
% 0.16/0.43 => ( ord_less_real @ X @ Z ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_trans
% 0.16/0.43 thf(fact_1778_order__le__less__subst1,axiom,
% 0.16/0.43 ! [A: rat,F: real > rat,B: real,C: real] :
% 0.16/0.43 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_real @ B @ C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst1
% 0.16/0.43 thf(fact_1779_order__le__less__subst1,axiom,
% 0.16/0.43 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_rat @ B @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst1
% 0.16/0.43 thf(fact_1780_order__le__less__subst1,axiom,
% 0.16/0.43 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.16/0.43 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_num @ B @ C )
% 0.16/0.43 => ( ! [X4: num,Y: num] :
% 0.16/0.43 ( ( ord_less_num @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst1
% 0.16/0.43 thf(fact_1781_order__le__less__subst1,axiom,
% 0.16/0.43 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_nat @ B @ C )
% 0.16/0.43 => ( ! [X4: nat,Y: nat] :
% 0.16/0.43 ( ( ord_less_nat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst1
% 0.16/0.43 thf(fact_1782_order__le__less__subst1,axiom,
% 0.16/0.43 ! [A: rat,F: int > rat,B: int,C: int] :
% 0.16/0.43 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_int @ B @ C )
% 0.16/0.43 => ( ! [X4: int,Y: int] :
% 0.16/0.43 ( ( ord_less_int @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst1
% 0.16/0.43 thf(fact_1783_order__le__less__subst1,axiom,
% 0.16/0.43 ! [A: num,F: real > num,B: real,C: real] :
% 0.16/0.43 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_real @ B @ C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst1
% 0.16/0.43 thf(fact_1784_order__le__less__subst1,axiom,
% 0.16/0.43 ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.16/0.43 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_rat @ B @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst1
% 0.16/0.43 thf(fact_1785_order__le__less__subst1,axiom,
% 0.16/0.43 ! [A: num,F: num > num,B: num,C: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_num @ B @ C )
% 0.16/0.43 => ( ! [X4: num,Y: num] :
% 0.16/0.43 ( ( ord_less_num @ X4 @ Y )
% 0.16/0.43 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst1
% 0.16/0.43 thf(fact_1786_order__le__less__subst1,axiom,
% 0.16/0.43 ! [A: num,F: nat > num,B: nat,C: nat] :
% 0.16/0.43 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_nat @ B @ C )
% 0.16/0.43 => ( ! [X4: nat,Y: nat] :
% 0.16/0.43 ( ( ord_less_nat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst1
% 0.16/0.43 thf(fact_1787_order__le__less__subst1,axiom,
% 0.16/0.43 ! [A: num,F: int > num,B: int,C: int] :
% 0.16/0.43 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_int @ B @ C )
% 0.16/0.43 => ( ! [X4: int,Y: int] :
% 0.16/0.43 ( ( ord_less_int @ X4 @ Y )
% 0.16/0.43 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst1
% 0.16/0.43 thf(fact_1788_order__le__less__subst2,axiom,
% 0.16/0.43 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.43 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst2
% 0.16/0.43 thf(fact_1789_order__le__less__subst2,axiom,
% 0.16/0.43 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.16/0.43 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.43 => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst2
% 0.16/0.43 thf(fact_1790_order__le__less__subst2,axiom,
% 0.16/0.43 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.43 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst2
% 0.16/0.43 thf(fact_1791_order__le__less__subst2,axiom,
% 0.16/0.43 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.16/0.43 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.43 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst2
% 0.16/0.43 thf(fact_1792_order__le__less__subst2,axiom,
% 0.16/0.43 ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.16/0.43 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.43 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst2
% 0.16/0.43 thf(fact_1793_order__le__less__subst2,axiom,
% 0.16/0.43 ! [A: num,B: num,F: num > rat,C: rat] :
% 0.16/0.43 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.43 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: num,Y: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst2
% 0.16/0.43 thf(fact_1794_order__le__less__subst2,axiom,
% 0.16/0.43 ! [A: num,B: num,F: num > num,C: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.43 => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: num,Y: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst2
% 0.16/0.43 thf(fact_1795_order__le__less__subst2,axiom,
% 0.16/0.43 ! [A: num,B: num,F: num > nat,C: nat] :
% 0.16/0.43 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.43 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: num,Y: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst2
% 0.16/0.43 thf(fact_1796_order__le__less__subst2,axiom,
% 0.16/0.43 ! [A: num,B: num,F: num > int,C: int] :
% 0.16/0.43 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.43 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: num,Y: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst2
% 0.16/0.43 thf(fact_1797_order__le__less__subst2,axiom,
% 0.16/0.43 ! [A: num,B: num,F: num > real,C: real] :
% 0.16/0.43 ( ( ord_less_eq_num @ A @ B )
% 0.16/0.43 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: num,Y: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_less_subst2
% 0.16/0.43 thf(fact_1798_order__less__le__subst1,axiom,
% 0.16/0.43 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.16/0.43 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_eq_rat @ B @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst1
% 0.16/0.43 thf(fact_1799_order__less__le__subst1,axiom,
% 0.16/0.43 ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.16/0.43 ( ( ord_less_num @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_eq_rat @ B @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst1
% 0.16/0.43 thf(fact_1800_order__less__le__subst1,axiom,
% 0.16/0.43 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.16/0.43 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_eq_rat @ B @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst1
% 0.16/0.43 thf(fact_1801_order__less__le__subst1,axiom,
% 0.16/0.43 ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.16/0.43 ( ( ord_less_int @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_eq_rat @ B @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst1
% 0.16/0.43 thf(fact_1802_order__less__le__subst1,axiom,
% 0.16/0.43 ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.16/0.43 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_eq_rat @ B @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst1
% 0.16/0.43 thf(fact_1803_order__less__le__subst1,axiom,
% 0.16/0.43 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.16/0.43 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_eq_num @ B @ C )
% 0.16/0.43 => ( ! [X4: num,Y: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst1
% 0.16/0.43 thf(fact_1804_order__less__le__subst1,axiom,
% 0.16/0.43 ! [A: num,F: num > num,B: num,C: num] :
% 0.16/0.43 ( ( ord_less_num @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_eq_num @ B @ C )
% 0.16/0.43 => ( ! [X4: num,Y: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst1
% 0.16/0.43 thf(fact_1805_order__less__le__subst1,axiom,
% 0.16/0.43 ! [A: nat,F: num > nat,B: num,C: num] :
% 0.16/0.43 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_eq_num @ B @ C )
% 0.16/0.43 => ( ! [X4: num,Y: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst1
% 0.16/0.43 thf(fact_1806_order__less__le__subst1,axiom,
% 0.16/0.43 ! [A: int,F: num > int,B: num,C: num] :
% 0.16/0.43 ( ( ord_less_int @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_eq_num @ B @ C )
% 0.16/0.43 => ( ! [X4: num,Y: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst1
% 0.16/0.43 thf(fact_1807_order__less__le__subst1,axiom,
% 0.16/0.43 ! [A: real,F: num > real,B: num,C: num] :
% 0.16/0.43 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.16/0.43 => ( ( ord_less_eq_num @ B @ C )
% 0.16/0.43 => ( ! [X4: num,Y: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ X4 @ Y )
% 0.16/0.43 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst1
% 0.16/0.43 thf(fact_1808_order__less__le__subst2,axiom,
% 0.16/0.43 ! [A: real,B: real,F: real > rat,C: rat] :
% 0.16/0.43 ( ( ord_less_real @ A @ B )
% 0.16/0.43 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst2
% 0.16/0.43 thf(fact_1809_order__less__le__subst2,axiom,
% 0.16/0.43 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.16/0.43 ( ( ord_less_rat @ A @ B )
% 0.16/0.43 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst2
% 0.16/0.43 thf(fact_1810_order__less__le__subst2,axiom,
% 0.16/0.43 ! [A: num,B: num,F: num > rat,C: rat] :
% 0.16/0.43 ( ( ord_less_num @ A @ B )
% 0.16/0.43 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: num,Y: num] :
% 0.16/0.43 ( ( ord_less_num @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst2
% 0.16/0.43 thf(fact_1811_order__less__le__subst2,axiom,
% 0.16/0.43 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 0.16/0.43 ( ( ord_less_nat @ A @ B )
% 0.16/0.43 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: nat,Y: nat] :
% 0.16/0.43 ( ( ord_less_nat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst2
% 0.16/0.43 thf(fact_1812_order__less__le__subst2,axiom,
% 0.16/0.43 ! [A: int,B: int,F: int > rat,C: rat] :
% 0.16/0.43 ( ( ord_less_int @ A @ B )
% 0.16/0.43 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: int,Y: int] :
% 0.16/0.43 ( ( ord_less_int @ X4 @ Y )
% 0.16/0.43 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst2
% 0.16/0.43 thf(fact_1813_order__less__le__subst2,axiom,
% 0.16/0.43 ! [A: real,B: real,F: real > num,C: num] :
% 0.16/0.43 ( ( ord_less_real @ A @ B )
% 0.16/0.43 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: real,Y: real] :
% 0.16/0.43 ( ( ord_less_real @ X4 @ Y )
% 0.16/0.43 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst2
% 0.16/0.43 thf(fact_1814_order__less__le__subst2,axiom,
% 0.16/0.43 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.16/0.43 ( ( ord_less_rat @ A @ B )
% 0.16/0.43 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: rat,Y: rat] :
% 0.16/0.43 ( ( ord_less_rat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst2
% 0.16/0.43 thf(fact_1815_order__less__le__subst2,axiom,
% 0.16/0.43 ! [A: num,B: num,F: num > num,C: num] :
% 0.16/0.43 ( ( ord_less_num @ A @ B )
% 0.16/0.43 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: num,Y: num] :
% 0.16/0.43 ( ( ord_less_num @ X4 @ Y )
% 0.16/0.43 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst2
% 0.16/0.43 thf(fact_1816_order__less__le__subst2,axiom,
% 0.16/0.43 ! [A: nat,B: nat,F: nat > num,C: num] :
% 0.16/0.43 ( ( ord_less_nat @ A @ B )
% 0.16/0.43 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: nat,Y: nat] :
% 0.16/0.43 ( ( ord_less_nat @ X4 @ Y )
% 0.16/0.43 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst2
% 0.16/0.43 thf(fact_1817_order__less__le__subst2,axiom,
% 0.16/0.43 ! [A: int,B: int,F: int > num,C: num] :
% 0.16/0.43 ( ( ord_less_int @ A @ B )
% 0.16/0.43 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.16/0.43 => ( ! [X4: int,Y: int] :
% 0.16/0.43 ( ( ord_less_int @ X4 @ Y )
% 0.16/0.43 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 0.16/0.43 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_less_le_subst2
% 0.16/0.43 thf(fact_1818_linorder__le__less__linear,axiom,
% 0.16/0.43 ! [X: rat,Y4: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ X @ Y4 )
% 0.16/0.43 | ( ord_less_rat @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_le_less_linear
% 0.16/0.43 thf(fact_1819_linorder__le__less__linear,axiom,
% 0.16/0.43 ! [X: num,Y4: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ X @ Y4 )
% 0.16/0.43 | ( ord_less_num @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_le_less_linear
% 0.16/0.43 thf(fact_1820_linorder__le__less__linear,axiom,
% 0.16/0.43 ! [X: nat,Y4: nat] :
% 0.16/0.43 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.16/0.43 | ( ord_less_nat @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_le_less_linear
% 0.16/0.43 thf(fact_1821_linorder__le__less__linear,axiom,
% 0.16/0.43 ! [X: int,Y4: int] :
% 0.16/0.43 ( ( ord_less_eq_int @ X @ Y4 )
% 0.16/0.43 | ( ord_less_int @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_le_less_linear
% 0.16/0.43 thf(fact_1822_linorder__le__less__linear,axiom,
% 0.16/0.43 ! [X: real,Y4: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ X @ Y4 )
% 0.16/0.43 | ( ord_less_real @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % linorder_le_less_linear
% 0.16/0.43 thf(fact_1823_order__le__imp__less__or__eq,axiom,
% 0.16/0.43 ! [X: set_int,Y4: set_int] :
% 0.16/0.43 ( ( ord_less_eq_set_int @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_set_int @ X @ Y4 )
% 0.16/0.43 | ( X = Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_imp_less_or_eq
% 0.16/0.43 thf(fact_1824_order__le__imp__less__or__eq,axiom,
% 0.16/0.43 ! [X: rat,Y4: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_rat @ X @ Y4 )
% 0.16/0.43 | ( X = Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_imp_less_or_eq
% 0.16/0.43 thf(fact_1825_order__le__imp__less__or__eq,axiom,
% 0.16/0.43 ! [X: num,Y4: num] :
% 0.16/0.43 ( ( ord_less_eq_num @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_num @ X @ Y4 )
% 0.16/0.43 | ( X = Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_imp_less_or_eq
% 0.16/0.43 thf(fact_1826_order__le__imp__less__or__eq,axiom,
% 0.16/0.43 ! [X: nat,Y4: nat] :
% 0.16/0.43 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_nat @ X @ Y4 )
% 0.16/0.43 | ( X = Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_imp_less_or_eq
% 0.16/0.43 thf(fact_1827_order__le__imp__less__or__eq,axiom,
% 0.16/0.43 ! [X: int,Y4: int] :
% 0.16/0.43 ( ( ord_less_eq_int @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_int @ X @ Y4 )
% 0.16/0.43 | ( X = Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_imp_less_or_eq
% 0.16/0.43 thf(fact_1828_order__le__imp__less__or__eq,axiom,
% 0.16/0.43 ! [X: real,Y4: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ X @ Y4 )
% 0.16/0.43 => ( ( ord_less_real @ X @ Y4 )
% 0.16/0.43 | ( X = Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % order_le_imp_less_or_eq
% 0.16/0.43 thf(fact_1829_less__numeral__extra_I3_J,axiom,
% 0.16/0.43 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 0.16/0.43
% 0.16/0.43 % less_numeral_extra(3)
% 0.16/0.43 thf(fact_1830_less__numeral__extra_I3_J,axiom,
% 0.16/0.43 ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 0.16/0.43
% 0.16/0.43 % less_numeral_extra(3)
% 0.16/0.43 thf(fact_1831_less__numeral__extra_I3_J,axiom,
% 0.16/0.43 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.16/0.43
% 0.16/0.43 % less_numeral_extra(3)
% 0.16/0.43 thf(fact_1832_less__numeral__extra_I3_J,axiom,
% 0.16/0.43 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.16/0.43
% 0.16/0.43 % less_numeral_extra(3)
% 0.16/0.43 thf(fact_1833_field__lbound__gt__zero,axiom,
% 0.16/0.43 ! [D1: real,D22: real] :
% 0.16/0.43 ( ( ord_less_real @ zero_zero_real @ D1 )
% 0.16/0.43 => ( ( ord_less_real @ zero_zero_real @ D22 )
% 0.16/0.43 => ? [E: real] :
% 0.16/0.43 ( ( ord_less_real @ zero_zero_real @ E )
% 0.16/0.43 & ( ord_less_real @ E @ D1 )
% 0.16/0.43 & ( ord_less_real @ E @ D22 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % field_lbound_gt_zero
% 0.16/0.43 thf(fact_1834_field__lbound__gt__zero,axiom,
% 0.16/0.43 ! [D1: rat,D22: rat] :
% 0.16/0.43 ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 0.16/0.43 => ( ( ord_less_rat @ zero_zero_rat @ D22 )
% 0.16/0.43 => ? [E: rat] :
% 0.16/0.43 ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.16/0.43 & ( ord_less_rat @ E @ D1 )
% 0.16/0.43 & ( ord_less_rat @ E @ D22 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % field_lbound_gt_zero
% 0.16/0.43 thf(fact_1835_zero__less__iff__neq__zero,axiom,
% 0.16/0.43 ! [N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.43 = ( N != zero_zero_nat ) ) ).
% 0.16/0.43
% 0.16/0.43 % zero_less_iff_neq_zero
% 0.16/0.43 thf(fact_1836_gr__implies__not__zero,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ M @ N )
% 0.16/0.43 => ( N != zero_zero_nat ) ) ).
% 0.16/0.43
% 0.16/0.43 % gr_implies_not_zero
% 0.16/0.43 thf(fact_1837_not__less__zero,axiom,
% 0.16/0.43 ! [N: nat] :
% 0.16/0.43 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.16/0.43
% 0.16/0.43 % not_less_zero
% 0.16/0.43 thf(fact_1838_gr__zeroI,axiom,
% 0.16/0.43 ! [N: nat] :
% 0.16/0.43 ( ( N != zero_zero_nat )
% 0.16/0.43 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.16/0.43
% 0.16/0.43 % gr_zeroI
% 0.16/0.43 thf(fact_1839_zdiff__int__split,axiom,
% 0.16/0.43 ! [P: int > $o,X: nat,Y4: nat] :
% 0.16/0.43 ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y4 ) ) )
% 0.16/0.43 = ( ( ( ord_less_eq_nat @ Y4 @ X )
% 0.16/0.43 => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y4 ) ) ) )
% 0.16/0.43 & ( ( ord_less_nat @ X @ Y4 )
% 0.16/0.43 => ( P @ zero_zero_int ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % zdiff_int_split
% 0.16/0.43 thf(fact_1840_diff__strict__right__mono,axiom,
% 0.16/0.43 ! [A: real,B: real,C: real] :
% 0.16/0.43 ( ( ord_less_real @ A @ B )
% 0.16/0.43 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % diff_strict_right_mono
% 0.16/0.43 thf(fact_1841_diff__strict__right__mono,axiom,
% 0.16/0.43 ! [A: rat,B: rat,C: rat] :
% 0.16/0.43 ( ( ord_less_rat @ A @ B )
% 0.16/0.43 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % diff_strict_right_mono
% 0.16/0.43 thf(fact_1842_diff__strict__right__mono,axiom,
% 0.16/0.43 ! [A: int,B: int,C: int] :
% 0.16/0.43 ( ( ord_less_int @ A @ B )
% 0.16/0.43 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % diff_strict_right_mono
% 0.16/0.43 thf(fact_1843_diff__strict__left__mono,axiom,
% 0.16/0.43 ! [B: real,A: real,C: real] :
% 0.16/0.43 ( ( ord_less_real @ B @ A )
% 0.16/0.43 => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % diff_strict_left_mono
% 0.16/0.43 thf(fact_1844_diff__strict__left__mono,axiom,
% 0.16/0.43 ! [B: rat,A: rat,C: rat] :
% 0.16/0.43 ( ( ord_less_rat @ B @ A )
% 0.16/0.43 => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % diff_strict_left_mono
% 0.16/0.43 thf(fact_1845_diff__strict__left__mono,axiom,
% 0.16/0.43 ! [B: int,A: int,C: int] :
% 0.16/0.43 ( ( ord_less_int @ B @ A )
% 0.16/0.43 => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % diff_strict_left_mono
% 0.16/0.43 thf(fact_1846_diff__eq__diff__less,axiom,
% 0.16/0.43 ! [A: real,B: real,C: real,D: real] :
% 0.16/0.43 ( ( ( minus_minus_real @ A @ B )
% 0.16/0.43 = ( minus_minus_real @ C @ D ) )
% 0.16/0.43 => ( ( ord_less_real @ A @ B )
% 0.16/0.43 = ( ord_less_real @ C @ D ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % diff_eq_diff_less
% 0.16/0.43 thf(fact_1847_diff__eq__diff__less,axiom,
% 0.16/0.43 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.16/0.43 ( ( ( minus_minus_rat @ A @ B )
% 0.16/0.43 = ( minus_minus_rat @ C @ D ) )
% 0.16/0.43 => ( ( ord_less_rat @ A @ B )
% 0.16/0.43 = ( ord_less_rat @ C @ D ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % diff_eq_diff_less
% 0.16/0.43 thf(fact_1848_diff__eq__diff__less,axiom,
% 0.16/0.43 ! [A: int,B: int,C: int,D: int] :
% 0.16/0.43 ( ( ( minus_minus_int @ A @ B )
% 0.16/0.43 = ( minus_minus_int @ C @ D ) )
% 0.16/0.43 => ( ( ord_less_int @ A @ B )
% 0.16/0.43 = ( ord_less_int @ C @ D ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % diff_eq_diff_less
% 0.16/0.43 thf(fact_1849_diff__strict__mono,axiom,
% 0.16/0.43 ! [A: real,B: real,D: real,C: real] :
% 0.16/0.43 ( ( ord_less_real @ A @ B )
% 0.16/0.43 => ( ( ord_less_real @ D @ C )
% 0.16/0.43 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % diff_strict_mono
% 0.16/0.43 thf(fact_1850_diff__strict__mono,axiom,
% 0.16/0.43 ! [A: rat,B: rat,D: rat,C: rat] :
% 0.16/0.43 ( ( ord_less_rat @ A @ B )
% 0.16/0.43 => ( ( ord_less_rat @ D @ C )
% 0.16/0.43 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % diff_strict_mono
% 0.16/0.43 thf(fact_1851_diff__strict__mono,axiom,
% 0.16/0.43 ! [A: int,B: int,D: int,C: int] :
% 0.16/0.43 ( ( ord_less_int @ A @ B )
% 0.16/0.43 => ( ( ord_less_int @ D @ C )
% 0.16/0.43 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % diff_strict_mono
% 0.16/0.43 thf(fact_1852_double__diff,axiom,
% 0.16/0.43 ! [A2: set_nat,B4: set_nat,C2: set_nat] :
% 0.16/0.43 ( ( ord_less_eq_set_nat @ A2 @ B4 )
% 0.16/0.43 => ( ( ord_less_eq_set_nat @ B4 @ C2 )
% 0.16/0.43 => ( ( minus_minus_set_nat @ B4 @ ( minus_minus_set_nat @ C2 @ A2 ) )
% 0.16/0.43 = A2 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % double_diff
% 0.16/0.43 thf(fact_1853_double__diff,axiom,
% 0.16/0.43 ! [A2: set_int,B4: set_int,C2: set_int] :
% 0.16/0.43 ( ( ord_less_eq_set_int @ A2 @ B4 )
% 0.16/0.43 => ( ( ord_less_eq_set_int @ B4 @ C2 )
% 0.16/0.43 => ( ( minus_minus_set_int @ B4 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 0.16/0.43 = A2 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % double_diff
% 0.16/0.43 thf(fact_1854_Diff__subset,axiom,
% 0.16/0.43 ! [A2: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B4 ) @ A2 ) ).
% 0.16/0.43
% 0.16/0.43 % Diff_subset
% 0.16/0.43 thf(fact_1855_Diff__subset,axiom,
% 0.16/0.43 ! [A2: set_int,B4: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B4 ) @ A2 ) ).
% 0.16/0.43
% 0.16/0.43 % Diff_subset
% 0.16/0.43 thf(fact_1856_Diff__mono,axiom,
% 0.16/0.43 ! [A2: set_nat,C2: set_nat,D3: set_nat,B4: set_nat] :
% 0.16/0.43 ( ( ord_less_eq_set_nat @ A2 @ C2 )
% 0.16/0.43 => ( ( ord_less_eq_set_nat @ D3 @ B4 )
% 0.16/0.43 => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B4 ) @ ( minus_minus_set_nat @ C2 @ D3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % Diff_mono
% 0.16/0.43 thf(fact_1857_Diff__mono,axiom,
% 0.16/0.43 ! [A2: set_int,C2: set_int,D3: set_int,B4: set_int] :
% 0.16/0.43 ( ( ord_less_eq_set_int @ A2 @ C2 )
% 0.16/0.43 => ( ( ord_less_eq_set_int @ D3 @ B4 )
% 0.16/0.43 => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B4 ) @ ( minus_minus_set_int @ C2 @ D3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % Diff_mono
% 0.16/0.43 thf(fact_1858_le__numeral__extra_I4_J,axiom,
% 0.16/0.43 ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 0.16/0.43
% 0.16/0.43 % le_numeral_extra(4)
% 0.16/0.43 thf(fact_1859_le__numeral__extra_I4_J,axiom,
% 0.16/0.43 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 0.16/0.43
% 0.16/0.43 % le_numeral_extra(4)
% 0.16/0.43 thf(fact_1860_le__numeral__extra_I4_J,axiom,
% 0.16/0.43 ord_less_eq_int @ one_one_int @ one_one_int ).
% 0.16/0.43
% 0.16/0.43 % le_numeral_extra(4)
% 0.16/0.43 thf(fact_1861_le__numeral__extra_I4_J,axiom,
% 0.16/0.43 ord_less_eq_real @ one_one_real @ one_one_real ).
% 0.16/0.43
% 0.16/0.43 % le_numeral_extra(4)
% 0.16/0.43 thf(fact_1862_Nat_OlessE,axiom,
% 0.16/0.43 ! [I: nat,K: nat] :
% 0.16/0.43 ( ( ord_less_nat @ I @ K )
% 0.16/0.43 => ( ( K
% 0.16/0.43 != ( suc @ I ) )
% 0.16/0.43 => ~ ! [J2: nat] :
% 0.16/0.43 ( ( ord_less_nat @ I @ J2 )
% 0.16/0.43 => ( K
% 0.16/0.43 != ( suc @ J2 ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % Nat.lessE
% 0.16/0.43 thf(fact_1863_Suc__lessD,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 0.16/0.43 => ( ord_less_nat @ M @ N ) ) ).
% 0.16/0.43
% 0.16/0.43 % Suc_lessD
% 0.16/0.43 thf(fact_1864_Suc__lessE,axiom,
% 0.16/0.43 ! [I: nat,K: nat] :
% 0.16/0.43 ( ( ord_less_nat @ ( suc @ I ) @ K )
% 0.16/0.43 => ~ ! [J2: nat] :
% 0.16/0.43 ( ( ord_less_nat @ I @ J2 )
% 0.16/0.43 => ( K
% 0.16/0.43 != ( suc @ J2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % Suc_lessE
% 0.16/0.43 thf(fact_1865_Suc__lessI,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ M @ N )
% 0.16/0.43 => ( ( ( suc @ M )
% 0.16/0.43 != N )
% 0.16/0.43 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % Suc_lessI
% 0.16/0.43 thf(fact_1866_less__SucE,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.16/0.43 => ( ~ ( ord_less_nat @ M @ N )
% 0.16/0.43 => ( M = N ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_SucE
% 0.16/0.43 thf(fact_1867_less__SucI,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ M @ N )
% 0.16/0.43 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_SucI
% 0.16/0.43 thf(fact_1868_Ex__less__Suc,axiom,
% 0.16/0.43 ! [N: nat,P: nat > $o] :
% 0.16/0.43 ( ( ? [I2: nat] :
% 0.16/0.43 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 0.16/0.43 & ( P @ I2 ) ) )
% 0.16/0.43 = ( ( P @ N )
% 0.16/0.43 | ? [I2: nat] :
% 0.16/0.43 ( ( ord_less_nat @ I2 @ N )
% 0.16/0.43 & ( P @ I2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % Ex_less_Suc
% 0.16/0.43 thf(fact_1869_less__Suc__eq,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.16/0.43 = ( ( ord_less_nat @ M @ N )
% 0.16/0.43 | ( M = N ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_Suc_eq
% 0.16/0.43 thf(fact_1870_not__less__eq,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ~ ( ord_less_nat @ M @ N ) )
% 0.16/0.43 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % not_less_eq
% 0.16/0.43 thf(fact_1871_All__less__Suc,axiom,
% 0.16/0.43 ! [N: nat,P: nat > $o] :
% 0.16/0.43 ( ( ! [I2: nat] :
% 0.16/0.43 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 0.16/0.43 => ( P @ I2 ) ) )
% 0.16/0.43 = ( ( P @ N )
% 0.16/0.43 & ! [I2: nat] :
% 0.16/0.43 ( ( ord_less_nat @ I2 @ N )
% 0.16/0.43 => ( P @ I2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % All_less_Suc
% 0.16/0.43 thf(fact_1872_Suc__less__eq2,axiom,
% 0.16/0.43 ! [N: nat,M: nat] :
% 0.16/0.43 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.16/0.43 = ( ? [M6: nat] :
% 0.16/0.43 ( ( M
% 0.16/0.43 = ( suc @ M6 ) )
% 0.16/0.43 & ( ord_less_nat @ N @ M6 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % Suc_less_eq2
% 0.16/0.43 thf(fact_1873_less__antisym,axiom,
% 0.16/0.43 ! [N: nat,M: nat] :
% 0.16/0.43 ( ~ ( ord_less_nat @ N @ M )
% 0.16/0.43 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.16/0.43 => ( M = N ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_antisym
% 0.16/0.43 thf(fact_1874_Suc__less__SucD,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.16/0.43 => ( ord_less_nat @ M @ N ) ) ).
% 0.16/0.43
% 0.16/0.43 % Suc_less_SucD
% 0.16/0.43 thf(fact_1875_less__trans__Suc,axiom,
% 0.16/0.43 ! [I: nat,J: nat,K: nat] :
% 0.16/0.43 ( ( ord_less_nat @ I @ J )
% 0.16/0.43 => ( ( ord_less_nat @ J @ K )
% 0.16/0.43 => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_trans_Suc
% 0.16/0.43 thf(fact_1876_less__Suc__induct,axiom,
% 0.16/0.43 ! [I: nat,J: nat,P: nat > nat > $o] :
% 0.16/0.43 ( ( ord_less_nat @ I @ J )
% 0.16/0.43 => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
% 0.16/0.43 => ( ! [I3: nat,J2: nat,K2: nat] :
% 0.16/0.43 ( ( ord_less_nat @ I3 @ J2 )
% 0.16/0.43 => ( ( ord_less_nat @ J2 @ K2 )
% 0.16/0.43 => ( ( P @ I3 @ J2 )
% 0.16/0.43 => ( ( P @ J2 @ K2 )
% 0.16/0.43 => ( P @ I3 @ K2 ) ) ) ) )
% 0.16/0.43 => ( P @ I @ J ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_Suc_induct
% 0.16/0.43 thf(fact_1877_strict__inc__induct,axiom,
% 0.16/0.43 ! [I: nat,J: nat,P: nat > $o] :
% 0.16/0.43 ( ( ord_less_nat @ I @ J )
% 0.16/0.43 => ( ! [I3: nat] :
% 0.16/0.43 ( ( J
% 0.16/0.43 = ( suc @ I3 ) )
% 0.16/0.43 => ( P @ I3 ) )
% 0.16/0.43 => ( ! [I3: nat] :
% 0.16/0.43 ( ( ord_less_nat @ I3 @ J )
% 0.16/0.43 => ( ( P @ ( suc @ I3 ) )
% 0.16/0.43 => ( P @ I3 ) ) )
% 0.16/0.43 => ( P @ I ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % strict_inc_induct
% 0.16/0.43 thf(fact_1878_not__less__less__Suc__eq,axiom,
% 0.16/0.43 ! [N: nat,M: nat] :
% 0.16/0.43 ( ~ ( ord_less_nat @ N @ M )
% 0.16/0.43 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.16/0.43 = ( N = M ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % not_less_less_Suc_eq
% 0.16/0.43 thf(fact_1879_zero__neq__one,axiom,
% 0.16/0.43 zero_zero_complex != one_one_complex ).
% 0.16/0.43
% 0.16/0.43 % zero_neq_one
% 0.16/0.43 thf(fact_1880_zero__neq__one,axiom,
% 0.16/0.43 zero_zero_real != one_one_real ).
% 0.16/0.43
% 0.16/0.43 % zero_neq_one
% 0.16/0.43 thf(fact_1881_zero__neq__one,axiom,
% 0.16/0.43 zero_zero_rat != one_one_rat ).
% 0.16/0.43
% 0.16/0.43 % zero_neq_one
% 0.16/0.43 thf(fact_1882_zero__neq__one,axiom,
% 0.16/0.43 zero_zero_nat != one_one_nat ).
% 0.16/0.43
% 0.16/0.43 % zero_neq_one
% 0.16/0.43 thf(fact_1883_zero__neq__one,axiom,
% 0.16/0.43 zero_zero_int != one_one_int ).
% 0.16/0.43
% 0.16/0.43 % zero_neq_one
% 0.16/0.43 thf(fact_1884_bot__nat__0_Oextremum__strict,axiom,
% 0.16/0.43 ! [A: nat] :
% 0.16/0.43 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.16/0.43
% 0.16/0.43 % bot_nat_0.extremum_strict
% 0.16/0.43 thf(fact_1885_gr0I,axiom,
% 0.16/0.43 ! [N: nat] :
% 0.16/0.43 ( ( N != zero_zero_nat )
% 0.16/0.43 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.16/0.43
% 0.16/0.43 % gr0I
% 0.16/0.43 thf(fact_1886_not__gr0,axiom,
% 0.16/0.43 ! [N: nat] :
% 0.16/0.43 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.16/0.43 = ( N = zero_zero_nat ) ) ).
% 0.16/0.43
% 0.16/0.43 % not_gr0
% 0.16/0.43 thf(fact_1887_not__less0,axiom,
% 0.16/0.43 ! [N: nat] :
% 0.16/0.43 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.16/0.43
% 0.16/0.43 % not_less0
% 0.16/0.43 thf(fact_1888_less__zeroE,axiom,
% 0.16/0.43 ! [N: nat] :
% 0.16/0.43 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.16/0.43
% 0.16/0.43 % less_zeroE
% 0.16/0.43 thf(fact_1889_gr__implies__not0,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ M @ N )
% 0.16/0.43 => ( N != zero_zero_nat ) ) ).
% 0.16/0.43
% 0.16/0.43 % gr_implies_not0
% 0.16/0.43 thf(fact_1890_infinite__descent0,axiom,
% 0.16/0.43 ! [P: nat > $o,N: nat] :
% 0.16/0.43 ( ( P @ zero_zero_nat )
% 0.16/0.43 => ( ! [N2: nat] :
% 0.16/0.43 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.16/0.43 => ( ~ ( P @ N2 )
% 0.16/0.43 => ? [M2: nat] :
% 0.16/0.43 ( ( ord_less_nat @ M2 @ N2 )
% 0.16/0.43 & ~ ( P @ M2 ) ) ) )
% 0.16/0.43 => ( P @ N ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % infinite_descent0
% 0.16/0.43 thf(fact_1891_comm__monoid__mult__class_Omult__1,axiom,
% 0.16/0.43 ! [A: complex] :
% 0.16/0.43 ( ( times_times_complex @ one_one_complex @ A )
% 0.16/0.43 = A ) ).
% 0.16/0.43
% 0.16/0.43 % comm_monoid_mult_class.mult_1
% 0.16/0.43 thf(fact_1892_comm__monoid__mult__class_Omult__1,axiom,
% 0.16/0.43 ! [A: real] :
% 0.16/0.43 ( ( times_times_real @ one_one_real @ A )
% 0.16/0.43 = A ) ).
% 0.16/0.43
% 0.16/0.43 % comm_monoid_mult_class.mult_1
% 0.16/0.43 thf(fact_1893_comm__monoid__mult__class_Omult__1,axiom,
% 0.16/0.43 ! [A: rat] :
% 0.16/0.43 ( ( times_times_rat @ one_one_rat @ A )
% 0.16/0.43 = A ) ).
% 0.16/0.43
% 0.16/0.43 % comm_monoid_mult_class.mult_1
% 0.16/0.43 thf(fact_1894_comm__monoid__mult__class_Omult__1,axiom,
% 0.16/0.43 ! [A: nat] :
% 0.16/0.43 ( ( times_times_nat @ one_one_nat @ A )
% 0.16/0.43 = A ) ).
% 0.16/0.43
% 0.16/0.43 % comm_monoid_mult_class.mult_1
% 0.16/0.43 thf(fact_1895_comm__monoid__mult__class_Omult__1,axiom,
% 0.16/0.43 ! [A: int] :
% 0.16/0.43 ( ( times_times_int @ one_one_int @ A )
% 0.16/0.43 = A ) ).
% 0.16/0.43
% 0.16/0.43 % comm_monoid_mult_class.mult_1
% 0.16/0.43 thf(fact_1896_mult_Ocomm__neutral,axiom,
% 0.16/0.43 ! [A: complex] :
% 0.16/0.43 ( ( times_times_complex @ A @ one_one_complex )
% 0.16/0.43 = A ) ).
% 0.16/0.43
% 0.16/0.43 % mult.comm_neutral
% 0.16/0.43 thf(fact_1897_mult_Ocomm__neutral,axiom,
% 0.16/0.43 ! [A: real] :
% 0.16/0.43 ( ( times_times_real @ A @ one_one_real )
% 0.16/0.43 = A ) ).
% 0.16/0.43
% 0.16/0.43 % mult.comm_neutral
% 0.16/0.43 thf(fact_1898_mult_Ocomm__neutral,axiom,
% 0.16/0.43 ! [A: rat] :
% 0.16/0.43 ( ( times_times_rat @ A @ one_one_rat )
% 0.16/0.43 = A ) ).
% 0.16/0.43
% 0.16/0.43 % mult.comm_neutral
% 0.16/0.43 thf(fact_1899_mult_Ocomm__neutral,axiom,
% 0.16/0.43 ! [A: nat] :
% 0.16/0.43 ( ( times_times_nat @ A @ one_one_nat )
% 0.16/0.43 = A ) ).
% 0.16/0.43
% 0.16/0.43 % mult.comm_neutral
% 0.16/0.43 thf(fact_1900_mult_Ocomm__neutral,axiom,
% 0.16/0.43 ! [A: int] :
% 0.16/0.43 ( ( times_times_int @ A @ one_one_int )
% 0.16/0.43 = A ) ).
% 0.16/0.43
% 0.16/0.43 % mult.comm_neutral
% 0.16/0.43 thf(fact_1901_int__distrib_I4_J,axiom,
% 0.16/0.43 ! [W: int,Z1: int,Z22: int] :
% 0.16/0.43 ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
% 0.16/0.43 = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % int_distrib(4)
% 0.16/0.43 thf(fact_1902_int__distrib_I3_J,axiom,
% 0.16/0.43 ! [Z1: int,Z22: int,W: int] :
% 0.16/0.43 ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
% 0.16/0.43 = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % int_distrib(3)
% 0.16/0.43 thf(fact_1903_less__mono__imp__le__mono,axiom,
% 0.16/0.43 ! [F: nat > nat,I: nat,J: nat] :
% 0.16/0.43 ( ! [I3: nat,J2: nat] :
% 0.16/0.43 ( ( ord_less_nat @ I3 @ J2 )
% 0.16/0.43 => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
% 0.16/0.43 => ( ( ord_less_eq_nat @ I @ J )
% 0.16/0.43 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_mono_imp_le_mono
% 0.16/0.43 thf(fact_1904_le__neq__implies__less,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.43 => ( ( M != N )
% 0.16/0.43 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % le_neq_implies_less
% 0.16/0.43 thf(fact_1905_less__or__eq__imp__le,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ( ord_less_nat @ M @ N )
% 0.16/0.43 | ( M = N ) )
% 0.16/0.43 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_or_eq_imp_le
% 0.16/0.43 thf(fact_1906_le__eq__less__or__eq,axiom,
% 0.16/0.43 ( ord_less_eq_nat
% 0.16/0.43 = ( ^ [M5: nat,N4: nat] :
% 0.16/0.43 ( ( ord_less_nat @ M5 @ N4 )
% 0.16/0.43 | ( M5 = N4 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % le_eq_less_or_eq
% 0.16/0.43 thf(fact_1907_less__imp__le__nat,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ M @ N )
% 0.16/0.43 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_imp_le_nat
% 0.16/0.43 thf(fact_1908_nat__less__le,axiom,
% 0.16/0.43 ( ord_less_nat
% 0.16/0.43 = ( ^ [M5: nat,N4: nat] :
% 0.16/0.43 ( ( ord_less_eq_nat @ M5 @ N4 )
% 0.16/0.43 & ( M5 != N4 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % nat_less_le
% 0.16/0.43 thf(fact_1909_int__diff__cases,axiom,
% 0.16/0.43 ! [Z: int] :
% 0.16/0.43 ~ ! [M4: nat,N2: nat] :
% 0.16/0.43 ( Z
% 0.16/0.43 != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % int_diff_cases
% 0.16/0.43 thf(fact_1910_diff__less__mono2,axiom,
% 0.16/0.43 ! [M: nat,N: nat,L: nat] :
% 0.16/0.43 ( ( ord_less_nat @ M @ N )
% 0.16/0.43 => ( ( ord_less_nat @ M @ L )
% 0.16/0.43 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % diff_less_mono2
% 0.16/0.43 thf(fact_1911_less__imp__diff__less,axiom,
% 0.16/0.43 ! [J: nat,K: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ J @ K )
% 0.16/0.43 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 0.16/0.43
% 0.16/0.43 % less_imp_diff_less
% 0.16/0.43 thf(fact_1912_dvd__unit__imp__unit,axiom,
% 0.16/0.43 ! [A: code_integer,B: code_integer] :
% 0.16/0.43 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.16/0.43 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.16/0.43 => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dvd_unit_imp_unit
% 0.16/0.43 thf(fact_1913_dvd__unit__imp__unit,axiom,
% 0.16/0.43 ! [A: nat,B: nat] :
% 0.16/0.43 ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.43 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.16/0.43 => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dvd_unit_imp_unit
% 0.16/0.43 thf(fact_1914_dvd__unit__imp__unit,axiom,
% 0.16/0.43 ! [A: int,B: int] :
% 0.16/0.43 ( ( dvd_dvd_int @ A @ B )
% 0.16/0.43 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.16/0.43 => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dvd_unit_imp_unit
% 0.16/0.43 thf(fact_1915_unit__imp__dvd,axiom,
% 0.16/0.43 ! [B: code_integer,A: code_integer] :
% 0.16/0.43 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.16/0.43 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.16/0.43
% 0.16/0.43 % unit_imp_dvd
% 0.16/0.43 thf(fact_1916_unit__imp__dvd,axiom,
% 0.16/0.43 ! [B: nat,A: nat] :
% 0.16/0.43 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.16/0.43 => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.16/0.43
% 0.16/0.43 % unit_imp_dvd
% 0.16/0.43 thf(fact_1917_unit__imp__dvd,axiom,
% 0.16/0.43 ! [B: int,A: int] :
% 0.16/0.43 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.16/0.43 => ( dvd_dvd_int @ B @ A ) ) ).
% 0.16/0.43
% 0.16/0.43 % unit_imp_dvd
% 0.16/0.43 thf(fact_1918_one__dvd,axiom,
% 0.16/0.43 ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% 0.16/0.43
% 0.16/0.43 % one_dvd
% 0.16/0.43 thf(fact_1919_one__dvd,axiom,
% 0.16/0.43 ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% 0.16/0.43
% 0.16/0.43 % one_dvd
% 0.16/0.43 thf(fact_1920_one__dvd,axiom,
% 0.16/0.43 ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% 0.16/0.43
% 0.16/0.43 % one_dvd
% 0.16/0.43 thf(fact_1921_one__dvd,axiom,
% 0.16/0.43 ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% 0.16/0.43
% 0.16/0.43 % one_dvd
% 0.16/0.43 thf(fact_1922_one__dvd,axiom,
% 0.16/0.43 ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% 0.16/0.43
% 0.16/0.43 % one_dvd
% 0.16/0.43 thf(fact_1923_one__dvd,axiom,
% 0.16/0.43 ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% 0.16/0.43
% 0.16/0.43 % one_dvd
% 0.16/0.43 thf(fact_1924_zdvd__zdiffD,axiom,
% 0.16/0.43 ! [K: int,M: int,N: int] :
% 0.16/0.43 ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
% 0.16/0.43 => ( ( dvd_dvd_int @ K @ N )
% 0.16/0.43 => ( dvd_dvd_int @ K @ M ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % zdvd_zdiffD
% 0.16/0.43 thf(fact_1925_nat__mult__1__right,axiom,
% 0.16/0.43 ! [N: nat] :
% 0.16/0.43 ( ( times_times_nat @ N @ one_one_nat )
% 0.16/0.43 = N ) ).
% 0.16/0.43
% 0.16/0.43 % nat_mult_1_right
% 0.16/0.43 thf(fact_1926_nat__mult__1,axiom,
% 0.16/0.43 ! [N: nat] :
% 0.16/0.43 ( ( times_times_nat @ one_one_nat @ N )
% 0.16/0.43 = N ) ).
% 0.16/0.43
% 0.16/0.43 % nat_mult_1
% 0.16/0.43 thf(fact_1927_dvd__pos__nat,axiom,
% 0.16/0.43 ! [N: nat,M: nat] :
% 0.16/0.43 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.43 => ( ( dvd_dvd_nat @ M @ N )
% 0.16/0.43 => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dvd_pos_nat
% 0.16/0.43 thf(fact_1928_mult__le__cancel__left1,axiom,
% 0.16/0.43 ! [C: rat,B: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
% 0.16/0.43 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.43 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 0.16/0.43 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.43 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_le_cancel_left1
% 0.16/0.43 thf(fact_1929_mult__le__cancel__left1,axiom,
% 0.16/0.43 ! [C: int,B: int] :
% 0.16/0.43 ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
% 0.16/0.43 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.16/0.43 => ( ord_less_eq_int @ one_one_int @ B ) )
% 0.16/0.43 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.16/0.43 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_le_cancel_left1
% 0.16/0.43 thf(fact_1930_mult__le__cancel__left1,axiom,
% 0.16/0.43 ! [C: real,B: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
% 0.16/0.43 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.43 => ( ord_less_eq_real @ one_one_real @ B ) )
% 0.16/0.43 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.43 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_le_cancel_left1
% 0.16/0.43 thf(fact_1931_mult__le__cancel__left2,axiom,
% 0.16/0.43 ! [C: rat,A: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.16/0.43 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.43 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.16/0.43 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.43 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_le_cancel_left2
% 0.16/0.43 thf(fact_1932_mult__le__cancel__left2,axiom,
% 0.16/0.43 ! [C: int,A: int] :
% 0.16/0.43 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 0.16/0.43 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.16/0.43 => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.16/0.43 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.16/0.43 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_le_cancel_left2
% 0.16/0.43 thf(fact_1933_mult__le__cancel__left2,axiom,
% 0.16/0.43 ! [C: real,A: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 0.16/0.43 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.43 => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.16/0.43 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.43 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_le_cancel_left2
% 0.16/0.43 thf(fact_1934_mult__le__cancel__right1,axiom,
% 0.16/0.43 ! [C: rat,B: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
% 0.16/0.43 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.43 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 0.16/0.43 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.43 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_le_cancel_right1
% 0.16/0.43 thf(fact_1935_mult__le__cancel__right1,axiom,
% 0.16/0.43 ! [C: int,B: int] :
% 0.16/0.43 ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
% 0.16/0.43 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.16/0.43 => ( ord_less_eq_int @ one_one_int @ B ) )
% 0.16/0.43 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.16/0.43 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_le_cancel_right1
% 0.16/0.43 thf(fact_1936_mult__le__cancel__right1,axiom,
% 0.16/0.43 ! [C: real,B: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
% 0.16/0.43 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.43 => ( ord_less_eq_real @ one_one_real @ B ) )
% 0.16/0.43 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.43 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_le_cancel_right1
% 0.16/0.43 thf(fact_1937_mult__le__cancel__right2,axiom,
% 0.16/0.43 ! [A: rat,C: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.16/0.43 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.43 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.16/0.43 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.43 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_le_cancel_right2
% 0.16/0.43 thf(fact_1938_mult__le__cancel__right2,axiom,
% 0.16/0.43 ! [A: int,C: int] :
% 0.16/0.43 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 0.16/0.43 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.16/0.43 => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.16/0.43 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.16/0.43 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_le_cancel_right2
% 0.16/0.43 thf(fact_1939_mult__le__cancel__right2,axiom,
% 0.16/0.43 ! [A: real,C: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 0.16/0.43 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.43 => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.16/0.43 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.43 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_le_cancel_right2
% 0.16/0.43 thf(fact_1940_mult__less__cancel__left1,axiom,
% 0.16/0.43 ! [C: rat,B: rat] :
% 0.16/0.43 ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
% 0.16/0.43 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.16/0.43 => ( ord_less_rat @ one_one_rat @ B ) )
% 0.16/0.43 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.16/0.43 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_less_cancel_left1
% 0.16/0.43 thf(fact_1941_mult__less__cancel__left1,axiom,
% 0.16/0.43 ! [C: int,B: int] :
% 0.16/0.43 ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
% 0.16/0.43 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.16/0.43 => ( ord_less_int @ one_one_int @ B ) )
% 0.16/0.43 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.16/0.43 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_less_cancel_left1
% 0.16/0.43 thf(fact_1942_mult__less__cancel__left1,axiom,
% 0.16/0.43 ! [C: real,B: real] :
% 0.16/0.43 ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
% 0.16/0.43 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.43 => ( ord_less_real @ one_one_real @ B ) )
% 0.16/0.43 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.16/0.43 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_less_cancel_left1
% 0.16/0.43 thf(fact_1943_mult__less__cancel__left2,axiom,
% 0.16/0.43 ! [C: rat,A: rat] :
% 0.16/0.43 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.16/0.43 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.16/0.43 => ( ord_less_rat @ A @ one_one_rat ) )
% 0.16/0.43 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.16/0.43 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_less_cancel_left2
% 0.16/0.43 thf(fact_1944_mult__less__cancel__left2,axiom,
% 0.16/0.43 ! [C: int,A: int] :
% 0.16/0.43 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 0.16/0.43 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.16/0.43 => ( ord_less_int @ A @ one_one_int ) )
% 0.16/0.43 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.16/0.43 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_less_cancel_left2
% 0.16/0.43 thf(fact_1945_mult__less__cancel__left2,axiom,
% 0.16/0.43 ! [C: real,A: real] :
% 0.16/0.43 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 0.16/0.43 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.43 => ( ord_less_real @ A @ one_one_real ) )
% 0.16/0.43 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.16/0.43 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_less_cancel_left2
% 0.16/0.43 thf(fact_1946_mult__less__cancel__right1,axiom,
% 0.16/0.43 ! [C: rat,B: rat] :
% 0.16/0.43 ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
% 0.16/0.43 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.16/0.43 => ( ord_less_rat @ one_one_rat @ B ) )
% 0.16/0.43 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.16/0.43 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_less_cancel_right1
% 0.16/0.43 thf(fact_1947_mult__less__cancel__right1,axiom,
% 0.16/0.43 ! [C: int,B: int] :
% 0.16/0.43 ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
% 0.16/0.43 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.16/0.43 => ( ord_less_int @ one_one_int @ B ) )
% 0.16/0.43 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.16/0.43 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_less_cancel_right1
% 0.16/0.43 thf(fact_1948_mult__less__cancel__right1,axiom,
% 0.16/0.43 ! [C: real,B: real] :
% 0.16/0.43 ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
% 0.16/0.43 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.43 => ( ord_less_real @ one_one_real @ B ) )
% 0.16/0.43 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.16/0.43 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_less_cancel_right1
% 0.16/0.43 thf(fact_1949_mult__less__cancel__right2,axiom,
% 0.16/0.43 ! [A: rat,C: rat] :
% 0.16/0.43 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.16/0.43 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.16/0.43 => ( ord_less_rat @ A @ one_one_rat ) )
% 0.16/0.43 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.16/0.43 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_less_cancel_right2
% 0.16/0.43 thf(fact_1950_mult__less__cancel__right2,axiom,
% 0.16/0.43 ! [A: int,C: int] :
% 0.16/0.43 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 0.16/0.43 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.16/0.43 => ( ord_less_int @ A @ one_one_int ) )
% 0.16/0.43 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.16/0.43 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_less_cancel_right2
% 0.16/0.43 thf(fact_1951_mult__less__cancel__right2,axiom,
% 0.16/0.43 ! [A: real,C: real] :
% 0.16/0.43 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 0.16/0.43 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.43 => ( ord_less_real @ A @ one_one_real ) )
% 0.16/0.43 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.16/0.43 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mult_less_cancel_right2
% 0.16/0.43 thf(fact_1952_field__le__mult__one__interval,axiom,
% 0.16/0.43 ! [X: rat,Y4: rat] :
% 0.16/0.43 ( ! [Z2: rat] :
% 0.16/0.43 ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 0.16/0.43 => ( ( ord_less_rat @ Z2 @ one_one_rat )
% 0.16/0.43 => ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X ) @ Y4 ) ) )
% 0.16/0.43 => ( ord_less_eq_rat @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % field_le_mult_one_interval
% 0.16/0.43 thf(fact_1953_field__le__mult__one__interval,axiom,
% 0.16/0.43 ! [X: real,Y4: real] :
% 0.16/0.43 ( ! [Z2: real] :
% 0.16/0.43 ( ( ord_less_real @ zero_zero_real @ Z2 )
% 0.16/0.43 => ( ( ord_less_real @ Z2 @ one_one_real )
% 0.16/0.43 => ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ Y4 ) ) )
% 0.16/0.43 => ( ord_less_eq_real @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % field_le_mult_one_interval
% 0.16/0.43 thf(fact_1954_divide__le__eq__1,axiom,
% 0.16/0.43 ! [B: rat,A: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.16/0.43 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.43 & ( ord_less_eq_rat @ B @ A ) )
% 0.16/0.43 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.16/0.43 & ( ord_less_eq_rat @ A @ B ) )
% 0.16/0.43 | ( A = zero_zero_rat ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % divide_le_eq_1
% 0.16/0.43 thf(fact_1955_divide__le__eq__1,axiom,
% 0.16/0.43 ! [B: real,A: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.16/0.43 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.43 & ( ord_less_eq_real @ B @ A ) )
% 0.16/0.43 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.16/0.43 & ( ord_less_eq_real @ A @ B ) )
% 0.16/0.43 | ( A = zero_zero_real ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % divide_le_eq_1
% 0.16/0.43 thf(fact_1956_le__divide__eq__1,axiom,
% 0.16/0.43 ! [B: rat,A: rat] :
% 0.16/0.43 ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.16/0.43 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.43 & ( ord_less_eq_rat @ A @ B ) )
% 0.16/0.43 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.16/0.43 & ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % le_divide_eq_1
% 0.16/0.43 thf(fact_1957_le__divide__eq__1,axiom,
% 0.16/0.43 ! [B: real,A: real] :
% 0.16/0.43 ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.16/0.43 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.43 & ( ord_less_eq_real @ A @ B ) )
% 0.16/0.43 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.16/0.43 & ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % le_divide_eq_1
% 0.16/0.43 thf(fact_1958_Suc__pred_H,axiom,
% 0.16/0.43 ! [N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.43 => ( N
% 0.16/0.43 = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % Suc_pred'
% 0.16/0.43 thf(fact_1959_Suc__diff__eq__diff__pred,axiom,
% 0.16/0.43 ! [N: nat,M: nat] :
% 0.16/0.43 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.43 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.16/0.43 = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % Suc_diff_eq_diff_pred
% 0.16/0.43 thf(fact_1960_dvd__mult__cancel2,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.16/0.43 => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
% 0.16/0.43 = ( N = one_one_nat ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dvd_mult_cancel2
% 0.16/0.43 thf(fact_1961_dvd__mult__cancel1,axiom,
% 0.16/0.43 ! [M: nat,N: nat] :
% 0.16/0.43 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.16/0.43 => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
% 0.16/0.43 = ( N = one_one_nat ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % dvd_mult_cancel1
% 0.16/0.43 thf(fact_1962_of__nat__less__two__power,axiom,
% 0.16/0.43 ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 0.16/0.43
% 0.16/0.43 % of_nat_less_two_power
% 0.16/0.43 thf(fact_1963_of__nat__less__two__power,axiom,
% 0.16/0.43 ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.16/0.43
% 0.16/0.43 % of_nat_less_two_power
% 0.16/0.43 thf(fact_1964_of__nat__less__two__power,axiom,
% 0.16/0.43 ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.16/0.43
% 0.16/0.43 % of_nat_less_two_power
% 0.16/0.43 thf(fact_1965_power__diff__power__eq,axiom,
% 0.16/0.43 ! [A: nat,N: nat,M: nat] :
% 0.16/0.43 ( ( A != zero_zero_nat )
% 0.16/0.43 => ( ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.43 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.16/0.43 = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 0.16/0.43 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.16/0.43 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.16/0.43 = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % power_diff_power_eq
% 0.16/0.43 thf(fact_1966_power__diff__power__eq,axiom,
% 0.16/0.43 ! [A: int,N: nat,M: nat] :
% 0.16/0.43 ( ( A != zero_zero_int )
% 0.16/0.43 => ( ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.43 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.16/0.43 = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 0.16/0.43 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.16/0.43 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.16/0.43 = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % power_diff_power_eq
% 0.16/0.43 thf(fact_1967_not__numeral__less__zero,axiom,
% 0.16/0.43 ! [N: num] :
% 0.16/0.43 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 0.16/0.43
% 0.16/0.43 % not_numeral_less_zero
% 0.16/0.43 thf(fact_1968_not__numeral__less__zero,axiom,
% 0.16/0.43 ! [N: num] :
% 0.16/0.43 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 0.16/0.44
% 0.16/0.44 % not_numeral_less_zero
% 0.16/0.44 thf(fact_1969_not__numeral__less__zero,axiom,
% 0.16/0.44 ! [N: num] :
% 0.16/0.44 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 0.16/0.44
% 0.16/0.44 % not_numeral_less_zero
% 0.16/0.44 thf(fact_1970_not__numeral__less__zero,axiom,
% 0.16/0.44 ! [N: num] :
% 0.16/0.44 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 0.16/0.44
% 0.16/0.44 % not_numeral_less_zero
% 0.16/0.44 thf(fact_1971_zero__less__numeral,axiom,
% 0.16/0.44 ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_numeral
% 0.16/0.44 thf(fact_1972_zero__less__numeral,axiom,
% 0.16/0.44 ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_numeral
% 0.16/0.44 thf(fact_1973_zero__less__numeral,axiom,
% 0.16/0.44 ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_numeral
% 0.16/0.44 thf(fact_1974_zero__less__numeral,axiom,
% 0.16/0.44 ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_numeral
% 0.16/0.44 thf(fact_1975_mult__neg__neg,axiom,
% 0.16/0.44 ! [A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ A @ zero_zero_real )
% 0.16/0.44 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.16/0.44 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_neg_neg
% 0.16/0.44 thf(fact_1976_mult__neg__neg,axiom,
% 0.16/0.44 ! [A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.16/0.44 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_neg_neg
% 0.16/0.44 thf(fact_1977_mult__neg__neg,axiom,
% 0.16/0.44 ! [A: int,B: int] :
% 0.16/0.44 ( ( ord_less_int @ A @ zero_zero_int )
% 0.16/0.44 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.16/0.44 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_neg_neg
% 0.16/0.44 thf(fact_1978_not__square__less__zero,axiom,
% 0.16/0.44 ! [A: real] :
% 0.16/0.44 ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 0.16/0.44
% 0.16/0.44 % not_square_less_zero
% 0.16/0.44 thf(fact_1979_not__square__less__zero,axiom,
% 0.16/0.44 ! [A: rat] :
% 0.16/0.44 ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 0.16/0.44
% 0.16/0.44 % not_square_less_zero
% 0.16/0.44 thf(fact_1980_not__square__less__zero,axiom,
% 0.16/0.44 ! [A: int] :
% 0.16/0.44 ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 0.16/0.44
% 0.16/0.44 % not_square_less_zero
% 0.16/0.44 thf(fact_1981_mult__less__0__iff,axiom,
% 0.16/0.44 ! [A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.16/0.44 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.44 & ( ord_less_real @ B @ zero_zero_real ) )
% 0.16/0.44 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.16/0.44 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_0_iff
% 0.16/0.44 thf(fact_1982_mult__less__0__iff,axiom,
% 0.16/0.44 ! [A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.16/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.44 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 0.16/0.44 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.16/0.44 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_0_iff
% 0.16/0.44 thf(fact_1983_mult__less__0__iff,axiom,
% 0.16/0.44 ! [A: int,B: int] :
% 0.16/0.44 ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 0.16/0.44 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.16/0.44 & ( ord_less_int @ B @ zero_zero_int ) )
% 0.16/0.44 | ( ( ord_less_int @ A @ zero_zero_int )
% 0.16/0.44 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_0_iff
% 0.16/0.44 thf(fact_1984_mult__neg__pos,axiom,
% 0.16/0.44 ! [A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ A @ zero_zero_real )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.16/0.44 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_neg_pos
% 0.16/0.44 thf(fact_1985_mult__neg__pos,axiom,
% 0.16/0.44 ! [A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.16/0.44 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_neg_pos
% 0.16/0.44 thf(fact_1986_mult__neg__pos,axiom,
% 0.16/0.44 ! [A: nat,B: nat] :
% 0.16/0.44 ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.16/0.44 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_neg_pos
% 0.16/0.44 thf(fact_1987_mult__neg__pos,axiom,
% 0.16/0.44 ! [A: int,B: int] :
% 0.16/0.44 ( ( ord_less_int @ A @ zero_zero_int )
% 0.16/0.44 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.16/0.44 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_neg_pos
% 0.16/0.44 thf(fact_1988_mult__pos__neg,axiom,
% 0.16/0.44 ! [A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.44 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.16/0.44 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_pos_neg
% 0.16/0.44 thf(fact_1989_mult__pos__neg,axiom,
% 0.16/0.44 ! [A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.44 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_pos_neg
% 0.16/0.44 thf(fact_1990_mult__pos__neg,axiom,
% 0.16/0.44 ! [A: nat,B: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.16/0.44 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.16/0.44 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_pos_neg
% 0.16/0.44 thf(fact_1991_mult__pos__neg,axiom,
% 0.16/0.44 ! [A: int,B: int] :
% 0.16/0.44 ( ( ord_less_int @ zero_zero_int @ A )
% 0.16/0.44 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.16/0.44 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_pos_neg
% 0.16/0.44 thf(fact_1992_mult__pos__pos,axiom,
% 0.16/0.44 ! [A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.16/0.44 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_pos_pos
% 0.16/0.44 thf(fact_1993_mult__pos__pos,axiom,
% 0.16/0.44 ! [A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.16/0.44 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_pos_pos
% 0.16/0.44 thf(fact_1994_mult__pos__pos,axiom,
% 0.16/0.44 ! [A: nat,B: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.16/0.44 => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_pos_pos
% 0.16/0.44 thf(fact_1995_mult__pos__pos,axiom,
% 0.16/0.44 ! [A: int,B: int] :
% 0.16/0.44 ( ( ord_less_int @ zero_zero_int @ A )
% 0.16/0.44 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.16/0.44 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_pos_pos
% 0.16/0.44 thf(fact_1996_mult__pos__neg2,axiom,
% 0.16/0.44 ! [A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.44 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.16/0.44 => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_pos_neg2
% 0.16/0.44 thf(fact_1997_mult__pos__neg2,axiom,
% 0.16/0.44 ! [A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.44 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_pos_neg2
% 0.16/0.44 thf(fact_1998_mult__pos__neg2,axiom,
% 0.16/0.44 ! [A: nat,B: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.16/0.44 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.16/0.44 => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_pos_neg2
% 0.16/0.44 thf(fact_1999_mult__pos__neg2,axiom,
% 0.16/0.44 ! [A: int,B: int] :
% 0.16/0.44 ( ( ord_less_int @ zero_zero_int @ A )
% 0.16/0.44 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.16/0.44 => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_pos_neg2
% 0.16/0.44 thf(fact_2000_zero__less__mult__iff,axiom,
% 0.16/0.44 ! [A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.16/0.44 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.44 & ( ord_less_real @ zero_zero_real @ B ) )
% 0.16/0.44 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.16/0.44 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_mult_iff
% 0.16/0.44 thf(fact_2001_zero__less__mult__iff,axiom,
% 0.16/0.44 ! [A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.16/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.44 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 0.16/0.44 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.16/0.44 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_mult_iff
% 0.16/0.44 thf(fact_2002_zero__less__mult__iff,axiom,
% 0.16/0.44 ! [A: int,B: int] :
% 0.16/0.44 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.16/0.44 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.16/0.44 & ( ord_less_int @ zero_zero_int @ B ) )
% 0.16/0.44 | ( ( ord_less_int @ A @ zero_zero_int )
% 0.16/0.44 & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_mult_iff
% 0.16/0.44 thf(fact_2003_zero__less__mult__pos,axiom,
% 0.16/0.44 ! [A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.44 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_mult_pos
% 0.16/0.44 thf(fact_2004_zero__less__mult__pos,axiom,
% 0.16/0.44 ! [A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.44 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_mult_pos
% 0.16/0.44 thf(fact_2005_zero__less__mult__pos,axiom,
% 0.16/0.44 ! [A: nat,B: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.16/0.44 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_mult_pos
% 0.16/0.44 thf(fact_2006_zero__less__mult__pos,axiom,
% 0.16/0.44 ! [A: int,B: int] :
% 0.16/0.44 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.16/0.44 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.16/0.44 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_mult_pos
% 0.16/0.44 thf(fact_2007_zero__less__mult__pos2,axiom,
% 0.16/0.44 ! [B: real,A: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.44 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_mult_pos2
% 0.16/0.44 thf(fact_2008_zero__less__mult__pos2,axiom,
% 0.16/0.44 ! [B: rat,A: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.44 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_mult_pos2
% 0.16/0.44 thf(fact_2009_zero__less__mult__pos2,axiom,
% 0.16/0.44 ! [B: nat,A: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.16/0.44 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_mult_pos2
% 0.16/0.44 thf(fact_2010_zero__less__mult__pos2,axiom,
% 0.16/0.44 ! [B: int,A: int] :
% 0.16/0.44 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
% 0.16/0.44 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.16/0.44 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_mult_pos2
% 0.16/0.44 thf(fact_2011_mult__less__cancel__left__neg,axiom,
% 0.16/0.44 ! [C: real,A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.16/0.44 = ( ord_less_real @ B @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_cancel_left_neg
% 0.16/0.44 thf(fact_2012_mult__less__cancel__left__neg,axiom,
% 0.16/0.44 ! [C: rat,A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.16/0.44 = ( ord_less_rat @ B @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_cancel_left_neg
% 0.16/0.44 thf(fact_2013_mult__less__cancel__left__neg,axiom,
% 0.16/0.44 ! [C: int,A: int,B: int] :
% 0.16/0.44 ( ( ord_less_int @ C @ zero_zero_int )
% 0.16/0.44 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.16/0.44 = ( ord_less_int @ B @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_cancel_left_neg
% 0.16/0.44 thf(fact_2014_mult__less__cancel__left__pos,axiom,
% 0.16/0.44 ! [C: real,A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.16/0.44 = ( ord_less_real @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_cancel_left_pos
% 0.16/0.44 thf(fact_2015_mult__less__cancel__left__pos,axiom,
% 0.16/0.44 ! [C: rat,A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.16/0.44 = ( ord_less_rat @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_cancel_left_pos
% 0.16/0.44 thf(fact_2016_mult__less__cancel__left__pos,axiom,
% 0.16/0.44 ! [C: int,A: int,B: int] :
% 0.16/0.44 ( ( ord_less_int @ zero_zero_int @ C )
% 0.16/0.44 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.16/0.44 = ( ord_less_int @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_cancel_left_pos
% 0.16/0.44 thf(fact_2017_mult__strict__left__mono__neg,axiom,
% 0.16/0.44 ! [B: real,A: real,C: real] :
% 0.16/0.44 ( ( ord_less_real @ B @ A )
% 0.16/0.44 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_left_mono_neg
% 0.16/0.44 thf(fact_2018_mult__strict__left__mono__neg,axiom,
% 0.16/0.44 ! [B: rat,A: rat,C: rat] :
% 0.16/0.44 ( ( ord_less_rat @ B @ A )
% 0.16/0.44 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_left_mono_neg
% 0.16/0.44 thf(fact_2019_mult__strict__left__mono__neg,axiom,
% 0.16/0.44 ! [B: int,A: int,C: int] :
% 0.16/0.44 ( ( ord_less_int @ B @ A )
% 0.16/0.44 => ( ( ord_less_int @ C @ zero_zero_int )
% 0.16/0.44 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_left_mono_neg
% 0.16/0.44 thf(fact_2020_mult__strict__left__mono,axiom,
% 0.16/0.44 ! [A: real,B: real,C: real] :
% 0.16/0.44 ( ( ord_less_real @ A @ B )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_left_mono
% 0.16/0.44 thf(fact_2021_mult__strict__left__mono,axiom,
% 0.16/0.44 ! [A: rat,B: rat,C: rat] :
% 0.16/0.44 ( ( ord_less_rat @ A @ B )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_left_mono
% 0.16/0.44 thf(fact_2022_mult__strict__left__mono,axiom,
% 0.16/0.44 ! [A: nat,B: nat,C: nat] :
% 0.16/0.44 ( ( ord_less_nat @ A @ B )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.16/0.44 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_left_mono
% 0.16/0.44 thf(fact_2023_mult__strict__left__mono,axiom,
% 0.16/0.44 ! [A: int,B: int,C: int] :
% 0.16/0.44 ( ( ord_less_int @ A @ B )
% 0.16/0.44 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.16/0.44 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_left_mono
% 0.16/0.44 thf(fact_2024_mult__less__cancel__left__disj,axiom,
% 0.16/0.44 ! [C: real,A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.16/0.44 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 & ( ord_less_real @ A @ B ) )
% 0.16/0.44 | ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_cancel_left_disj
% 0.16/0.44 thf(fact_2025_mult__less__cancel__left__disj,axiom,
% 0.16/0.44 ! [C: rat,A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.16/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 & ( ord_less_rat @ A @ B ) )
% 0.16/0.44 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_cancel_left_disj
% 0.16/0.44 thf(fact_2026_mult__less__cancel__left__disj,axiom,
% 0.16/0.44 ! [C: int,A: int,B: int] :
% 0.16/0.44 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.16/0.44 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.16/0.44 & ( ord_less_int @ A @ B ) )
% 0.16/0.44 | ( ( ord_less_int @ C @ zero_zero_int )
% 0.16/0.44 & ( ord_less_int @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_cancel_left_disj
% 0.16/0.44 thf(fact_2027_mult__strict__right__mono__neg,axiom,
% 0.16/0.44 ! [B: real,A: real,C: real] :
% 0.16/0.44 ( ( ord_less_real @ B @ A )
% 0.16/0.44 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_right_mono_neg
% 0.16/0.44 thf(fact_2028_mult__strict__right__mono__neg,axiom,
% 0.16/0.44 ! [B: rat,A: rat,C: rat] :
% 0.16/0.44 ( ( ord_less_rat @ B @ A )
% 0.16/0.44 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_right_mono_neg
% 0.16/0.44 thf(fact_2029_mult__strict__right__mono__neg,axiom,
% 0.16/0.44 ! [B: int,A: int,C: int] :
% 0.16/0.44 ( ( ord_less_int @ B @ A )
% 0.16/0.44 => ( ( ord_less_int @ C @ zero_zero_int )
% 0.16/0.44 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_right_mono_neg
% 0.16/0.44 thf(fact_2030_mult__strict__right__mono,axiom,
% 0.16/0.44 ! [A: real,B: real,C: real] :
% 0.16/0.44 ( ( ord_less_real @ A @ B )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_right_mono
% 0.16/0.44 thf(fact_2031_mult__strict__right__mono,axiom,
% 0.16/0.44 ! [A: rat,B: rat,C: rat] :
% 0.16/0.44 ( ( ord_less_rat @ A @ B )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_right_mono
% 0.16/0.44 thf(fact_2032_mult__strict__right__mono,axiom,
% 0.16/0.44 ! [A: nat,B: nat,C: nat] :
% 0.16/0.44 ( ( ord_less_nat @ A @ B )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.16/0.44 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_right_mono
% 0.16/0.44 thf(fact_2033_mult__strict__right__mono,axiom,
% 0.16/0.44 ! [A: int,B: int,C: int] :
% 0.16/0.44 ( ( ord_less_int @ A @ B )
% 0.16/0.44 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.16/0.44 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_right_mono
% 0.16/0.44 thf(fact_2034_mult__less__cancel__right__disj,axiom,
% 0.16/0.44 ! [A: real,C: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 & ( ord_less_real @ A @ B ) )
% 0.16/0.44 | ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_cancel_right_disj
% 0.16/0.44 thf(fact_2035_mult__less__cancel__right__disj,axiom,
% 0.16/0.44 ! [A: rat,C: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 & ( ord_less_rat @ A @ B ) )
% 0.16/0.44 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_cancel_right_disj
% 0.16/0.44 thf(fact_2036_mult__less__cancel__right__disj,axiom,
% 0.16/0.44 ! [A: int,C: int,B: int] :
% 0.16/0.44 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.16/0.44 & ( ord_less_int @ A @ B ) )
% 0.16/0.44 | ( ( ord_less_int @ C @ zero_zero_int )
% 0.16/0.44 & ( ord_less_int @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_cancel_right_disj
% 0.16/0.44 thf(fact_2037_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.16/0.44 ! [A: real,B: real,C: real] :
% 0.16/0.44 ( ( ord_less_real @ A @ B )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.16/0.44 thf(fact_2038_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.16/0.44 ! [A: rat,B: rat,C: rat] :
% 0.16/0.44 ( ( ord_less_rat @ A @ B )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.16/0.44 thf(fact_2039_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.16/0.44 ! [A: nat,B: nat,C: nat] :
% 0.16/0.44 ( ( ord_less_nat @ A @ B )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.16/0.44 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.16/0.44 thf(fact_2040_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.16/0.44 ! [A: int,B: int,C: int] :
% 0.16/0.44 ( ( ord_less_int @ A @ B )
% 0.16/0.44 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.16/0.44 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.16/0.44 thf(fact_2041_less__iff__diff__less__0,axiom,
% 0.16/0.44 ( ord_less_real
% 0.16/0.44 = ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_iff_diff_less_0
% 0.16/0.44 thf(fact_2042_less__iff__diff__less__0,axiom,
% 0.16/0.44 ( ord_less_rat
% 0.16/0.44 = ( ^ [A3: rat,B2: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_iff_diff_less_0
% 0.16/0.44 thf(fact_2043_less__iff__diff__less__0,axiom,
% 0.16/0.44 ( ord_less_int
% 0.16/0.44 = ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_iff_diff_less_0
% 0.16/0.44 thf(fact_2044_divide__neg__neg,axiom,
% 0.16/0.44 ! [X: real,Y4: real] :
% 0.16/0.44 ( ( ord_less_real @ X @ zero_zero_real )
% 0.16/0.44 => ( ( ord_less_real @ Y4 @ zero_zero_real )
% 0.16/0.44 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_neg_neg
% 0.16/0.44 thf(fact_2045_divide__neg__neg,axiom,
% 0.16/0.44 ! [X: rat,Y4: rat] :
% 0.16/0.44 ( ( ord_less_rat @ X @ zero_zero_rat )
% 0.16/0.44 => ( ( ord_less_rat @ Y4 @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_neg_neg
% 0.16/0.44 thf(fact_2046_divide__neg__pos,axiom,
% 0.16/0.44 ! [X: real,Y4: real] :
% 0.16/0.44 ( ( ord_less_real @ X @ zero_zero_real )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.16/0.44 => ( ord_less_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_neg_pos
% 0.16/0.44 thf(fact_2047_divide__neg__pos,axiom,
% 0.16/0.44 ! [X: rat,Y4: rat] :
% 0.16/0.44 ( ( ord_less_rat @ X @ zero_zero_rat )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ Y4 )
% 0.16/0.44 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y4 ) @ zero_zero_rat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_neg_pos
% 0.16/0.44 thf(fact_2048_divide__pos__neg,axiom,
% 0.16/0.44 ! [X: real,Y4: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ X )
% 0.16/0.44 => ( ( ord_less_real @ Y4 @ zero_zero_real )
% 0.16/0.44 => ( ord_less_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_pos_neg
% 0.16/0.44 thf(fact_2049_divide__pos__neg,axiom,
% 0.16/0.44 ! [X: rat,Y4: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.16/0.44 => ( ( ord_less_rat @ Y4 @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y4 ) @ zero_zero_rat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_pos_neg
% 0.16/0.44 thf(fact_2050_divide__pos__pos,axiom,
% 0.16/0.44 ! [X: real,Y4: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ X )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.16/0.44 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_pos_pos
% 0.16/0.44 thf(fact_2051_divide__pos__pos,axiom,
% 0.16/0.44 ! [X: rat,Y4: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ Y4 )
% 0.16/0.44 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_pos_pos
% 0.16/0.44 thf(fact_2052_divide__less__0__iff,axiom,
% 0.16/0.44 ! [A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.16/0.44 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.44 & ( ord_less_real @ B @ zero_zero_real ) )
% 0.16/0.44 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.16/0.44 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_less_0_iff
% 0.16/0.44 thf(fact_2053_divide__less__0__iff,axiom,
% 0.16/0.44 ! [A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 0.16/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.44 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 0.16/0.44 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.16/0.44 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_less_0_iff
% 0.16/0.44 thf(fact_2054_divide__less__cancel,axiom,
% 0.16/0.44 ! [A: real,C: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_real @ A @ B ) )
% 0.16/0.44 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_real @ B @ A ) )
% 0.16/0.44 & ( C != zero_zero_real ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_less_cancel
% 0.16/0.44 thf(fact_2055_divide__less__cancel,axiom,
% 0.16/0.44 ! [A: rat,C: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_rat @ A @ B ) )
% 0.16/0.44 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_rat @ B @ A ) )
% 0.16/0.44 & ( C != zero_zero_rat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_less_cancel
% 0.16/0.44 thf(fact_2056_zero__less__divide__iff,axiom,
% 0.16/0.44 ! [A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 0.16/0.44 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.44 & ( ord_less_real @ zero_zero_real @ B ) )
% 0.16/0.44 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.16/0.44 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_divide_iff
% 0.16/0.44 thf(fact_2057_zero__less__divide__iff,axiom,
% 0.16/0.44 ! [A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 0.16/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.44 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 0.16/0.44 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.16/0.44 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_divide_iff
% 0.16/0.44 thf(fact_2058_divide__strict__right__mono,axiom,
% 0.16/0.44 ! [A: real,B: real,C: real] :
% 0.16/0.44 ( ( ord_less_real @ A @ B )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_strict_right_mono
% 0.16/0.44 thf(fact_2059_divide__strict__right__mono,axiom,
% 0.16/0.44 ! [A: rat,B: rat,C: rat] :
% 0.16/0.44 ( ( ord_less_rat @ A @ B )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_strict_right_mono
% 0.16/0.44 thf(fact_2060_divide__strict__right__mono__neg,axiom,
% 0.16/0.44 ! [B: real,A: real,C: real] :
% 0.16/0.44 ( ( ord_less_real @ B @ A )
% 0.16/0.44 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_strict_right_mono_neg
% 0.16/0.44 thf(fact_2061_divide__strict__right__mono__neg,axiom,
% 0.16/0.44 ! [B: rat,A: rat,C: rat] :
% 0.16/0.44 ( ( ord_less_rat @ B @ A )
% 0.16/0.44 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_strict_right_mono_neg
% 0.16/0.44 thf(fact_2062_of__nat__less__0__iff,axiom,
% 0.16/0.44 ! [M: nat] :
% 0.16/0.44 ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_less_0_iff
% 0.16/0.44 thf(fact_2063_of__nat__less__0__iff,axiom,
% 0.16/0.44 ! [M: nat] :
% 0.16/0.44 ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_less_0_iff
% 0.16/0.44 thf(fact_2064_of__nat__less__0__iff,axiom,
% 0.16/0.44 ! [M: nat] :
% 0.16/0.44 ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_less_0_iff
% 0.16/0.44 thf(fact_2065_of__nat__less__0__iff,axiom,
% 0.16/0.44 ! [M: nat] :
% 0.16/0.44 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_less_0_iff
% 0.16/0.44 thf(fact_2066_zero__less__one__class_Ozero__le__one,axiom,
% 0.16/0.44 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.16/0.44
% 0.16/0.44 % zero_less_one_class.zero_le_one
% 0.16/0.44 thf(fact_2067_zero__less__one__class_Ozero__le__one,axiom,
% 0.16/0.44 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.16/0.44
% 0.16/0.44 % zero_less_one_class.zero_le_one
% 0.16/0.44 thf(fact_2068_zero__less__one__class_Ozero__le__one,axiom,
% 0.16/0.44 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.16/0.44
% 0.16/0.44 % zero_less_one_class.zero_le_one
% 0.16/0.44 thf(fact_2069_zero__less__one__class_Ozero__le__one,axiom,
% 0.16/0.44 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.16/0.44
% 0.16/0.44 % zero_less_one_class.zero_le_one
% 0.16/0.44 thf(fact_2070_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.16/0.44 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.16/0.44
% 0.16/0.44 % linordered_nonzero_semiring_class.zero_le_one
% 0.16/0.44 thf(fact_2071_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.16/0.44 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.16/0.44
% 0.16/0.44 % linordered_nonzero_semiring_class.zero_le_one
% 0.16/0.44 thf(fact_2072_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.16/0.44 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.16/0.44
% 0.16/0.44 % linordered_nonzero_semiring_class.zero_le_one
% 0.16/0.44 thf(fact_2073_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.16/0.44 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.16/0.44
% 0.16/0.44 % linordered_nonzero_semiring_class.zero_le_one
% 0.16/0.44 thf(fact_2074_not__one__le__zero,axiom,
% 0.16/0.44 ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.16/0.44
% 0.16/0.44 % not_one_le_zero
% 0.16/0.44 thf(fact_2075_not__one__le__zero,axiom,
% 0.16/0.44 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.16/0.44
% 0.16/0.44 % not_one_le_zero
% 0.16/0.44 thf(fact_2076_not__one__le__zero,axiom,
% 0.16/0.44 ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 0.16/0.44
% 0.16/0.44 % not_one_le_zero
% 0.16/0.44 thf(fact_2077_not__one__le__zero,axiom,
% 0.16/0.44 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 0.16/0.44
% 0.16/0.44 % not_one_le_zero
% 0.16/0.44 thf(fact_2078_one__le__numeral,axiom,
% 0.16/0.44 ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % one_le_numeral
% 0.16/0.44 thf(fact_2079_one__le__numeral,axiom,
% 0.16/0.44 ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % one_le_numeral
% 0.16/0.44 thf(fact_2080_one__le__numeral,axiom,
% 0.16/0.44 ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % one_le_numeral
% 0.16/0.44 thf(fact_2081_one__le__numeral,axiom,
% 0.16/0.44 ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % one_le_numeral
% 0.16/0.44 thf(fact_2082_less__Suc__eq__0__disj,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.16/0.44 = ( ( M = zero_zero_nat )
% 0.16/0.44 | ? [J3: nat] :
% 0.16/0.44 ( ( M
% 0.16/0.44 = ( suc @ J3 ) )
% 0.16/0.44 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_Suc_eq_0_disj
% 0.16/0.44 thf(fact_2083_gr0__implies__Suc,axiom,
% 0.16/0.44 ! [N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.44 => ? [M4: nat] :
% 0.16/0.44 ( N
% 0.16/0.44 = ( suc @ M4 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % gr0_implies_Suc
% 0.16/0.44 thf(fact_2084_All__less__Suc2,axiom,
% 0.16/0.44 ! [N: nat,P: nat > $o] :
% 0.16/0.44 ( ( ! [I2: nat] :
% 0.16/0.44 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 0.16/0.44 => ( P @ I2 ) ) )
% 0.16/0.44 = ( ( P @ zero_zero_nat )
% 0.16/0.44 & ! [I2: nat] :
% 0.16/0.44 ( ( ord_less_nat @ I2 @ N )
% 0.16/0.44 => ( P @ ( suc @ I2 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % All_less_Suc2
% 0.16/0.44 thf(fact_2085_gr0__conv__Suc,axiom,
% 0.16/0.44 ! [N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.44 = ( ? [M5: nat] :
% 0.16/0.44 ( N
% 0.16/0.44 = ( suc @ M5 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % gr0_conv_Suc
% 0.16/0.44 thf(fact_2086_Ex__less__Suc2,axiom,
% 0.16/0.44 ! [N: nat,P: nat > $o] :
% 0.16/0.44 ( ( ? [I2: nat] :
% 0.16/0.44 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 0.16/0.44 & ( P @ I2 ) ) )
% 0.16/0.44 = ( ( P @ zero_zero_nat )
% 0.16/0.44 | ? [I2: nat] :
% 0.16/0.44 ( ( ord_less_nat @ I2 @ N )
% 0.16/0.44 & ( P @ ( suc @ I2 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % Ex_less_Suc2
% 0.16/0.44 thf(fact_2087_le__imp__less__Suc,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.44 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_imp_less_Suc
% 0.16/0.44 thf(fact_2088_less__eq__Suc__le,axiom,
% 0.16/0.44 ( ord_less_nat
% 0.16/0.44 = ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_eq_Suc_le
% 0.16/0.44 thf(fact_2089_less__Suc__eq__le,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.16/0.44 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_Suc_eq_le
% 0.16/0.44 thf(fact_2090_le__less__Suc__eq,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.44 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.16/0.44 = ( N = M ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_less_Suc_eq
% 0.16/0.44 thf(fact_2091_Suc__le__lessD,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.16/0.44 => ( ord_less_nat @ M @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % Suc_le_lessD
% 0.16/0.44 thf(fact_2092_inc__induct,axiom,
% 0.16/0.44 ! [I: nat,J: nat,P: nat > $o] :
% 0.16/0.44 ( ( ord_less_eq_nat @ I @ J )
% 0.16/0.44 => ( ( P @ J )
% 0.16/0.44 => ( ! [N2: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ I @ N2 )
% 0.16/0.44 => ( ( ord_less_nat @ N2 @ J )
% 0.16/0.44 => ( ( P @ ( suc @ N2 ) )
% 0.16/0.44 => ( P @ N2 ) ) ) )
% 0.16/0.44 => ( P @ I ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inc_induct
% 0.16/0.44 thf(fact_2093_dec__induct,axiom,
% 0.16/0.44 ! [I: nat,J: nat,P: nat > $o] :
% 0.16/0.44 ( ( ord_less_eq_nat @ I @ J )
% 0.16/0.44 => ( ( P @ I )
% 0.16/0.44 => ( ! [N2: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ I @ N2 )
% 0.16/0.44 => ( ( ord_less_nat @ N2 @ J )
% 0.16/0.44 => ( ( P @ N2 )
% 0.16/0.44 => ( P @ ( suc @ N2 ) ) ) ) )
% 0.16/0.44 => ( P @ J ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dec_induct
% 0.16/0.44 thf(fact_2094_Suc__le__eq,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.16/0.44 = ( ord_less_nat @ M @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % Suc_le_eq
% 0.16/0.44 thf(fact_2095_Suc__leI,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ M @ N )
% 0.16/0.44 => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % Suc_leI
% 0.16/0.44 thf(fact_2096_right__inverse__eq,axiom,
% 0.16/0.44 ! [B: complex,A: complex] :
% 0.16/0.44 ( ( B != zero_zero_complex )
% 0.16/0.44 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.16/0.44 = one_one_complex )
% 0.16/0.44 = ( A = B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % right_inverse_eq
% 0.16/0.44 thf(fact_2097_right__inverse__eq,axiom,
% 0.16/0.44 ! [B: real,A: real] :
% 0.16/0.44 ( ( B != zero_zero_real )
% 0.16/0.44 => ( ( ( divide_divide_real @ A @ B )
% 0.16/0.44 = one_one_real )
% 0.16/0.44 = ( A = B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % right_inverse_eq
% 0.16/0.44 thf(fact_2098_right__inverse__eq,axiom,
% 0.16/0.44 ! [B: rat,A: rat] :
% 0.16/0.44 ( ( B != zero_zero_rat )
% 0.16/0.44 => ( ( ( divide_divide_rat @ A @ B )
% 0.16/0.44 = one_one_rat )
% 0.16/0.44 = ( A = B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % right_inverse_eq
% 0.16/0.44 thf(fact_2099_numeral__One,axiom,
% 0.16/0.44 ( ( numera6690914467698888265omplex @ one )
% 0.16/0.44 = one_one_complex ) ).
% 0.16/0.44
% 0.16/0.44 % numeral_One
% 0.16/0.44 thf(fact_2100_numeral__One,axiom,
% 0.16/0.44 ( ( numeral_numeral_real @ one )
% 0.16/0.44 = one_one_real ) ).
% 0.16/0.44
% 0.16/0.44 % numeral_One
% 0.16/0.44 thf(fact_2101_numeral__One,axiom,
% 0.16/0.44 ( ( numeral_numeral_rat @ one )
% 0.16/0.44 = one_one_rat ) ).
% 0.16/0.44
% 0.16/0.44 % numeral_One
% 0.16/0.44 thf(fact_2102_numeral__One,axiom,
% 0.16/0.44 ( ( numeral_numeral_nat @ one )
% 0.16/0.44 = one_one_nat ) ).
% 0.16/0.44
% 0.16/0.44 % numeral_One
% 0.16/0.44 thf(fact_2103_numeral__One,axiom,
% 0.16/0.44 ( ( numeral_numeral_int @ one )
% 0.16/0.44 = one_one_int ) ).
% 0.16/0.44
% 0.16/0.44 % numeral_One
% 0.16/0.44 thf(fact_2104_ex__least__nat__le,axiom,
% 0.16/0.44 ! [P: nat > $o,N: nat] :
% 0.16/0.44 ( ( P @ N )
% 0.16/0.44 => ( ~ ( P @ zero_zero_nat )
% 0.16/0.44 => ? [K2: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ K2 @ N )
% 0.16/0.44 & ! [I4: nat] :
% 0.16/0.44 ( ( ord_less_nat @ I4 @ K2 )
% 0.16/0.44 => ~ ( P @ I4 ) )
% 0.16/0.44 & ( P @ K2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % ex_least_nat_le
% 0.16/0.44 thf(fact_2105_not__is__unit__0,axiom,
% 0.16/0.44 ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 0.16/0.44
% 0.16/0.44 % not_is_unit_0
% 0.16/0.44 thf(fact_2106_not__is__unit__0,axiom,
% 0.16/0.44 ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 0.16/0.44
% 0.16/0.44 % not_is_unit_0
% 0.16/0.44 thf(fact_2107_not__is__unit__0,axiom,
% 0.16/0.44 ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 0.16/0.44
% 0.16/0.44 % not_is_unit_0
% 0.16/0.44 thf(fact_2108_diff__less__Suc,axiom,
% 0.16/0.44 ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 0.16/0.44
% 0.16/0.44 % diff_less_Suc
% 0.16/0.44 thf(fact_2109_Suc__diff__Suc,axiom,
% 0.16/0.44 ! [N: nat,M: nat] :
% 0.16/0.44 ( ( ord_less_nat @ N @ M )
% 0.16/0.44 => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 0.16/0.44 = ( minus_minus_nat @ M @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % Suc_diff_Suc
% 0.16/0.44 thf(fact_2110_Suc__mult__less__cancel1,axiom,
% 0.16/0.44 ! [K: nat,M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.16/0.44 = ( ord_less_nat @ M @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % Suc_mult_less_cancel1
% 0.16/0.44 thf(fact_2111_diff__less,axiom,
% 0.16/0.44 ! [N: nat,M: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.16/0.44 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % diff_less
% 0.16/0.44 thf(fact_2112_nat__mult__less__cancel1,axiom,
% 0.16/0.44 ! [K: nat,M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.16/0.44 => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.16/0.44 = ( ord_less_nat @ M @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % nat_mult_less_cancel1
% 0.16/0.44 thf(fact_2113_nat__mult__eq__cancel1,axiom,
% 0.16/0.44 ! [K: nat,M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.16/0.44 => ( ( ( times_times_nat @ K @ M )
% 0.16/0.44 = ( times_times_nat @ K @ N ) )
% 0.16/0.44 = ( M = N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % nat_mult_eq_cancel1
% 0.16/0.44 thf(fact_2114_mult__less__mono2,axiom,
% 0.16/0.44 ! [I: nat,J: nat,K: nat] :
% 0.16/0.44 ( ( ord_less_nat @ I @ J )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.16/0.44 => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_mono2
% 0.16/0.44 thf(fact_2115_mult__less__mono1,axiom,
% 0.16/0.44 ! [I: nat,J: nat,K: nat] :
% 0.16/0.44 ( ( ord_less_nat @ I @ J )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.16/0.44 => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_mono1
% 0.16/0.44 thf(fact_2116_is__unit__mult__iff,axiom,
% 0.16/0.44 ! [A: code_integer,B: code_integer] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
% 0.16/0.44 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.16/0.44 & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % is_unit_mult_iff
% 0.16/0.44 thf(fact_2117_is__unit__mult__iff,axiom,
% 0.16/0.44 ! [A: nat,B: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
% 0.16/0.44 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.16/0.44 & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % is_unit_mult_iff
% 0.16/0.44 thf(fact_2118_is__unit__mult__iff,axiom,
% 0.16/0.44 ! [A: int,B: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
% 0.16/0.44 = ( ( dvd_dvd_int @ A @ one_one_int )
% 0.16/0.44 & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % is_unit_mult_iff
% 0.16/0.44 thf(fact_2119_dvd__mult__unit__iff,axiom,
% 0.16/0.44 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.16/0.44 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.16/0.44 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dvd_mult_unit_iff
% 0.16/0.44 thf(fact_2120_dvd__mult__unit__iff,axiom,
% 0.16/0.44 ! [B: nat,A: nat,C: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.16/0.44 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
% 0.16/0.44 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dvd_mult_unit_iff
% 0.16/0.44 thf(fact_2121_dvd__mult__unit__iff,axiom,
% 0.16/0.44 ! [B: int,A: int,C: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.16/0.44 => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
% 0.16/0.44 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dvd_mult_unit_iff
% 0.16/0.44 thf(fact_2122_mult__unit__dvd__iff,axiom,
% 0.16/0.44 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.16/0.44 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.16/0.44 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_unit_dvd_iff
% 0.16/0.44 thf(fact_2123_mult__unit__dvd__iff,axiom,
% 0.16/0.44 ! [B: nat,A: nat,C: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.16/0.44 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.16/0.44 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_unit_dvd_iff
% 0.16/0.44 thf(fact_2124_mult__unit__dvd__iff,axiom,
% 0.16/0.44 ! [B: int,A: int,C: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.16/0.44 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.16/0.44 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_unit_dvd_iff
% 0.16/0.44 thf(fact_2125_dvd__mult__unit__iff_H,axiom,
% 0.16/0.44 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.16/0.44 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.16/0.44 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dvd_mult_unit_iff'
% 0.16/0.44 thf(fact_2126_dvd__mult__unit__iff_H,axiom,
% 0.16/0.44 ! [B: nat,A: nat,C: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.16/0.44 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.16/0.44 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dvd_mult_unit_iff'
% 0.16/0.44 thf(fact_2127_dvd__mult__unit__iff_H,axiom,
% 0.16/0.44 ! [B: int,A: int,C: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.16/0.44 => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 0.16/0.44 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dvd_mult_unit_iff'
% 0.16/0.44 thf(fact_2128_mult__unit__dvd__iff_H,axiom,
% 0.16/0.44 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.16/0.44 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.16/0.44 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_unit_dvd_iff'
% 0.16/0.44 thf(fact_2129_mult__unit__dvd__iff_H,axiom,
% 0.16/0.44 ! [A: nat,B: nat,C: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.16/0.44 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.16/0.44 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_unit_dvd_iff'
% 0.16/0.44 thf(fact_2130_mult__unit__dvd__iff_H,axiom,
% 0.16/0.44 ! [A: int,B: int,C: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.16/0.44 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.16/0.44 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_unit_dvd_iff'
% 0.16/0.44 thf(fact_2131_unit__mult__left__cancel,axiom,
% 0.16/0.44 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.16/0.44 => ( ( ( times_3573771949741848930nteger @ A @ B )
% 0.16/0.44 = ( times_3573771949741848930nteger @ A @ C ) )
% 0.16/0.44 = ( B = C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_mult_left_cancel
% 0.16/0.44 thf(fact_2132_unit__mult__left__cancel,axiom,
% 0.16/0.44 ! [A: nat,B: nat,C: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.16/0.44 => ( ( ( times_times_nat @ A @ B )
% 0.16/0.44 = ( times_times_nat @ A @ C ) )
% 0.16/0.44 = ( B = C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_mult_left_cancel
% 0.16/0.44 thf(fact_2133_unit__mult__left__cancel,axiom,
% 0.16/0.44 ! [A: int,B: int,C: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.16/0.44 => ( ( ( times_times_int @ A @ B )
% 0.16/0.44 = ( times_times_int @ A @ C ) )
% 0.16/0.44 = ( B = C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_mult_left_cancel
% 0.16/0.44 thf(fact_2134_unit__mult__right__cancel,axiom,
% 0.16/0.44 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.16/0.44 => ( ( ( times_3573771949741848930nteger @ B @ A )
% 0.16/0.44 = ( times_3573771949741848930nteger @ C @ A ) )
% 0.16/0.44 = ( B = C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_mult_right_cancel
% 0.16/0.44 thf(fact_2135_unit__mult__right__cancel,axiom,
% 0.16/0.44 ! [A: nat,B: nat,C: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.16/0.44 => ( ( ( times_times_nat @ B @ A )
% 0.16/0.44 = ( times_times_nat @ C @ A ) )
% 0.16/0.44 = ( B = C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_mult_right_cancel
% 0.16/0.44 thf(fact_2136_unit__mult__right__cancel,axiom,
% 0.16/0.44 ! [A: int,B: int,C: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.16/0.44 => ( ( ( times_times_int @ B @ A )
% 0.16/0.44 = ( times_times_int @ C @ A ) )
% 0.16/0.44 = ( B = C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_mult_right_cancel
% 0.16/0.44 thf(fact_2137_diff__less__mono,axiom,
% 0.16/0.44 ! [A: nat,B: nat,C: nat] :
% 0.16/0.44 ( ( ord_less_nat @ A @ B )
% 0.16/0.44 => ( ( ord_less_eq_nat @ C @ A )
% 0.16/0.44 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % diff_less_mono
% 0.16/0.44 thf(fact_2138_less__diff__iff,axiom,
% 0.16/0.44 ! [K: nat,M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ K @ M )
% 0.16/0.44 => ( ( ord_less_eq_nat @ K @ N )
% 0.16/0.44 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.16/0.44 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_diff_iff
% 0.16/0.44 thf(fact_2139_nat__dvd__not__less,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.16/0.44 => ( ( ord_less_nat @ M @ N )
% 0.16/0.44 => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % nat_dvd_not_less
% 0.16/0.44 thf(fact_2140_unit__div__cancel,axiom,
% 0.16/0.44 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.16/0.44 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.16/0.44 = ( divide6298287555418463151nteger @ C @ A ) )
% 0.16/0.44 = ( B = C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_div_cancel
% 0.16/0.44 thf(fact_2141_unit__div__cancel,axiom,
% 0.16/0.44 ! [A: nat,B: nat,C: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.16/0.44 => ( ( ( divide_divide_nat @ B @ A )
% 0.16/0.44 = ( divide_divide_nat @ C @ A ) )
% 0.16/0.44 = ( B = C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_div_cancel
% 0.16/0.44 thf(fact_2142_unit__div__cancel,axiom,
% 0.16/0.44 ! [A: int,B: int,C: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.16/0.44 => ( ( ( divide_divide_int @ B @ A )
% 0.16/0.44 = ( divide_divide_int @ C @ A ) )
% 0.16/0.44 = ( B = C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_div_cancel
% 0.16/0.44 thf(fact_2143_div__unit__dvd__iff,axiom,
% 0.16/0.44 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.16/0.44 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.16/0.44 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % div_unit_dvd_iff
% 0.16/0.44 thf(fact_2144_div__unit__dvd__iff,axiom,
% 0.16/0.44 ! [B: nat,A: nat,C: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.16/0.44 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.16/0.44 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % div_unit_dvd_iff
% 0.16/0.44 thf(fact_2145_div__unit__dvd__iff,axiom,
% 0.16/0.44 ! [B: int,A: int,C: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.16/0.44 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.16/0.44 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % div_unit_dvd_iff
% 0.16/0.44 thf(fact_2146_dvd__div__unit__iff,axiom,
% 0.16/0.44 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.16/0.44 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
% 0.16/0.44 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dvd_div_unit_iff
% 0.16/0.44 thf(fact_2147_dvd__div__unit__iff,axiom,
% 0.16/0.44 ! [B: nat,A: nat,C: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.16/0.44 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
% 0.16/0.44 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dvd_div_unit_iff
% 0.16/0.44 thf(fact_2148_dvd__div__unit__iff,axiom,
% 0.16/0.44 ! [B: int,A: int,C: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.16/0.44 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
% 0.16/0.44 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dvd_div_unit_iff
% 0.16/0.44 thf(fact_2149_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ( divide_divide_nat @ M @ N )
% 0.16/0.44 = zero_zero_nat )
% 0.16/0.44 = ( ( ord_less_nat @ M @ N )
% 0.16/0.44 | ( N = zero_zero_nat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % Euclidean_Division.div_eq_0_iff
% 0.16/0.44 thf(fact_2150_numerals_I1_J,axiom,
% 0.16/0.44 ( ( numeral_numeral_nat @ one )
% 0.16/0.44 = one_one_nat ) ).
% 0.16/0.44
% 0.16/0.44 % numerals(1)
% 0.16/0.44 thf(fact_2151_One__nat__def,axiom,
% 0.16/0.44 ( one_one_nat
% 0.16/0.44 = ( suc @ zero_zero_nat ) ) ).
% 0.16/0.44
% 0.16/0.44 % One_nat_def
% 0.16/0.44 thf(fact_2152_dvd__minus__self,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
% 0.16/0.44 = ( ( ord_less_nat @ N @ M )
% 0.16/0.44 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dvd_minus_self
% 0.16/0.44 thf(fact_2153_less__mult__imp__div__less,axiom,
% 0.16/0.44 ! [M: nat,I: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
% 0.16/0.44 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_mult_imp_div_less
% 0.16/0.44 thf(fact_2154_diff__Suc__eq__diff__pred,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.16/0.44 = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % diff_Suc_eq_diff_pred
% 0.16/0.44 thf(fact_2155_mult__eq__self__implies__10,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( M
% 0.16/0.44 = ( times_times_nat @ M @ N ) )
% 0.16/0.44 => ( ( N = one_one_nat )
% 0.16/0.44 | ( M = zero_zero_nat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_eq_self_implies_10
% 0.16/0.44 thf(fact_2156_zero__less__power__eq,axiom,
% 0.16/0.44 ! [A: real,N: nat] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.16/0.44 = ( ( N = zero_zero_nat )
% 0.16/0.44 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 & ( A != zero_zero_real ) )
% 0.16/0.44 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_power_eq
% 0.16/0.44 thf(fact_2157_zero__less__power__eq,axiom,
% 0.16/0.44 ! [A: rat,N: nat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.16/0.44 = ( ( N = zero_zero_nat )
% 0.16/0.44 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 & ( A != zero_zero_rat ) )
% 0.16/0.44 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_power_eq
% 0.16/0.44 thf(fact_2158_zero__less__power__eq,axiom,
% 0.16/0.44 ! [A: int,N: nat] :
% 0.16/0.44 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.16/0.44 = ( ( N = zero_zero_nat )
% 0.16/0.44 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 & ( A != zero_zero_int ) )
% 0.16/0.44 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_power_eq
% 0.16/0.44 thf(fact_2159_mult__le__cancel__left,axiom,
% 0.16/0.44 ! [C: rat,A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.16/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_eq_rat @ A @ B ) )
% 0.16/0.44 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_cancel_left
% 0.16/0.44 thf(fact_2160_mult__le__cancel__left,axiom,
% 0.16/0.44 ! [C: int,A: int,B: int] :
% 0.16/0.44 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.16/0.44 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.16/0.44 => ( ord_less_eq_int @ A @ B ) )
% 0.16/0.44 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.16/0.44 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_cancel_left
% 0.16/0.44 thf(fact_2161_mult__le__cancel__left,axiom,
% 0.16/0.44 ! [C: real,A: real,B: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.16/0.44 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_eq_real @ A @ B ) )
% 0.16/0.44 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_cancel_left
% 0.16/0.44 thf(fact_2162_mult__le__cancel__right,axiom,
% 0.16/0.44 ! [A: rat,C: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_eq_rat @ A @ B ) )
% 0.16/0.44 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_cancel_right
% 0.16/0.44 thf(fact_2163_mult__le__cancel__right,axiom,
% 0.16/0.44 ! [A: int,C: int,B: int] :
% 0.16/0.44 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.16/0.44 => ( ord_less_eq_int @ A @ B ) )
% 0.16/0.44 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.16/0.44 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_cancel_right
% 0.16/0.44 thf(fact_2164_mult__le__cancel__right,axiom,
% 0.16/0.44 ! [A: real,C: real,B: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_eq_real @ A @ B ) )
% 0.16/0.44 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_cancel_right
% 0.16/0.44 thf(fact_2165_mult__left__less__imp__less,axiom,
% 0.16/0.44 ! [C: rat,A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.16/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_left_less_imp_less
% 0.16/0.44 thf(fact_2166_mult__left__less__imp__less,axiom,
% 0.16/0.44 ! [C: nat,A: nat,B: nat] :
% 0.16/0.44 ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.16/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.16/0.44 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_left_less_imp_less
% 0.16/0.44 thf(fact_2167_mult__left__less__imp__less,axiom,
% 0.16/0.44 ! [C: int,A: int,B: int] :
% 0.16/0.44 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.16/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.16/0.44 => ( ord_less_int @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_left_less_imp_less
% 0.16/0.44 thf(fact_2168_mult__left__less__imp__less,axiom,
% 0.16/0.44 ! [C: real,A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.16/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_real @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_left_less_imp_less
% 0.16/0.44 thf(fact_2169_mult__strict__mono,axiom,
% 0.16/0.44 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.16/0.44 ( ( ord_less_rat @ A @ B )
% 0.16/0.44 => ( ( ord_less_rat @ C @ D )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.16/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_mono
% 0.16/0.44 thf(fact_2170_mult__strict__mono,axiom,
% 0.16/0.44 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.16/0.44 ( ( ord_less_nat @ A @ B )
% 0.16/0.44 => ( ( ord_less_nat @ C @ D )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.16/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.16/0.44 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_mono
% 0.16/0.44 thf(fact_2171_mult__strict__mono,axiom,
% 0.16/0.44 ! [A: int,B: int,C: int,D: int] :
% 0.16/0.44 ( ( ord_less_int @ A @ B )
% 0.16/0.44 => ( ( ord_less_int @ C @ D )
% 0.16/0.44 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.16/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.16/0.44 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_mono
% 0.16/0.44 thf(fact_2172_mult__strict__mono,axiom,
% 0.16/0.44 ! [A: real,B: real,C: real,D: real] :
% 0.16/0.44 ( ( ord_less_real @ A @ B )
% 0.16/0.44 => ( ( ord_less_real @ C @ D )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.16/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_mono
% 0.16/0.44 thf(fact_2173_mult__less__cancel__left,axiom,
% 0.16/0.44 ! [C: rat,A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.16/0.44 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_rat @ A @ B ) )
% 0.16/0.44 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_cancel_left
% 0.16/0.44 thf(fact_2174_mult__less__cancel__left,axiom,
% 0.16/0.44 ! [C: int,A: int,B: int] :
% 0.16/0.44 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.16/0.44 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.16/0.44 => ( ord_less_int @ A @ B ) )
% 0.16/0.44 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.16/0.44 => ( ord_less_int @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_cancel_left
% 0.16/0.44 thf(fact_2175_mult__less__cancel__left,axiom,
% 0.16/0.44 ! [C: real,A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.16/0.44 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_real @ A @ B ) )
% 0.16/0.44 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_real @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_cancel_left
% 0.16/0.44 thf(fact_2176_mult__right__less__imp__less,axiom,
% 0.16/0.44 ! [A: rat,C: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.16/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_right_less_imp_less
% 0.16/0.44 thf(fact_2177_mult__right__less__imp__less,axiom,
% 0.16/0.44 ! [A: nat,C: nat,B: nat] :
% 0.16/0.44 ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.16/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.16/0.44 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_right_less_imp_less
% 0.16/0.44 thf(fact_2178_mult__right__less__imp__less,axiom,
% 0.16/0.44 ! [A: int,C: int,B: int] :
% 0.16/0.44 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.16/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.16/0.44 => ( ord_less_int @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_right_less_imp_less
% 0.16/0.44 thf(fact_2179_mult__right__less__imp__less,axiom,
% 0.16/0.44 ! [A: real,C: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.16/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_real @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_right_less_imp_less
% 0.16/0.44 thf(fact_2180_mult__strict__mono_H,axiom,
% 0.16/0.44 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.16/0.44 ( ( ord_less_rat @ A @ B )
% 0.16/0.44 => ( ( ord_less_rat @ C @ D )
% 0.16/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_mono'
% 0.16/0.44 thf(fact_2181_mult__strict__mono_H,axiom,
% 0.16/0.44 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.16/0.44 ( ( ord_less_nat @ A @ B )
% 0.16/0.44 => ( ( ord_less_nat @ C @ D )
% 0.16/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.16/0.44 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_mono'
% 0.16/0.44 thf(fact_2182_mult__strict__mono_H,axiom,
% 0.16/0.44 ! [A: int,B: int,C: int,D: int] :
% 0.16/0.44 ( ( ord_less_int @ A @ B )
% 0.16/0.44 => ( ( ord_less_int @ C @ D )
% 0.16/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.16/0.44 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_mono'
% 0.16/0.44 thf(fact_2183_mult__strict__mono_H,axiom,
% 0.16/0.44 ! [A: real,B: real,C: real,D: real] :
% 0.16/0.44 ( ( ord_less_real @ A @ B )
% 0.16/0.44 => ( ( ord_less_real @ C @ D )
% 0.16/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_strict_mono'
% 0.16/0.44 thf(fact_2184_mult__less__cancel__right,axiom,
% 0.16/0.44 ! [A: rat,C: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_rat @ A @ B ) )
% 0.16/0.44 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_cancel_right
% 0.16/0.44 thf(fact_2185_mult__less__cancel__right,axiom,
% 0.16/0.44 ! [A: int,C: int,B: int] :
% 0.16/0.44 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.16/0.44 => ( ord_less_int @ A @ B ) )
% 0.16/0.44 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.16/0.44 => ( ord_less_int @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_cancel_right
% 0.16/0.44 thf(fact_2186_mult__less__cancel__right,axiom,
% 0.16/0.44 ! [A: real,C: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_real @ A @ B ) )
% 0.16/0.44 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_real @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_cancel_right
% 0.16/0.44 thf(fact_2187_mult__le__cancel__left__neg,axiom,
% 0.16/0.44 ! [C: rat,A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.16/0.44 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_cancel_left_neg
% 0.16/0.44 thf(fact_2188_mult__le__cancel__left__neg,axiom,
% 0.16/0.44 ! [C: int,A: int,B: int] :
% 0.16/0.44 ( ( ord_less_int @ C @ zero_zero_int )
% 0.16/0.44 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.16/0.44 = ( ord_less_eq_int @ B @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_cancel_left_neg
% 0.16/0.44 thf(fact_2189_mult__le__cancel__left__neg,axiom,
% 0.16/0.44 ! [C: real,A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.16/0.44 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_cancel_left_neg
% 0.16/0.44 thf(fact_2190_mult__le__cancel__left__pos,axiom,
% 0.16/0.44 ! [C: rat,A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.16/0.44 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_cancel_left_pos
% 0.16/0.44 thf(fact_2191_mult__le__cancel__left__pos,axiom,
% 0.16/0.44 ! [C: int,A: int,B: int] :
% 0.16/0.44 ( ( ord_less_int @ zero_zero_int @ C )
% 0.16/0.44 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.16/0.44 = ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_cancel_left_pos
% 0.16/0.44 thf(fact_2192_mult__le__cancel__left__pos,axiom,
% 0.16/0.44 ! [C: real,A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.16/0.44 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_cancel_left_pos
% 0.16/0.44 thf(fact_2193_mult__left__le__imp__le,axiom,
% 0.16/0.44 ! [C: rat,A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_left_le_imp_le
% 0.16/0.44 thf(fact_2194_mult__left__le__imp__le,axiom,
% 0.16/0.44 ! [C: nat,A: nat,B: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.16/0.44 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_left_le_imp_le
% 0.16/0.44 thf(fact_2195_mult__left__le__imp__le,axiom,
% 0.16/0.44 ! [C: int,A: int,B: int] :
% 0.16/0.44 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.16/0.44 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.16/0.44 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_left_le_imp_le
% 0.16/0.44 thf(fact_2196_mult__left__le__imp__le,axiom,
% 0.16/0.44 ! [C: real,A: real,B: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_left_le_imp_le
% 0.16/0.44 thf(fact_2197_mult__right__le__imp__le,axiom,
% 0.16/0.44 ! [A: rat,C: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_right_le_imp_le
% 0.16/0.44 thf(fact_2198_mult__right__le__imp__le,axiom,
% 0.16/0.44 ! [A: nat,C: nat,B: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.16/0.44 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_right_le_imp_le
% 0.16/0.44 thf(fact_2199_mult__right__le__imp__le,axiom,
% 0.16/0.44 ! [A: int,C: int,B: int] :
% 0.16/0.44 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.16/0.44 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.16/0.44 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_right_le_imp_le
% 0.16/0.44 thf(fact_2200_mult__right__le__imp__le,axiom,
% 0.16/0.44 ! [A: real,C: real,B: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_right_le_imp_le
% 0.16/0.44 thf(fact_2201_mult__le__less__imp__less,axiom,
% 0.16/0.44 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.44 => ( ( ord_less_rat @ C @ D )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_less_imp_less
% 0.16/0.44 thf(fact_2202_mult__le__less__imp__less,axiom,
% 0.16/0.44 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.44 => ( ( ord_less_nat @ C @ D )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.16/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.16/0.44 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_less_imp_less
% 0.16/0.44 thf(fact_2203_mult__le__less__imp__less,axiom,
% 0.16/0.44 ! [A: int,B: int,C: int,D: int] :
% 0.16/0.44 ( ( ord_less_eq_int @ A @ B )
% 0.16/0.44 => ( ( ord_less_int @ C @ D )
% 0.16/0.44 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.16/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.16/0.44 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_less_imp_less
% 0.16/0.44 thf(fact_2204_mult__le__less__imp__less,axiom,
% 0.16/0.44 ! [A: real,B: real,C: real,D: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.44 => ( ( ord_less_real @ C @ D )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_less_imp_less
% 0.16/0.44 thf(fact_2205_mult__less__le__imp__less,axiom,
% 0.16/0.44 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.16/0.44 ( ( ord_less_rat @ A @ B )
% 0.16/0.44 => ( ( ord_less_eq_rat @ C @ D )
% 0.16/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_le_imp_less
% 0.16/0.44 thf(fact_2206_mult__less__le__imp__less,axiom,
% 0.16/0.44 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.16/0.44 ( ( ord_less_nat @ A @ B )
% 0.16/0.44 => ( ( ord_less_eq_nat @ C @ D )
% 0.16/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.16/0.44 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_le_imp_less
% 0.16/0.44 thf(fact_2207_mult__less__le__imp__less,axiom,
% 0.16/0.44 ! [A: int,B: int,C: int,D: int] :
% 0.16/0.44 ( ( ord_less_int @ A @ B )
% 0.16/0.44 => ( ( ord_less_eq_int @ C @ D )
% 0.16/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.44 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.16/0.44 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_le_imp_less
% 0.16/0.44 thf(fact_2208_mult__less__le__imp__less,axiom,
% 0.16/0.44 ! [A: real,B: real,C: real,D: real] :
% 0.16/0.44 ( ( ord_less_real @ A @ B )
% 0.16/0.44 => ( ( ord_less_eq_real @ C @ D )
% 0.16/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_less_le_imp_less
% 0.16/0.44 thf(fact_2209_frac__le,axiom,
% 0.16/0.44 ! [Y4: rat,X: rat,W: rat,Z: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 0.16/0.44 => ( ( ord_less_eq_rat @ X @ Y4 )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.16/0.44 => ( ( ord_less_eq_rat @ W @ Z )
% 0.16/0.44 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y4 @ W ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % frac_le
% 0.16/0.44 thf(fact_2210_frac__le,axiom,
% 0.16/0.44 ! [Y4: real,X: real,W: real,Z: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.16/0.44 => ( ( ord_less_eq_real @ X @ Y4 )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.16/0.44 => ( ( ord_less_eq_real @ W @ Z )
% 0.16/0.44 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y4 @ W ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % frac_le
% 0.16/0.44 thf(fact_2211_frac__less,axiom,
% 0.16/0.44 ! [X: rat,Y4: rat,W: rat,Z: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.16/0.44 => ( ( ord_less_rat @ X @ Y4 )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.16/0.44 => ( ( ord_less_eq_rat @ W @ Z )
% 0.16/0.44 => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y4 @ W ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % frac_less
% 0.16/0.44 thf(fact_2212_frac__less,axiom,
% 0.16/0.44 ! [X: real,Y4: real,W: real,Z: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.16/0.44 => ( ( ord_less_real @ X @ Y4 )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.16/0.44 => ( ( ord_less_eq_real @ W @ Z )
% 0.16/0.44 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y4 @ W ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % frac_less
% 0.16/0.44 thf(fact_2213_frac__less2,axiom,
% 0.16/0.44 ! [X: rat,Y4: rat,W: rat,Z: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.16/0.44 => ( ( ord_less_eq_rat @ X @ Y4 )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.16/0.44 => ( ( ord_less_rat @ W @ Z )
% 0.16/0.44 => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y4 @ W ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % frac_less2
% 0.16/0.44 thf(fact_2214_frac__less2,axiom,
% 0.16/0.44 ! [X: real,Y4: real,W: real,Z: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ X )
% 0.16/0.44 => ( ( ord_less_eq_real @ X @ Y4 )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.16/0.44 => ( ( ord_less_real @ W @ Z )
% 0.16/0.44 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y4 @ W ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % frac_less2
% 0.16/0.44 thf(fact_2215_divide__le__cancel,axiom,
% 0.16/0.44 ! [A: rat,C: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_eq_rat @ A @ B ) )
% 0.16/0.44 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_le_cancel
% 0.16/0.44 thf(fact_2216_divide__le__cancel,axiom,
% 0.16/0.44 ! [A: real,C: real,B: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_eq_real @ A @ B ) )
% 0.16/0.44 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_le_cancel
% 0.16/0.44 thf(fact_2217_divide__nonneg__neg,axiom,
% 0.16/0.44 ! [X: rat,Y4: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.16/0.44 => ( ( ord_less_rat @ Y4 @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y4 ) @ zero_zero_rat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_nonneg_neg
% 0.16/0.44 thf(fact_2218_divide__nonneg__neg,axiom,
% 0.16/0.44 ! [X: real,Y4: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.16/0.44 => ( ( ord_less_real @ Y4 @ zero_zero_real )
% 0.16/0.44 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_nonneg_neg
% 0.16/0.44 thf(fact_2219_divide__nonneg__pos,axiom,
% 0.16/0.44 ! [X: rat,Y4: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ Y4 )
% 0.16/0.44 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_nonneg_pos
% 0.16/0.44 thf(fact_2220_divide__nonneg__pos,axiom,
% 0.16/0.44 ! [X: real,Y4: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.16/0.44 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_nonneg_pos
% 0.16/0.44 thf(fact_2221_divide__nonpos__neg,axiom,
% 0.16/0.44 ! [X: rat,Y4: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.16/0.44 => ( ( ord_less_rat @ Y4 @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_nonpos_neg
% 0.16/0.44 thf(fact_2222_divide__nonpos__neg,axiom,
% 0.16/0.44 ! [X: real,Y4: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.16/0.44 => ( ( ord_less_real @ Y4 @ zero_zero_real )
% 0.16/0.44 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_nonpos_neg
% 0.16/0.44 thf(fact_2223_divide__nonpos__pos,axiom,
% 0.16/0.44 ! [X: rat,Y4: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ Y4 )
% 0.16/0.44 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y4 ) @ zero_zero_rat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_nonpos_pos
% 0.16/0.44 thf(fact_2224_divide__nonpos__pos,axiom,
% 0.16/0.44 ! [X: real,Y4: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.16/0.44 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_nonpos_pos
% 0.16/0.44 thf(fact_2225_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.16/0.44 ! [A: nat,B: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.44 => ( ( ord_less_nat @ A @ B )
% 0.16/0.44 => ( ( divide_divide_nat @ A @ B )
% 0.16/0.44 = zero_zero_nat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unique_euclidean_semiring_numeral_class.div_less
% 0.16/0.44 thf(fact_2226_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.16/0.44 ! [A: int,B: int] :
% 0.16/0.44 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.44 => ( ( ord_less_int @ A @ B )
% 0.16/0.44 => ( ( divide_divide_int @ A @ B )
% 0.16/0.44 = zero_zero_int ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unique_euclidean_semiring_numeral_class.div_less
% 0.16/0.44 thf(fact_2227_div__positive,axiom,
% 0.16/0.44 ! [B: nat,A: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.16/0.44 => ( ( ord_less_eq_nat @ B @ A )
% 0.16/0.44 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % div_positive
% 0.16/0.44 thf(fact_2228_div__positive,axiom,
% 0.16/0.44 ! [B: int,A: int] :
% 0.16/0.44 ( ( ord_less_int @ zero_zero_int @ B )
% 0.16/0.44 => ( ( ord_less_eq_int @ B @ A )
% 0.16/0.44 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % div_positive
% 0.16/0.44 thf(fact_2229_divide__less__eq,axiom,
% 0.16/0.44 ! [B: real,C: real,A: real] :
% 0.16/0.44 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.16/0.44 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.16/0.44 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.16/0.44 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_less_eq
% 0.16/0.44 thf(fact_2230_divide__less__eq,axiom,
% 0.16/0.44 ! [B: rat,C: rat,A: rat] :
% 0.16/0.44 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.16/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.16/0.44 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.16/0.44 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_less_eq
% 0.16/0.44 thf(fact_2231_less__divide__eq,axiom,
% 0.16/0.44 ! [A: real,B: real,C: real] :
% 0.16/0.44 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.16/0.44 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.16/0.44 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_divide_eq
% 0.16/0.44 thf(fact_2232_less__divide__eq,axiom,
% 0.16/0.44 ! [A: rat,B: rat,C: rat] :
% 0.16/0.44 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.16/0.44 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.16/0.44 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_divide_eq
% 0.16/0.44 thf(fact_2233_neg__divide__less__eq,axiom,
% 0.16/0.44 ! [C: real,B: real,A: real] :
% 0.16/0.44 ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.16/0.44 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % neg_divide_less_eq
% 0.16/0.44 thf(fact_2234_neg__divide__less__eq,axiom,
% 0.16/0.44 ! [C: rat,B: rat,A: rat] :
% 0.16/0.44 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.16/0.44 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % neg_divide_less_eq
% 0.16/0.44 thf(fact_2235_neg__less__divide__eq,axiom,
% 0.16/0.44 ! [C: real,A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.16/0.44 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % neg_less_divide_eq
% 0.16/0.44 thf(fact_2236_neg__less__divide__eq,axiom,
% 0.16/0.44 ! [C: rat,A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.16/0.44 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % neg_less_divide_eq
% 0.16/0.44 thf(fact_2237_pos__divide__less__eq,axiom,
% 0.16/0.44 ! [C: real,B: real,A: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.16/0.44 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % pos_divide_less_eq
% 0.16/0.44 thf(fact_2238_pos__divide__less__eq,axiom,
% 0.16/0.44 ! [C: rat,B: rat,A: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.16/0.44 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % pos_divide_less_eq
% 0.16/0.44 thf(fact_2239_pos__less__divide__eq,axiom,
% 0.16/0.44 ! [C: real,A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.16/0.44 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % pos_less_divide_eq
% 0.16/0.44 thf(fact_2240_pos__less__divide__eq,axiom,
% 0.16/0.44 ! [C: rat,A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.16/0.44 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % pos_less_divide_eq
% 0.16/0.44 thf(fact_2241_mult__imp__div__pos__less,axiom,
% 0.16/0.44 ! [Y4: real,X: real,Z: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.16/0.44 => ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y4 ) )
% 0.16/0.44 => ( ord_less_real @ ( divide_divide_real @ X @ Y4 ) @ Z ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_imp_div_pos_less
% 0.16/0.44 thf(fact_2242_mult__imp__div__pos__less,axiom,
% 0.16/0.44 ! [Y4: rat,X: rat,Z: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ Y4 )
% 0.16/0.44 => ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y4 ) )
% 0.16/0.44 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y4 ) @ Z ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_imp_div_pos_less
% 0.16/0.44 thf(fact_2243_mult__imp__less__div__pos,axiom,
% 0.16/0.44 ! [Y4: real,Z: real,X: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.16/0.44 => ( ( ord_less_real @ ( times_times_real @ Z @ Y4 ) @ X )
% 0.16/0.44 => ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_imp_less_div_pos
% 0.16/0.44 thf(fact_2244_mult__imp__less__div__pos,axiom,
% 0.16/0.44 ! [Y4: rat,Z: rat,X: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ Y4 )
% 0.16/0.44 => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y4 ) @ X )
% 0.16/0.44 => ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_imp_less_div_pos
% 0.16/0.44 thf(fact_2245_divide__strict__left__mono,axiom,
% 0.16/0.44 ! [B: real,A: real,C: real] :
% 0.16/0.44 ( ( ord_less_real @ B @ A )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.16/0.44 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_strict_left_mono
% 0.16/0.44 thf(fact_2246_divide__strict__left__mono,axiom,
% 0.16/0.44 ! [B: rat,A: rat,C: rat] :
% 0.16/0.44 ( ( ord_less_rat @ B @ A )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.16/0.44 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_strict_left_mono
% 0.16/0.44 thf(fact_2247_divide__strict__left__mono__neg,axiom,
% 0.16/0.44 ! [A: real,B: real,C: real] :
% 0.16/0.44 ( ( ord_less_real @ A @ B )
% 0.16/0.44 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.16/0.44 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_strict_left_mono_neg
% 0.16/0.44 thf(fact_2248_divide__strict__left__mono__neg,axiom,
% 0.16/0.44 ! [A: rat,B: rat,C: rat] :
% 0.16/0.44 ( ( ord_less_rat @ A @ B )
% 0.16/0.44 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.16/0.44 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_strict_left_mono_neg
% 0.16/0.44 thf(fact_2249_mult__left__le__one__le,axiom,
% 0.16/0.44 ! [X: rat,Y4: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.16/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 0.16/0.44 => ( ( ord_less_eq_rat @ Y4 @ one_one_rat )
% 0.16/0.44 => ( ord_less_eq_rat @ ( times_times_rat @ Y4 @ X ) @ X ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_left_le_one_le
% 0.16/0.44 thf(fact_2250_mult__left__le__one__le,axiom,
% 0.16/0.44 ! [X: int,Y4: int] :
% 0.16/0.44 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.16/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.16/0.44 => ( ( ord_less_eq_int @ Y4 @ one_one_int )
% 0.16/0.44 => ( ord_less_eq_int @ ( times_times_int @ Y4 @ X ) @ X ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_left_le_one_le
% 0.16/0.44 thf(fact_2251_mult__left__le__one__le,axiom,
% 0.16/0.44 ! [X: real,Y4: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.16/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.16/0.44 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.16/0.44 => ( ord_less_eq_real @ ( times_times_real @ Y4 @ X ) @ X ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_left_le_one_le
% 0.16/0.44 thf(fact_2252_mult__right__le__one__le,axiom,
% 0.16/0.44 ! [X: rat,Y4: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.16/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 0.16/0.44 => ( ( ord_less_eq_rat @ Y4 @ one_one_rat )
% 0.16/0.44 => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y4 ) @ X ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_right_le_one_le
% 0.16/0.44 thf(fact_2253_mult__right__le__one__le,axiom,
% 0.16/0.44 ! [X: int,Y4: int] :
% 0.16/0.44 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.16/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.16/0.44 => ( ( ord_less_eq_int @ Y4 @ one_one_int )
% 0.16/0.44 => ( ord_less_eq_int @ ( times_times_int @ X @ Y4 ) @ X ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_right_le_one_le
% 0.16/0.44 thf(fact_2254_mult__right__le__one__le,axiom,
% 0.16/0.44 ! [X: real,Y4: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.16/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.16/0.44 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.16/0.44 => ( ord_less_eq_real @ ( times_times_real @ X @ Y4 ) @ X ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_right_le_one_le
% 0.16/0.44 thf(fact_2255_mult__le__one,axiom,
% 0.16/0.44 ! [A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.16/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.16/0.44 => ( ( ord_less_eq_rat @ B @ one_one_rat )
% 0.16/0.44 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_one
% 0.16/0.44 thf(fact_2256_mult__le__one,axiom,
% 0.16/0.44 ! [A: nat,B: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.16/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.16/0.44 => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 0.16/0.44 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_one
% 0.16/0.44 thf(fact_2257_mult__le__one,axiom,
% 0.16/0.44 ! [A: int,B: int] :
% 0.16/0.44 ( ( ord_less_eq_int @ A @ one_one_int )
% 0.16/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.16/0.44 => ( ( ord_less_eq_int @ B @ one_one_int )
% 0.16/0.44 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_one
% 0.16/0.44 thf(fact_2258_mult__le__one,axiom,
% 0.16/0.44 ! [A: real,B: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ A @ one_one_real )
% 0.16/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.16/0.44 => ( ( ord_less_eq_real @ B @ one_one_real )
% 0.16/0.44 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_le_one
% 0.16/0.44 thf(fact_2259_mult__left__le,axiom,
% 0.16/0.44 ! [C: rat,A: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ C @ one_one_rat )
% 0.16/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.44 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_left_le
% 0.16/0.44 thf(fact_2260_mult__left__le,axiom,
% 0.16/0.44 ! [C: nat,A: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 0.16/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.44 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_left_le
% 0.16/0.44 thf(fact_2261_mult__left__le,axiom,
% 0.16/0.44 ! [C: int,A: int] :
% 0.16/0.44 ( ( ord_less_eq_int @ C @ one_one_int )
% 0.16/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.44 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_left_le
% 0.16/0.44 thf(fact_2262_mult__left__le,axiom,
% 0.16/0.44 ! [C: real,A: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ C @ one_one_real )
% 0.16/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.44 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_left_le
% 0.16/0.44 thf(fact_2263_ex__least__nat__less,axiom,
% 0.16/0.44 ! [P: nat > $o,N: nat] :
% 0.16/0.44 ( ( P @ N )
% 0.16/0.44 => ( ~ ( P @ zero_zero_nat )
% 0.16/0.44 => ? [K2: nat] :
% 0.16/0.44 ( ( ord_less_nat @ K2 @ N )
% 0.16/0.44 & ! [I4: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ I4 @ K2 )
% 0.16/0.44 => ~ ( P @ I4 ) )
% 0.16/0.44 & ( P @ ( suc @ K2 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % ex_least_nat_less
% 0.16/0.44 thf(fact_2264_unit__dvdE,axiom,
% 0.16/0.44 ! [A: code_integer,B: code_integer] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.16/0.44 => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.16/0.44 => ! [C3: code_integer] :
% 0.16/0.44 ( B
% 0.16/0.44 != ( times_3573771949741848930nteger @ A @ C3 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_dvdE
% 0.16/0.44 thf(fact_2265_unit__dvdE,axiom,
% 0.16/0.44 ! [A: nat,B: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.16/0.44 => ~ ( ( A != zero_zero_nat )
% 0.16/0.44 => ! [C3: nat] :
% 0.16/0.44 ( B
% 0.16/0.44 != ( times_times_nat @ A @ C3 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_dvdE
% 0.16/0.44 thf(fact_2266_unit__dvdE,axiom,
% 0.16/0.44 ! [A: int,B: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.16/0.44 => ~ ( ( A != zero_zero_int )
% 0.16/0.44 => ! [C3: int] :
% 0.16/0.44 ( B
% 0.16/0.44 != ( times_times_int @ A @ C3 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_dvdE
% 0.16/0.44 thf(fact_2267_diff__Suc__less,axiom,
% 0.16/0.44 ! [N: nat,I: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.44 => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % diff_Suc_less
% 0.16/0.44 thf(fact_2268_n__less__n__mult__m,axiom,
% 0.16/0.44 ! [N: nat,M: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.44 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.16/0.44 => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % n_less_n_mult_m
% 0.16/0.44 thf(fact_2269_n__less__m__mult__n,axiom,
% 0.16/0.44 ! [N: nat,M: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.44 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.16/0.44 => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % n_less_m_mult_n
% 0.16/0.44 thf(fact_2270_one__less__mult,axiom,
% 0.16/0.44 ! [N: nat,M: nat] :
% 0.16/0.44 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.16/0.44 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.16/0.44 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % one_less_mult
% 0.16/0.44 thf(fact_2271_unit__div__eq__0__iff,axiom,
% 0.16/0.44 ! [B: code_integer,A: code_integer] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.16/0.44 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.16/0.44 = zero_z3403309356797280102nteger )
% 0.16/0.44 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_div_eq_0_iff
% 0.16/0.44 thf(fact_2272_unit__div__eq__0__iff,axiom,
% 0.16/0.44 ! [B: nat,A: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.16/0.44 => ( ( ( divide_divide_nat @ A @ B )
% 0.16/0.44 = zero_zero_nat )
% 0.16/0.44 = ( A = zero_zero_nat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_div_eq_0_iff
% 0.16/0.44 thf(fact_2273_unit__div__eq__0__iff,axiom,
% 0.16/0.44 ! [B: int,A: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.16/0.44 => ( ( ( divide_divide_int @ A @ B )
% 0.16/0.44 = zero_zero_int )
% 0.16/0.44 = ( A = zero_zero_int ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_div_eq_0_iff
% 0.16/0.44 thf(fact_2274_nat__mult__le__cancel1,axiom,
% 0.16/0.44 ! [K: nat,M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.16/0.44 => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.16/0.44 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % nat_mult_le_cancel1
% 0.16/0.44 thf(fact_2275_unit__eq__div1,axiom,
% 0.16/0.44 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.16/0.44 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.16/0.44 = C )
% 0.16/0.44 = ( A
% 0.16/0.44 = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_eq_div1
% 0.16/0.44 thf(fact_2276_unit__eq__div1,axiom,
% 0.16/0.44 ! [B: nat,A: nat,C: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.16/0.44 => ( ( ( divide_divide_nat @ A @ B )
% 0.16/0.44 = C )
% 0.16/0.44 = ( A
% 0.16/0.44 = ( times_times_nat @ C @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_eq_div1
% 0.16/0.44 thf(fact_2277_unit__eq__div1,axiom,
% 0.16/0.44 ! [B: int,A: int,C: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.16/0.44 => ( ( ( divide_divide_int @ A @ B )
% 0.16/0.44 = C )
% 0.16/0.44 = ( A
% 0.16/0.44 = ( times_times_int @ C @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_eq_div1
% 0.16/0.44 thf(fact_2278_unit__eq__div2,axiom,
% 0.16/0.44 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.16/0.44 => ( ( A
% 0.16/0.44 = ( divide6298287555418463151nteger @ C @ B ) )
% 0.16/0.44 = ( ( times_3573771949741848930nteger @ A @ B )
% 0.16/0.44 = C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_eq_div2
% 0.16/0.44 thf(fact_2279_unit__eq__div2,axiom,
% 0.16/0.44 ! [B: nat,A: nat,C: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.16/0.44 => ( ( A
% 0.16/0.44 = ( divide_divide_nat @ C @ B ) )
% 0.16/0.44 = ( ( times_times_nat @ A @ B )
% 0.16/0.44 = C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_eq_div2
% 0.16/0.44 thf(fact_2280_unit__eq__div2,axiom,
% 0.16/0.44 ! [B: int,A: int,C: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.16/0.44 => ( ( A
% 0.16/0.44 = ( divide_divide_int @ C @ B ) )
% 0.16/0.44 = ( ( times_times_int @ A @ B )
% 0.16/0.44 = C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_eq_div2
% 0.16/0.44 thf(fact_2281_div__mult__unit2,axiom,
% 0.16/0.44 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.16/0.44 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.16/0.44 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.16/0.44 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % div_mult_unit2
% 0.16/0.44 thf(fact_2282_div__mult__unit2,axiom,
% 0.16/0.44 ! [C: nat,B: nat,A: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.16/0.44 => ( ( dvd_dvd_nat @ B @ A )
% 0.16/0.44 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.16/0.44 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % div_mult_unit2
% 0.16/0.44 thf(fact_2283_div__mult__unit2,axiom,
% 0.16/0.44 ! [C: int,B: int,A: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ C @ one_one_int )
% 0.16/0.44 => ( ( dvd_dvd_int @ B @ A )
% 0.16/0.44 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.16/0.44 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % div_mult_unit2
% 0.16/0.44 thf(fact_2284_unit__div__commute,axiom,
% 0.16/0.44 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.16/0.44 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.16/0.44 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_div_commute
% 0.16/0.44 thf(fact_2285_unit__div__commute,axiom,
% 0.16/0.44 ! [B: nat,A: nat,C: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.16/0.44 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.16/0.44 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_div_commute
% 0.16/0.44 thf(fact_2286_unit__div__commute,axiom,
% 0.16/0.44 ! [B: int,A: int,C: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.16/0.44 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.16/0.44 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_div_commute
% 0.16/0.44 thf(fact_2287_unit__div__mult__swap,axiom,
% 0.16/0.44 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.16/0.44 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.16/0.44 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_div_mult_swap
% 0.16/0.44 thf(fact_2288_unit__div__mult__swap,axiom,
% 0.16/0.44 ! [C: nat,A: nat,B: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.16/0.44 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.16/0.44 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_div_mult_swap
% 0.16/0.44 thf(fact_2289_unit__div__mult__swap,axiom,
% 0.16/0.44 ! [C: int,A: int,B: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ C @ one_one_int )
% 0.16/0.44 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.16/0.44 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % unit_div_mult_swap
% 0.16/0.44 thf(fact_2290_is__unit__div__mult2__eq,axiom,
% 0.16/0.44 ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.16/0.44 => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.16/0.44 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.16/0.44 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % is_unit_div_mult2_eq
% 0.16/0.44 thf(fact_2291_is__unit__div__mult2__eq,axiom,
% 0.16/0.44 ! [B: nat,C: nat,A: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.16/0.44 => ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.16/0.44 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.16/0.44 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % is_unit_div_mult2_eq
% 0.16/0.44 thf(fact_2292_is__unit__div__mult2__eq,axiom,
% 0.16/0.44 ! [B: int,C: int,A: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.16/0.44 => ( ( dvd_dvd_int @ C @ one_one_int )
% 0.16/0.44 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.16/0.44 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % is_unit_div_mult2_eq
% 0.16/0.44 thf(fact_2293_dvd__imp__le,axiom,
% 0.16/0.44 ! [K: nat,N: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ K @ N )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.44 => ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dvd_imp_le
% 0.16/0.44 thf(fact_2294_div__greater__zero__iff,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 0.16/0.44 = ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.44 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % div_greater_zero_iff
% 0.16/0.44 thf(fact_2295_div__le__mono2,axiom,
% 0.16/0.44 ! [M: nat,N: nat,K: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.16/0.44 => ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.44 => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % div_le_mono2
% 0.16/0.44 thf(fact_2296_nat__mult__dvd__cancel1,axiom,
% 0.16/0.44 ! [K: nat,M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.16/0.44 => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.16/0.44 = ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % nat_mult_dvd_cancel1
% 0.16/0.44 thf(fact_2297_dvd__mult__cancel,axiom,
% 0.16/0.44 ! [K: nat,M: nat,N: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.16/0.44 => ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dvd_mult_cancel
% 0.16/0.44 thf(fact_2298_div__less__iff__less__mult,axiom,
% 0.16/0.44 ! [Q2: nat,M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 0.16/0.44 => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
% 0.16/0.44 = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % div_less_iff_less_mult
% 0.16/0.44 thf(fact_2299_nat__mult__div__cancel1,axiom,
% 0.16/0.44 ! [K: nat,M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.16/0.44 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.16/0.44 = ( divide_divide_nat @ M @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % nat_mult_div_cancel1
% 0.16/0.44 thf(fact_2300_power__le__zero__eq,axiom,
% 0.16/0.44 ! [A: rat,N: nat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 0.16/0.44 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.44 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.16/0.44 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_le_zero_eq
% 0.16/0.44 thf(fact_2301_power__le__zero__eq,axiom,
% 0.16/0.44 ! [A: int,N: nat] :
% 0.16/0.44 ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 0.16/0.44 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.44 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.16/0.44 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 & ( A = zero_zero_int ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_le_zero_eq
% 0.16/0.44 thf(fact_2302_power__le__zero__eq,axiom,
% 0.16/0.44 ! [A: real,N: nat] :
% 0.16/0.44 ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 0.16/0.44 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.44 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.16/0.44 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 & ( A = zero_zero_real ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_le_zero_eq
% 0.16/0.44 thf(fact_2303_even__mask__div__iff_H,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( 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 ) ) )
% 0.16/0.44 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % even_mask_div_iff'
% 0.16/0.44 thf(fact_2304_even__mask__div__iff_H,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( 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 ) ) )
% 0.16/0.44 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % even_mask_div_iff'
% 0.16/0.44 thf(fact_2305_even__mask__div__iff_H,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( 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 ) ) )
% 0.16/0.44 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % even_mask_div_iff'
% 0.16/0.44 thf(fact_2306_conj__le__cong,axiom,
% 0.16/0.44 ! [X: int,X7: int,P: $o,P4: $o] :
% 0.16/0.44 ( ( X = X7 )
% 0.16/0.44 => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 0.16/0.44 => ( P = P4 ) )
% 0.16/0.44 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.16/0.44 & P )
% 0.16/0.44 = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 0.16/0.44 & P4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % conj_le_cong
% 0.16/0.44 thf(fact_2307_imp__le__cong,axiom,
% 0.16/0.44 ! [X: int,X7: int,P: $o,P4: $o] :
% 0.16/0.44 ( ( X = X7 )
% 0.16/0.44 => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 0.16/0.44 => ( P = P4 ) )
% 0.16/0.44 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.16/0.44 => P )
% 0.16/0.44 = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 0.16/0.44 => P4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % imp_le_cong
% 0.16/0.44 thf(fact_2308_gcd__nat_Onot__eq__order__implies__strict,axiom,
% 0.16/0.44 ! [A: nat,B: nat] :
% 0.16/0.44 ( ( A != B )
% 0.16/0.44 => ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.44 => ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.44 & ( A != B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % gcd_nat.not_eq_order_implies_strict
% 0.16/0.44 thf(fact_2309_gcd__nat_Ostrict__implies__not__eq,axiom,
% 0.16/0.44 ! [A: nat,B: nat] :
% 0.16/0.44 ( ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.44 & ( A != B ) )
% 0.16/0.44 => ( A != B ) ) ).
% 0.16/0.44
% 0.16/0.44 % gcd_nat.strict_implies_not_eq
% 0.16/0.44 thf(fact_2310_gcd__nat_Ostrict__implies__order,axiom,
% 0.16/0.44 ! [A: nat,B: nat] :
% 0.16/0.44 ( ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.44 & ( A != B ) )
% 0.16/0.44 => ( dvd_dvd_nat @ A @ B ) ) ).
% 0.16/0.44
% 0.16/0.44 % gcd_nat.strict_implies_order
% 0.16/0.44 thf(fact_2311_gcd__nat_Ostrict__iff__order,axiom,
% 0.16/0.44 ! [A: nat,B: nat] :
% 0.16/0.44 ( ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.44 & ( A != B ) )
% 0.16/0.44 = ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.44 & ( A != B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % gcd_nat.strict_iff_order
% 0.16/0.44 thf(fact_2312_gcd__nat_Oorder__iff__strict,axiom,
% 0.16/0.44 ( dvd_dvd_nat
% 0.16/0.44 = ( ^ [A3: nat,B2: nat] :
% 0.16/0.44 ( ( ( dvd_dvd_nat @ A3 @ B2 )
% 0.16/0.44 & ( A3 != B2 ) )
% 0.16/0.44 | ( A3 = B2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % gcd_nat.order_iff_strict
% 0.16/0.44 thf(fact_2313_gcd__nat_Ostrict__iff__not,axiom,
% 0.16/0.44 ! [A: nat,B: nat] :
% 0.16/0.44 ( ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.44 & ( A != B ) )
% 0.16/0.44 = ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.44 & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % gcd_nat.strict_iff_not
% 0.16/0.44 thf(fact_2314_gcd__nat_Ostrict__trans2,axiom,
% 0.16/0.44 ! [A: nat,B: nat,C: nat] :
% 0.16/0.44 ( ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.44 & ( A != B ) )
% 0.16/0.44 => ( ( dvd_dvd_nat @ B @ C )
% 0.16/0.44 => ( ( dvd_dvd_nat @ A @ C )
% 0.16/0.44 & ( A != C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % gcd_nat.strict_trans2
% 0.16/0.44 thf(fact_2315_gcd__nat_Ostrict__trans1,axiom,
% 0.16/0.44 ! [A: nat,B: nat,C: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.44 => ( ( ( dvd_dvd_nat @ B @ C )
% 0.16/0.44 & ( B != C ) )
% 0.16/0.44 => ( ( dvd_dvd_nat @ A @ C )
% 0.16/0.44 & ( A != C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % gcd_nat.strict_trans1
% 0.16/0.44 thf(fact_2316_gcd__nat_Ostrict__trans,axiom,
% 0.16/0.44 ! [A: nat,B: nat,C: nat] :
% 0.16/0.44 ( ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.44 & ( A != B ) )
% 0.16/0.44 => ( ( ( dvd_dvd_nat @ B @ C )
% 0.16/0.44 & ( B != C ) )
% 0.16/0.44 => ( ( dvd_dvd_nat @ A @ C )
% 0.16/0.44 & ( A != C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % gcd_nat.strict_trans
% 0.16/0.44 thf(fact_2317_gcd__nat_Oantisym,axiom,
% 0.16/0.44 ! [A: nat,B: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.44 => ( ( dvd_dvd_nat @ B @ A )
% 0.16/0.44 => ( A = B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % gcd_nat.antisym
% 0.16/0.44 thf(fact_2318_gcd__nat_Oirrefl,axiom,
% 0.16/0.44 ! [A: nat] :
% 0.16/0.44 ~ ( ( dvd_dvd_nat @ A @ A )
% 0.16/0.44 & ( A != A ) ) ).
% 0.16/0.44
% 0.16/0.44 % gcd_nat.irrefl
% 0.16/0.44 thf(fact_2319_gcd__nat_Oeq__iff,axiom,
% 0.16/0.44 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.16/0.44 = ( ^ [A3: nat,B2: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ A3 @ B2 )
% 0.16/0.44 & ( dvd_dvd_nat @ B2 @ A3 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % gcd_nat.eq_iff
% 0.16/0.44 thf(fact_2320_gcd__nat_Otrans,axiom,
% 0.16/0.44 ! [A: nat,B: nat,C: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.44 => ( ( dvd_dvd_nat @ B @ C )
% 0.16/0.44 => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % gcd_nat.trans
% 0.16/0.44 thf(fact_2321_gcd__nat_Orefl,axiom,
% 0.16/0.44 ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 0.16/0.44
% 0.16/0.44 % gcd_nat.refl
% 0.16/0.44 thf(fact_2322_gcd__nat_Oasym,axiom,
% 0.16/0.44 ! [A: nat,B: nat] :
% 0.16/0.44 ( ( ( dvd_dvd_nat @ A @ B )
% 0.16/0.44 & ( A != B ) )
% 0.16/0.44 => ~ ( ( dvd_dvd_nat @ B @ A )
% 0.16/0.44 & ( B != A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % gcd_nat.asym
% 0.16/0.44 thf(fact_2323_even__mask__div__iff,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( 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 ) ) )
% 0.16/0.44 = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 = zero_z3403309356797280102nteger )
% 0.16/0.44 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % even_mask_div_iff
% 0.16/0.44 thf(fact_2324_even__mask__div__iff,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( 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 ) ) )
% 0.16/0.44 = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 = zero_zero_nat )
% 0.16/0.44 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % even_mask_div_iff
% 0.16/0.44 thf(fact_2325_even__mask__div__iff,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( 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 ) ) )
% 0.16/0.44 = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 = zero_zero_int )
% 0.16/0.44 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % even_mask_div_iff
% 0.16/0.44 thf(fact_2326_divide__le__eq,axiom,
% 0.16/0.44 ! [B: rat,C: rat,A: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.16/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.16/0.44 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.16/0.44 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_le_eq
% 0.16/0.44 thf(fact_2327_divide__le__eq,axiom,
% 0.16/0.44 ! [B: real,C: real,A: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.16/0.44 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.16/0.44 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.16/0.44 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_le_eq
% 0.16/0.44 thf(fact_2328_le__divide__eq,axiom,
% 0.16/0.44 ! [A: rat,B: rat,C: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.16/0.44 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.16/0.44 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_divide_eq
% 0.16/0.44 thf(fact_2329_le__divide__eq,axiom,
% 0.16/0.44 ! [A: real,B: real,C: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.16/0.44 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.16/0.44 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_divide_eq
% 0.16/0.44 thf(fact_2330_divide__left__mono,axiom,
% 0.16/0.44 ! [B: rat,A: rat,C: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ B @ A )
% 0.16/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.16/0.44 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_left_mono
% 0.16/0.44 thf(fact_2331_divide__left__mono,axiom,
% 0.16/0.44 ! [B: real,A: real,C: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ B @ A )
% 0.16/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.16/0.44 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_left_mono
% 0.16/0.44 thf(fact_2332_neg__divide__le__eq,axiom,
% 0.16/0.44 ! [C: rat,B: rat,A: rat] :
% 0.16/0.44 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.16/0.44 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % neg_divide_le_eq
% 0.16/0.44 thf(fact_2333_neg__divide__le__eq,axiom,
% 0.16/0.44 ! [C: real,B: real,A: real] :
% 0.16/0.44 ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.16/0.44 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % neg_divide_le_eq
% 0.16/0.44 thf(fact_2334_neg__le__divide__eq,axiom,
% 0.16/0.44 ! [C: rat,A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.16/0.44 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % neg_le_divide_eq
% 0.16/0.44 thf(fact_2335_neg__le__divide__eq,axiom,
% 0.16/0.44 ! [C: real,A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.16/0.44 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % neg_le_divide_eq
% 0.16/0.44 thf(fact_2336_pos__divide__le__eq,axiom,
% 0.16/0.44 ! [C: rat,B: rat,A: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.16/0.44 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % pos_divide_le_eq
% 0.16/0.44 thf(fact_2337_pos__divide__le__eq,axiom,
% 0.16/0.44 ! [C: real,B: real,A: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.16/0.44 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % pos_divide_le_eq
% 0.16/0.44 thf(fact_2338_pos__le__divide__eq,axiom,
% 0.16/0.44 ! [C: rat,A: rat,B: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.16/0.44 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % pos_le_divide_eq
% 0.16/0.44 thf(fact_2339_pos__le__divide__eq,axiom,
% 0.16/0.44 ! [C: real,A: real,B: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.16/0.44 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % pos_le_divide_eq
% 0.16/0.44 thf(fact_2340_mult__imp__div__pos__le,axiom,
% 0.16/0.44 ! [Y4: rat,X: rat,Z: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ Y4 )
% 0.16/0.44 => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y4 ) )
% 0.16/0.44 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y4 ) @ Z ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_imp_div_pos_le
% 0.16/0.44 thf(fact_2341_mult__imp__div__pos__le,axiom,
% 0.16/0.44 ! [Y4: real,X: real,Z: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.16/0.44 => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y4 ) )
% 0.16/0.44 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ Z ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_imp_div_pos_le
% 0.16/0.44 thf(fact_2342_mult__imp__le__div__pos,axiom,
% 0.16/0.44 ! [Y4: rat,Z: rat,X: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ Y4 )
% 0.16/0.44 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y4 ) @ X )
% 0.16/0.44 => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_imp_le_div_pos
% 0.16/0.44 thf(fact_2343_mult__imp__le__div__pos,axiom,
% 0.16/0.44 ! [Y4: real,Z: real,X: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.16/0.44 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y4 ) @ X )
% 0.16/0.44 => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % mult_imp_le_div_pos
% 0.16/0.44 thf(fact_2344_divide__left__mono__neg,axiom,
% 0.16/0.44 ! [A: rat,B: rat,C: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.44 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.16/0.44 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_left_mono_neg
% 0.16/0.44 thf(fact_2345_divide__left__mono__neg,axiom,
% 0.16/0.44 ! [A: real,B: real,C: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.44 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.16/0.44 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_left_mono_neg
% 0.16/0.44 thf(fact_2346_divide__less__eq__numeral_I1_J,axiom,
% 0.16/0.44 ! [B: real,C: real,W: num] :
% 0.16/0.44 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 0.16/0.44 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.16/0.44 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.16/0.44 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_less_eq_numeral(1)
% 0.16/0.44 thf(fact_2347_divide__less__eq__numeral_I1_J,axiom,
% 0.16/0.44 ! [B: rat,C: rat,W: num] :
% 0.16/0.44 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 0.16/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.16/0.44 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.16/0.44 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_less_eq_numeral(1)
% 0.16/0.44 thf(fact_2348_less__divide__eq__numeral_I1_J,axiom,
% 0.16/0.44 ! [W: num,B: real,C: real] :
% 0.16/0.44 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.16/0.44 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.16/0.44 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_divide_eq_numeral(1)
% 0.16/0.44 thf(fact_2349_less__divide__eq__numeral_I1_J,axiom,
% 0.16/0.44 ! [W: num,B: rat,C: rat] :
% 0.16/0.44 ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.16/0.44 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.16/0.44 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_divide_eq_numeral(1)
% 0.16/0.44 thf(fact_2350_frac__less__eq,axiom,
% 0.16/0.44 ! [Y4: real,Z: real,X: real,W: real] :
% 0.16/0.44 ( ( Y4 != zero_zero_real )
% 0.16/0.44 => ( ( Z != zero_zero_real )
% 0.16/0.44 => ( ( ord_less_real @ ( divide_divide_real @ X @ Y4 ) @ ( divide_divide_real @ W @ Z ) )
% 0.16/0.44 = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y4 ) ) @ ( times_times_real @ Y4 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % frac_less_eq
% 0.16/0.44 thf(fact_2351_frac__less__eq,axiom,
% 0.16/0.44 ! [Y4: rat,Z: rat,X: rat,W: rat] :
% 0.16/0.44 ( ( Y4 != zero_zero_rat )
% 0.16/0.44 => ( ( Z != zero_zero_rat )
% 0.16/0.44 => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y4 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.16/0.44 = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y4 ) ) @ ( times_times_rat @ Y4 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % frac_less_eq
% 0.16/0.44 thf(fact_2352_not__exp__less__eq__0__int,axiom,
% 0.16/0.44 ! [N: nat] :
% 0.16/0.44 ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% 0.16/0.44
% 0.16/0.44 % not_exp_less_eq_0_int
% 0.16/0.44 thf(fact_2353_pos2,axiom,
% 0.16/0.44 ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 0.16/0.44
% 0.16/0.44 % pos2
% 0.16/0.44 thf(fact_2354_is__unitE,axiom,
% 0.16/0.44 ! [A: code_integer,C: code_integer] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.16/0.44 => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.16/0.44 => ! [B3: code_integer] :
% 0.16/0.44 ( ( B3 != zero_z3403309356797280102nteger )
% 0.16/0.44 => ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
% 0.16/0.44 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 0.16/0.44 = B3 )
% 0.16/0.44 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B3 )
% 0.16/0.44 = A )
% 0.16/0.44 => ( ( ( times_3573771949741848930nteger @ A @ B3 )
% 0.16/0.44 = one_one_Code_integer )
% 0.16/0.44 => ( ( divide6298287555418463151nteger @ C @ A )
% 0.16/0.44 != ( times_3573771949741848930nteger @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % is_unitE
% 0.16/0.44 thf(fact_2355_is__unitE,axiom,
% 0.16/0.44 ! [A: nat,C: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.16/0.44 => ~ ( ( A != zero_zero_nat )
% 0.16/0.44 => ! [B3: nat] :
% 0.16/0.44 ( ( B3 != zero_zero_nat )
% 0.16/0.44 => ( ( dvd_dvd_nat @ B3 @ one_one_nat )
% 0.16/0.44 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 0.16/0.44 = B3 )
% 0.16/0.44 => ( ( ( divide_divide_nat @ one_one_nat @ B3 )
% 0.16/0.44 = A )
% 0.16/0.44 => ( ( ( times_times_nat @ A @ B3 )
% 0.16/0.44 = one_one_nat )
% 0.16/0.44 => ( ( divide_divide_nat @ C @ A )
% 0.16/0.44 != ( times_times_nat @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % is_unitE
% 0.16/0.44 thf(fact_2356_is__unitE,axiom,
% 0.16/0.44 ! [A: int,C: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.16/0.44 => ~ ( ( A != zero_zero_int )
% 0.16/0.44 => ! [B3: int] :
% 0.16/0.44 ( ( B3 != zero_zero_int )
% 0.16/0.44 => ( ( dvd_dvd_int @ B3 @ one_one_int )
% 0.16/0.44 => ( ( ( divide_divide_int @ one_one_int @ A )
% 0.16/0.44 = B3 )
% 0.16/0.44 => ( ( ( divide_divide_int @ one_one_int @ B3 )
% 0.16/0.44 = A )
% 0.16/0.44 => ( ( ( times_times_int @ A @ B3 )
% 0.16/0.44 = one_one_int )
% 0.16/0.44 => ( ( divide_divide_int @ C @ A )
% 0.16/0.44 != ( times_times_int @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % is_unitE
% 0.16/0.44 thf(fact_2357_is__unit__div__mult__cancel__left,axiom,
% 0.16/0.44 ! [A: code_integer,B: code_integer] :
% 0.16/0.44 ( ( A != zero_z3403309356797280102nteger )
% 0.16/0.44 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.16/0.44 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.16/0.44 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % is_unit_div_mult_cancel_left
% 0.16/0.44 thf(fact_2358_is__unit__div__mult__cancel__left,axiom,
% 0.16/0.44 ! [A: nat,B: nat] :
% 0.16/0.44 ( ( A != zero_zero_nat )
% 0.16/0.44 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.16/0.44 => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
% 0.16/0.44 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % is_unit_div_mult_cancel_left
% 0.16/0.44 thf(fact_2359_is__unit__div__mult__cancel__left,axiom,
% 0.16/0.44 ! [A: int,B: int] :
% 0.16/0.44 ( ( A != zero_zero_int )
% 0.16/0.44 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.16/0.44 => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
% 0.16/0.44 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % is_unit_div_mult_cancel_left
% 0.16/0.44 thf(fact_2360_is__unit__div__mult__cancel__right,axiom,
% 0.16/0.44 ! [A: code_integer,B: code_integer] :
% 0.16/0.44 ( ( A != zero_z3403309356797280102nteger )
% 0.16/0.44 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.16/0.44 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
% 0.16/0.44 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % is_unit_div_mult_cancel_right
% 0.16/0.44 thf(fact_2361_is__unit__div__mult__cancel__right,axiom,
% 0.16/0.44 ! [A: nat,B: nat] :
% 0.16/0.44 ( ( A != zero_zero_nat )
% 0.16/0.44 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.16/0.44 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
% 0.16/0.44 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % is_unit_div_mult_cancel_right
% 0.16/0.44 thf(fact_2362_is__unit__div__mult__cancel__right,axiom,
% 0.16/0.44 ! [A: int,B: int] :
% 0.16/0.44 ( ( A != zero_zero_int )
% 0.16/0.44 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.16/0.44 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
% 0.16/0.44 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % is_unit_div_mult_cancel_right
% 0.16/0.44 thf(fact_2363_odd__one,axiom,
% 0.16/0.44 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% 0.16/0.44
% 0.16/0.44 % odd_one
% 0.16/0.44 thf(fact_2364_odd__one,axiom,
% 0.16/0.44 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% 0.16/0.44
% 0.16/0.44 % odd_one
% 0.16/0.44 thf(fact_2365_odd__one,axiom,
% 0.16/0.44 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% 0.16/0.44
% 0.16/0.44 % odd_one
% 0.16/0.44 thf(fact_2366_div__if,axiom,
% 0.16/0.44 ( divide_divide_nat
% 0.16/0.44 = ( ^ [M5: nat,N4: nat] :
% 0.16/0.44 ( if_nat
% 0.16/0.44 @ ( ( ord_less_nat @ M5 @ N4 )
% 0.16/0.44 | ( N4 = zero_zero_nat ) )
% 0.16/0.44 @ zero_zero_nat
% 0.16/0.44 @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M5 @ N4 ) @ N4 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % div_if
% 0.16/0.44 thf(fact_2367_div__geq,axiom,
% 0.16/0.44 ! [N: nat,M: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.44 => ( ~ ( ord_less_nat @ M @ N )
% 0.16/0.44 => ( ( divide_divide_nat @ M @ N )
% 0.16/0.44 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % div_geq
% 0.16/0.44 thf(fact_2368_div__nat__eqI,axiom,
% 0.16/0.44 ! [N: nat,Q2: nat,M: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
% 0.16/0.44 => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
% 0.16/0.44 => ( ( divide_divide_nat @ M @ N )
% 0.16/0.44 = Q2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % div_nat_eqI
% 0.16/0.44 thf(fact_2369_less__eq__div__iff__mult__less__eq,axiom,
% 0.16/0.44 ! [Q2: nat,M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 0.16/0.44 => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
% 0.16/0.44 = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_eq_div_iff_mult_less_eq
% 0.16/0.44 thf(fact_2370_even__mult__exp__div__exp__iff,axiom,
% 0.16/0.44 ! [A: code_integer,M: nat,N: nat] :
% 0.16/0.44 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 0.16/0.44 = ( ( ord_less_nat @ N @ M )
% 0.16/0.44 | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 = zero_z3403309356797280102nteger )
% 0.16/0.44 | ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.44 & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % even_mult_exp_div_exp_iff
% 0.16/0.44 thf(fact_2371_even__mult__exp__div__exp__iff,axiom,
% 0.16/0.44 ! [A: nat,M: nat,N: nat] :
% 0.16/0.44 ( ( 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 ) ) )
% 0.16/0.44 = ( ( ord_less_nat @ N @ M )
% 0.16/0.44 | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 = zero_zero_nat )
% 0.16/0.44 | ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.44 & ( 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 ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % even_mult_exp_div_exp_iff
% 0.16/0.44 thf(fact_2372_even__mult__exp__div__exp__iff,axiom,
% 0.16/0.44 ! [A: int,M: nat,N: nat] :
% 0.16/0.44 ( ( 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 ) ) )
% 0.16/0.44 = ( ( ord_less_nat @ N @ M )
% 0.16/0.44 | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 = zero_zero_int )
% 0.16/0.44 | ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.44 & ( 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 ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % even_mult_exp_div_exp_iff
% 0.16/0.44 thf(fact_2373_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.16/0.44 ! [N: nat,M: nat] :
% 0.16/0.44 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 != zero_zero_nat )
% 0.16/0.44 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 0.16/0.44 != zero_zero_nat ) ) ).
% 0.16/0.44
% 0.16/0.44 % exp_not_zero_imp_exp_diff_not_zero
% 0.16/0.44 thf(fact_2374_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.16/0.44 ! [N: nat,M: nat] :
% 0.16/0.44 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 != zero_zero_int )
% 0.16/0.44 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 0.16/0.44 != zero_zero_int ) ) ).
% 0.16/0.44
% 0.16/0.44 % exp_not_zero_imp_exp_diff_not_zero
% 0.16/0.44 thf(fact_2375_power__mono__odd,axiom,
% 0.16/0.44 ! [N: nat,A: rat,B: rat] :
% 0.16/0.44 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 => ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.44 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_mono_odd
% 0.16/0.44 thf(fact_2376_power__mono__odd,axiom,
% 0.16/0.44 ! [N: nat,A: int,B: int] :
% 0.16/0.44 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 => ( ( ord_less_eq_int @ A @ B )
% 0.16/0.44 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_mono_odd
% 0.16/0.44 thf(fact_2377_power__mono__odd,axiom,
% 0.16/0.44 ! [N: nat,A: real,B: real] :
% 0.16/0.44 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 => ( ( ord_less_eq_real @ A @ B )
% 0.16/0.44 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_mono_odd
% 0.16/0.44 thf(fact_2378_divide__le__eq__numeral_I1_J,axiom,
% 0.16/0.44 ! [B: rat,C: rat,W: num] :
% 0.16/0.44 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 0.16/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.16/0.44 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.16/0.44 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_le_eq_numeral(1)
% 0.16/0.44 thf(fact_2379_divide__le__eq__numeral_I1_J,axiom,
% 0.16/0.44 ! [B: real,C: real,W: num] :
% 0.16/0.44 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 0.16/0.44 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.16/0.44 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.16/0.44 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % divide_le_eq_numeral(1)
% 0.16/0.44 thf(fact_2380_le__divide__eq__numeral_I1_J,axiom,
% 0.16/0.44 ! [W: num,B: rat,C: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.16/0.44 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.16/0.44 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.16/0.44 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.16/0.44 => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_divide_eq_numeral(1)
% 0.16/0.44 thf(fact_2381_le__divide__eq__numeral_I1_J,axiom,
% 0.16/0.44 ! [W: num,B: real,C: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 0.16/0.44 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.16/0.44 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.16/0.44 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.16/0.44 => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_divide_eq_numeral(1)
% 0.16/0.44 thf(fact_2382_half__gt__zero,axiom,
% 0.16/0.44 ! [A: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.44 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % half_gt_zero
% 0.16/0.44 thf(fact_2383_half__gt__zero,axiom,
% 0.16/0.44 ! [A: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.44 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % half_gt_zero
% 0.16/0.44 thf(fact_2384_half__gt__zero__iff,axiom,
% 0.16/0.44 ! [A: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.16/0.44 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.16/0.44
% 0.16/0.44 % half_gt_zero_iff
% 0.16/0.44 thf(fact_2385_half__gt__zero__iff,axiom,
% 0.16/0.44 ! [A: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.16/0.44 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.16/0.44
% 0.16/0.44 % half_gt_zero_iff
% 0.16/0.44 thf(fact_2386_four__x__squared,axiom,
% 0.16/0.44 ! [X: real] :
% 0.16/0.44 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.44 = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % four_x_squared
% 0.16/0.44 thf(fact_2387_less__2__cases,axiom,
% 0.16/0.44 ! [N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.44 => ( ( N = zero_zero_nat )
% 0.16/0.44 | ( N
% 0.16/0.44 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_2_cases
% 0.16/0.44 thf(fact_2388_less__2__cases__iff,axiom,
% 0.16/0.44 ! [N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.44 = ( ( N = zero_zero_nat )
% 0.16/0.44 | ( N
% 0.16/0.44 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_2_cases_iff
% 0.16/0.44 thf(fact_2389_real__of__nat__div2,axiom,
% 0.16/0.44 ! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % real_of_nat_div2
% 0.16/0.44 thf(fact_2390_odd__pos,axiom,
% 0.16/0.44 ! [N: nat] :
% 0.16/0.44 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % odd_pos
% 0.16/0.44 thf(fact_2391_le__div__geq,axiom,
% 0.16/0.44 ! [N: nat,M: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.44 => ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.44 => ( ( divide_divide_nat @ M @ N )
% 0.16/0.44 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_div_geq
% 0.16/0.44 thf(fact_2392_split__div_H,axiom,
% 0.16/0.44 ! [P: nat > $o,M: nat,N: nat] :
% 0.16/0.44 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 0.16/0.44 = ( ( ( N = zero_zero_nat )
% 0.16/0.44 & ( P @ zero_zero_nat ) )
% 0.16/0.44 | ? [Q3: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
% 0.16/0.44 & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
% 0.16/0.44 & ( P @ Q3 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % split_div'
% 0.16/0.44 thf(fact_2393_inverse__of__nat__le,axiom,
% 0.16/0.44 ! [N: nat,M: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.44 => ( ( N != zero_zero_nat )
% 0.16/0.44 => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_of_nat_le
% 0.16/0.44 thf(fact_2394_inverse__of__nat__le,axiom,
% 0.16/0.44 ! [N: nat,M: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.44 => ( ( N != zero_zero_nat )
% 0.16/0.44 => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_of_nat_le
% 0.16/0.44 thf(fact_2395_real__archimedian__rdiv__eq__0,axiom,
% 0.16/0.44 ! [X: real,C: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.16/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.16/0.44 => ( ! [M4: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ M4 )
% 0.16/0.44 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
% 0.16/0.44 => ( X = zero_zero_real ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % real_archimedian_rdiv_eq_0
% 0.16/0.44 thf(fact_2396_zero__le__power__eq,axiom,
% 0.16/0.44 ! [A: rat,N: nat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.16/0.44 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_le_power_eq
% 0.16/0.44 thf(fact_2397_zero__le__power__eq,axiom,
% 0.16/0.44 ! [A: int,N: nat] :
% 0.16/0.44 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.16/0.44 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_le_power_eq
% 0.16/0.44 thf(fact_2398_zero__le__power__eq,axiom,
% 0.16/0.44 ! [A: real,N: nat] :
% 0.16/0.44 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.16/0.44 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_le_power_eq
% 0.16/0.44 thf(fact_2399_zero__le__odd__power,axiom,
% 0.16/0.44 ! [N: nat,A: rat] :
% 0.16/0.44 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.16/0.44 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_le_odd_power
% 0.16/0.44 thf(fact_2400_zero__le__odd__power,axiom,
% 0.16/0.44 ! [N: nat,A: int] :
% 0.16/0.44 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.16/0.44 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_le_odd_power
% 0.16/0.44 thf(fact_2401_zero__le__odd__power,axiom,
% 0.16/0.44 ! [N: nat,A: real] :
% 0.16/0.44 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.16/0.44 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_le_odd_power
% 0.16/0.44 thf(fact_2402_zero__le__even__power,axiom,
% 0.16/0.44 ! [N: nat,A: rat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_le_even_power
% 0.16/0.44 thf(fact_2403_zero__le__even__power,axiom,
% 0.16/0.44 ! [N: nat,A: int] :
% 0.16/0.44 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_le_even_power
% 0.16/0.44 thf(fact_2404_zero__le__even__power,axiom,
% 0.16/0.44 ! [N: nat,A: real] :
% 0.16/0.44 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.16/0.44 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_le_even_power
% 0.16/0.44 thf(fact_2405_nat__bit__induct,axiom,
% 0.16/0.44 ! [P: nat > $o,N: nat] :
% 0.16/0.44 ( ( P @ zero_zero_nat )
% 0.16/0.44 => ( ! [N2: nat] :
% 0.16/0.44 ( ( P @ N2 )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.16/0.44 => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.16/0.44 => ( ! [N2: nat] :
% 0.16/0.44 ( ( P @ N2 )
% 0.16/0.44 => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.16/0.44 => ( P @ N ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % nat_bit_induct
% 0.16/0.44 thf(fact_2406_div__2__gt__zero,axiom,
% 0.16/0.44 ! [N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.16/0.44 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % div_2_gt_zero
% 0.16/0.44 thf(fact_2407_Suc__n__div__2__gt__zero,axiom,
% 0.16/0.44 ! [N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.44 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % Suc_n_div_2_gt_zero
% 0.16/0.44 thf(fact_2408_inf__period_I2_J,axiom,
% 0.16/0.44 ! [P: real > $o,D3: real,Q: real > $o] :
% 0.16/0.44 ( ! [X4: real,K2: real] :
% 0.16/0.44 ( ( P @ X4 )
% 0.16/0.44 = ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D3 ) ) ) )
% 0.16/0.44 => ( ! [X4: real,K2: real] :
% 0.16/0.44 ( ( Q @ X4 )
% 0.16/0.44 = ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D3 ) ) ) )
% 0.16/0.44 => ! [X5: real,K4: real] :
% 0.16/0.44 ( ( ( P @ X5 )
% 0.16/0.44 | ( Q @ X5 ) )
% 0.16/0.44 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) )
% 0.16/0.44 | ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inf_period(2)
% 0.16/0.44 thf(fact_2409_inf__period_I2_J,axiom,
% 0.16/0.44 ! [P: rat > $o,D3: rat,Q: rat > $o] :
% 0.16/0.44 ( ! [X4: rat,K2: rat] :
% 0.16/0.44 ( ( P @ X4 )
% 0.16/0.44 = ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D3 ) ) ) )
% 0.16/0.44 => ( ! [X4: rat,K2: rat] :
% 0.16/0.44 ( ( Q @ X4 )
% 0.16/0.44 = ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D3 ) ) ) )
% 0.16/0.44 => ! [X5: rat,K4: rat] :
% 0.16/0.44 ( ( ( P @ X5 )
% 0.16/0.44 | ( Q @ X5 ) )
% 0.16/0.44 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) )
% 0.16/0.44 | ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inf_period(2)
% 0.16/0.44 thf(fact_2410_inf__period_I2_J,axiom,
% 0.16/0.44 ! [P: int > $o,D3: int,Q: int > $o] :
% 0.16/0.44 ( ! [X4: int,K2: int] :
% 0.16/0.44 ( ( P @ X4 )
% 0.16/0.44 = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 0.16/0.44 => ( ! [X4: int,K2: int] :
% 0.16/0.44 ( ( Q @ X4 )
% 0.16/0.44 = ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 0.16/0.44 => ! [X5: int,K4: int] :
% 0.16/0.44 ( ( ( P @ X5 )
% 0.16/0.44 | ( Q @ X5 ) )
% 0.16/0.44 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
% 0.16/0.44 | ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inf_period(2)
% 0.16/0.44 thf(fact_2411_inf__period_I1_J,axiom,
% 0.16/0.44 ! [P: real > $o,D3: real,Q: real > $o] :
% 0.16/0.44 ( ! [X4: real,K2: real] :
% 0.16/0.44 ( ( P @ X4 )
% 0.16/0.44 = ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D3 ) ) ) )
% 0.16/0.44 => ( ! [X4: real,K2: real] :
% 0.16/0.44 ( ( Q @ X4 )
% 0.16/0.44 = ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D3 ) ) ) )
% 0.16/0.44 => ! [X5: real,K4: real] :
% 0.16/0.44 ( ( ( P @ X5 )
% 0.16/0.44 & ( Q @ X5 ) )
% 0.16/0.44 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) )
% 0.16/0.44 & ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inf_period(1)
% 0.16/0.44 thf(fact_2412_inf__period_I1_J,axiom,
% 0.16/0.44 ! [P: rat > $o,D3: rat,Q: rat > $o] :
% 0.16/0.44 ( ! [X4: rat,K2: rat] :
% 0.16/0.44 ( ( P @ X4 )
% 0.16/0.44 = ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D3 ) ) ) )
% 0.16/0.44 => ( ! [X4: rat,K2: rat] :
% 0.16/0.44 ( ( Q @ X4 )
% 0.16/0.44 = ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D3 ) ) ) )
% 0.16/0.44 => ! [X5: rat,K4: rat] :
% 0.16/0.44 ( ( ( P @ X5 )
% 0.16/0.44 & ( Q @ X5 ) )
% 0.16/0.44 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) )
% 0.16/0.44 & ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inf_period(1)
% 0.16/0.44 thf(fact_2413_inf__period_I1_J,axiom,
% 0.16/0.44 ! [P: int > $o,D3: int,Q: int > $o] :
% 0.16/0.44 ( ! [X4: int,K2: int] :
% 0.16/0.44 ( ( P @ X4 )
% 0.16/0.44 = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 0.16/0.44 => ( ! [X4: int,K2: int] :
% 0.16/0.44 ( ( Q @ X4 )
% 0.16/0.44 = ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 0.16/0.44 => ! [X5: int,K4: int] :
% 0.16/0.44 ( ( ( P @ X5 )
% 0.16/0.44 & ( Q @ X5 ) )
% 0.16/0.44 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
% 0.16/0.44 & ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inf_period(1)
% 0.16/0.44 thf(fact_2414_dvd__productE,axiom,
% 0.16/0.44 ! [P5: nat,A: nat,B: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ P5 @ ( times_times_nat @ A @ B ) )
% 0.16/0.44 => ~ ! [X4: nat,Y: nat] :
% 0.16/0.44 ( ( P5
% 0.16/0.44 = ( times_times_nat @ X4 @ Y ) )
% 0.16/0.44 => ( ( dvd_dvd_nat @ X4 @ A )
% 0.16/0.44 => ~ ( dvd_dvd_nat @ Y @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dvd_productE
% 0.16/0.44 thf(fact_2415_dvd__productE,axiom,
% 0.16/0.44 ! [P5: int,A: int,B: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ P5 @ ( times_times_int @ A @ B ) )
% 0.16/0.44 => ~ ! [X4: int,Y: int] :
% 0.16/0.44 ( ( P5
% 0.16/0.44 = ( times_times_int @ X4 @ Y ) )
% 0.16/0.44 => ( ( dvd_dvd_int @ X4 @ A )
% 0.16/0.44 => ~ ( dvd_dvd_int @ Y @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dvd_productE
% 0.16/0.44 thf(fact_2416_division__decomp,axiom,
% 0.16/0.44 ! [A: nat,B: nat,C: nat] :
% 0.16/0.44 ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.16/0.44 => ? [B7: nat,C4: nat] :
% 0.16/0.44 ( ( A
% 0.16/0.44 = ( times_times_nat @ B7 @ C4 ) )
% 0.16/0.44 & ( dvd_dvd_nat @ B7 @ B )
% 0.16/0.44 & ( dvd_dvd_nat @ C4 @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % division_decomp
% 0.16/0.44 thf(fact_2417_division__decomp,axiom,
% 0.16/0.44 ! [A: int,B: int,C: int] :
% 0.16/0.44 ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 0.16/0.44 => ? [B7: int,C4: int] :
% 0.16/0.44 ( ( A
% 0.16/0.44 = ( times_times_int @ B7 @ C4 ) )
% 0.16/0.44 & ( dvd_dvd_int @ B7 @ B )
% 0.16/0.44 & ( dvd_dvd_int @ C4 @ C ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % division_decomp
% 0.16/0.44 thf(fact_2418_space__space_H,axiom,
% 0.16/0.44 ! [T: vEBT_VEBT] : ( ord_less_nat @ ( vEBT_VEBT_space @ T ) @ ( vEBT_VEBT_space2 @ T ) ) ).
% 0.16/0.44
% 0.16/0.44 % space_space'
% 0.16/0.44 thf(fact_2419_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.16/0.44 ! [I: num,N: nat,X: nat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.16/0.44 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.16/0.44
% 0.16/0.44 % numeral_power_le_of_nat_cancel_iff
% 0.16/0.44 thf(fact_2420_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.16/0.44 ! [I: num,N: nat,X: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.16/0.44 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.16/0.44
% 0.16/0.44 % numeral_power_le_of_nat_cancel_iff
% 0.16/0.44 thf(fact_2421_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.16/0.44 ! [I: num,N: nat,X: nat] :
% 0.16/0.44 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.16/0.44 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.16/0.44
% 0.16/0.44 % numeral_power_le_of_nat_cancel_iff
% 0.16/0.44 thf(fact_2422_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.16/0.44 ! [I: num,N: nat,X: nat] :
% 0.16/0.44 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.16/0.44 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.16/0.44
% 0.16/0.44 % numeral_power_le_of_nat_cancel_iff
% 0.16/0.44 thf(fact_2423_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.16/0.44 ! [X: nat,I: num,N: nat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 0.16/0.44 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_le_numeral_power_cancel_iff
% 0.16/0.44 thf(fact_2424_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.16/0.44 ! [X: nat,I: num,N: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 0.16/0.44 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_le_numeral_power_cancel_iff
% 0.16/0.44 thf(fact_2425_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.16/0.44 ! [X: nat,I: num,N: nat] :
% 0.16/0.44 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 0.16/0.44 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_le_numeral_power_cancel_iff
% 0.16/0.44 thf(fact_2426_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.16/0.44 ! [X: nat,I: num,N: nat] :
% 0.16/0.44 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 0.16/0.44 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_le_numeral_power_cancel_iff
% 0.16/0.44 thf(fact_2427_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.16/0.44 ! [I: num,N: nat,X: nat] :
% 0.16/0.44 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.16/0.44 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.16/0.44
% 0.16/0.44 % numeral_power_less_of_nat_cancel_iff
% 0.16/0.44 thf(fact_2428_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.16/0.44 ! [I: num,N: nat,X: nat] :
% 0.16/0.44 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.16/0.44 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.16/0.44
% 0.16/0.44 % numeral_power_less_of_nat_cancel_iff
% 0.16/0.44 thf(fact_2429_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.16/0.44 ! [I: num,N: nat,X: nat] :
% 0.16/0.44 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.16/0.44 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.16/0.44
% 0.16/0.44 % numeral_power_less_of_nat_cancel_iff
% 0.16/0.44 thf(fact_2430_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.16/0.44 ! [I: num,N: nat,X: nat] :
% 0.16/0.44 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.16/0.44 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.16/0.44
% 0.16/0.44 % numeral_power_less_of_nat_cancel_iff
% 0.16/0.44 thf(fact_2431_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.16/0.44 ! [X: nat,I: num,N: nat] :
% 0.16/0.44 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 0.16/0.44 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_less_numeral_power_cancel_iff
% 0.16/0.44 thf(fact_2432_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.16/0.44 ! [X: nat,I: num,N: nat] :
% 0.16/0.44 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 0.16/0.44 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_less_numeral_power_cancel_iff
% 0.16/0.44 thf(fact_2433_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.16/0.44 ! [X: nat,I: num,N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 0.16/0.44 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_less_numeral_power_cancel_iff
% 0.16/0.44 thf(fact_2434_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.16/0.44 ! [X: nat,I: num,N: nat] :
% 0.16/0.44 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 0.16/0.44 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_less_numeral_power_cancel_iff
% 0.16/0.44 thf(fact_2435_of__nat__zero__less__power__iff,axiom,
% 0.16/0.44 ! [X: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
% 0.16/0.44 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.16/0.44 | ( N = zero_zero_nat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_zero_less_power_iff
% 0.16/0.44 thf(fact_2436_of__nat__zero__less__power__iff,axiom,
% 0.16/0.44 ! [X: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
% 0.16/0.44 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.16/0.44 | ( N = zero_zero_nat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_zero_less_power_iff
% 0.16/0.44 thf(fact_2437_of__nat__zero__less__power__iff,axiom,
% 0.16/0.44 ! [X: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
% 0.16/0.44 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.16/0.44 | ( N = zero_zero_nat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_zero_less_power_iff
% 0.16/0.44 thf(fact_2438_of__nat__zero__less__power__iff,axiom,
% 0.16/0.44 ! [X: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
% 0.16/0.44 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.16/0.44 | ( N = zero_zero_nat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_zero_less_power_iff
% 0.16/0.44 thf(fact_2439_zero__less__power2,axiom,
% 0.16/0.44 ! [A: real] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.44 = ( A != zero_zero_real ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_power2
% 0.16/0.44 thf(fact_2440_zero__less__power2,axiom,
% 0.16/0.44 ! [A: rat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.44 = ( A != zero_zero_rat ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_power2
% 0.16/0.44 thf(fact_2441_zero__less__power2,axiom,
% 0.16/0.44 ! [A: int] :
% 0.16/0.44 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.44 = ( A != zero_zero_int ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_power2
% 0.16/0.44 thf(fact_2442_power2__less__eq__zero__iff,axiom,
% 0.16/0.44 ! [A: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 0.16/0.44 = ( A = zero_zero_rat ) ) ).
% 0.16/0.44
% 0.16/0.44 % power2_less_eq_zero_iff
% 0.16/0.44 thf(fact_2443_power2__less__eq__zero__iff,axiom,
% 0.16/0.44 ! [A: int] :
% 0.16/0.44 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.16/0.44 = ( A = zero_zero_int ) ) ).
% 0.16/0.44
% 0.16/0.44 % power2_less_eq_zero_iff
% 0.16/0.44 thf(fact_2444_power2__less__eq__zero__iff,axiom,
% 0.16/0.44 ! [A: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 0.16/0.44 = ( A = zero_zero_real ) ) ).
% 0.16/0.44
% 0.16/0.44 % power2_less_eq_zero_iff
% 0.16/0.44 thf(fact_2445_power2__eq__iff__nonneg,axiom,
% 0.16/0.44 ! [X: rat,Y4: rat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.16/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 0.16/0.44 => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.44 = ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.44 = ( X = Y4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power2_eq_iff_nonneg
% 0.16/0.44 thf(fact_2446_power2__eq__iff__nonneg,axiom,
% 0.16/0.44 ! [X: nat,Y4: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.16/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
% 0.16/0.44 => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.44 = ( power_power_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.44 = ( X = Y4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power2_eq_iff_nonneg
% 0.16/0.44 thf(fact_2447_power2__eq__iff__nonneg,axiom,
% 0.16/0.44 ! [X: int,Y4: int] :
% 0.16/0.44 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.16/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.16/0.44 => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.44 = ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.44 = ( X = Y4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power2_eq_iff_nonneg
% 0.16/0.44 thf(fact_2448_power2__eq__iff__nonneg,axiom,
% 0.16/0.44 ! [X: real,Y4: real] :
% 0.16/0.44 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.16/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.16/0.44 => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.44 = ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.44 = ( X = Y4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power2_eq_iff_nonneg
% 0.16/0.44 thf(fact_2449_power__decreasing__iff,axiom,
% 0.16/0.44 ! [B: rat,M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.16/0.44 => ( ( ord_less_rat @ B @ one_one_rat )
% 0.16/0.44 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 0.16/0.44 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_decreasing_iff
% 0.16/0.44 thf(fact_2450_power__decreasing__iff,axiom,
% 0.16/0.44 ! [B: nat,M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.16/0.44 => ( ( ord_less_nat @ B @ one_one_nat )
% 0.16/0.44 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 0.16/0.44 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_decreasing_iff
% 0.16/0.44 thf(fact_2451_power__decreasing__iff,axiom,
% 0.16/0.44 ! [B: int,M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_int @ zero_zero_int @ B )
% 0.16/0.44 => ( ( ord_less_int @ B @ one_one_int )
% 0.16/0.44 => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 0.16/0.44 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_decreasing_iff
% 0.16/0.44 thf(fact_2452_power__decreasing__iff,axiom,
% 0.16/0.44 ! [B: real,M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_real @ zero_zero_real @ B )
% 0.16/0.44 => ( ( ord_less_real @ B @ one_one_real )
% 0.16/0.44 => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 0.16/0.44 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_decreasing_iff
% 0.16/0.44 thf(fact_2453_power__mono__iff,axiom,
% 0.16/0.44 ! [A: rat,B: rat,N: nat] :
% 0.16/0.44 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.44 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.44 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 0.16/0.44 = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_mono_iff
% 0.16/0.44 thf(fact_2454_power__mono__iff,axiom,
% 0.16/0.44 ! [A: nat,B: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.44 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.16/0.44 = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_mono_iff
% 0.16/0.44 thf(fact_2455_power__mono__iff,axiom,
% 0.16/0.44 ! [A: int,B: int,N: nat] :
% 0.16/0.44 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.44 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.44 => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.16/0.44 = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_mono_iff
% 0.16/0.44 thf(fact_2456_power__mono__iff,axiom,
% 0.16/0.44 ! [A: real,B: real,N: nat] :
% 0.16/0.44 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.44 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.16/0.44 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.44 => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 0.16/0.44 = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_mono_iff
% 0.16/0.44 thf(fact_2457_psubsetI,axiom,
% 0.16/0.44 ! [A2: set_int,B4: set_int] :
% 0.16/0.44 ( ( ord_less_eq_set_int @ A2 @ B4 )
% 0.16/0.44 => ( ( A2 != B4 )
% 0.16/0.44 => ( ord_less_set_int @ A2 @ B4 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % psubsetI
% 0.16/0.44 thf(fact_2458_DiffI,axiom,
% 0.16/0.44 ! [C: complex,A2: set_complex,B4: set_complex] :
% 0.16/0.44 ( ( member_complex @ C @ A2 )
% 0.16/0.44 => ( ~ ( member_complex @ C @ B4 )
% 0.16/0.44 => ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % DiffI
% 0.16/0.44 thf(fact_2459_DiffI,axiom,
% 0.16/0.44 ! [C: real,A2: set_real,B4: set_real] :
% 0.16/0.44 ( ( member_real @ C @ A2 )
% 0.16/0.44 => ( ~ ( member_real @ C @ B4 )
% 0.16/0.44 => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % DiffI
% 0.16/0.45 thf(fact_2460_DiffI,axiom,
% 0.16/0.45 ! [C: set_nat,A2: set_set_nat,B4: set_set_nat] :
% 0.16/0.45 ( ( member_set_nat @ C @ A2 )
% 0.16/0.45 => ( ~ ( member_set_nat @ C @ B4 )
% 0.16/0.45 => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B4 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % DiffI
% 0.16/0.45 thf(fact_2461_DiffI,axiom,
% 0.16/0.45 ! [C: int,A2: set_int,B4: set_int] :
% 0.16/0.45 ( ( member_int @ C @ A2 )
% 0.16/0.45 => ( ~ ( member_int @ C @ B4 )
% 0.16/0.45 => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B4 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % DiffI
% 0.16/0.45 thf(fact_2462_DiffI,axiom,
% 0.16/0.45 ! [C: nat,A2: set_nat,B4: set_nat] :
% 0.16/0.45 ( ( member_nat @ C @ A2 )
% 0.16/0.45 => ( ~ ( member_nat @ C @ B4 )
% 0.16/0.45 => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B4 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % DiffI
% 0.16/0.45 thf(fact_2463_Diff__iff,axiom,
% 0.16/0.45 ! [C: complex,A2: set_complex,B4: set_complex] :
% 0.16/0.45 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B4 ) )
% 0.16/0.45 = ( ( member_complex @ C @ A2 )
% 0.16/0.45 & ~ ( member_complex @ C @ B4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % Diff_iff
% 0.16/0.45 thf(fact_2464_Diff__iff,axiom,
% 0.16/0.45 ! [C: real,A2: set_real,B4: set_real] :
% 0.16/0.45 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) )
% 0.16/0.45 = ( ( member_real @ C @ A2 )
% 0.16/0.45 & ~ ( member_real @ C @ B4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % Diff_iff
% 0.16/0.45 thf(fact_2465_Diff__iff,axiom,
% 0.16/0.45 ! [C: set_nat,A2: set_set_nat,B4: set_set_nat] :
% 0.16/0.45 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B4 ) )
% 0.16/0.45 = ( ( member_set_nat @ C @ A2 )
% 0.16/0.45 & ~ ( member_set_nat @ C @ B4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % Diff_iff
% 0.16/0.45 thf(fact_2466_Diff__iff,axiom,
% 0.16/0.45 ! [C: int,A2: set_int,B4: set_int] :
% 0.16/0.45 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B4 ) )
% 0.16/0.45 = ( ( member_int @ C @ A2 )
% 0.16/0.45 & ~ ( member_int @ C @ B4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % Diff_iff
% 0.16/0.45 thf(fact_2467_Diff__iff,axiom,
% 0.16/0.45 ! [C: nat,A2: set_nat,B4: set_nat] :
% 0.16/0.45 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B4 ) )
% 0.16/0.45 = ( ( member_nat @ C @ A2 )
% 0.16/0.45 & ~ ( member_nat @ C @ B4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % Diff_iff
% 0.16/0.45 thf(fact_2468_Diff__idemp,axiom,
% 0.16/0.45 ! [A2: set_nat,B4: set_nat] :
% 0.16/0.45 ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B4 ) @ B4 )
% 0.16/0.45 = ( minus_minus_set_nat @ A2 @ B4 ) ) ).
% 0.16/0.45
% 0.16/0.45 % Diff_idemp
% 0.16/0.45 thf(fact_2469_i0__less,axiom,
% 0.16/0.45 ! [N: extended_enat] :
% 0.16/0.45 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.16/0.45 = ( N != zero_z5237406670263579293d_enat ) ) ).
% 0.16/0.45
% 0.16/0.45 % i0_less
% 0.16/0.45 thf(fact_2470_two__realpow__ge__two,axiom,
% 0.16/0.45 ! [N: nat] :
% 0.16/0.45 ( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.16/0.45 => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % two_realpow_ge_two
% 0.16/0.45 thf(fact_2471_not__real__square__gt__zero,axiom,
% 0.16/0.45 ! [X: real] :
% 0.16/0.45 ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
% 0.16/0.45 = ( X = zero_zero_real ) ) ).
% 0.16/0.45
% 0.16/0.45 % not_real_square_gt_zero
% 0.16/0.45 thf(fact_2472_zle__diff1__eq,axiom,
% 0.16/0.45 ! [W: int,Z: int] :
% 0.16/0.45 ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
% 0.16/0.45 = ( ord_less_int @ W @ Z ) ) ).
% 0.16/0.45
% 0.16/0.45 % zle_diff1_eq
% 0.16/0.45 thf(fact_2473_semiring__norm_I78_J,axiom,
% 0.16/0.45 ! [M: num,N: num] :
% 0.16/0.45 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.16/0.45 = ( ord_less_num @ M @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % semiring_norm(78)
% 0.16/0.45 thf(fact_2474_semiring__norm_I75_J,axiom,
% 0.16/0.45 ! [M: num] :
% 0.16/0.45 ~ ( ord_less_num @ M @ one ) ).
% 0.16/0.45
% 0.16/0.45 % semiring_norm(75)
% 0.16/0.45 thf(fact_2475_semiring__norm_I80_J,axiom,
% 0.16/0.45 ! [M: num,N: num] :
% 0.16/0.45 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.16/0.45 = ( ord_less_num @ M @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % semiring_norm(80)
% 0.16/0.45 thf(fact_2476_unset__bit__negative__int__iff,axiom,
% 0.16/0.45 ! [N: nat,K: int] :
% 0.16/0.45 ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
% 0.16/0.45 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.16/0.45
% 0.16/0.45 % unset_bit_negative_int_iff
% 0.16/0.45 thf(fact_2477_set__bit__negative__int__iff,axiom,
% 0.16/0.45 ! [N: nat,K: int] :
% 0.16/0.45 ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
% 0.16/0.45 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.16/0.45
% 0.16/0.45 % set_bit_negative_int_iff
% 0.16/0.45 thf(fact_2478_flip__bit__negative__int__iff,axiom,
% 0.16/0.45 ! [N: nat,K: int] :
% 0.16/0.45 ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
% 0.16/0.45 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.16/0.45
% 0.16/0.45 % flip_bit_negative_int_iff
% 0.16/0.45 thf(fact_2479_power__0__Suc,axiom,
% 0.16/0.45 ! [N: nat] :
% 0.16/0.45 ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
% 0.16/0.45 = zero_zero_rat ) ).
% 0.16/0.45
% 0.16/0.45 % power_0_Suc
% 0.16/0.45 thf(fact_2480_power__0__Suc,axiom,
% 0.16/0.45 ! [N: nat] :
% 0.16/0.45 ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 0.16/0.45 = zero_zero_int ) ).
% 0.16/0.45
% 0.16/0.45 % power_0_Suc
% 0.16/0.45 thf(fact_2481_power__0__Suc,axiom,
% 0.16/0.45 ! [N: nat] :
% 0.16/0.45 ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 0.16/0.45 = zero_zero_real ) ).
% 0.16/0.45
% 0.16/0.45 % power_0_Suc
% 0.16/0.45 thf(fact_2482_power__0__Suc,axiom,
% 0.16/0.45 ! [N: nat] :
% 0.16/0.45 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.16/0.45 = zero_zero_nat ) ).
% 0.16/0.45
% 0.16/0.45 % power_0_Suc
% 0.16/0.45 thf(fact_2483_power__0__Suc,axiom,
% 0.16/0.45 ! [N: nat] :
% 0.16/0.45 ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
% 0.16/0.45 = zero_zero_complex ) ).
% 0.16/0.45
% 0.16/0.45 % power_0_Suc
% 0.16/0.45 thf(fact_2484_power__zero__numeral,axiom,
% 0.16/0.45 ! [K: num] :
% 0.16/0.45 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 0.16/0.45 = zero_zero_rat ) ).
% 0.16/0.45
% 0.16/0.45 % power_zero_numeral
% 0.16/0.45 thf(fact_2485_power__zero__numeral,axiom,
% 0.16/0.45 ! [K: num] :
% 0.16/0.45 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 0.16/0.45 = zero_zero_int ) ).
% 0.16/0.45
% 0.16/0.45 % power_zero_numeral
% 0.16/0.45 thf(fact_2486_power__zero__numeral,axiom,
% 0.16/0.45 ! [K: num] :
% 0.16/0.45 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 0.16/0.45 = zero_zero_real ) ).
% 0.16/0.45
% 0.16/0.45 % power_zero_numeral
% 0.16/0.45 thf(fact_2487_power__zero__numeral,axiom,
% 0.16/0.45 ! [K: num] :
% 0.16/0.45 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 0.16/0.45 = zero_zero_nat ) ).
% 0.16/0.45
% 0.16/0.45 % power_zero_numeral
% 0.16/0.45 thf(fact_2488_power__zero__numeral,axiom,
% 0.16/0.45 ! [K: num] :
% 0.16/0.45 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 0.16/0.45 = zero_zero_complex ) ).
% 0.16/0.45
% 0.16/0.45 % power_zero_numeral
% 0.16/0.45 thf(fact_2489_power__Suc0__right,axiom,
% 0.16/0.45 ! [A: int] :
% 0.16/0.45 ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 0.16/0.45 = A ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc0_right
% 0.16/0.45 thf(fact_2490_power__Suc0__right,axiom,
% 0.16/0.45 ! [A: real] :
% 0.16/0.45 ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 0.16/0.45 = A ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc0_right
% 0.16/0.45 thf(fact_2491_power__Suc0__right,axiom,
% 0.16/0.45 ! [A: nat] :
% 0.16/0.45 ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 0.16/0.45 = A ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc0_right
% 0.16/0.45 thf(fact_2492_power__Suc0__right,axiom,
% 0.16/0.45 ! [A: complex] :
% 0.16/0.45 ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 0.16/0.45 = A ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc0_right
% 0.16/0.45 thf(fact_2493_power__Suc__0,axiom,
% 0.16/0.45 ! [N: nat] :
% 0.16/0.45 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.16/0.45 = ( suc @ zero_zero_nat ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc_0
% 0.16/0.45 thf(fact_2494_nat__power__eq__Suc__0__iff,axiom,
% 0.16/0.45 ! [X: nat,M: nat] :
% 0.16/0.45 ( ( ( power_power_nat @ X @ M )
% 0.16/0.45 = ( suc @ zero_zero_nat ) )
% 0.16/0.45 = ( ( M = zero_zero_nat )
% 0.16/0.45 | ( X
% 0.16/0.45 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % nat_power_eq_Suc_0_iff
% 0.16/0.45 thf(fact_2495_nat__zero__less__power__iff,axiom,
% 0.16/0.45 ! [X: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
% 0.16/0.45 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.16/0.45 | ( N = zero_zero_nat ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % nat_zero_less_power_iff
% 0.16/0.45 thf(fact_2496_of__nat__power,axiom,
% 0.16/0.45 ! [M: nat,N: nat] :
% 0.16/0.45 ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
% 0.16/0.45 = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_power
% 0.16/0.45 thf(fact_2497_of__nat__power,axiom,
% 0.16/0.45 ! [M: nat,N: nat] :
% 0.16/0.45 ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
% 0.16/0.45 = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_power
% 0.16/0.45 thf(fact_2498_of__nat__power,axiom,
% 0.16/0.45 ! [M: nat,N: nat] :
% 0.16/0.45 ( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N ) )
% 0.16/0.45 = ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_power
% 0.16/0.45 thf(fact_2499_of__nat__power,axiom,
% 0.16/0.45 ! [M: nat,N: nat] :
% 0.16/0.45 ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
% 0.16/0.45 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_power
% 0.16/0.45 thf(fact_2500_of__nat__power,axiom,
% 0.16/0.45 ! [M: nat,N: nat] :
% 0.16/0.45 ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
% 0.16/0.45 = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_power
% 0.16/0.45 thf(fact_2501_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.16/0.45 ! [B: nat,W: nat,X: nat] :
% 0.16/0.45 ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
% 0.16/0.45 = ( semiri8010041392384452111omplex @ X ) )
% 0.16/0.45 = ( ( power_power_nat @ B @ W )
% 0.16/0.45 = X ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_eq_of_nat_power_cancel_iff
% 0.16/0.45 thf(fact_2502_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.16/0.45 ! [B: nat,W: nat,X: nat] :
% 0.16/0.45 ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
% 0.16/0.45 = ( semiri5074537144036343181t_real @ X ) )
% 0.16/0.45 = ( ( power_power_nat @ B @ W )
% 0.16/0.45 = X ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_eq_of_nat_power_cancel_iff
% 0.16/0.45 thf(fact_2503_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.16/0.45 ! [B: nat,W: nat,X: nat] :
% 0.16/0.45 ( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W )
% 0.16/0.45 = ( semiri681578069525770553at_rat @ X ) )
% 0.16/0.45 = ( ( power_power_nat @ B @ W )
% 0.16/0.45 = X ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_eq_of_nat_power_cancel_iff
% 0.16/0.45 thf(fact_2504_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.16/0.45 ! [B: nat,W: nat,X: nat] :
% 0.16/0.45 ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
% 0.16/0.45 = ( semiri1316708129612266289at_nat @ X ) )
% 0.16/0.45 = ( ( power_power_nat @ B @ W )
% 0.16/0.45 = X ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_eq_of_nat_power_cancel_iff
% 0.16/0.45 thf(fact_2505_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.16/0.45 ! [B: nat,W: nat,X: nat] :
% 0.16/0.45 ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
% 0.16/0.45 = ( semiri1314217659103216013at_int @ X ) )
% 0.16/0.45 = ( ( power_power_nat @ B @ W )
% 0.16/0.45 = X ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_eq_of_nat_power_cancel_iff
% 0.16/0.45 thf(fact_2506_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.16/0.45 ! [X: nat,B: nat,W: nat] :
% 0.16/0.45 ( ( ( semiri8010041392384452111omplex @ X )
% 0.16/0.45 = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
% 0.16/0.45 = ( X
% 0.16/0.45 = ( power_power_nat @ B @ W ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_power_eq_of_nat_cancel_iff
% 0.16/0.45 thf(fact_2507_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.16/0.45 ! [X: nat,B: nat,W: nat] :
% 0.16/0.45 ( ( ( semiri5074537144036343181t_real @ X )
% 0.16/0.45 = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 0.16/0.45 = ( X
% 0.16/0.45 = ( power_power_nat @ B @ W ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_power_eq_of_nat_cancel_iff
% 0.16/0.45 thf(fact_2508_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.16/0.45 ! [X: nat,B: nat,W: nat] :
% 0.16/0.45 ( ( ( semiri681578069525770553at_rat @ X )
% 0.16/0.45 = ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 0.16/0.45 = ( X
% 0.16/0.45 = ( power_power_nat @ B @ W ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_power_eq_of_nat_cancel_iff
% 0.16/0.45 thf(fact_2509_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.16/0.45 ! [X: nat,B: nat,W: nat] :
% 0.16/0.45 ( ( ( semiri1316708129612266289at_nat @ X )
% 0.16/0.45 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 0.16/0.45 = ( X
% 0.16/0.45 = ( power_power_nat @ B @ W ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_power_eq_of_nat_cancel_iff
% 0.16/0.45 thf(fact_2510_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.16/0.45 ! [X: nat,B: nat,W: nat] :
% 0.16/0.45 ( ( ( semiri1314217659103216013at_int @ X )
% 0.16/0.45 = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 0.16/0.45 = ( X
% 0.16/0.45 = ( power_power_nat @ B @ W ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_power_eq_of_nat_cancel_iff
% 0.16/0.45 thf(fact_2511_div__neg__neg__trivial,axiom,
% 0.16/0.45 ! [K: int,L: int] :
% 0.16/0.45 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.16/0.45 => ( ( ord_less_int @ L @ K )
% 0.16/0.45 => ( ( divide_divide_int @ K @ L )
% 0.16/0.45 = zero_zero_int ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % div_neg_neg_trivial
% 0.16/0.45 thf(fact_2512_div__pos__pos__trivial,axiom,
% 0.16/0.45 ! [K: int,L: int] :
% 0.16/0.45 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.16/0.45 => ( ( ord_less_int @ K @ L )
% 0.16/0.45 => ( ( divide_divide_int @ K @ L )
% 0.16/0.45 = zero_zero_int ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % div_pos_pos_trivial
% 0.16/0.45 thf(fact_2513_power__mult__numeral,axiom,
% 0.16/0.45 ! [A: int,M: num,N: num] :
% 0.16/0.45 ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.16/0.45 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_mult_numeral
% 0.16/0.45 thf(fact_2514_power__mult__numeral,axiom,
% 0.16/0.45 ! [A: real,M: num,N: num] :
% 0.16/0.45 ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.16/0.45 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_mult_numeral
% 0.16/0.45 thf(fact_2515_power__mult__numeral,axiom,
% 0.16/0.45 ! [A: nat,M: num,N: num] :
% 0.16/0.45 ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.16/0.45 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_mult_numeral
% 0.16/0.45 thf(fact_2516_power__mult__numeral,axiom,
% 0.16/0.45 ! [A: complex,M: num,N: num] :
% 0.16/0.45 ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.16/0.45 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_mult_numeral
% 0.16/0.45 thf(fact_2517_semiring__norm_I76_J,axiom,
% 0.16/0.45 ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % semiring_norm(76)
% 0.16/0.45 thf(fact_2518_semiring__norm_I81_J,axiom,
% 0.16/0.45 ! [M: num,N: num] :
% 0.16/0.45 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.16/0.45 = ( ord_less_num @ M @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % semiring_norm(81)
% 0.16/0.45 thf(fact_2519_semiring__norm_I77_J,axiom,
% 0.16/0.45 ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % semiring_norm(77)
% 0.16/0.45 thf(fact_2520_enat__ord__number_I2_J,axiom,
% 0.16/0.45 ! [M: num,N: num] :
% 0.16/0.45 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.16/0.45 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % enat_ord_number(2)
% 0.16/0.45 thf(fact_2521_power__eq__0__iff,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( ( power_power_rat @ A @ N )
% 0.16/0.45 = zero_zero_rat )
% 0.16/0.45 = ( ( A = zero_zero_rat )
% 0.16/0.45 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_eq_0_iff
% 0.16/0.45 thf(fact_2522_power__eq__0__iff,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( ( power_power_int @ A @ N )
% 0.16/0.45 = zero_zero_int )
% 0.16/0.45 = ( ( A = zero_zero_int )
% 0.16/0.45 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_eq_0_iff
% 0.16/0.45 thf(fact_2523_power__eq__0__iff,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( ( power_power_real @ A @ N )
% 0.16/0.45 = zero_zero_real )
% 0.16/0.45 = ( ( A = zero_zero_real )
% 0.16/0.45 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_eq_0_iff
% 0.16/0.45 thf(fact_2524_power__eq__0__iff,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( ( power_power_nat @ A @ N )
% 0.16/0.45 = zero_zero_nat )
% 0.16/0.45 = ( ( A = zero_zero_nat )
% 0.16/0.45 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_eq_0_iff
% 0.16/0.45 thf(fact_2525_power__eq__0__iff,axiom,
% 0.16/0.45 ! [A: complex,N: nat] :
% 0.16/0.45 ( ( ( power_power_complex @ A @ N )
% 0.16/0.45 = zero_zero_complex )
% 0.16/0.45 = ( ( A = zero_zero_complex )
% 0.16/0.45 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_eq_0_iff
% 0.16/0.45 thf(fact_2526_real__of__nat__less__numeral__iff,axiom,
% 0.16/0.45 ! [N: nat,W: num] :
% 0.16/0.45 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
% 0.16/0.45 = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % real_of_nat_less_numeral_iff
% 0.16/0.45 thf(fact_2527_numeral__less__real__of__nat__iff,axiom,
% 0.16/0.45 ! [W: num,N: nat] :
% 0.16/0.45 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.16/0.45 = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % numeral_less_real_of_nat_iff
% 0.16/0.45 thf(fact_2528_semiring__norm_I74_J,axiom,
% 0.16/0.45 ! [M: num,N: num] :
% 0.16/0.45 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.16/0.45 = ( ord_less_num @ M @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % semiring_norm(74)
% 0.16/0.45 thf(fact_2529_semiring__norm_I79_J,axiom,
% 0.16/0.45 ! [M: num,N: num] :
% 0.16/0.45 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.16/0.45 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % semiring_norm(79)
% 0.16/0.45 thf(fact_2530_power__strict__decreasing__iff,axiom,
% 0.16/0.45 ! [B: real,M: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_real @ zero_zero_real @ B )
% 0.16/0.45 => ( ( ord_less_real @ B @ one_one_real )
% 0.16/0.45 => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 0.16/0.45 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_strict_decreasing_iff
% 0.16/0.45 thf(fact_2531_power__strict__decreasing__iff,axiom,
% 0.16/0.45 ! [B: rat,M: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.16/0.45 => ( ( ord_less_rat @ B @ one_one_rat )
% 0.16/0.45 => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 0.16/0.45 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_strict_decreasing_iff
% 0.16/0.45 thf(fact_2532_power__strict__decreasing__iff,axiom,
% 0.16/0.45 ! [B: nat,M: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.16/0.45 => ( ( ord_less_nat @ B @ one_one_nat )
% 0.16/0.45 => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 0.16/0.45 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_strict_decreasing_iff
% 0.16/0.45 thf(fact_2533_power__strict__decreasing__iff,axiom,
% 0.16/0.45 ! [B: int,M: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_int @ zero_zero_int @ B )
% 0.16/0.45 => ( ( ord_less_int @ B @ one_one_int )
% 0.16/0.45 => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 0.16/0.45 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_strict_decreasing_iff
% 0.16/0.45 thf(fact_2534_power__increasing__iff,axiom,
% 0.16/0.45 ! [B: rat,X: nat,Y4: nat] :
% 0.16/0.45 ( ( ord_less_rat @ one_one_rat @ B )
% 0.16/0.45 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y4 ) )
% 0.16/0.45 = ( ord_less_eq_nat @ X @ Y4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_increasing_iff
% 0.16/0.45 thf(fact_2535_power__increasing__iff,axiom,
% 0.16/0.45 ! [B: nat,X: nat,Y4: nat] :
% 0.16/0.45 ( ( ord_less_nat @ one_one_nat @ B )
% 0.16/0.45 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y4 ) )
% 0.16/0.45 = ( ord_less_eq_nat @ X @ Y4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_increasing_iff
% 0.16/0.45 thf(fact_2536_power__increasing__iff,axiom,
% 0.16/0.45 ! [B: int,X: nat,Y4: nat] :
% 0.16/0.45 ( ( ord_less_int @ one_one_int @ B )
% 0.16/0.45 => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y4 ) )
% 0.16/0.45 = ( ord_less_eq_nat @ X @ Y4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_increasing_iff
% 0.16/0.45 thf(fact_2537_power__increasing__iff,axiom,
% 0.16/0.45 ! [B: real,X: nat,Y4: nat] :
% 0.16/0.45 ( ( ord_less_real @ one_one_real @ B )
% 0.16/0.45 => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y4 ) )
% 0.16/0.45 = ( ord_less_eq_nat @ X @ Y4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_increasing_iff
% 0.16/0.45 thf(fact_2538_zero__eq__power2,axiom,
% 0.16/0.45 ! [A: rat] :
% 0.16/0.45 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = zero_zero_rat )
% 0.16/0.45 = ( A = zero_zero_rat ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_eq_power2
% 0.16/0.45 thf(fact_2539_zero__eq__power2,axiom,
% 0.16/0.45 ! [A: int] :
% 0.16/0.45 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = zero_zero_int )
% 0.16/0.45 = ( A = zero_zero_int ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_eq_power2
% 0.16/0.45 thf(fact_2540_zero__eq__power2,axiom,
% 0.16/0.45 ! [A: real] :
% 0.16/0.45 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = zero_zero_real )
% 0.16/0.45 = ( A = zero_zero_real ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_eq_power2
% 0.16/0.45 thf(fact_2541_zero__eq__power2,axiom,
% 0.16/0.45 ! [A: nat] :
% 0.16/0.45 ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = zero_zero_nat )
% 0.16/0.45 = ( A = zero_zero_nat ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_eq_power2
% 0.16/0.45 thf(fact_2542_zero__eq__power2,axiom,
% 0.16/0.45 ! [A: complex] :
% 0.16/0.45 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = zero_zero_complex )
% 0.16/0.45 = ( A = zero_zero_complex ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_eq_power2
% 0.16/0.45 thf(fact_2543_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.16/0.45 ! [B: nat,W: nat,X: nat] :
% 0.16/0.45 ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.16/0.45 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_less_of_nat_power_cancel_iff
% 0.16/0.45 thf(fact_2544_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.16/0.45 ! [B: nat,W: nat,X: nat] :
% 0.16/0.45 ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.16/0.45 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_less_of_nat_power_cancel_iff
% 0.16/0.45 thf(fact_2545_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.16/0.45 ! [B: nat,W: nat,X: nat] :
% 0.16/0.45 ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.16/0.45 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_less_of_nat_power_cancel_iff
% 0.16/0.45 thf(fact_2546_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.16/0.45 ! [B: nat,W: nat,X: nat] :
% 0.16/0.45 ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.16/0.45 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_less_of_nat_power_cancel_iff
% 0.16/0.45 thf(fact_2547_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.16/0.45 ! [X: nat,B: nat,W: nat] :
% 0.16/0.45 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 0.16/0.45 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_power_less_of_nat_cancel_iff
% 0.16/0.45 thf(fact_2548_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.16/0.45 ! [X: nat,B: nat,W: nat] :
% 0.16/0.45 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 0.16/0.45 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_power_less_of_nat_cancel_iff
% 0.16/0.45 thf(fact_2549_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.16/0.45 ! [X: nat,B: nat,W: nat] :
% 0.16/0.45 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 0.16/0.45 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_power_less_of_nat_cancel_iff
% 0.16/0.45 thf(fact_2550_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.16/0.45 ! [X: nat,B: nat,W: nat] :
% 0.16/0.45 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 0.16/0.45 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_power_less_of_nat_cancel_iff
% 0.16/0.45 thf(fact_2551_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.16/0.45 ! [X: nat,B: nat,W: nat] :
% 0.16/0.45 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 0.16/0.45 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_power_le_of_nat_cancel_iff
% 0.16/0.45 thf(fact_2552_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.16/0.45 ! [X: nat,B: nat,W: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 0.16/0.45 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_power_le_of_nat_cancel_iff
% 0.16/0.45 thf(fact_2553_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.16/0.45 ! [X: nat,B: nat,W: nat] :
% 0.16/0.45 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 0.16/0.45 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_power_le_of_nat_cancel_iff
% 0.16/0.45 thf(fact_2554_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.16/0.45 ! [X: nat,B: nat,W: nat] :
% 0.16/0.45 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 0.16/0.45 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_power_le_of_nat_cancel_iff
% 0.16/0.45 thf(fact_2555_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.16/0.45 ! [B: nat,W: nat,X: nat] :
% 0.16/0.45 ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.16/0.45 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_le_of_nat_power_cancel_iff
% 0.16/0.45 thf(fact_2556_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.16/0.45 ! [B: nat,W: nat,X: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.16/0.45 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_le_of_nat_power_cancel_iff
% 0.16/0.45 thf(fact_2557_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.16/0.45 ! [B: nat,W: nat,X: nat] :
% 0.16/0.45 ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.16/0.45 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_le_of_nat_power_cancel_iff
% 0.16/0.45 thf(fact_2558_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.16/0.45 ! [B: nat,W: nat,X: nat] :
% 0.16/0.45 ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.16/0.45 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.16/0.45
% 0.16/0.45 % of_nat_le_of_nat_power_cancel_iff
% 0.16/0.45 thf(fact_2559_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.16/0.45 ! [Y4: nat,X: num,N: nat] :
% 0.16/0.45 ( ( ( semiri8010041392384452111omplex @ Y4 )
% 0.16/0.45 = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
% 0.16/0.45 = ( Y4
% 0.16/0.45 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % real_of_nat_eq_numeral_power_cancel_iff
% 0.16/0.45 thf(fact_2560_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.16/0.45 ! [Y4: nat,X: num,N: nat] :
% 0.16/0.45 ( ( ( semiri5074537144036343181t_real @ Y4 )
% 0.16/0.45 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.16/0.45 = ( Y4
% 0.16/0.45 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % real_of_nat_eq_numeral_power_cancel_iff
% 0.16/0.45 thf(fact_2561_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.16/0.45 ! [Y4: nat,X: num,N: nat] :
% 0.16/0.45 ( ( ( semiri681578069525770553at_rat @ Y4 )
% 0.16/0.45 = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.16/0.45 = ( Y4
% 0.16/0.45 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % real_of_nat_eq_numeral_power_cancel_iff
% 0.16/0.45 thf(fact_2562_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.16/0.45 ! [Y4: nat,X: num,N: nat] :
% 0.16/0.45 ( ( ( semiri1316708129612266289at_nat @ Y4 )
% 0.16/0.45 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.16/0.45 = ( Y4
% 0.16/0.45 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % real_of_nat_eq_numeral_power_cancel_iff
% 0.16/0.45 thf(fact_2563_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.16/0.45 ! [Y4: nat,X: num,N: nat] :
% 0.16/0.45 ( ( ( semiri1314217659103216013at_int @ Y4 )
% 0.16/0.45 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.16/0.45 = ( Y4
% 0.16/0.45 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % real_of_nat_eq_numeral_power_cancel_iff
% 0.16/0.45 thf(fact_2564_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.16/0.45 ! [X: num,N: nat,Y4: nat] :
% 0.16/0.45 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
% 0.16/0.45 = ( semiri8010041392384452111omplex @ Y4 ) )
% 0.16/0.45 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.16/0.45 = Y4 ) ) ).
% 0.16/0.45
% 0.16/0.45 % numeral_power_eq_of_nat_cancel_iff
% 0.16/0.45 thf(fact_2565_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.16/0.45 ! [X: num,N: nat,Y4: nat] :
% 0.16/0.45 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 0.16/0.45 = ( semiri5074537144036343181t_real @ Y4 ) )
% 0.16/0.45 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.16/0.45 = Y4 ) ) ).
% 0.16/0.45
% 0.16/0.45 % numeral_power_eq_of_nat_cancel_iff
% 0.16/0.45 thf(fact_2566_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.16/0.45 ! [X: num,N: nat,Y4: nat] :
% 0.16/0.45 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
% 0.16/0.45 = ( semiri681578069525770553at_rat @ Y4 ) )
% 0.16/0.45 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.16/0.45 = Y4 ) ) ).
% 0.16/0.45
% 0.16/0.45 % numeral_power_eq_of_nat_cancel_iff
% 0.16/0.45 thf(fact_2567_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.16/0.45 ! [X: num,N: nat,Y4: nat] :
% 0.16/0.45 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.16/0.45 = ( semiri1316708129612266289at_nat @ Y4 ) )
% 0.16/0.45 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.16/0.45 = Y4 ) ) ).
% 0.16/0.45
% 0.16/0.45 % numeral_power_eq_of_nat_cancel_iff
% 0.16/0.45 thf(fact_2568_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.16/0.45 ! [X: num,N: nat,Y4: nat] :
% 0.16/0.45 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.16/0.45 = ( semiri1314217659103216013at_int @ Y4 ) )
% 0.16/0.45 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.16/0.45 = Y4 ) ) ).
% 0.16/0.45
% 0.16/0.45 % numeral_power_eq_of_nat_cancel_iff
% 0.16/0.45 thf(fact_2569_half__negative__int__iff,axiom,
% 0.16/0.45 ! [K: int] :
% 0.16/0.45 ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.16/0.45 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.16/0.45
% 0.16/0.45 % half_negative_int_iff
% 0.16/0.45 thf(fact_2570_DiffE,axiom,
% 0.16/0.45 ! [C: complex,A2: set_complex,B4: set_complex] :
% 0.16/0.45 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B4 ) )
% 0.16/0.45 => ~ ( ( member_complex @ C @ A2 )
% 0.16/0.45 => ( member_complex @ C @ B4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % DiffE
% 0.16/0.45 thf(fact_2571_DiffE,axiom,
% 0.16/0.45 ! [C: real,A2: set_real,B4: set_real] :
% 0.16/0.45 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) )
% 0.16/0.45 => ~ ( ( member_real @ C @ A2 )
% 0.16/0.45 => ( member_real @ C @ B4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % DiffE
% 0.16/0.45 thf(fact_2572_DiffE,axiom,
% 0.16/0.45 ! [C: set_nat,A2: set_set_nat,B4: set_set_nat] :
% 0.16/0.45 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B4 ) )
% 0.16/0.45 => ~ ( ( member_set_nat @ C @ A2 )
% 0.16/0.45 => ( member_set_nat @ C @ B4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % DiffE
% 0.16/0.45 thf(fact_2573_DiffE,axiom,
% 0.16/0.45 ! [C: int,A2: set_int,B4: set_int] :
% 0.16/0.45 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B4 ) )
% 0.16/0.45 => ~ ( ( member_int @ C @ A2 )
% 0.16/0.45 => ( member_int @ C @ B4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % DiffE
% 0.16/0.45 thf(fact_2574_DiffE,axiom,
% 0.16/0.45 ! [C: nat,A2: set_nat,B4: set_nat] :
% 0.16/0.45 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B4 ) )
% 0.16/0.45 => ~ ( ( member_nat @ C @ A2 )
% 0.16/0.45 => ( member_nat @ C @ B4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % DiffE
% 0.16/0.45 thf(fact_2575_DiffD1,axiom,
% 0.16/0.45 ! [C: complex,A2: set_complex,B4: set_complex] :
% 0.16/0.45 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B4 ) )
% 0.16/0.45 => ( member_complex @ C @ A2 ) ) ).
% 0.16/0.45
% 0.16/0.45 % DiffD1
% 0.16/0.45 thf(fact_2576_DiffD1,axiom,
% 0.16/0.45 ! [C: real,A2: set_real,B4: set_real] :
% 0.16/0.45 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) )
% 0.16/0.45 => ( member_real @ C @ A2 ) ) ).
% 0.16/0.45
% 0.16/0.45 % DiffD1
% 0.16/0.45 thf(fact_2577_DiffD1,axiom,
% 0.16/0.45 ! [C: set_nat,A2: set_set_nat,B4: set_set_nat] :
% 0.16/0.45 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B4 ) )
% 0.16/0.45 => ( member_set_nat @ C @ A2 ) ) ).
% 0.16/0.45
% 0.16/0.45 % DiffD1
% 0.16/0.45 thf(fact_2578_DiffD1,axiom,
% 0.16/0.45 ! [C: int,A2: set_int,B4: set_int] :
% 0.16/0.45 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B4 ) )
% 0.16/0.45 => ( member_int @ C @ A2 ) ) ).
% 0.16/0.45
% 0.16/0.45 % DiffD1
% 0.16/0.45 thf(fact_2579_DiffD1,axiom,
% 0.16/0.45 ! [C: nat,A2: set_nat,B4: set_nat] :
% 0.16/0.45 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B4 ) )
% 0.16/0.45 => ( member_nat @ C @ A2 ) ) ).
% 0.16/0.45
% 0.16/0.45 % DiffD1
% 0.16/0.45 thf(fact_2580_DiffD2,axiom,
% 0.16/0.45 ! [C: complex,A2: set_complex,B4: set_complex] :
% 0.16/0.45 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B4 ) )
% 0.16/0.45 => ~ ( member_complex @ C @ B4 ) ) ).
% 0.16/0.45
% 0.16/0.45 % DiffD2
% 0.16/0.45 thf(fact_2581_DiffD2,axiom,
% 0.16/0.45 ! [C: real,A2: set_real,B4: set_real] :
% 0.16/0.45 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) )
% 0.16/0.45 => ~ ( member_real @ C @ B4 ) ) ).
% 0.16/0.45
% 0.16/0.45 % DiffD2
% 0.16/0.45 thf(fact_2582_DiffD2,axiom,
% 0.16/0.45 ! [C: set_nat,A2: set_set_nat,B4: set_set_nat] :
% 0.16/0.45 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B4 ) )
% 0.16/0.45 => ~ ( member_set_nat @ C @ B4 ) ) ).
% 0.16/0.45
% 0.16/0.45 % DiffD2
% 0.16/0.45 thf(fact_2583_DiffD2,axiom,
% 0.16/0.45 ! [C: int,A2: set_int,B4: set_int] :
% 0.16/0.45 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B4 ) )
% 0.16/0.45 => ~ ( member_int @ C @ B4 ) ) ).
% 0.16/0.45
% 0.16/0.45 % DiffD2
% 0.16/0.45 thf(fact_2584_DiffD2,axiom,
% 0.16/0.45 ! [C: nat,A2: set_nat,B4: set_nat] :
% 0.16/0.45 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B4 ) )
% 0.16/0.45 => ~ ( member_nat @ C @ B4 ) ) ).
% 0.16/0.45
% 0.16/0.45 % DiffD2
% 0.16/0.45 thf(fact_2585_psubset__imp__ex__mem,axiom,
% 0.16/0.45 ! [A2: set_complex,B4: set_complex] :
% 0.16/0.45 ( ( ord_less_set_complex @ A2 @ B4 )
% 0.16/0.45 => ? [B3: complex] : ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B4 @ A2 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % psubset_imp_ex_mem
% 0.16/0.45 thf(fact_2586_psubset__imp__ex__mem,axiom,
% 0.16/0.45 ! [A2: set_real,B4: set_real] :
% 0.16/0.45 ( ( ord_less_set_real @ A2 @ B4 )
% 0.16/0.45 => ? [B3: real] : ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A2 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % psubset_imp_ex_mem
% 0.16/0.45 thf(fact_2587_psubset__imp__ex__mem,axiom,
% 0.16/0.45 ! [A2: set_set_nat,B4: set_set_nat] :
% 0.16/0.45 ( ( ord_less_set_set_nat @ A2 @ B4 )
% 0.16/0.45 => ? [B3: set_nat] : ( member_set_nat @ B3 @ ( minus_2163939370556025621et_nat @ B4 @ A2 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % psubset_imp_ex_mem
% 0.16/0.45 thf(fact_2588_psubset__imp__ex__mem,axiom,
% 0.16/0.45 ! [A2: set_int,B4: set_int] :
% 0.16/0.45 ( ( ord_less_set_int @ A2 @ B4 )
% 0.16/0.45 => ? [B3: int] : ( member_int @ B3 @ ( minus_minus_set_int @ B4 @ A2 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % psubset_imp_ex_mem
% 0.16/0.45 thf(fact_2589_psubset__imp__ex__mem,axiom,
% 0.16/0.45 ! [A2: set_nat,B4: set_nat] :
% 0.16/0.45 ( ( ord_less_set_nat @ A2 @ B4 )
% 0.16/0.45 => ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B4 @ A2 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % psubset_imp_ex_mem
% 0.16/0.45 thf(fact_2590_int__less__induct,axiom,
% 0.16/0.45 ! [I: int,K: int,P: int > $o] :
% 0.16/0.45 ( ( ord_less_int @ I @ K )
% 0.16/0.45 => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 0.16/0.45 => ( ! [I3: int] :
% 0.16/0.45 ( ( ord_less_int @ I3 @ K )
% 0.16/0.45 => ( ( P @ I3 )
% 0.16/0.45 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 0.16/0.45 => ( P @ I ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % int_less_induct
% 0.16/0.45 thf(fact_2591_real__arch__pow,axiom,
% 0.16/0.45 ! [X: real,Y4: real] :
% 0.16/0.45 ( ( ord_less_real @ one_one_real @ X )
% 0.16/0.45 => ? [N2: nat] : ( ord_less_real @ Y4 @ ( power_power_real @ X @ N2 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % real_arch_pow
% 0.16/0.45 thf(fact_2592_real__arch__pow__inv,axiom,
% 0.16/0.45 ! [Y4: real,X: real] :
% 0.16/0.45 ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.16/0.45 => ( ( ord_less_real @ X @ one_one_real )
% 0.16/0.45 => ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X @ N2 ) @ Y4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % real_arch_pow_inv
% 0.16/0.45 thf(fact_2593_int__one__le__iff__zero__less,axiom,
% 0.16/0.45 ! [Z: int] :
% 0.16/0.45 ( ( ord_less_eq_int @ one_one_int @ Z )
% 0.16/0.45 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.16/0.45
% 0.16/0.45 % int_one_le_iff_zero_less
% 0.16/0.45 thf(fact_2594_pos__zmult__eq__1__iff,axiom,
% 0.16/0.45 ! [M: int,N: int] :
% 0.16/0.45 ( ( ord_less_int @ zero_zero_int @ M )
% 0.16/0.45 => ( ( ( times_times_int @ M @ N )
% 0.16/0.45 = one_one_int )
% 0.16/0.45 = ( ( M = one_one_int )
% 0.16/0.45 & ( N = one_one_int ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % pos_zmult_eq_1_iff
% 0.16/0.45 thf(fact_2595_int__div__less__self,axiom,
% 0.16/0.45 ! [X: int,K: int] :
% 0.16/0.45 ( ( ord_less_int @ zero_zero_int @ X )
% 0.16/0.45 => ( ( ord_less_int @ one_one_int @ K )
% 0.16/0.45 => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % int_div_less_self
% 0.16/0.45 thf(fact_2596_psubsetE,axiom,
% 0.16/0.45 ! [A2: set_int,B4: set_int] :
% 0.16/0.45 ( ( ord_less_set_int @ A2 @ B4 )
% 0.16/0.45 => ~ ( ( ord_less_eq_set_int @ A2 @ B4 )
% 0.16/0.45 => ( ord_less_eq_set_int @ B4 @ A2 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % psubsetE
% 0.16/0.45 thf(fact_2597_psubset__eq,axiom,
% 0.16/0.45 ( ord_less_set_int
% 0.16/0.45 = ( ^ [A5: set_int,B5: set_int] :
% 0.16/0.45 ( ( ord_less_eq_set_int @ A5 @ B5 )
% 0.16/0.45 & ( A5 != B5 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % psubset_eq
% 0.16/0.45 thf(fact_2598_psubset__imp__subset,axiom,
% 0.16/0.45 ! [A2: set_int,B4: set_int] :
% 0.16/0.45 ( ( ord_less_set_int @ A2 @ B4 )
% 0.16/0.45 => ( ord_less_eq_set_int @ A2 @ B4 ) ) ).
% 0.16/0.45
% 0.16/0.45 % psubset_imp_subset
% 0.16/0.45 thf(fact_2599_psubset__subset__trans,axiom,
% 0.16/0.45 ! [A2: set_int,B4: set_int,C2: set_int] :
% 0.16/0.45 ( ( ord_less_set_int @ A2 @ B4 )
% 0.16/0.45 => ( ( ord_less_eq_set_int @ B4 @ C2 )
% 0.16/0.45 => ( ord_less_set_int @ A2 @ C2 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % psubset_subset_trans
% 0.16/0.45 thf(fact_2600_subset__not__subset__eq,axiom,
% 0.16/0.45 ( ord_less_set_int
% 0.16/0.45 = ( ^ [A5: set_int,B5: set_int] :
% 0.16/0.45 ( ( ord_less_eq_set_int @ A5 @ B5 )
% 0.16/0.45 & ~ ( ord_less_eq_set_int @ B5 @ A5 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % subset_not_subset_eq
% 0.16/0.45 thf(fact_2601_subset__psubset__trans,axiom,
% 0.16/0.45 ! [A2: set_int,B4: set_int,C2: set_int] :
% 0.16/0.45 ( ( ord_less_eq_set_int @ A2 @ B4 )
% 0.16/0.45 => ( ( ord_less_set_int @ B4 @ C2 )
% 0.16/0.45 => ( ord_less_set_int @ A2 @ C2 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % subset_psubset_trans
% 0.16/0.45 thf(fact_2602_subset__iff__psubset__eq,axiom,
% 0.16/0.45 ( ord_less_eq_set_int
% 0.16/0.45 = ( ^ [A5: set_int,B5: set_int] :
% 0.16/0.45 ( ( ord_less_set_int @ A5 @ B5 )
% 0.16/0.45 | ( A5 = B5 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % subset_iff_psubset_eq
% 0.16/0.45 thf(fact_2603_less__int__code_I1_J,axiom,
% 0.16/0.45 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.16/0.45
% 0.16/0.45 % less_int_code(1)
% 0.16/0.45 thf(fact_2604_less__eq__real__def,axiom,
% 0.16/0.45 ( ord_less_eq_real
% 0.16/0.45 = ( ^ [X3: real,Y6: real] :
% 0.16/0.45 ( ( ord_less_real @ X3 @ Y6 )
% 0.16/0.45 | ( X3 = Y6 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % less_eq_real_def
% 0.16/0.45 thf(fact_2605_int__le__induct,axiom,
% 0.16/0.45 ! [I: int,K: int,P: int > $o] :
% 0.16/0.45 ( ( ord_less_eq_int @ I @ K )
% 0.16/0.45 => ( ( P @ K )
% 0.16/0.45 => ( ! [I3: int] :
% 0.16/0.45 ( ( ord_less_eq_int @ I3 @ K )
% 0.16/0.45 => ( ( P @ I3 )
% 0.16/0.45 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 0.16/0.45 => ( P @ I ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % int_le_induct
% 0.16/0.45 thf(fact_2606_enat__0__less__mult__iff,axiom,
% 0.16/0.45 ! [M: extended_enat,N: extended_enat] :
% 0.16/0.45 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
% 0.16/0.45 = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 0.16/0.45 & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % enat_0_less_mult_iff
% 0.16/0.45 thf(fact_2607_not__iless0,axiom,
% 0.16/0.45 ! [N: extended_enat] :
% 0.16/0.45 ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% 0.16/0.45
% 0.16/0.45 % not_iless0
% 0.16/0.45 thf(fact_2608_zero__one__enat__neq_I1_J,axiom,
% 0.16/0.45 zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% 0.16/0.45
% 0.16/0.45 % zero_one_enat_neq(1)
% 0.16/0.45 thf(fact_2609_realpow__pos__nth2,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.45 => ? [R3: real] :
% 0.16/0.45 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.16/0.45 & ( ( power_power_real @ R3 @ ( suc @ N ) )
% 0.16/0.45 = A ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % realpow_pos_nth2
% 0.16/0.45 thf(fact_2610_nat__int__comparison_I2_J,axiom,
% 0.16/0.45 ( ord_less_nat
% 0.16/0.45 = ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % nat_int_comparison(2)
% 0.16/0.45 thf(fact_2611_zmult__zless__mono2,axiom,
% 0.16/0.45 ! [I: int,J: int,K: int] :
% 0.16/0.45 ( ( ord_less_int @ I @ J )
% 0.16/0.45 => ( ( ord_less_int @ zero_zero_int @ K )
% 0.16/0.45 => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zmult_zless_mono2
% 0.16/0.45 thf(fact_2612_plusinfinity,axiom,
% 0.16/0.45 ! [D: int,P4: int > $o,P: int > $o] :
% 0.16/0.45 ( ( ord_less_int @ zero_zero_int @ D )
% 0.16/0.45 => ( ! [X4: int,K2: int] :
% 0.16/0.45 ( ( P4 @ X4 )
% 0.16/0.45 = ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 0.16/0.45 => ( ? [Z3: int] :
% 0.16/0.45 ! [X4: int] :
% 0.16/0.45 ( ( ord_less_int @ Z3 @ X4 )
% 0.16/0.45 => ( ( P @ X4 )
% 0.16/0.45 = ( P4 @ X4 ) ) )
% 0.16/0.45 => ( ? [X_12: int] : ( P4 @ X_12 )
% 0.16/0.45 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % plusinfinity
% 0.16/0.45 thf(fact_2613_minusinfinity,axiom,
% 0.16/0.45 ! [D: int,P1: int > $o,P: int > $o] :
% 0.16/0.45 ( ( ord_less_int @ zero_zero_int @ D )
% 0.16/0.45 => ( ! [X4: int,K2: int] :
% 0.16/0.45 ( ( P1 @ X4 )
% 0.16/0.45 = ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 0.16/0.45 => ( ? [Z3: int] :
% 0.16/0.45 ! [X4: int] :
% 0.16/0.45 ( ( ord_less_int @ X4 @ Z3 )
% 0.16/0.45 => ( ( P @ X4 )
% 0.16/0.45 = ( P1 @ X4 ) ) )
% 0.16/0.45 => ( ? [X_12: int] : ( P1 @ X_12 )
% 0.16/0.45 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % minusinfinity
% 0.16/0.45 thf(fact_2614_pos__imp__zdiv__neg__iff,axiom,
% 0.16/0.45 ! [B: int,A: int] :
% 0.16/0.45 ( ( ord_less_int @ zero_zero_int @ B )
% 0.16/0.45 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 0.16/0.45 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % pos_imp_zdiv_neg_iff
% 0.16/0.45 thf(fact_2615_neg__imp__zdiv__neg__iff,axiom,
% 0.16/0.45 ! [B: int,A: int] :
% 0.16/0.45 ( ( ord_less_int @ B @ zero_zero_int )
% 0.16/0.45 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 0.16/0.45 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % neg_imp_zdiv_neg_iff
% 0.16/0.45 thf(fact_2616_div__neg__pos__less0,axiom,
% 0.16/0.45 ! [A: int,B: int] :
% 0.16/0.45 ( ( ord_less_int @ A @ zero_zero_int )
% 0.16/0.45 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.16/0.45 => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % div_neg_pos_less0
% 0.16/0.45 thf(fact_2617_zdvd__not__zless,axiom,
% 0.16/0.45 ! [M: int,N: int] :
% 0.16/0.45 ( ( ord_less_int @ zero_zero_int @ M )
% 0.16/0.45 => ( ( ord_less_int @ M @ N )
% 0.16/0.45 => ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zdvd_not_zless
% 0.16/0.45 thf(fact_2618_int__ops_I2_J,axiom,
% 0.16/0.45 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 0.16/0.45 = one_one_int ) ).
% 0.16/0.45
% 0.16/0.45 % int_ops(2)
% 0.16/0.45 thf(fact_2619_realpow__pos__nth__unique,axiom,
% 0.16/0.45 ! [N: nat,A: real] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.45 => ? [X4: real] :
% 0.16/0.45 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.16/0.45 & ( ( power_power_real @ X4 @ N )
% 0.16/0.45 = A )
% 0.16/0.45 & ! [Y3: real] :
% 0.16/0.45 ( ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.16/0.45 & ( ( power_power_real @ Y3 @ N )
% 0.16/0.45 = A ) )
% 0.16/0.45 => ( Y3 = X4 ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % realpow_pos_nth_unique
% 0.16/0.45 thf(fact_2620_realpow__pos__nth,axiom,
% 0.16/0.45 ! [N: nat,A: real] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.45 => ? [R3: real] :
% 0.16/0.45 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.16/0.45 & ( ( power_power_real @ R3 @ N )
% 0.16/0.45 = A ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % realpow_pos_nth
% 0.16/0.45 thf(fact_2621_int__ops_I6_J,axiom,
% 0.16/0.45 ! [A: nat,B: nat] :
% 0.16/0.45 ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 0.16/0.45 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 0.16/0.45 = zero_zero_int ) )
% 0.16/0.45 & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 0.16/0.45 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 0.16/0.45 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % int_ops(6)
% 0.16/0.45 thf(fact_2622_decr__mult__lemma,axiom,
% 0.16/0.45 ! [D: int,P: int > $o,K: int] :
% 0.16/0.45 ( ( ord_less_int @ zero_zero_int @ D )
% 0.16/0.45 => ( ! [X4: int] :
% 0.16/0.45 ( ( P @ X4 )
% 0.16/0.45 => ( P @ ( minus_minus_int @ X4 @ D ) ) )
% 0.16/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.16/0.45 => ! [X5: int] :
% 0.16/0.45 ( ( P @ X5 )
% 0.16/0.45 => ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % decr_mult_lemma
% 0.16/0.45 thf(fact_2623_reals__Archimedean3,axiom,
% 0.16/0.45 ! [X: real] :
% 0.16/0.45 ( ( ord_less_real @ zero_zero_real @ X )
% 0.16/0.45 => ! [Y3: real] :
% 0.16/0.45 ? [N2: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % reals_Archimedean3
% 0.16/0.45 thf(fact_2624_zdiv__mono1,axiom,
% 0.16/0.45 ! [A: int,A6: int,B: int] :
% 0.16/0.45 ( ( ord_less_eq_int @ A @ A6 )
% 0.16/0.45 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.16/0.45 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A6 @ B ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zdiv_mono1
% 0.16/0.45 thf(fact_2625_zdiv__mono2,axiom,
% 0.16/0.45 ! [A: int,B6: int,B: int] :
% 0.16/0.45 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.45 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.16/0.45 => ( ( ord_less_eq_int @ B6 @ B )
% 0.16/0.45 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B6 ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zdiv_mono2
% 0.16/0.45 thf(fact_2626_zdiv__eq__0__iff,axiom,
% 0.16/0.45 ! [I: int,K: int] :
% 0.16/0.45 ( ( ( divide_divide_int @ I @ K )
% 0.16/0.45 = zero_zero_int )
% 0.16/0.45 = ( ( K = zero_zero_int )
% 0.16/0.45 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.16/0.45 & ( ord_less_int @ I @ K ) )
% 0.16/0.45 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 0.16/0.45 & ( ord_less_int @ K @ I ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zdiv_eq_0_iff
% 0.16/0.45 thf(fact_2627_zdiv__mono1__neg,axiom,
% 0.16/0.45 ! [A: int,A6: int,B: int] :
% 0.16/0.45 ( ( ord_less_eq_int @ A @ A6 )
% 0.16/0.45 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.16/0.45 => ( ord_less_eq_int @ ( divide_divide_int @ A6 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zdiv_mono1_neg
% 0.16/0.45 thf(fact_2628_zdiv__mono2__neg,axiom,
% 0.16/0.45 ! [A: int,B6: int,B: int] :
% 0.16/0.45 ( ( ord_less_int @ A @ zero_zero_int )
% 0.16/0.45 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.16/0.45 => ( ( ord_less_eq_int @ B6 @ B )
% 0.16/0.45 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B6 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zdiv_mono2_neg
% 0.16/0.45 thf(fact_2629_div__int__pos__iff,axiom,
% 0.16/0.45 ! [K: int,L: int] :
% 0.16/0.45 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
% 0.16/0.45 = ( ( K = zero_zero_int )
% 0.16/0.45 | ( L = zero_zero_int )
% 0.16/0.45 | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.16/0.45 & ( ord_less_eq_int @ zero_zero_int @ L ) )
% 0.16/0.45 | ( ( ord_less_int @ K @ zero_zero_int )
% 0.16/0.45 & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % div_int_pos_iff
% 0.16/0.45 thf(fact_2630_div__positive__int,axiom,
% 0.16/0.45 ! [L: int,K: int] :
% 0.16/0.45 ( ( ord_less_eq_int @ L @ K )
% 0.16/0.45 => ( ( ord_less_int @ zero_zero_int @ L )
% 0.16/0.45 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % div_positive_int
% 0.16/0.45 thf(fact_2631_div__nonneg__neg__le0,axiom,
% 0.16/0.45 ! [A: int,B: int] :
% 0.16/0.45 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.45 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.16/0.45 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % div_nonneg_neg_le0
% 0.16/0.45 thf(fact_2632_div__nonpos__pos__le0,axiom,
% 0.16/0.45 ! [A: int,B: int] :
% 0.16/0.45 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.16/0.45 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.16/0.45 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % div_nonpos_pos_le0
% 0.16/0.45 thf(fact_2633_pos__imp__zdiv__pos__iff,axiom,
% 0.16/0.45 ! [K: int,I: int] :
% 0.16/0.45 ( ( ord_less_int @ zero_zero_int @ K )
% 0.16/0.45 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
% 0.16/0.45 = ( ord_less_eq_int @ K @ I ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % pos_imp_zdiv_pos_iff
% 0.16/0.45 thf(fact_2634_neg__imp__zdiv__nonneg__iff,axiom,
% 0.16/0.45 ! [B: int,A: int] :
% 0.16/0.45 ( ( ord_less_int @ B @ zero_zero_int )
% 0.16/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.16/0.45 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % neg_imp_zdiv_nonneg_iff
% 0.16/0.45 thf(fact_2635_pos__imp__zdiv__nonneg__iff,axiom,
% 0.16/0.45 ! [B: int,A: int] :
% 0.16/0.45 ( ( ord_less_int @ zero_zero_int @ B )
% 0.16/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.16/0.45 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % pos_imp_zdiv_nonneg_iff
% 0.16/0.45 thf(fact_2636_nonneg1__imp__zdiv__pos__iff,axiom,
% 0.16/0.45 ! [A: int,B: int] :
% 0.16/0.45 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.45 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.16/0.45 = ( ( ord_less_eq_int @ B @ A )
% 0.16/0.45 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % nonneg1_imp_zdiv_pos_iff
% 0.16/0.45 thf(fact_2637_zdvd__imp__le,axiom,
% 0.16/0.45 ! [Z: int,N: int] :
% 0.16/0.45 ( ( dvd_dvd_int @ Z @ N )
% 0.16/0.45 => ( ( ord_less_int @ zero_zero_int @ N )
% 0.16/0.45 => ( ord_less_eq_int @ Z @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zdvd_imp_le
% 0.16/0.45 thf(fact_2638_pos__int__cases,axiom,
% 0.16/0.45 ! [K: int] :
% 0.16/0.45 ( ( ord_less_int @ zero_zero_int @ K )
% 0.16/0.45 => ~ ! [N2: nat] :
% 0.16/0.45 ( ( K
% 0.16/0.45 = ( semiri1314217659103216013at_int @ N2 ) )
% 0.16/0.45 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % pos_int_cases
% 0.16/0.45 thf(fact_2639_zero__less__imp__eq__int,axiom,
% 0.16/0.45 ! [K: int] :
% 0.16/0.45 ( ( ord_less_int @ zero_zero_int @ K )
% 0.16/0.45 => ? [N2: nat] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.16/0.45 & ( K
% 0.16/0.45 = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_less_imp_eq_int
% 0.16/0.45 thf(fact_2640_zmult__zless__mono2__lemma,axiom,
% 0.16/0.45 ! [I: int,J: int,K: nat] :
% 0.16/0.45 ( ( ord_less_int @ I @ J )
% 0.16/0.45 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.16/0.45 => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zmult_zless_mono2_lemma
% 0.16/0.45 thf(fact_2641_two__realpow__ge__one,axiom,
% 0.16/0.45 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % two_realpow_ge_one
% 0.16/0.45 thf(fact_2642_real__of__nat__div3,axiom,
% 0.16/0.45 ! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) @ one_one_real ) ).
% 0.16/0.45
% 0.16/0.45 % real_of_nat_div3
% 0.16/0.45 thf(fact_2643_int__power__div__base,axiom,
% 0.16/0.45 ! [M: nat,K: int] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.16/0.45 => ( ( ord_less_int @ zero_zero_int @ K )
% 0.16/0.45 => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 0.16/0.45 = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % int_power_div_base
% 0.16/0.45 thf(fact_2644_power__not__zero,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( A != zero_zero_rat )
% 0.16/0.45 => ( ( power_power_rat @ A @ N )
% 0.16/0.45 != zero_zero_rat ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_not_zero
% 0.16/0.45 thf(fact_2645_power__not__zero,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( A != zero_zero_int )
% 0.16/0.45 => ( ( power_power_int @ A @ N )
% 0.16/0.45 != zero_zero_int ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_not_zero
% 0.16/0.45 thf(fact_2646_power__not__zero,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( A != zero_zero_real )
% 0.16/0.45 => ( ( power_power_real @ A @ N )
% 0.16/0.45 != zero_zero_real ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_not_zero
% 0.16/0.45 thf(fact_2647_power__not__zero,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( A != zero_zero_nat )
% 0.16/0.45 => ( ( power_power_nat @ A @ N )
% 0.16/0.45 != zero_zero_nat ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_not_zero
% 0.16/0.45 thf(fact_2648_power__not__zero,axiom,
% 0.16/0.45 ! [A: complex,N: nat] :
% 0.16/0.45 ( ( A != zero_zero_complex )
% 0.16/0.45 => ( ( power_power_complex @ A @ N )
% 0.16/0.45 != zero_zero_complex ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_not_zero
% 0.16/0.45 thf(fact_2649_power__commuting__commutes,axiom,
% 0.16/0.45 ! [X: complex,Y4: complex,N: nat] :
% 0.16/0.45 ( ( ( times_times_complex @ X @ Y4 )
% 0.16/0.45 = ( times_times_complex @ Y4 @ X ) )
% 0.16/0.45 => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ Y4 )
% 0.16/0.45 = ( times_times_complex @ Y4 @ ( power_power_complex @ X @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_commuting_commutes
% 0.16/0.45 thf(fact_2650_power__commuting__commutes,axiom,
% 0.16/0.45 ! [X: real,Y4: real,N: nat] :
% 0.16/0.45 ( ( ( times_times_real @ X @ Y4 )
% 0.16/0.45 = ( times_times_real @ Y4 @ X ) )
% 0.16/0.45 => ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y4 )
% 0.16/0.45 = ( times_times_real @ Y4 @ ( power_power_real @ X @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_commuting_commutes
% 0.16/0.45 thf(fact_2651_power__commuting__commutes,axiom,
% 0.16/0.45 ! [X: rat,Y4: rat,N: nat] :
% 0.16/0.45 ( ( ( times_times_rat @ X @ Y4 )
% 0.16/0.45 = ( times_times_rat @ Y4 @ X ) )
% 0.16/0.45 => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ Y4 )
% 0.16/0.45 = ( times_times_rat @ Y4 @ ( power_power_rat @ X @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_commuting_commutes
% 0.16/0.45 thf(fact_2652_power__commuting__commutes,axiom,
% 0.16/0.45 ! [X: nat,Y4: nat,N: nat] :
% 0.16/0.45 ( ( ( times_times_nat @ X @ Y4 )
% 0.16/0.45 = ( times_times_nat @ Y4 @ X ) )
% 0.16/0.45 => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y4 )
% 0.16/0.45 = ( times_times_nat @ Y4 @ ( power_power_nat @ X @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_commuting_commutes
% 0.16/0.45 thf(fact_2653_power__commuting__commutes,axiom,
% 0.16/0.45 ! [X: int,Y4: int,N: nat] :
% 0.16/0.45 ( ( ( times_times_int @ X @ Y4 )
% 0.16/0.45 = ( times_times_int @ Y4 @ X ) )
% 0.16/0.45 => ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y4 )
% 0.16/0.45 = ( times_times_int @ Y4 @ ( power_power_int @ X @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_commuting_commutes
% 0.16/0.45 thf(fact_2654_power__mult__distrib,axiom,
% 0.16/0.45 ! [A: complex,B: complex,N: nat] :
% 0.16/0.45 ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
% 0.16/0.45 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_mult_distrib
% 0.16/0.45 thf(fact_2655_power__mult__distrib,axiom,
% 0.16/0.45 ! [A: real,B: real,N: nat] :
% 0.16/0.45 ( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
% 0.16/0.45 = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_mult_distrib
% 0.16/0.45 thf(fact_2656_power__mult__distrib,axiom,
% 0.16/0.45 ! [A: rat,B: rat,N: nat] :
% 0.16/0.45 ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
% 0.16/0.45 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_mult_distrib
% 0.16/0.45 thf(fact_2657_power__mult__distrib,axiom,
% 0.16/0.45 ! [A: nat,B: nat,N: nat] :
% 0.16/0.45 ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
% 0.16/0.45 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_mult_distrib
% 0.16/0.45 thf(fact_2658_power__mult__distrib,axiom,
% 0.16/0.45 ! [A: int,B: int,N: nat] :
% 0.16/0.45 ( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
% 0.16/0.45 = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_mult_distrib
% 0.16/0.45 thf(fact_2659_power__commutes,axiom,
% 0.16/0.45 ! [A: complex,N: nat] :
% 0.16/0.45 ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
% 0.16/0.45 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_commutes
% 0.16/0.45 thf(fact_2660_power__commutes,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
% 0.16/0.45 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_commutes
% 0.16/0.45 thf(fact_2661_power__commutes,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
% 0.16/0.45 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_commutes
% 0.16/0.45 thf(fact_2662_power__commutes,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
% 0.16/0.45 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_commutes
% 0.16/0.45 thf(fact_2663_power__commutes,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
% 0.16/0.45 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_commutes
% 0.16/0.45 thf(fact_2664_power__divide,axiom,
% 0.16/0.45 ! [A: complex,B: complex,N: nat] :
% 0.16/0.45 ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N )
% 0.16/0.45 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_divide
% 0.16/0.45 thf(fact_2665_power__divide,axiom,
% 0.16/0.45 ! [A: real,B: real,N: nat] :
% 0.16/0.45 ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
% 0.16/0.45 = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_divide
% 0.16/0.45 thf(fact_2666_power__divide,axiom,
% 0.16/0.45 ! [A: rat,B: rat,N: nat] :
% 0.16/0.45 ( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N )
% 0.16/0.45 = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_divide
% 0.16/0.45 thf(fact_2667_dvd__power__same,axiom,
% 0.16/0.45 ! [X: code_integer,Y4: code_integer,N: nat] :
% 0.16/0.45 ( ( dvd_dvd_Code_integer @ X @ Y4 )
% 0.16/0.45 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y4 @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power_same
% 0.16/0.45 thf(fact_2668_dvd__power__same,axiom,
% 0.16/0.45 ! [X: int,Y4: int,N: nat] :
% 0.16/0.45 ( ( dvd_dvd_int @ X @ Y4 )
% 0.16/0.45 => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y4 @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power_same
% 0.16/0.45 thf(fact_2669_dvd__power__same,axiom,
% 0.16/0.45 ! [X: real,Y4: real,N: nat] :
% 0.16/0.45 ( ( dvd_dvd_real @ X @ Y4 )
% 0.16/0.45 => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y4 @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power_same
% 0.16/0.45 thf(fact_2670_dvd__power__same,axiom,
% 0.16/0.45 ! [X: nat,Y4: nat,N: nat] :
% 0.16/0.45 ( ( dvd_dvd_nat @ X @ Y4 )
% 0.16/0.45 => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y4 @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power_same
% 0.16/0.45 thf(fact_2671_dvd__power__same,axiom,
% 0.16/0.45 ! [X: complex,Y4: complex,N: nat] :
% 0.16/0.45 ( ( dvd_dvd_complex @ X @ Y4 )
% 0.16/0.45 => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y4 @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power_same
% 0.16/0.45 thf(fact_2672_nat__power__less__imp__less,axiom,
% 0.16/0.45 ! [I: nat,M: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ I )
% 0.16/0.45 => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 0.16/0.45 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % nat_power_less_imp_less
% 0.16/0.45 thf(fact_2673_power__mult,axiom,
% 0.16/0.45 ! [A: int,M: nat,N: nat] :
% 0.16/0.45 ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
% 0.16/0.45 = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_mult
% 0.16/0.45 thf(fact_2674_power__mult,axiom,
% 0.16/0.45 ! [A: real,M: nat,N: nat] :
% 0.16/0.45 ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
% 0.16/0.45 = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_mult
% 0.16/0.45 thf(fact_2675_power__mult,axiom,
% 0.16/0.45 ! [A: nat,M: nat,N: nat] :
% 0.16/0.45 ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
% 0.16/0.45 = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_mult
% 0.16/0.45 thf(fact_2676_power__mult,axiom,
% 0.16/0.45 ! [A: complex,M: nat,N: nat] :
% 0.16/0.45 ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
% 0.16/0.45 = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_mult
% 0.16/0.45 thf(fact_2677_power__mono,axiom,
% 0.16/0.45 ! [A: rat,B: rat,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_rat @ A @ B )
% 0.16/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.45 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_mono
% 0.16/0.45 thf(fact_2678_power__mono,axiom,
% 0.16/0.45 ! [A: nat,B: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.16/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.45 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_mono
% 0.16/0.45 thf(fact_2679_power__mono,axiom,
% 0.16/0.45 ! [A: int,B: int,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_int @ A @ B )
% 0.16/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.45 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_mono
% 0.16/0.45 thf(fact_2680_power__mono,axiom,
% 0.16/0.45 ! [A: real,B: real,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.16/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.45 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_mono
% 0.16/0.45 thf(fact_2681_zero__le__power,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.45 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_le_power
% 0.16/0.45 thf(fact_2682_zero__le__power,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.45 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_le_power
% 0.16/0.45 thf(fact_2683_zero__le__power,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.45 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_le_power
% 0.16/0.45 thf(fact_2684_zero__le__power,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.45 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_le_power
% 0.16/0.45 thf(fact_2685_zero__less__power,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.45 => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_less_power
% 0.16/0.45 thf(fact_2686_zero__less__power,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.45 => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_less_power
% 0.16/0.45 thf(fact_2687_zero__less__power,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.16/0.45 => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_less_power
% 0.16/0.45 thf(fact_2688_zero__less__power,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( ord_less_int @ zero_zero_int @ A )
% 0.16/0.45 => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_less_power
% 0.16/0.45 thf(fact_2689_one__le__power,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.16/0.45 => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % one_le_power
% 0.16/0.45 thf(fact_2690_one__le__power,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.16/0.45 => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % one_le_power
% 0.16/0.45 thf(fact_2691_one__le__power,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_int @ one_one_int @ A )
% 0.16/0.45 => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % one_le_power
% 0.16/0.45 thf(fact_2692_one__le__power,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_real @ one_one_real @ A )
% 0.16/0.45 => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % one_le_power
% 0.16/0.45 thf(fact_2693_left__right__inverse__power,axiom,
% 0.16/0.45 ! [X: complex,Y4: complex,N: nat] :
% 0.16/0.45 ( ( ( times_times_complex @ X @ Y4 )
% 0.16/0.45 = one_one_complex )
% 0.16/0.45 => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y4 @ N ) )
% 0.16/0.45 = one_one_complex ) ) ).
% 0.16/0.45
% 0.16/0.45 % left_right_inverse_power
% 0.16/0.45 thf(fact_2694_left__right__inverse__power,axiom,
% 0.16/0.45 ! [X: real,Y4: real,N: nat] :
% 0.16/0.45 ( ( ( times_times_real @ X @ Y4 )
% 0.16/0.45 = one_one_real )
% 0.16/0.45 => ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y4 @ N ) )
% 0.16/0.45 = one_one_real ) ) ).
% 0.16/0.45
% 0.16/0.45 % left_right_inverse_power
% 0.16/0.45 thf(fact_2695_left__right__inverse__power,axiom,
% 0.16/0.45 ! [X: rat,Y4: rat,N: nat] :
% 0.16/0.45 ( ( ( times_times_rat @ X @ Y4 )
% 0.16/0.45 = one_one_rat )
% 0.16/0.45 => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y4 @ N ) )
% 0.16/0.45 = one_one_rat ) ) ).
% 0.16/0.45
% 0.16/0.45 % left_right_inverse_power
% 0.16/0.45 thf(fact_2696_left__right__inverse__power,axiom,
% 0.16/0.45 ! [X: nat,Y4: nat,N: nat] :
% 0.16/0.45 ( ( ( times_times_nat @ X @ Y4 )
% 0.16/0.45 = one_one_nat )
% 0.16/0.45 => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y4 @ N ) )
% 0.16/0.45 = one_one_nat ) ) ).
% 0.16/0.45
% 0.16/0.45 % left_right_inverse_power
% 0.16/0.45 thf(fact_2697_left__right__inverse__power,axiom,
% 0.16/0.45 ! [X: int,Y4: int,N: nat] :
% 0.16/0.45 ( ( ( times_times_int @ X @ Y4 )
% 0.16/0.45 = one_one_int )
% 0.16/0.45 => ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y4 @ N ) )
% 0.16/0.45 = one_one_int ) ) ).
% 0.16/0.45
% 0.16/0.45 % left_right_inverse_power
% 0.16/0.45 thf(fact_2698_power__Suc2,axiom,
% 0.16/0.45 ! [A: complex,N: nat] :
% 0.16/0.45 ( ( power_power_complex @ A @ ( suc @ N ) )
% 0.16/0.45 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc2
% 0.16/0.45 thf(fact_2699_power__Suc2,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( power_power_real @ A @ ( suc @ N ) )
% 0.16/0.45 = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc2
% 0.16/0.45 thf(fact_2700_power__Suc2,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.16/0.45 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc2
% 0.16/0.45 thf(fact_2701_power__Suc2,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.16/0.45 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc2
% 0.16/0.45 thf(fact_2702_power__Suc2,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( power_power_int @ A @ ( suc @ N ) )
% 0.16/0.45 = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc2
% 0.16/0.45 thf(fact_2703_power__Suc,axiom,
% 0.16/0.45 ! [A: complex,N: nat] :
% 0.16/0.45 ( ( power_power_complex @ A @ ( suc @ N ) )
% 0.16/0.45 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc
% 0.16/0.45 thf(fact_2704_power__Suc,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( power_power_real @ A @ ( suc @ N ) )
% 0.16/0.45 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc
% 0.16/0.45 thf(fact_2705_power__Suc,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.16/0.45 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc
% 0.16/0.45 thf(fact_2706_power__Suc,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.16/0.45 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc
% 0.16/0.45 thf(fact_2707_power__Suc,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( power_power_int @ A @ ( suc @ N ) )
% 0.16/0.45 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc
% 0.16/0.45 thf(fact_2708_power__one__over,axiom,
% 0.16/0.45 ! [A: complex,N: nat] :
% 0.16/0.45 ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
% 0.16/0.45 = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_one_over
% 0.16/0.45 thf(fact_2709_power__one__over,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
% 0.16/0.45 = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_one_over
% 0.16/0.45 thf(fact_2710_power__one__over,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
% 0.16/0.45 = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_one_over
% 0.16/0.45 thf(fact_2711_power__0,axiom,
% 0.16/0.45 ! [A: rat] :
% 0.16/0.45 ( ( power_power_rat @ A @ zero_zero_nat )
% 0.16/0.45 = one_one_rat ) ).
% 0.16/0.45
% 0.16/0.45 % power_0
% 0.16/0.45 thf(fact_2712_power__0,axiom,
% 0.16/0.45 ! [A: int] :
% 0.16/0.45 ( ( power_power_int @ A @ zero_zero_nat )
% 0.16/0.45 = one_one_int ) ).
% 0.16/0.45
% 0.16/0.45 % power_0
% 0.16/0.45 thf(fact_2713_power__0,axiom,
% 0.16/0.45 ! [A: real] :
% 0.16/0.45 ( ( power_power_real @ A @ zero_zero_nat )
% 0.16/0.45 = one_one_real ) ).
% 0.16/0.45
% 0.16/0.45 % power_0
% 0.16/0.45 thf(fact_2714_power__0,axiom,
% 0.16/0.45 ! [A: nat] :
% 0.16/0.45 ( ( power_power_nat @ A @ zero_zero_nat )
% 0.16/0.45 = one_one_nat ) ).
% 0.16/0.45
% 0.16/0.45 % power_0
% 0.16/0.45 thf(fact_2715_power__0,axiom,
% 0.16/0.45 ! [A: complex] :
% 0.16/0.45 ( ( power_power_complex @ A @ zero_zero_nat )
% 0.16/0.45 = one_one_complex ) ).
% 0.16/0.45
% 0.16/0.45 % power_0
% 0.16/0.45 thf(fact_2716_div__power,axiom,
% 0.16/0.45 ! [B: code_integer,A: code_integer,N: nat] :
% 0.16/0.45 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.16/0.45 => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N )
% 0.16/0.45 = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % div_power
% 0.16/0.45 thf(fact_2717_div__power,axiom,
% 0.16/0.45 ! [B: nat,A: nat,N: nat] :
% 0.16/0.45 ( ( dvd_dvd_nat @ B @ A )
% 0.16/0.45 => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N )
% 0.16/0.45 = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % div_power
% 0.16/0.45 thf(fact_2718_div__power,axiom,
% 0.16/0.45 ! [B: int,A: int,N: nat] :
% 0.16/0.45 ( ( dvd_dvd_int @ B @ A )
% 0.16/0.45 => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N )
% 0.16/0.45 = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % div_power
% 0.16/0.45 thf(fact_2719_power__gt__expt,axiom,
% 0.16/0.45 ! [N: nat,K: nat] :
% 0.16/0.45 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.16/0.45 => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_gt_expt
% 0.16/0.45 thf(fact_2720_nat__one__le__power,axiom,
% 0.16/0.45 ! [I: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 0.16/0.45 => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % nat_one_le_power
% 0.16/0.45 thf(fact_2721_dvd__power__le,axiom,
% 0.16/0.45 ! [X: code_integer,Y4: code_integer,N: nat,M: nat] :
% 0.16/0.45 ( ( dvd_dvd_Code_integer @ X @ Y4 )
% 0.16/0.45 => ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.45 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y4 @ M ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power_le
% 0.16/0.45 thf(fact_2722_dvd__power__le,axiom,
% 0.16/0.45 ! [X: int,Y4: int,N: nat,M: nat] :
% 0.16/0.45 ( ( dvd_dvd_int @ X @ Y4 )
% 0.16/0.45 => ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.45 => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y4 @ M ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power_le
% 0.16/0.45 thf(fact_2723_dvd__power__le,axiom,
% 0.16/0.45 ! [X: real,Y4: real,N: nat,M: nat] :
% 0.16/0.45 ( ( dvd_dvd_real @ X @ Y4 )
% 0.16/0.45 => ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.45 => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y4 @ M ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power_le
% 0.16/0.45 thf(fact_2724_dvd__power__le,axiom,
% 0.16/0.45 ! [X: nat,Y4: nat,N: nat,M: nat] :
% 0.16/0.45 ( ( dvd_dvd_nat @ X @ Y4 )
% 0.16/0.45 => ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.45 => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y4 @ M ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power_le
% 0.16/0.45 thf(fact_2725_dvd__power__le,axiom,
% 0.16/0.45 ! [X: complex,Y4: complex,N: nat,M: nat] :
% 0.16/0.45 ( ( dvd_dvd_complex @ X @ Y4 )
% 0.16/0.45 => ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.45 => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y4 @ M ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power_le
% 0.16/0.45 thf(fact_2726_power__le__dvd,axiom,
% 0.16/0.45 ! [A: code_integer,N: nat,B: code_integer,M: nat] :
% 0.16/0.45 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
% 0.16/0.45 => ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.45 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_le_dvd
% 0.16/0.45 thf(fact_2727_power__le__dvd,axiom,
% 0.16/0.45 ! [A: int,N: nat,B: int,M: nat] :
% 0.16/0.45 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
% 0.16/0.45 => ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.45 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_le_dvd
% 0.16/0.45 thf(fact_2728_power__le__dvd,axiom,
% 0.16/0.45 ! [A: real,N: nat,B: real,M: nat] :
% 0.16/0.45 ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
% 0.16/0.45 => ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.45 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_le_dvd
% 0.16/0.45 thf(fact_2729_power__le__dvd,axiom,
% 0.16/0.45 ! [A: nat,N: nat,B: nat,M: nat] :
% 0.16/0.45 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
% 0.16/0.45 => ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.45 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_le_dvd
% 0.16/0.45 thf(fact_2730_power__le__dvd,axiom,
% 0.16/0.45 ! [A: complex,N: nat,B: complex,M: nat] :
% 0.16/0.45 ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
% 0.16/0.45 => ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.45 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_le_dvd
% 0.16/0.45 thf(fact_2731_le__imp__power__dvd,axiom,
% 0.16/0.45 ! [M: nat,N: nat,A: code_integer] :
% 0.16/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.45 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % le_imp_power_dvd
% 0.16/0.45 thf(fact_2732_le__imp__power__dvd,axiom,
% 0.16/0.45 ! [M: nat,N: nat,A: int] :
% 0.16/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.45 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % le_imp_power_dvd
% 0.16/0.45 thf(fact_2733_le__imp__power__dvd,axiom,
% 0.16/0.45 ! [M: nat,N: nat,A: real] :
% 0.16/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.45 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % le_imp_power_dvd
% 0.16/0.45 thf(fact_2734_le__imp__power__dvd,axiom,
% 0.16/0.45 ! [M: nat,N: nat,A: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.45 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % le_imp_power_dvd
% 0.16/0.45 thf(fact_2735_le__imp__power__dvd,axiom,
% 0.16/0.45 ! [M: nat,N: nat,A: complex] :
% 0.16/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.16/0.45 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % le_imp_power_dvd
% 0.16/0.45 thf(fact_2736_power__less__imp__less__base,axiom,
% 0.16/0.45 ! [A: rat,N: nat,B: rat] :
% 0.16/0.45 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 0.16/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.16/0.45 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_less_imp_less_base
% 0.16/0.45 thf(fact_2737_power__less__imp__less__base,axiom,
% 0.16/0.45 ! [A: nat,N: nat,B: nat] :
% 0.16/0.45 ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.16/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.16/0.45 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_less_imp_less_base
% 0.16/0.45 thf(fact_2738_power__less__imp__less__base,axiom,
% 0.16/0.45 ! [A: int,N: nat,B: int] :
% 0.16/0.45 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.16/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.16/0.45 => ( ord_less_int @ A @ B ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_less_imp_less_base
% 0.16/0.45 thf(fact_2739_power__less__imp__less__base,axiom,
% 0.16/0.45 ! [A: real,N: nat,B: real] :
% 0.16/0.45 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 0.16/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.16/0.45 => ( ord_less_real @ A @ B ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_less_imp_less_base
% 0.16/0.45 thf(fact_2740_power__le__one,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.45 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.16/0.45 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_le_one
% 0.16/0.45 thf(fact_2741_power__le__one,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.45 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.16/0.45 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_le_one
% 0.16/0.45 thf(fact_2742_power__le__one,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.45 => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.16/0.45 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_le_one
% 0.16/0.45 thf(fact_2743_power__le__one,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.45 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.16/0.45 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_le_one
% 0.16/0.45 thf(fact_2744_power__le__imp__le__base,axiom,
% 0.16/0.45 ! [A: rat,N: nat,B: rat] :
% 0.16/0.45 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
% 0.16/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.16/0.45 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_le_imp_le_base
% 0.16/0.45 thf(fact_2745_power__le__imp__le__base,axiom,
% 0.16/0.45 ! [A: nat,N: nat,B: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
% 0.16/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.16/0.45 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_le_imp_le_base
% 0.16/0.45 thf(fact_2746_power__le__imp__le__base,axiom,
% 0.16/0.45 ! [A: int,N: nat,B: int] :
% 0.16/0.45 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
% 0.16/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.16/0.45 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_le_imp_le_base
% 0.16/0.45 thf(fact_2747_power__le__imp__le__base,axiom,
% 0.16/0.45 ! [A: real,N: nat,B: real] :
% 0.16/0.45 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
% 0.16/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.16/0.45 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_le_imp_le_base
% 0.16/0.45 thf(fact_2748_power__inject__base,axiom,
% 0.16/0.45 ! [A: rat,N: nat,B: rat] :
% 0.16/0.45 ( ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.16/0.45 = ( power_power_rat @ B @ ( suc @ N ) ) )
% 0.16/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.16/0.45 => ( A = B ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_inject_base
% 0.16/0.45 thf(fact_2749_power__inject__base,axiom,
% 0.16/0.45 ! [A: nat,N: nat,B: nat] :
% 0.16/0.45 ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.16/0.45 = ( power_power_nat @ B @ ( suc @ N ) ) )
% 0.16/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.16/0.45 => ( A = B ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_inject_base
% 0.16/0.45 thf(fact_2750_power__inject__base,axiom,
% 0.16/0.45 ! [A: int,N: nat,B: int] :
% 0.16/0.45 ( ( ( power_power_int @ A @ ( suc @ N ) )
% 0.16/0.45 = ( power_power_int @ B @ ( suc @ N ) ) )
% 0.16/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.16/0.45 => ( A = B ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_inject_base
% 0.16/0.45 thf(fact_2751_power__inject__base,axiom,
% 0.16/0.45 ! [A: real,N: nat,B: real] :
% 0.16/0.45 ( ( ( power_power_real @ A @ ( suc @ N ) )
% 0.16/0.45 = ( power_power_real @ B @ ( suc @ N ) ) )
% 0.16/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.16/0.45 => ( A = B ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_inject_base
% 0.16/0.45 thf(fact_2752_power__less__power__Suc,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( ord_less_real @ one_one_real @ A )
% 0.16/0.45 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_less_power_Suc
% 0.16/0.45 thf(fact_2753_power__less__power__Suc,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( ord_less_rat @ one_one_rat @ A )
% 0.16/0.45 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_less_power_Suc
% 0.16/0.45 thf(fact_2754_power__less__power__Suc,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_nat @ one_one_nat @ A )
% 0.16/0.45 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_less_power_Suc
% 0.16/0.45 thf(fact_2755_power__less__power__Suc,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( ord_less_int @ one_one_int @ A )
% 0.16/0.45 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_less_power_Suc
% 0.16/0.45 thf(fact_2756_power__gt1__lemma,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( ord_less_real @ one_one_real @ A )
% 0.16/0.45 => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_gt1_lemma
% 0.16/0.45 thf(fact_2757_power__gt1__lemma,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( ord_less_rat @ one_one_rat @ A )
% 0.16/0.45 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_gt1_lemma
% 0.16/0.45 thf(fact_2758_power__gt1__lemma,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_nat @ one_one_nat @ A )
% 0.16/0.45 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_gt1_lemma
% 0.16/0.45 thf(fact_2759_power__gt1__lemma,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( ord_less_int @ one_one_int @ A )
% 0.16/0.45 => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_gt1_lemma
% 0.16/0.45 thf(fact_2760_power__0__left,axiom,
% 0.16/0.45 ! [N: nat] :
% 0.16/0.45 ( ( ( N = zero_zero_nat )
% 0.16/0.45 => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.16/0.45 = one_one_rat ) )
% 0.16/0.45 & ( ( N != zero_zero_nat )
% 0.16/0.45 => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.16/0.45 = zero_zero_rat ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_0_left
% 0.16/0.45 thf(fact_2761_power__0__left,axiom,
% 0.16/0.45 ! [N: nat] :
% 0.16/0.45 ( ( ( N = zero_zero_nat )
% 0.16/0.45 => ( ( power_power_int @ zero_zero_int @ N )
% 0.16/0.45 = one_one_int ) )
% 0.16/0.45 & ( ( N != zero_zero_nat )
% 0.16/0.45 => ( ( power_power_int @ zero_zero_int @ N )
% 0.16/0.45 = zero_zero_int ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_0_left
% 0.16/0.45 thf(fact_2762_power__0__left,axiom,
% 0.16/0.45 ! [N: nat] :
% 0.16/0.45 ( ( ( N = zero_zero_nat )
% 0.16/0.45 => ( ( power_power_real @ zero_zero_real @ N )
% 0.16/0.45 = one_one_real ) )
% 0.16/0.45 & ( ( N != zero_zero_nat )
% 0.16/0.45 => ( ( power_power_real @ zero_zero_real @ N )
% 0.16/0.45 = zero_zero_real ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_0_left
% 0.16/0.45 thf(fact_2763_power__0__left,axiom,
% 0.16/0.45 ! [N: nat] :
% 0.16/0.45 ( ( ( N = zero_zero_nat )
% 0.16/0.45 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.16/0.45 = one_one_nat ) )
% 0.16/0.45 & ( ( N != zero_zero_nat )
% 0.16/0.45 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.16/0.45 = zero_zero_nat ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_0_left
% 0.16/0.45 thf(fact_2764_power__0__left,axiom,
% 0.16/0.45 ! [N: nat] :
% 0.16/0.45 ( ( ( N = zero_zero_nat )
% 0.16/0.45 => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.16/0.45 = one_one_complex ) )
% 0.16/0.45 & ( ( N != zero_zero_nat )
% 0.16/0.45 => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.16/0.45 = zero_zero_complex ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_0_left
% 0.16/0.45 thf(fact_2765_power__gt1,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( ord_less_real @ one_one_real @ A )
% 0.16/0.45 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_gt1
% 0.16/0.45 thf(fact_2766_power__gt1,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( ord_less_rat @ one_one_rat @ A )
% 0.16/0.45 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_gt1
% 0.16/0.45 thf(fact_2767_power__gt1,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_nat @ one_one_nat @ A )
% 0.16/0.45 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_gt1
% 0.16/0.45 thf(fact_2768_power__gt1,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( ord_less_int @ one_one_int @ A )
% 0.16/0.45 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_gt1
% 0.16/0.45 thf(fact_2769_power__increasing,axiom,
% 0.16/0.45 ! [N: nat,N5: nat,A: rat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ N @ N5 )
% 0.16/0.45 => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.16/0.45 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_increasing
% 0.16/0.45 thf(fact_2770_power__increasing,axiom,
% 0.16/0.45 ! [N: nat,N5: nat,A: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ N @ N5 )
% 0.16/0.45 => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.16/0.45 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_increasing
% 0.16/0.45 thf(fact_2771_power__increasing,axiom,
% 0.16/0.45 ! [N: nat,N5: nat,A: int] :
% 0.16/0.45 ( ( ord_less_eq_nat @ N @ N5 )
% 0.16/0.45 => ( ( ord_less_eq_int @ one_one_int @ A )
% 0.16/0.45 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_increasing
% 0.16/0.45 thf(fact_2772_power__increasing,axiom,
% 0.16/0.45 ! [N: nat,N5: nat,A: real] :
% 0.16/0.45 ( ( ord_less_eq_nat @ N @ N5 )
% 0.16/0.45 => ( ( ord_less_eq_real @ one_one_real @ A )
% 0.16/0.45 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_increasing
% 0.16/0.45 thf(fact_2773_zero__power,axiom,
% 0.16/0.45 ! [N: nat] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.16/0.45 = zero_zero_rat ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_power
% 0.16/0.45 thf(fact_2774_zero__power,axiom,
% 0.16/0.45 ! [N: nat] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ( power_power_int @ zero_zero_int @ N )
% 0.16/0.45 = zero_zero_int ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_power
% 0.16/0.45 thf(fact_2775_zero__power,axiom,
% 0.16/0.45 ! [N: nat] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ( power_power_real @ zero_zero_real @ N )
% 0.16/0.45 = zero_zero_real ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_power
% 0.16/0.45 thf(fact_2776_zero__power,axiom,
% 0.16/0.45 ! [N: nat] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.16/0.45 = zero_zero_nat ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_power
% 0.16/0.45 thf(fact_2777_zero__power,axiom,
% 0.16/0.45 ! [N: nat] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.16/0.45 = zero_zero_complex ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_power
% 0.16/0.45 thf(fact_2778_less__exp,axiom,
% 0.16/0.45 ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % less_exp
% 0.16/0.45 thf(fact_2779_power2__nat__le__imp__le,axiom,
% 0.16/0.45 ! [M: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 0.16/0.45 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_nat_le_imp_le
% 0.16/0.45 thf(fact_2780_power2__nat__le__eq__le,axiom,
% 0.16/0.45 ! [M: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.45 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_nat_le_eq_le
% 0.16/0.45 thf(fact_2781_self__le__ge2__pow,axiom,
% 0.16/0.45 ! [K: nat,M: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.16/0.45 => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % self_le_ge2_pow
% 0.16/0.45 thf(fact_2782_is__unit__power__iff,axiom,
% 0.16/0.45 ! [A: code_integer,N: nat] :
% 0.16/0.45 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
% 0.16/0.45 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.16/0.45 | ( N = zero_zero_nat ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % is_unit_power_iff
% 0.16/0.45 thf(fact_2783_is__unit__power__iff,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
% 0.16/0.45 = ( ( dvd_dvd_int @ A @ one_one_int )
% 0.16/0.45 | ( N = zero_zero_nat ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % is_unit_power_iff
% 0.16/0.45 thf(fact_2784_is__unit__power__iff,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
% 0.16/0.45 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.16/0.45 | ( N = zero_zero_nat ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % is_unit_power_iff
% 0.16/0.45 thf(fact_2785_power__dvd__imp__le,axiom,
% 0.16/0.45 ! [I: nat,M: nat,N: nat] :
% 0.16/0.45 ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 0.16/0.45 => ( ( ord_less_nat @ one_one_nat @ I )
% 0.16/0.45 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_dvd_imp_le
% 0.16/0.45 thf(fact_2786_power__Suc__less,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.45 => ( ( ord_less_real @ A @ one_one_real )
% 0.16/0.45 => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc_less
% 0.16/0.45 thf(fact_2787_power__Suc__less,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.45 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.16/0.45 => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc_less
% 0.16/0.45 thf(fact_2788_power__Suc__less,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.16/0.45 => ( ( ord_less_nat @ A @ one_one_nat )
% 0.16/0.45 => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc_less
% 0.16/0.45 thf(fact_2789_power__Suc__less,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( ord_less_int @ zero_zero_int @ A )
% 0.16/0.45 => ( ( ord_less_int @ A @ one_one_int )
% 0.16/0.45 => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc_less
% 0.16/0.45 thf(fact_2790_power__Suc__le__self,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.45 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.16/0.45 => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc_le_self
% 0.16/0.45 thf(fact_2791_power__Suc__le__self,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.45 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.16/0.45 => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc_le_self
% 0.16/0.45 thf(fact_2792_power__Suc__le__self,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.45 => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.16/0.45 => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc_le_self
% 0.16/0.45 thf(fact_2793_power__Suc__le__self,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.45 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.16/0.45 => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc_le_self
% 0.16/0.45 thf(fact_2794_power__Suc__less__one,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.45 => ( ( ord_less_real @ A @ one_one_real )
% 0.16/0.45 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc_less_one
% 0.16/0.45 thf(fact_2795_power__Suc__less__one,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.45 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.16/0.45 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc_less_one
% 0.16/0.45 thf(fact_2796_power__Suc__less__one,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.16/0.45 => ( ( ord_less_nat @ A @ one_one_nat )
% 0.16/0.45 => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc_less_one
% 0.16/0.45 thf(fact_2797_power__Suc__less__one,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( ord_less_int @ zero_zero_int @ A )
% 0.16/0.45 => ( ( ord_less_int @ A @ one_one_int )
% 0.16/0.45 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_Suc_less_one
% 0.16/0.45 thf(fact_2798_power__strict__decreasing,axiom,
% 0.16/0.45 ! [N: nat,N5: nat,A: real] :
% 0.16/0.45 ( ( ord_less_nat @ N @ N5 )
% 0.16/0.45 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.16/0.45 => ( ( ord_less_real @ A @ one_one_real )
% 0.16/0.45 => ( ord_less_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_strict_decreasing
% 0.16/0.45 thf(fact_2799_power__strict__decreasing,axiom,
% 0.16/0.45 ! [N: nat,N5: nat,A: rat] :
% 0.16/0.45 ( ( ord_less_nat @ N @ N5 )
% 0.16/0.45 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.16/0.45 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.16/0.45 => ( ord_less_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_strict_decreasing
% 0.16/0.45 thf(fact_2800_power__strict__decreasing,axiom,
% 0.16/0.45 ! [N: nat,N5: nat,A: nat] :
% 0.16/0.45 ( ( ord_less_nat @ N @ N5 )
% 0.16/0.45 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.16/0.45 => ( ( ord_less_nat @ A @ one_one_nat )
% 0.16/0.45 => ( ord_less_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_strict_decreasing
% 0.16/0.45 thf(fact_2801_power__strict__decreasing,axiom,
% 0.16/0.45 ! [N: nat,N5: nat,A: int] :
% 0.16/0.45 ( ( ord_less_nat @ N @ N5 )
% 0.16/0.45 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.16/0.45 => ( ( ord_less_int @ A @ one_one_int )
% 0.16/0.45 => ( ord_less_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_strict_decreasing
% 0.16/0.45 thf(fact_2802_power__decreasing,axiom,
% 0.16/0.45 ! [N: nat,N5: nat,A: rat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ N @ N5 )
% 0.16/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.45 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.16/0.45 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_decreasing
% 0.16/0.45 thf(fact_2803_power__decreasing,axiom,
% 0.16/0.45 ! [N: nat,N5: nat,A: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ N @ N5 )
% 0.16/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.45 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.16/0.45 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_decreasing
% 0.16/0.45 thf(fact_2804_power__decreasing,axiom,
% 0.16/0.45 ! [N: nat,N5: nat,A: int] :
% 0.16/0.45 ( ( ord_less_eq_nat @ N @ N5 )
% 0.16/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.45 => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.16/0.45 => ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_decreasing
% 0.16/0.45 thf(fact_2805_power__decreasing,axiom,
% 0.16/0.45 ! [N: nat,N5: nat,A: real] :
% 0.16/0.45 ( ( ord_less_eq_nat @ N @ N5 )
% 0.16/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.45 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.16/0.45 => ( ord_less_eq_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_decreasing
% 0.16/0.45 thf(fact_2806_power__le__imp__le__exp,axiom,
% 0.16/0.45 ! [A: rat,M: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_rat @ one_one_rat @ A )
% 0.16/0.45 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 0.16/0.45 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_le_imp_le_exp
% 0.16/0.45 thf(fact_2807_power__le__imp__le__exp,axiom,
% 0.16/0.45 ! [A: nat,M: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_nat @ one_one_nat @ A )
% 0.16/0.45 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.16/0.45 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_le_imp_le_exp
% 0.16/0.45 thf(fact_2808_power__le__imp__le__exp,axiom,
% 0.16/0.45 ! [A: int,M: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_int @ one_one_int @ A )
% 0.16/0.45 => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.16/0.45 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_le_imp_le_exp
% 0.16/0.45 thf(fact_2809_power__le__imp__le__exp,axiom,
% 0.16/0.45 ! [A: real,M: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_real @ one_one_real @ A )
% 0.16/0.45 => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 0.16/0.45 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_le_imp_le_exp
% 0.16/0.45 thf(fact_2810_zero__power2,axiom,
% 0.16/0.45 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = zero_zero_rat ) ).
% 0.16/0.45
% 0.16/0.45 % zero_power2
% 0.16/0.45 thf(fact_2811_zero__power2,axiom,
% 0.16/0.45 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = zero_zero_int ) ).
% 0.16/0.45
% 0.16/0.45 % zero_power2
% 0.16/0.45 thf(fact_2812_zero__power2,axiom,
% 0.16/0.45 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = zero_zero_real ) ).
% 0.16/0.45
% 0.16/0.45 % zero_power2
% 0.16/0.45 thf(fact_2813_zero__power2,axiom,
% 0.16/0.45 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = zero_zero_nat ) ).
% 0.16/0.45
% 0.16/0.45 % zero_power2
% 0.16/0.45 thf(fact_2814_zero__power2,axiom,
% 0.16/0.45 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = zero_zero_complex ) ).
% 0.16/0.45
% 0.16/0.45 % zero_power2
% 0.16/0.45 thf(fact_2815_power__eq__iff__eq__base,axiom,
% 0.16/0.45 ! [N: nat,A: rat,B: rat] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.16/0.45 => ( ( ( power_power_rat @ A @ N )
% 0.16/0.45 = ( power_power_rat @ B @ N ) )
% 0.16/0.45 = ( A = B ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_eq_iff_eq_base
% 0.16/0.45 thf(fact_2816_power__eq__iff__eq__base,axiom,
% 0.16/0.45 ! [N: nat,A: nat,B: nat] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.16/0.45 => ( ( ( power_power_nat @ A @ N )
% 0.16/0.45 = ( power_power_nat @ B @ N ) )
% 0.16/0.45 = ( A = B ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_eq_iff_eq_base
% 0.16/0.45 thf(fact_2817_power__eq__iff__eq__base,axiom,
% 0.16/0.45 ! [N: nat,A: int,B: int] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.16/0.45 => ( ( ( power_power_int @ A @ N )
% 0.16/0.45 = ( power_power_int @ B @ N ) )
% 0.16/0.45 = ( A = B ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_eq_iff_eq_base
% 0.16/0.45 thf(fact_2818_power__eq__iff__eq__base,axiom,
% 0.16/0.45 ! [N: nat,A: real,B: real] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.16/0.45 => ( ( ( power_power_real @ A @ N )
% 0.16/0.45 = ( power_power_real @ B @ N ) )
% 0.16/0.45 = ( A = B ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_eq_iff_eq_base
% 0.16/0.45 thf(fact_2819_power__eq__imp__eq__base,axiom,
% 0.16/0.45 ! [A: rat,N: nat,B: rat] :
% 0.16/0.45 ( ( ( power_power_rat @ A @ N )
% 0.16/0.45 = ( power_power_rat @ B @ N ) )
% 0.16/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.16/0.45 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( A = B ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_eq_imp_eq_base
% 0.16/0.45 thf(fact_2820_power__eq__imp__eq__base,axiom,
% 0.16/0.45 ! [A: nat,N: nat,B: nat] :
% 0.16/0.45 ( ( ( power_power_nat @ A @ N )
% 0.16/0.45 = ( power_power_nat @ B @ N ) )
% 0.16/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.16/0.45 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( A = B ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_eq_imp_eq_base
% 0.16/0.45 thf(fact_2821_power__eq__imp__eq__base,axiom,
% 0.16/0.45 ! [A: int,N: nat,B: int] :
% 0.16/0.45 ( ( ( power_power_int @ A @ N )
% 0.16/0.45 = ( power_power_int @ B @ N ) )
% 0.16/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.16/0.45 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( A = B ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_eq_imp_eq_base
% 0.16/0.45 thf(fact_2822_power__eq__imp__eq__base,axiom,
% 0.16/0.45 ! [A: real,N: nat,B: real] :
% 0.16/0.45 ( ( ( power_power_real @ A @ N )
% 0.16/0.45 = ( power_power_real @ B @ N ) )
% 0.16/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.16/0.45 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( A = B ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_eq_imp_eq_base
% 0.16/0.45 thf(fact_2823_power2__eq__square,axiom,
% 0.16/0.45 ! [A: complex] :
% 0.16/0.45 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = ( times_times_complex @ A @ A ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_eq_square
% 0.16/0.45 thf(fact_2824_power2__eq__square,axiom,
% 0.16/0.45 ! [A: real] :
% 0.16/0.45 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = ( times_times_real @ A @ A ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_eq_square
% 0.16/0.45 thf(fact_2825_power2__eq__square,axiom,
% 0.16/0.45 ! [A: rat] :
% 0.16/0.45 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = ( times_times_rat @ A @ A ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_eq_square
% 0.16/0.45 thf(fact_2826_power2__eq__square,axiom,
% 0.16/0.45 ! [A: nat] :
% 0.16/0.45 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = ( times_times_nat @ A @ A ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_eq_square
% 0.16/0.45 thf(fact_2827_power2__eq__square,axiom,
% 0.16/0.45 ! [A: int] :
% 0.16/0.45 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = ( times_times_int @ A @ A ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_eq_square
% 0.16/0.45 thf(fact_2828_power4__eq__xxxx,axiom,
% 0.16/0.45 ! [X: complex] :
% 0.16/0.45 ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.16/0.45 = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X @ X ) @ X ) @ X ) ) ).
% 0.16/0.45
% 0.16/0.45 % power4_eq_xxxx
% 0.16/0.45 thf(fact_2829_power4__eq__xxxx,axiom,
% 0.16/0.45 ! [X: real] :
% 0.16/0.45 ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.16/0.45 = ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).
% 0.16/0.45
% 0.16/0.45 % power4_eq_xxxx
% 0.16/0.45 thf(fact_2830_power4__eq__xxxx,axiom,
% 0.16/0.45 ! [X: rat] :
% 0.16/0.45 ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.16/0.45 = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).
% 0.16/0.45
% 0.16/0.45 % power4_eq_xxxx
% 0.16/0.45 thf(fact_2831_power4__eq__xxxx,axiom,
% 0.16/0.45 ! [X: nat] :
% 0.16/0.45 ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.16/0.45 = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% 0.16/0.45
% 0.16/0.45 % power4_eq_xxxx
% 0.16/0.45 thf(fact_2832_power4__eq__xxxx,axiom,
% 0.16/0.45 ! [X: int] :
% 0.16/0.45 ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.16/0.45 = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% 0.16/0.45
% 0.16/0.45 % power4_eq_xxxx
% 0.16/0.45 thf(fact_2833_one__power2,axiom,
% 0.16/0.45 ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = one_one_rat ) ).
% 0.16/0.45
% 0.16/0.45 % one_power2
% 0.16/0.45 thf(fact_2834_one__power2,axiom,
% 0.16/0.45 ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = one_one_int ) ).
% 0.16/0.45
% 0.16/0.45 % one_power2
% 0.16/0.45 thf(fact_2835_one__power2,axiom,
% 0.16/0.45 ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = one_one_real ) ).
% 0.16/0.45
% 0.16/0.45 % one_power2
% 0.16/0.45 thf(fact_2836_one__power2,axiom,
% 0.16/0.45 ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = one_one_nat ) ).
% 0.16/0.45
% 0.16/0.45 % one_power2
% 0.16/0.45 thf(fact_2837_one__power2,axiom,
% 0.16/0.45 ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = one_one_complex ) ).
% 0.16/0.45
% 0.16/0.45 % one_power2
% 0.16/0.45 thf(fact_2838_self__le__power,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.16/0.45 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % self_le_power
% 0.16/0.45 thf(fact_2839_self__le__power,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.16/0.45 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % self_le_power
% 0.16/0.45 thf(fact_2840_self__le__power,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_int @ one_one_int @ A )
% 0.16/0.45 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % self_le_power
% 0.16/0.45 thf(fact_2841_self__le__power,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_real @ one_one_real @ A )
% 0.16/0.45 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % self_le_power
% 0.16/0.45 thf(fact_2842_power2__commute,axiom,
% 0.16/0.45 ! [X: complex,Y4: complex] :
% 0.16/0.45 ( ( power_power_complex @ ( minus_minus_complex @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = ( power_power_complex @ ( minus_minus_complex @ Y4 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_commute
% 0.16/0.45 thf(fact_2843_power2__commute,axiom,
% 0.16/0.45 ! [X: real,Y4: real] :
% 0.16/0.45 ( ( power_power_real @ ( minus_minus_real @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = ( power_power_real @ ( minus_minus_real @ Y4 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_commute
% 0.16/0.45 thf(fact_2844_power2__commute,axiom,
% 0.16/0.45 ! [X: rat,Y4: rat] :
% 0.16/0.45 ( ( power_power_rat @ ( minus_minus_rat @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = ( power_power_rat @ ( minus_minus_rat @ Y4 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_commute
% 0.16/0.45 thf(fact_2845_power2__commute,axiom,
% 0.16/0.45 ! [X: int,Y4: int] :
% 0.16/0.45 ( ( power_power_int @ ( minus_minus_int @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = ( power_power_int @ ( minus_minus_int @ Y4 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_commute
% 0.16/0.45 thf(fact_2846_one__less__power,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( ord_less_real @ one_one_real @ A )
% 0.16/0.45 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % one_less_power
% 0.16/0.45 thf(fact_2847_one__less__power,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( ord_less_rat @ one_one_rat @ A )
% 0.16/0.45 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % one_less_power
% 0.16/0.45 thf(fact_2848_one__less__power,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_nat @ one_one_nat @ A )
% 0.16/0.45 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % one_less_power
% 0.16/0.45 thf(fact_2849_one__less__power,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( ord_less_int @ one_one_int @ A )
% 0.16/0.45 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % one_less_power
% 0.16/0.45 thf(fact_2850_dvd__power__iff,axiom,
% 0.16/0.45 ! [X: code_integer,M: nat,N: nat] :
% 0.16/0.45 ( ( X != zero_z3403309356797280102nteger )
% 0.16/0.45 => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N ) )
% 0.16/0.45 = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
% 0.16/0.45 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power_iff
% 0.16/0.45 thf(fact_2851_dvd__power__iff,axiom,
% 0.16/0.45 ! [X: int,M: nat,N: nat] :
% 0.16/0.45 ( ( X != zero_zero_int )
% 0.16/0.45 => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
% 0.16/0.45 = ( ( dvd_dvd_int @ X @ one_one_int )
% 0.16/0.45 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power_iff
% 0.16/0.45 thf(fact_2852_dvd__power__iff,axiom,
% 0.16/0.45 ! [X: nat,M: nat,N: nat] :
% 0.16/0.45 ( ( X != zero_zero_nat )
% 0.16/0.45 => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
% 0.16/0.45 = ( ( dvd_dvd_nat @ X @ one_one_nat )
% 0.16/0.45 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power_iff
% 0.16/0.45 thf(fact_2853_power3__eq__cube,axiom,
% 0.16/0.45 ! [A: complex] :
% 0.16/0.45 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.16/0.45 = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).
% 0.16/0.45
% 0.16/0.45 % power3_eq_cube
% 0.16/0.45 thf(fact_2854_power3__eq__cube,axiom,
% 0.16/0.45 ! [A: real] :
% 0.16/0.45 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.16/0.45 = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).
% 0.16/0.45
% 0.16/0.45 % power3_eq_cube
% 0.16/0.45 thf(fact_2855_power3__eq__cube,axiom,
% 0.16/0.45 ! [A: rat] :
% 0.16/0.45 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.16/0.45 = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% 0.16/0.45
% 0.16/0.45 % power3_eq_cube
% 0.16/0.45 thf(fact_2856_power3__eq__cube,axiom,
% 0.16/0.45 ! [A: nat] :
% 0.16/0.45 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.16/0.45 = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% 0.16/0.45
% 0.16/0.45 % power3_eq_cube
% 0.16/0.45 thf(fact_2857_power3__eq__cube,axiom,
% 0.16/0.45 ! [A: int] :
% 0.16/0.45 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.16/0.45 = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% 0.16/0.45
% 0.16/0.45 % power3_eq_cube
% 0.16/0.45 thf(fact_2858_dvd__power,axiom,
% 0.16/0.45 ! [N: nat,X: code_integer] :
% 0.16/0.45 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 | ( X = one_one_Code_integer ) )
% 0.16/0.45 => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power
% 0.16/0.45 thf(fact_2859_dvd__power,axiom,
% 0.16/0.45 ! [N: nat,X: rat] :
% 0.16/0.45 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 | ( X = one_one_rat ) )
% 0.16/0.45 => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power
% 0.16/0.45 thf(fact_2860_dvd__power,axiom,
% 0.16/0.45 ! [N: nat,X: int] :
% 0.16/0.45 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 | ( X = one_one_int ) )
% 0.16/0.45 => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power
% 0.16/0.45 thf(fact_2861_dvd__power,axiom,
% 0.16/0.45 ! [N: nat,X: real] :
% 0.16/0.45 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 | ( X = one_one_real ) )
% 0.16/0.45 => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power
% 0.16/0.45 thf(fact_2862_dvd__power,axiom,
% 0.16/0.45 ! [N: nat,X: nat] :
% 0.16/0.45 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 | ( X = one_one_nat ) )
% 0.16/0.45 => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power
% 0.16/0.45 thf(fact_2863_dvd__power,axiom,
% 0.16/0.45 ! [N: nat,X: complex] :
% 0.16/0.45 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 | ( X = one_one_complex ) )
% 0.16/0.45 => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power
% 0.16/0.45 thf(fact_2864_power__diff,axiom,
% 0.16/0.45 ! [A: complex,N: nat,M: nat] :
% 0.16/0.45 ( ( A != zero_zero_complex )
% 0.16/0.45 => ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.45 => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
% 0.16/0.45 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_diff
% 0.16/0.45 thf(fact_2865_power__diff,axiom,
% 0.16/0.45 ! [A: real,N: nat,M: nat] :
% 0.16/0.45 ( ( A != zero_zero_real )
% 0.16/0.45 => ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.45 => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
% 0.16/0.45 = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_diff
% 0.16/0.45 thf(fact_2866_power__diff,axiom,
% 0.16/0.45 ! [A: rat,N: nat,M: nat] :
% 0.16/0.45 ( ( A != zero_zero_rat )
% 0.16/0.45 => ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.45 => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
% 0.16/0.45 = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_diff
% 0.16/0.45 thf(fact_2867_power__diff,axiom,
% 0.16/0.45 ! [A: nat,N: nat,M: nat] :
% 0.16/0.45 ( ( A != zero_zero_nat )
% 0.16/0.45 => ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.45 => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
% 0.16/0.45 = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_diff
% 0.16/0.45 thf(fact_2868_power__diff,axiom,
% 0.16/0.45 ! [A: int,N: nat,M: nat] :
% 0.16/0.45 ( ( A != zero_zero_int )
% 0.16/0.45 => ( ( ord_less_eq_nat @ N @ M )
% 0.16/0.45 => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
% 0.16/0.45 = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_diff
% 0.16/0.45 thf(fact_2869_power__even__eq,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.16/0.45 = ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_even_eq
% 0.16/0.45 thf(fact_2870_power__even__eq,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.16/0.45 = ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_even_eq
% 0.16/0.45 thf(fact_2871_power__even__eq,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.16/0.45 = ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_even_eq
% 0.16/0.45 thf(fact_2872_power__even__eq,axiom,
% 0.16/0.45 ! [A: complex,N: nat] :
% 0.16/0.45 ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.16/0.45 = ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_even_eq
% 0.16/0.45 thf(fact_2873_diff__le__diff__pow,axiom,
% 0.16/0.45 ! [K: nat,M: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.16/0.45 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % diff_le_diff_pow
% 0.16/0.45 thf(fact_2874_dvd__power__iff__le,axiom,
% 0.16/0.45 ! [K: nat,M: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.16/0.45 => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
% 0.16/0.45 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % dvd_power_iff_le
% 0.16/0.45 thf(fact_2875_power2__le__imp__le,axiom,
% 0.16/0.45 ! [X: rat,Y4: rat] :
% 0.16/0.45 ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 0.16/0.45 => ( ord_less_eq_rat @ X @ Y4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_le_imp_le
% 0.16/0.45 thf(fact_2876_power2__le__imp__le,axiom,
% 0.16/0.45 ! [X: nat,Y4: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
% 0.16/0.45 => ( ord_less_eq_nat @ X @ Y4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_le_imp_le
% 0.16/0.45 thf(fact_2877_power2__le__imp__le,axiom,
% 0.16/0.45 ! [X: int,Y4: int] :
% 0.16/0.45 ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.16/0.45 => ( ord_less_eq_int @ X @ Y4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_le_imp_le
% 0.16/0.45 thf(fact_2878_power2__le__imp__le,axiom,
% 0.16/0.45 ! [X: real,Y4: real] :
% 0.16/0.45 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.16/0.45 => ( ord_less_eq_real @ X @ Y4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_le_imp_le
% 0.16/0.45 thf(fact_2879_power2__eq__imp__eq,axiom,
% 0.16/0.45 ! [X: rat,Y4: rat] :
% 0.16/0.45 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.16/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 0.16/0.45 => ( X = Y4 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_eq_imp_eq
% 0.16/0.45 thf(fact_2880_power2__eq__imp__eq,axiom,
% 0.16/0.45 ! [X: nat,Y4: nat] :
% 0.16/0.45 ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = ( power_power_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.16/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
% 0.16/0.45 => ( X = Y4 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_eq_imp_eq
% 0.16/0.45 thf(fact_2881_power2__eq__imp__eq,axiom,
% 0.16/0.45 ! [X: int,Y4: int] :
% 0.16/0.45 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.16/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.16/0.45 => ( X = Y4 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_eq_imp_eq
% 0.16/0.45 thf(fact_2882_power2__eq__imp__eq,axiom,
% 0.16/0.45 ! [X: real,Y4: real] :
% 0.16/0.45 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.16/0.45 = ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.16/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.16/0.45 => ( X = Y4 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_eq_imp_eq
% 0.16/0.45 thf(fact_2883_zero__le__power2,axiom,
% 0.16/0.45 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_le_power2
% 0.16/0.45 thf(fact_2884_zero__le__power2,axiom,
% 0.16/0.45 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_le_power2
% 0.16/0.45 thf(fact_2885_zero__le__power2,axiom,
% 0.16/0.45 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_le_power2
% 0.16/0.45 thf(fact_2886_power2__less__0,axiom,
% 0.16/0.45 ! [A: real] :
% 0.16/0.45 ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 0.16/0.45
% 0.16/0.45 % power2_less_0
% 0.16/0.45 thf(fact_2887_power2__less__0,axiom,
% 0.16/0.45 ! [A: rat] :
% 0.16/0.45 ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 0.16/0.45
% 0.16/0.45 % power2_less_0
% 0.16/0.45 thf(fact_2888_power2__less__0,axiom,
% 0.16/0.45 ! [A: int] :
% 0.16/0.45 ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 0.16/0.45
% 0.16/0.45 % power2_less_0
% 0.16/0.45 thf(fact_2889_power__strict__mono,axiom,
% 0.16/0.45 ! [A: rat,B: rat,N: nat] :
% 0.16/0.45 ( ( ord_less_rat @ A @ B )
% 0.16/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.16/0.45 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_strict_mono
% 0.16/0.45 thf(fact_2890_power__strict__mono,axiom,
% 0.16/0.45 ! [A: nat,B: nat,N: nat] :
% 0.16/0.45 ( ( ord_less_nat @ A @ B )
% 0.16/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.16/0.45 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_strict_mono
% 0.16/0.45 thf(fact_2891_power__strict__mono,axiom,
% 0.16/0.45 ! [A: int,B: int,N: nat] :
% 0.16/0.45 ( ( ord_less_int @ A @ B )
% 0.16/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.16/0.45 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_strict_mono
% 0.16/0.45 thf(fact_2892_power__strict__mono,axiom,
% 0.16/0.45 ! [A: real,B: real,N: nat] :
% 0.16/0.45 ( ( ord_less_real @ A @ B )
% 0.16/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.16/0.45 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_strict_mono
% 0.16/0.45 thf(fact_2893_power__eq__if,axiom,
% 0.16/0.45 ( power_power_complex
% 0.16/0.45 = ( ^ [P6: complex,M5: nat] : ( if_complex @ ( M5 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P6 @ ( power_power_complex @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_eq_if
% 0.16/0.45 thf(fact_2894_power__eq__if,axiom,
% 0.16/0.45 ( power_power_real
% 0.16/0.45 = ( ^ [P6: real,M5: nat] : ( if_real @ ( M5 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P6 @ ( power_power_real @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_eq_if
% 0.16/0.45 thf(fact_2895_power__eq__if,axiom,
% 0.16/0.45 ( power_power_rat
% 0.16/0.45 = ( ^ [P6: rat,M5: nat] : ( if_rat @ ( M5 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P6 @ ( power_power_rat @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_eq_if
% 0.16/0.45 thf(fact_2896_power__eq__if,axiom,
% 0.16/0.45 ( power_power_nat
% 0.16/0.45 = ( ^ [P6: nat,M5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_eq_if
% 0.16/0.45 thf(fact_2897_power__eq__if,axiom,
% 0.16/0.45 ( power_power_int
% 0.16/0.45 = ( ^ [P6: int,M5: nat] : ( if_int @ ( M5 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P6 @ ( power_power_int @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_eq_if
% 0.16/0.45 thf(fact_2898_power__minus__mult,axiom,
% 0.16/0.45 ! [N: nat,A: complex] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.16/0.45 = ( power_power_complex @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_minus_mult
% 0.16/0.45 thf(fact_2899_power__minus__mult,axiom,
% 0.16/0.45 ! [N: nat,A: real] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.16/0.45 = ( power_power_real @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_minus_mult
% 0.16/0.45 thf(fact_2900_power__minus__mult,axiom,
% 0.16/0.45 ! [N: nat,A: rat] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.16/0.45 = ( power_power_rat @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_minus_mult
% 0.16/0.45 thf(fact_2901_power__minus__mult,axiom,
% 0.16/0.45 ! [N: nat,A: nat] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.16/0.45 = ( power_power_nat @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_minus_mult
% 0.16/0.45 thf(fact_2902_power__minus__mult,axiom,
% 0.16/0.45 ! [N: nat,A: int] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.16/0.45 = ( power_power_int @ A @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_minus_mult
% 0.16/0.45 thf(fact_2903_power2__less__imp__less,axiom,
% 0.16/0.45 ! [X: rat,Y4: rat] :
% 0.16/0.45 ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.45 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 0.16/0.45 => ( ord_less_rat @ X @ Y4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_less_imp_less
% 0.16/0.45 thf(fact_2904_power2__less__imp__less,axiom,
% 0.16/0.45 ! [X: nat,Y4: nat] :
% 0.16/0.45 ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
% 0.16/0.45 => ( ord_less_nat @ X @ Y4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_less_imp_less
% 0.16/0.45 thf(fact_2905_power2__less__imp__less,axiom,
% 0.16/0.45 ! [X: int,Y4: int] :
% 0.16/0.45 ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.45 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.16/0.45 => ( ord_less_int @ X @ Y4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_less_imp_less
% 0.16/0.45 thf(fact_2906_power2__less__imp__less,axiom,
% 0.16/0.45 ! [X: real,Y4: real] :
% 0.16/0.45 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.16/0.45 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.16/0.45 => ( ord_less_real @ X @ Y4 ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power2_less_imp_less
% 0.16/0.45 thf(fact_2907_zero__le__even__power_H,axiom,
% 0.16/0.45 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_le_even_power'
% 0.16/0.45 thf(fact_2908_zero__le__even__power_H,axiom,
% 0.16/0.45 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_le_even_power'
% 0.16/0.45 thf(fact_2909_zero__le__even__power_H,axiom,
% 0.16/0.45 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_le_even_power'
% 0.16/0.45 thf(fact_2910_power__odd__eq,axiom,
% 0.16/0.45 ! [A: complex,N: nat] :
% 0.16/0.45 ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.16/0.45 = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_odd_eq
% 0.16/0.45 thf(fact_2911_power__odd__eq,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.16/0.45 = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_odd_eq
% 0.16/0.45 thf(fact_2912_power__odd__eq,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.16/0.45 = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_odd_eq
% 0.16/0.45 thf(fact_2913_power__odd__eq,axiom,
% 0.16/0.45 ! [A: nat,N: nat] :
% 0.16/0.45 ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.16/0.45 = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_odd_eq
% 0.16/0.45 thf(fact_2914_power__odd__eq,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.16/0.45 = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % power_odd_eq
% 0.16/0.45 thf(fact_2915_odd__0__le__power__imp__0__le,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.16/0.45 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.16/0.45
% 0.16/0.45 % odd_0_le_power_imp_0_le
% 0.16/0.45 thf(fact_2916_odd__0__le__power__imp__0__le,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.16/0.45 => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.16/0.45
% 0.16/0.45 % odd_0_le_power_imp_0_le
% 0.16/0.45 thf(fact_2917_odd__0__le__power__imp__0__le,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.16/0.45 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.16/0.45
% 0.16/0.45 % odd_0_le_power_imp_0_le
% 0.16/0.45 thf(fact_2918_odd__power__less__zero,axiom,
% 0.16/0.45 ! [A: real,N: nat] :
% 0.16/0.45 ( ( ord_less_real @ A @ zero_zero_real )
% 0.16/0.45 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% 0.16/0.45
% 0.16/0.45 % odd_power_less_zero
% 0.16/0.45 thf(fact_2919_odd__power__less__zero,axiom,
% 0.16/0.45 ! [A: rat,N: nat] :
% 0.16/0.45 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.16/0.45 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% 0.16/0.45
% 0.16/0.45 % odd_power_less_zero
% 0.16/0.45 thf(fact_2920_odd__power__less__zero,axiom,
% 0.16/0.45 ! [A: int,N: nat] :
% 0.16/0.45 ( ( ord_less_int @ A @ zero_zero_int )
% 0.16/0.45 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% 0.16/0.45
% 0.16/0.45 % odd_power_less_zero
% 0.16/0.45 thf(fact_2921_space__2__pow__bound,axiom,
% 0.16/0.45 ! [T: vEBT_VEBT,N: nat] :
% 0.16/0.45 ( ( vEBT_invar_vebt @ T @ N )
% 0.16/0.45 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_space2 @ T ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( minus_minus_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) @ one_one_real ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % space_2_pow_bound
% 0.16/0.45 thf(fact_2922_cnt__bound,axiom,
% 0.16/0.45 ! [T: vEBT_VEBT,N: nat] :
% 0.16/0.45 ( ( vEBT_invar_vebt @ T @ N )
% 0.16/0.45 => ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ T ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( minus_minus_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % cnt_bound
% 0.16/0.45 thf(fact_2923_space__bound,axiom,
% 0.16/0.45 ! [T: vEBT_VEBT,N: nat,U: nat] :
% 0.16/0.45 ( ( vEBT_invar_vebt @ T @ N )
% 0.16/0.45 => ( ( U
% 0.16/0.45 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.16/0.45 => ( ord_less_eq_nat @ ( vEBT_VEBT_space @ T ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ U ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % space_bound
% 0.16/0.45 thf(fact_2924_cnt__bound_H,axiom,
% 0.16/0.45 ! [T: vEBT_VEBT,N: nat] :
% 0.16/0.45 ( ( vEBT_invar_vebt @ T @ N )
% 0.16/0.45 => ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ T ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( minus_minus_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) @ one_one_real ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % cnt_bound'
% 0.16/0.45 thf(fact_2925_space_H__bound,axiom,
% 0.16/0.45 ! [T: vEBT_VEBT,N: nat,U: nat] :
% 0.16/0.45 ( ( vEBT_invar_vebt @ T @ N )
% 0.16/0.45 => ( ( U
% 0.16/0.45 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.16/0.45 => ( ord_less_eq_nat @ ( vEBT_VEBT_space2 @ T ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ U ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % space'_bound
% 0.16/0.45 thf(fact_2926_nat__approx__posE,axiom,
% 0.16/0.45 ! [E2: real] :
% 0.16/0.45 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.16/0.45 => ~ ! [N2: nat] :
% 0.16/0.45 ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ E2 ) ) ).
% 0.16/0.45
% 0.16/0.45 % nat_approx_posE
% 0.16/0.45 thf(fact_2927_nat__approx__posE,axiom,
% 0.16/0.45 ! [E2: rat] :
% 0.16/0.45 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.16/0.45 => ~ ! [N2: nat] :
% 0.16/0.45 ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ E2 ) ) ).
% 0.16/0.45
% 0.16/0.45 % nat_approx_posE
% 0.16/0.45 thf(fact_2928_high__def,axiom,
% 0.16/0.45 ( vEBT_VEBT_high
% 0.16/0.45 = ( ^ [X3: nat,N4: nat] : ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % high_def
% 0.16/0.45 thf(fact_2929_pow__sum,axiom,
% 0.16/0.45 ! [A: nat,B: nat] :
% 0.16/0.45 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.16/0.45 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % pow_sum
% 0.16/0.45 thf(fact_2930_even__odd__cases,axiom,
% 0.16/0.45 ! [X: nat] :
% 0.16/0.45 ( ! [N2: nat] :
% 0.16/0.45 ( X
% 0.16/0.45 != ( plus_plus_nat @ N2 @ N2 ) )
% 0.16/0.45 => ~ ! [N2: nat] :
% 0.16/0.45 ( X
% 0.16/0.45 != ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % even_odd_cases
% 0.16/0.45 thf(fact_2931_deg__not__0,axiom,
% 0.16/0.45 ! [T: vEBT_VEBT,N: nat] :
% 0.16/0.45 ( ( vEBT_invar_vebt @ T @ N )
% 0.16/0.45 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % deg_not_0
% 0.16/0.45 thf(fact_2932_buildup__gives__valid,axiom,
% 0.16/0.45 ! [N: nat] :
% 0.16/0.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.16/0.45 => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% 0.16/0.45
% 0.16/0.45 % buildup_gives_valid
% 0.16/0.45 thf(fact_2933_add__left__cancel,axiom,
% 0.16/0.45 ! [A: real,B: real,C: real] :
% 0.16/0.45 ( ( ( plus_plus_real @ A @ B )
% 0.16/0.45 = ( plus_plus_real @ A @ C ) )
% 0.16/0.45 = ( B = C ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_left_cancel
% 0.16/0.45 thf(fact_2934_add__left__cancel,axiom,
% 0.16/0.45 ! [A: rat,B: rat,C: rat] :
% 0.16/0.45 ( ( ( plus_plus_rat @ A @ B )
% 0.16/0.45 = ( plus_plus_rat @ A @ C ) )
% 0.16/0.45 = ( B = C ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_left_cancel
% 0.16/0.45 thf(fact_2935_add__left__cancel,axiom,
% 0.16/0.45 ! [A: nat,B: nat,C: nat] :
% 0.16/0.45 ( ( ( plus_plus_nat @ A @ B )
% 0.16/0.45 = ( plus_plus_nat @ A @ C ) )
% 0.16/0.45 = ( B = C ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_left_cancel
% 0.16/0.45 thf(fact_2936_add__left__cancel,axiom,
% 0.16/0.45 ! [A: int,B: int,C: int] :
% 0.16/0.45 ( ( ( plus_plus_int @ A @ B )
% 0.16/0.45 = ( plus_plus_int @ A @ C ) )
% 0.16/0.45 = ( B = C ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_left_cancel
% 0.16/0.45 thf(fact_2937_add__right__cancel,axiom,
% 0.16/0.45 ! [B: real,A: real,C: real] :
% 0.16/0.45 ( ( ( plus_plus_real @ B @ A )
% 0.16/0.45 = ( plus_plus_real @ C @ A ) )
% 0.16/0.45 = ( B = C ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_right_cancel
% 0.16/0.45 thf(fact_2938_add__right__cancel,axiom,
% 0.16/0.45 ! [B: rat,A: rat,C: rat] :
% 0.16/0.45 ( ( ( plus_plus_rat @ B @ A )
% 0.16/0.45 = ( plus_plus_rat @ C @ A ) )
% 0.16/0.45 = ( B = C ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_right_cancel
% 0.16/0.45 thf(fact_2939_add__right__cancel,axiom,
% 0.16/0.45 ! [B: nat,A: nat,C: nat] :
% 0.16/0.45 ( ( ( plus_plus_nat @ B @ A )
% 0.16/0.45 = ( plus_plus_nat @ C @ A ) )
% 0.16/0.45 = ( B = C ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_right_cancel
% 0.16/0.45 thf(fact_2940_add__right__cancel,axiom,
% 0.16/0.45 ! [B: int,A: int,C: int] :
% 0.16/0.45 ( ( ( plus_plus_int @ B @ A )
% 0.16/0.45 = ( plus_plus_int @ C @ A ) )
% 0.16/0.45 = ( B = C ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_right_cancel
% 0.16/0.45 thf(fact_2941_high__bound__aux,axiom,
% 0.16/0.45 ! [Ma: nat,N: nat,M: nat] :
% 0.16/0.45 ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.16/0.45 => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % high_bound_aux
% 0.16/0.45 thf(fact_2942_add__le__cancel__left,axiom,
% 0.16/0.45 ! [C: rat,A: rat,B: rat] :
% 0.16/0.45 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.16/0.45 = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_le_cancel_left
% 0.16/0.45 thf(fact_2943_add__le__cancel__left,axiom,
% 0.16/0.45 ! [C: nat,A: nat,B: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.16/0.45 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_le_cancel_left
% 0.16/0.45 thf(fact_2944_add__le__cancel__left,axiom,
% 0.16/0.45 ! [C: int,A: int,B: int] :
% 0.16/0.45 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.16/0.45 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_le_cancel_left
% 0.16/0.45 thf(fact_2945_add__le__cancel__left,axiom,
% 0.16/0.45 ! [C: real,A: real,B: real] :
% 0.16/0.45 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.16/0.45 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_le_cancel_left
% 0.16/0.45 thf(fact_2946_add__le__cancel__right,axiom,
% 0.16/0.45 ! [A: rat,C: rat,B: rat] :
% 0.16/0.45 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.16/0.45 = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_le_cancel_right
% 0.16/0.45 thf(fact_2947_add__le__cancel__right,axiom,
% 0.16/0.45 ! [A: nat,C: nat,B: nat] :
% 0.16/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.16/0.45 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_le_cancel_right
% 0.16/0.45 thf(fact_2948_add__le__cancel__right,axiom,
% 0.16/0.45 ! [A: int,C: int,B: int] :
% 0.16/0.45 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.16/0.45 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_le_cancel_right
% 0.16/0.45 thf(fact_2949_add__le__cancel__right,axiom,
% 0.16/0.45 ! [A: real,C: real,B: real] :
% 0.16/0.45 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.16/0.45 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_le_cancel_right
% 0.16/0.45 thf(fact_2950_double__eq__0__iff,axiom,
% 0.16/0.45 ! [A: real] :
% 0.16/0.45 ( ( ( plus_plus_real @ A @ A )
% 0.16/0.45 = zero_zero_real )
% 0.16/0.45 = ( A = zero_zero_real ) ) ).
% 0.16/0.45
% 0.16/0.45 % double_eq_0_iff
% 0.16/0.45 thf(fact_2951_double__eq__0__iff,axiom,
% 0.16/0.45 ! [A: rat] :
% 0.16/0.45 ( ( ( plus_plus_rat @ A @ A )
% 0.16/0.45 = zero_zero_rat )
% 0.16/0.45 = ( A = zero_zero_rat ) ) ).
% 0.16/0.45
% 0.16/0.45 % double_eq_0_iff
% 0.16/0.45 thf(fact_2952_double__eq__0__iff,axiom,
% 0.16/0.45 ! [A: int] :
% 0.16/0.45 ( ( ( plus_plus_int @ A @ A )
% 0.16/0.45 = zero_zero_int )
% 0.16/0.45 = ( A = zero_zero_int ) ) ).
% 0.16/0.45
% 0.16/0.45 % double_eq_0_iff
% 0.16/0.45 thf(fact_2953_add__0,axiom,
% 0.16/0.45 ! [A: complex] :
% 0.16/0.45 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.16/0.45 = A ) ).
% 0.16/0.45
% 0.16/0.45 % add_0
% 0.16/0.45 thf(fact_2954_add__0,axiom,
% 0.16/0.45 ! [A: real] :
% 0.16/0.45 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.16/0.45 = A ) ).
% 0.16/0.45
% 0.16/0.45 % add_0
% 0.16/0.45 thf(fact_2955_add__0,axiom,
% 0.16/0.45 ! [A: rat] :
% 0.16/0.45 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.16/0.45 = A ) ).
% 0.16/0.45
% 0.16/0.45 % add_0
% 0.16/0.45 thf(fact_2956_add__0,axiom,
% 0.16/0.45 ! [A: nat] :
% 0.16/0.45 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.16/0.45 = A ) ).
% 0.16/0.45
% 0.16/0.45 % add_0
% 0.16/0.45 thf(fact_2957_add__0,axiom,
% 0.16/0.45 ! [A: int] :
% 0.16/0.45 ( ( plus_plus_int @ zero_zero_int @ A )
% 0.16/0.45 = A ) ).
% 0.16/0.45
% 0.16/0.45 % add_0
% 0.16/0.45 thf(fact_2958_zero__eq__add__iff__both__eq__0,axiom,
% 0.16/0.45 ! [X: nat,Y4: nat] :
% 0.16/0.45 ( ( zero_zero_nat
% 0.16/0.45 = ( plus_plus_nat @ X @ Y4 ) )
% 0.16/0.45 = ( ( X = zero_zero_nat )
% 0.16/0.45 & ( Y4 = zero_zero_nat ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % zero_eq_add_iff_both_eq_0
% 0.16/0.45 thf(fact_2959_add__eq__0__iff__both__eq__0,axiom,
% 0.16/0.45 ! [X: nat,Y4: nat] :
% 0.16/0.45 ( ( ( plus_plus_nat @ X @ Y4 )
% 0.16/0.45 = zero_zero_nat )
% 0.16/0.45 = ( ( X = zero_zero_nat )
% 0.16/0.45 & ( Y4 = zero_zero_nat ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_eq_0_iff_both_eq_0
% 0.16/0.45 thf(fact_2960_add__cancel__right__right,axiom,
% 0.16/0.45 ! [A: complex,B: complex] :
% 0.16/0.45 ( ( A
% 0.16/0.45 = ( plus_plus_complex @ A @ B ) )
% 0.16/0.45 = ( B = zero_zero_complex ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_right_right
% 0.16/0.45 thf(fact_2961_add__cancel__right__right,axiom,
% 0.16/0.45 ! [A: real,B: real] :
% 0.16/0.45 ( ( A
% 0.16/0.45 = ( plus_plus_real @ A @ B ) )
% 0.16/0.45 = ( B = zero_zero_real ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_right_right
% 0.16/0.45 thf(fact_2962_add__cancel__right__right,axiom,
% 0.16/0.45 ! [A: rat,B: rat] :
% 0.16/0.45 ( ( A
% 0.16/0.45 = ( plus_plus_rat @ A @ B ) )
% 0.16/0.45 = ( B = zero_zero_rat ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_right_right
% 0.16/0.45 thf(fact_2963_add__cancel__right__right,axiom,
% 0.16/0.45 ! [A: nat,B: nat] :
% 0.16/0.45 ( ( A
% 0.16/0.45 = ( plus_plus_nat @ A @ B ) )
% 0.16/0.45 = ( B = zero_zero_nat ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_right_right
% 0.16/0.45 thf(fact_2964_add__cancel__right__right,axiom,
% 0.16/0.45 ! [A: int,B: int] :
% 0.16/0.45 ( ( A
% 0.16/0.45 = ( plus_plus_int @ A @ B ) )
% 0.16/0.45 = ( B = zero_zero_int ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_right_right
% 0.16/0.45 thf(fact_2965_add__cancel__right__left,axiom,
% 0.16/0.45 ! [A: complex,B: complex] :
% 0.16/0.45 ( ( A
% 0.16/0.45 = ( plus_plus_complex @ B @ A ) )
% 0.16/0.45 = ( B = zero_zero_complex ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_right_left
% 0.16/0.45 thf(fact_2966_add__cancel__right__left,axiom,
% 0.16/0.45 ! [A: real,B: real] :
% 0.16/0.45 ( ( A
% 0.16/0.45 = ( plus_plus_real @ B @ A ) )
% 0.16/0.45 = ( B = zero_zero_real ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_right_left
% 0.16/0.45 thf(fact_2967_add__cancel__right__left,axiom,
% 0.16/0.45 ! [A: rat,B: rat] :
% 0.16/0.45 ( ( A
% 0.16/0.45 = ( plus_plus_rat @ B @ A ) )
% 0.16/0.45 = ( B = zero_zero_rat ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_right_left
% 0.16/0.45 thf(fact_2968_add__cancel__right__left,axiom,
% 0.16/0.45 ! [A: nat,B: nat] :
% 0.16/0.45 ( ( A
% 0.16/0.45 = ( plus_plus_nat @ B @ A ) )
% 0.16/0.45 = ( B = zero_zero_nat ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_right_left
% 0.16/0.45 thf(fact_2969_add__cancel__right__left,axiom,
% 0.16/0.45 ! [A: int,B: int] :
% 0.16/0.45 ( ( A
% 0.16/0.45 = ( plus_plus_int @ B @ A ) )
% 0.16/0.45 = ( B = zero_zero_int ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_right_left
% 0.16/0.45 thf(fact_2970_add__cancel__left__right,axiom,
% 0.16/0.45 ! [A: complex,B: complex] :
% 0.16/0.45 ( ( ( plus_plus_complex @ A @ B )
% 0.16/0.45 = A )
% 0.16/0.45 = ( B = zero_zero_complex ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_left_right
% 0.16/0.45 thf(fact_2971_add__cancel__left__right,axiom,
% 0.16/0.45 ! [A: real,B: real] :
% 0.16/0.45 ( ( ( plus_plus_real @ A @ B )
% 0.16/0.45 = A )
% 0.16/0.45 = ( B = zero_zero_real ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_left_right
% 0.16/0.45 thf(fact_2972_add__cancel__left__right,axiom,
% 0.16/0.45 ! [A: rat,B: rat] :
% 0.16/0.45 ( ( ( plus_plus_rat @ A @ B )
% 0.16/0.45 = A )
% 0.16/0.45 = ( B = zero_zero_rat ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_left_right
% 0.16/0.45 thf(fact_2973_add__cancel__left__right,axiom,
% 0.16/0.45 ! [A: nat,B: nat] :
% 0.16/0.45 ( ( ( plus_plus_nat @ A @ B )
% 0.16/0.45 = A )
% 0.16/0.45 = ( B = zero_zero_nat ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_left_right
% 0.16/0.45 thf(fact_2974_add__cancel__left__right,axiom,
% 0.16/0.45 ! [A: int,B: int] :
% 0.16/0.45 ( ( ( plus_plus_int @ A @ B )
% 0.16/0.45 = A )
% 0.16/0.45 = ( B = zero_zero_int ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_left_right
% 0.16/0.45 thf(fact_2975_add__cancel__left__left,axiom,
% 0.16/0.45 ! [B: complex,A: complex] :
% 0.16/0.45 ( ( ( plus_plus_complex @ B @ A )
% 0.16/0.45 = A )
% 0.16/0.45 = ( B = zero_zero_complex ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_left_left
% 0.16/0.45 thf(fact_2976_add__cancel__left__left,axiom,
% 0.16/0.45 ! [B: real,A: real] :
% 0.16/0.45 ( ( ( plus_plus_real @ B @ A )
% 0.16/0.45 = A )
% 0.16/0.45 = ( B = zero_zero_real ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_left_left
% 0.16/0.45 thf(fact_2977_add__cancel__left__left,axiom,
% 0.16/0.45 ! [B: rat,A: rat] :
% 0.16/0.45 ( ( ( plus_plus_rat @ B @ A )
% 0.16/0.45 = A )
% 0.16/0.45 = ( B = zero_zero_rat ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_left_left
% 0.16/0.45 thf(fact_2978_add__cancel__left__left,axiom,
% 0.16/0.45 ! [B: nat,A: nat] :
% 0.16/0.45 ( ( ( plus_plus_nat @ B @ A )
% 0.16/0.45 = A )
% 0.16/0.45 = ( B = zero_zero_nat ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_left_left
% 0.16/0.45 thf(fact_2979_add__cancel__left__left,axiom,
% 0.16/0.45 ! [B: int,A: int] :
% 0.16/0.45 ( ( ( plus_plus_int @ B @ A )
% 0.16/0.45 = A )
% 0.16/0.45 = ( B = zero_zero_int ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_cancel_left_left
% 0.16/0.45 thf(fact_2980_double__zero__sym,axiom,
% 0.16/0.45 ! [A: real] :
% 0.16/0.45 ( ( zero_zero_real
% 0.16/0.45 = ( plus_plus_real @ A @ A ) )
% 0.16/0.45 = ( A = zero_zero_real ) ) ).
% 0.16/0.45
% 0.16/0.45 % double_zero_sym
% 0.16/0.45 thf(fact_2981_double__zero__sym,axiom,
% 0.16/0.45 ! [A: rat] :
% 0.16/0.45 ( ( zero_zero_rat
% 0.16/0.45 = ( plus_plus_rat @ A @ A ) )
% 0.16/0.45 = ( A = zero_zero_rat ) ) ).
% 0.16/0.45
% 0.16/0.45 % double_zero_sym
% 0.16/0.45 thf(fact_2982_double__zero__sym,axiom,
% 0.16/0.45 ! [A: int] :
% 0.16/0.45 ( ( zero_zero_int
% 0.16/0.45 = ( plus_plus_int @ A @ A ) )
% 0.16/0.45 = ( A = zero_zero_int ) ) ).
% 0.16/0.45
% 0.16/0.45 % double_zero_sym
% 0.16/0.45 thf(fact_2983_add_Oright__neutral,axiom,
% 0.16/0.45 ! [A: complex] :
% 0.16/0.45 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.16/0.45 = A ) ).
% 0.16/0.45
% 0.16/0.45 % add.right_neutral
% 0.16/0.45 thf(fact_2984_add_Oright__neutral,axiom,
% 0.16/0.45 ! [A: real] :
% 0.16/0.45 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.16/0.45 = A ) ).
% 0.16/0.45
% 0.16/0.45 % add.right_neutral
% 0.16/0.45 thf(fact_2985_add_Oright__neutral,axiom,
% 0.16/0.45 ! [A: rat] :
% 0.16/0.45 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.16/0.45 = A ) ).
% 0.16/0.45
% 0.16/0.45 % add.right_neutral
% 0.16/0.45 thf(fact_2986_add_Oright__neutral,axiom,
% 0.16/0.45 ! [A: nat] :
% 0.16/0.45 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.16/0.45 = A ) ).
% 0.16/0.45
% 0.16/0.45 % add.right_neutral
% 0.16/0.45 thf(fact_2987_add_Oright__neutral,axiom,
% 0.16/0.45 ! [A: int] :
% 0.16/0.45 ( ( plus_plus_int @ A @ zero_zero_int )
% 0.16/0.45 = A ) ).
% 0.16/0.45
% 0.16/0.45 % add.right_neutral
% 0.16/0.45 thf(fact_2988_add__less__cancel__left,axiom,
% 0.16/0.45 ! [C: real,A: real,B: real] :
% 0.16/0.45 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.16/0.45 = ( ord_less_real @ A @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_less_cancel_left
% 0.16/0.45 thf(fact_2989_add__less__cancel__left,axiom,
% 0.16/0.45 ! [C: rat,A: rat,B: rat] :
% 0.16/0.45 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.16/0.45 = ( ord_less_rat @ A @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_less_cancel_left
% 0.16/0.45 thf(fact_2990_add__less__cancel__left,axiom,
% 0.16/0.45 ! [C: nat,A: nat,B: nat] :
% 0.16/0.45 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.16/0.45 = ( ord_less_nat @ A @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_less_cancel_left
% 0.16/0.45 thf(fact_2991_add__less__cancel__left,axiom,
% 0.16/0.45 ! [C: int,A: int,B: int] :
% 0.16/0.45 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.16/0.45 = ( ord_less_int @ A @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_less_cancel_left
% 0.16/0.45 thf(fact_2992_add__less__cancel__right,axiom,
% 0.16/0.45 ! [A: real,C: real,B: real] :
% 0.16/0.45 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.16/0.45 = ( ord_less_real @ A @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_less_cancel_right
% 0.16/0.45 thf(fact_2993_add__less__cancel__right,axiom,
% 0.16/0.45 ! [A: rat,C: rat,B: rat] :
% 0.16/0.45 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.16/0.45 = ( ord_less_rat @ A @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_less_cancel_right
% 0.16/0.45 thf(fact_2994_add__less__cancel__right,axiom,
% 0.16/0.45 ! [A: nat,C: nat,B: nat] :
% 0.16/0.45 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.16/0.45 = ( ord_less_nat @ A @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_less_cancel_right
% 0.16/0.45 thf(fact_2995_add__less__cancel__right,axiom,
% 0.16/0.45 ! [A: int,C: int,B: int] :
% 0.16/0.45 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.16/0.45 = ( ord_less_int @ A @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_less_cancel_right
% 0.16/0.45 thf(fact_2996_numeral__plus__numeral,axiom,
% 0.16/0.45 ! [M: num,N: num] :
% 0.16/0.45 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 0.16/0.45 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % numeral_plus_numeral
% 0.16/0.45 thf(fact_2997_numeral__plus__numeral,axiom,
% 0.16/0.45 ! [M: num,N: num] :
% 0.16/0.45 ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.16/0.45 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % numeral_plus_numeral
% 0.16/0.45 thf(fact_2998_numeral__plus__numeral,axiom,
% 0.16/0.45 ! [M: num,N: num] :
% 0.16/0.45 ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.16/0.45 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % numeral_plus_numeral
% 0.16/0.45 thf(fact_2999_numeral__plus__numeral,axiom,
% 0.16/0.45 ! [M: num,N: num] :
% 0.16/0.45 ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.16/0.45 = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % numeral_plus_numeral
% 0.16/0.45 thf(fact_3000_numeral__plus__numeral,axiom,
% 0.16/0.45 ! [M: num,N: num] :
% 0.16/0.45 ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.16/0.45 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.16/0.45
% 0.16/0.45 % numeral_plus_numeral
% 0.16/0.45 thf(fact_3001_add__numeral__left,axiom,
% 0.16/0.45 ! [V: num,W: num,Z: complex] :
% 0.16/0.45 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 0.16/0.45 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_numeral_left
% 0.16/0.45 thf(fact_3002_add__numeral__left,axiom,
% 0.16/0.45 ! [V: num,W: num,Z: real] :
% 0.16/0.45 ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 0.16/0.45 = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_numeral_left
% 0.16/0.45 thf(fact_3003_add__numeral__left,axiom,
% 0.16/0.45 ! [V: num,W: num,Z: rat] :
% 0.16/0.45 ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 0.16/0.45 = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_numeral_left
% 0.16/0.45 thf(fact_3004_add__numeral__left,axiom,
% 0.16/0.45 ! [V: num,W: num,Z: nat] :
% 0.16/0.45 ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 0.16/0.45 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_numeral_left
% 0.16/0.45 thf(fact_3005_add__numeral__left,axiom,
% 0.16/0.45 ! [V: num,W: num,Z: int] :
% 0.16/0.45 ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 0.16/0.45 = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_numeral_left
% 0.16/0.45 thf(fact_3006_add__diff__cancel__right_H,axiom,
% 0.16/0.45 ! [A: real,B: real] :
% 0.16/0.45 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.16/0.45 = A ) ).
% 0.16/0.45
% 0.16/0.45 % add_diff_cancel_right'
% 0.16/0.45 thf(fact_3007_add__diff__cancel__right_H,axiom,
% 0.16/0.45 ! [A: rat,B: rat] :
% 0.16/0.45 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.16/0.45 = A ) ).
% 0.16/0.45
% 0.16/0.45 % add_diff_cancel_right'
% 0.16/0.45 thf(fact_3008_add__diff__cancel__right_H,axiom,
% 0.16/0.45 ! [A: nat,B: nat] :
% 0.16/0.45 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.16/0.45 = A ) ).
% 0.16/0.45
% 0.16/0.45 % add_diff_cancel_right'
% 0.16/0.45 thf(fact_3009_add__diff__cancel__right_H,axiom,
% 0.16/0.45 ! [A: int,B: int] :
% 0.16/0.45 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.16/0.45 = A ) ).
% 0.16/0.45
% 0.16/0.45 % add_diff_cancel_right'
% 0.16/0.45 thf(fact_3010_add__diff__cancel__right,axiom,
% 0.16/0.45 ! [A: real,C: real,B: real] :
% 0.16/0.45 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.16/0.45 = ( minus_minus_real @ A @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_diff_cancel_right
% 0.16/0.45 thf(fact_3011_add__diff__cancel__right,axiom,
% 0.16/0.45 ! [A: rat,C: rat,B: rat] :
% 0.16/0.45 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.16/0.45 = ( minus_minus_rat @ A @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_diff_cancel_right
% 0.16/0.45 thf(fact_3012_add__diff__cancel__right,axiom,
% 0.16/0.45 ! [A: nat,C: nat,B: nat] :
% 0.16/0.45 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.16/0.45 = ( minus_minus_nat @ A @ B ) ) ).
% 0.16/0.45
% 0.16/0.45 % add_diff_cancel_right
% 0.16/0.45 thf(fact_3013_add__diff__cancel__right,axiom,
% 0.16/0.45 ! [A: int,C: int,B: int] :
% 0.16/0.45 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.16/0.46 = ( minus_minus_int @ A @ B ) ) ).
% 0.16/0.46
% 0.16/0.46 % add_diff_cancel_right
% 0.16/0.46 thf(fact_3014_add__diff__cancel__left_H,axiom,
% 0.16/0.46 ! [A: real,B: real] :
% 0.16/0.46 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 0.16/0.46 = B ) ).
% 0.16/0.46
% 0.16/0.46 % add_diff_cancel_left'
% 0.16/0.46 thf(fact_3015_add__diff__cancel__left_H,axiom,
% 0.16/0.46 ! [A: rat,B: rat] :
% 0.16/0.46 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
% 0.16/0.46 = B ) ).
% 0.16/0.46
% 0.16/0.46 % add_diff_cancel_left'
% 0.16/0.46 thf(fact_3016_add__diff__cancel__left_H,axiom,
% 0.16/0.46 ! [A: nat,B: nat] :
% 0.16/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 0.16/0.46 = B ) ).
% 0.16/0.46
% 0.16/0.46 % add_diff_cancel_left'
% 0.16/0.46 thf(fact_3017_add__diff__cancel__left_H,axiom,
% 0.16/0.46 ! [A: int,B: int] :
% 0.16/0.46 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 0.16/0.46 = B ) ).
% 0.16/0.46
% 0.16/0.46 % add_diff_cancel_left'
% 0.16/0.46 thf(fact_3018_add__diff__cancel__left,axiom,
% 0.16/0.46 ! [C: real,A: real,B: real] :
% 0.16/0.46 ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.16/0.46 = ( minus_minus_real @ A @ B ) ) ).
% 0.16/0.46
% 0.16/0.46 % add_diff_cancel_left
% 0.16/0.46 thf(fact_3019_add__diff__cancel__left,axiom,
% 0.16/0.46 ! [C: rat,A: rat,B: rat] :
% 0.16/0.46 ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.16/0.46 = ( minus_minus_rat @ A @ B ) ) ).
% 0.16/0.46
% 0.16/0.46 % add_diff_cancel_left
% 0.16/0.46 thf(fact_3020_add__diff__cancel__left,axiom,
% 0.16/0.46 ! [C: nat,A: nat,B: nat] :
% 0.16/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.16/0.46 = ( minus_minus_nat @ A @ B ) ) ).
% 0.16/0.46
% 0.16/0.46 % add_diff_cancel_left
% 0.16/0.46 thf(fact_3021_add__diff__cancel__left,axiom,
% 0.16/0.46 ! [C: int,A: int,B: int] :
% 0.16/0.46 ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.16/0.46 = ( minus_minus_int @ A @ B ) ) ).
% 0.16/0.46
% 0.16/0.46 % add_diff_cancel_left
% 0.16/0.46 thf(fact_3022_diff__add__cancel,axiom,
% 0.16/0.46 ! [A: real,B: real] :
% 0.16/0.46 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 0.16/0.46 = A ) ).
% 0.16/0.46
% 0.16/0.46 % diff_add_cancel
% 0.16/0.46 thf(fact_3023_diff__add__cancel,axiom,
% 0.16/0.46 ! [A: rat,B: rat] :
% 0.16/0.46 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 0.16/0.46 = A ) ).
% 0.16/0.46
% 0.16/0.46 % diff_add_cancel
% 0.16/0.46 thf(fact_3024_diff__add__cancel,axiom,
% 0.16/0.46 ! [A: int,B: int] :
% 0.16/0.46 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.16/0.46 = A ) ).
% 0.16/0.46
% 0.16/0.46 % diff_add_cancel
% 0.16/0.46 thf(fact_3025_add__diff__cancel,axiom,
% 0.16/0.46 ! [A: real,B: real] :
% 0.16/0.46 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.16/0.46 = A ) ).
% 0.16/0.46
% 0.16/0.46 % add_diff_cancel
% 0.16/0.46 thf(fact_3026_add__diff__cancel,axiom,
% 0.16/0.46 ! [A: rat,B: rat] :
% 0.16/0.46 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.16/0.46 = A ) ).
% 0.16/0.46
% 0.16/0.46 % add_diff_cancel
% 0.16/0.46 thf(fact_3027_add__diff__cancel,axiom,
% 0.16/0.46 ! [A: int,B: int] :
% 0.16/0.46 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.16/0.46 = A ) ).
% 0.16/0.46
% 0.16/0.46 % add_diff_cancel
% 0.16/0.46 thf(fact_3028_high__inv,axiom,
% 0.16/0.46 ! [X: nat,N: nat,Y4: nat] :
% 0.16/0.46 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.16/0.46 => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
% 0.16/0.46 = Y4 ) ) ).
% 0.16/0.46
% 0.16/0.46 % high_inv
% 0.16/0.46 thf(fact_3029_dvd__add__triv__right__iff,axiom,
% 0.16/0.46 ! [A: code_integer,B: code_integer] :
% 0.16/0.46 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
% 0.16/0.46 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.16/0.46
% 0.16/0.46 % dvd_add_triv_right_iff
% 0.16/0.46 thf(fact_3030_dvd__add__triv__right__iff,axiom,
% 0.16/0.46 ! [A: real,B: real] :
% 0.16/0.46 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.16/0.46 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.16/0.46
% 0.16/0.46 % dvd_add_triv_right_iff
% 0.16/0.46 thf(fact_3031_dvd__add__triv__right__iff,axiom,
% 0.16/0.46 ! [A: rat,B: rat] :
% 0.16/0.46 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.16/0.46 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.16/0.46
% 0.16/0.46 % dvd_add_triv_right_iff
% 0.16/0.46 thf(fact_3032_dvd__add__triv__right__iff,axiom,
% 0.16/0.46 ! [A: nat,B: nat] :
% 0.16/0.46 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.16/0.46 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.16/0.46
% 0.16/0.46 % dvd_add_triv_right_iff
% 0.16/0.46 thf(fact_3033_dvd__add__triv__right__iff,axiom,
% 0.16/0.46 ! [A: int,B: int] :
% 0.16/0.46 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.16/0.46 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.16/0.46
% 0.16/0.46 % dvd_add_triv_right_iff
% 0.16/0.46 thf(fact_3034_dvd__add__triv__left__iff,axiom,
% 0.16/0.46 ! [A: code_integer,B: code_integer] :
% 0.16/0.46 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.16/0.46 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.16/0.46
% 0.16/0.46 % dvd_add_triv_left_iff
% 0.16/0.46 thf(fact_3035_dvd__add__triv__left__iff,axiom,
% 0.16/0.46 ! [A: real,B: real] :
% 0.16/0.46 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.16/0.46 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.16/0.46
% 0.16/0.46 % dvd_add_triv_left_iff
% 0.16/0.46 thf(fact_3036_dvd__add__triv__left__iff,axiom,
% 0.16/0.46 ! [A: rat,B: rat] :
% 0.16/0.46 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.16/0.46 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.16/0.46
% 0.16/0.46 % dvd_add_triv_left_iff
% 0.16/0.46 thf(fact_3037_dvd__add__triv__left__iff,axiom,
% 0.16/0.46 ! [A: nat,B: nat] :
% 0.16/0.46 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.16/0.46 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.16/0.46
% 0.16/0.46 % dvd_add_triv_left_iff
% 0.16/0.46 thf(fact_3038_dvd__add__triv__left__iff,axiom,
% 0.16/0.46 ! [A: int,B: int] :
% 0.16/0.46 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.16/0.46 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.16/0.46
% 0.16/0.46 % dvd_add_triv_left_iff
% 0.16/0.46 thf(fact_3039_add__Suc__right,axiom,
% 0.16/0.46 ! [M: nat,N: nat] :
% 0.16/0.46 ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 0.16/0.46 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.16/0.46
% 0.16/0.46 % add_Suc_right
% 0.16/0.46 thf(fact_3040_add__is__0,axiom,
% 0.16/0.46 ! [M: nat,N: nat] :
% 0.16/0.46 ( ( ( plus_plus_nat @ M @ N )
% 0.16/0.46 = zero_zero_nat )
% 0.16/0.46 = ( ( M = zero_zero_nat )
% 0.16/0.46 & ( N = zero_zero_nat ) ) ) ).
% 0.16/0.46
% 0.16/0.46 % add_is_0
% 0.16/0.46 thf(fact_3041_Nat_Oadd__0__right,axiom,
% 0.16/0.46 ! [M: nat] :
% 0.16/0.46 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 0.16/0.46 = M ) ).
% 0.16/0.46
% 0.16/0.46 % Nat.add_0_right
% 0.16/0.46 thf(fact_3042_nat__add__left__cancel__less,axiom,
% 0.16/0.46 ! [K: nat,M: nat,N: nat] :
% 0.16/0.46 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.16/0.46 = ( ord_less_nat @ M @ N ) ) ).
% 0.16/0.46
% 0.16/0.46 % nat_add_left_cancel_less
% 0.16/0.46 thf(fact_3043_nat__add__left__cancel__le,axiom,
% 0.16/0.46 ! [K: nat,M: nat,N: nat] :
% 0.16/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.16/0.46 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.16/0.46
% 0.16/0.46 % nat_add_left_cancel_le
% 0.16/0.46 thf(fact_3044_diff__diff__left,axiom,
% 0.16/0.46 ! [I: nat,J: nat,K: nat] :
% 0.16/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.16/0.46 = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.16/0.46
% 0.16/0.46 % diff_diff_left
% 0.16/0.46 thf(fact_3045_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.16/0.46 ! [A: rat] :
% 0.16/0.46 ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.16/0.46 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.16/0.46
% 0.16/0.46 % zero_le_double_add_iff_zero_le_single_add
% 0.16/0.46 thf(fact_3046_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.16/0.46 ! [A: int] :
% 0.16/0.46 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.16/0.46 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.16/0.46
% 0.16/0.46 % zero_le_double_add_iff_zero_le_single_add
% 0.16/0.46 thf(fact_3047_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.16/0.46 ! [A: real] :
% 0.16/0.46 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.16/0.46 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.16/0.46
% 0.16/0.46 % zero_le_double_add_iff_zero_le_single_add
% 0.16/0.46 thf(fact_3048_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.16/0.46 ! [A: rat] :
% 0.16/0.46 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.16/0.46 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.16/0.46
% 0.16/0.46 % double_add_le_zero_iff_single_add_le_zero
% 0.16/0.46 thf(fact_3049_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.16/0.46 ! [A: int] :
% 0.16/0.46 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.16/0.46 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.16/0.46
% 0.16/0.46 % double_add_le_zero_iff_single_add_le_zero
% 0.16/0.46 thf(fact_3050_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.16/0.46 ! [A: real] :
% 0.16/0.46 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.16/0.46 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.16/0.46
% 0.16/0.46 % double_add_le_zero_iff_single_add_le_zero
% 0.16/0.46 thf(fact_3051_le__add__same__cancel2,axiom,
% 0.16/0.46 ! [A: rat,B: rat] :
% 0.16/0.46 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.16/0.46 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 0.16/0.46
% 0.16/0.46 % le_add_same_cancel2
% 0.16/0.46 thf(fact_3052_le__add__same__cancel2,axiom,
% 0.16/0.46 ! [A: nat,B: nat] :
% 0.16/0.46 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.16/0.46 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.16/0.46
% 0.16/0.46 % le_add_same_cancel2
% 0.16/0.46 thf(fact_3053_le__add__same__cancel2,axiom,
% 0.16/0.46 ! [A: int,B: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.22/0.46 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_add_same_cancel2
% 0.22/0.46 thf(fact_3054_le__add__same__cancel2,axiom,
% 0.22/0.46 ! [A: real,B: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.22/0.46 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_add_same_cancel2
% 0.22/0.46 thf(fact_3055_le__add__same__cancel1,axiom,
% 0.22/0.46 ! [A: rat,B: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.22/0.46 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_add_same_cancel1
% 0.22/0.46 thf(fact_3056_le__add__same__cancel1,axiom,
% 0.22/0.46 ! [A: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.22/0.46 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_add_same_cancel1
% 0.22/0.46 thf(fact_3057_le__add__same__cancel1,axiom,
% 0.22/0.46 ! [A: int,B: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.22/0.46 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_add_same_cancel1
% 0.22/0.46 thf(fact_3058_le__add__same__cancel1,axiom,
% 0.22/0.46 ! [A: real,B: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.22/0.46 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_add_same_cancel1
% 0.22/0.46 thf(fact_3059_add__le__same__cancel2,axiom,
% 0.22/0.46 ! [A: rat,B: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.22/0.46 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_same_cancel2
% 0.22/0.46 thf(fact_3060_add__le__same__cancel2,axiom,
% 0.22/0.46 ! [A: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.22/0.46 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_same_cancel2
% 0.22/0.46 thf(fact_3061_add__le__same__cancel2,axiom,
% 0.22/0.46 ! [A: int,B: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.22/0.46 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_same_cancel2
% 0.22/0.46 thf(fact_3062_add__le__same__cancel2,axiom,
% 0.22/0.46 ! [A: real,B: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.22/0.46 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_same_cancel2
% 0.22/0.46 thf(fact_3063_add__le__same__cancel1,axiom,
% 0.22/0.46 ! [B: rat,A: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 0.22/0.46 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_same_cancel1
% 0.22/0.46 thf(fact_3064_add__le__same__cancel1,axiom,
% 0.22/0.46 ! [B: nat,A: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.22/0.46 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_same_cancel1
% 0.22/0.46 thf(fact_3065_add__le__same__cancel1,axiom,
% 0.22/0.46 ! [B: int,A: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.22/0.46 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_same_cancel1
% 0.22/0.46 thf(fact_3066_add__le__same__cancel1,axiom,
% 0.22/0.46 ! [B: real,A: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.22/0.46 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_same_cancel1
% 0.22/0.46 thf(fact_3067_add__less__same__cancel1,axiom,
% 0.22/0.46 ! [B: real,A: real] :
% 0.22/0.46 ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.22/0.46 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_same_cancel1
% 0.22/0.46 thf(fact_3068_add__less__same__cancel1,axiom,
% 0.22/0.46 ! [B: rat,A: rat] :
% 0.22/0.46 ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 0.22/0.46 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_same_cancel1
% 0.22/0.46 thf(fact_3069_add__less__same__cancel1,axiom,
% 0.22/0.46 ! [B: nat,A: nat] :
% 0.22/0.46 ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.22/0.46 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_same_cancel1
% 0.22/0.46 thf(fact_3070_add__less__same__cancel1,axiom,
% 0.22/0.46 ! [B: int,A: int] :
% 0.22/0.46 ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.22/0.46 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_same_cancel1
% 0.22/0.46 thf(fact_3071_add__less__same__cancel2,axiom,
% 0.22/0.46 ! [A: real,B: real] :
% 0.22/0.46 ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.22/0.46 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_same_cancel2
% 0.22/0.46 thf(fact_3072_add__less__same__cancel2,axiom,
% 0.22/0.46 ! [A: rat,B: rat] :
% 0.22/0.46 ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.22/0.46 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_same_cancel2
% 0.22/0.46 thf(fact_3073_add__less__same__cancel2,axiom,
% 0.22/0.46 ! [A: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.22/0.46 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_same_cancel2
% 0.22/0.46 thf(fact_3074_add__less__same__cancel2,axiom,
% 0.22/0.46 ! [A: int,B: int] :
% 0.22/0.46 ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.22/0.46 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_same_cancel2
% 0.22/0.46 thf(fact_3075_less__add__same__cancel1,axiom,
% 0.22/0.46 ! [A: real,B: real] :
% 0.22/0.46 ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.22/0.46 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % less_add_same_cancel1
% 0.22/0.46 thf(fact_3076_less__add__same__cancel1,axiom,
% 0.22/0.46 ! [A: rat,B: rat] :
% 0.22/0.46 ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.22/0.46 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % less_add_same_cancel1
% 0.22/0.46 thf(fact_3077_less__add__same__cancel1,axiom,
% 0.22/0.46 ! [A: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.22/0.46 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % less_add_same_cancel1
% 0.22/0.46 thf(fact_3078_less__add__same__cancel1,axiom,
% 0.22/0.46 ! [A: int,B: int] :
% 0.22/0.46 ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.22/0.46 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % less_add_same_cancel1
% 0.22/0.46 thf(fact_3079_less__add__same__cancel2,axiom,
% 0.22/0.46 ! [A: real,B: real] :
% 0.22/0.46 ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.22/0.46 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % less_add_same_cancel2
% 0.22/0.46 thf(fact_3080_less__add__same__cancel2,axiom,
% 0.22/0.46 ! [A: rat,B: rat] :
% 0.22/0.46 ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.22/0.46 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % less_add_same_cancel2
% 0.22/0.46 thf(fact_3081_less__add__same__cancel2,axiom,
% 0.22/0.46 ! [A: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.22/0.46 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % less_add_same_cancel2
% 0.22/0.46 thf(fact_3082_less__add__same__cancel2,axiom,
% 0.22/0.46 ! [A: int,B: int] :
% 0.22/0.46 ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.22/0.46 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % less_add_same_cancel2
% 0.22/0.46 thf(fact_3083_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.22/0.46 ! [A: real] :
% 0.22/0.46 ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.22/0.46 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.22/0.46
% 0.22/0.46 % double_add_less_zero_iff_single_add_less_zero
% 0.22/0.46 thf(fact_3084_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.22/0.46 ! [A: rat] :
% 0.22/0.46 ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.22/0.46 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.22/0.46
% 0.22/0.46 % double_add_less_zero_iff_single_add_less_zero
% 0.22/0.46 thf(fact_3085_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.22/0.46 ! [A: int] :
% 0.22/0.46 ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.22/0.46 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.22/0.46
% 0.22/0.46 % double_add_less_zero_iff_single_add_less_zero
% 0.22/0.46 thf(fact_3086_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.22/0.46 ! [A: real] :
% 0.22/0.46 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.22/0.46 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.22/0.46
% 0.22/0.46 % zero_less_double_add_iff_zero_less_single_add
% 0.22/0.46 thf(fact_3087_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.22/0.46 ! [A: rat] :
% 0.22/0.46 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.22/0.46 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.22/0.46
% 0.22/0.46 % zero_less_double_add_iff_zero_less_single_add
% 0.22/0.46 thf(fact_3088_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.22/0.46 ! [A: int] :
% 0.22/0.46 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.22/0.46 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.22/0.46
% 0.22/0.46 % zero_less_double_add_iff_zero_less_single_add
% 0.22/0.46 thf(fact_3089_sum__squares__eq__zero__iff,axiom,
% 0.22/0.46 ! [X: real,Y4: real] :
% 0.22/0.46 ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) )
% 0.22/0.46 = zero_zero_real )
% 0.22/0.46 = ( ( X = zero_zero_real )
% 0.22/0.46 & ( Y4 = zero_zero_real ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % sum_squares_eq_zero_iff
% 0.22/0.46 thf(fact_3090_sum__squares__eq__zero__iff,axiom,
% 0.22/0.46 ! [X: rat,Y4: rat] :
% 0.22/0.46 ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y4 @ Y4 ) )
% 0.22/0.46 = zero_zero_rat )
% 0.22/0.46 = ( ( X = zero_zero_rat )
% 0.22/0.46 & ( Y4 = zero_zero_rat ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % sum_squares_eq_zero_iff
% 0.22/0.46 thf(fact_3091_sum__squares__eq__zero__iff,axiom,
% 0.22/0.46 ! [X: int,Y4: int] :
% 0.22/0.46 ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y4 @ Y4 ) )
% 0.22/0.46 = zero_zero_int )
% 0.22/0.46 = ( ( X = zero_zero_int )
% 0.22/0.46 & ( Y4 = zero_zero_int ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % sum_squares_eq_zero_iff
% 0.22/0.46 thf(fact_3092_le__add__diff__inverse,axiom,
% 0.22/0.46 ! [B: rat,A: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ B @ A )
% 0.22/0.46 => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 0.22/0.46 = A ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_add_diff_inverse
% 0.22/0.46 thf(fact_3093_le__add__diff__inverse,axiom,
% 0.22/0.46 ! [B: nat,A: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ B @ A )
% 0.22/0.46 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 0.22/0.46 = A ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_add_diff_inverse
% 0.22/0.46 thf(fact_3094_le__add__diff__inverse,axiom,
% 0.22/0.46 ! [B: int,A: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ B @ A )
% 0.22/0.46 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 0.22/0.46 = A ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_add_diff_inverse
% 0.22/0.46 thf(fact_3095_le__add__diff__inverse,axiom,
% 0.22/0.46 ! [B: real,A: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ B @ A )
% 0.22/0.46 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 0.22/0.46 = A ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_add_diff_inverse
% 0.22/0.46 thf(fact_3096_le__add__diff__inverse2,axiom,
% 0.22/0.46 ! [B: rat,A: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ B @ A )
% 0.22/0.46 => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 0.22/0.46 = A ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_add_diff_inverse2
% 0.22/0.46 thf(fact_3097_le__add__diff__inverse2,axiom,
% 0.22/0.46 ! [B: nat,A: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ B @ A )
% 0.22/0.46 => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
% 0.22/0.46 = A ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_add_diff_inverse2
% 0.22/0.46 thf(fact_3098_le__add__diff__inverse2,axiom,
% 0.22/0.46 ! [B: int,A: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ B @ A )
% 0.22/0.46 => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.22/0.46 = A ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_add_diff_inverse2
% 0.22/0.46 thf(fact_3099_le__add__diff__inverse2,axiom,
% 0.22/0.46 ! [B: real,A: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ B @ A )
% 0.22/0.46 => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 0.22/0.46 = A ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_add_diff_inverse2
% 0.22/0.46 thf(fact_3100_diff__add__zero,axiom,
% 0.22/0.46 ! [A: nat,B: nat] :
% 0.22/0.46 ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.22/0.46 = zero_zero_nat ) ).
% 0.22/0.46
% 0.22/0.46 % diff_add_zero
% 0.22/0.46 thf(fact_3101_distrib__left__numeral,axiom,
% 0.22/0.46 ! [V: num,B: complex,C: complex] :
% 0.22/0.46 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
% 0.22/0.46 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % distrib_left_numeral
% 0.22/0.46 thf(fact_3102_distrib__left__numeral,axiom,
% 0.22/0.46 ! [V: num,B: real,C: real] :
% 0.22/0.46 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
% 0.22/0.46 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % distrib_left_numeral
% 0.22/0.46 thf(fact_3103_distrib__left__numeral,axiom,
% 0.22/0.46 ! [V: num,B: rat,C: rat] :
% 0.22/0.46 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
% 0.22/0.46 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % distrib_left_numeral
% 0.22/0.46 thf(fact_3104_distrib__left__numeral,axiom,
% 0.22/0.46 ! [V: num,B: nat,C: nat] :
% 0.22/0.46 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
% 0.22/0.46 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % distrib_left_numeral
% 0.22/0.46 thf(fact_3105_distrib__left__numeral,axiom,
% 0.22/0.46 ! [V: num,B: int,C: int] :
% 0.22/0.46 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
% 0.22/0.46 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % distrib_left_numeral
% 0.22/0.46 thf(fact_3106_distrib__right__numeral,axiom,
% 0.22/0.46 ! [A: complex,B: complex,V: num] :
% 0.22/0.46 ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 0.22/0.46 = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % distrib_right_numeral
% 0.22/0.46 thf(fact_3107_distrib__right__numeral,axiom,
% 0.22/0.46 ! [A: real,B: real,V: num] :
% 0.22/0.46 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 0.22/0.46 = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % distrib_right_numeral
% 0.22/0.46 thf(fact_3108_distrib__right__numeral,axiom,
% 0.22/0.46 ! [A: rat,B: rat,V: num] :
% 0.22/0.46 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 0.22/0.46 = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % distrib_right_numeral
% 0.22/0.46 thf(fact_3109_distrib__right__numeral,axiom,
% 0.22/0.46 ! [A: nat,B: nat,V: num] :
% 0.22/0.46 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
% 0.22/0.46 = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % distrib_right_numeral
% 0.22/0.46 thf(fact_3110_distrib__right__numeral,axiom,
% 0.22/0.46 ! [A: int,B: int,V: num] :
% 0.22/0.46 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 0.22/0.46 = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % distrib_right_numeral
% 0.22/0.46 thf(fact_3111_dvd__add__times__triv__left__iff,axiom,
% 0.22/0.46 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.22/0.46 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
% 0.22/0.46 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_times_triv_left_iff
% 0.22/0.46 thf(fact_3112_dvd__add__times__triv__left__iff,axiom,
% 0.22/0.46 ! [A: real,C: real,B: real] :
% 0.22/0.46 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
% 0.22/0.46 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_times_triv_left_iff
% 0.22/0.46 thf(fact_3113_dvd__add__times__triv__left__iff,axiom,
% 0.22/0.46 ! [A: rat,C: rat,B: rat] :
% 0.22/0.46 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
% 0.22/0.46 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_times_triv_left_iff
% 0.22/0.46 thf(fact_3114_dvd__add__times__triv__left__iff,axiom,
% 0.22/0.46 ! [A: nat,C: nat,B: nat] :
% 0.22/0.46 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
% 0.22/0.46 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_times_triv_left_iff
% 0.22/0.46 thf(fact_3115_dvd__add__times__triv__left__iff,axiom,
% 0.22/0.46 ! [A: int,C: int,B: int] :
% 0.22/0.46 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
% 0.22/0.46 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_times_triv_left_iff
% 0.22/0.46 thf(fact_3116_dvd__add__times__triv__right__iff,axiom,
% 0.22/0.46 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.22/0.46 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
% 0.22/0.46 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_times_triv_right_iff
% 0.22/0.46 thf(fact_3117_dvd__add__times__triv__right__iff,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
% 0.22/0.46 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_times_triv_right_iff
% 0.22/0.46 thf(fact_3118_dvd__add__times__triv__right__iff,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
% 0.22/0.46 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_times_triv_right_iff
% 0.22/0.46 thf(fact_3119_dvd__add__times__triv__right__iff,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
% 0.22/0.46 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_times_triv_right_iff
% 0.22/0.46 thf(fact_3120_dvd__add__times__triv__right__iff,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
% 0.22/0.46 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_times_triv_right_iff
% 0.22/0.46 thf(fact_3121_div__add,axiom,
% 0.22/0.46 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.22/0.46 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.22/0.46 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.22/0.46 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.22/0.46 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % div_add
% 0.22/0.46 thf(fact_3122_div__add,axiom,
% 0.22/0.46 ! [C: nat,A: nat,B: nat] :
% 0.22/0.46 ( ( dvd_dvd_nat @ C @ A )
% 0.22/0.46 => ( ( dvd_dvd_nat @ C @ B )
% 0.22/0.46 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % div_add
% 0.22/0.46 thf(fact_3123_div__add,axiom,
% 0.22/0.46 ! [C: int,A: int,B: int] :
% 0.22/0.46 ( ( dvd_dvd_int @ C @ A )
% 0.22/0.46 => ( ( dvd_dvd_int @ C @ B )
% 0.22/0.46 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % div_add
% 0.22/0.46 thf(fact_3124_of__nat__add,axiom,
% 0.22/0.46 ! [M: nat,N: nat] :
% 0.22/0.46 ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N ) )
% 0.22/0.46 = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % of_nat_add
% 0.22/0.46 thf(fact_3125_of__nat__add,axiom,
% 0.22/0.46 ! [M: nat,N: nat] :
% 0.22/0.46 ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
% 0.22/0.46 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % of_nat_add
% 0.22/0.46 thf(fact_3126_of__nat__add,axiom,
% 0.22/0.46 ! [M: nat,N: nat] :
% 0.22/0.46 ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
% 0.22/0.46 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % of_nat_add
% 0.22/0.46 thf(fact_3127_of__nat__add,axiom,
% 0.22/0.46 ! [M: nat,N: nat] :
% 0.22/0.46 ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
% 0.22/0.46 = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % of_nat_add
% 0.22/0.46 thf(fact_3128_of__nat__add,axiom,
% 0.22/0.46 ! [M: nat,N: nat] :
% 0.22/0.46 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
% 0.22/0.46 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % of_nat_add
% 0.22/0.46 thf(fact_3129_add__gr__0,axiom,
% 0.22/0.46 ! [M: nat,N: nat] :
% 0.22/0.46 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 0.22/0.46 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.22/0.46 | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_gr_0
% 0.22/0.46 thf(fact_3130_mult__Suc__right,axiom,
% 0.22/0.46 ! [M: nat,N: nat] :
% 0.22/0.46 ( ( times_times_nat @ M @ ( suc @ N ) )
% 0.22/0.46 = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % mult_Suc_right
% 0.22/0.46 thf(fact_3131_Nat_Odiff__diff__right,axiom,
% 0.22/0.46 ! [K: nat,J: nat,I: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ K @ J )
% 0.22/0.46 => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.22/0.46 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % Nat.diff_diff_right
% 0.22/0.46 thf(fact_3132_Nat_Oadd__diff__assoc2,axiom,
% 0.22/0.46 ! [K: nat,J: nat,I: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ K @ J )
% 0.22/0.46 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.22/0.46 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % Nat.add_diff_assoc2
% 0.22/0.46 thf(fact_3133_Nat_Oadd__diff__assoc,axiom,
% 0.22/0.46 ! [K: nat,J: nat,I: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ K @ J )
% 0.22/0.46 => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.22/0.46 = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % Nat.add_diff_assoc
% 0.22/0.46 thf(fact_3134_triangle__Suc,axiom,
% 0.22/0.46 ! [N: nat] :
% 0.22/0.46 ( ( nat_triangle @ ( suc @ N ) )
% 0.22/0.46 = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % triangle_Suc
% 0.22/0.46 thf(fact_3135_div__mult__self1,axiom,
% 0.22/0.46 ! [B: nat,A: nat,C: nat] :
% 0.22/0.46 ( ( B != zero_zero_nat )
% 0.22/0.46 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 0.22/0.46 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % div_mult_self1
% 0.22/0.46 thf(fact_3136_div__mult__self1,axiom,
% 0.22/0.46 ! [B: int,A: int,C: int] :
% 0.22/0.46 ( ( B != zero_zero_int )
% 0.22/0.46 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 0.22/0.46 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % div_mult_self1
% 0.22/0.46 thf(fact_3137_div__mult__self2,axiom,
% 0.22/0.46 ! [B: nat,A: nat,C: nat] :
% 0.22/0.46 ( ( B != zero_zero_nat )
% 0.22/0.46 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 0.22/0.46 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % div_mult_self2
% 0.22/0.46 thf(fact_3138_div__mult__self2,axiom,
% 0.22/0.46 ! [B: int,A: int,C: int] :
% 0.22/0.46 ( ( B != zero_zero_int )
% 0.22/0.46 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 0.22/0.46 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % div_mult_self2
% 0.22/0.46 thf(fact_3139_div__mult__self3,axiom,
% 0.22/0.46 ! [B: nat,C: nat,A: nat] :
% 0.22/0.46 ( ( B != zero_zero_nat )
% 0.22/0.46 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 0.22/0.46 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % div_mult_self3
% 0.22/0.46 thf(fact_3140_div__mult__self3,axiom,
% 0.22/0.46 ! [B: int,C: int,A: int] :
% 0.22/0.46 ( ( B != zero_zero_int )
% 0.22/0.46 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 0.22/0.46 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % div_mult_self3
% 0.22/0.46 thf(fact_3141_div__mult__self4,axiom,
% 0.22/0.46 ! [B: nat,C: nat,A: nat] :
% 0.22/0.46 ( ( B != zero_zero_nat )
% 0.22/0.46 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 0.22/0.46 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % div_mult_self4
% 0.22/0.46 thf(fact_3142_div__mult__self4,axiom,
% 0.22/0.46 ! [B: int,C: int,A: int] :
% 0.22/0.46 ( ( B != zero_zero_int )
% 0.22/0.46 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 0.22/0.46 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % div_mult_self4
% 0.22/0.46 thf(fact_3143_numeral__plus__one,axiom,
% 0.22/0.46 ! [N: num] :
% 0.22/0.46 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
% 0.22/0.46 = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % numeral_plus_one
% 0.22/0.46 thf(fact_3144_numeral__plus__one,axiom,
% 0.22/0.46 ! [N: num] :
% 0.22/0.46 ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 0.22/0.46 = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % numeral_plus_one
% 0.22/0.46 thf(fact_3145_numeral__plus__one,axiom,
% 0.22/0.46 ! [N: num] :
% 0.22/0.46 ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 0.22/0.46 = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % numeral_plus_one
% 0.22/0.46 thf(fact_3146_numeral__plus__one,axiom,
% 0.22/0.46 ! [N: num] :
% 0.22/0.46 ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 0.22/0.46 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % numeral_plus_one
% 0.22/0.46 thf(fact_3147_numeral__plus__one,axiom,
% 0.22/0.46 ! [N: num] :
% 0.22/0.46 ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 0.22/0.46 = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % numeral_plus_one
% 0.22/0.46 thf(fact_3148_one__plus__numeral,axiom,
% 0.22/0.46 ! [N: num] :
% 0.22/0.46 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
% 0.22/0.46 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % one_plus_numeral
% 0.22/0.46 thf(fact_3149_one__plus__numeral,axiom,
% 0.22/0.46 ! [N: num] :
% 0.22/0.46 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 0.22/0.46 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % one_plus_numeral
% 0.22/0.46 thf(fact_3150_one__plus__numeral,axiom,
% 0.22/0.46 ! [N: num] :
% 0.22/0.46 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 0.22/0.46 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % one_plus_numeral
% 0.22/0.46 thf(fact_3151_one__plus__numeral,axiom,
% 0.22/0.46 ! [N: num] :
% 0.22/0.46 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 0.22/0.46 = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % one_plus_numeral
% 0.22/0.46 thf(fact_3152_one__plus__numeral,axiom,
% 0.22/0.46 ! [N: num] :
% 0.22/0.46 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 0.22/0.46 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % one_plus_numeral
% 0.22/0.46 thf(fact_3153_of__nat__Suc,axiom,
% 0.22/0.46 ! [M: nat] :
% 0.22/0.46 ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 0.22/0.46 = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % of_nat_Suc
% 0.22/0.46 thf(fact_3154_of__nat__Suc,axiom,
% 0.22/0.46 ! [M: nat] :
% 0.22/0.46 ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 0.22/0.46 = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % of_nat_Suc
% 0.22/0.46 thf(fact_3155_of__nat__Suc,axiom,
% 0.22/0.46 ! [M: nat] :
% 0.22/0.46 ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 0.22/0.46 = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % of_nat_Suc
% 0.22/0.46 thf(fact_3156_of__nat__Suc,axiom,
% 0.22/0.46 ! [M: nat] :
% 0.22/0.46 ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 0.22/0.46 = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % of_nat_Suc
% 0.22/0.46 thf(fact_3157_of__nat__Suc,axiom,
% 0.22/0.46 ! [M: nat] :
% 0.22/0.46 ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 0.22/0.46 = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % of_nat_Suc
% 0.22/0.46 thf(fact_3158_diff__Suc__diff__eq2,axiom,
% 0.22/0.46 ! [K: nat,J: nat,I: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ K @ J )
% 0.22/0.46 => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
% 0.22/0.46 = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_Suc_diff_eq2
% 0.22/0.46 thf(fact_3159_diff__Suc__diff__eq1,axiom,
% 0.22/0.46 ! [K: nat,J: nat,I: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ K @ J )
% 0.22/0.46 => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 0.22/0.46 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_Suc_diff_eq1
% 0.22/0.46 thf(fact_3160_one__add__one,axiom,
% 0.22/0.46 ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
% 0.22/0.46 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % one_add_one
% 0.22/0.46 thf(fact_3161_one__add__one,axiom,
% 0.22/0.46 ( ( plus_plus_real @ one_one_real @ one_one_real )
% 0.22/0.46 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % one_add_one
% 0.22/0.46 thf(fact_3162_one__add__one,axiom,
% 0.22/0.46 ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
% 0.22/0.46 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % one_add_one
% 0.22/0.46 thf(fact_3163_one__add__one,axiom,
% 0.22/0.46 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 0.22/0.46 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % one_add_one
% 0.22/0.46 thf(fact_3164_one__add__one,axiom,
% 0.22/0.46 ( ( plus_plus_int @ one_one_int @ one_one_int )
% 0.22/0.46 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % one_add_one
% 0.22/0.46 thf(fact_3165_odd__add,axiom,
% 0.22/0.46 ! [A: code_integer,B: code_integer] :
% 0.22/0.46 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
% 0.22/0.46 = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.22/0.46 != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % odd_add
% 0.22/0.46 thf(fact_3166_odd__add,axiom,
% 0.22/0.46 ! [A: nat,B: nat] :
% 0.22/0.46 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
% 0.22/0.46 = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.22/0.46 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % odd_add
% 0.22/0.46 thf(fact_3167_odd__add,axiom,
% 0.22/0.46 ! [A: int,B: int] :
% 0.22/0.46 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
% 0.22/0.46 = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.22/0.46 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % odd_add
% 0.22/0.46 thf(fact_3168_even__add,axiom,
% 0.22/0.46 ! [A: code_integer,B: code_integer] :
% 0.22/0.46 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.22/0.46 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.46 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % even_add
% 0.22/0.46 thf(fact_3169_even__add,axiom,
% 0.22/0.46 ! [A: nat,B: nat] :
% 0.22/0.46 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
% 0.22/0.46 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.46 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % even_add
% 0.22/0.46 thf(fact_3170_even__add,axiom,
% 0.22/0.46 ! [A: int,B: int] :
% 0.22/0.46 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
% 0.22/0.46 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.46 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % even_add
% 0.22/0.46 thf(fact_3171_add__2__eq__Suc,axiom,
% 0.22/0.46 ! [N: nat] :
% 0.22/0.46 ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.46 = ( suc @ ( suc @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_2_eq_Suc
% 0.22/0.46 thf(fact_3172_add__2__eq__Suc_H,axiom,
% 0.22/0.46 ! [N: nat] :
% 0.22/0.46 ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.46 = ( suc @ ( suc @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_2_eq_Suc'
% 0.22/0.46 thf(fact_3173_add__self__div__2,axiom,
% 0.22/0.46 ! [M: nat] :
% 0.22/0.46 ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.46 = M ) ).
% 0.22/0.46
% 0.22/0.46 % add_self_div_2
% 0.22/0.46 thf(fact_3174_even__plus__one__iff,axiom,
% 0.22/0.46 ! [A: code_integer] :
% 0.22/0.46 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
% 0.22/0.46 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % even_plus_one_iff
% 0.22/0.46 thf(fact_3175_even__plus__one__iff,axiom,
% 0.22/0.46 ! [A: nat] :
% 0.22/0.46 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
% 0.22/0.46 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % even_plus_one_iff
% 0.22/0.46 thf(fact_3176_even__plus__one__iff,axiom,
% 0.22/0.46 ! [A: int] :
% 0.22/0.46 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
% 0.22/0.46 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % even_plus_one_iff
% 0.22/0.46 thf(fact_3177_sum__power2__eq__zero__iff,axiom,
% 0.22/0.46 ! [X: rat,Y4: rat] :
% 0.22/0.46 ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.46 = zero_zero_rat )
% 0.22/0.46 = ( ( X = zero_zero_rat )
% 0.22/0.46 & ( Y4 = zero_zero_rat ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % sum_power2_eq_zero_iff
% 0.22/0.46 thf(fact_3178_sum__power2__eq__zero__iff,axiom,
% 0.22/0.46 ! [X: int,Y4: int] :
% 0.22/0.46 ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.46 = zero_zero_int )
% 0.22/0.46 = ( ( X = zero_zero_int )
% 0.22/0.46 & ( Y4 = zero_zero_int ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % sum_power2_eq_zero_iff
% 0.22/0.46 thf(fact_3179_sum__power2__eq__zero__iff,axiom,
% 0.22/0.46 ! [X: real,Y4: real] :
% 0.22/0.46 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.46 = zero_zero_real )
% 0.22/0.46 = ( ( X = zero_zero_real )
% 0.22/0.46 & ( Y4 = zero_zero_real ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % sum_power2_eq_zero_iff
% 0.22/0.46 thf(fact_3180_even__diff,axiom,
% 0.22/0.46 ! [A: code_integer,B: code_integer] :
% 0.22/0.46 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.22/0.46 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % even_diff
% 0.22/0.46 thf(fact_3181_even__diff,axiom,
% 0.22/0.46 ! [A: int,B: int] :
% 0.22/0.46 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
% 0.22/0.46 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % even_diff
% 0.22/0.46 thf(fact_3182_div__Suc__eq__div__add3,axiom,
% 0.22/0.46 ! [M: nat,N: nat] :
% 0.22/0.46 ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 0.22/0.46 = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % div_Suc_eq_div_add3
% 0.22/0.46 thf(fact_3183_Suc__div__eq__add3__div__numeral,axiom,
% 0.22/0.46 ! [M: nat,V: num] :
% 0.22/0.46 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 0.22/0.46 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % Suc_div_eq_add3_div_numeral
% 0.22/0.46 thf(fact_3184_odd__succ__div__two,axiom,
% 0.22/0.46 ! [A: code_integer] :
% 0.22/0.46 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.46 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.46 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % odd_succ_div_two
% 0.22/0.46 thf(fact_3185_odd__succ__div__two,axiom,
% 0.22/0.46 ! [A: nat] :
% 0.22/0.46 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.46 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.46 = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % odd_succ_div_two
% 0.22/0.46 thf(fact_3186_odd__succ__div__two,axiom,
% 0.22/0.46 ! [A: int] :
% 0.22/0.46 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.46 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.46 = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % odd_succ_div_two
% 0.22/0.46 thf(fact_3187_even__succ__div__two,axiom,
% 0.22/0.46 ! [A: code_integer] :
% 0.22/0.46 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.46 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.46 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % even_succ_div_two
% 0.22/0.46 thf(fact_3188_even__succ__div__two,axiom,
% 0.22/0.46 ! [A: nat] :
% 0.22/0.46 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.46 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.46 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % even_succ_div_two
% 0.22/0.46 thf(fact_3189_even__succ__div__two,axiom,
% 0.22/0.46 ! [A: int] :
% 0.22/0.46 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.46 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.46 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % even_succ_div_two
% 0.22/0.46 thf(fact_3190_even__succ__div__2,axiom,
% 0.22/0.46 ! [A: code_integer] :
% 0.22/0.46 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.46 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.46 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % even_succ_div_2
% 0.22/0.46 thf(fact_3191_even__succ__div__2,axiom,
% 0.22/0.46 ! [A: nat] :
% 0.22/0.46 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.46 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.46 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % even_succ_div_2
% 0.22/0.46 thf(fact_3192_even__succ__div__2,axiom,
% 0.22/0.46 ! [A: int] :
% 0.22/0.46 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.46 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.46 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % even_succ_div_2
% 0.22/0.46 thf(fact_3193_even__diff__nat,axiom,
% 0.22/0.46 ! [M: nat,N: nat] :
% 0.22/0.46 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
% 0.22/0.46 = ( ( ord_less_nat @ M @ N )
% 0.22/0.46 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % even_diff_nat
% 0.22/0.46 thf(fact_3194_odd__two__times__div__two__succ,axiom,
% 0.22/0.46 ! [A: code_integer] :
% 0.22/0.46 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.46 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
% 0.22/0.46 = A ) ) ).
% 0.22/0.46
% 0.22/0.46 % odd_two_times_div_two_succ
% 0.22/0.46 thf(fact_3195_odd__two__times__div__two__succ,axiom,
% 0.22/0.46 ! [A: nat] :
% 0.22/0.46 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.46 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
% 0.22/0.46 = A ) ) ).
% 0.22/0.46
% 0.22/0.46 % odd_two_times_div_two_succ
% 0.22/0.46 thf(fact_3196_odd__two__times__div__two__succ,axiom,
% 0.22/0.46 ! [A: int] :
% 0.22/0.46 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.46 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
% 0.22/0.46 = A ) ) ).
% 0.22/0.46
% 0.22/0.46 % odd_two_times_div_two_succ
% 0.22/0.46 thf(fact_3197_set__bit__0,axiom,
% 0.22/0.46 ! [A: int] :
% 0.22/0.46 ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
% 0.22/0.46 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % set_bit_0
% 0.22/0.46 thf(fact_3198_set__bit__0,axiom,
% 0.22/0.46 ! [A: nat] :
% 0.22/0.46 ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
% 0.22/0.46 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % set_bit_0
% 0.22/0.46 thf(fact_3199_set__bit__0,axiom,
% 0.22/0.46 ! [A: code_integer] :
% 0.22/0.46 ( ( bit_se2793503036327961859nteger @ zero_zero_nat @ A )
% 0.22/0.46 = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % set_bit_0
% 0.22/0.46 thf(fact_3200_even__succ__div__exp,axiom,
% 0.22/0.46 ! [A: code_integer,N: nat] :
% 0.22/0.46 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.46 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.46 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.46 = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % even_succ_div_exp
% 0.22/0.46 thf(fact_3201_even__succ__div__exp,axiom,
% 0.22/0.46 ! [A: nat,N: nat] :
% 0.22/0.46 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.46 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.46 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.46 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % even_succ_div_exp
% 0.22/0.46 thf(fact_3202_even__succ__div__exp,axiom,
% 0.22/0.46 ! [A: int,N: nat] :
% 0.22/0.46 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.46 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.46 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.46 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % even_succ_div_exp
% 0.22/0.46 thf(fact_3203_psubsetD,axiom,
% 0.22/0.46 ! [A2: set_complex,B4: set_complex,C: complex] :
% 0.22/0.46 ( ( ord_less_set_complex @ A2 @ B4 )
% 0.22/0.46 => ( ( member_complex @ C @ A2 )
% 0.22/0.46 => ( member_complex @ C @ B4 ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % psubsetD
% 0.22/0.46 thf(fact_3204_psubsetD,axiom,
% 0.22/0.46 ! [A2: set_real,B4: set_real,C: real] :
% 0.22/0.46 ( ( ord_less_set_real @ A2 @ B4 )
% 0.22/0.46 => ( ( member_real @ C @ A2 )
% 0.22/0.46 => ( member_real @ C @ B4 ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % psubsetD
% 0.22/0.46 thf(fact_3205_psubsetD,axiom,
% 0.22/0.46 ! [A2: set_set_nat,B4: set_set_nat,C: set_nat] :
% 0.22/0.46 ( ( ord_less_set_set_nat @ A2 @ B4 )
% 0.22/0.46 => ( ( member_set_nat @ C @ A2 )
% 0.22/0.46 => ( member_set_nat @ C @ B4 ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % psubsetD
% 0.22/0.46 thf(fact_3206_psubsetD,axiom,
% 0.22/0.46 ! [A2: set_nat,B4: set_nat,C: nat] :
% 0.22/0.46 ( ( ord_less_set_nat @ A2 @ B4 )
% 0.22/0.46 => ( ( member_nat @ C @ A2 )
% 0.22/0.46 => ( member_nat @ C @ B4 ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % psubsetD
% 0.22/0.46 thf(fact_3207_psubsetD,axiom,
% 0.22/0.46 ! [A2: set_int,B4: set_int,C: int] :
% 0.22/0.46 ( ( ord_less_set_int @ A2 @ B4 )
% 0.22/0.46 => ( ( member_int @ C @ A2 )
% 0.22/0.46 => ( member_int @ C @ B4 ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % psubsetD
% 0.22/0.46 thf(fact_3208_enat__less__induct,axiom,
% 0.22/0.46 ! [P: extended_enat > $o,N: extended_enat] :
% 0.22/0.46 ( ! [N2: extended_enat] :
% 0.22/0.46 ( ! [M2: extended_enat] :
% 0.22/0.46 ( ( ord_le72135733267957522d_enat @ M2 @ N2 )
% 0.22/0.46 => ( P @ M2 ) )
% 0.22/0.46 => ( P @ N2 ) )
% 0.22/0.46 => ( P @ N ) ) ).
% 0.22/0.46
% 0.22/0.46 % enat_less_induct
% 0.22/0.46 thf(fact_3209_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % ab_semigroup_add_class.add_ac(1)
% 0.22/0.46 thf(fact_3210_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % ab_semigroup_add_class.add_ac(1)
% 0.22/0.46 thf(fact_3211_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % ab_semigroup_add_class.add_ac(1)
% 0.22/0.46 thf(fact_3212_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % ab_semigroup_add_class.add_ac(1)
% 0.22/0.46 thf(fact_3213_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.22/0.46 ! [I: real,J: real,K: real,L: real] :
% 0.22/0.46 ( ( ( I = J )
% 0.22/0.46 & ( K = L ) )
% 0.22/0.46 => ( ( plus_plus_real @ I @ K )
% 0.22/0.46 = ( plus_plus_real @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_semiring(4)
% 0.22/0.46 thf(fact_3214_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.22/0.46 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.22/0.46 ( ( ( I = J )
% 0.22/0.46 & ( K = L ) )
% 0.22/0.46 => ( ( plus_plus_rat @ I @ K )
% 0.22/0.46 = ( plus_plus_rat @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_semiring(4)
% 0.22/0.46 thf(fact_3215_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.22/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.22/0.46 ( ( ( I = J )
% 0.22/0.46 & ( K = L ) )
% 0.22/0.46 => ( ( plus_plus_nat @ I @ K )
% 0.22/0.46 = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_semiring(4)
% 0.22/0.46 thf(fact_3216_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.22/0.46 ! [I: int,J: int,K: int,L: int] :
% 0.22/0.46 ( ( ( I = J )
% 0.22/0.46 & ( K = L ) )
% 0.22/0.46 => ( ( plus_plus_int @ I @ K )
% 0.22/0.46 = ( plus_plus_int @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_semiring(4)
% 0.22/0.46 thf(fact_3217_group__cancel_Oadd1,axiom,
% 0.22/0.46 ! [A2: real,K: real,A: real,B: real] :
% 0.22/0.46 ( ( A2
% 0.22/0.46 = ( plus_plus_real @ K @ A ) )
% 0.22/0.46 => ( ( plus_plus_real @ A2 @ B )
% 0.22/0.46 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % group_cancel.add1
% 0.22/0.46 thf(fact_3218_group__cancel_Oadd1,axiom,
% 0.22/0.46 ! [A2: rat,K: rat,A: rat,B: rat] :
% 0.22/0.46 ( ( A2
% 0.22/0.46 = ( plus_plus_rat @ K @ A ) )
% 0.22/0.46 => ( ( plus_plus_rat @ A2 @ B )
% 0.22/0.46 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % group_cancel.add1
% 0.22/0.46 thf(fact_3219_group__cancel_Oadd1,axiom,
% 0.22/0.46 ! [A2: nat,K: nat,A: nat,B: nat] :
% 0.22/0.46 ( ( A2
% 0.22/0.46 = ( plus_plus_nat @ K @ A ) )
% 0.22/0.46 => ( ( plus_plus_nat @ A2 @ B )
% 0.22/0.46 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % group_cancel.add1
% 0.22/0.46 thf(fact_3220_group__cancel_Oadd1,axiom,
% 0.22/0.46 ! [A2: int,K: int,A: int,B: int] :
% 0.22/0.46 ( ( A2
% 0.22/0.46 = ( plus_plus_int @ K @ A ) )
% 0.22/0.46 => ( ( plus_plus_int @ A2 @ B )
% 0.22/0.46 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % group_cancel.add1
% 0.22/0.46 thf(fact_3221_group__cancel_Oadd2,axiom,
% 0.22/0.46 ! [B4: real,K: real,B: real,A: real] :
% 0.22/0.46 ( ( B4
% 0.22/0.46 = ( plus_plus_real @ K @ B ) )
% 0.22/0.46 => ( ( plus_plus_real @ A @ B4 )
% 0.22/0.46 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % group_cancel.add2
% 0.22/0.46 thf(fact_3222_group__cancel_Oadd2,axiom,
% 0.22/0.46 ! [B4: rat,K: rat,B: rat,A: rat] :
% 0.22/0.46 ( ( B4
% 0.22/0.46 = ( plus_plus_rat @ K @ B ) )
% 0.22/0.46 => ( ( plus_plus_rat @ A @ B4 )
% 0.22/0.46 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % group_cancel.add2
% 0.22/0.46 thf(fact_3223_group__cancel_Oadd2,axiom,
% 0.22/0.46 ! [B4: nat,K: nat,B: nat,A: nat] :
% 0.22/0.46 ( ( B4
% 0.22/0.46 = ( plus_plus_nat @ K @ B ) )
% 0.22/0.46 => ( ( plus_plus_nat @ A @ B4 )
% 0.22/0.46 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % group_cancel.add2
% 0.22/0.46 thf(fact_3224_group__cancel_Oadd2,axiom,
% 0.22/0.46 ! [B4: int,K: int,B: int,A: int] :
% 0.22/0.46 ( ( B4
% 0.22/0.46 = ( plus_plus_int @ K @ B ) )
% 0.22/0.46 => ( ( plus_plus_int @ A @ B4 )
% 0.22/0.46 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % group_cancel.add2
% 0.22/0.46 thf(fact_3225_add_Oassoc,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add.assoc
% 0.22/0.46 thf(fact_3226_add_Oassoc,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add.assoc
% 0.22/0.46 thf(fact_3227_add_Oassoc,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add.assoc
% 0.22/0.46 thf(fact_3228_add_Oassoc,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add.assoc
% 0.22/0.46 thf(fact_3229_add_Oleft__cancel,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( ( plus_plus_real @ A @ B )
% 0.22/0.46 = ( plus_plus_real @ A @ C ) )
% 0.22/0.46 = ( B = C ) ) ).
% 0.22/0.46
% 0.22/0.46 % add.left_cancel
% 0.22/0.46 thf(fact_3230_add_Oleft__cancel,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( ( plus_plus_rat @ A @ B )
% 0.22/0.46 = ( plus_plus_rat @ A @ C ) )
% 0.22/0.46 = ( B = C ) ) ).
% 0.22/0.46
% 0.22/0.46 % add.left_cancel
% 0.22/0.46 thf(fact_3231_add_Oleft__cancel,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( ( plus_plus_int @ A @ B )
% 0.22/0.46 = ( plus_plus_int @ A @ C ) )
% 0.22/0.46 = ( B = C ) ) ).
% 0.22/0.46
% 0.22/0.46 % add.left_cancel
% 0.22/0.46 thf(fact_3232_add_Oright__cancel,axiom,
% 0.22/0.46 ! [B: real,A: real,C: real] :
% 0.22/0.46 ( ( ( plus_plus_real @ B @ A )
% 0.22/0.46 = ( plus_plus_real @ C @ A ) )
% 0.22/0.46 = ( B = C ) ) ).
% 0.22/0.46
% 0.22/0.46 % add.right_cancel
% 0.22/0.46 thf(fact_3233_add_Oright__cancel,axiom,
% 0.22/0.46 ! [B: rat,A: rat,C: rat] :
% 0.22/0.46 ( ( ( plus_plus_rat @ B @ A )
% 0.22/0.46 = ( plus_plus_rat @ C @ A ) )
% 0.22/0.46 = ( B = C ) ) ).
% 0.22/0.46
% 0.22/0.46 % add.right_cancel
% 0.22/0.46 thf(fact_3234_add_Oright__cancel,axiom,
% 0.22/0.46 ! [B: int,A: int,C: int] :
% 0.22/0.46 ( ( ( plus_plus_int @ B @ A )
% 0.22/0.46 = ( plus_plus_int @ C @ A ) )
% 0.22/0.46 = ( B = C ) ) ).
% 0.22/0.46
% 0.22/0.46 % add.right_cancel
% 0.22/0.46 thf(fact_3235_add_Ocommute,axiom,
% 0.22/0.46 ( plus_plus_real
% 0.22/0.46 = ( ^ [A3: real,B2: real] : ( plus_plus_real @ B2 @ A3 ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add.commute
% 0.22/0.46 thf(fact_3236_add_Ocommute,axiom,
% 0.22/0.46 ( plus_plus_rat
% 0.22/0.46 = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ B2 @ A3 ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add.commute
% 0.22/0.46 thf(fact_3237_add_Ocommute,axiom,
% 0.22/0.46 ( plus_plus_nat
% 0.22/0.46 = ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add.commute
% 0.22/0.46 thf(fact_3238_add_Ocommute,axiom,
% 0.22/0.46 ( plus_plus_int
% 0.22/0.46 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add.commute
% 0.22/0.46 thf(fact_3239_add_Oleft__commute,axiom,
% 0.22/0.46 ! [B: real,A: real,C: real] :
% 0.22/0.46 ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 0.22/0.46 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add.left_commute
% 0.22/0.46 thf(fact_3240_add_Oleft__commute,axiom,
% 0.22/0.46 ! [B: rat,A: rat,C: rat] :
% 0.22/0.46 ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
% 0.22/0.46 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add.left_commute
% 0.22/0.46 thf(fact_3241_add_Oleft__commute,axiom,
% 0.22/0.46 ! [B: nat,A: nat,C: nat] :
% 0.22/0.46 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 0.22/0.46 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add.left_commute
% 0.22/0.46 thf(fact_3242_add_Oleft__commute,axiom,
% 0.22/0.46 ! [B: int,A: int,C: int] :
% 0.22/0.46 ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 0.22/0.46 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add.left_commute
% 0.22/0.46 thf(fact_3243_add__left__imp__eq,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( ( plus_plus_real @ A @ B )
% 0.22/0.46 = ( plus_plus_real @ A @ C ) )
% 0.22/0.46 => ( B = C ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_left_imp_eq
% 0.22/0.46 thf(fact_3244_add__left__imp__eq,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( ( plus_plus_rat @ A @ B )
% 0.22/0.46 = ( plus_plus_rat @ A @ C ) )
% 0.22/0.46 => ( B = C ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_left_imp_eq
% 0.22/0.46 thf(fact_3245_add__left__imp__eq,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( ( plus_plus_nat @ A @ B )
% 0.22/0.46 = ( plus_plus_nat @ A @ C ) )
% 0.22/0.46 => ( B = C ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_left_imp_eq
% 0.22/0.46 thf(fact_3246_add__left__imp__eq,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( ( plus_plus_int @ A @ B )
% 0.22/0.46 = ( plus_plus_int @ A @ C ) )
% 0.22/0.46 => ( B = C ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_left_imp_eq
% 0.22/0.46 thf(fact_3247_add__right__imp__eq,axiom,
% 0.22/0.46 ! [B: real,A: real,C: real] :
% 0.22/0.46 ( ( ( plus_plus_real @ B @ A )
% 0.22/0.46 = ( plus_plus_real @ C @ A ) )
% 0.22/0.46 => ( B = C ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_right_imp_eq
% 0.22/0.46 thf(fact_3248_add__right__imp__eq,axiom,
% 0.22/0.46 ! [B: rat,A: rat,C: rat] :
% 0.22/0.46 ( ( ( plus_plus_rat @ B @ A )
% 0.22/0.46 = ( plus_plus_rat @ C @ A ) )
% 0.22/0.46 => ( B = C ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_right_imp_eq
% 0.22/0.46 thf(fact_3249_add__right__imp__eq,axiom,
% 0.22/0.46 ! [B: nat,A: nat,C: nat] :
% 0.22/0.46 ( ( ( plus_plus_nat @ B @ A )
% 0.22/0.46 = ( plus_plus_nat @ C @ A ) )
% 0.22/0.46 => ( B = C ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_right_imp_eq
% 0.22/0.46 thf(fact_3250_add__right__imp__eq,axiom,
% 0.22/0.46 ! [B: int,A: int,C: int] :
% 0.22/0.46 ( ( ( plus_plus_int @ B @ A )
% 0.22/0.46 = ( plus_plus_int @ C @ A ) )
% 0.22/0.46 => ( B = C ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_right_imp_eq
% 0.22/0.46 thf(fact_3251_is__num__normalize_I1_J,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % is_num_normalize(1)
% 0.22/0.46 thf(fact_3252_is__num__normalize_I1_J,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % is_num_normalize(1)
% 0.22/0.46 thf(fact_3253_is__num__normalize_I1_J,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % is_num_normalize(1)
% 0.22/0.46 thf(fact_3254_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.22/0.46 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.22/0.46 ( ( ( ord_less_eq_rat @ I @ J )
% 0.22/0.46 & ( K = L ) )
% 0.22/0.46 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_semiring(3)
% 0.22/0.46 thf(fact_3255_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.22/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.22/0.46 ( ( ( ord_less_eq_nat @ I @ J )
% 0.22/0.46 & ( K = L ) )
% 0.22/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_semiring(3)
% 0.22/0.46 thf(fact_3256_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.22/0.46 ! [I: int,J: int,K: int,L: int] :
% 0.22/0.46 ( ( ( ord_less_eq_int @ I @ J )
% 0.22/0.46 & ( K = L ) )
% 0.22/0.46 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_semiring(3)
% 0.22/0.46 thf(fact_3257_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.22/0.46 ! [I: real,J: real,K: real,L: real] :
% 0.22/0.46 ( ( ( ord_less_eq_real @ I @ J )
% 0.22/0.46 & ( K = L ) )
% 0.22/0.46 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_semiring(3)
% 0.22/0.46 thf(fact_3258_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.22/0.46 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.22/0.46 ( ( ( I = J )
% 0.22/0.46 & ( ord_less_eq_rat @ K @ L ) )
% 0.22/0.46 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_semiring(2)
% 0.22/0.46 thf(fact_3259_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.22/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.22/0.46 ( ( ( I = J )
% 0.22/0.46 & ( ord_less_eq_nat @ K @ L ) )
% 0.22/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_semiring(2)
% 0.22/0.46 thf(fact_3260_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.22/0.46 ! [I: int,J: int,K: int,L: int] :
% 0.22/0.46 ( ( ( I = J )
% 0.22/0.46 & ( ord_less_eq_int @ K @ L ) )
% 0.22/0.46 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_semiring(2)
% 0.22/0.46 thf(fact_3261_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.22/0.46 ! [I: real,J: real,K: real,L: real] :
% 0.22/0.46 ( ( ( I = J )
% 0.22/0.46 & ( ord_less_eq_real @ K @ L ) )
% 0.22/0.46 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_semiring(2)
% 0.22/0.46 thf(fact_3262_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.22/0.46 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.22/0.46 ( ( ( ord_less_eq_rat @ I @ J )
% 0.22/0.46 & ( ord_less_eq_rat @ K @ L ) )
% 0.22/0.46 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_semiring(1)
% 0.22/0.46 thf(fact_3263_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.22/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.22/0.46 ( ( ( ord_less_eq_nat @ I @ J )
% 0.22/0.46 & ( ord_less_eq_nat @ K @ L ) )
% 0.22/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_semiring(1)
% 0.22/0.46 thf(fact_3264_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.22/0.46 ! [I: int,J: int,K: int,L: int] :
% 0.22/0.46 ( ( ( ord_less_eq_int @ I @ J )
% 0.22/0.46 & ( ord_less_eq_int @ K @ L ) )
% 0.22/0.46 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_semiring(1)
% 0.22/0.46 thf(fact_3265_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.22/0.46 ! [I: real,J: real,K: real,L: real] :
% 0.22/0.46 ( ( ( ord_less_eq_real @ I @ J )
% 0.22/0.46 & ( ord_less_eq_real @ K @ L ) )
% 0.22/0.46 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_semiring(1)
% 0.22/0.46 thf(fact_3266_add__mono,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ A @ B )
% 0.22/0.46 => ( ( ord_less_eq_rat @ C @ D )
% 0.22/0.46 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono
% 0.22/0.46 thf(fact_3267_add__mono,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.46 => ( ( ord_less_eq_nat @ C @ D )
% 0.22/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono
% 0.22/0.46 thf(fact_3268_add__mono,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int,D: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.22/0.46 => ( ( ord_less_eq_int @ C @ D )
% 0.22/0.46 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono
% 0.22/0.46 thf(fact_3269_add__mono,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real,D: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.22/0.46 => ( ( ord_less_eq_real @ C @ D )
% 0.22/0.46 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono
% 0.22/0.46 thf(fact_3270_add__left__mono,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ A @ B )
% 0.22/0.46 => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_left_mono
% 0.22/0.46 thf(fact_3271_add__left__mono,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_left_mono
% 0.22/0.46 thf(fact_3272_add__left__mono,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.22/0.46 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_left_mono
% 0.22/0.46 thf(fact_3273_add__left__mono,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.22/0.46 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_left_mono
% 0.22/0.46 thf(fact_3274_less__eqE,axiom,
% 0.22/0.46 ! [A: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.46 => ~ ! [C3: nat] :
% 0.22/0.46 ( B
% 0.22/0.46 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % less_eqE
% 0.22/0.46 thf(fact_3275_add__right__mono,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ A @ B )
% 0.22/0.46 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_right_mono
% 0.22/0.46 thf(fact_3276_add__right__mono,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_right_mono
% 0.22/0.46 thf(fact_3277_add__right__mono,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.22/0.46 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_right_mono
% 0.22/0.46 thf(fact_3278_add__right__mono,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.22/0.46 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_right_mono
% 0.22/0.46 thf(fact_3279_le__iff__add,axiom,
% 0.22/0.46 ( ord_less_eq_nat
% 0.22/0.46 = ( ^ [A3: nat,B2: nat] :
% 0.22/0.46 ? [C5: nat] :
% 0.22/0.46 ( B2
% 0.22/0.46 = ( plus_plus_nat @ A3 @ C5 ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_iff_add
% 0.22/0.46 thf(fact_3280_add__le__imp__le__left,axiom,
% 0.22/0.46 ! [C: rat,A: rat,B: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.22/0.46 => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_imp_le_left
% 0.22/0.46 thf(fact_3281_add__le__imp__le__left,axiom,
% 0.22/0.46 ! [C: nat,A: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.22/0.46 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_imp_le_left
% 0.22/0.46 thf(fact_3282_add__le__imp__le__left,axiom,
% 0.22/0.46 ! [C: int,A: int,B: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.22/0.46 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_imp_le_left
% 0.22/0.46 thf(fact_3283_add__le__imp__le__left,axiom,
% 0.22/0.46 ! [C: real,A: real,B: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.22/0.46 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_imp_le_left
% 0.22/0.46 thf(fact_3284_add__le__imp__le__right,axiom,
% 0.22/0.46 ! [A: rat,C: rat,B: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.22/0.46 => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_imp_le_right
% 0.22/0.46 thf(fact_3285_add__le__imp__le__right,axiom,
% 0.22/0.46 ! [A: nat,C: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.22/0.46 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_imp_le_right
% 0.22/0.46 thf(fact_3286_add__le__imp__le__right,axiom,
% 0.22/0.46 ! [A: int,C: int,B: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.22/0.46 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_imp_le_right
% 0.22/0.46 thf(fact_3287_add__le__imp__le__right,axiom,
% 0.22/0.46 ! [A: real,C: real,B: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.22/0.46 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_imp_le_right
% 0.22/0.46 thf(fact_3288_verit__sum__simplify,axiom,
% 0.22/0.46 ! [A: complex] :
% 0.22/0.46 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.22/0.46 = A ) ).
% 0.22/0.46
% 0.22/0.46 % verit_sum_simplify
% 0.22/0.46 thf(fact_3289_verit__sum__simplify,axiom,
% 0.22/0.46 ! [A: real] :
% 0.22/0.46 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.22/0.46 = A ) ).
% 0.22/0.46
% 0.22/0.46 % verit_sum_simplify
% 0.22/0.46 thf(fact_3290_verit__sum__simplify,axiom,
% 0.22/0.46 ! [A: rat] :
% 0.22/0.46 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.22/0.46 = A ) ).
% 0.22/0.46
% 0.22/0.46 % verit_sum_simplify
% 0.22/0.46 thf(fact_3291_verit__sum__simplify,axiom,
% 0.22/0.46 ! [A: nat] :
% 0.22/0.46 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.22/0.46 = A ) ).
% 0.22/0.46
% 0.22/0.46 % verit_sum_simplify
% 0.22/0.46 thf(fact_3292_verit__sum__simplify,axiom,
% 0.22/0.46 ! [A: int] :
% 0.22/0.46 ( ( plus_plus_int @ A @ zero_zero_int )
% 0.22/0.46 = A ) ).
% 0.22/0.46
% 0.22/0.46 % verit_sum_simplify
% 0.22/0.46 thf(fact_3293_add_Ogroup__left__neutral,axiom,
% 0.22/0.46 ! [A: complex] :
% 0.22/0.46 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.22/0.46 = A ) ).
% 0.22/0.46
% 0.22/0.46 % add.group_left_neutral
% 0.22/0.46 thf(fact_3294_add_Ogroup__left__neutral,axiom,
% 0.22/0.46 ! [A: real] :
% 0.22/0.46 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.22/0.46 = A ) ).
% 0.22/0.46
% 0.22/0.46 % add.group_left_neutral
% 0.22/0.46 thf(fact_3295_add_Ogroup__left__neutral,axiom,
% 0.22/0.46 ! [A: rat] :
% 0.22/0.46 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.22/0.46 = A ) ).
% 0.22/0.46
% 0.22/0.46 % add.group_left_neutral
% 0.22/0.46 thf(fact_3296_add_Ogroup__left__neutral,axiom,
% 0.22/0.46 ! [A: int] :
% 0.22/0.46 ( ( plus_plus_int @ zero_zero_int @ A )
% 0.22/0.46 = A ) ).
% 0.22/0.46
% 0.22/0.46 % add.group_left_neutral
% 0.22/0.46 thf(fact_3297_add_Ocomm__neutral,axiom,
% 0.22/0.46 ! [A: complex] :
% 0.22/0.46 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.22/0.46 = A ) ).
% 0.22/0.46
% 0.22/0.46 % add.comm_neutral
% 0.22/0.46 thf(fact_3298_add_Ocomm__neutral,axiom,
% 0.22/0.46 ! [A: real] :
% 0.22/0.46 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.22/0.46 = A ) ).
% 0.22/0.46
% 0.22/0.46 % add.comm_neutral
% 0.22/0.46 thf(fact_3299_add_Ocomm__neutral,axiom,
% 0.22/0.46 ! [A: rat] :
% 0.22/0.46 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.22/0.46 = A ) ).
% 0.22/0.46
% 0.22/0.46 % add.comm_neutral
% 0.22/0.46 thf(fact_3300_add_Ocomm__neutral,axiom,
% 0.22/0.46 ! [A: nat] :
% 0.22/0.46 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.22/0.46 = A ) ).
% 0.22/0.46
% 0.22/0.46 % add.comm_neutral
% 0.22/0.46 thf(fact_3301_add_Ocomm__neutral,axiom,
% 0.22/0.46 ! [A: int] :
% 0.22/0.46 ( ( plus_plus_int @ A @ zero_zero_int )
% 0.22/0.46 = A ) ).
% 0.22/0.46
% 0.22/0.46 % add.comm_neutral
% 0.22/0.46 thf(fact_3302_comm__monoid__add__class_Oadd__0,axiom,
% 0.22/0.46 ! [A: complex] :
% 0.22/0.46 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.22/0.46 = A ) ).
% 0.22/0.46
% 0.22/0.46 % comm_monoid_add_class.add_0
% 0.22/0.46 thf(fact_3303_comm__monoid__add__class_Oadd__0,axiom,
% 0.22/0.46 ! [A: real] :
% 0.22/0.46 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.22/0.46 = A ) ).
% 0.22/0.46
% 0.22/0.46 % comm_monoid_add_class.add_0
% 0.22/0.46 thf(fact_3304_comm__monoid__add__class_Oadd__0,axiom,
% 0.22/0.46 ! [A: rat] :
% 0.22/0.46 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.22/0.46 = A ) ).
% 0.22/0.46
% 0.22/0.46 % comm_monoid_add_class.add_0
% 0.22/0.46 thf(fact_3305_comm__monoid__add__class_Oadd__0,axiom,
% 0.22/0.46 ! [A: nat] :
% 0.22/0.46 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.22/0.46 = A ) ).
% 0.22/0.46
% 0.22/0.46 % comm_monoid_add_class.add_0
% 0.22/0.46 thf(fact_3306_comm__monoid__add__class_Oadd__0,axiom,
% 0.22/0.46 ! [A: int] :
% 0.22/0.46 ( ( plus_plus_int @ zero_zero_int @ A )
% 0.22/0.46 = A ) ).
% 0.22/0.46
% 0.22/0.46 % comm_monoid_add_class.add_0
% 0.22/0.46 thf(fact_3307_add__mono__thms__linordered__field_I5_J,axiom,
% 0.22/0.46 ! [I: real,J: real,K: real,L: real] :
% 0.22/0.46 ( ( ( ord_less_real @ I @ J )
% 0.22/0.46 & ( ord_less_real @ K @ L ) )
% 0.22/0.46 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(5)
% 0.22/0.46 thf(fact_3308_add__mono__thms__linordered__field_I5_J,axiom,
% 0.22/0.46 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.22/0.46 ( ( ( ord_less_rat @ I @ J )
% 0.22/0.46 & ( ord_less_rat @ K @ L ) )
% 0.22/0.46 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(5)
% 0.22/0.46 thf(fact_3309_add__mono__thms__linordered__field_I5_J,axiom,
% 0.22/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.22/0.46 ( ( ( ord_less_nat @ I @ J )
% 0.22/0.46 & ( ord_less_nat @ K @ L ) )
% 0.22/0.46 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(5)
% 0.22/0.46 thf(fact_3310_add__mono__thms__linordered__field_I5_J,axiom,
% 0.22/0.46 ! [I: int,J: int,K: int,L: int] :
% 0.22/0.46 ( ( ( ord_less_int @ I @ J )
% 0.22/0.46 & ( ord_less_int @ K @ L ) )
% 0.22/0.46 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(5)
% 0.22/0.46 thf(fact_3311_add__mono__thms__linordered__field_I2_J,axiom,
% 0.22/0.46 ! [I: real,J: real,K: real,L: real] :
% 0.22/0.46 ( ( ( I = J )
% 0.22/0.46 & ( ord_less_real @ K @ L ) )
% 0.22/0.46 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(2)
% 0.22/0.46 thf(fact_3312_add__mono__thms__linordered__field_I2_J,axiom,
% 0.22/0.46 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.22/0.46 ( ( ( I = J )
% 0.22/0.46 & ( ord_less_rat @ K @ L ) )
% 0.22/0.46 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(2)
% 0.22/0.46 thf(fact_3313_add__mono__thms__linordered__field_I2_J,axiom,
% 0.22/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.22/0.46 ( ( ( I = J )
% 0.22/0.46 & ( ord_less_nat @ K @ L ) )
% 0.22/0.46 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(2)
% 0.22/0.46 thf(fact_3314_add__mono__thms__linordered__field_I2_J,axiom,
% 0.22/0.46 ! [I: int,J: int,K: int,L: int] :
% 0.22/0.46 ( ( ( I = J )
% 0.22/0.46 & ( ord_less_int @ K @ L ) )
% 0.22/0.46 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(2)
% 0.22/0.46 thf(fact_3315_add__mono__thms__linordered__field_I1_J,axiom,
% 0.22/0.46 ! [I: real,J: real,K: real,L: real] :
% 0.22/0.46 ( ( ( ord_less_real @ I @ J )
% 0.22/0.46 & ( K = L ) )
% 0.22/0.46 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(1)
% 0.22/0.46 thf(fact_3316_add__mono__thms__linordered__field_I1_J,axiom,
% 0.22/0.46 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.22/0.46 ( ( ( ord_less_rat @ I @ J )
% 0.22/0.46 & ( K = L ) )
% 0.22/0.46 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(1)
% 0.22/0.46 thf(fact_3317_add__mono__thms__linordered__field_I1_J,axiom,
% 0.22/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.22/0.46 ( ( ( ord_less_nat @ I @ J )
% 0.22/0.46 & ( K = L ) )
% 0.22/0.46 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(1)
% 0.22/0.46 thf(fact_3318_add__mono__thms__linordered__field_I1_J,axiom,
% 0.22/0.46 ! [I: int,J: int,K: int,L: int] :
% 0.22/0.46 ( ( ( ord_less_int @ I @ J )
% 0.22/0.46 & ( K = L ) )
% 0.22/0.46 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(1)
% 0.22/0.46 thf(fact_3319_add__strict__mono,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real,D: real] :
% 0.22/0.46 ( ( ord_less_real @ A @ B )
% 0.22/0.46 => ( ( ord_less_real @ C @ D )
% 0.22/0.46 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_strict_mono
% 0.22/0.46 thf(fact_3320_add__strict__mono,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.22/0.46 ( ( ord_less_rat @ A @ B )
% 0.22/0.46 => ( ( ord_less_rat @ C @ D )
% 0.22/0.46 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_strict_mono
% 0.22/0.46 thf(fact_3321_add__strict__mono,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.22/0.46 ( ( ord_less_nat @ A @ B )
% 0.22/0.46 => ( ( ord_less_nat @ C @ D )
% 0.22/0.46 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_strict_mono
% 0.22/0.46 thf(fact_3322_add__strict__mono,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int,D: int] :
% 0.22/0.46 ( ( ord_less_int @ A @ B )
% 0.22/0.46 => ( ( ord_less_int @ C @ D )
% 0.22/0.46 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_strict_mono
% 0.22/0.46 thf(fact_3323_add__strict__left__mono,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( ord_less_real @ A @ B )
% 0.22/0.46 => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_strict_left_mono
% 0.22/0.46 thf(fact_3324_add__strict__left__mono,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( ord_less_rat @ A @ B )
% 0.22/0.46 => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_strict_left_mono
% 0.22/0.46 thf(fact_3325_add__strict__left__mono,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( ord_less_nat @ A @ B )
% 0.22/0.46 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_strict_left_mono
% 0.22/0.46 thf(fact_3326_add__strict__left__mono,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( ord_less_int @ A @ B )
% 0.22/0.46 => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_strict_left_mono
% 0.22/0.46 thf(fact_3327_add__strict__right__mono,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( ord_less_real @ A @ B )
% 0.22/0.46 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_strict_right_mono
% 0.22/0.46 thf(fact_3328_add__strict__right__mono,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( ord_less_rat @ A @ B )
% 0.22/0.46 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_strict_right_mono
% 0.22/0.46 thf(fact_3329_add__strict__right__mono,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( ord_less_nat @ A @ B )
% 0.22/0.46 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_strict_right_mono
% 0.22/0.46 thf(fact_3330_add__strict__right__mono,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( ord_less_int @ A @ B )
% 0.22/0.46 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_strict_right_mono
% 0.22/0.46 thf(fact_3331_add__less__imp__less__left,axiom,
% 0.22/0.46 ! [C: real,A: real,B: real] :
% 0.22/0.46 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.22/0.46 => ( ord_less_real @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_imp_less_left
% 0.22/0.46 thf(fact_3332_add__less__imp__less__left,axiom,
% 0.22/0.46 ! [C: rat,A: rat,B: rat] :
% 0.22/0.46 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.22/0.46 => ( ord_less_rat @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_imp_less_left
% 0.22/0.46 thf(fact_3333_add__less__imp__less__left,axiom,
% 0.22/0.46 ! [C: nat,A: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.22/0.46 => ( ord_less_nat @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_imp_less_left
% 0.22/0.46 thf(fact_3334_add__less__imp__less__left,axiom,
% 0.22/0.46 ! [C: int,A: int,B: int] :
% 0.22/0.46 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.22/0.46 => ( ord_less_int @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_imp_less_left
% 0.22/0.46 thf(fact_3335_add__less__imp__less__right,axiom,
% 0.22/0.46 ! [A: real,C: real,B: real] :
% 0.22/0.46 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.22/0.46 => ( ord_less_real @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_imp_less_right
% 0.22/0.46 thf(fact_3336_add__less__imp__less__right,axiom,
% 0.22/0.46 ! [A: rat,C: rat,B: rat] :
% 0.22/0.46 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.22/0.46 => ( ord_less_rat @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_imp_less_right
% 0.22/0.46 thf(fact_3337_add__less__imp__less__right,axiom,
% 0.22/0.46 ! [A: nat,C: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.22/0.46 => ( ord_less_nat @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_imp_less_right
% 0.22/0.46 thf(fact_3338_add__less__imp__less__right,axiom,
% 0.22/0.46 ! [A: int,C: int,B: int] :
% 0.22/0.46 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.22/0.46 => ( ord_less_int @ A @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_imp_less_right
% 0.22/0.46 thf(fact_3339_ring__class_Oring__distribs_I2_J,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % ring_class.ring_distribs(2)
% 0.22/0.46 thf(fact_3340_ring__class_Oring__distribs_I2_J,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % ring_class.ring_distribs(2)
% 0.22/0.46 thf(fact_3341_ring__class_Oring__distribs_I2_J,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % ring_class.ring_distribs(2)
% 0.22/0.46 thf(fact_3342_ring__class_Oring__distribs_I1_J,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.22/0.46 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % ring_class.ring_distribs(1)
% 0.22/0.46 thf(fact_3343_ring__class_Oring__distribs_I1_J,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.22/0.46 = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % ring_class.ring_distribs(1)
% 0.22/0.46 thf(fact_3344_ring__class_Oring__distribs_I1_J,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.22/0.46 = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % ring_class.ring_distribs(1)
% 0.22/0.46 thf(fact_3345_comm__semiring__class_Odistrib,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % comm_semiring_class.distrib
% 0.22/0.46 thf(fact_3346_comm__semiring__class_Odistrib,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % comm_semiring_class.distrib
% 0.22/0.46 thf(fact_3347_comm__semiring__class_Odistrib,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % comm_semiring_class.distrib
% 0.22/0.46 thf(fact_3348_comm__semiring__class_Odistrib,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % comm_semiring_class.distrib
% 0.22/0.46 thf(fact_3349_distrib__left,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.22/0.46 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % distrib_left
% 0.22/0.46 thf(fact_3350_distrib__left,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.22/0.46 = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % distrib_left
% 0.22/0.46 thf(fact_3351_distrib__left,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.22/0.46 = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % distrib_left
% 0.22/0.46 thf(fact_3352_distrib__left,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.22/0.46 = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % distrib_left
% 0.22/0.46 thf(fact_3353_distrib__right,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % distrib_right
% 0.22/0.46 thf(fact_3354_distrib__right,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % distrib_right
% 0.22/0.46 thf(fact_3355_distrib__right,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % distrib_right
% 0.22/0.46 thf(fact_3356_distrib__right,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % distrib_right
% 0.22/0.46 thf(fact_3357_combine__common__factor,axiom,
% 0.22/0.46 ! [A: real,E2: real,B: real,C: real] :
% 0.22/0.46 ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C ) )
% 0.22/0.46 = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E2 ) @ C ) ) ).
% 0.22/0.46
% 0.22/0.46 % combine_common_factor
% 0.22/0.46 thf(fact_3358_combine__common__factor,axiom,
% 0.22/0.46 ! [A: rat,E2: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ C ) )
% 0.22/0.46 = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E2 ) @ C ) ) ).
% 0.22/0.46
% 0.22/0.46 % combine_common_factor
% 0.22/0.46 thf(fact_3359_combine__common__factor,axiom,
% 0.22/0.46 ! [A: nat,E2: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
% 0.22/0.46 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% 0.22/0.46
% 0.22/0.46 % combine_common_factor
% 0.22/0.46 thf(fact_3360_combine__common__factor,axiom,
% 0.22/0.46 ! [A: int,E2: int,B: int,C: int] :
% 0.22/0.46 ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
% 0.22/0.46 = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).
% 0.22/0.46
% 0.22/0.46 % combine_common_factor
% 0.22/0.46 thf(fact_3361_diff__diff__eq,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.22/0.46 = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_diff_eq
% 0.22/0.46 thf(fact_3362_diff__diff__eq,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.22/0.46 = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_diff_eq
% 0.22/0.46 thf(fact_3363_diff__diff__eq,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 0.22/0.46 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_diff_eq
% 0.22/0.46 thf(fact_3364_diff__diff__eq,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.22/0.46 = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_diff_eq
% 0.22/0.46 thf(fact_3365_add__implies__diff,axiom,
% 0.22/0.46 ! [C: real,B: real,A: real] :
% 0.22/0.46 ( ( ( plus_plus_real @ C @ B )
% 0.22/0.46 = A )
% 0.22/0.46 => ( C
% 0.22/0.46 = ( minus_minus_real @ A @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_implies_diff
% 0.22/0.46 thf(fact_3366_add__implies__diff,axiom,
% 0.22/0.46 ! [C: rat,B: rat,A: rat] :
% 0.22/0.46 ( ( ( plus_plus_rat @ C @ B )
% 0.22/0.46 = A )
% 0.22/0.46 => ( C
% 0.22/0.46 = ( minus_minus_rat @ A @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_implies_diff
% 0.22/0.46 thf(fact_3367_add__implies__diff,axiom,
% 0.22/0.46 ! [C: nat,B: nat,A: nat] :
% 0.22/0.46 ( ( ( plus_plus_nat @ C @ B )
% 0.22/0.46 = A )
% 0.22/0.46 => ( C
% 0.22/0.46 = ( minus_minus_nat @ A @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_implies_diff
% 0.22/0.46 thf(fact_3368_add__implies__diff,axiom,
% 0.22/0.46 ! [C: int,B: int,A: int] :
% 0.22/0.46 ( ( ( plus_plus_int @ C @ B )
% 0.22/0.46 = A )
% 0.22/0.46 => ( C
% 0.22/0.46 = ( minus_minus_int @ A @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_implies_diff
% 0.22/0.46 thf(fact_3369_diff__add__eq__diff__diff__swap,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.22/0.46 = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_add_eq_diff_diff_swap
% 0.22/0.46 thf(fact_3370_diff__add__eq__diff__diff__swap,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.22/0.46 = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_add_eq_diff_diff_swap
% 0.22/0.46 thf(fact_3371_diff__add__eq__diff__diff__swap,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.22/0.46 = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_add_eq_diff_diff_swap
% 0.22/0.46 thf(fact_3372_diff__add__eq,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.22/0.46 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_add_eq
% 0.22/0.46 thf(fact_3373_diff__add__eq,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.22/0.46 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_add_eq
% 0.22/0.46 thf(fact_3374_diff__add__eq,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.22/0.46 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_add_eq
% 0.22/0.46 thf(fact_3375_diff__diff__eq2,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.22/0.46 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_diff_eq2
% 0.22/0.46 thf(fact_3376_diff__diff__eq2,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.22/0.46 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_diff_eq2
% 0.22/0.46 thf(fact_3377_diff__diff__eq2,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.22/0.46 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_diff_eq2
% 0.22/0.46 thf(fact_3378_add__diff__eq,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.22/0.46 = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_diff_eq
% 0.22/0.46 thf(fact_3379_add__diff__eq,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.22/0.46 = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_diff_eq
% 0.22/0.46 thf(fact_3380_add__diff__eq,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.22/0.46 = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_diff_eq
% 0.22/0.46 thf(fact_3381_eq__diff__eq,axiom,
% 0.22/0.46 ! [A: real,C: real,B: real] :
% 0.22/0.46 ( ( A
% 0.22/0.46 = ( minus_minus_real @ C @ B ) )
% 0.22/0.46 = ( ( plus_plus_real @ A @ B )
% 0.22/0.46 = C ) ) ).
% 0.22/0.46
% 0.22/0.46 % eq_diff_eq
% 0.22/0.46 thf(fact_3382_eq__diff__eq,axiom,
% 0.22/0.46 ! [A: rat,C: rat,B: rat] :
% 0.22/0.46 ( ( A
% 0.22/0.46 = ( minus_minus_rat @ C @ B ) )
% 0.22/0.46 = ( ( plus_plus_rat @ A @ B )
% 0.22/0.46 = C ) ) ).
% 0.22/0.46
% 0.22/0.46 % eq_diff_eq
% 0.22/0.46 thf(fact_3383_eq__diff__eq,axiom,
% 0.22/0.46 ! [A: int,C: int,B: int] :
% 0.22/0.46 ( ( A
% 0.22/0.46 = ( minus_minus_int @ C @ B ) )
% 0.22/0.46 = ( ( plus_plus_int @ A @ B )
% 0.22/0.46 = C ) ) ).
% 0.22/0.46
% 0.22/0.46 % eq_diff_eq
% 0.22/0.46 thf(fact_3384_diff__eq__eq,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( ( minus_minus_real @ A @ B )
% 0.22/0.46 = C )
% 0.22/0.46 = ( A
% 0.22/0.46 = ( plus_plus_real @ C @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_eq_eq
% 0.22/0.46 thf(fact_3385_diff__eq__eq,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( ( minus_minus_rat @ A @ B )
% 0.22/0.46 = C )
% 0.22/0.46 = ( A
% 0.22/0.46 = ( plus_plus_rat @ C @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_eq_eq
% 0.22/0.46 thf(fact_3386_diff__eq__eq,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( ( minus_minus_int @ A @ B )
% 0.22/0.46 = C )
% 0.22/0.46 = ( A
% 0.22/0.46 = ( plus_plus_int @ C @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_eq_eq
% 0.22/0.46 thf(fact_3387_group__cancel_Osub1,axiom,
% 0.22/0.46 ! [A2: real,K: real,A: real,B: real] :
% 0.22/0.46 ( ( A2
% 0.22/0.46 = ( plus_plus_real @ K @ A ) )
% 0.22/0.46 => ( ( minus_minus_real @ A2 @ B )
% 0.22/0.46 = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % group_cancel.sub1
% 0.22/0.46 thf(fact_3388_group__cancel_Osub1,axiom,
% 0.22/0.46 ! [A2: rat,K: rat,A: rat,B: rat] :
% 0.22/0.46 ( ( A2
% 0.22/0.46 = ( plus_plus_rat @ K @ A ) )
% 0.22/0.46 => ( ( minus_minus_rat @ A2 @ B )
% 0.22/0.46 = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % group_cancel.sub1
% 0.22/0.46 thf(fact_3389_group__cancel_Osub1,axiom,
% 0.22/0.46 ! [A2: int,K: int,A: int,B: int] :
% 0.22/0.46 ( ( A2
% 0.22/0.46 = ( plus_plus_int @ K @ A ) )
% 0.22/0.46 => ( ( minus_minus_int @ A2 @ B )
% 0.22/0.46 = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % group_cancel.sub1
% 0.22/0.46 thf(fact_3390_add__diff__add,axiom,
% 0.22/0.46 ! [A: real,C: real,B: real,D: real] :
% 0.22/0.46 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
% 0.22/0.46 = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_diff_add
% 0.22/0.46 thf(fact_3391_add__diff__add,axiom,
% 0.22/0.46 ! [A: rat,C: rat,B: rat,D: rat] :
% 0.22/0.46 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
% 0.22/0.46 = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_diff_add
% 0.22/0.46 thf(fact_3392_add__diff__add,axiom,
% 0.22/0.46 ! [A: int,C: int,B: int,D: int] :
% 0.22/0.46 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
% 0.22/0.46 = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_diff_add
% 0.22/0.46 thf(fact_3393_add__divide__distrib,axiom,
% 0.22/0.46 ! [A: complex,B: complex,C: complex] :
% 0.22/0.46 ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_divide_distrib
% 0.22/0.46 thf(fact_3394_add__divide__distrib,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_divide_distrib
% 0.22/0.46 thf(fact_3395_add__divide__distrib,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_divide_distrib
% 0.22/0.46 thf(fact_3396_dvd__add__right__iff,axiom,
% 0.22/0.46 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.22/0.46 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.22/0.46 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.22/0.46 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_right_iff
% 0.22/0.46 thf(fact_3397_dvd__add__right__iff,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( dvd_dvd_real @ A @ B )
% 0.22/0.46 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.22/0.46 = ( dvd_dvd_real @ A @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_right_iff
% 0.22/0.46 thf(fact_3398_dvd__add__right__iff,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( dvd_dvd_rat @ A @ B )
% 0.22/0.46 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.22/0.46 = ( dvd_dvd_rat @ A @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_right_iff
% 0.22/0.46 thf(fact_3399_dvd__add__right__iff,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( dvd_dvd_nat @ A @ B )
% 0.22/0.46 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.22/0.46 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_right_iff
% 0.22/0.46 thf(fact_3400_dvd__add__right__iff,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( dvd_dvd_int @ A @ B )
% 0.22/0.46 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.22/0.46 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_right_iff
% 0.22/0.46 thf(fact_3401_dvd__add__left__iff,axiom,
% 0.22/0.46 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.22/0.46 ( ( dvd_dvd_Code_integer @ A @ C )
% 0.22/0.46 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.22/0.46 = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_left_iff
% 0.22/0.46 thf(fact_3402_dvd__add__left__iff,axiom,
% 0.22/0.46 ! [A: real,C: real,B: real] :
% 0.22/0.46 ( ( dvd_dvd_real @ A @ C )
% 0.22/0.46 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.22/0.46 = ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_left_iff
% 0.22/0.46 thf(fact_3403_dvd__add__left__iff,axiom,
% 0.22/0.46 ! [A: rat,C: rat,B: rat] :
% 0.22/0.46 ( ( dvd_dvd_rat @ A @ C )
% 0.22/0.46 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.22/0.46 = ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_left_iff
% 0.22/0.46 thf(fact_3404_dvd__add__left__iff,axiom,
% 0.22/0.46 ! [A: nat,C: nat,B: nat] :
% 0.22/0.46 ( ( dvd_dvd_nat @ A @ C )
% 0.22/0.46 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.22/0.46 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_left_iff
% 0.22/0.46 thf(fact_3405_dvd__add__left__iff,axiom,
% 0.22/0.46 ! [A: int,C: int,B: int] :
% 0.22/0.46 ( ( dvd_dvd_int @ A @ C )
% 0.22/0.46 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.22/0.46 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add_left_iff
% 0.22/0.46 thf(fact_3406_dvd__add,axiom,
% 0.22/0.46 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.22/0.46 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.22/0.46 => ( ( dvd_dvd_Code_integer @ A @ C )
% 0.22/0.46 => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add
% 0.22/0.46 thf(fact_3407_dvd__add,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( dvd_dvd_real @ A @ B )
% 0.22/0.46 => ( ( dvd_dvd_real @ A @ C )
% 0.22/0.46 => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add
% 0.22/0.46 thf(fact_3408_dvd__add,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( dvd_dvd_rat @ A @ B )
% 0.22/0.46 => ( ( dvd_dvd_rat @ A @ C )
% 0.22/0.46 => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add
% 0.22/0.46 thf(fact_3409_dvd__add,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( dvd_dvd_nat @ A @ B )
% 0.22/0.46 => ( ( dvd_dvd_nat @ A @ C )
% 0.22/0.46 => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add
% 0.22/0.46 thf(fact_3410_dvd__add,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( dvd_dvd_int @ A @ B )
% 0.22/0.46 => ( ( dvd_dvd_int @ A @ C )
% 0.22/0.46 => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % dvd_add
% 0.22/0.46 thf(fact_3411_nat__arith_Osuc1,axiom,
% 0.22/0.46 ! [A2: nat,K: nat,A: nat] :
% 0.22/0.46 ( ( A2
% 0.22/0.46 = ( plus_plus_nat @ K @ A ) )
% 0.22/0.46 => ( ( suc @ A2 )
% 0.22/0.46 = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % nat_arith.suc1
% 0.22/0.46 thf(fact_3412_add__Suc,axiom,
% 0.22/0.46 ! [M: nat,N: nat] :
% 0.22/0.46 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.22/0.46 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_Suc
% 0.22/0.46 thf(fact_3413_add__Suc__shift,axiom,
% 0.22/0.46 ! [M: nat,N: nat] :
% 0.22/0.46 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.22/0.46 = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_Suc_shift
% 0.22/0.46 thf(fact_3414_Euclid__induct,axiom,
% 0.22/0.46 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.22/0.46 ( ! [A4: nat,B3: nat] :
% 0.22/0.46 ( ( P @ A4 @ B3 )
% 0.22/0.46 = ( P @ B3 @ A4 ) )
% 0.22/0.46 => ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
% 0.22/0.46 => ( ! [A4: nat,B3: nat] :
% 0.22/0.46 ( ( P @ A4 @ B3 )
% 0.22/0.46 => ( P @ A4 @ ( plus_plus_nat @ A4 @ B3 ) ) )
% 0.22/0.46 => ( P @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % Euclid_induct
% 0.22/0.46 thf(fact_3415_plus__nat_Oadd__0,axiom,
% 0.22/0.46 ! [N: nat] :
% 0.22/0.46 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 0.22/0.46 = N ) ).
% 0.22/0.46
% 0.22/0.46 % plus_nat.add_0
% 0.22/0.46 thf(fact_3416_add__eq__self__zero,axiom,
% 0.22/0.46 ! [M: nat,N: nat] :
% 0.22/0.46 ( ( ( plus_plus_nat @ M @ N )
% 0.22/0.46 = M )
% 0.22/0.46 => ( N = zero_zero_nat ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_eq_self_zero
% 0.22/0.46 thf(fact_3417_add__lessD1,axiom,
% 0.22/0.46 ! [I: nat,J: nat,K: nat] :
% 0.22/0.46 ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.22/0.46 => ( ord_less_nat @ I @ K ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_lessD1
% 0.22/0.46 thf(fact_3418_add__less__mono,axiom,
% 0.22/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.22/0.46 ( ( ord_less_nat @ I @ J )
% 0.22/0.46 => ( ( ord_less_nat @ K @ L )
% 0.22/0.46 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_mono
% 0.22/0.46 thf(fact_3419_not__add__less1,axiom,
% 0.22/0.46 ! [I: nat,J: nat] :
% 0.22/0.46 ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 0.22/0.46
% 0.22/0.46 % not_add_less1
% 0.22/0.46 thf(fact_3420_not__add__less2,axiom,
% 0.22/0.46 ! [J: nat,I: nat] :
% 0.22/0.46 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 0.22/0.46
% 0.22/0.46 % not_add_less2
% 0.22/0.46 thf(fact_3421_add__less__mono1,axiom,
% 0.22/0.46 ! [I: nat,J: nat,K: nat] :
% 0.22/0.46 ( ( ord_less_nat @ I @ J )
% 0.22/0.46 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_mono1
% 0.22/0.46 thf(fact_3422_trans__less__add1,axiom,
% 0.22/0.46 ! [I: nat,J: nat,M: nat] :
% 0.22/0.46 ( ( ord_less_nat @ I @ J )
% 0.22/0.46 => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % trans_less_add1
% 0.22/0.46 thf(fact_3423_trans__less__add2,axiom,
% 0.22/0.46 ! [I: nat,J: nat,M: nat] :
% 0.22/0.46 ( ( ord_less_nat @ I @ J )
% 0.22/0.46 => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % trans_less_add2
% 0.22/0.46 thf(fact_3424_less__add__eq__less,axiom,
% 0.22/0.46 ! [K: nat,L: nat,M: nat,N: nat] :
% 0.22/0.46 ( ( ord_less_nat @ K @ L )
% 0.22/0.46 => ( ( ( plus_plus_nat @ M @ L )
% 0.22/0.46 = ( plus_plus_nat @ K @ N ) )
% 0.22/0.46 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % less_add_eq_less
% 0.22/0.46 thf(fact_3425_nat__le__iff__add,axiom,
% 0.22/0.46 ( ord_less_eq_nat
% 0.22/0.46 = ( ^ [M5: nat,N4: nat] :
% 0.22/0.46 ? [K3: nat] :
% 0.22/0.46 ( N4
% 0.22/0.46 = ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % nat_le_iff_add
% 0.22/0.46 thf(fact_3426_trans__le__add2,axiom,
% 0.22/0.46 ! [I: nat,J: nat,M: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ I @ J )
% 0.22/0.46 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % trans_le_add2
% 0.22/0.46 thf(fact_3427_trans__le__add1,axiom,
% 0.22/0.46 ! [I: nat,J: nat,M: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ I @ J )
% 0.22/0.46 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % trans_le_add1
% 0.22/0.46 thf(fact_3428_add__le__mono1,axiom,
% 0.22/0.46 ! [I: nat,J: nat,K: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ I @ J )
% 0.22/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_mono1
% 0.22/0.46 thf(fact_3429_add__le__mono,axiom,
% 0.22/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ I @ J )
% 0.22/0.46 => ( ( ord_less_eq_nat @ K @ L )
% 0.22/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_mono
% 0.22/0.46 thf(fact_3430_le__Suc__ex,axiom,
% 0.22/0.46 ! [K: nat,L: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ K @ L )
% 0.22/0.46 => ? [N2: nat] :
% 0.22/0.46 ( L
% 0.22/0.46 = ( plus_plus_nat @ K @ N2 ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_Suc_ex
% 0.22/0.46 thf(fact_3431_add__leD2,axiom,
% 0.22/0.46 ! [M: nat,K: nat,N: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.22/0.46 => ( ord_less_eq_nat @ K @ N ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_leD2
% 0.22/0.46 thf(fact_3432_add__leD1,axiom,
% 0.22/0.46 ! [M: nat,K: nat,N: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.22/0.46 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_leD1
% 0.22/0.46 thf(fact_3433_le__add2,axiom,
% 0.22/0.46 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_add2
% 0.22/0.46 thf(fact_3434_le__add1,axiom,
% 0.22/0.46 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_add1
% 0.22/0.46 thf(fact_3435_add__leE,axiom,
% 0.22/0.46 ! [M: nat,K: nat,N: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.22/0.46 => ~ ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.46 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_leE
% 0.22/0.46 thf(fact_3436_Nat_Odiff__cancel,axiom,
% 0.22/0.46 ! [K: nat,M: nat,N: nat] :
% 0.22/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.22/0.46 = ( minus_minus_nat @ M @ N ) ) ).
% 0.22/0.46
% 0.22/0.46 % Nat.diff_cancel
% 0.22/0.46 thf(fact_3437_diff__cancel2,axiom,
% 0.22/0.46 ! [M: nat,K: nat,N: nat] :
% 0.22/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 0.22/0.46 = ( minus_minus_nat @ M @ N ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_cancel2
% 0.22/0.46 thf(fact_3438_diff__add__inverse,axiom,
% 0.22/0.46 ! [N: nat,M: nat] :
% 0.22/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 0.22/0.46 = M ) ).
% 0.22/0.46
% 0.22/0.46 % diff_add_inverse
% 0.22/0.46 thf(fact_3439_diff__add__inverse2,axiom,
% 0.22/0.46 ! [M: nat,N: nat] :
% 0.22/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 0.22/0.46 = M ) ).
% 0.22/0.46
% 0.22/0.46 % diff_add_inverse2
% 0.22/0.46 thf(fact_3440_left__add__mult__distrib,axiom,
% 0.22/0.46 ! [I: nat,U: nat,J: nat,K: nat] :
% 0.22/0.46 ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 0.22/0.46 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% 0.22/0.46
% 0.22/0.46 % left_add_mult_distrib
% 0.22/0.46 thf(fact_3441_add__mult__distrib2,axiom,
% 0.22/0.46 ! [K: nat,M: nat,N: nat] :
% 0.22/0.46 ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 0.22/0.46 = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mult_distrib2
% 0.22/0.46 thf(fact_3442_add__mult__distrib,axiom,
% 0.22/0.46 ! [M: nat,N: nat,K: nat] :
% 0.22/0.46 ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 0.22/0.46 = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mult_distrib
% 0.22/0.46 thf(fact_3443_add__nonpos__eq__0__iff,axiom,
% 0.22/0.46 ! [X: rat,Y4: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.22/0.46 => ( ( ord_less_eq_rat @ Y4 @ zero_zero_rat )
% 0.22/0.46 => ( ( ( plus_plus_rat @ X @ Y4 )
% 0.22/0.46 = zero_zero_rat )
% 0.22/0.46 = ( ( X = zero_zero_rat )
% 0.22/0.46 & ( Y4 = zero_zero_rat ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_nonpos_eq_0_iff
% 0.22/0.46 thf(fact_3444_add__nonpos__eq__0__iff,axiom,
% 0.22/0.46 ! [X: nat,Y4: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ X @ zero_zero_nat )
% 0.22/0.46 => ( ( ord_less_eq_nat @ Y4 @ zero_zero_nat )
% 0.22/0.46 => ( ( ( plus_plus_nat @ X @ Y4 )
% 0.22/0.46 = zero_zero_nat )
% 0.22/0.46 = ( ( X = zero_zero_nat )
% 0.22/0.46 & ( Y4 = zero_zero_nat ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_nonpos_eq_0_iff
% 0.22/0.46 thf(fact_3445_add__nonpos__eq__0__iff,axiom,
% 0.22/0.46 ! [X: int,Y4: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ X @ zero_zero_int )
% 0.22/0.46 => ( ( ord_less_eq_int @ Y4 @ zero_zero_int )
% 0.22/0.46 => ( ( ( plus_plus_int @ X @ Y4 )
% 0.22/0.46 = zero_zero_int )
% 0.22/0.46 = ( ( X = zero_zero_int )
% 0.22/0.46 & ( Y4 = zero_zero_int ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_nonpos_eq_0_iff
% 0.22/0.46 thf(fact_3446_add__nonpos__eq__0__iff,axiom,
% 0.22/0.46 ! [X: real,Y4: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.22/0.46 => ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
% 0.22/0.46 => ( ( ( plus_plus_real @ X @ Y4 )
% 0.22/0.46 = zero_zero_real )
% 0.22/0.46 = ( ( X = zero_zero_real )
% 0.22/0.46 & ( Y4 = zero_zero_real ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_nonpos_eq_0_iff
% 0.22/0.46 thf(fact_3447_add__nonneg__eq__0__iff,axiom,
% 0.22/0.46 ! [X: rat,Y4: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.22/0.46 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 0.22/0.46 => ( ( ( plus_plus_rat @ X @ Y4 )
% 0.22/0.46 = zero_zero_rat )
% 0.22/0.46 = ( ( X = zero_zero_rat )
% 0.22/0.46 & ( Y4 = zero_zero_rat ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_nonneg_eq_0_iff
% 0.22/0.46 thf(fact_3448_add__nonneg__eq__0__iff,axiom,
% 0.22/0.46 ! [X: nat,Y4: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.22/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
% 0.22/0.46 => ( ( ( plus_plus_nat @ X @ Y4 )
% 0.22/0.46 = zero_zero_nat )
% 0.22/0.46 = ( ( X = zero_zero_nat )
% 0.22/0.46 & ( Y4 = zero_zero_nat ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_nonneg_eq_0_iff
% 0.22/0.46 thf(fact_3449_add__nonneg__eq__0__iff,axiom,
% 0.22/0.46 ! [X: int,Y4: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.22/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.22/0.46 => ( ( ( plus_plus_int @ X @ Y4 )
% 0.22/0.46 = zero_zero_int )
% 0.22/0.46 = ( ( X = zero_zero_int )
% 0.22/0.46 & ( Y4 = zero_zero_int ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_nonneg_eq_0_iff
% 0.22/0.46 thf(fact_3450_add__nonneg__eq__0__iff,axiom,
% 0.22/0.46 ! [X: real,Y4: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.46 => ( ( ( plus_plus_real @ X @ Y4 )
% 0.22/0.46 = zero_zero_real )
% 0.22/0.46 = ( ( X = zero_zero_real )
% 0.22/0.46 & ( Y4 = zero_zero_real ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_nonneg_eq_0_iff
% 0.22/0.46 thf(fact_3451_add__nonpos__nonpos,axiom,
% 0.22/0.46 ! [A: rat,B: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.22/0.46 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.22/0.46 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_nonpos_nonpos
% 0.22/0.46 thf(fact_3452_add__nonpos__nonpos,axiom,
% 0.22/0.46 ! [A: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.22/0.46 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.22/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_nonpos_nonpos
% 0.22/0.46 thf(fact_3453_add__nonpos__nonpos,axiom,
% 0.22/0.46 ! [A: int,B: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.22/0.46 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.22/0.46 => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_nonpos_nonpos
% 0.22/0.46 thf(fact_3454_add__nonpos__nonpos,axiom,
% 0.22/0.46 ! [A: real,B: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.22/0.46 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.22/0.46 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_nonpos_nonpos
% 0.22/0.46 thf(fact_3455_add__nonneg__nonneg,axiom,
% 0.22/0.46 ! [A: rat,B: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.22/0.46 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.22/0.46 => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_nonneg_nonneg
% 0.22/0.46 thf(fact_3456_add__nonneg__nonneg,axiom,
% 0.22/0.46 ! [A: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.22/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.22/0.46 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_nonneg_nonneg
% 0.22/0.46 thf(fact_3457_add__nonneg__nonneg,axiom,
% 0.22/0.46 ! [A: int,B: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.22/0.46 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.22/0.46 => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_nonneg_nonneg
% 0.22/0.46 thf(fact_3458_add__nonneg__nonneg,axiom,
% 0.22/0.46 ! [A: real,B: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.22/0.46 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_nonneg_nonneg
% 0.22/0.46 thf(fact_3459_add__increasing2,axiom,
% 0.22/0.46 ! [C: rat,B: rat,A: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.22/0.46 => ( ( ord_less_eq_rat @ B @ A )
% 0.22/0.46 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_increasing2
% 0.22/0.46 thf(fact_3460_add__increasing2,axiom,
% 0.22/0.46 ! [C: nat,B: nat,A: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.22/0.46 => ( ( ord_less_eq_nat @ B @ A )
% 0.22/0.46 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_increasing2
% 0.22/0.46 thf(fact_3461_add__increasing2,axiom,
% 0.22/0.46 ! [C: int,B: int,A: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.22/0.46 => ( ( ord_less_eq_int @ B @ A )
% 0.22/0.46 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_increasing2
% 0.22/0.46 thf(fact_3462_add__increasing2,axiom,
% 0.22/0.46 ! [C: real,B: real,A: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.22/0.46 => ( ( ord_less_eq_real @ B @ A )
% 0.22/0.46 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_increasing2
% 0.22/0.46 thf(fact_3463_add__decreasing2,axiom,
% 0.22/0.46 ! [C: rat,A: rat,B: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.22/0.46 => ( ( ord_less_eq_rat @ A @ B )
% 0.22/0.46 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_decreasing2
% 0.22/0.46 thf(fact_3464_add__decreasing2,axiom,
% 0.22/0.46 ! [C: nat,A: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 0.22/0.46 => ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_decreasing2
% 0.22/0.46 thf(fact_3465_add__decreasing2,axiom,
% 0.22/0.46 ! [C: int,A: int,B: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.22/0.46 => ( ( ord_less_eq_int @ A @ B )
% 0.22/0.46 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_decreasing2
% 0.22/0.46 thf(fact_3466_add__decreasing2,axiom,
% 0.22/0.46 ! [C: real,A: real,B: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.22/0.46 => ( ( ord_less_eq_real @ A @ B )
% 0.22/0.46 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_decreasing2
% 0.22/0.46 thf(fact_3467_add__increasing,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.22/0.46 => ( ( ord_less_eq_rat @ B @ C )
% 0.22/0.46 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_increasing
% 0.22/0.46 thf(fact_3468_add__increasing,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.22/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.22/0.46 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_increasing
% 0.22/0.46 thf(fact_3469_add__increasing,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.22/0.46 => ( ( ord_less_eq_int @ B @ C )
% 0.22/0.46 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_increasing
% 0.22/0.46 thf(fact_3470_add__increasing,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.22/0.46 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_increasing
% 0.22/0.46 thf(fact_3471_add__decreasing,axiom,
% 0.22/0.46 ! [A: rat,C: rat,B: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.22/0.46 => ( ( ord_less_eq_rat @ C @ B )
% 0.22/0.46 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_decreasing
% 0.22/0.46 thf(fact_3472_add__decreasing,axiom,
% 0.22/0.46 ! [A: nat,C: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.22/0.46 => ( ( ord_less_eq_nat @ C @ B )
% 0.22/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_decreasing
% 0.22/0.46 thf(fact_3473_add__decreasing,axiom,
% 0.22/0.46 ! [A: int,C: int,B: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.22/0.46 => ( ( ord_less_eq_int @ C @ B )
% 0.22/0.46 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_decreasing
% 0.22/0.46 thf(fact_3474_add__decreasing,axiom,
% 0.22/0.46 ! [A: real,C: real,B: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.22/0.46 => ( ( ord_less_eq_real @ C @ B )
% 0.22/0.46 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_decreasing
% 0.22/0.46 thf(fact_3475_add__mono__thms__linordered__field_I4_J,axiom,
% 0.22/0.46 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.22/0.46 ( ( ( ord_less_eq_rat @ I @ J )
% 0.22/0.46 & ( ord_less_rat @ K @ L ) )
% 0.22/0.46 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(4)
% 0.22/0.46 thf(fact_3476_add__mono__thms__linordered__field_I4_J,axiom,
% 0.22/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.22/0.46 ( ( ( ord_less_eq_nat @ I @ J )
% 0.22/0.46 & ( ord_less_nat @ K @ L ) )
% 0.22/0.46 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(4)
% 0.22/0.46 thf(fact_3477_add__mono__thms__linordered__field_I4_J,axiom,
% 0.22/0.46 ! [I: int,J: int,K: int,L: int] :
% 0.22/0.46 ( ( ( ord_less_eq_int @ I @ J )
% 0.22/0.46 & ( ord_less_int @ K @ L ) )
% 0.22/0.46 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(4)
% 0.22/0.46 thf(fact_3478_add__mono__thms__linordered__field_I4_J,axiom,
% 0.22/0.46 ! [I: real,J: real,K: real,L: real] :
% 0.22/0.46 ( ( ( ord_less_eq_real @ I @ J )
% 0.22/0.46 & ( ord_less_real @ K @ L ) )
% 0.22/0.46 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(4)
% 0.22/0.46 thf(fact_3479_add__mono__thms__linordered__field_I3_J,axiom,
% 0.22/0.46 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.22/0.46 ( ( ( ord_less_rat @ I @ J )
% 0.22/0.46 & ( ord_less_eq_rat @ K @ L ) )
% 0.22/0.46 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(3)
% 0.22/0.46 thf(fact_3480_add__mono__thms__linordered__field_I3_J,axiom,
% 0.22/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.22/0.46 ( ( ( ord_less_nat @ I @ J )
% 0.22/0.46 & ( ord_less_eq_nat @ K @ L ) )
% 0.22/0.46 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(3)
% 0.22/0.46 thf(fact_3481_add__mono__thms__linordered__field_I3_J,axiom,
% 0.22/0.46 ! [I: int,J: int,K: int,L: int] :
% 0.22/0.46 ( ( ( ord_less_int @ I @ J )
% 0.22/0.46 & ( ord_less_eq_int @ K @ L ) )
% 0.22/0.46 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(3)
% 0.22/0.46 thf(fact_3482_add__mono__thms__linordered__field_I3_J,axiom,
% 0.22/0.46 ! [I: real,J: real,K: real,L: real] :
% 0.22/0.46 ( ( ( ord_less_real @ I @ J )
% 0.22/0.46 & ( ord_less_eq_real @ K @ L ) )
% 0.22/0.46 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono_thms_linordered_field(3)
% 0.22/0.46 thf(fact_3483_add__le__less__mono,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ A @ B )
% 0.22/0.46 => ( ( ord_less_rat @ C @ D )
% 0.22/0.46 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_less_mono
% 0.22/0.46 thf(fact_3484_add__le__less__mono,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.46 => ( ( ord_less_nat @ C @ D )
% 0.22/0.46 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_less_mono
% 0.22/0.46 thf(fact_3485_add__le__less__mono,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int,D: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.22/0.46 => ( ( ord_less_int @ C @ D )
% 0.22/0.46 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_less_mono
% 0.22/0.46 thf(fact_3486_add__le__less__mono,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real,D: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.22/0.46 => ( ( ord_less_real @ C @ D )
% 0.22/0.46 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_less_mono
% 0.22/0.46 thf(fact_3487_add__less__le__mono,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.22/0.46 ( ( ord_less_rat @ A @ B )
% 0.22/0.46 => ( ( ord_less_eq_rat @ C @ D )
% 0.22/0.46 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_le_mono
% 0.22/0.46 thf(fact_3488_add__less__le__mono,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.22/0.46 ( ( ord_less_nat @ A @ B )
% 0.22/0.46 => ( ( ord_less_eq_nat @ C @ D )
% 0.22/0.46 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_le_mono
% 0.22/0.46 thf(fact_3489_add__less__le__mono,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int,D: int] :
% 0.22/0.46 ( ( ord_less_int @ A @ B )
% 0.22/0.46 => ( ( ord_less_eq_int @ C @ D )
% 0.22/0.46 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_le_mono
% 0.22/0.46 thf(fact_3490_add__less__le__mono,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real,D: real] :
% 0.22/0.46 ( ( ord_less_real @ A @ B )
% 0.22/0.46 => ( ( ord_less_eq_real @ C @ D )
% 0.22/0.46 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_le_mono
% 0.22/0.46 thf(fact_3491_add__less__zeroD,axiom,
% 0.22/0.46 ! [X: real,Y4: real] :
% 0.22/0.46 ( ( ord_less_real @ ( plus_plus_real @ X @ Y4 ) @ zero_zero_real )
% 0.22/0.46 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.22/0.46 | ( ord_less_real @ Y4 @ zero_zero_real ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_zeroD
% 0.22/0.46 thf(fact_3492_add__less__zeroD,axiom,
% 0.22/0.46 ! [X: rat,Y4: rat] :
% 0.22/0.46 ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y4 ) @ zero_zero_rat )
% 0.22/0.46 => ( ( ord_less_rat @ X @ zero_zero_rat )
% 0.22/0.46 | ( ord_less_rat @ Y4 @ zero_zero_rat ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_zeroD
% 0.22/0.46 thf(fact_3493_add__less__zeroD,axiom,
% 0.22/0.46 ! [X: int,Y4: int] :
% 0.22/0.46 ( ( ord_less_int @ ( plus_plus_int @ X @ Y4 ) @ zero_zero_int )
% 0.22/0.46 => ( ( ord_less_int @ X @ zero_zero_int )
% 0.22/0.46 | ( ord_less_int @ Y4 @ zero_zero_int ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_less_zeroD
% 0.22/0.46 thf(fact_3494_add__neg__neg,axiom,
% 0.22/0.46 ! [A: real,B: real] :
% 0.22/0.46 ( ( ord_less_real @ A @ zero_zero_real )
% 0.22/0.46 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.22/0.46 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_neg_neg
% 0.22/0.46 thf(fact_3495_add__neg__neg,axiom,
% 0.22/0.46 ! [A: rat,B: rat] :
% 0.22/0.46 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.22/0.46 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.22/0.46 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_neg_neg
% 0.22/0.46 thf(fact_3496_add__neg__neg,axiom,
% 0.22/0.46 ! [A: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.22/0.46 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.22/0.46 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_neg_neg
% 0.22/0.46 thf(fact_3497_add__neg__neg,axiom,
% 0.22/0.46 ! [A: int,B: int] :
% 0.22/0.46 ( ( ord_less_int @ A @ zero_zero_int )
% 0.22/0.46 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.22/0.46 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_neg_neg
% 0.22/0.46 thf(fact_3498_add__pos__pos,axiom,
% 0.22/0.46 ! [A: real,B: real] :
% 0.22/0.46 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.46 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.22/0.46 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_pos_pos
% 0.22/0.46 thf(fact_3499_add__pos__pos,axiom,
% 0.22/0.46 ! [A: rat,B: rat] :
% 0.22/0.46 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.22/0.46 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.22/0.46 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_pos_pos
% 0.22/0.46 thf(fact_3500_add__pos__pos,axiom,
% 0.22/0.46 ! [A: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.22/0.46 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.22/0.46 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_pos_pos
% 0.22/0.46 thf(fact_3501_add__pos__pos,axiom,
% 0.22/0.46 ! [A: int,B: int] :
% 0.22/0.46 ( ( ord_less_int @ zero_zero_int @ A )
% 0.22/0.46 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.46 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_pos_pos
% 0.22/0.46 thf(fact_3502_canonically__ordered__monoid__add__class_OlessE,axiom,
% 0.22/0.46 ! [A: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_nat @ A @ B )
% 0.22/0.46 => ~ ! [C3: nat] :
% 0.22/0.46 ( ( B
% 0.22/0.46 = ( plus_plus_nat @ A @ C3 ) )
% 0.22/0.46 => ( C3 = zero_zero_nat ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % canonically_ordered_monoid_add_class.lessE
% 0.22/0.46 thf(fact_3503_pos__add__strict,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.46 => ( ( ord_less_real @ B @ C )
% 0.22/0.46 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % pos_add_strict
% 0.22/0.46 thf(fact_3504_pos__add__strict,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.22/0.46 => ( ( ord_less_rat @ B @ C )
% 0.22/0.46 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % pos_add_strict
% 0.22/0.46 thf(fact_3505_pos__add__strict,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.22/0.46 => ( ( ord_less_nat @ B @ C )
% 0.22/0.46 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % pos_add_strict
% 0.22/0.46 thf(fact_3506_pos__add__strict,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( ord_less_int @ zero_zero_int @ A )
% 0.22/0.46 => ( ( ord_less_int @ B @ C )
% 0.22/0.46 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % pos_add_strict
% 0.22/0.46 thf(fact_3507_add__le__imp__le__diff,axiom,
% 0.22/0.46 ! [I: rat,K: rat,N: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.22/0.46 => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_imp_le_diff
% 0.22/0.46 thf(fact_3508_add__le__imp__le__diff,axiom,
% 0.22/0.46 ! [I: nat,K: nat,N: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.22/0.46 => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_imp_le_diff
% 0.22/0.46 thf(fact_3509_add__le__imp__le__diff,axiom,
% 0.22/0.46 ! [I: int,K: int,N: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.22/0.46 => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_imp_le_diff
% 0.22/0.46 thf(fact_3510_add__le__imp__le__diff,axiom,
% 0.22/0.46 ! [I: real,K: real,N: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.22/0.46 => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_imp_le_diff
% 0.22/0.46 thf(fact_3511_add__le__add__imp__diff__le,axiom,
% 0.22/0.46 ! [I: rat,K: rat,N: rat,J: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.22/0.46 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 0.22/0.46 => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.22/0.46 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 0.22/0.46 => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_add_imp_diff_le
% 0.22/0.46 thf(fact_3512_add__le__add__imp__diff__le,axiom,
% 0.22/0.46 ! [I: nat,K: nat,N: nat,J: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.22/0.46 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.22/0.46 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.22/0.46 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.22/0.46 => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_add_imp_diff_le
% 0.22/0.46 thf(fact_3513_add__le__add__imp__diff__le,axiom,
% 0.22/0.46 ! [I: int,K: int,N: int,J: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.22/0.46 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.22/0.46 => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.22/0.46 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.22/0.46 => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_add_imp_diff_le
% 0.22/0.46 thf(fact_3514_add__le__add__imp__diff__le,axiom,
% 0.22/0.46 ! [I: real,K: real,N: real,J: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.22/0.46 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.22/0.46 => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.22/0.46 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.22/0.46 => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_le_add_imp_diff_le
% 0.22/0.46 thf(fact_3515_diff__le__eq,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.22/0.46 = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_le_eq
% 0.22/0.46 thf(fact_3516_diff__le__eq,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.22/0.46 = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_le_eq
% 0.22/0.46 thf(fact_3517_diff__le__eq,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.22/0.46 = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_le_eq
% 0.22/0.46 thf(fact_3518_le__diff__eq,axiom,
% 0.22/0.46 ! [A: rat,C: rat,B: rat] :
% 0.22/0.46 ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 0.22/0.46 = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_diff_eq
% 0.22/0.46 thf(fact_3519_le__diff__eq,axiom,
% 0.22/0.46 ! [A: int,C: int,B: int] :
% 0.22/0.46 ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.22/0.46 = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_diff_eq
% 0.22/0.46 thf(fact_3520_le__diff__eq,axiom,
% 0.22/0.46 ! [A: real,C: real,B: real] :
% 0.22/0.46 ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 0.22/0.46 = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_diff_eq
% 0.22/0.46 thf(fact_3521_diff__add,axiom,
% 0.22/0.46 ! [A: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.46 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 0.22/0.46 = B ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_add
% 0.22/0.46 thf(fact_3522_le__add__diff,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.46 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % le_add_diff
% 0.22/0.46 thf(fact_3523_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.46 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.22/0.46 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.22/0.46 thf(fact_3524_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.46 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.22/0.46 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.22/0.46 thf(fact_3525_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.46 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 0.22/0.46 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.22/0.46 thf(fact_3526_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.46 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 0.22/0.46 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.22/0.46 thf(fact_3527_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.46 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 0.22/0.46 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.22/0.46 thf(fact_3528_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.46 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.22/0.46 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.22/0.46 thf(fact_3529_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.22/0.46 ! [A: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.46 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 0.22/0.46 = B ) ) ).
% 0.22/0.46
% 0.22/0.46 % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.22/0.46 thf(fact_3530_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.22/0.46 ! [A: nat,B: nat,C: nat] :
% 0.22/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.46 => ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.46 => ( ( ( minus_minus_nat @ B @ A )
% 0.22/0.46 = C )
% 0.22/0.46 = ( B
% 0.22/0.46 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.22/0.46 thf(fact_3531_less__add__one,axiom,
% 0.22/0.46 ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 0.22/0.46
% 0.22/0.46 % less_add_one
% 0.22/0.46 thf(fact_3532_less__add__one,axiom,
% 0.22/0.46 ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% 0.22/0.46
% 0.22/0.46 % less_add_one
% 0.22/0.46 thf(fact_3533_less__add__one,axiom,
% 0.22/0.46 ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 0.22/0.46
% 0.22/0.46 % less_add_one
% 0.22/0.46 thf(fact_3534_less__add__one,axiom,
% 0.22/0.46 ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% 0.22/0.46
% 0.22/0.46 % less_add_one
% 0.22/0.46 thf(fact_3535_add__mono1,axiom,
% 0.22/0.46 ! [A: real,B: real] :
% 0.22/0.46 ( ( ord_less_real @ A @ B )
% 0.22/0.46 => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono1
% 0.22/0.46 thf(fact_3536_add__mono1,axiom,
% 0.22/0.46 ! [A: rat,B: rat] :
% 0.22/0.46 ( ( ord_less_rat @ A @ B )
% 0.22/0.46 => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono1
% 0.22/0.46 thf(fact_3537_add__mono1,axiom,
% 0.22/0.46 ! [A: nat,B: nat] :
% 0.22/0.46 ( ( ord_less_nat @ A @ B )
% 0.22/0.46 => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono1
% 0.22/0.46 thf(fact_3538_add__mono1,axiom,
% 0.22/0.46 ! [A: int,B: int] :
% 0.22/0.46 ( ( ord_less_int @ A @ B )
% 0.22/0.46 => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % add_mono1
% 0.22/0.46 thf(fact_3539_numeral__Bit0,axiom,
% 0.22/0.46 ! [N: num] :
% 0.22/0.46 ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 0.22/0.46 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % numeral_Bit0
% 0.22/0.46 thf(fact_3540_numeral__Bit0,axiom,
% 0.22/0.46 ! [N: num] :
% 0.22/0.46 ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 0.22/0.46 = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % numeral_Bit0
% 0.22/0.46 thf(fact_3541_numeral__Bit0,axiom,
% 0.22/0.46 ! [N: num] :
% 0.22/0.46 ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 0.22/0.46 = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % numeral_Bit0
% 0.22/0.46 thf(fact_3542_numeral__Bit0,axiom,
% 0.22/0.46 ! [N: num] :
% 0.22/0.46 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 0.22/0.46 = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % numeral_Bit0
% 0.22/0.46 thf(fact_3543_numeral__Bit0,axiom,
% 0.22/0.46 ! [N: num] :
% 0.22/0.46 ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 0.22/0.46 = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % numeral_Bit0
% 0.22/0.46 thf(fact_3544_less__diff__eq,axiom,
% 0.22/0.46 ! [A: real,C: real,B: real] :
% 0.22/0.46 ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
% 0.22/0.46 = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.22/0.46
% 0.22/0.46 % less_diff_eq
% 0.22/0.46 thf(fact_3545_less__diff__eq,axiom,
% 0.22/0.46 ! [A: rat,C: rat,B: rat] :
% 0.22/0.46 ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 0.22/0.46 = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.22/0.46
% 0.22/0.46 % less_diff_eq
% 0.22/0.46 thf(fact_3546_less__diff__eq,axiom,
% 0.22/0.46 ! [A: int,C: int,B: int] :
% 0.22/0.46 ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.22/0.46 = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.22/0.46
% 0.22/0.46 % less_diff_eq
% 0.22/0.46 thf(fact_3547_diff__less__eq,axiom,
% 0.22/0.46 ! [A: real,B: real,C: real] :
% 0.22/0.46 ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.22/0.46 = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_less_eq
% 0.22/0.46 thf(fact_3548_diff__less__eq,axiom,
% 0.22/0.46 ! [A: rat,B: rat,C: rat] :
% 0.22/0.46 ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.22/0.46 = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_less_eq
% 0.22/0.46 thf(fact_3549_diff__less__eq,axiom,
% 0.22/0.46 ! [A: int,B: int,C: int] :
% 0.22/0.46 ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.22/0.46 = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % diff_less_eq
% 0.22/0.46 thf(fact_3550_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.22/0.46 ! [A: real,B: real] :
% 0.22/0.46 ( ~ ( ord_less_real @ A @ B )
% 0.22/0.46 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 0.22/0.46 = A ) ) ).
% 0.22/0.46
% 0.22/0.46 % linordered_semidom_class.add_diff_inverse
% 0.22/0.46 thf(fact_3551_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.22/0.46 ! [A: rat,B: rat] :
% 0.22/0.46 ( ~ ( ord_less_rat @ A @ B )
% 0.22/0.46 => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 0.22/0.46 = A ) ) ).
% 0.22/0.46
% 0.22/0.46 % linordered_semidom_class.add_diff_inverse
% 0.22/0.46 thf(fact_3552_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.22/0.46 ! [A: nat,B: nat] :
% 0.22/0.46 ( ~ ( ord_less_nat @ A @ B )
% 0.22/0.46 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 0.22/0.46 = A ) ) ).
% 0.22/0.46
% 0.22/0.46 % linordered_semidom_class.add_diff_inverse
% 0.22/0.46 thf(fact_3553_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.22/0.46 ! [A: int,B: int] :
% 0.22/0.46 ( ~ ( ord_less_int @ A @ B )
% 0.22/0.46 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 0.22/0.46 = A ) ) ).
% 0.22/0.46
% 0.22/0.46 % linordered_semidom_class.add_diff_inverse
% 0.22/0.46 thf(fact_3554_one__plus__numeral__commute,axiom,
% 0.22/0.46 ! [X: num] :
% 0.22/0.46 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
% 0.22/0.46 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% 0.22/0.46
% 0.22/0.46 % one_plus_numeral_commute
% 0.22/0.46 thf(fact_3555_one__plus__numeral__commute,axiom,
% 0.22/0.46 ! [X: num] :
% 0.22/0.46 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
% 0.22/0.46 = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% 0.22/0.46
% 0.22/0.46 % one_plus_numeral_commute
% 0.22/0.46 thf(fact_3556_one__plus__numeral__commute,axiom,
% 0.22/0.46 ! [X: num] :
% 0.22/0.46 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
% 0.22/0.46 = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% 0.22/0.46
% 0.22/0.46 % one_plus_numeral_commute
% 0.22/0.46 thf(fact_3557_one__plus__numeral__commute,axiom,
% 0.22/0.46 ! [X: num] :
% 0.22/0.46 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
% 0.22/0.46 = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% 0.22/0.46
% 0.22/0.46 % one_plus_numeral_commute
% 0.22/0.46 thf(fact_3558_one__plus__numeral__commute,axiom,
% 0.22/0.46 ! [X: num] :
% 0.22/0.46 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
% 0.22/0.46 = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% 0.22/0.46
% 0.22/0.46 % one_plus_numeral_commute
% 0.22/0.46 thf(fact_3559_eq__add__iff1,axiom,
% 0.22/0.46 ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.22/0.46 ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
% 0.22/0.46 = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.22/0.46 = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C )
% 0.22/0.46 = D ) ) ).
% 0.22/0.46
% 0.22/0.46 % eq_add_iff1
% 0.22/0.46 thf(fact_3560_eq__add__iff1,axiom,
% 0.22/0.46 ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.22/0.46 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
% 0.22/0.46 = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.22/0.46 = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C )
% 0.22/0.46 = D ) ) ).
% 0.22/0.46
% 0.22/0.46 % eq_add_iff1
% 0.22/0.46 thf(fact_3561_eq__add__iff1,axiom,
% 0.22/0.46 ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.22/0.46 ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
% 0.22/0.46 = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.22/0.46 = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
% 0.22/0.46 = D ) ) ).
% 0.22/0.46
% 0.22/0.46 % eq_add_iff1
% 0.22/0.46 thf(fact_3562_eq__add__iff2,axiom,
% 0.22/0.46 ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.22/0.46 ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
% 0.22/0.46 = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.22/0.46 = ( C
% 0.22/0.46 = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % eq_add_iff2
% 0.22/0.46 thf(fact_3563_eq__add__iff2,axiom,
% 0.22/0.46 ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.22/0.46 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
% 0.22/0.46 = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.22/0.46 = ( C
% 0.22/0.46 = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % eq_add_iff2
% 0.22/0.46 thf(fact_3564_eq__add__iff2,axiom,
% 0.22/0.46 ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.22/0.46 ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
% 0.22/0.46 = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.22/0.46 = ( C
% 0.22/0.46 = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % eq_add_iff2
% 0.22/0.46 thf(fact_3565_square__diff__square__factored,axiom,
% 0.22/0.46 ! [X: real,Y4: real] :
% 0.22/0.46 ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) )
% 0.22/0.46 = ( times_times_real @ ( plus_plus_real @ X @ Y4 ) @ ( minus_minus_real @ X @ Y4 ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % square_diff_square_factored
% 0.22/0.46 thf(fact_3566_square__diff__square__factored,axiom,
% 0.22/0.46 ! [X: rat,Y4: rat] :
% 0.22/0.46 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y4 @ Y4 ) )
% 0.22/0.46 = ( times_times_rat @ ( plus_plus_rat @ X @ Y4 ) @ ( minus_minus_rat @ X @ Y4 ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % square_diff_square_factored
% 0.22/0.46 thf(fact_3567_square__diff__square__factored,axiom,
% 0.22/0.46 ! [X: int,Y4: int] :
% 0.22/0.46 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y4 @ Y4 ) )
% 0.22/0.46 = ( times_times_int @ ( plus_plus_int @ X @ Y4 ) @ ( minus_minus_int @ X @ Y4 ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % square_diff_square_factored
% 0.22/0.46 thf(fact_3568_mult__diff__mult,axiom,
% 0.22/0.46 ! [X: real,Y4: real,A: real,B: real] :
% 0.22/0.46 ( ( minus_minus_real @ ( times_times_real @ X @ Y4 ) @ ( times_times_real @ A @ B ) )
% 0.22/0.46 = ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y4 @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % mult_diff_mult
% 0.22/0.46 thf(fact_3569_mult__diff__mult,axiom,
% 0.22/0.46 ! [X: rat,Y4: rat,A: rat,B: rat] :
% 0.22/0.46 ( ( minus_minus_rat @ ( times_times_rat @ X @ Y4 ) @ ( times_times_rat @ A @ B ) )
% 0.22/0.46 = ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y4 @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % mult_diff_mult
% 0.22/0.46 thf(fact_3570_mult__diff__mult,axiom,
% 0.22/0.46 ! [X: int,Y4: int,A: int,B: int] :
% 0.22/0.46 ( ( minus_minus_int @ ( times_times_int @ X @ Y4 ) @ ( times_times_int @ A @ B ) )
% 0.22/0.46 = ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y4 @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % mult_diff_mult
% 0.22/0.46 thf(fact_3571_minf_I10_J,axiom,
% 0.22/0.46 ! [D: code_integer,S2: code_integer] :
% 0.22/0.46 ? [Z2: code_integer] :
% 0.22/0.46 ! [X5: code_integer] :
% 0.22/0.46 ( ( ord_le6747313008572928689nteger @ X5 @ Z2 )
% 0.22/0.46 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) )
% 0.22/0.46 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % minf(10)
% 0.22/0.46 thf(fact_3572_minf_I10_J,axiom,
% 0.22/0.46 ! [D: real,S2: real] :
% 0.22/0.46 ? [Z2: real] :
% 0.22/0.46 ! [X5: real] :
% 0.22/0.46 ( ( ord_less_real @ X5 @ Z2 )
% 0.22/0.46 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) )
% 0.22/0.46 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % minf(10)
% 0.22/0.46 thf(fact_3573_minf_I10_J,axiom,
% 0.22/0.46 ! [D: rat,S2: rat] :
% 0.22/0.46 ? [Z2: rat] :
% 0.22/0.46 ! [X5: rat] :
% 0.22/0.46 ( ( ord_less_rat @ X5 @ Z2 )
% 0.22/0.46 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) )
% 0.22/0.46 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % minf(10)
% 0.22/0.46 thf(fact_3574_minf_I10_J,axiom,
% 0.22/0.46 ! [D: nat,S2: nat] :
% 0.22/0.46 ? [Z2: nat] :
% 0.22/0.46 ! [X5: nat] :
% 0.22/0.46 ( ( ord_less_nat @ X5 @ Z2 )
% 0.22/0.46 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) )
% 0.22/0.46 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % minf(10)
% 0.22/0.46 thf(fact_3575_minf_I10_J,axiom,
% 0.22/0.46 ! [D: int,S2: int] :
% 0.22/0.46 ? [Z2: int] :
% 0.22/0.46 ! [X5: int] :
% 0.22/0.46 ( ( ord_less_int @ X5 @ Z2 )
% 0.22/0.46 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) )
% 0.22/0.46 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % minf(10)
% 0.22/0.46 thf(fact_3576_minf_I9_J,axiom,
% 0.22/0.46 ! [D: code_integer,S2: code_integer] :
% 0.22/0.46 ? [Z2: code_integer] :
% 0.22/0.46 ! [X5: code_integer] :
% 0.22/0.46 ( ( ord_le6747313008572928689nteger @ X5 @ Z2 )
% 0.22/0.46 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) )
% 0.22/0.46 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % minf(9)
% 0.22/0.46 thf(fact_3577_minf_I9_J,axiom,
% 0.22/0.46 ! [D: real,S2: real] :
% 0.22/0.46 ? [Z2: real] :
% 0.22/0.46 ! [X5: real] :
% 0.22/0.46 ( ( ord_less_real @ X5 @ Z2 )
% 0.22/0.46 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) )
% 0.22/0.46 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % minf(9)
% 0.22/0.46 thf(fact_3578_minf_I9_J,axiom,
% 0.22/0.46 ! [D: rat,S2: rat] :
% 0.22/0.46 ? [Z2: rat] :
% 0.22/0.46 ! [X5: rat] :
% 0.22/0.46 ( ( ord_less_rat @ X5 @ Z2 )
% 0.22/0.46 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) )
% 0.22/0.46 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % minf(9)
% 0.22/0.46 thf(fact_3579_minf_I9_J,axiom,
% 0.22/0.46 ! [D: nat,S2: nat] :
% 0.22/0.46 ? [Z2: nat] :
% 0.22/0.46 ! [X5: nat] :
% 0.22/0.46 ( ( ord_less_nat @ X5 @ Z2 )
% 0.22/0.46 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) )
% 0.22/0.46 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % minf(9)
% 0.22/0.46 thf(fact_3580_minf_I9_J,axiom,
% 0.22/0.46 ! [D: int,S2: int] :
% 0.22/0.46 ? [Z2: int] :
% 0.22/0.46 ! [X5: int] :
% 0.22/0.46 ( ( ord_less_int @ X5 @ Z2 )
% 0.22/0.46 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) )
% 0.22/0.46 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % minf(9)
% 0.22/0.46 thf(fact_3581_pinf_I10_J,axiom,
% 0.22/0.46 ! [D: code_integer,S2: code_integer] :
% 0.22/0.46 ? [Z2: code_integer] :
% 0.22/0.46 ! [X5: code_integer] :
% 0.22/0.46 ( ( ord_le6747313008572928689nteger @ Z2 @ X5 )
% 0.22/0.46 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) )
% 0.22/0.46 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % pinf(10)
% 0.22/0.46 thf(fact_3582_pinf_I10_J,axiom,
% 0.22/0.46 ! [D: real,S2: real] :
% 0.22/0.46 ? [Z2: real] :
% 0.22/0.46 ! [X5: real] :
% 0.22/0.46 ( ( ord_less_real @ Z2 @ X5 )
% 0.22/0.46 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) )
% 0.22/0.46 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % pinf(10)
% 0.22/0.46 thf(fact_3583_pinf_I10_J,axiom,
% 0.22/0.46 ! [D: rat,S2: rat] :
% 0.22/0.46 ? [Z2: rat] :
% 0.22/0.46 ! [X5: rat] :
% 0.22/0.46 ( ( ord_less_rat @ Z2 @ X5 )
% 0.22/0.46 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) )
% 0.22/0.46 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % pinf(10)
% 0.22/0.46 thf(fact_3584_pinf_I10_J,axiom,
% 0.22/0.46 ! [D: nat,S2: nat] :
% 0.22/0.46 ? [Z2: nat] :
% 0.22/0.46 ! [X5: nat] :
% 0.22/0.46 ( ( ord_less_nat @ Z2 @ X5 )
% 0.22/0.46 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) )
% 0.22/0.46 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % pinf(10)
% 0.22/0.46 thf(fact_3585_pinf_I10_J,axiom,
% 0.22/0.46 ! [D: int,S2: int] :
% 0.22/0.46 ? [Z2: int] :
% 0.22/0.46 ! [X5: int] :
% 0.22/0.46 ( ( ord_less_int @ Z2 @ X5 )
% 0.22/0.46 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) )
% 0.22/0.46 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % pinf(10)
% 0.22/0.46 thf(fact_3586_pinf_I9_J,axiom,
% 0.22/0.46 ! [D: code_integer,S2: code_integer] :
% 0.22/0.46 ? [Z2: code_integer] :
% 0.22/0.46 ! [X5: code_integer] :
% 0.22/0.46 ( ( ord_le6747313008572928689nteger @ Z2 @ X5 )
% 0.22/0.46 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) )
% 0.22/0.46 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % pinf(9)
% 0.22/0.46 thf(fact_3587_pinf_I9_J,axiom,
% 0.22/0.46 ! [D: real,S2: real] :
% 0.22/0.46 ? [Z2: real] :
% 0.22/0.46 ! [X5: real] :
% 0.22/0.46 ( ( ord_less_real @ Z2 @ X5 )
% 0.22/0.46 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) )
% 0.22/0.46 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % pinf(9)
% 0.22/0.46 thf(fact_3588_pinf_I9_J,axiom,
% 0.22/0.46 ! [D: rat,S2: rat] :
% 0.22/0.46 ? [Z2: rat] :
% 0.22/0.46 ! [X5: rat] :
% 0.22/0.46 ( ( ord_less_rat @ Z2 @ X5 )
% 0.22/0.46 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) )
% 0.22/0.46 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % pinf(9)
% 0.22/0.46 thf(fact_3589_pinf_I9_J,axiom,
% 0.22/0.46 ! [D: nat,S2: nat] :
% 0.22/0.46 ? [Z2: nat] :
% 0.22/0.46 ! [X5: nat] :
% 0.22/0.46 ( ( ord_less_nat @ Z2 @ X5 )
% 0.22/0.46 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) )
% 0.22/0.46 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % pinf(9)
% 0.22/0.46 thf(fact_3590_pinf_I9_J,axiom,
% 0.22/0.46 ! [D: int,S2: int] :
% 0.22/0.46 ? [Z2: int] :
% 0.22/0.46 ! [X5: int] :
% 0.22/0.46 ( ( ord_less_int @ Z2 @ X5 )
% 0.22/0.46 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) )
% 0.22/0.46 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % pinf(9)
% 0.22/0.46 thf(fact_3591_div__plus__div__distrib__dvd__left,axiom,
% 0.22/0.46 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.22/0.46 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.22/0.46 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.22/0.46 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % div_plus_div_distrib_dvd_left
% 0.22/0.46 thf(fact_3592_div__plus__div__distrib__dvd__left,axiom,
% 0.22/0.46 ! [C: nat,A: nat,B: nat] :
% 0.22/0.46 ( ( dvd_dvd_nat @ C @ A )
% 0.22/0.46 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.22/0.46 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 0.22/0.46
% 0.22/0.46 % div_plus_div_distrib_dvd_left
% 0.22/0.46 thf(fact_3593_div__plus__div__distrib__dvd__left,axiom,
% 0.22/0.47 ! [C: int,A: int,B: int] :
% 0.22/0.47 ( ( dvd_dvd_int @ C @ A )
% 0.22/0.47 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.22/0.47 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % div_plus_div_distrib_dvd_left
% 0.22/0.47 thf(fact_3594_div__plus__div__distrib__dvd__right,axiom,
% 0.22/0.47 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.22/0.47 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.22/0.47 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.22/0.47 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % div_plus_div_distrib_dvd_right
% 0.22/0.47 thf(fact_3595_div__plus__div__distrib__dvd__right,axiom,
% 0.22/0.47 ! [C: nat,B: nat,A: nat] :
% 0.22/0.47 ( ( dvd_dvd_nat @ C @ B )
% 0.22/0.47 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.22/0.47 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % div_plus_div_distrib_dvd_right
% 0.22/0.47 thf(fact_3596_div__plus__div__distrib__dvd__right,axiom,
% 0.22/0.47 ! [C: int,B: int,A: int] :
% 0.22/0.47 ( ( dvd_dvd_int @ C @ B )
% 0.22/0.47 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.22/0.47 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % div_plus_div_distrib_dvd_right
% 0.22/0.47 thf(fact_3597_power__add,axiom,
% 0.22/0.47 ! [A: complex,M: nat,N: nat] :
% 0.22/0.47 ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
% 0.22/0.47 = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power_add
% 0.22/0.47 thf(fact_3598_power__add,axiom,
% 0.22/0.47 ! [A: real,M: nat,N: nat] :
% 0.22/0.47 ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
% 0.22/0.47 = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power_add
% 0.22/0.47 thf(fact_3599_power__add,axiom,
% 0.22/0.47 ! [A: rat,M: nat,N: nat] :
% 0.22/0.47 ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
% 0.22/0.47 = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power_add
% 0.22/0.47 thf(fact_3600_power__add,axiom,
% 0.22/0.47 ! [A: nat,M: nat,N: nat] :
% 0.22/0.47 ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
% 0.22/0.47 = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power_add
% 0.22/0.47 thf(fact_3601_power__add,axiom,
% 0.22/0.47 ! [A: int,M: nat,N: nat] :
% 0.22/0.47 ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
% 0.22/0.47 = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power_add
% 0.22/0.47 thf(fact_3602_one__is__add,axiom,
% 0.22/0.47 ! [M: nat,N: nat] :
% 0.22/0.47 ( ( ( suc @ zero_zero_nat )
% 0.22/0.47 = ( plus_plus_nat @ M @ N ) )
% 0.22/0.47 = ( ( ( M
% 0.22/0.47 = ( suc @ zero_zero_nat ) )
% 0.22/0.47 & ( N = zero_zero_nat ) )
% 0.22/0.47 | ( ( M = zero_zero_nat )
% 0.22/0.47 & ( N
% 0.22/0.47 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % one_is_add
% 0.22/0.47 thf(fact_3603_add__is__1,axiom,
% 0.22/0.47 ! [M: nat,N: nat] :
% 0.22/0.47 ( ( ( plus_plus_nat @ M @ N )
% 0.22/0.47 = ( suc @ zero_zero_nat ) )
% 0.22/0.47 = ( ( ( M
% 0.22/0.47 = ( suc @ zero_zero_nat ) )
% 0.22/0.47 & ( N = zero_zero_nat ) )
% 0.22/0.47 | ( ( M = zero_zero_nat )
% 0.22/0.47 & ( N
% 0.22/0.47 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_is_1
% 0.22/0.47 thf(fact_3604_less__imp__Suc__add,axiom,
% 0.22/0.47 ! [M: nat,N: nat] :
% 0.22/0.47 ( ( ord_less_nat @ M @ N )
% 0.22/0.47 => ? [K2: nat] :
% 0.22/0.47 ( N
% 0.22/0.47 = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % less_imp_Suc_add
% 0.22/0.47 thf(fact_3605_less__iff__Suc__add,axiom,
% 0.22/0.47 ( ord_less_nat
% 0.22/0.47 = ( ^ [M5: nat,N4: nat] :
% 0.22/0.47 ? [K3: nat] :
% 0.22/0.47 ( N4
% 0.22/0.47 = ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % less_iff_Suc_add
% 0.22/0.47 thf(fact_3606_less__add__Suc2,axiom,
% 0.22/0.47 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % less_add_Suc2
% 0.22/0.47 thf(fact_3607_less__add__Suc1,axiom,
% 0.22/0.47 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % less_add_Suc1
% 0.22/0.47 thf(fact_3608_less__natE,axiom,
% 0.22/0.47 ! [M: nat,N: nat] :
% 0.22/0.47 ( ( ord_less_nat @ M @ N )
% 0.22/0.47 => ~ ! [Q4: nat] :
% 0.22/0.47 ( N
% 0.22/0.47 != ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % less_natE
% 0.22/0.47 thf(fact_3609_less__imp__add__positive,axiom,
% 0.22/0.47 ! [I: nat,J: nat] :
% 0.22/0.47 ( ( ord_less_nat @ I @ J )
% 0.22/0.47 => ? [K2: nat] :
% 0.22/0.47 ( ( ord_less_nat @ zero_zero_nat @ K2 )
% 0.22/0.47 & ( ( plus_plus_nat @ I @ K2 )
% 0.22/0.47 = J ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % less_imp_add_positive
% 0.22/0.47 thf(fact_3610_mono__nat__linear__lb,axiom,
% 0.22/0.47 ! [F: nat > nat,M: nat,K: nat] :
% 0.22/0.47 ( ! [M4: nat,N2: nat] :
% 0.22/0.47 ( ( ord_less_nat @ M4 @ N2 )
% 0.22/0.47 => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
% 0.22/0.47 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % mono_nat_linear_lb
% 0.22/0.47 thf(fact_3611_mult__Suc,axiom,
% 0.22/0.47 ! [M: nat,N: nat] :
% 0.22/0.47 ( ( times_times_nat @ ( suc @ M ) @ N )
% 0.22/0.47 = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_Suc
% 0.22/0.47 thf(fact_3612_diff__add__0,axiom,
% 0.22/0.47 ! [N: nat,M: nat] :
% 0.22/0.47 ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 0.22/0.47 = zero_zero_nat ) ).
% 0.22/0.47
% 0.22/0.47 % diff_add_0
% 0.22/0.47 thf(fact_3613_less__diff__conv,axiom,
% 0.22/0.47 ! [I: nat,J: nat,K: nat] :
% 0.22/0.47 ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.22/0.47 = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 0.22/0.47
% 0.22/0.47 % less_diff_conv
% 0.22/0.47 thf(fact_3614_add__diff__inverse__nat,axiom,
% 0.22/0.47 ! [M: nat,N: nat] :
% 0.22/0.47 ( ~ ( ord_less_nat @ M @ N )
% 0.22/0.47 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 0.22/0.47 = M ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_diff_inverse_nat
% 0.22/0.47 thf(fact_3615_Suc__eq__plus1__left,axiom,
% 0.22/0.47 ( suc
% 0.22/0.47 = ( plus_plus_nat @ one_one_nat ) ) ).
% 0.22/0.47
% 0.22/0.47 % Suc_eq_plus1_left
% 0.22/0.47 thf(fact_3616_plus__1__eq__Suc,axiom,
% 0.22/0.47 ( ( plus_plus_nat @ one_one_nat )
% 0.22/0.47 = suc ) ).
% 0.22/0.47
% 0.22/0.47 % plus_1_eq_Suc
% 0.22/0.47 thf(fact_3617_Suc__eq__plus1,axiom,
% 0.22/0.47 ( suc
% 0.22/0.47 = ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % Suc_eq_plus1
% 0.22/0.47 thf(fact_3618_Nat_Ole__imp__diff__is__add,axiom,
% 0.22/0.47 ! [I: nat,J: nat,K: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ I @ J )
% 0.22/0.47 => ( ( ( minus_minus_nat @ J @ I )
% 0.22/0.47 = K )
% 0.22/0.47 = ( J
% 0.22/0.47 = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % Nat.le_imp_diff_is_add
% 0.22/0.47 thf(fact_3619_Nat_Odiff__add__assoc2,axiom,
% 0.22/0.47 ! [K: nat,J: nat,I: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ K @ J )
% 0.22/0.47 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 0.22/0.47 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % Nat.diff_add_assoc2
% 0.22/0.47 thf(fact_3620_Nat_Odiff__add__assoc,axiom,
% 0.22/0.47 ! [K: nat,J: nat,I: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ K @ J )
% 0.22/0.47 => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.22/0.47 = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % Nat.diff_add_assoc
% 0.22/0.47 thf(fact_3621_Nat_Ole__diff__conv2,axiom,
% 0.22/0.47 ! [K: nat,J: nat,I: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ K @ J )
% 0.22/0.47 => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.22/0.47 = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % Nat.le_diff_conv2
% 0.22/0.47 thf(fact_3622_le__diff__conv,axiom,
% 0.22/0.47 ! [J: nat,K: nat,I: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.22/0.47 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % le_diff_conv
% 0.22/0.47 thf(fact_3623_bezout__lemma__nat,axiom,
% 0.22/0.47 ! [D: nat,A: nat,B: nat,X: nat,Y4: nat] :
% 0.22/0.47 ( ( dvd_dvd_nat @ D @ A )
% 0.22/0.47 => ( ( dvd_dvd_nat @ D @ B )
% 0.22/0.47 => ( ( ( ( times_times_nat @ A @ X )
% 0.22/0.47 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D ) )
% 0.22/0.47 | ( ( times_times_nat @ B @ X )
% 0.22/0.47 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) )
% 0.22/0.47 => ? [X4: nat,Y: nat] :
% 0.22/0.47 ( ( dvd_dvd_nat @ D @ A )
% 0.22/0.47 & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
% 0.22/0.47 & ( ( ( times_times_nat @ A @ X4 )
% 0.22/0.47 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y ) @ D ) )
% 0.22/0.47 | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X4 )
% 0.22/0.47 = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % bezout_lemma_nat
% 0.22/0.47 thf(fact_3624_bezout__add__nat,axiom,
% 0.22/0.47 ! [A: nat,B: nat] :
% 0.22/0.47 ? [D2: nat,X4: nat,Y: nat] :
% 0.22/0.47 ( ( dvd_dvd_nat @ D2 @ A )
% 0.22/0.47 & ( dvd_dvd_nat @ D2 @ B )
% 0.22/0.47 & ( ( ( times_times_nat @ A @ X4 )
% 0.22/0.47 = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D2 ) )
% 0.22/0.47 | ( ( times_times_nat @ B @ X4 )
% 0.22/0.47 = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D2 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % bezout_add_nat
% 0.22/0.47 thf(fact_3625_field__le__epsilon,axiom,
% 0.22/0.47 ! [X: rat,Y4: rat] :
% 0.22/0.47 ( ! [E: rat] :
% 0.22/0.47 ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.22/0.47 => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y4 @ E ) ) )
% 0.22/0.47 => ( ord_less_eq_rat @ X @ Y4 ) ) ).
% 0.22/0.47
% 0.22/0.47 % field_le_epsilon
% 0.22/0.47 thf(fact_3626_field__le__epsilon,axiom,
% 0.22/0.47 ! [X: real,Y4: real] :
% 0.22/0.47 ( ! [E: real] :
% 0.22/0.47 ( ( ord_less_real @ zero_zero_real @ E )
% 0.22/0.47 => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y4 @ E ) ) )
% 0.22/0.47 => ( ord_less_eq_real @ X @ Y4 ) ) ).
% 0.22/0.47
% 0.22/0.47 % field_le_epsilon
% 0.22/0.47 thf(fact_3627_add__strict__increasing2,axiom,
% 0.22/0.47 ! [A: rat,B: rat,C: rat] :
% 0.22/0.47 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.22/0.47 => ( ( ord_less_rat @ B @ C )
% 0.22/0.47 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_strict_increasing2
% 0.22/0.47 thf(fact_3628_add__strict__increasing2,axiom,
% 0.22/0.47 ! [A: nat,B: nat,C: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.22/0.47 => ( ( ord_less_nat @ B @ C )
% 0.22/0.47 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_strict_increasing2
% 0.22/0.47 thf(fact_3629_add__strict__increasing2,axiom,
% 0.22/0.47 ! [A: int,B: int,C: int] :
% 0.22/0.47 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.22/0.47 => ( ( ord_less_int @ B @ C )
% 0.22/0.47 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_strict_increasing2
% 0.22/0.47 thf(fact_3630_add__strict__increasing2,axiom,
% 0.22/0.47 ! [A: real,B: real,C: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.47 => ( ( ord_less_real @ B @ C )
% 0.22/0.47 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_strict_increasing2
% 0.22/0.47 thf(fact_3631_add__strict__increasing,axiom,
% 0.22/0.47 ! [A: rat,B: rat,C: rat] :
% 0.22/0.47 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.22/0.47 => ( ( ord_less_eq_rat @ B @ C )
% 0.22/0.47 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_strict_increasing
% 0.22/0.47 thf(fact_3632_add__strict__increasing,axiom,
% 0.22/0.47 ! [A: nat,B: nat,C: nat] :
% 0.22/0.47 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.22/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.22/0.47 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_strict_increasing
% 0.22/0.47 thf(fact_3633_add__strict__increasing,axiom,
% 0.22/0.47 ! [A: int,B: int,C: int] :
% 0.22/0.47 ( ( ord_less_int @ zero_zero_int @ A )
% 0.22/0.47 => ( ( ord_less_eq_int @ B @ C )
% 0.22/0.47 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_strict_increasing
% 0.22/0.47 thf(fact_3634_add__strict__increasing,axiom,
% 0.22/0.47 ! [A: real,B: real,C: real] :
% 0.22/0.47 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.47 => ( ( ord_less_eq_real @ B @ C )
% 0.22/0.47 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_strict_increasing
% 0.22/0.47 thf(fact_3635_add__pos__nonneg,axiom,
% 0.22/0.47 ! [A: rat,B: rat] :
% 0.22/0.47 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.22/0.47 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.22/0.47 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_pos_nonneg
% 0.22/0.47 thf(fact_3636_add__pos__nonneg,axiom,
% 0.22/0.47 ! [A: nat,B: nat] :
% 0.22/0.47 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.22/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.22/0.47 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_pos_nonneg
% 0.22/0.47 thf(fact_3637_add__pos__nonneg,axiom,
% 0.22/0.47 ! [A: int,B: int] :
% 0.22/0.47 ( ( ord_less_int @ zero_zero_int @ A )
% 0.22/0.47 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.22/0.47 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_pos_nonneg
% 0.22/0.47 thf(fact_3638_add__pos__nonneg,axiom,
% 0.22/0.47 ! [A: real,B: real] :
% 0.22/0.47 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.22/0.47 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_pos_nonneg
% 0.22/0.47 thf(fact_3639_add__nonpos__neg,axiom,
% 0.22/0.47 ! [A: rat,B: rat] :
% 0.22/0.47 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.22/0.47 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.22/0.47 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_nonpos_neg
% 0.22/0.47 thf(fact_3640_add__nonpos__neg,axiom,
% 0.22/0.47 ! [A: nat,B: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.22/0.47 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.22/0.47 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_nonpos_neg
% 0.22/0.47 thf(fact_3641_add__nonpos__neg,axiom,
% 0.22/0.47 ! [A: int,B: int] :
% 0.22/0.47 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.22/0.47 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.22/0.47 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_nonpos_neg
% 0.22/0.47 thf(fact_3642_add__nonpos__neg,axiom,
% 0.22/0.47 ! [A: real,B: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.22/0.47 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.22/0.47 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_nonpos_neg
% 0.22/0.47 thf(fact_3643_add__nonneg__pos,axiom,
% 0.22/0.47 ! [A: rat,B: rat] :
% 0.22/0.47 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.22/0.47 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.22/0.47 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_nonneg_pos
% 0.22/0.47 thf(fact_3644_add__nonneg__pos,axiom,
% 0.22/0.47 ! [A: nat,B: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.22/0.47 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.22/0.47 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_nonneg_pos
% 0.22/0.47 thf(fact_3645_add__nonneg__pos,axiom,
% 0.22/0.47 ! [A: int,B: int] :
% 0.22/0.47 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.22/0.47 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.47 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_nonneg_pos
% 0.22/0.47 thf(fact_3646_add__nonneg__pos,axiom,
% 0.22/0.47 ! [A: real,B: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.47 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.22/0.47 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_nonneg_pos
% 0.22/0.47 thf(fact_3647_add__neg__nonpos,axiom,
% 0.22/0.47 ! [A: rat,B: rat] :
% 0.22/0.47 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.22/0.47 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.22/0.47 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_neg_nonpos
% 0.22/0.47 thf(fact_3648_add__neg__nonpos,axiom,
% 0.22/0.47 ! [A: nat,B: nat] :
% 0.22/0.47 ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.22/0.47 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.22/0.47 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_neg_nonpos
% 0.22/0.47 thf(fact_3649_add__neg__nonpos,axiom,
% 0.22/0.47 ! [A: int,B: int] :
% 0.22/0.47 ( ( ord_less_int @ A @ zero_zero_int )
% 0.22/0.47 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.22/0.47 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_neg_nonpos
% 0.22/0.47 thf(fact_3650_add__neg__nonpos,axiom,
% 0.22/0.47 ! [A: real,B: real] :
% 0.22/0.47 ( ( ord_less_real @ A @ zero_zero_real )
% 0.22/0.47 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.22/0.47 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_neg_nonpos
% 0.22/0.47 thf(fact_3651_sum__squares__ge__zero,axiom,
% 0.22/0.47 ! [X: rat,Y4: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y4 @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_squares_ge_zero
% 0.22/0.47 thf(fact_3652_sum__squares__ge__zero,axiom,
% 0.22/0.47 ! [X: int,Y4: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y4 @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_squares_ge_zero
% 0.22/0.47 thf(fact_3653_sum__squares__ge__zero,axiom,
% 0.22/0.47 ! [X: real,Y4: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_squares_ge_zero
% 0.22/0.47 thf(fact_3654_sum__squares__le__zero__iff,axiom,
% 0.22/0.47 ! [X: rat,Y4: rat] :
% 0.22/0.47 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y4 @ Y4 ) ) @ zero_zero_rat )
% 0.22/0.47 = ( ( X = zero_zero_rat )
% 0.22/0.47 & ( Y4 = zero_zero_rat ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_squares_le_zero_iff
% 0.22/0.47 thf(fact_3655_sum__squares__le__zero__iff,axiom,
% 0.22/0.47 ! [X: int,Y4: int] :
% 0.22/0.47 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y4 @ Y4 ) ) @ zero_zero_int )
% 0.22/0.47 = ( ( X = zero_zero_int )
% 0.22/0.47 & ( Y4 = zero_zero_int ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_squares_le_zero_iff
% 0.22/0.47 thf(fact_3656_sum__squares__le__zero__iff,axiom,
% 0.22/0.47 ! [X: real,Y4: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) ) @ zero_zero_real )
% 0.22/0.47 = ( ( X = zero_zero_real )
% 0.22/0.47 & ( Y4 = zero_zero_real ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_squares_le_zero_iff
% 0.22/0.47 thf(fact_3657_not__sum__squares__lt__zero,axiom,
% 0.22/0.47 ! [X: real,Y4: real] :
% 0.22/0.47 ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) ) @ zero_zero_real ) ).
% 0.22/0.47
% 0.22/0.47 % not_sum_squares_lt_zero
% 0.22/0.47 thf(fact_3658_not__sum__squares__lt__zero,axiom,
% 0.22/0.47 ! [X: rat,Y4: rat] :
% 0.22/0.47 ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y4 @ Y4 ) ) @ zero_zero_rat ) ).
% 0.22/0.47
% 0.22/0.47 % not_sum_squares_lt_zero
% 0.22/0.47 thf(fact_3659_not__sum__squares__lt__zero,axiom,
% 0.22/0.47 ! [X: int,Y4: int] :
% 0.22/0.47 ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y4 @ Y4 ) ) @ zero_zero_int ) ).
% 0.22/0.47
% 0.22/0.47 % not_sum_squares_lt_zero
% 0.22/0.47 thf(fact_3660_sum__squares__gt__zero__iff,axiom,
% 0.22/0.47 ! [X: real,Y4: real] :
% 0.22/0.47 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) ) )
% 0.22/0.47 = ( ( X != zero_zero_real )
% 0.22/0.47 | ( Y4 != zero_zero_real ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_squares_gt_zero_iff
% 0.22/0.47 thf(fact_3661_sum__squares__gt__zero__iff,axiom,
% 0.22/0.47 ! [X: rat,Y4: rat] :
% 0.22/0.47 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y4 @ Y4 ) ) )
% 0.22/0.47 = ( ( X != zero_zero_rat )
% 0.22/0.47 | ( Y4 != zero_zero_rat ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_squares_gt_zero_iff
% 0.22/0.47 thf(fact_3662_sum__squares__gt__zero__iff,axiom,
% 0.22/0.47 ! [X: int,Y4: int] :
% 0.22/0.47 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y4 @ Y4 ) ) )
% 0.22/0.47 = ( ( X != zero_zero_int )
% 0.22/0.47 | ( Y4 != zero_zero_int ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_squares_gt_zero_iff
% 0.22/0.47 thf(fact_3663_discrete,axiom,
% 0.22/0.47 ( ord_less_nat
% 0.22/0.47 = ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % discrete
% 0.22/0.47 thf(fact_3664_discrete,axiom,
% 0.22/0.47 ( ord_less_int
% 0.22/0.47 = ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % discrete
% 0.22/0.47 thf(fact_3665_zero__less__two,axiom,
% 0.22/0.47 ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 0.22/0.47
% 0.22/0.47 % zero_less_two
% 0.22/0.47 thf(fact_3666_zero__less__two,axiom,
% 0.22/0.47 ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 0.22/0.47
% 0.22/0.47 % zero_less_two
% 0.22/0.47 thf(fact_3667_zero__less__two,axiom,
% 0.22/0.47 ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 0.22/0.47
% 0.22/0.47 % zero_less_two
% 0.22/0.47 thf(fact_3668_zero__less__two,axiom,
% 0.22/0.47 ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 0.22/0.47
% 0.22/0.47 % zero_less_two
% 0.22/0.47 thf(fact_3669_ordered__ring__class_Ole__add__iff1,axiom,
% 0.22/0.47 ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.22/0.47 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.22/0.47 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.22/0.47
% 0.22/0.47 % ordered_ring_class.le_add_iff1
% 0.22/0.47 thf(fact_3670_ordered__ring__class_Ole__add__iff1,axiom,
% 0.22/0.47 ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.22/0.47 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.22/0.47 = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.22/0.47
% 0.22/0.47 % ordered_ring_class.le_add_iff1
% 0.22/0.47 thf(fact_3671_ordered__ring__class_Ole__add__iff1,axiom,
% 0.22/0.47 ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.22/0.47 = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.22/0.47
% 0.22/0.47 % ordered_ring_class.le_add_iff1
% 0.22/0.47 thf(fact_3672_ordered__ring__class_Ole__add__iff2,axiom,
% 0.22/0.47 ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.22/0.47 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.22/0.47 = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % ordered_ring_class.le_add_iff2
% 0.22/0.47 thf(fact_3673_ordered__ring__class_Ole__add__iff2,axiom,
% 0.22/0.47 ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.22/0.47 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.22/0.47 = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % ordered_ring_class.le_add_iff2
% 0.22/0.47 thf(fact_3674_ordered__ring__class_Ole__add__iff2,axiom,
% 0.22/0.47 ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.22/0.47 = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % ordered_ring_class.le_add_iff2
% 0.22/0.47 thf(fact_3675_add__divide__eq__if__simps_I2_J,axiom,
% 0.22/0.47 ! [Z: complex,A: complex,B: complex] :
% 0.22/0.47 ( ( ( Z = zero_zero_complex )
% 0.22/0.47 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.22/0.47 = B ) )
% 0.22/0.47 & ( ( Z != zero_zero_complex )
% 0.22/0.47 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.22/0.47 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_divide_eq_if_simps(2)
% 0.22/0.47 thf(fact_3676_add__divide__eq__if__simps_I2_J,axiom,
% 0.22/0.47 ! [Z: real,A: real,B: real] :
% 0.22/0.47 ( ( ( Z = zero_zero_real )
% 0.22/0.47 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.22/0.47 = B ) )
% 0.22/0.47 & ( ( Z != zero_zero_real )
% 0.22/0.47 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.22/0.47 = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_divide_eq_if_simps(2)
% 0.22/0.47 thf(fact_3677_add__divide__eq__if__simps_I2_J,axiom,
% 0.22/0.47 ! [Z: rat,A: rat,B: rat] :
% 0.22/0.47 ( ( ( Z = zero_zero_rat )
% 0.22/0.47 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.22/0.47 = B ) )
% 0.22/0.47 & ( ( Z != zero_zero_rat )
% 0.22/0.47 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.22/0.47 = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_divide_eq_if_simps(2)
% 0.22/0.47 thf(fact_3678_add__divide__eq__if__simps_I1_J,axiom,
% 0.22/0.47 ! [Z: complex,A: complex,B: complex] :
% 0.22/0.47 ( ( ( Z = zero_zero_complex )
% 0.22/0.47 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.22/0.47 = A ) )
% 0.22/0.47 & ( ( Z != zero_zero_complex )
% 0.22/0.47 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.22/0.47 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_divide_eq_if_simps(1)
% 0.22/0.47 thf(fact_3679_add__divide__eq__if__simps_I1_J,axiom,
% 0.22/0.47 ! [Z: real,A: real,B: real] :
% 0.22/0.47 ( ( ( Z = zero_zero_real )
% 0.22/0.47 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.22/0.47 = A ) )
% 0.22/0.47 & ( ( Z != zero_zero_real )
% 0.22/0.47 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.22/0.47 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_divide_eq_if_simps(1)
% 0.22/0.47 thf(fact_3680_add__divide__eq__if__simps_I1_J,axiom,
% 0.22/0.47 ! [Z: rat,A: rat,B: rat] :
% 0.22/0.47 ( ( ( Z = zero_zero_rat )
% 0.22/0.47 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.22/0.47 = A ) )
% 0.22/0.47 & ( ( Z != zero_zero_rat )
% 0.22/0.47 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.22/0.47 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_divide_eq_if_simps(1)
% 0.22/0.47 thf(fact_3681_add__frac__eq,axiom,
% 0.22/0.47 ! [Y4: complex,Z: complex,X: complex,W: complex] :
% 0.22/0.47 ( ( Y4 != zero_zero_complex )
% 0.22/0.47 => ( ( Z != zero_zero_complex )
% 0.22/0.47 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y4 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 0.22/0.47 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y4 ) ) @ ( times_times_complex @ Y4 @ Z ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_frac_eq
% 0.22/0.47 thf(fact_3682_add__frac__eq,axiom,
% 0.22/0.47 ! [Y4: real,Z: real,X: real,W: real] :
% 0.22/0.47 ( ( Y4 != zero_zero_real )
% 0.22/0.47 => ( ( Z != zero_zero_real )
% 0.22/0.47 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y4 ) @ ( divide_divide_real @ W @ Z ) )
% 0.22/0.47 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y4 ) ) @ ( times_times_real @ Y4 @ Z ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_frac_eq
% 0.22/0.47 thf(fact_3683_add__frac__eq,axiom,
% 0.22/0.47 ! [Y4: rat,Z: rat,X: rat,W: rat] :
% 0.22/0.47 ( ( Y4 != zero_zero_rat )
% 0.22/0.47 => ( ( Z != zero_zero_rat )
% 0.22/0.47 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y4 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.22/0.47 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y4 ) ) @ ( times_times_rat @ Y4 @ Z ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_frac_eq
% 0.22/0.47 thf(fact_3684_add__frac__num,axiom,
% 0.22/0.47 ! [Y4: complex,X: complex,Z: complex] :
% 0.22/0.47 ( ( Y4 != zero_zero_complex )
% 0.22/0.47 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y4 ) @ Z )
% 0.22/0.47 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y4 ) ) @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_frac_num
% 0.22/0.47 thf(fact_3685_add__frac__num,axiom,
% 0.22/0.47 ! [Y4: real,X: real,Z: real] :
% 0.22/0.47 ( ( Y4 != zero_zero_real )
% 0.22/0.47 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y4 ) @ Z )
% 0.22/0.47 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y4 ) ) @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_frac_num
% 0.22/0.47 thf(fact_3686_add__frac__num,axiom,
% 0.22/0.47 ! [Y4: rat,X: rat,Z: rat] :
% 0.22/0.47 ( ( Y4 != zero_zero_rat )
% 0.22/0.47 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y4 ) @ Z )
% 0.22/0.47 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y4 ) ) @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_frac_num
% 0.22/0.47 thf(fact_3687_add__num__frac,axiom,
% 0.22/0.47 ! [Y4: complex,Z: complex,X: complex] :
% 0.22/0.47 ( ( Y4 != zero_zero_complex )
% 0.22/0.47 => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X @ Y4 ) )
% 0.22/0.47 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y4 ) ) @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_num_frac
% 0.22/0.47 thf(fact_3688_add__num__frac,axiom,
% 0.22/0.47 ! [Y4: real,Z: real,X: real] :
% 0.22/0.47 ( ( Y4 != zero_zero_real )
% 0.22/0.47 => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y4 ) )
% 0.22/0.47 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y4 ) ) @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_num_frac
% 0.22/0.47 thf(fact_3689_add__num__frac,axiom,
% 0.22/0.47 ! [Y4: rat,Z: rat,X: rat] :
% 0.22/0.47 ( ( Y4 != zero_zero_rat )
% 0.22/0.47 => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y4 ) )
% 0.22/0.47 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y4 ) ) @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_num_frac
% 0.22/0.47 thf(fact_3690_add__divide__eq__iff,axiom,
% 0.22/0.47 ! [Z: complex,X: complex,Y4: complex] :
% 0.22/0.47 ( ( Z != zero_zero_complex )
% 0.22/0.47 => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y4 @ Z ) )
% 0.22/0.47 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ Y4 ) @ Z ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_divide_eq_iff
% 0.22/0.47 thf(fact_3691_add__divide__eq__iff,axiom,
% 0.22/0.47 ! [Z: real,X: real,Y4: real] :
% 0.22/0.47 ( ( Z != zero_zero_real )
% 0.22/0.47 => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y4 @ Z ) )
% 0.22/0.47 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y4 ) @ Z ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_divide_eq_iff
% 0.22/0.47 thf(fact_3692_add__divide__eq__iff,axiom,
% 0.22/0.47 ! [Z: rat,X: rat,Y4: rat] :
% 0.22/0.47 ( ( Z != zero_zero_rat )
% 0.22/0.47 => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y4 @ Z ) )
% 0.22/0.47 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y4 ) @ Z ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_divide_eq_iff
% 0.22/0.47 thf(fact_3693_divide__add__eq__iff,axiom,
% 0.22/0.47 ! [Z: complex,X: complex,Y4: complex] :
% 0.22/0.47 ( ( Z != zero_zero_complex )
% 0.22/0.47 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y4 )
% 0.22/0.47 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y4 @ Z ) ) @ Z ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % divide_add_eq_iff
% 0.22/0.47 thf(fact_3694_divide__add__eq__iff,axiom,
% 0.22/0.47 ! [Z: real,X: real,Y4: real] :
% 0.22/0.47 ( ( Z != zero_zero_real )
% 0.22/0.47 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y4 )
% 0.22/0.47 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y4 @ Z ) ) @ Z ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % divide_add_eq_iff
% 0.22/0.47 thf(fact_3695_divide__add__eq__iff,axiom,
% 0.22/0.47 ! [Z: rat,X: rat,Y4: rat] :
% 0.22/0.47 ( ( Z != zero_zero_rat )
% 0.22/0.47 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y4 )
% 0.22/0.47 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y4 @ Z ) ) @ Z ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % divide_add_eq_iff
% 0.22/0.47 thf(fact_3696_less__add__iff1,axiom,
% 0.22/0.47 ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.22/0.47 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.22/0.47 = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.22/0.47
% 0.22/0.47 % less_add_iff1
% 0.22/0.47 thf(fact_3697_less__add__iff1,axiom,
% 0.22/0.47 ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.22/0.47 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.22/0.47 = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.22/0.47
% 0.22/0.47 % less_add_iff1
% 0.22/0.47 thf(fact_3698_less__add__iff1,axiom,
% 0.22/0.47 ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.22/0.47 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.22/0.47 = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.22/0.47
% 0.22/0.47 % less_add_iff1
% 0.22/0.47 thf(fact_3699_less__add__iff2,axiom,
% 0.22/0.47 ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.22/0.47 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.22/0.47 = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % less_add_iff2
% 0.22/0.47 thf(fact_3700_less__add__iff2,axiom,
% 0.22/0.47 ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.22/0.47 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.22/0.47 = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % less_add_iff2
% 0.22/0.47 thf(fact_3701_less__add__iff2,axiom,
% 0.22/0.47 ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.22/0.47 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.22/0.47 = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % less_add_iff2
% 0.22/0.47 thf(fact_3702_div__add__self2,axiom,
% 0.22/0.47 ! [B: nat,A: nat] :
% 0.22/0.47 ( ( B != zero_zero_nat )
% 0.22/0.47 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.22/0.47 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % div_add_self2
% 0.22/0.47 thf(fact_3703_div__add__self2,axiom,
% 0.22/0.47 ! [B: int,A: int] :
% 0.22/0.47 ( ( B != zero_zero_int )
% 0.22/0.47 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.22/0.47 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % div_add_self2
% 0.22/0.47 thf(fact_3704_div__add__self1,axiom,
% 0.22/0.47 ! [B: nat,A: nat] :
% 0.22/0.47 ( ( B != zero_zero_nat )
% 0.22/0.47 => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.22/0.47 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % div_add_self1
% 0.22/0.47 thf(fact_3705_div__add__self1,axiom,
% 0.22/0.47 ! [B: int,A: int] :
% 0.22/0.47 ( ( B != zero_zero_int )
% 0.22/0.47 => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.22/0.47 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % div_add_self1
% 0.22/0.47 thf(fact_3706_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 0.22/0.47 ! [X: nat,N: nat,M: nat] :
% 0.22/0.47 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.22/0.47 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.47 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.22/0.47 => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % VEBT_internal.exp_split_high_low(1)
% 0.22/0.47 thf(fact_3707_less__half__sum,axiom,
% 0.22/0.47 ! [A: real,B: real] :
% 0.22/0.47 ( ( ord_less_real @ A @ B )
% 0.22/0.47 => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % less_half_sum
% 0.22/0.47 thf(fact_3708_less__half__sum,axiom,
% 0.22/0.47 ! [A: rat,B: rat] :
% 0.22/0.47 ( ( ord_less_rat @ A @ B )
% 0.22/0.47 => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % less_half_sum
% 0.22/0.47 thf(fact_3709_gt__half__sum,axiom,
% 0.22/0.47 ! [A: real,B: real] :
% 0.22/0.47 ( ( ord_less_real @ A @ B )
% 0.22/0.47 => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % gt_half_sum
% 0.22/0.47 thf(fact_3710_gt__half__sum,axiom,
% 0.22/0.47 ! [A: rat,B: rat] :
% 0.22/0.47 ( ( ord_less_rat @ A @ B )
% 0.22/0.47 => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % gt_half_sum
% 0.22/0.47 thf(fact_3711_square__diff__one__factored,axiom,
% 0.22/0.47 ! [X: complex] :
% 0.22/0.47 ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
% 0.22/0.47 = ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % square_diff_one_factored
% 0.22/0.47 thf(fact_3712_square__diff__one__factored,axiom,
% 0.22/0.47 ! [X: real] :
% 0.22/0.47 ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
% 0.22/0.47 = ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % square_diff_one_factored
% 0.22/0.47 thf(fact_3713_square__diff__one__factored,axiom,
% 0.22/0.47 ! [X: rat] :
% 0.22/0.47 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
% 0.22/0.47 = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % square_diff_one_factored
% 0.22/0.47 thf(fact_3714_square__diff__one__factored,axiom,
% 0.22/0.47 ! [X: int] :
% 0.22/0.47 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
% 0.22/0.47 = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % square_diff_one_factored
% 0.22/0.47 thf(fact_3715_unity__coeff__ex,axiom,
% 0.22/0.47 ! [P: code_integer > $o,L: code_integer] :
% 0.22/0.47 ( ( ? [X3: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X3 ) ) )
% 0.22/0.47 = ( ? [X3: code_integer] :
% 0.22/0.47 ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X3 @ zero_z3403309356797280102nteger ) )
% 0.22/0.47 & ( P @ X3 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % unity_coeff_ex
% 0.22/0.47 thf(fact_3716_unity__coeff__ex,axiom,
% 0.22/0.47 ! [P: complex > $o,L: complex] :
% 0.22/0.47 ( ( ? [X3: complex] : ( P @ ( times_times_complex @ L @ X3 ) ) )
% 0.22/0.47 = ( ? [X3: complex] :
% 0.22/0.47 ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X3 @ zero_zero_complex ) )
% 0.22/0.47 & ( P @ X3 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % unity_coeff_ex
% 0.22/0.47 thf(fact_3717_unity__coeff__ex,axiom,
% 0.22/0.47 ! [P: real > $o,L: real] :
% 0.22/0.47 ( ( ? [X3: real] : ( P @ ( times_times_real @ L @ X3 ) ) )
% 0.22/0.47 = ( ? [X3: real] :
% 0.22/0.47 ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X3 @ zero_zero_real ) )
% 0.22/0.47 & ( P @ X3 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % unity_coeff_ex
% 0.22/0.47 thf(fact_3718_unity__coeff__ex,axiom,
% 0.22/0.47 ! [P: rat > $o,L: rat] :
% 0.22/0.47 ( ( ? [X3: rat] : ( P @ ( times_times_rat @ L @ X3 ) ) )
% 0.22/0.47 = ( ? [X3: rat] :
% 0.22/0.47 ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X3 @ zero_zero_rat ) )
% 0.22/0.47 & ( P @ X3 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % unity_coeff_ex
% 0.22/0.47 thf(fact_3719_unity__coeff__ex,axiom,
% 0.22/0.47 ! [P: nat > $o,L: nat] :
% 0.22/0.47 ( ( ? [X3: nat] : ( P @ ( times_times_nat @ L @ X3 ) ) )
% 0.22/0.47 = ( ? [X3: nat] :
% 0.22/0.47 ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X3 @ zero_zero_nat ) )
% 0.22/0.47 & ( P @ X3 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % unity_coeff_ex
% 0.22/0.47 thf(fact_3720_unity__coeff__ex,axiom,
% 0.22/0.47 ! [P: int > $o,L: int] :
% 0.22/0.47 ( ( ? [X3: int] : ( P @ ( times_times_int @ L @ X3 ) ) )
% 0.22/0.47 = ( ? [X3: int] :
% 0.22/0.47 ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X3 @ zero_zero_int ) )
% 0.22/0.47 & ( P @ X3 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % unity_coeff_ex
% 0.22/0.47 thf(fact_3721_inf__period_I3_J,axiom,
% 0.22/0.47 ! [D: code_integer,D3: code_integer,T: code_integer] :
% 0.22/0.47 ( ( dvd_dvd_Code_integer @ D @ D3 )
% 0.22/0.47 => ! [X5: code_integer,K4: code_integer] :
% 0.22/0.47 ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) )
% 0.22/0.47 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % inf_period(3)
% 0.22/0.47 thf(fact_3722_inf__period_I3_J,axiom,
% 0.22/0.47 ! [D: real,D3: real,T: real] :
% 0.22/0.47 ( ( dvd_dvd_real @ D @ D3 )
% 0.22/0.47 => ! [X5: real,K4: real] :
% 0.22/0.47 ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) )
% 0.22/0.47 = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % inf_period(3)
% 0.22/0.47 thf(fact_3723_inf__period_I3_J,axiom,
% 0.22/0.47 ! [D: rat,D3: rat,T: rat] :
% 0.22/0.47 ( ( dvd_dvd_rat @ D @ D3 )
% 0.22/0.47 => ! [X5: rat,K4: rat] :
% 0.22/0.47 ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) )
% 0.22/0.47 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % inf_period(3)
% 0.22/0.47 thf(fact_3724_inf__period_I3_J,axiom,
% 0.22/0.47 ! [D: int,D3: int,T: int] :
% 0.22/0.47 ( ( dvd_dvd_int @ D @ D3 )
% 0.22/0.47 => ! [X5: int,K4: int] :
% 0.22/0.47 ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.22/0.47 = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % inf_period(3)
% 0.22/0.47 thf(fact_3725_inf__period_I4_J,axiom,
% 0.22/0.47 ! [D: code_integer,D3: code_integer,T: code_integer] :
% 0.22/0.47 ( ( dvd_dvd_Code_integer @ D @ D3 )
% 0.22/0.47 => ! [X5: code_integer,K4: code_integer] :
% 0.22/0.47 ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) ) )
% 0.22/0.47 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % inf_period(4)
% 0.22/0.47 thf(fact_3726_inf__period_I4_J,axiom,
% 0.22/0.47 ! [D: real,D3: real,T: real] :
% 0.22/0.47 ( ( dvd_dvd_real @ D @ D3 )
% 0.22/0.47 => ! [X5: real,K4: real] :
% 0.22/0.47 ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) ) )
% 0.22/0.47 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % inf_period(4)
% 0.22/0.47 thf(fact_3727_inf__period_I4_J,axiom,
% 0.22/0.47 ! [D: rat,D3: rat,T: rat] :
% 0.22/0.47 ( ( dvd_dvd_rat @ D @ D3 )
% 0.22/0.47 => ! [X5: rat,K4: rat] :
% 0.22/0.47 ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) ) )
% 0.22/0.47 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % inf_period(4)
% 0.22/0.47 thf(fact_3728_inf__period_I4_J,axiom,
% 0.22/0.47 ! [D: int,D3: int,T: int] :
% 0.22/0.47 ( ( dvd_dvd_int @ D @ D3 )
% 0.22/0.47 => ! [X5: int,K4: int] :
% 0.22/0.47 ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) ) )
% 0.22/0.47 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % inf_period(4)
% 0.22/0.47 thf(fact_3729_numeral__Bit1,axiom,
% 0.22/0.47 ! [N: num] :
% 0.22/0.47 ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 0.22/0.47 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 0.22/0.47
% 0.22/0.47 % numeral_Bit1
% 0.22/0.47 thf(fact_3730_numeral__Bit1,axiom,
% 0.22/0.47 ! [N: num] :
% 0.22/0.47 ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 0.22/0.47 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 0.22/0.47
% 0.22/0.47 % numeral_Bit1
% 0.22/0.47 thf(fact_3731_numeral__Bit1,axiom,
% 0.22/0.47 ! [N: num] :
% 0.22/0.47 ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 0.22/0.47 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 0.22/0.47
% 0.22/0.47 % numeral_Bit1
% 0.22/0.47 thf(fact_3732_numeral__Bit1,axiom,
% 0.22/0.47 ! [N: num] :
% 0.22/0.47 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 0.22/0.47 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 0.22/0.47
% 0.22/0.47 % numeral_Bit1
% 0.22/0.47 thf(fact_3733_numeral__Bit1,axiom,
% 0.22/0.47 ! [N: num] :
% 0.22/0.47 ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 0.22/0.47 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 0.22/0.47
% 0.22/0.47 % numeral_Bit1
% 0.22/0.47 thf(fact_3734_nat__diff__split__asm,axiom,
% 0.22/0.47 ! [P: nat > $o,A: nat,B: nat] :
% 0.22/0.47 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 0.22/0.47 = ( ~ ( ( ( ord_less_nat @ A @ B )
% 0.22/0.47 & ~ ( P @ zero_zero_nat ) )
% 0.22/0.47 | ? [D4: nat] :
% 0.22/0.47 ( ( A
% 0.22/0.47 = ( plus_plus_nat @ B @ D4 ) )
% 0.22/0.47 & ~ ( P @ D4 ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % nat_diff_split_asm
% 0.22/0.47 thf(fact_3735_nat__diff__split,axiom,
% 0.22/0.47 ! [P: nat > $o,A: nat,B: nat] :
% 0.22/0.47 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 0.22/0.47 = ( ( ( ord_less_nat @ A @ B )
% 0.22/0.47 => ( P @ zero_zero_nat ) )
% 0.22/0.47 & ! [D4: nat] :
% 0.22/0.47 ( ( A
% 0.22/0.47 = ( plus_plus_nat @ B @ D4 ) )
% 0.22/0.47 => ( P @ D4 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % nat_diff_split
% 0.22/0.47 thf(fact_3736_less__diff__conv2,axiom,
% 0.22/0.47 ! [K: nat,J: nat,I: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ K @ J )
% 0.22/0.47 => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.22/0.47 = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % less_diff_conv2
% 0.22/0.47 thf(fact_3737_nat__diff__add__eq2,axiom,
% 0.22/0.47 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ I @ J )
% 0.22/0.47 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.22/0.47 = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % nat_diff_add_eq2
% 0.22/0.47 thf(fact_3738_nat__diff__add__eq1,axiom,
% 0.22/0.47 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ J @ I )
% 0.22/0.47 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.22/0.47 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % nat_diff_add_eq1
% 0.22/0.47 thf(fact_3739_nat__le__add__iff2,axiom,
% 0.22/0.47 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ I @ J )
% 0.22/0.47 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.22/0.47 = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % nat_le_add_iff2
% 0.22/0.47 thf(fact_3740_nat__le__add__iff1,axiom,
% 0.22/0.47 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ J @ I )
% 0.22/0.47 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.22/0.47 = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % nat_le_add_iff1
% 0.22/0.47 thf(fact_3741_nat__eq__add__iff2,axiom,
% 0.22/0.47 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ I @ J )
% 0.22/0.47 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 0.22/0.47 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.22/0.47 = ( M
% 0.22/0.47 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % nat_eq_add_iff2
% 0.22/0.47 thf(fact_3742_nat__eq__add__iff1,axiom,
% 0.22/0.47 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ J @ I )
% 0.22/0.47 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 0.22/0.47 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.22/0.47 = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
% 0.22/0.47 = N ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % nat_eq_add_iff1
% 0.22/0.47 thf(fact_3743_bezout__add__strong__nat,axiom,
% 0.22/0.47 ! [A: nat,B: nat] :
% 0.22/0.47 ( ( A != zero_zero_nat )
% 0.22/0.47 => ? [D2: nat,X4: nat,Y: nat] :
% 0.22/0.47 ( ( dvd_dvd_nat @ D2 @ A )
% 0.22/0.47 & ( dvd_dvd_nat @ D2 @ B )
% 0.22/0.47 & ( ( times_times_nat @ A @ X4 )
% 0.22/0.47 = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D2 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % bezout_add_strong_nat
% 0.22/0.47 thf(fact_3744_convex__bound__le,axiom,
% 0.22/0.47 ! [X: rat,A: rat,Y4: rat,U: rat,V: rat] :
% 0.22/0.47 ( ( ord_less_eq_rat @ X @ A )
% 0.22/0.47 => ( ( ord_less_eq_rat @ Y4 @ A )
% 0.22/0.47 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.22/0.47 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.22/0.47 => ( ( ( plus_plus_rat @ U @ V )
% 0.22/0.47 = one_one_rat )
% 0.22/0.47 => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y4 ) ) @ A ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % convex_bound_le
% 0.22/0.47 thf(fact_3745_convex__bound__le,axiom,
% 0.22/0.47 ! [X: int,A: int,Y4: int,U: int,V: int] :
% 0.22/0.47 ( ( ord_less_eq_int @ X @ A )
% 0.22/0.47 => ( ( ord_less_eq_int @ Y4 @ A )
% 0.22/0.47 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.22/0.47 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.22/0.47 => ( ( ( plus_plus_int @ U @ V )
% 0.22/0.47 = one_one_int )
% 0.22/0.47 => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y4 ) ) @ A ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % convex_bound_le
% 0.22/0.47 thf(fact_3746_convex__bound__le,axiom,
% 0.22/0.47 ! [X: real,A: real,Y4: real,U: real,V: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ X @ A )
% 0.22/0.47 => ( ( ord_less_eq_real @ Y4 @ A )
% 0.22/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.22/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.22/0.47 => ( ( ( plus_plus_real @ U @ V )
% 0.22/0.47 = one_one_real )
% 0.22/0.47 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y4 ) ) @ A ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % convex_bound_le
% 0.22/0.47 thf(fact_3747_mult__2,axiom,
% 0.22/0.47 ! [Z: complex] :
% 0.22/0.47 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
% 0.22/0.47 = ( plus_plus_complex @ Z @ Z ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_2
% 0.22/0.47 thf(fact_3748_mult__2,axiom,
% 0.22/0.47 ! [Z: real] :
% 0.22/0.47 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
% 0.22/0.47 = ( plus_plus_real @ Z @ Z ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_2
% 0.22/0.47 thf(fact_3749_mult__2,axiom,
% 0.22/0.47 ! [Z: rat] :
% 0.22/0.47 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
% 0.22/0.47 = ( plus_plus_rat @ Z @ Z ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_2
% 0.22/0.47 thf(fact_3750_mult__2,axiom,
% 0.22/0.47 ! [Z: nat] :
% 0.22/0.47 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
% 0.22/0.47 = ( plus_plus_nat @ Z @ Z ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_2
% 0.22/0.47 thf(fact_3751_mult__2,axiom,
% 0.22/0.47 ! [Z: int] :
% 0.22/0.47 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
% 0.22/0.47 = ( plus_plus_int @ Z @ Z ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_2
% 0.22/0.47 thf(fact_3752_mult__2__right,axiom,
% 0.22/0.47 ! [Z: complex] :
% 0.22/0.47 ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
% 0.22/0.47 = ( plus_plus_complex @ Z @ Z ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_2_right
% 0.22/0.47 thf(fact_3753_mult__2__right,axiom,
% 0.22/0.47 ! [Z: real] :
% 0.22/0.47 ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.22/0.47 = ( plus_plus_real @ Z @ Z ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_2_right
% 0.22/0.47 thf(fact_3754_mult__2__right,axiom,
% 0.22/0.47 ! [Z: rat] :
% 0.22/0.47 ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
% 0.22/0.47 = ( plus_plus_rat @ Z @ Z ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_2_right
% 0.22/0.47 thf(fact_3755_mult__2__right,axiom,
% 0.22/0.47 ! [Z: nat] :
% 0.22/0.47 ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = ( plus_plus_nat @ Z @ Z ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_2_right
% 0.22/0.47 thf(fact_3756_mult__2__right,axiom,
% 0.22/0.47 ! [Z: int] :
% 0.22/0.47 ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.47 = ( plus_plus_int @ Z @ Z ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_2_right
% 0.22/0.47 thf(fact_3757_left__add__twice,axiom,
% 0.22/0.47 ! [A: complex,B: complex] :
% 0.22/0.47 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
% 0.22/0.47 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % left_add_twice
% 0.22/0.47 thf(fact_3758_left__add__twice,axiom,
% 0.22/0.47 ! [A: real,B: real] :
% 0.22/0.47 ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.22/0.47 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % left_add_twice
% 0.22/0.47 thf(fact_3759_left__add__twice,axiom,
% 0.22/0.47 ! [A: rat,B: rat] :
% 0.22/0.47 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.22/0.47 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % left_add_twice
% 0.22/0.47 thf(fact_3760_left__add__twice,axiom,
% 0.22/0.47 ! [A: nat,B: nat] :
% 0.22/0.47 ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.22/0.47 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % left_add_twice
% 0.22/0.47 thf(fact_3761_left__add__twice,axiom,
% 0.22/0.47 ! [A: int,B: int] :
% 0.22/0.47 ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.22/0.47 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % left_add_twice
% 0.22/0.47 thf(fact_3762_field__sum__of__halves,axiom,
% 0.22/0.47 ! [X: real] :
% 0.22/0.47 ( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.47 = X ) ).
% 0.22/0.47
% 0.22/0.47 % field_sum_of_halves
% 0.22/0.47 thf(fact_3763_field__sum__of__halves,axiom,
% 0.22/0.47 ! [X: rat] :
% 0.22/0.47 ( ( plus_plus_rat @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.22/0.47 = X ) ).
% 0.22/0.47
% 0.22/0.47 % field_sum_of_halves
% 0.22/0.47 thf(fact_3764_odd__even__add,axiom,
% 0.22/0.47 ! [A: code_integer,B: code_integer] :
% 0.22/0.47 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.47 => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
% 0.22/0.47 => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % odd_even_add
% 0.22/0.47 thf(fact_3765_odd__even__add,axiom,
% 0.22/0.47 ! [A: nat,B: nat] :
% 0.22/0.47 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.47 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.22/0.47 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % odd_even_add
% 0.22/0.47 thf(fact_3766_odd__even__add,axiom,
% 0.22/0.47 ! [A: int,B: int] :
% 0.22/0.47 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.47 => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
% 0.22/0.47 => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % odd_even_add
% 0.22/0.47 thf(fact_3767_Suc3__eq__add__3,axiom,
% 0.22/0.47 ! [N: nat] :
% 0.22/0.47 ( ( suc @ ( suc @ ( suc @ N ) ) )
% 0.22/0.47 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% 0.22/0.47
% 0.22/0.47 % Suc3_eq_add_3
% 0.22/0.47 thf(fact_3768_nat__1__add__1,axiom,
% 0.22/0.47 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 0.22/0.47 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % nat_1_add_1
% 0.22/0.47 thf(fact_3769_add__eq__if,axiom,
% 0.22/0.47 ( plus_plus_nat
% 0.22/0.47 = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_eq_if
% 0.22/0.47 thf(fact_3770_nat__less__add__iff2,axiom,
% 0.22/0.47 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ I @ J )
% 0.22/0.47 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.22/0.47 = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % nat_less_add_iff2
% 0.22/0.47 thf(fact_3771_nat__less__add__iff1,axiom,
% 0.22/0.47 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ J @ I )
% 0.22/0.47 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.22/0.47 = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % nat_less_add_iff1
% 0.22/0.47 thf(fact_3772_dividend__less__times__div,axiom,
% 0.22/0.47 ! [N: nat,M: nat] :
% 0.22/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.47 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dividend_less_times_div
% 0.22/0.47 thf(fact_3773_dividend__less__div__times,axiom,
% 0.22/0.47 ! [N: nat,M: nat] :
% 0.22/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.47 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dividend_less_div_times
% 0.22/0.47 thf(fact_3774_split__div,axiom,
% 0.22/0.47 ! [P: nat > $o,M: nat,N: nat] :
% 0.22/0.47 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 0.22/0.47 = ( ( ( N = zero_zero_nat )
% 0.22/0.47 => ( P @ zero_zero_nat ) )
% 0.22/0.47 & ( ( N != zero_zero_nat )
% 0.22/0.47 => ! [I2: nat,J3: nat] :
% 0.22/0.47 ( ( ord_less_nat @ J3 @ N )
% 0.22/0.47 => ( ( M
% 0.22/0.47 = ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J3 ) )
% 0.22/0.47 => ( P @ I2 ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % split_div
% 0.22/0.47 thf(fact_3775_mult__eq__if,axiom,
% 0.22/0.47 ( times_times_nat
% 0.22/0.47 = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_eq_if
% 0.22/0.47 thf(fact_3776_dvd__minus__add,axiom,
% 0.22/0.47 ! [Q2: nat,N: nat,R2: nat,M: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ Q2 @ N )
% 0.22/0.47 => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R2 @ M ) )
% 0.22/0.47 => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q2 ) )
% 0.22/0.47 = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q2 ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dvd_minus_add
% 0.22/0.47 thf(fact_3777_convex__bound__lt,axiom,
% 0.22/0.47 ! [X: rat,A: rat,Y4: rat,U: rat,V: rat] :
% 0.22/0.47 ( ( ord_less_rat @ X @ A )
% 0.22/0.47 => ( ( ord_less_rat @ Y4 @ A )
% 0.22/0.47 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.22/0.47 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.22/0.47 => ( ( ( plus_plus_rat @ U @ V )
% 0.22/0.47 = one_one_rat )
% 0.22/0.47 => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y4 ) ) @ A ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % convex_bound_lt
% 0.22/0.47 thf(fact_3778_convex__bound__lt,axiom,
% 0.22/0.47 ! [X: int,A: int,Y4: int,U: int,V: int] :
% 0.22/0.47 ( ( ord_less_int @ X @ A )
% 0.22/0.47 => ( ( ord_less_int @ Y4 @ A )
% 0.22/0.47 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.22/0.47 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.22/0.47 => ( ( ( plus_plus_int @ U @ V )
% 0.22/0.47 = one_one_int )
% 0.22/0.47 => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y4 ) ) @ A ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % convex_bound_lt
% 0.22/0.47 thf(fact_3779_convex__bound__lt,axiom,
% 0.22/0.47 ! [X: real,A: real,Y4: real,U: real,V: real] :
% 0.22/0.47 ( ( ord_less_real @ X @ A )
% 0.22/0.47 => ( ( ord_less_real @ Y4 @ A )
% 0.22/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.22/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.22/0.47 => ( ( ( plus_plus_real @ U @ V )
% 0.22/0.47 = one_one_real )
% 0.22/0.47 => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y4 ) ) @ A ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % convex_bound_lt
% 0.22/0.47 thf(fact_3780_scaling__mono,axiom,
% 0.22/0.47 ! [U: rat,V: rat,R2: rat,S2: rat] :
% 0.22/0.47 ( ( ord_less_eq_rat @ U @ V )
% 0.22/0.47 => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 0.22/0.47 => ( ( ord_less_eq_rat @ R2 @ S2 )
% 0.22/0.47 => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % scaling_mono
% 0.22/0.47 thf(fact_3781_scaling__mono,axiom,
% 0.22/0.47 ! [U: real,V: real,R2: real,S2: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ U @ V )
% 0.22/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 0.22/0.47 => ( ( ord_less_eq_real @ R2 @ S2 )
% 0.22/0.47 => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % scaling_mono
% 0.22/0.47 thf(fact_3782_field__less__half__sum,axiom,
% 0.22/0.47 ! [X: real,Y4: real] :
% 0.22/0.47 ( ( ord_less_real @ X @ Y4 )
% 0.22/0.47 => ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % field_less_half_sum
% 0.22/0.47 thf(fact_3783_field__less__half__sum,axiom,
% 0.22/0.47 ! [X: rat,Y4: rat] :
% 0.22/0.47 ( ( ord_less_rat @ X @ Y4 )
% 0.22/0.47 => ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y4 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % field_less_half_sum
% 0.22/0.47 thf(fact_3784_exp__add__not__zero__imp__left,axiom,
% 0.22/0.47 ! [M: nat,N: nat] :
% 0.22/0.47 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.22/0.47 != zero_zero_nat )
% 0.22/0.47 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.22/0.47 != zero_zero_nat ) ) ).
% 0.22/0.47
% 0.22/0.47 % exp_add_not_zero_imp_left
% 0.22/0.47 thf(fact_3785_exp__add__not__zero__imp__left,axiom,
% 0.22/0.47 ! [M: nat,N: nat] :
% 0.22/0.47 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.22/0.47 != zero_zero_int )
% 0.22/0.47 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.22/0.47 != zero_zero_int ) ) ).
% 0.22/0.47
% 0.22/0.47 % exp_add_not_zero_imp_left
% 0.22/0.47 thf(fact_3786_exp__add__not__zero__imp__right,axiom,
% 0.22/0.47 ! [M: nat,N: nat] :
% 0.22/0.47 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.22/0.47 != zero_zero_nat )
% 0.22/0.47 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.47 != zero_zero_nat ) ) ).
% 0.22/0.47
% 0.22/0.47 % exp_add_not_zero_imp_right
% 0.22/0.47 thf(fact_3787_exp__add__not__zero__imp__right,axiom,
% 0.22/0.47 ! [M: nat,N: nat] :
% 0.22/0.47 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.22/0.47 != zero_zero_int )
% 0.22/0.47 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.22/0.47 != zero_zero_int ) ) ).
% 0.22/0.47
% 0.22/0.47 % exp_add_not_zero_imp_right
% 0.22/0.47 thf(fact_3788_div__exp__eq,axiom,
% 0.22/0.47 ! [A: nat,M: nat,N: nat] :
% 0.22/0.47 ( ( 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 ) )
% 0.22/0.47 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % div_exp_eq
% 0.22/0.47 thf(fact_3789_div__exp__eq,axiom,
% 0.22/0.47 ! [A: int,M: nat,N: nat] :
% 0.22/0.47 ( ( 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 ) )
% 0.22/0.47 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % div_exp_eq
% 0.22/0.47 thf(fact_3790_nat__induct2,axiom,
% 0.22/0.47 ! [P: nat > $o,N: nat] :
% 0.22/0.47 ( ( P @ zero_zero_nat )
% 0.22/0.47 => ( ( P @ one_one_nat )
% 0.22/0.47 => ( ! [N2: nat] :
% 0.22/0.47 ( ( P @ N2 )
% 0.22/0.47 => ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.47 => ( P @ N ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % nat_induct2
% 0.22/0.47 thf(fact_3791_Suc__div__eq__add3__div,axiom,
% 0.22/0.47 ! [M: nat,N: nat] :
% 0.22/0.47 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 0.22/0.47 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 0.22/0.47
% 0.22/0.47 % Suc_div_eq_add3_div
% 0.22/0.47 thf(fact_3792_sum__power2__ge__zero,axiom,
% 0.22/0.47 ! [X: rat,Y4: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_power2_ge_zero
% 0.22/0.47 thf(fact_3793_sum__power2__ge__zero,axiom,
% 0.22/0.47 ! [X: int,Y4: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_power2_ge_zero
% 0.22/0.47 thf(fact_3794_sum__power2__ge__zero,axiom,
% 0.22/0.47 ! [X: real,Y4: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_power2_ge_zero
% 0.22/0.47 thf(fact_3795_sum__power2__le__zero__iff,axiom,
% 0.22/0.47 ! [X: rat,Y4: rat] :
% 0.22/0.47 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 0.22/0.47 = ( ( X = zero_zero_rat )
% 0.22/0.47 & ( Y4 = zero_zero_rat ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_power2_le_zero_iff
% 0.22/0.47 thf(fact_3796_sum__power2__le__zero__iff,axiom,
% 0.22/0.47 ! [X: int,Y4: int] :
% 0.22/0.47 ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 0.22/0.47 = ( ( X = zero_zero_int )
% 0.22/0.47 & ( Y4 = zero_zero_int ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_power2_le_zero_iff
% 0.22/0.47 thf(fact_3797_sum__power2__le__zero__iff,axiom,
% 0.22/0.47 ! [X: real,Y4: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 0.22/0.47 = ( ( X = zero_zero_real )
% 0.22/0.47 & ( Y4 = zero_zero_real ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_power2_le_zero_iff
% 0.22/0.47 thf(fact_3798_not__sum__power2__lt__zero,axiom,
% 0.22/0.47 ! [X: real,Y4: real] :
% 0.22/0.47 ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 0.22/0.47
% 0.22/0.47 % not_sum_power2_lt_zero
% 0.22/0.47 thf(fact_3799_not__sum__power2__lt__zero,axiom,
% 0.22/0.47 ! [X: rat,Y4: rat] :
% 0.22/0.47 ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 0.22/0.47
% 0.22/0.47 % not_sum_power2_lt_zero
% 0.22/0.47 thf(fact_3800_not__sum__power2__lt__zero,axiom,
% 0.22/0.47 ! [X: int,Y4: int] :
% 0.22/0.47 ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 0.22/0.47
% 0.22/0.47 % not_sum_power2_lt_zero
% 0.22/0.47 thf(fact_3801_sum__power2__gt__zero__iff,axiom,
% 0.22/0.47 ! [X: real,Y4: real] :
% 0.22/0.47 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.47 = ( ( X != zero_zero_real )
% 0.22/0.47 | ( Y4 != zero_zero_real ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_power2_gt_zero_iff
% 0.22/0.47 thf(fact_3802_sum__power2__gt__zero__iff,axiom,
% 0.22/0.47 ! [X: rat,Y4: rat] :
% 0.22/0.47 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.47 = ( ( X != zero_zero_rat )
% 0.22/0.47 | ( Y4 != zero_zero_rat ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_power2_gt_zero_iff
% 0.22/0.47 thf(fact_3803_sum__power2__gt__zero__iff,axiom,
% 0.22/0.47 ! [X: int,Y4: int] :
% 0.22/0.47 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.47 = ( ( X != zero_zero_int )
% 0.22/0.47 | ( Y4 != zero_zero_int ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_power2_gt_zero_iff
% 0.22/0.47 thf(fact_3804_oddE,axiom,
% 0.22/0.47 ! [A: code_integer] :
% 0.22/0.47 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.47 => ~ ! [B3: code_integer] :
% 0.22/0.47 ( A
% 0.22/0.47 != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) @ one_one_Code_integer ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % oddE
% 0.22/0.47 thf(fact_3805_oddE,axiom,
% 0.22/0.47 ! [A: nat] :
% 0.22/0.47 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.47 => ~ ! [B3: nat] :
% 0.22/0.47 ( A
% 0.22/0.47 != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) @ one_one_nat ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % oddE
% 0.22/0.47 thf(fact_3806_oddE,axiom,
% 0.22/0.47 ! [A: int] :
% 0.22/0.47 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.47 => ~ ! [B3: int] :
% 0.22/0.47 ( A
% 0.22/0.47 != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ one_one_int ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % oddE
% 0.22/0.47 thf(fact_3807_power2__sum,axiom,
% 0.22/0.47 ! [X: complex,Y4: complex] :
% 0.22/0.47 ( ( power_power_complex @ ( plus_plus_complex @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power2_sum
% 0.22/0.47 thf(fact_3808_power2__sum,axiom,
% 0.22/0.47 ! [X: real,Y4: real] :
% 0.22/0.47 ( ( power_power_real @ ( plus_plus_real @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power2_sum
% 0.22/0.47 thf(fact_3809_power2__sum,axiom,
% 0.22/0.47 ! [X: rat,Y4: rat] :
% 0.22/0.47 ( ( power_power_rat @ ( plus_plus_rat @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power2_sum
% 0.22/0.47 thf(fact_3810_power2__sum,axiom,
% 0.22/0.47 ! [X: nat,Y4: nat] :
% 0.22/0.47 ( ( power_power_nat @ ( plus_plus_nat @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power2_sum
% 0.22/0.47 thf(fact_3811_power2__sum,axiom,
% 0.22/0.47 ! [X: int,Y4: int] :
% 0.22/0.47 ( ( power_power_int @ ( plus_plus_int @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power2_sum
% 0.22/0.47 thf(fact_3812_real__arch__simple,axiom,
% 0.22/0.47 ! [X: rat] :
% 0.22/0.47 ? [N2: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 0.22/0.47
% 0.22/0.47 % real_arch_simple
% 0.22/0.47 thf(fact_3813_real__arch__simple,axiom,
% 0.22/0.47 ! [X: real] :
% 0.22/0.47 ? [N2: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 0.22/0.47
% 0.22/0.47 % real_arch_simple
% 0.22/0.47 thf(fact_3814_reals__Archimedean2,axiom,
% 0.22/0.47 ! [X: real] :
% 0.22/0.47 ? [N2: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 0.22/0.47
% 0.22/0.47 % reals_Archimedean2
% 0.22/0.47 thf(fact_3815_reals__Archimedean2,axiom,
% 0.22/0.47 ! [X: rat] :
% 0.22/0.47 ? [N2: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 0.22/0.47
% 0.22/0.47 % reals_Archimedean2
% 0.22/0.47 thf(fact_3816_exists__least__lemma,axiom,
% 0.22/0.47 ! [P: nat > $o] :
% 0.22/0.47 ( ~ ( P @ zero_zero_nat )
% 0.22/0.47 => ( ? [X_12: nat] : ( P @ X_12 )
% 0.22/0.47 => ? [N2: nat] :
% 0.22/0.47 ( ~ ( P @ N2 )
% 0.22/0.47 & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % exists_least_lemma
% 0.22/0.47 thf(fact_3817_sum__squares__bound,axiom,
% 0.22/0.47 ! [X: rat,Y4: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y4 ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_squares_bound
% 0.22/0.47 thf(fact_3818_sum__squares__bound,axiom,
% 0.22/0.47 ! [X: real,Y4: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y4 ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % sum_squares_bound
% 0.22/0.47 thf(fact_3819_power2__diff,axiom,
% 0.22/0.47 ! [X: complex,Y4: complex] :
% 0.22/0.47 ( ( power_power_complex @ ( minus_minus_complex @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power2_diff
% 0.22/0.47 thf(fact_3820_power2__diff,axiom,
% 0.22/0.47 ! [X: real,Y4: real] :
% 0.22/0.47 ( ( power_power_real @ ( minus_minus_real @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power2_diff
% 0.22/0.47 thf(fact_3821_power2__diff,axiom,
% 0.22/0.47 ! [X: rat,Y4: rat] :
% 0.22/0.47 ( ( power_power_rat @ ( minus_minus_rat @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power2_diff
% 0.22/0.47 thf(fact_3822_power2__diff,axiom,
% 0.22/0.47 ! [X: int,Y4: int] :
% 0.22/0.47 ( ( power_power_int @ ( minus_minus_int @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power2_diff
% 0.22/0.47 thf(fact_3823_ex__power__ivl1,axiom,
% 0.22/0.47 ! [B: nat,K: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.22/0.47 => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 0.22/0.47 => ? [N2: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 0.22/0.47 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % ex_power_ivl1
% 0.22/0.47 thf(fact_3824_ex__power__ivl2,axiom,
% 0.22/0.47 ! [B: nat,K: nat] :
% 0.22/0.47 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.22/0.47 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.22/0.47 => ? [N2: nat] :
% 0.22/0.47 ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 0.22/0.47 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % ex_power_ivl2
% 0.22/0.47 thf(fact_3825_arith__geo__mean,axiom,
% 0.22/0.47 ! [U: rat,X: rat,Y4: rat] :
% 0.22/0.47 ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = ( times_times_rat @ X @ Y4 ) )
% 0.22/0.47 => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.22/0.47 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 0.22/0.47 => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y4 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % arith_geo_mean
% 0.22/0.47 thf(fact_3826_arith__geo__mean,axiom,
% 0.22/0.47 ! [U: real,X: real,Y4: real] :
% 0.22/0.47 ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = ( times_times_real @ X @ Y4 ) )
% 0.22/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.47 => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % arith_geo_mean
% 0.22/0.47 thf(fact_3827_ex__less__of__nat__mult,axiom,
% 0.22/0.47 ! [X: real,Y4: real] :
% 0.22/0.47 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.47 => ? [N2: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % ex_less_of_nat_mult
% 0.22/0.47 thf(fact_3828_ex__less__of__nat__mult,axiom,
% 0.22/0.47 ! [X: rat,Y4: rat] :
% 0.22/0.47 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.22/0.47 => ? [N2: nat] : ( ord_less_rat @ Y4 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ X ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % ex_less_of_nat_mult
% 0.22/0.47 thf(fact_3829_low__inv,axiom,
% 0.22/0.47 ! [X: nat,N: nat,Y4: nat] :
% 0.22/0.47 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.47 => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
% 0.22/0.47 = X ) ) ).
% 0.22/0.47
% 0.22/0.47 % low_inv
% 0.22/0.47 thf(fact_3830_Bolzano,axiom,
% 0.22/0.47 ! [A: real,B: real,P: real > real > $o] :
% 0.22/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.22/0.47 => ( ! [A4: real,B3: real,C3: real] :
% 0.22/0.47 ( ( P @ A4 @ B3 )
% 0.22/0.47 => ( ( P @ B3 @ C3 )
% 0.22/0.47 => ( ( ord_less_eq_real @ A4 @ B3 )
% 0.22/0.47 => ( ( ord_less_eq_real @ B3 @ C3 )
% 0.22/0.47 => ( P @ A4 @ C3 ) ) ) ) )
% 0.22/0.47 => ( ! [X4: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ A @ X4 )
% 0.22/0.47 => ( ( ord_less_eq_real @ X4 @ B )
% 0.22/0.47 => ? [D5: real] :
% 0.22/0.47 ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.22/0.47 & ! [A4: real,B3: real] :
% 0.22/0.47 ( ( ( ord_less_eq_real @ A4 @ X4 )
% 0.22/0.47 & ( ord_less_eq_real @ X4 @ B3 )
% 0.22/0.47 & ( ord_less_real @ ( minus_minus_real @ B3 @ A4 ) @ D5 ) )
% 0.22/0.47 => ( P @ A4 @ B3 ) ) ) ) )
% 0.22/0.47 => ( P @ A @ B ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % Bolzano
% 0.22/0.47 thf(fact_3831_mult__le__cancel__iff1,axiom,
% 0.22/0.47 ! [Z: rat,X: rat,Y4: rat] :
% 0.22/0.47 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.22/0.47 => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y4 @ Z ) )
% 0.22/0.47 = ( ord_less_eq_rat @ X @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_le_cancel_iff1
% 0.22/0.47 thf(fact_3832_mult__le__cancel__iff1,axiom,
% 0.22/0.47 ! [Z: int,X: int,Y4: int] :
% 0.22/0.47 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.22/0.47 => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y4 @ Z ) )
% 0.22/0.47 = ( ord_less_eq_int @ X @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_le_cancel_iff1
% 0.22/0.47 thf(fact_3833_mult__le__cancel__iff1,axiom,
% 0.22/0.47 ! [Z: real,X: real,Y4: real] :
% 0.22/0.47 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.22/0.47 => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y4 @ Z ) )
% 0.22/0.47 = ( ord_less_eq_real @ X @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_le_cancel_iff1
% 0.22/0.47 thf(fact_3834_mult__le__cancel__iff2,axiom,
% 0.22/0.47 ! [Z: rat,X: rat,Y4: rat] :
% 0.22/0.47 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.22/0.47 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y4 ) )
% 0.22/0.47 = ( ord_less_eq_rat @ X @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_le_cancel_iff2
% 0.22/0.47 thf(fact_3835_mult__le__cancel__iff2,axiom,
% 0.22/0.47 ! [Z: int,X: int,Y4: int] :
% 0.22/0.47 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.22/0.47 => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y4 ) )
% 0.22/0.47 = ( ord_less_eq_int @ X @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_le_cancel_iff2
% 0.22/0.47 thf(fact_3836_mult__le__cancel__iff2,axiom,
% 0.22/0.47 ! [Z: real,X: real,Y4: real] :
% 0.22/0.47 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.22/0.47 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y4 ) )
% 0.22/0.47 = ( ord_less_eq_real @ X @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_le_cancel_iff2
% 0.22/0.47 thf(fact_3837_lemma__termdiff3,axiom,
% 0.22/0.47 ! [H: real,Z: real,K5: real,N: nat] :
% 0.22/0.47 ( ( H != zero_zero_real )
% 0.22/0.47 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
% 0.22/0.47 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H ) ) @ K5 )
% 0.22/0.47 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % lemma_termdiff3
% 0.22/0.47 thf(fact_3838_lemma__termdiff3,axiom,
% 0.22/0.47 ! [H: complex,Z: complex,K5: real,N: nat] :
% 0.22/0.47 ( ( H != zero_zero_complex )
% 0.22/0.47 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
% 0.22/0.47 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H ) ) @ K5 )
% 0.22/0.47 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % lemma_termdiff3
% 0.22/0.47 thf(fact_3839_dbl__inc__simps_I3_J,axiom,
% 0.22/0.47 ( ( neg_nu8557863876264182079omplex @ one_one_complex )
% 0.22/0.47 = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_inc_simps(3)
% 0.22/0.47 thf(fact_3840_dbl__inc__simps_I3_J,axiom,
% 0.22/0.47 ( ( neg_nu8295874005876285629c_real @ one_one_real )
% 0.22/0.47 = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_inc_simps(3)
% 0.22/0.47 thf(fact_3841_dbl__inc__simps_I3_J,axiom,
% 0.22/0.47 ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
% 0.22/0.47 = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_inc_simps(3)
% 0.22/0.47 thf(fact_3842_dbl__inc__simps_I3_J,axiom,
% 0.22/0.47 ( ( neg_nu5851722552734809277nc_int @ one_one_int )
% 0.22/0.47 = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_inc_simps(3)
% 0.22/0.47 thf(fact_3843_dbl__simps_I3_J,axiom,
% 0.22/0.47 ( ( neg_nu7009210354673126013omplex @ one_one_complex )
% 0.22/0.47 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_simps(3)
% 0.22/0.47 thf(fact_3844_dbl__simps_I3_J,axiom,
% 0.22/0.47 ( ( neg_numeral_dbl_real @ one_one_real )
% 0.22/0.47 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_simps(3)
% 0.22/0.47 thf(fact_3845_dbl__simps_I3_J,axiom,
% 0.22/0.47 ( ( neg_numeral_dbl_rat @ one_one_rat )
% 0.22/0.47 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_simps(3)
% 0.22/0.47 thf(fact_3846_dbl__simps_I3_J,axiom,
% 0.22/0.47 ( ( neg_numeral_dbl_int @ one_one_int )
% 0.22/0.47 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_simps(3)
% 0.22/0.47 thf(fact_3847_semiring__norm_I6_J,axiom,
% 0.22/0.47 ! [M: num,N: num] :
% 0.22/0.47 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.22/0.47 = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % semiring_norm(6)
% 0.22/0.47 thf(fact_3848_dbl__simps_I2_J,axiom,
% 0.22/0.47 ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
% 0.22/0.47 = zero_zero_complex ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_simps(2)
% 0.22/0.47 thf(fact_3849_dbl__simps_I2_J,axiom,
% 0.22/0.47 ( ( neg_numeral_dbl_real @ zero_zero_real )
% 0.22/0.47 = zero_zero_real ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_simps(2)
% 0.22/0.47 thf(fact_3850_dbl__simps_I2_J,axiom,
% 0.22/0.47 ( ( neg_numeral_dbl_rat @ zero_zero_rat )
% 0.22/0.47 = zero_zero_rat ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_simps(2)
% 0.22/0.47 thf(fact_3851_dbl__simps_I2_J,axiom,
% 0.22/0.47 ( ( neg_numeral_dbl_int @ zero_zero_int )
% 0.22/0.47 = zero_zero_int ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_simps(2)
% 0.22/0.47 thf(fact_3852_semiring__norm_I2_J,axiom,
% 0.22/0.47 ( ( plus_plus_num @ one @ one )
% 0.22/0.47 = ( bit0 @ one ) ) ).
% 0.22/0.47
% 0.22/0.47 % semiring_norm(2)
% 0.22/0.47 thf(fact_3853_zle__add1__eq__le,axiom,
% 0.22/0.47 ! [W: int,Z: int] :
% 0.22/0.47 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 0.22/0.47 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.22/0.47
% 0.22/0.47 % zle_add1_eq_le
% 0.22/0.47 thf(fact_3854_semiring__norm_I7_J,axiom,
% 0.22/0.47 ! [M: num,N: num] :
% 0.22/0.47 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.22/0.47 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % semiring_norm(7)
% 0.22/0.47 thf(fact_3855_semiring__norm_I9_J,axiom,
% 0.22/0.47 ! [M: num,N: num] :
% 0.22/0.47 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.22/0.47 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % semiring_norm(9)
% 0.22/0.47 thf(fact_3856_dbl__simps_I5_J,axiom,
% 0.22/0.47 ! [K: num] :
% 0.22/0.47 ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.22/0.47 = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_simps(5)
% 0.22/0.47 thf(fact_3857_dbl__simps_I5_J,axiom,
% 0.22/0.47 ! [K: num] :
% 0.22/0.47 ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
% 0.22/0.47 = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_simps(5)
% 0.22/0.47 thf(fact_3858_dbl__simps_I5_J,axiom,
% 0.22/0.47 ! [K: num] :
% 0.22/0.47 ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
% 0.22/0.47 = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_simps(5)
% 0.22/0.47 thf(fact_3859_dbl__simps_I5_J,axiom,
% 0.22/0.47 ! [K: num] :
% 0.22/0.47 ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
% 0.22/0.47 = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_simps(5)
% 0.22/0.47 thf(fact_3860_dbl__inc__simps_I2_J,axiom,
% 0.22/0.47 ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
% 0.22/0.47 = one_one_complex ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_inc_simps(2)
% 0.22/0.47 thf(fact_3861_dbl__inc__simps_I2_J,axiom,
% 0.22/0.47 ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
% 0.22/0.47 = one_one_real ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_inc_simps(2)
% 0.22/0.47 thf(fact_3862_dbl__inc__simps_I2_J,axiom,
% 0.22/0.47 ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
% 0.22/0.47 = one_one_rat ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_inc_simps(2)
% 0.22/0.47 thf(fact_3863_dbl__inc__simps_I2_J,axiom,
% 0.22/0.47 ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 0.22/0.47 = one_one_int ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_inc_simps(2)
% 0.22/0.47 thf(fact_3864_dbl__inc__simps_I5_J,axiom,
% 0.22/0.47 ! [K: num] :
% 0.22/0.47 ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.22/0.47 = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_inc_simps(5)
% 0.22/0.47 thf(fact_3865_dbl__inc__simps_I5_J,axiom,
% 0.22/0.47 ! [K: num] :
% 0.22/0.47 ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
% 0.22/0.47 = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_inc_simps(5)
% 0.22/0.47 thf(fact_3866_dbl__inc__simps_I5_J,axiom,
% 0.22/0.47 ! [K: num] :
% 0.22/0.47 ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
% 0.22/0.47 = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_inc_simps(5)
% 0.22/0.47 thf(fact_3867_dbl__inc__simps_I5_J,axiom,
% 0.22/0.47 ! [K: num] :
% 0.22/0.47 ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
% 0.22/0.47 = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_inc_simps(5)
% 0.22/0.47 thf(fact_3868_power__add__numeral2,axiom,
% 0.22/0.47 ! [A: complex,M: num,N: num,B: complex] :
% 0.22/0.47 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.22/0.47 = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % power_add_numeral2
% 0.22/0.47 thf(fact_3869_power__add__numeral2,axiom,
% 0.22/0.47 ! [A: real,M: num,N: num,B: real] :
% 0.22/0.47 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.22/0.47 = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % power_add_numeral2
% 0.22/0.47 thf(fact_3870_power__add__numeral2,axiom,
% 0.22/0.47 ! [A: rat,M: num,N: num,B: rat] :
% 0.22/0.47 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.22/0.47 = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % power_add_numeral2
% 0.22/0.47 thf(fact_3871_power__add__numeral2,axiom,
% 0.22/0.47 ! [A: nat,M: num,N: num,B: nat] :
% 0.22/0.47 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.22/0.47 = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % power_add_numeral2
% 0.22/0.47 thf(fact_3872_power__add__numeral2,axiom,
% 0.22/0.47 ! [A: int,M: num,N: num,B: int] :
% 0.22/0.47 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.22/0.47 = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % power_add_numeral2
% 0.22/0.47 thf(fact_3873_power__add__numeral,axiom,
% 0.22/0.47 ! [A: complex,M: num,N: num] :
% 0.22/0.47 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.22/0.47 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power_add_numeral
% 0.22/0.47 thf(fact_3874_power__add__numeral,axiom,
% 0.22/0.47 ! [A: real,M: num,N: num] :
% 0.22/0.47 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.22/0.47 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power_add_numeral
% 0.22/0.47 thf(fact_3875_power__add__numeral,axiom,
% 0.22/0.47 ! [A: rat,M: num,N: num] :
% 0.22/0.47 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.22/0.47 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power_add_numeral
% 0.22/0.47 thf(fact_3876_power__add__numeral,axiom,
% 0.22/0.47 ! [A: nat,M: num,N: num] :
% 0.22/0.47 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.22/0.47 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power_add_numeral
% 0.22/0.47 thf(fact_3877_power__add__numeral,axiom,
% 0.22/0.47 ! [A: int,M: num,N: num] :
% 0.22/0.47 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.22/0.47 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power_add_numeral
% 0.22/0.47 thf(fact_3878_Suc__numeral,axiom,
% 0.22/0.47 ! [N: num] :
% 0.22/0.47 ( ( suc @ ( numeral_numeral_nat @ N ) )
% 0.22/0.47 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % Suc_numeral
% 0.22/0.47 thf(fact_3879_semiring__norm_I3_J,axiom,
% 0.22/0.47 ! [N: num] :
% 0.22/0.47 ( ( plus_plus_num @ one @ ( bit0 @ N ) )
% 0.22/0.47 = ( bit1 @ N ) ) ).
% 0.22/0.47
% 0.22/0.47 % semiring_norm(3)
% 0.22/0.47 thf(fact_3880_semiring__norm_I4_J,axiom,
% 0.22/0.47 ! [N: num] :
% 0.22/0.47 ( ( plus_plus_num @ one @ ( bit1 @ N ) )
% 0.22/0.47 = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % semiring_norm(4)
% 0.22/0.47 thf(fact_3881_semiring__norm_I5_J,axiom,
% 0.22/0.47 ! [M: num] :
% 0.22/0.47 ( ( plus_plus_num @ ( bit0 @ M ) @ one )
% 0.22/0.47 = ( bit1 @ M ) ) ).
% 0.22/0.47
% 0.22/0.47 % semiring_norm(5)
% 0.22/0.47 thf(fact_3882_semiring__norm_I8_J,axiom,
% 0.22/0.47 ! [M: num] :
% 0.22/0.47 ( ( plus_plus_num @ ( bit1 @ M ) @ one )
% 0.22/0.47 = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % semiring_norm(8)
% 0.22/0.47 thf(fact_3883_semiring__norm_I10_J,axiom,
% 0.22/0.47 ! [M: num,N: num] :
% 0.22/0.47 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.22/0.47 = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % semiring_norm(10)
% 0.22/0.47 thf(fact_3884_semiring__norm_I16_J,axiom,
% 0.22/0.47 ! [M: num,N: num] :
% 0.22/0.47 ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.22/0.47 = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % semiring_norm(16)
% 0.22/0.47 thf(fact_3885_add__One__commute,axiom,
% 0.22/0.47 ! [N: num] :
% 0.22/0.47 ( ( plus_plus_num @ one @ N )
% 0.22/0.47 = ( plus_plus_num @ N @ one ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_One_commute
% 0.22/0.47 thf(fact_3886_plus__int__code_I1_J,axiom,
% 0.22/0.47 ! [K: int] :
% 0.22/0.47 ( ( plus_plus_int @ K @ zero_zero_int )
% 0.22/0.47 = K ) ).
% 0.22/0.47
% 0.22/0.47 % plus_int_code(1)
% 0.22/0.47 thf(fact_3887_plus__int__code_I2_J,axiom,
% 0.22/0.47 ! [L: int] :
% 0.22/0.47 ( ( plus_plus_int @ zero_zero_int @ L )
% 0.22/0.47 = L ) ).
% 0.22/0.47
% 0.22/0.47 % plus_int_code(2)
% 0.22/0.47 thf(fact_3888_int__distrib_I1_J,axiom,
% 0.22/0.47 ! [Z1: int,Z22: int,W: int] :
% 0.22/0.47 ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
% 0.22/0.47 = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % int_distrib(1)
% 0.22/0.47 thf(fact_3889_int__distrib_I2_J,axiom,
% 0.22/0.47 ! [W: int,Z1: int,Z22: int] :
% 0.22/0.47 ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
% 0.22/0.47 = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % int_distrib(2)
% 0.22/0.47 thf(fact_3890_iadd__is__0,axiom,
% 0.22/0.47 ! [M: extended_enat,N: extended_enat] :
% 0.22/0.47 ( ( ( plus_p3455044024723400733d_enat @ M @ N )
% 0.22/0.47 = zero_z5237406670263579293d_enat )
% 0.22/0.47 = ( ( M = zero_z5237406670263579293d_enat )
% 0.22/0.47 & ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % iadd_is_0
% 0.22/0.47 thf(fact_3891_dbl__def,axiom,
% 0.22/0.47 ( neg_numeral_dbl_real
% 0.22/0.47 = ( ^ [X3: real] : ( plus_plus_real @ X3 @ X3 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_def
% 0.22/0.47 thf(fact_3892_dbl__def,axiom,
% 0.22/0.47 ( neg_numeral_dbl_rat
% 0.22/0.47 = ( ^ [X3: rat] : ( plus_plus_rat @ X3 @ X3 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_def
% 0.22/0.47 thf(fact_3893_dbl__def,axiom,
% 0.22/0.47 ( neg_numeral_dbl_int
% 0.22/0.47 = ( ^ [X3: int] : ( plus_plus_int @ X3 @ X3 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_def
% 0.22/0.47 thf(fact_3894_odd__nonzero,axiom,
% 0.22/0.47 ! [Z: int] :
% 0.22/0.47 ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
% 0.22/0.47 != zero_zero_int ) ).
% 0.22/0.47
% 0.22/0.47 % odd_nonzero
% 0.22/0.47 thf(fact_3895_int__plus,axiom,
% 0.22/0.47 ! [N: nat,M: nat] :
% 0.22/0.47 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
% 0.22/0.47 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % int_plus
% 0.22/0.47 thf(fact_3896_int__ops_I5_J,axiom,
% 0.22/0.47 ! [A: nat,B: nat] :
% 0.22/0.47 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
% 0.22/0.47 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % int_ops(5)
% 0.22/0.47 thf(fact_3897_zadd__int__left,axiom,
% 0.22/0.47 ! [M: nat,N: nat,Z: int] :
% 0.22/0.47 ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
% 0.22/0.47 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% 0.22/0.47
% 0.22/0.47 % zadd_int_left
% 0.22/0.47 thf(fact_3898_int__ge__induct,axiom,
% 0.22/0.47 ! [K: int,I: int,P: int > $o] :
% 0.22/0.47 ( ( ord_less_eq_int @ K @ I )
% 0.22/0.47 => ( ( P @ K )
% 0.22/0.47 => ( ! [I3: int] :
% 0.22/0.47 ( ( ord_less_eq_int @ K @ I3 )
% 0.22/0.47 => ( ( P @ I3 )
% 0.22/0.47 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 0.22/0.47 => ( P @ I ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % int_ge_induct
% 0.22/0.47 thf(fact_3899_zless__add1__eq,axiom,
% 0.22/0.47 ! [W: int,Z: int] :
% 0.22/0.47 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 0.22/0.47 = ( ( ord_less_int @ W @ Z )
% 0.22/0.47 | ( W = Z ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % zless_add1_eq
% 0.22/0.47 thf(fact_3900_int__gr__induct,axiom,
% 0.22/0.47 ! [K: int,I: int,P: int > $o] :
% 0.22/0.47 ( ( ord_less_int @ K @ I )
% 0.22/0.47 => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 0.22/0.47 => ( ! [I3: int] :
% 0.22/0.47 ( ( ord_less_int @ K @ I3 )
% 0.22/0.47 => ( ( P @ I3 )
% 0.22/0.47 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 0.22/0.47 => ( P @ I ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % int_gr_induct
% 0.22/0.47 thf(fact_3901_zle__iff__zadd,axiom,
% 0.22/0.47 ( ord_less_eq_int
% 0.22/0.47 = ( ^ [W3: int,Z5: int] :
% 0.22/0.47 ? [N4: nat] :
% 0.22/0.47 ( Z5
% 0.22/0.47 = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % zle_iff_zadd
% 0.22/0.47 thf(fact_3902_zdvd__period,axiom,
% 0.22/0.47 ! [A: int,D: int,X: int,T: int,C: int] :
% 0.22/0.47 ( ( dvd_dvd_int @ A @ D )
% 0.22/0.47 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
% 0.22/0.47 = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % zdvd_period
% 0.22/0.47 thf(fact_3903_zdvd__reduce,axiom,
% 0.22/0.47 ! [K: int,N: int,M: int] :
% 0.22/0.47 ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
% 0.22/0.47 = ( dvd_dvd_int @ K @ N ) ) ).
% 0.22/0.47
% 0.22/0.47 % zdvd_reduce
% 0.22/0.47 thf(fact_3904_add__diff__assoc__enat,axiom,
% 0.22/0.47 ! [Z: extended_enat,Y4: extended_enat,X: extended_enat] :
% 0.22/0.47 ( ( ord_le2932123472753598470d_enat @ Z @ Y4 )
% 0.22/0.47 => ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y4 @ Z ) )
% 0.22/0.47 = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y4 ) @ Z ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % add_diff_assoc_enat
% 0.22/0.47 thf(fact_3905_int__Suc,axiom,
% 0.22/0.47 ! [N: nat] :
% 0.22/0.47 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 0.22/0.47 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% 0.22/0.47
% 0.22/0.47 % int_Suc
% 0.22/0.47 thf(fact_3906_int__ops_I4_J,axiom,
% 0.22/0.47 ! [A: nat] :
% 0.22/0.47 ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
% 0.22/0.47 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% 0.22/0.47
% 0.22/0.47 % int_ops(4)
% 0.22/0.47 thf(fact_3907_zless__iff__Suc__zadd,axiom,
% 0.22/0.47 ( ord_less_int
% 0.22/0.47 = ( ^ [W3: int,Z5: int] :
% 0.22/0.47 ? [N4: nat] :
% 0.22/0.47 ( Z5
% 0.22/0.47 = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % zless_iff_Suc_zadd
% 0.22/0.47 thf(fact_3908_odd__less__0__iff,axiom,
% 0.22/0.47 ! [Z: int] :
% 0.22/0.47 ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
% 0.22/0.47 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.22/0.47
% 0.22/0.47 % odd_less_0_iff
% 0.22/0.47 thf(fact_3909_add1__zle__eq,axiom,
% 0.22/0.47 ! [W: int,Z: int] :
% 0.22/0.47 ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
% 0.22/0.47 = ( ord_less_int @ W @ Z ) ) ).
% 0.22/0.47
% 0.22/0.47 % add1_zle_eq
% 0.22/0.47 thf(fact_3910_zless__imp__add1__zle,axiom,
% 0.22/0.47 ! [W: int,Z: int] :
% 0.22/0.47 ( ( ord_less_int @ W @ Z )
% 0.22/0.47 => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% 0.22/0.47
% 0.22/0.47 % zless_imp_add1_zle
% 0.22/0.47 thf(fact_3911_int__induct,axiom,
% 0.22/0.47 ! [P: int > $o,K: int,I: int] :
% 0.22/0.47 ( ( P @ K )
% 0.22/0.47 => ( ! [I3: int] :
% 0.22/0.47 ( ( ord_less_eq_int @ K @ I3 )
% 0.22/0.47 => ( ( P @ I3 )
% 0.22/0.47 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 0.22/0.47 => ( ! [I3: int] :
% 0.22/0.47 ( ( ord_less_eq_int @ I3 @ K )
% 0.22/0.47 => ( ( P @ I3 )
% 0.22/0.47 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 0.22/0.47 => ( P @ I ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % int_induct
% 0.22/0.47 thf(fact_3912_Suc__nat__number__of__add,axiom,
% 0.22/0.47 ! [V: num,N: nat] :
% 0.22/0.47 ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
% 0.22/0.47 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% 0.22/0.47
% 0.22/0.47 % Suc_nat_number_of_add
% 0.22/0.47 thf(fact_3913_dbl__inc__def,axiom,
% 0.22/0.47 ( neg_nu8557863876264182079omplex
% 0.22/0.47 = ( ^ [X3: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X3 @ X3 ) @ one_one_complex ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_inc_def
% 0.22/0.47 thf(fact_3914_dbl__inc__def,axiom,
% 0.22/0.47 ( neg_nu8295874005876285629c_real
% 0.22/0.47 = ( ^ [X3: real] : ( plus_plus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_inc_def
% 0.22/0.47 thf(fact_3915_dbl__inc__def,axiom,
% 0.22/0.47 ( neg_nu5219082963157363817nc_rat
% 0.22/0.47 = ( ^ [X3: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_inc_def
% 0.22/0.47 thf(fact_3916_dbl__inc__def,axiom,
% 0.22/0.47 ( neg_nu5851722552734809277nc_int
% 0.22/0.47 = ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % dbl_inc_def
% 0.22/0.47 thf(fact_3917_nat__less__real__le,axiom,
% 0.22/0.47 ( ord_less_nat
% 0.22/0.47 = ( ^ [N4: nat,M5: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N4 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M5 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % nat_less_real_le
% 0.22/0.47 thf(fact_3918_nat__le__real__less,axiom,
% 0.22/0.47 ( ord_less_eq_nat
% 0.22/0.47 = ( ^ [N4: nat,M5: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M5 ) @ one_one_real ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % nat_le_real_less
% 0.22/0.47 thf(fact_3919_le__imp__0__less,axiom,
% 0.22/0.47 ! [Z: int] :
% 0.22/0.47 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.22/0.47 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % le_imp_0_less
% 0.22/0.47 thf(fact_3920_unique__quotient__lemma__neg,axiom,
% 0.22/0.47 ! [B: int,Q5: int,R4: int,Q2: int,R2: int] :
% 0.22/0.47 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
% 0.22/0.47 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.22/0.47 => ( ( ord_less_int @ B @ R2 )
% 0.22/0.47 => ( ( ord_less_int @ B @ R4 )
% 0.22/0.47 => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % unique_quotient_lemma_neg
% 0.22/0.47 thf(fact_3921_unique__quotient__lemma,axiom,
% 0.22/0.47 ! [B: int,Q5: int,R4: int,Q2: int,R2: int] :
% 0.22/0.47 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
% 0.22/0.47 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 0.22/0.47 => ( ( ord_less_int @ R4 @ B )
% 0.22/0.47 => ( ( ord_less_int @ R2 @ B )
% 0.22/0.47 => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % unique_quotient_lemma
% 0.22/0.47 thf(fact_3922_zdiv__mono2__neg__lemma,axiom,
% 0.22/0.47 ! [B: int,Q2: int,R2: int,B6: int,Q5: int,R4: int] :
% 0.22/0.47 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 )
% 0.22/0.47 = ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) )
% 0.22/0.47 => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) @ zero_zero_int )
% 0.22/0.47 => ( ( ord_less_int @ R2 @ B )
% 0.22/0.47 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 0.22/0.47 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.22/0.47 => ( ( ord_less_eq_int @ B6 @ B )
% 0.22/0.47 => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % zdiv_mono2_neg_lemma
% 0.22/0.47 thf(fact_3923_zdiv__mono2__lemma,axiom,
% 0.22/0.47 ! [B: int,Q2: int,R2: int,B6: int,Q5: int,R4: int] :
% 0.22/0.47 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 )
% 0.22/0.47 = ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) )
% 0.22/0.47 => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) )
% 0.22/0.47 => ( ( ord_less_int @ R4 @ B6 )
% 0.22/0.47 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.22/0.47 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.22/0.47 => ( ( ord_less_eq_int @ B6 @ B )
% 0.22/0.47 => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % zdiv_mono2_lemma
% 0.22/0.47 thf(fact_3924_q__pos__lemma,axiom,
% 0.22/0.47 ! [B6: int,Q5: int,R4: int] :
% 0.22/0.47 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) )
% 0.22/0.47 => ( ( ord_less_int @ R4 @ B6 )
% 0.22/0.47 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.22/0.47 => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % q_pos_lemma
% 0.22/0.47 thf(fact_3925_incr__mult__lemma,axiom,
% 0.22/0.47 ! [D: int,P: int > $o,K: int] :
% 0.22/0.47 ( ( ord_less_int @ zero_zero_int @ D )
% 0.22/0.47 => ( ! [X4: int] :
% 0.22/0.47 ( ( P @ X4 )
% 0.22/0.47 => ( P @ ( plus_plus_int @ X4 @ D ) ) )
% 0.22/0.47 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.22/0.47 => ! [X5: int] :
% 0.22/0.47 ( ( P @ X5 )
% 0.22/0.47 => ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % incr_mult_lemma
% 0.22/0.47 thf(fact_3926_linear__plus__1__le__power,axiom,
% 0.22/0.47 ! [X: real,N: nat] :
% 0.22/0.47 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.47 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % linear_plus_1_le_power
% 0.22/0.47 thf(fact_3927_even__diff__iff,axiom,
% 0.22/0.47 ! [K: int,L: int] :
% 0.22/0.47 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
% 0.22/0.47 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % even_diff_iff
% 0.22/0.47 thf(fact_3928_int__div__pos__eq,axiom,
% 0.22/0.47 ! [A: int,B: int,Q2: int,R2: int] :
% 0.22/0.47 ( ( A
% 0.22/0.47 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
% 0.22/0.47 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.22/0.47 => ( ( ord_less_int @ R2 @ B )
% 0.22/0.47 => ( ( divide_divide_int @ A @ B )
% 0.22/0.47 = Q2 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % int_div_pos_eq
% 0.22/0.47 thf(fact_3929_int__div__neg__eq,axiom,
% 0.22/0.47 ! [A: int,B: int,Q2: int,R2: int] :
% 0.22/0.47 ( ( A
% 0.22/0.47 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
% 0.22/0.47 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.22/0.47 => ( ( ord_less_int @ B @ R2 )
% 0.22/0.47 => ( ( divide_divide_int @ A @ B )
% 0.22/0.47 = Q2 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % int_div_neg_eq
% 0.22/0.47 thf(fact_3930_split__zdiv,axiom,
% 0.22/0.47 ! [P: int > $o,N: int,K: int] :
% 0.22/0.47 ( ( P @ ( divide_divide_int @ N @ K ) )
% 0.22/0.47 = ( ( ( K = zero_zero_int )
% 0.22/0.47 => ( P @ zero_zero_int ) )
% 0.22/0.47 & ( ( ord_less_int @ zero_zero_int @ K )
% 0.22/0.47 => ! [I2: int,J3: int] :
% 0.22/0.47 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.22/0.47 & ( ord_less_int @ J3 @ K )
% 0.22/0.47 & ( N
% 0.22/0.47 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 0.22/0.47 => ( P @ I2 ) ) )
% 0.22/0.47 & ( ( ord_less_int @ K @ zero_zero_int )
% 0.22/0.47 => ! [I2: int,J3: int] :
% 0.22/0.47 ( ( ( ord_less_int @ K @ J3 )
% 0.22/0.47 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.22/0.47 & ( N
% 0.22/0.47 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 0.22/0.47 => ( P @ I2 ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % split_zdiv
% 0.22/0.47 thf(fact_3931_div__pos__geq,axiom,
% 0.22/0.47 ! [L: int,K: int] :
% 0.22/0.47 ( ( ord_less_int @ zero_zero_int @ L )
% 0.22/0.47 => ( ( ord_less_eq_int @ L @ K )
% 0.22/0.47 => ( ( divide_divide_int @ K @ L )
% 0.22/0.47 = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % div_pos_geq
% 0.22/0.47 thf(fact_3932_L2__set__mult__ineq__lemma,axiom,
% 0.22/0.47 ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % L2_set_mult_ineq_lemma
% 0.22/0.47 thf(fact_3933_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 0.22/0.47 ! [X: nat,N: nat,M: nat] :
% 0.22/0.47 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.22/0.47 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.47 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.22/0.47 => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % VEBT_internal.exp_split_high_low(2)
% 0.22/0.47 thf(fact_3934_neg__zdiv__mult__2,axiom,
% 0.22/0.47 ! [A: int,B: int] :
% 0.22/0.47 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.22/0.47 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.22/0.47 = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % neg_zdiv_mult_2
% 0.22/0.47 thf(fact_3935_pos__zdiv__mult__2,axiom,
% 0.22/0.47 ! [A: int,B: int] :
% 0.22/0.47 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.22/0.47 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.22/0.47 = ( divide_divide_int @ B @ A ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % pos_zdiv_mult_2
% 0.22/0.47 thf(fact_3936_Bernoulli__inequality__even,axiom,
% 0.22/0.47 ! [N: nat,X: real] :
% 0.22/0.47 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.47 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % Bernoulli_inequality_even
% 0.22/0.47 thf(fact_3937_mult__less__iff1,axiom,
% 0.22/0.47 ! [Z: real,X: real,Y4: real] :
% 0.22/0.47 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.22/0.47 => ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y4 @ Z ) )
% 0.22/0.47 = ( ord_less_real @ X @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_less_iff1
% 0.22/0.47 thf(fact_3938_mult__less__iff1,axiom,
% 0.22/0.47 ! [Z: rat,X: rat,Y4: rat] :
% 0.22/0.47 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.22/0.47 => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y4 @ Z ) )
% 0.22/0.47 = ( ord_less_rat @ X @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_less_iff1
% 0.22/0.47 thf(fact_3939_mult__less__iff1,axiom,
% 0.22/0.47 ! [Z: int,X: int,Y4: int] :
% 0.22/0.47 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.22/0.47 => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y4 @ Z ) )
% 0.22/0.47 = ( ord_less_int @ X @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % mult_less_iff1
% 0.22/0.47 thf(fact_3940_real__average__minus__second,axiom,
% 0.22/0.47 ! [B: real,A: real] :
% 0.22/0.47 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 0.22/0.47 = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % real_average_minus_second
% 0.22/0.47 thf(fact_3941_real__average__minus__first,axiom,
% 0.22/0.47 ! [A: real,B: real] :
% 0.22/0.47 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 0.22/0.47 = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % real_average_minus_first
% 0.22/0.47 thf(fact_3942_norm__divide__numeral,axiom,
% 0.22/0.47 ! [A: real,W: num] :
% 0.22/0.47 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
% 0.22/0.47 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_divide_numeral
% 0.22/0.47 thf(fact_3943_norm__divide__numeral,axiom,
% 0.22/0.47 ! [A: complex,W: num] :
% 0.22/0.47 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 0.22/0.47 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_divide_numeral
% 0.22/0.47 thf(fact_3944_norm__mult__numeral2,axiom,
% 0.22/0.47 ! [A: real,W: num] :
% 0.22/0.47 ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
% 0.22/0.47 = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_mult_numeral2
% 0.22/0.47 thf(fact_3945_norm__mult__numeral2,axiom,
% 0.22/0.47 ! [A: complex,W: num] :
% 0.22/0.47 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 0.22/0.47 = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_mult_numeral2
% 0.22/0.47 thf(fact_3946_norm__mult__numeral1,axiom,
% 0.22/0.47 ! [W: num,A: real] :
% 0.22/0.47 ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 0.22/0.47 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_mult_numeral1
% 0.22/0.47 thf(fact_3947_norm__mult__numeral1,axiom,
% 0.22/0.47 ! [W: num,A: complex] :
% 0.22/0.47 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 0.22/0.47 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_mult_numeral1
% 0.22/0.47 thf(fact_3948_norm__le__zero__iff,axiom,
% 0.22/0.47 ! [X: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
% 0.22/0.47 = ( X = zero_zero_real ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_le_zero_iff
% 0.22/0.47 thf(fact_3949_norm__le__zero__iff,axiom,
% 0.22/0.47 ! [X: complex] :
% 0.22/0.47 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
% 0.22/0.47 = ( X = zero_zero_complex ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_le_zero_iff
% 0.22/0.47 thf(fact_3950_zero__less__norm__iff,axiom,
% 0.22/0.47 ! [X: real] :
% 0.22/0.47 ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
% 0.22/0.47 = ( X != zero_zero_real ) ) ).
% 0.22/0.47
% 0.22/0.47 % zero_less_norm_iff
% 0.22/0.47 thf(fact_3951_zero__less__norm__iff,axiom,
% 0.22/0.47 ! [X: complex] :
% 0.22/0.47 ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
% 0.22/0.47 = ( X != zero_zero_complex ) ) ).
% 0.22/0.47
% 0.22/0.47 % zero_less_norm_iff
% 0.22/0.47 thf(fact_3952_norm__of__nat,axiom,
% 0.22/0.47 ! [N: nat] :
% 0.22/0.47 ( ( real_V7735802525324610683m_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.22/0.47 = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_of_nat
% 0.22/0.47 thf(fact_3953_norm__of__nat,axiom,
% 0.22/0.47 ! [N: nat] :
% 0.22/0.47 ( ( real_V1022390504157884413omplex @ ( semiri8010041392384452111omplex @ N ) )
% 0.22/0.47 = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_of_nat
% 0.22/0.47 thf(fact_3954_norm__eq__zero,axiom,
% 0.22/0.47 ! [X: real] :
% 0.22/0.47 ( ( ( real_V7735802525324610683m_real @ X )
% 0.22/0.47 = zero_zero_real )
% 0.22/0.47 = ( X = zero_zero_real ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_eq_zero
% 0.22/0.47 thf(fact_3955_norm__eq__zero,axiom,
% 0.22/0.47 ! [X: complex] :
% 0.22/0.47 ( ( ( real_V1022390504157884413omplex @ X )
% 0.22/0.47 = zero_zero_real )
% 0.22/0.47 = ( X = zero_zero_complex ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_eq_zero
% 0.22/0.47 thf(fact_3956_norm__zero,axiom,
% 0.22/0.47 ( ( real_V7735802525324610683m_real @ zero_zero_real )
% 0.22/0.47 = zero_zero_real ) ).
% 0.22/0.47
% 0.22/0.47 % norm_zero
% 0.22/0.47 thf(fact_3957_norm__zero,axiom,
% 0.22/0.47 ( ( real_V1022390504157884413omplex @ zero_zero_complex )
% 0.22/0.47 = zero_zero_real ) ).
% 0.22/0.47
% 0.22/0.47 % norm_zero
% 0.22/0.47 thf(fact_3958_norm__numeral,axiom,
% 0.22/0.47 ! [W: num] :
% 0.22/0.47 ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
% 0.22/0.47 = ( numeral_numeral_real @ W ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_numeral
% 0.22/0.47 thf(fact_3959_norm__numeral,axiom,
% 0.22/0.47 ! [W: num] :
% 0.22/0.47 ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.22/0.47 = ( numeral_numeral_real @ W ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_numeral
% 0.22/0.47 thf(fact_3960_norm__minus__commute,axiom,
% 0.22/0.47 ! [A: real,B: real] :
% 0.22/0.47 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) )
% 0.22/0.47 = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_minus_commute
% 0.22/0.47 thf(fact_3961_norm__minus__commute,axiom,
% 0.22/0.47 ! [A: complex,B: complex] :
% 0.22/0.47 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) )
% 0.22/0.47 = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ A ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_minus_commute
% 0.22/0.47 thf(fact_3962_norm__triangle__mono,axiom,
% 0.22/0.47 ! [A: real,R2: real,B: real,S2: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
% 0.22/0.47 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S2 )
% 0.22/0.47 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_triangle_mono
% 0.22/0.47 thf(fact_3963_norm__triangle__mono,axiom,
% 0.22/0.47 ! [A: complex,R2: real,B: complex,S2: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
% 0.22/0.47 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S2 )
% 0.22/0.47 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_triangle_mono
% 0.22/0.47 thf(fact_3964_norm__triangle__ineq,axiom,
% 0.22/0.47 ! [X: real,Y4: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y4 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_triangle_ineq
% 0.22/0.47 thf(fact_3965_norm__triangle__ineq,axiom,
% 0.22/0.47 ! [X: complex,Y4: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y4 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_triangle_ineq
% 0.22/0.47 thf(fact_3966_norm__triangle__le,axiom,
% 0.22/0.47 ! [X: real,Y4: real,E2: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y4 ) ) @ E2 )
% 0.22/0.47 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y4 ) ) @ E2 ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_triangle_le
% 0.22/0.47 thf(fact_3967_norm__triangle__le,axiom,
% 0.22/0.47 ! [X: complex,Y4: complex,E2: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) ) @ E2 )
% 0.22/0.47 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y4 ) ) @ E2 ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_triangle_le
% 0.22/0.47 thf(fact_3968_norm__add__leD,axiom,
% 0.22/0.47 ! [A: real,B: real,C: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
% 0.22/0.47 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_add_leD
% 0.22/0.47 thf(fact_3969_norm__add__leD,axiom,
% 0.22/0.47 ! [A: complex,B: complex,C: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
% 0.22/0.47 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_add_leD
% 0.22/0.47 thf(fact_3970_norm__diff__triangle__less,axiom,
% 0.22/0.47 ! [X: real,Y4: real,E1: real,Z: real,E22: real] :
% 0.22/0.47 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y4 ) ) @ E1 )
% 0.22/0.47 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y4 @ Z ) ) @ E22 )
% 0.22/0.47 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_diff_triangle_less
% 0.22/0.47 thf(fact_3971_norm__diff__triangle__less,axiom,
% 0.22/0.47 ! [X: complex,Y4: complex,E1: real,Z: complex,E22: real] :
% 0.22/0.47 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y4 ) ) @ E1 )
% 0.22/0.47 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y4 @ Z ) ) @ E22 )
% 0.22/0.47 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_diff_triangle_less
% 0.22/0.47 thf(fact_3972_norm__triangle__sub,axiom,
% 0.22/0.47 ! [X: real,Y4: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y4 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y4 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_triangle_sub
% 0.22/0.47 thf(fact_3973_norm__triangle__sub,axiom,
% 0.22/0.47 ! [X: complex,Y4: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y4 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y4 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_triangle_sub
% 0.22/0.47 thf(fact_3974_norm__triangle__ineq4,axiom,
% 0.22/0.47 ! [A: real,B: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_triangle_ineq4
% 0.22/0.47 thf(fact_3975_norm__triangle__ineq4,axiom,
% 0.22/0.47 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_triangle_ineq4
% 0.22/0.47 thf(fact_3976_norm__diff__triangle__le,axiom,
% 0.22/0.47 ! [X: real,Y4: real,E1: real,Z: real,E22: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y4 ) ) @ E1 )
% 0.22/0.47 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y4 @ Z ) ) @ E22 )
% 0.22/0.47 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_diff_triangle_le
% 0.22/0.47 thf(fact_3977_norm__diff__triangle__le,axiom,
% 0.22/0.47 ! [X: complex,Y4: complex,E1: real,Z: complex,E22: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y4 ) ) @ E1 )
% 0.22/0.47 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y4 @ Z ) ) @ E22 )
% 0.22/0.47 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_diff_triangle_le
% 0.22/0.47 thf(fact_3978_norm__triangle__le__diff,axiom,
% 0.22/0.47 ! [X: real,Y4: real,E2: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y4 ) ) @ E2 )
% 0.22/0.47 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y4 ) ) @ E2 ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_triangle_le_diff
% 0.22/0.47 thf(fact_3979_norm__triangle__le__diff,axiom,
% 0.22/0.47 ! [X: complex,Y4: complex,E2: real] :
% 0.22/0.47 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) ) @ E2 )
% 0.22/0.47 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y4 ) ) @ E2 ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_triangle_le_diff
% 0.22/0.47 thf(fact_3980_norm__not__less__zero,axiom,
% 0.22/0.47 ! [X: complex] :
% 0.22/0.47 ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).
% 0.22/0.47
% 0.22/0.47 % norm_not_less_zero
% 0.22/0.47 thf(fact_3981_norm__ge__zero,axiom,
% 0.22/0.47 ! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_ge_zero
% 0.22/0.47 thf(fact_3982_norm__mult,axiom,
% 0.22/0.47 ! [X: real,Y4: real] :
% 0.22/0.47 ( ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y4 ) )
% 0.22/0.47 = ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_mult
% 0.22/0.47 thf(fact_3983_norm__mult,axiom,
% 0.22/0.47 ! [X: complex,Y4: complex] :
% 0.22/0.47 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y4 ) )
% 0.22/0.47 = ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_mult
% 0.22/0.47 thf(fact_3984_norm__divide,axiom,
% 0.22/0.47 ! [A: real,B: real] :
% 0.22/0.47 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 0.22/0.47 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_divide
% 0.22/0.47 thf(fact_3985_norm__divide,axiom,
% 0.22/0.47 ! [A: complex,B: complex] :
% 0.22/0.47 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.22/0.47 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_divide
% 0.22/0.47 thf(fact_3986_norm__diff__triangle__ineq,axiom,
% 0.22/0.47 ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_diff_triangle_ineq
% 0.22/0.47 thf(fact_3987_norm__diff__triangle__ineq,axiom,
% 0.22/0.47 ! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_diff_triangle_ineq
% 0.22/0.47 thf(fact_3988_nonzero__norm__divide,axiom,
% 0.22/0.47 ! [B: real,A: real] :
% 0.22/0.47 ( ( B != zero_zero_real )
% 0.22/0.47 => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 0.22/0.47 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % nonzero_norm_divide
% 0.22/0.47 thf(fact_3989_nonzero__norm__divide,axiom,
% 0.22/0.47 ! [B: complex,A: complex] :
% 0.22/0.47 ( ( B != zero_zero_complex )
% 0.22/0.47 => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.22/0.47 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % nonzero_norm_divide
% 0.22/0.47 thf(fact_3990_power__eq__imp__eq__norm,axiom,
% 0.22/0.47 ! [W: real,N: nat,Z: real] :
% 0.22/0.47 ( ( ( power_power_real @ W @ N )
% 0.22/0.47 = ( power_power_real @ Z @ N ) )
% 0.22/0.47 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.47 => ( ( real_V7735802525324610683m_real @ W )
% 0.22/0.47 = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power_eq_imp_eq_norm
% 0.22/0.47 thf(fact_3991_power__eq__imp__eq__norm,axiom,
% 0.22/0.47 ! [W: complex,N: nat,Z: complex] :
% 0.22/0.47 ( ( ( power_power_complex @ W @ N )
% 0.22/0.47 = ( power_power_complex @ Z @ N ) )
% 0.22/0.47 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.47 => ( ( real_V1022390504157884413omplex @ W )
% 0.22/0.47 = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power_eq_imp_eq_norm
% 0.22/0.47 thf(fact_3992_norm__mult__less,axiom,
% 0.22/0.47 ! [X: real,R2: real,Y4: real,S2: real] :
% 0.22/0.47 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
% 0.22/0.47 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y4 ) @ S2 )
% 0.22/0.47 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y4 ) ) @ ( times_times_real @ R2 @ S2 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_mult_less
% 0.22/0.47 thf(fact_3993_norm__mult__less,axiom,
% 0.22/0.47 ! [X: complex,R2: real,Y4: complex,S2: real] :
% 0.22/0.47 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
% 0.22/0.47 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y4 ) @ S2 )
% 0.22/0.47 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y4 ) ) @ ( times_times_real @ R2 @ S2 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_mult_less
% 0.22/0.47 thf(fact_3994_norm__mult__ineq,axiom,
% 0.22/0.47 ! [X: real,Y4: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y4 ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_mult_ineq
% 0.22/0.47 thf(fact_3995_norm__mult__ineq,axiom,
% 0.22/0.47 ! [X: complex,Y4: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y4 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_mult_ineq
% 0.22/0.47 thf(fact_3996_norm__power__ineq,axiom,
% 0.22/0.47 ! [X: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_power_ineq
% 0.22/0.47 thf(fact_3997_norm__power__ineq,axiom,
% 0.22/0.47 ! [X: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_power_ineq
% 0.22/0.47 thf(fact_3998_norm__diff__ineq,axiom,
% 0.22/0.47 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_diff_ineq
% 0.22/0.47 thf(fact_3999_norm__diff__ineq,axiom,
% 0.22/0.47 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_diff_ineq
% 0.22/0.47 thf(fact_4000_norm__triangle__ineq2,axiom,
% 0.22/0.47 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_triangle_ineq2
% 0.22/0.47 thf(fact_4001_norm__triangle__ineq2,axiom,
% 0.22/0.47 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_triangle_ineq2
% 0.22/0.47 thf(fact_4002_power__eq__1__iff,axiom,
% 0.22/0.47 ! [W: real,N: nat] :
% 0.22/0.47 ( ( ( power_power_real @ W @ N )
% 0.22/0.47 = one_one_real )
% 0.22/0.47 => ( ( ( real_V7735802525324610683m_real @ W )
% 0.22/0.47 = one_one_real )
% 0.22/0.47 | ( N = zero_zero_nat ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power_eq_1_iff
% 0.22/0.47 thf(fact_4003_power__eq__1__iff,axiom,
% 0.22/0.47 ! [W: complex,N: nat] :
% 0.22/0.47 ( ( ( power_power_complex @ W @ N )
% 0.22/0.47 = one_one_complex )
% 0.22/0.47 => ( ( ( real_V1022390504157884413omplex @ W )
% 0.22/0.47 = one_one_real )
% 0.22/0.47 | ( N = zero_zero_nat ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % power_eq_1_iff
% 0.22/0.47 thf(fact_4004_square__norm__one,axiom,
% 0.22/0.47 ! [X: real] :
% 0.22/0.47 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = one_one_real )
% 0.22/0.47 => ( ( real_V7735802525324610683m_real @ X )
% 0.22/0.47 = one_one_real ) ) ).
% 0.22/0.47
% 0.22/0.47 % square_norm_one
% 0.22/0.47 thf(fact_4005_square__norm__one,axiom,
% 0.22/0.47 ! [X: complex] :
% 0.22/0.47 ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = one_one_complex )
% 0.22/0.47 => ( ( real_V1022390504157884413omplex @ X )
% 0.22/0.47 = one_one_real ) ) ).
% 0.22/0.47
% 0.22/0.47 % square_norm_one
% 0.22/0.47 thf(fact_4006_norm__power__diff,axiom,
% 0.22/0.47 ! [Z: real,W: real,M: nat] :
% 0.22/0.47 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 0.22/0.47 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
% 0.22/0.47 => ( 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 ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_power_diff
% 0.22/0.47 thf(fact_4007_norm__power__diff,axiom,
% 0.22/0.47 ! [Z: complex,W: complex,M: nat] :
% 0.22/0.47 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 0.22/0.47 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
% 0.22/0.47 => ( 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 ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % norm_power_diff
% 0.22/0.47 thf(fact_4008_arcosh__1,axiom,
% 0.22/0.47 ( ( arcosh_real @ one_one_real )
% 0.22/0.47 = zero_zero_real ) ).
% 0.22/0.47
% 0.22/0.47 % arcosh_1
% 0.22/0.47 thf(fact_4009_artanh__0,axiom,
% 0.22/0.47 ( ( artanh_real @ zero_zero_real )
% 0.22/0.47 = zero_zero_real ) ).
% 0.22/0.47
% 0.22/0.47 % artanh_0
% 0.22/0.47 thf(fact_4010_arsinh__0,axiom,
% 0.22/0.47 ( ( arsinh_real @ zero_zero_real )
% 0.22/0.47 = zero_zero_real ) ).
% 0.22/0.47
% 0.22/0.47 % arsinh_0
% 0.22/0.47 thf(fact_4011_low__def,axiom,
% 0.22/0.47 ( vEBT_VEBT_low
% 0.22/0.47 = ( ^ [X3: nat,N4: nat] : ( modulo_modulo_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % low_def
% 0.22/0.47 thf(fact_4012_pochhammer__double,axiom,
% 0.22/0.47 ! [Z: complex,N: nat] :
% 0.22/0.47 ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.47 = ( 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 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % pochhammer_double
% 0.22/0.47 thf(fact_4013_pochhammer__double,axiom,
% 0.22/0.47 ! [Z: real,N: nat] :
% 0.22/0.47 ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.47 = ( 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 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % pochhammer_double
% 0.22/0.47 thf(fact_4014_pochhammer__double,axiom,
% 0.22/0.47 ! [Z: rat,N: nat] :
% 0.22/0.47 ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.47 = ( 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 ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % pochhammer_double
% 0.22/0.47 thf(fact_4015_central__binomial__lower__bound,axiom,
% 0.22/0.47 ! [N: nat] :
% 0.22/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.47 => ( 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 ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % central_binomial_lower_bound
% 0.22/0.47 thf(fact_4016_flip__bit__0,axiom,
% 0.22/0.47 ! [A: int] :
% 0.22/0.47 ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 0.22/0.47 = ( 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 ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % flip_bit_0
% 0.22/0.47 thf(fact_4017_flip__bit__0,axiom,
% 0.22/0.47 ! [A: nat] :
% 0.22/0.47 ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 0.22/0.47 = ( 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 ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % flip_bit_0
% 0.22/0.47 thf(fact_4018_flip__bit__0,axiom,
% 0.22/0.47 ! [A: code_integer] :
% 0.22/0.47 ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 0.22/0.47 = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % flip_bit_0
% 0.22/0.47 thf(fact_4019_mod__mod__trivial,axiom,
% 0.22/0.47 ! [A: nat,B: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.22/0.47 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mod_trivial
% 0.22/0.47 thf(fact_4020_mod__mod__trivial,axiom,
% 0.22/0.47 ! [A: int,B: int] :
% 0.22/0.47 ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.22/0.47 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mod_trivial
% 0.22/0.47 thf(fact_4021_mod__mod__trivial,axiom,
% 0.22/0.47 ! [A: code_integer,B: code_integer] :
% 0.22/0.47 ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.22/0.47 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mod_trivial
% 0.22/0.47 thf(fact_4022_mod__self,axiom,
% 0.22/0.47 ! [A: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ A @ A )
% 0.22/0.47 = zero_zero_nat ) ).
% 0.22/0.47
% 0.22/0.47 % mod_self
% 0.22/0.47 thf(fact_4023_mod__self,axiom,
% 0.22/0.47 ! [A: int] :
% 0.22/0.47 ( ( modulo_modulo_int @ A @ A )
% 0.22/0.47 = zero_zero_int ) ).
% 0.22/0.47
% 0.22/0.47 % mod_self
% 0.22/0.47 thf(fact_4024_mod__self,axiom,
% 0.22/0.47 ! [A: code_integer] :
% 0.22/0.47 ( ( modulo364778990260209775nteger @ A @ A )
% 0.22/0.47 = zero_z3403309356797280102nteger ) ).
% 0.22/0.47
% 0.22/0.47 % mod_self
% 0.22/0.47 thf(fact_4025_mod__by__0,axiom,
% 0.22/0.47 ! [A: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ A @ zero_zero_nat )
% 0.22/0.47 = A ) ).
% 0.22/0.47
% 0.22/0.47 % mod_by_0
% 0.22/0.47 thf(fact_4026_mod__by__0,axiom,
% 0.22/0.47 ! [A: int] :
% 0.22/0.47 ( ( modulo_modulo_int @ A @ zero_zero_int )
% 0.22/0.47 = A ) ).
% 0.22/0.47
% 0.22/0.47 % mod_by_0
% 0.22/0.47 thf(fact_4027_mod__by__0,axiom,
% 0.22/0.47 ! [A: code_integer] :
% 0.22/0.47 ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
% 0.22/0.47 = A ) ).
% 0.22/0.47
% 0.22/0.47 % mod_by_0
% 0.22/0.47 thf(fact_4028_mod__0,axiom,
% 0.22/0.47 ! [A: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 0.22/0.47 = zero_zero_nat ) ).
% 0.22/0.47
% 0.22/0.47 % mod_0
% 0.22/0.47 thf(fact_4029_mod__0,axiom,
% 0.22/0.47 ! [A: int] :
% 0.22/0.47 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 0.22/0.47 = zero_zero_int ) ).
% 0.22/0.47
% 0.22/0.47 % mod_0
% 0.22/0.47 thf(fact_4030_mod__0,axiom,
% 0.22/0.47 ! [A: code_integer] :
% 0.22/0.47 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 0.22/0.47 = zero_z3403309356797280102nteger ) ).
% 0.22/0.47
% 0.22/0.47 % mod_0
% 0.22/0.47 thf(fact_4031_bits__mod__0,axiom,
% 0.22/0.47 ! [A: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 0.22/0.47 = zero_zero_nat ) ).
% 0.22/0.47
% 0.22/0.47 % bits_mod_0
% 0.22/0.47 thf(fact_4032_bits__mod__0,axiom,
% 0.22/0.47 ! [A: int] :
% 0.22/0.47 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 0.22/0.47 = zero_zero_int ) ).
% 0.22/0.47
% 0.22/0.47 % bits_mod_0
% 0.22/0.47 thf(fact_4033_bits__mod__0,axiom,
% 0.22/0.47 ! [A: code_integer] :
% 0.22/0.47 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 0.22/0.47 = zero_z3403309356797280102nteger ) ).
% 0.22/0.47
% 0.22/0.47 % bits_mod_0
% 0.22/0.47 thf(fact_4034_mod__add__self2,axiom,
% 0.22/0.47 ! [A: nat,B: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.22/0.47 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_add_self2
% 0.22/0.47 thf(fact_4035_mod__add__self2,axiom,
% 0.22/0.47 ! [A: int,B: int] :
% 0.22/0.47 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.22/0.47 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_add_self2
% 0.22/0.47 thf(fact_4036_mod__add__self2,axiom,
% 0.22/0.47 ! [A: code_integer,B: code_integer] :
% 0.22/0.47 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
% 0.22/0.47 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_add_self2
% 0.22/0.47 thf(fact_4037_mod__add__self1,axiom,
% 0.22/0.47 ! [B: nat,A: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.22/0.47 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_add_self1
% 0.22/0.47 thf(fact_4038_mod__add__self1,axiom,
% 0.22/0.47 ! [B: int,A: int] :
% 0.22/0.47 ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.22/0.47 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_add_self1
% 0.22/0.47 thf(fact_4039_mod__add__self1,axiom,
% 0.22/0.47 ! [B: code_integer,A: code_integer] :
% 0.22/0.47 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
% 0.22/0.47 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_add_self1
% 0.22/0.47 thf(fact_4040_minus__mod__self2,axiom,
% 0.22/0.47 ! [A: int,B: int] :
% 0.22/0.47 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.22/0.47 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % minus_mod_self2
% 0.22/0.47 thf(fact_4041_minus__mod__self2,axiom,
% 0.22/0.47 ! [A: code_integer,B: code_integer] :
% 0.22/0.47 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
% 0.22/0.47 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % minus_mod_self2
% 0.22/0.47 thf(fact_4042_of__bool__less__eq__iff,axiom,
% 0.22/0.47 ! [P: $o,Q: $o] :
% 0.22/0.47 ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.22/0.47 = ( P
% 0.22/0.47 => Q ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_less_eq_iff
% 0.22/0.47 thf(fact_4043_of__bool__less__eq__iff,axiom,
% 0.22/0.47 ! [P: $o,Q: $o] :
% 0.22/0.47 ( ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 0.22/0.47 = ( P
% 0.22/0.47 => Q ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_less_eq_iff
% 0.22/0.47 thf(fact_4044_of__bool__less__eq__iff,axiom,
% 0.22/0.47 ! [P: $o,Q: $o] :
% 0.22/0.47 ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.22/0.47 = ( P
% 0.22/0.47 => Q ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_less_eq_iff
% 0.22/0.47 thf(fact_4045_of__bool__less__eq__iff,axiom,
% 0.22/0.47 ! [P: $o,Q: $o] :
% 0.22/0.47 ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.22/0.47 = ( P
% 0.22/0.47 => Q ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_less_eq_iff
% 0.22/0.47 thf(fact_4046_of__bool__less__eq__iff,axiom,
% 0.22/0.47 ! [P: $o,Q: $o] :
% 0.22/0.47 ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 0.22/0.47 = ( P
% 0.22/0.47 => Q ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_less_eq_iff
% 0.22/0.47 thf(fact_4047_of__bool__eq__0__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ( zero_n1201886186963655149omplex @ P )
% 0.22/0.47 = zero_zero_complex )
% 0.22/0.47 = ( ~ P ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq_0_iff
% 0.22/0.47 thf(fact_4048_of__bool__eq__0__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ( zero_n3304061248610475627l_real @ P )
% 0.22/0.47 = zero_zero_real )
% 0.22/0.47 = ( ~ P ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq_0_iff
% 0.22/0.47 thf(fact_4049_of__bool__eq__0__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 0.22/0.47 = zero_zero_rat )
% 0.22/0.47 = ( ~ P ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq_0_iff
% 0.22/0.47 thf(fact_4050_of__bool__eq__0__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 0.22/0.47 = zero_zero_nat )
% 0.22/0.47 = ( ~ P ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq_0_iff
% 0.22/0.47 thf(fact_4051_of__bool__eq__0__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ( zero_n2684676970156552555ol_int @ P )
% 0.22/0.47 = zero_zero_int )
% 0.22/0.47 = ( ~ P ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq_0_iff
% 0.22/0.47 thf(fact_4052_of__bool__eq__0__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ( zero_n356916108424825756nteger @ P )
% 0.22/0.47 = zero_z3403309356797280102nteger )
% 0.22/0.47 = ( ~ P ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq_0_iff
% 0.22/0.47 thf(fact_4053_of__bool__eq_I1_J,axiom,
% 0.22/0.47 ( ( zero_n1201886186963655149omplex @ $false )
% 0.22/0.47 = zero_zero_complex ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq(1)
% 0.22/0.47 thf(fact_4054_of__bool__eq_I1_J,axiom,
% 0.22/0.47 ( ( zero_n3304061248610475627l_real @ $false )
% 0.22/0.47 = zero_zero_real ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq(1)
% 0.22/0.47 thf(fact_4055_of__bool__eq_I1_J,axiom,
% 0.22/0.47 ( ( zero_n2052037380579107095ol_rat @ $false )
% 0.22/0.47 = zero_zero_rat ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq(1)
% 0.22/0.47 thf(fact_4056_of__bool__eq_I1_J,axiom,
% 0.22/0.47 ( ( zero_n2687167440665602831ol_nat @ $false )
% 0.22/0.47 = zero_zero_nat ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq(1)
% 0.22/0.47 thf(fact_4057_of__bool__eq_I1_J,axiom,
% 0.22/0.47 ( ( zero_n2684676970156552555ol_int @ $false )
% 0.22/0.47 = zero_zero_int ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq(1)
% 0.22/0.47 thf(fact_4058_of__bool__eq_I1_J,axiom,
% 0.22/0.47 ( ( zero_n356916108424825756nteger @ $false )
% 0.22/0.47 = zero_z3403309356797280102nteger ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq(1)
% 0.22/0.47 thf(fact_4059_of__bool__less__iff,axiom,
% 0.22/0.47 ! [P: $o,Q: $o] :
% 0.22/0.47 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 0.22/0.47 = ( ~ P
% 0.22/0.47 & Q ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_less_iff
% 0.22/0.47 thf(fact_4060_of__bool__less__iff,axiom,
% 0.22/0.47 ! [P: $o,Q: $o] :
% 0.22/0.47 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.22/0.47 = ( ~ P
% 0.22/0.47 & Q ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_less_iff
% 0.22/0.47 thf(fact_4061_of__bool__less__iff,axiom,
% 0.22/0.47 ! [P: $o,Q: $o] :
% 0.22/0.47 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.22/0.47 = ( ~ P
% 0.22/0.47 & Q ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_less_iff
% 0.22/0.47 thf(fact_4062_of__bool__less__iff,axiom,
% 0.22/0.47 ! [P: $o,Q: $o] :
% 0.22/0.47 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.22/0.47 = ( ~ P
% 0.22/0.47 & Q ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_less_iff
% 0.22/0.47 thf(fact_4063_of__bool__less__iff,axiom,
% 0.22/0.47 ! [P: $o,Q: $o] :
% 0.22/0.47 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 0.22/0.47 = ( ~ P
% 0.22/0.47 & Q ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_less_iff
% 0.22/0.47 thf(fact_4064_of__bool__eq__1__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ( zero_n1201886186963655149omplex @ P )
% 0.22/0.47 = one_one_complex )
% 0.22/0.47 = P ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq_1_iff
% 0.22/0.47 thf(fact_4065_of__bool__eq__1__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ( zero_n3304061248610475627l_real @ P )
% 0.22/0.47 = one_one_real )
% 0.22/0.47 = P ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq_1_iff
% 0.22/0.47 thf(fact_4066_of__bool__eq__1__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 0.22/0.47 = one_one_rat )
% 0.22/0.47 = P ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq_1_iff
% 0.22/0.47 thf(fact_4067_of__bool__eq__1__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 0.22/0.47 = one_one_nat )
% 0.22/0.47 = P ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq_1_iff
% 0.22/0.47 thf(fact_4068_of__bool__eq__1__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ( zero_n2684676970156552555ol_int @ P )
% 0.22/0.47 = one_one_int )
% 0.22/0.47 = P ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq_1_iff
% 0.22/0.47 thf(fact_4069_of__bool__eq__1__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ( zero_n356916108424825756nteger @ P )
% 0.22/0.47 = one_one_Code_integer )
% 0.22/0.47 = P ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq_1_iff
% 0.22/0.47 thf(fact_4070_of__bool__eq_I2_J,axiom,
% 0.22/0.47 ( ( zero_n1201886186963655149omplex @ $true )
% 0.22/0.47 = one_one_complex ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq(2)
% 0.22/0.47 thf(fact_4071_of__bool__eq_I2_J,axiom,
% 0.22/0.47 ( ( zero_n3304061248610475627l_real @ $true )
% 0.22/0.47 = one_one_real ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq(2)
% 0.22/0.47 thf(fact_4072_of__bool__eq_I2_J,axiom,
% 0.22/0.47 ( ( zero_n2052037380579107095ol_rat @ $true )
% 0.22/0.47 = one_one_rat ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq(2)
% 0.22/0.47 thf(fact_4073_of__bool__eq_I2_J,axiom,
% 0.22/0.47 ( ( zero_n2687167440665602831ol_nat @ $true )
% 0.22/0.47 = one_one_nat ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq(2)
% 0.22/0.47 thf(fact_4074_of__bool__eq_I2_J,axiom,
% 0.22/0.47 ( ( zero_n2684676970156552555ol_int @ $true )
% 0.22/0.47 = one_one_int ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq(2)
% 0.22/0.47 thf(fact_4075_of__bool__eq_I2_J,axiom,
% 0.22/0.47 ( ( zero_n356916108424825756nteger @ $true )
% 0.22/0.47 = one_one_Code_integer ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_eq(2)
% 0.22/0.47 thf(fact_4076_mod__less,axiom,
% 0.22/0.47 ! [M: nat,N: nat] :
% 0.22/0.47 ( ( ord_less_nat @ M @ N )
% 0.22/0.47 => ( ( modulo_modulo_nat @ M @ N )
% 0.22/0.47 = M ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_less
% 0.22/0.47 thf(fact_4077_of__nat__of__bool,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.22/0.47 = ( zero_n1201886186963655149omplex @ P ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_nat_of_bool
% 0.22/0.47 thf(fact_4078_of__nat__of__bool,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.22/0.47 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_nat_of_bool
% 0.22/0.47 thf(fact_4079_of__nat__of__bool,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.22/0.47 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_nat_of_bool
% 0.22/0.47 thf(fact_4080_of__nat__of__bool,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.22/0.47 = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_nat_of_bool
% 0.22/0.47 thf(fact_4081_of__nat__of__bool,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.22/0.47 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_nat_of_bool
% 0.22/0.47 thf(fact_4082_of__nat__of__bool,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.22/0.47 = ( zero_n356916108424825756nteger @ P ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_nat_of_bool
% 0.22/0.47 thf(fact_4083_mod__mult__self1__is__0,axiom,
% 0.22/0.47 ! [B: nat,A: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
% 0.22/0.47 = zero_zero_nat ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mult_self1_is_0
% 0.22/0.47 thf(fact_4084_mod__mult__self1__is__0,axiom,
% 0.22/0.47 ! [B: int,A: int] :
% 0.22/0.47 ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
% 0.22/0.47 = zero_zero_int ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mult_self1_is_0
% 0.22/0.47 thf(fact_4085_mod__mult__self1__is__0,axiom,
% 0.22/0.47 ! [B: code_integer,A: code_integer] :
% 0.22/0.47 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
% 0.22/0.47 = zero_z3403309356797280102nteger ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mult_self1_is_0
% 0.22/0.47 thf(fact_4086_mod__mult__self2__is__0,axiom,
% 0.22/0.47 ! [A: nat,B: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
% 0.22/0.47 = zero_zero_nat ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mult_self2_is_0
% 0.22/0.47 thf(fact_4087_mod__mult__self2__is__0,axiom,
% 0.22/0.47 ! [A: int,B: int] :
% 0.22/0.47 ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
% 0.22/0.47 = zero_zero_int ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mult_self2_is_0
% 0.22/0.47 thf(fact_4088_mod__mult__self2__is__0,axiom,
% 0.22/0.47 ! [A: code_integer,B: code_integer] :
% 0.22/0.47 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 0.22/0.47 = zero_z3403309356797280102nteger ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mult_self2_is_0
% 0.22/0.47 thf(fact_4089_mod__by__1,axiom,
% 0.22/0.47 ! [A: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.22/0.47 = zero_zero_nat ) ).
% 0.22/0.47
% 0.22/0.47 % mod_by_1
% 0.22/0.47 thf(fact_4090_mod__by__1,axiom,
% 0.22/0.47 ! [A: int] :
% 0.22/0.47 ( ( modulo_modulo_int @ A @ one_one_int )
% 0.22/0.47 = zero_zero_int ) ).
% 0.22/0.47
% 0.22/0.47 % mod_by_1
% 0.22/0.47 thf(fact_4091_mod__by__1,axiom,
% 0.22/0.47 ! [A: code_integer] :
% 0.22/0.47 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.22/0.47 = zero_z3403309356797280102nteger ) ).
% 0.22/0.47
% 0.22/0.47 % mod_by_1
% 0.22/0.47 thf(fact_4092_bits__mod__by__1,axiom,
% 0.22/0.47 ! [A: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.22/0.47 = zero_zero_nat ) ).
% 0.22/0.47
% 0.22/0.47 % bits_mod_by_1
% 0.22/0.47 thf(fact_4093_bits__mod__by__1,axiom,
% 0.22/0.47 ! [A: int] :
% 0.22/0.47 ( ( modulo_modulo_int @ A @ one_one_int )
% 0.22/0.47 = zero_zero_int ) ).
% 0.22/0.47
% 0.22/0.47 % bits_mod_by_1
% 0.22/0.47 thf(fact_4094_bits__mod__by__1,axiom,
% 0.22/0.47 ! [A: code_integer] :
% 0.22/0.47 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.22/0.47 = zero_z3403309356797280102nteger ) ).
% 0.22/0.47
% 0.22/0.47 % bits_mod_by_1
% 0.22/0.47 thf(fact_4095_bits__mod__div__trivial,axiom,
% 0.22/0.47 ! [A: nat,B: nat] :
% 0.22/0.47 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.22/0.47 = zero_zero_nat ) ).
% 0.22/0.47
% 0.22/0.47 % bits_mod_div_trivial
% 0.22/0.47 thf(fact_4096_bits__mod__div__trivial,axiom,
% 0.22/0.47 ! [A: int,B: int] :
% 0.22/0.47 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.22/0.47 = zero_zero_int ) ).
% 0.22/0.47
% 0.22/0.47 % bits_mod_div_trivial
% 0.22/0.47 thf(fact_4097_bits__mod__div__trivial,axiom,
% 0.22/0.47 ! [A: code_integer,B: code_integer] :
% 0.22/0.47 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.22/0.47 = zero_z3403309356797280102nteger ) ).
% 0.22/0.47
% 0.22/0.47 % bits_mod_div_trivial
% 0.22/0.47 thf(fact_4098_mod__div__trivial,axiom,
% 0.22/0.47 ! [A: nat,B: nat] :
% 0.22/0.47 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.22/0.47 = zero_zero_nat ) ).
% 0.22/0.47
% 0.22/0.47 % mod_div_trivial
% 0.22/0.47 thf(fact_4099_mod__div__trivial,axiom,
% 0.22/0.47 ! [A: int,B: int] :
% 0.22/0.47 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.22/0.47 = zero_zero_int ) ).
% 0.22/0.47
% 0.22/0.47 % mod_div_trivial
% 0.22/0.47 thf(fact_4100_mod__div__trivial,axiom,
% 0.22/0.47 ! [A: code_integer,B: code_integer] :
% 0.22/0.47 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.22/0.47 = zero_z3403309356797280102nteger ) ).
% 0.22/0.47
% 0.22/0.47 % mod_div_trivial
% 0.22/0.47 thf(fact_4101_mod__mult__self4,axiom,
% 0.22/0.47 ! [B: nat,C: nat,A: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 0.22/0.47 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mult_self4
% 0.22/0.47 thf(fact_4102_mod__mult__self4,axiom,
% 0.22/0.47 ! [B: int,C: int,A: int] :
% 0.22/0.47 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 0.22/0.47 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mult_self4
% 0.22/0.47 thf(fact_4103_mod__mult__self4,axiom,
% 0.22/0.47 ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.22/0.47 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
% 0.22/0.47 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mult_self4
% 0.22/0.47 thf(fact_4104_mod__mult__self3,axiom,
% 0.22/0.47 ! [C: nat,B: nat,A: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 0.22/0.47 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mult_self3
% 0.22/0.47 thf(fact_4105_mod__mult__self3,axiom,
% 0.22/0.47 ! [C: int,B: int,A: int] :
% 0.22/0.47 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 0.22/0.47 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mult_self3
% 0.22/0.47 thf(fact_4106_mod__mult__self3,axiom,
% 0.22/0.47 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.22/0.47 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
% 0.22/0.47 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mult_self3
% 0.22/0.47 thf(fact_4107_mod__mult__self2,axiom,
% 0.22/0.47 ! [A: nat,B: nat,C: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 0.22/0.47 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mult_self2
% 0.22/0.47 thf(fact_4108_mod__mult__self2,axiom,
% 0.22/0.47 ! [A: int,B: int,C: int] :
% 0.22/0.47 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 0.22/0.47 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mult_self2
% 0.22/0.47 thf(fact_4109_mod__mult__self2,axiom,
% 0.22/0.47 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.22/0.47 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
% 0.22/0.47 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mult_self2
% 0.22/0.47 thf(fact_4110_mod__mult__self1,axiom,
% 0.22/0.47 ! [A: nat,C: nat,B: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 0.22/0.47 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mult_self1
% 0.22/0.47 thf(fact_4111_mod__mult__self1,axiom,
% 0.22/0.47 ! [A: int,C: int,B: int] :
% 0.22/0.47 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 0.22/0.47 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mult_self1
% 0.22/0.47 thf(fact_4112_mod__mult__self1,axiom,
% 0.22/0.47 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.22/0.47 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
% 0.22/0.47 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_mult_self1
% 0.22/0.47 thf(fact_4113_dvd__imp__mod__0,axiom,
% 0.22/0.47 ! [A: nat,B: nat] :
% 0.22/0.47 ( ( dvd_dvd_nat @ A @ B )
% 0.22/0.47 => ( ( modulo_modulo_nat @ B @ A )
% 0.22/0.47 = zero_zero_nat ) ) ).
% 0.22/0.47
% 0.22/0.47 % dvd_imp_mod_0
% 0.22/0.47 thf(fact_4114_dvd__imp__mod__0,axiom,
% 0.22/0.47 ! [A: int,B: int] :
% 0.22/0.47 ( ( dvd_dvd_int @ A @ B )
% 0.22/0.47 => ( ( modulo_modulo_int @ B @ A )
% 0.22/0.47 = zero_zero_int ) ) ).
% 0.22/0.47
% 0.22/0.47 % dvd_imp_mod_0
% 0.22/0.47 thf(fact_4115_dvd__imp__mod__0,axiom,
% 0.22/0.47 ! [A: code_integer,B: code_integer] :
% 0.22/0.47 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.22/0.47 => ( ( modulo364778990260209775nteger @ B @ A )
% 0.22/0.47 = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.47
% 0.22/0.47 % dvd_imp_mod_0
% 0.22/0.47 thf(fact_4116_zero__less__of__bool__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 0.22/0.47 = P ) ).
% 0.22/0.47
% 0.22/0.47 % zero_less_of_bool_iff
% 0.22/0.47 thf(fact_4117_zero__less__of__bool__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 0.22/0.47 = P ) ).
% 0.22/0.47
% 0.22/0.47 % zero_less_of_bool_iff
% 0.22/0.47 thf(fact_4118_zero__less__of__bool__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.22/0.47 = P ) ).
% 0.22/0.47
% 0.22/0.47 % zero_less_of_bool_iff
% 0.22/0.47 thf(fact_4119_zero__less__of__bool__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.22/0.47 = P ) ).
% 0.22/0.47
% 0.22/0.47 % zero_less_of_bool_iff
% 0.22/0.47 thf(fact_4120_zero__less__of__bool__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
% 0.22/0.47 = P ) ).
% 0.22/0.47
% 0.22/0.47 % zero_less_of_bool_iff
% 0.22/0.47 thf(fact_4121_of__bool__less__one__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 0.22/0.47 = ( ~ P ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_less_one_iff
% 0.22/0.47 thf(fact_4122_of__bool__less__one__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 0.22/0.47 = ( ~ P ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_less_one_iff
% 0.22/0.47 thf(fact_4123_of__bool__less__one__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 0.22/0.47 = ( ~ P ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_less_one_iff
% 0.22/0.47 thf(fact_4124_of__bool__less__one__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 0.22/0.47 = ( ~ P ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_less_one_iff
% 0.22/0.47 thf(fact_4125_of__bool__less__one__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
% 0.22/0.47 = ( ~ P ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_less_one_iff
% 0.22/0.47 thf(fact_4126_of__bool__not__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( zero_n1201886186963655149omplex @ ~ P )
% 0.22/0.47 = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_not_iff
% 0.22/0.47 thf(fact_4127_of__bool__not__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( zero_n3304061248610475627l_real @ ~ P )
% 0.22/0.47 = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_not_iff
% 0.22/0.47 thf(fact_4128_of__bool__not__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( zero_n2052037380579107095ol_rat @ ~ P )
% 0.22/0.47 = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_not_iff
% 0.22/0.47 thf(fact_4129_of__bool__not__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( zero_n2684676970156552555ol_int @ ~ P )
% 0.22/0.47 = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_not_iff
% 0.22/0.47 thf(fact_4130_of__bool__not__iff,axiom,
% 0.22/0.47 ! [P: $o] :
% 0.22/0.47 ( ( zero_n356916108424825756nteger @ ~ P )
% 0.22/0.47 = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_not_iff
% 0.22/0.47 thf(fact_4131_Suc__0__mod__eq,axiom,
% 0.22/0.47 ! [N: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.22/0.47 = ( zero_n2687167440665602831ol_nat
% 0.22/0.47 @ ( N
% 0.22/0.47 != ( suc @ zero_zero_nat ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % Suc_0_mod_eq
% 0.22/0.47 thf(fact_4132_mod__by__Suc__0,axiom,
% 0.22/0.47 ! [M: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.22/0.47 = zero_zero_nat ) ).
% 0.22/0.47
% 0.22/0.47 % mod_by_Suc_0
% 0.22/0.47 thf(fact_4133_pochhammer__0,axiom,
% 0.22/0.47 ! [A: complex] :
% 0.22/0.47 ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
% 0.22/0.47 = one_one_complex ) ).
% 0.22/0.47
% 0.22/0.47 % pochhammer_0
% 0.22/0.47 thf(fact_4134_pochhammer__0,axiom,
% 0.22/0.47 ! [A: real] :
% 0.22/0.47 ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
% 0.22/0.47 = one_one_real ) ).
% 0.22/0.47
% 0.22/0.47 % pochhammer_0
% 0.22/0.47 thf(fact_4135_pochhammer__0,axiom,
% 0.22/0.47 ! [A: rat] :
% 0.22/0.47 ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
% 0.22/0.47 = one_one_rat ) ).
% 0.22/0.47
% 0.22/0.47 % pochhammer_0
% 0.22/0.47 thf(fact_4136_pochhammer__0,axiom,
% 0.22/0.47 ! [A: nat] :
% 0.22/0.47 ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
% 0.22/0.47 = one_one_nat ) ).
% 0.22/0.47
% 0.22/0.47 % pochhammer_0
% 0.22/0.47 thf(fact_4137_pochhammer__0,axiom,
% 0.22/0.47 ! [A: int] :
% 0.22/0.47 ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
% 0.22/0.47 = one_one_int ) ).
% 0.22/0.47
% 0.22/0.47 % pochhammer_0
% 0.22/0.47 thf(fact_4138_Suc__mod__mult__self1,axiom,
% 0.22/0.47 ! [M: nat,K: nat,N: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
% 0.22/0.47 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.22/0.47
% 0.22/0.47 % Suc_mod_mult_self1
% 0.22/0.47 thf(fact_4139_Suc__mod__mult__self2,axiom,
% 0.22/0.47 ! [M: nat,N: nat,K: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
% 0.22/0.47 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.22/0.47
% 0.22/0.47 % Suc_mod_mult_self2
% 0.22/0.47 thf(fact_4140_Suc__mod__mult__self3,axiom,
% 0.22/0.47 ! [K: nat,N: nat,M: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
% 0.22/0.47 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.22/0.47
% 0.22/0.47 % Suc_mod_mult_self3
% 0.22/0.47 thf(fact_4141_Suc__mod__mult__self4,axiom,
% 0.22/0.47 ! [N: nat,K: nat,M: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
% 0.22/0.47 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.22/0.47
% 0.22/0.47 % Suc_mod_mult_self4
% 0.22/0.47 thf(fact_4142_bits__one__mod__two__eq__one,axiom,
% 0.22/0.47 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = one_one_nat ) ).
% 0.22/0.47
% 0.22/0.47 % bits_one_mod_two_eq_one
% 0.22/0.47 thf(fact_4143_bits__one__mod__two__eq__one,axiom,
% 0.22/0.47 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.47 = one_one_int ) ).
% 0.22/0.47
% 0.22/0.47 % bits_one_mod_two_eq_one
% 0.22/0.47 thf(fact_4144_bits__one__mod__two__eq__one,axiom,
% 0.22/0.47 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.47 = one_one_Code_integer ) ).
% 0.22/0.47
% 0.22/0.47 % bits_one_mod_two_eq_one
% 0.22/0.47 thf(fact_4145_one__mod__two__eq__one,axiom,
% 0.22/0.47 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = one_one_nat ) ).
% 0.22/0.47
% 0.22/0.47 % one_mod_two_eq_one
% 0.22/0.47 thf(fact_4146_one__mod__two__eq__one,axiom,
% 0.22/0.47 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.47 = one_one_int ) ).
% 0.22/0.47
% 0.22/0.47 % one_mod_two_eq_one
% 0.22/0.47 thf(fact_4147_one__mod__two__eq__one,axiom,
% 0.22/0.47 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.47 = one_one_Code_integer ) ).
% 0.22/0.47
% 0.22/0.47 % one_mod_two_eq_one
% 0.22/0.47 thf(fact_4148_even__mod__2__iff,axiom,
% 0.22/0.47 ! [A: nat] :
% 0.22/0.47 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.47 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).
% 0.22/0.47
% 0.22/0.47 % even_mod_2_iff
% 0.22/0.47 thf(fact_4149_even__mod__2__iff,axiom,
% 0.22/0.47 ! [A: int] :
% 0.22/0.47 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.22/0.47 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.22/0.47
% 0.22/0.47 % even_mod_2_iff
% 0.22/0.47 thf(fact_4150_even__mod__2__iff,axiom,
% 0.22/0.47 ! [A: code_integer] :
% 0.22/0.47 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.22/0.47 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.22/0.47
% 0.22/0.47 % even_mod_2_iff
% 0.22/0.47 thf(fact_4151_odd__of__bool__self,axiom,
% 0.22/0.47 ! [P5: $o] :
% 0.22/0.47 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P5 ) ) )
% 0.22/0.47 = P5 ) ).
% 0.22/0.47
% 0.22/0.47 % odd_of_bool_self
% 0.22/0.47 thf(fact_4152_odd__of__bool__self,axiom,
% 0.22/0.47 ! [P5: $o] :
% 0.22/0.47 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P5 ) ) )
% 0.22/0.47 = P5 ) ).
% 0.22/0.47
% 0.22/0.47 % odd_of_bool_self
% 0.22/0.47 thf(fact_4153_odd__of__bool__self,axiom,
% 0.22/0.47 ! [P5: $o] :
% 0.22/0.47 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P5 ) ) )
% 0.22/0.47 = P5 ) ).
% 0.22/0.47
% 0.22/0.47 % odd_of_bool_self
% 0.22/0.47 thf(fact_4154_mod2__Suc__Suc,axiom,
% 0.22/0.47 ! [M: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod2_Suc_Suc
% 0.22/0.47 thf(fact_4155_Suc__times__numeral__mod__eq,axiom,
% 0.22/0.47 ! [K: num,N: nat] :
% 0.22/0.47 ( ( ( numeral_numeral_nat @ K )
% 0.22/0.47 != one_one_nat )
% 0.22/0.47 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
% 0.22/0.47 = one_one_nat ) ) ).
% 0.22/0.47
% 0.22/0.47 % Suc_times_numeral_mod_eq
% 0.22/0.47 thf(fact_4156_not__mod__2__eq__1__eq__0,axiom,
% 0.22/0.47 ! [A: nat] :
% 0.22/0.47 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 != one_one_nat )
% 0.22/0.47 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = zero_zero_nat ) ) ).
% 0.22/0.47
% 0.22/0.47 % not_mod_2_eq_1_eq_0
% 0.22/0.47 thf(fact_4157_not__mod__2__eq__1__eq__0,axiom,
% 0.22/0.47 ! [A: int] :
% 0.22/0.47 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.47 != one_one_int )
% 0.22/0.47 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.47 = zero_zero_int ) ) ).
% 0.22/0.47
% 0.22/0.47 % not_mod_2_eq_1_eq_0
% 0.22/0.47 thf(fact_4158_not__mod__2__eq__1__eq__0,axiom,
% 0.22/0.47 ! [A: code_integer] :
% 0.22/0.47 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.47 != one_one_Code_integer )
% 0.22/0.47 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.47 = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.47
% 0.22/0.47 % not_mod_2_eq_1_eq_0
% 0.22/0.47 thf(fact_4159_not__mod__2__eq__0__eq__1,axiom,
% 0.22/0.47 ! [A: nat] :
% 0.22/0.47 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 != zero_zero_nat )
% 0.22/0.47 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = one_one_nat ) ) ).
% 0.22/0.47
% 0.22/0.47 % not_mod_2_eq_0_eq_1
% 0.22/0.47 thf(fact_4160_not__mod__2__eq__0__eq__1,axiom,
% 0.22/0.47 ! [A: int] :
% 0.22/0.47 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.47 != zero_zero_int )
% 0.22/0.47 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.47 = one_one_int ) ) ).
% 0.22/0.47
% 0.22/0.47 % not_mod_2_eq_0_eq_1
% 0.22/0.47 thf(fact_4161_not__mod__2__eq__0__eq__1,axiom,
% 0.22/0.47 ! [A: code_integer] :
% 0.22/0.47 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.47 != zero_z3403309356797280102nteger )
% 0.22/0.47 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.47 = one_one_Code_integer ) ) ).
% 0.22/0.47
% 0.22/0.47 % not_mod_2_eq_0_eq_1
% 0.22/0.47 thf(fact_4162_of__bool__half__eq__0,axiom,
% 0.22/0.47 ! [B: $o] :
% 0.22/0.47 ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = zero_zero_nat ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_half_eq_0
% 0.22/0.47 thf(fact_4163_of__bool__half__eq__0,axiom,
% 0.22/0.47 ! [B: $o] :
% 0.22/0.47 ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.47 = zero_zero_int ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_half_eq_0
% 0.22/0.47 thf(fact_4164_of__bool__half__eq__0,axiom,
% 0.22/0.47 ! [B: $o] :
% 0.22/0.47 ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.47 = zero_z3403309356797280102nteger ) ).
% 0.22/0.47
% 0.22/0.47 % of_bool_half_eq_0
% 0.22/0.47 thf(fact_4165_not__mod2__eq__Suc__0__eq__0,axiom,
% 0.22/0.47 ! [N: nat] :
% 0.22/0.47 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 != ( suc @ zero_zero_nat ) )
% 0.22/0.47 = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = zero_zero_nat ) ) ).
% 0.22/0.47
% 0.22/0.47 % not_mod2_eq_Suc_0_eq_0
% 0.22/0.47 thf(fact_4166_add__self__mod__2,axiom,
% 0.22/0.47 ! [M: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = zero_zero_nat ) ).
% 0.22/0.47
% 0.22/0.47 % add_self_mod_2
% 0.22/0.47 thf(fact_4167_Suc__mod__eq__add3__mod__numeral,axiom,
% 0.22/0.47 ! [M: nat,V: num] :
% 0.22/0.47 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 0.22/0.47 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % Suc_mod_eq_add3_mod_numeral
% 0.22/0.47 thf(fact_4168_mod__Suc__eq__mod__add3,axiom,
% 0.22/0.47 ! [M: nat,N: nat] :
% 0.22/0.47 ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 0.22/0.47 = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod_Suc_eq_mod_add3
% 0.22/0.47 thf(fact_4169_mod2__gr__0,axiom,
% 0.22/0.47 ! [M: nat] :
% 0.22/0.47 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.47 = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.47 = one_one_nat ) ) ).
% 0.22/0.47
% 0.22/0.47 % mod2_gr_0
% 0.22/0.47 thf(fact_4170_bits__1__div__exp,axiom,
% 0.22/0.47 ! [N: nat] :
% 0.22/0.47 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.47 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.22/0.47
% 0.22/0.47 % bits_1_div_exp
% 0.22/0.47 thf(fact_4171_bits__1__div__exp,axiom,
% 0.22/0.47 ! [N: nat] :
% 0.22/0.47 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.47 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % bits_1_div_exp
% 0.22/0.48 thf(fact_4172_bits__1__div__exp,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % bits_1_div_exp
% 0.22/0.48 thf(fact_4173_one__div__2__pow__eq,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % one_div_2_pow_eq
% 0.22/0.48 thf(fact_4174_one__div__2__pow__eq,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % one_div_2_pow_eq
% 0.22/0.48 thf(fact_4175_one__div__2__pow__eq,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % one_div_2_pow_eq
% 0.22/0.48 thf(fact_4176_one__mod__2__pow__eq,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % one_mod_2_pow_eq
% 0.22/0.48 thf(fact_4177_one__mod__2__pow__eq,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % one_mod_2_pow_eq
% 0.22/0.48 thf(fact_4178_one__mod__2__pow__eq,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % one_mod_2_pow_eq
% 0.22/0.48 thf(fact_4179_even__succ__mod__exp,axiom,
% 0.22/0.48 ! [A: nat,N: nat] :
% 0.22/0.48 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.48 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.48 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % even_succ_mod_exp
% 0.22/0.48 thf(fact_4180_even__succ__mod__exp,axiom,
% 0.22/0.48 ! [A: int,N: nat] :
% 0.22/0.48 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.48 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.48 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % even_succ_mod_exp
% 0.22/0.48 thf(fact_4181_even__succ__mod__exp,axiom,
% 0.22/0.48 ! [A: code_integer,N: nat] :
% 0.22/0.48 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.48 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.48 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % even_succ_mod_exp
% 0.22/0.48 thf(fact_4182_of__bool__eq__iff,axiom,
% 0.22/0.48 ! [P5: $o,Q2: $o] :
% 0.22/0.48 ( ( ( zero_n2687167440665602831ol_nat @ P5 )
% 0.22/0.48 = ( zero_n2687167440665602831ol_nat @ Q2 ) )
% 0.22/0.48 = ( P5 = Q2 ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_eq_iff
% 0.22/0.48 thf(fact_4183_of__bool__eq__iff,axiom,
% 0.22/0.48 ! [P5: $o,Q2: $o] :
% 0.22/0.48 ( ( ( zero_n2684676970156552555ol_int @ P5 )
% 0.22/0.48 = ( zero_n2684676970156552555ol_int @ Q2 ) )
% 0.22/0.48 = ( P5 = Q2 ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_eq_iff
% 0.22/0.48 thf(fact_4184_of__bool__eq__iff,axiom,
% 0.22/0.48 ! [P5: $o,Q2: $o] :
% 0.22/0.48 ( ( ( zero_n356916108424825756nteger @ P5 )
% 0.22/0.48 = ( zero_n356916108424825756nteger @ Q2 ) )
% 0.22/0.48 = ( P5 = Q2 ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_eq_iff
% 0.22/0.48 thf(fact_4185_of__nat__mod,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_nat_mod
% 0.22/0.48 thf(fact_4186_of__nat__mod,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
% 0.22/0.48 = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_nat_mod
% 0.22/0.48 thf(fact_4187_of__nat__mod,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
% 0.22/0.48 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_nat_mod
% 0.22/0.48 thf(fact_4188_pochhammer__of__nat,axiom,
% 0.22/0.48 ! [X: nat,N: nat] :
% 0.22/0.48 ( ( comm_s2602460028002588243omplex @ ( semiri8010041392384452111omplex @ X ) @ N )
% 0.22/0.48 = ( semiri8010041392384452111omplex @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_of_nat
% 0.22/0.48 thf(fact_4189_pochhammer__of__nat,axiom,
% 0.22/0.48 ! [X: nat,N: nat] :
% 0.22/0.48 ( ( comm_s7457072308508201937r_real @ ( semiri5074537144036343181t_real @ X ) @ N )
% 0.22/0.48 = ( semiri5074537144036343181t_real @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_of_nat
% 0.22/0.48 thf(fact_4190_pochhammer__of__nat,axiom,
% 0.22/0.48 ! [X: nat,N: nat] :
% 0.22/0.48 ( ( comm_s4028243227959126397er_rat @ ( semiri681578069525770553at_rat @ X ) @ N )
% 0.22/0.48 = ( semiri681578069525770553at_rat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_of_nat
% 0.22/0.48 thf(fact_4191_pochhammer__of__nat,axiom,
% 0.22/0.48 ! [X: nat,N: nat] :
% 0.22/0.48 ( ( comm_s4663373288045622133er_nat @ ( semiri1316708129612266289at_nat @ X ) @ N )
% 0.22/0.48 = ( semiri1316708129612266289at_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_of_nat
% 0.22/0.48 thf(fact_4192_pochhammer__of__nat,axiom,
% 0.22/0.48 ! [X: nat,N: nat] :
% 0.22/0.48 ( ( comm_s4660882817536571857er_int @ ( semiri1314217659103216013at_int @ X ) @ N )
% 0.22/0.48 = ( semiri1314217659103216013at_int @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_of_nat
% 0.22/0.48 thf(fact_4193_mod__mult__right__eq,axiom,
% 0.22/0.48 ! [A: nat,B: nat,C: nat] :
% 0.22/0.48 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.22/0.48 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult_right_eq
% 0.22/0.48 thf(fact_4194_mod__mult__right__eq,axiom,
% 0.22/0.48 ! [A: int,B: int,C: int] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.22/0.48 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult_right_eq
% 0.22/0.48 thf(fact_4195_mod__mult__right__eq,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult_right_eq
% 0.22/0.48 thf(fact_4196_mod__mult__left__eq,axiom,
% 0.22/0.48 ! [A: nat,C: nat,B: nat] :
% 0.22/0.48 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 0.22/0.48 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult_left_eq
% 0.22/0.48 thf(fact_4197_mod__mult__left__eq,axiom,
% 0.22/0.48 ! [A: int,C: int,B: int] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 0.22/0.48 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult_left_eq
% 0.22/0.48 thf(fact_4198_mod__mult__left__eq,axiom,
% 0.22/0.48 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult_left_eq
% 0.22/0.48 thf(fact_4199_mult__mod__right,axiom,
% 0.22/0.48 ! [C: nat,A: nat,B: nat] :
% 0.22/0.48 ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 0.22/0.48 = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mult_mod_right
% 0.22/0.48 thf(fact_4200_mult__mod__right,axiom,
% 0.22/0.48 ! [C: int,A: int,B: int] :
% 0.22/0.48 ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 0.22/0.48 = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mult_mod_right
% 0.22/0.48 thf(fact_4201_mult__mod__right,axiom,
% 0.22/0.48 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.22/0.48 ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mult_mod_right
% 0.22/0.48 thf(fact_4202_mod__mult__mult2,axiom,
% 0.22/0.48 ! [A: nat,C: nat,B: nat] :
% 0.22/0.48 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.22/0.48 = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult_mult2
% 0.22/0.48 thf(fact_4203_mod__mult__mult2,axiom,
% 0.22/0.48 ! [A: int,C: int,B: int] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.22/0.48 = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult_mult2
% 0.22/0.48 thf(fact_4204_mod__mult__mult2,axiom,
% 0.22/0.48 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.22/0.48 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult_mult2
% 0.22/0.48 thf(fact_4205_mod__mult__cong,axiom,
% 0.22/0.48 ! [A: nat,C: nat,A6: nat,B: nat,B6: nat] :
% 0.22/0.48 ( ( ( modulo_modulo_nat @ A @ C )
% 0.22/0.48 = ( modulo_modulo_nat @ A6 @ C ) )
% 0.22/0.48 => ( ( ( modulo_modulo_nat @ B @ C )
% 0.22/0.48 = ( modulo_modulo_nat @ B6 @ C ) )
% 0.22/0.48 => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.22/0.48 = ( modulo_modulo_nat @ ( times_times_nat @ A6 @ B6 ) @ C ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult_cong
% 0.22/0.48 thf(fact_4206_mod__mult__cong,axiom,
% 0.22/0.48 ! [A: int,C: int,A6: int,B: int,B6: int] :
% 0.22/0.48 ( ( ( modulo_modulo_int @ A @ C )
% 0.22/0.48 = ( modulo_modulo_int @ A6 @ C ) )
% 0.22/0.48 => ( ( ( modulo_modulo_int @ B @ C )
% 0.22/0.48 = ( modulo_modulo_int @ B6 @ C ) )
% 0.22/0.48 => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
% 0.22/0.48 = ( modulo_modulo_int @ ( times_times_int @ A6 @ B6 ) @ C ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult_cong
% 0.22/0.48 thf(fact_4207_mod__mult__cong,axiom,
% 0.22/0.48 ! [A: code_integer,C: code_integer,A6: code_integer,B: code_integer,B6: code_integer] :
% 0.22/0.48 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ A6 @ C ) )
% 0.22/0.48 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ B6 @ C ) )
% 0.22/0.48 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A6 @ B6 ) @ C ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult_cong
% 0.22/0.48 thf(fact_4208_mod__mult__eq,axiom,
% 0.22/0.48 ! [A: nat,C: nat,B: nat] :
% 0.22/0.48 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.22/0.48 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult_eq
% 0.22/0.48 thf(fact_4209_mod__mult__eq,axiom,
% 0.22/0.48 ! [A: int,C: int,B: int] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.22/0.48 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult_eq
% 0.22/0.48 thf(fact_4210_mod__mult__eq,axiom,
% 0.22/0.48 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult_eq
% 0.22/0.48 thf(fact_4211_mod__add__right__eq,axiom,
% 0.22/0.48 ! [A: nat,B: nat,C: nat] :
% 0.22/0.48 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.22/0.48 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_add_right_eq
% 0.22/0.48 thf(fact_4212_mod__add__right__eq,axiom,
% 0.22/0.48 ! [A: int,B: int,C: int] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.22/0.48 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_add_right_eq
% 0.22/0.48 thf(fact_4213_mod__add__right__eq,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_add_right_eq
% 0.22/0.48 thf(fact_4214_mod__add__left__eq,axiom,
% 0.22/0.48 ! [A: nat,C: nat,B: nat] :
% 0.22/0.48 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 0.22/0.48 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_add_left_eq
% 0.22/0.48 thf(fact_4215_mod__add__left__eq,axiom,
% 0.22/0.48 ! [A: int,C: int,B: int] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 0.22/0.48 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_add_left_eq
% 0.22/0.48 thf(fact_4216_mod__add__left__eq,axiom,
% 0.22/0.48 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_add_left_eq
% 0.22/0.48 thf(fact_4217_mod__add__cong,axiom,
% 0.22/0.48 ! [A: nat,C: nat,A6: nat,B: nat,B6: nat] :
% 0.22/0.48 ( ( ( modulo_modulo_nat @ A @ C )
% 0.22/0.48 = ( modulo_modulo_nat @ A6 @ C ) )
% 0.22/0.48 => ( ( ( modulo_modulo_nat @ B @ C )
% 0.22/0.48 = ( modulo_modulo_nat @ B6 @ C ) )
% 0.22/0.48 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.22/0.48 = ( modulo_modulo_nat @ ( plus_plus_nat @ A6 @ B6 ) @ C ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_add_cong
% 0.22/0.48 thf(fact_4218_mod__add__cong,axiom,
% 0.22/0.48 ! [A: int,C: int,A6: int,B: int,B6: int] :
% 0.22/0.48 ( ( ( modulo_modulo_int @ A @ C )
% 0.22/0.48 = ( modulo_modulo_int @ A6 @ C ) )
% 0.22/0.48 => ( ( ( modulo_modulo_int @ B @ C )
% 0.22/0.48 = ( modulo_modulo_int @ B6 @ C ) )
% 0.22/0.48 => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.22/0.48 = ( modulo_modulo_int @ ( plus_plus_int @ A6 @ B6 ) @ C ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_add_cong
% 0.22/0.48 thf(fact_4219_mod__add__cong,axiom,
% 0.22/0.48 ! [A: code_integer,C: code_integer,A6: code_integer,B: code_integer,B6: code_integer] :
% 0.22/0.48 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ A6 @ C ) )
% 0.22/0.48 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ B6 @ C ) )
% 0.22/0.48 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A6 @ B6 ) @ C ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_add_cong
% 0.22/0.48 thf(fact_4220_mod__add__eq,axiom,
% 0.22/0.48 ! [A: nat,C: nat,B: nat] :
% 0.22/0.48 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.22/0.48 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_add_eq
% 0.22/0.48 thf(fact_4221_mod__add__eq,axiom,
% 0.22/0.48 ! [A: int,C: int,B: int] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.22/0.48 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_add_eq
% 0.22/0.48 thf(fact_4222_mod__add__eq,axiom,
% 0.22/0.48 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_add_eq
% 0.22/0.48 thf(fact_4223_mod__diff__right__eq,axiom,
% 0.22/0.48 ! [A: int,B: int,C: int] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.22/0.48 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_diff_right_eq
% 0.22/0.48 thf(fact_4224_mod__diff__right__eq,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_diff_right_eq
% 0.22/0.48 thf(fact_4225_mod__diff__left__eq,axiom,
% 0.22/0.48 ! [A: int,C: int,B: int] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 0.22/0.48 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_diff_left_eq
% 0.22/0.48 thf(fact_4226_mod__diff__left__eq,axiom,
% 0.22/0.48 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_diff_left_eq
% 0.22/0.48 thf(fact_4227_mod__diff__cong,axiom,
% 0.22/0.48 ! [A: int,C: int,A6: int,B: int,B6: int] :
% 0.22/0.48 ( ( ( modulo_modulo_int @ A @ C )
% 0.22/0.48 = ( modulo_modulo_int @ A6 @ C ) )
% 0.22/0.48 => ( ( ( modulo_modulo_int @ B @ C )
% 0.22/0.48 = ( modulo_modulo_int @ B6 @ C ) )
% 0.22/0.48 => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.22/0.48 = ( modulo_modulo_int @ ( minus_minus_int @ A6 @ B6 ) @ C ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_diff_cong
% 0.22/0.48 thf(fact_4228_mod__diff__cong,axiom,
% 0.22/0.48 ! [A: code_integer,C: code_integer,A6: code_integer,B: code_integer,B6: code_integer] :
% 0.22/0.48 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ A6 @ C ) )
% 0.22/0.48 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ B6 @ C ) )
% 0.22/0.48 => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A6 @ B6 ) @ C ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_diff_cong
% 0.22/0.48 thf(fact_4229_mod__diff__eq,axiom,
% 0.22/0.48 ! [A: int,C: int,B: int] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.22/0.48 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_diff_eq
% 0.22/0.48 thf(fact_4230_mod__diff__eq,axiom,
% 0.22/0.48 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_diff_eq
% 0.22/0.48 thf(fact_4231_of__bool__conj,axiom,
% 0.22/0.48 ! [P: $o,Q: $o] :
% 0.22/0.48 ( ( zero_n3304061248610475627l_real
% 0.22/0.48 @ ( P
% 0.22/0.48 & Q ) )
% 0.22/0.48 = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_conj
% 0.22/0.48 thf(fact_4232_of__bool__conj,axiom,
% 0.22/0.48 ! [P: $o,Q: $o] :
% 0.22/0.48 ( ( zero_n2052037380579107095ol_rat
% 0.22/0.48 @ ( P
% 0.22/0.48 & Q ) )
% 0.22/0.48 = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_conj
% 0.22/0.48 thf(fact_4233_of__bool__conj,axiom,
% 0.22/0.48 ! [P: $o,Q: $o] :
% 0.22/0.48 ( ( zero_n2687167440665602831ol_nat
% 0.22/0.48 @ ( P
% 0.22/0.48 & Q ) )
% 0.22/0.48 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_conj
% 0.22/0.48 thf(fact_4234_of__bool__conj,axiom,
% 0.22/0.48 ! [P: $o,Q: $o] :
% 0.22/0.48 ( ( zero_n2684676970156552555ol_int
% 0.22/0.48 @ ( P
% 0.22/0.48 & Q ) )
% 0.22/0.48 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_conj
% 0.22/0.48 thf(fact_4235_of__bool__conj,axiom,
% 0.22/0.48 ! [P: $o,Q: $o] :
% 0.22/0.48 ( ( zero_n356916108424825756nteger
% 0.22/0.48 @ ( P
% 0.22/0.48 & Q ) )
% 0.22/0.48 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_conj
% 0.22/0.48 thf(fact_4236_power__mod,axiom,
% 0.22/0.48 ! [A: nat,B: nat,N: nat] :
% 0.22/0.48 ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N ) @ B )
% 0.22/0.48 = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B ) ) ).
% 0.22/0.48
% 0.22/0.48 % power_mod
% 0.22/0.48 thf(fact_4237_power__mod,axiom,
% 0.22/0.48 ! [A: int,B: int,N: nat] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N ) @ B )
% 0.22/0.48 = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B ) ) ).
% 0.22/0.48
% 0.22/0.48 % power_mod
% 0.22/0.48 thf(fact_4238_power__mod,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer,N: nat] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N ) @ B )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B ) ) ).
% 0.22/0.48
% 0.22/0.48 % power_mod
% 0.22/0.48 thf(fact_4239_dvd__mod,axiom,
% 0.22/0.48 ! [K: nat,M: nat,N: nat] :
% 0.22/0.48 ( ( dvd_dvd_nat @ K @ M )
% 0.22/0.48 => ( ( dvd_dvd_nat @ K @ N )
% 0.22/0.48 => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % dvd_mod
% 0.22/0.48 thf(fact_4240_dvd__mod,axiom,
% 0.22/0.48 ! [K: int,M: int,N: int] :
% 0.22/0.48 ( ( dvd_dvd_int @ K @ M )
% 0.22/0.48 => ( ( dvd_dvd_int @ K @ N )
% 0.22/0.48 => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % dvd_mod
% 0.22/0.48 thf(fact_4241_dvd__mod,axiom,
% 0.22/0.48 ! [K: code_integer,M: code_integer,N: code_integer] :
% 0.22/0.48 ( ( dvd_dvd_Code_integer @ K @ M )
% 0.22/0.48 => ( ( dvd_dvd_Code_integer @ K @ N )
% 0.22/0.48 => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % dvd_mod
% 0.22/0.48 thf(fact_4242_mod__mod__cancel,axiom,
% 0.22/0.48 ! [C: nat,B: nat,A: nat] :
% 0.22/0.48 ( ( dvd_dvd_nat @ C @ B )
% 0.22/0.48 => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ C )
% 0.22/0.48 = ( modulo_modulo_nat @ A @ C ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mod_cancel
% 0.22/0.48 thf(fact_4243_mod__mod__cancel,axiom,
% 0.22/0.48 ! [C: int,B: int,A: int] :
% 0.22/0.48 ( ( dvd_dvd_int @ C @ B )
% 0.22/0.48 => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ C )
% 0.22/0.48 = ( modulo_modulo_int @ A @ C ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mod_cancel
% 0.22/0.48 thf(fact_4244_mod__mod__cancel,axiom,
% 0.22/0.48 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.22/0.48 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.22/0.48 => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ A @ C ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mod_cancel
% 0.22/0.48 thf(fact_4245_dvd__mod__iff,axiom,
% 0.22/0.48 ! [C: nat,B: nat,A: nat] :
% 0.22/0.48 ( ( dvd_dvd_nat @ C @ B )
% 0.22/0.48 => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 0.22/0.48 = ( dvd_dvd_nat @ C @ A ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % dvd_mod_iff
% 0.22/0.48 thf(fact_4246_dvd__mod__iff,axiom,
% 0.22/0.48 ! [C: int,B: int,A: int] :
% 0.22/0.48 ( ( dvd_dvd_int @ C @ B )
% 0.22/0.48 => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 0.22/0.48 = ( dvd_dvd_int @ C @ A ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % dvd_mod_iff
% 0.22/0.48 thf(fact_4247_dvd__mod__iff,axiom,
% 0.22/0.48 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.22/0.48 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.22/0.48 => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.22/0.48 = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % dvd_mod_iff
% 0.22/0.48 thf(fact_4248_dvd__mod__imp__dvd,axiom,
% 0.22/0.48 ! [C: nat,A: nat,B: nat] :
% 0.22/0.48 ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 0.22/0.48 => ( ( dvd_dvd_nat @ C @ B )
% 0.22/0.48 => ( dvd_dvd_nat @ C @ A ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % dvd_mod_imp_dvd
% 0.22/0.48 thf(fact_4249_dvd__mod__imp__dvd,axiom,
% 0.22/0.48 ! [C: int,A: int,B: int] :
% 0.22/0.48 ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 0.22/0.48 => ( ( dvd_dvd_int @ C @ B )
% 0.22/0.48 => ( dvd_dvd_int @ C @ A ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % dvd_mod_imp_dvd
% 0.22/0.48 thf(fact_4250_dvd__mod__imp__dvd,axiom,
% 0.22/0.48 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.22/0.48 ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.22/0.48 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.22/0.48 => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % dvd_mod_imp_dvd
% 0.22/0.48 thf(fact_4251_mod__Suc__Suc__eq,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
% 0.22/0.48 = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_Suc_Suc_eq
% 0.22/0.48 thf(fact_4252_mod__Suc__eq,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
% 0.22/0.48 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_Suc_eq
% 0.22/0.48 thf(fact_4253_mod__less__eq__dividend,axiom,
% 0.22/0.48 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% 0.22/0.48
% 0.22/0.48 % mod_less_eq_dividend
% 0.22/0.48 thf(fact_4254_of__bool__odd__eq__mod__2,axiom,
% 0.22/0.48 ! [A: nat] :
% 0.22/0.48 ( ( zero_n2687167440665602831ol_nat
% 0.22/0.48 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.22/0.48 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_odd_eq_mod_2
% 0.22/0.48 thf(fact_4255_of__bool__odd__eq__mod__2,axiom,
% 0.22/0.48 ! [A: int] :
% 0.22/0.48 ( ( zero_n2684676970156552555ol_int
% 0.22/0.48 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.22/0.48 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_odd_eq_mod_2
% 0.22/0.48 thf(fact_4256_of__bool__odd__eq__mod__2,axiom,
% 0.22/0.48 ! [A: code_integer] :
% 0.22/0.48 ( ( zero_n356916108424825756nteger
% 0.22/0.48 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.22/0.48 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_odd_eq_mod_2
% 0.22/0.48 thf(fact_4257_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer] :
% 0.22/0.48 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.22/0.48 => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).
% 0.22/0.48
% 0.22/0.48 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.22/0.48 thf(fact_4258_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.22/0.48 ! [A: nat,B: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.22/0.48 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% 0.22/0.48
% 0.22/0.48 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.22/0.48 thf(fact_4259_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.22/0.48 ! [A: int,B: int] :
% 0.22/0.48 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.22/0.48 => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% 0.22/0.48
% 0.22/0.48 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.22/0.48 thf(fact_4260_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.22/0.48 ! [B: nat,A: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.22/0.48 => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% 0.22/0.48
% 0.22/0.48 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.22/0.48 thf(fact_4261_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.22/0.48 ! [B: int,A: int] :
% 0.22/0.48 ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.48 => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% 0.22/0.48
% 0.22/0.48 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.22/0.48 thf(fact_4262_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.22/0.48 ! [B: code_integer,A: code_integer] :
% 0.22/0.48 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.22/0.48 => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).
% 0.22/0.48
% 0.22/0.48 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.22/0.48 thf(fact_4263_zero__less__eq__of__bool,axiom,
% 0.22/0.48 ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.22/0.48
% 0.22/0.48 % zero_less_eq_of_bool
% 0.22/0.48 thf(fact_4264_zero__less__eq__of__bool,axiom,
% 0.22/0.48 ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.22/0.48
% 0.22/0.48 % zero_less_eq_of_bool
% 0.22/0.48 thf(fact_4265_zero__less__eq__of__bool,axiom,
% 0.22/0.48 ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.22/0.48
% 0.22/0.48 % zero_less_eq_of_bool
% 0.22/0.48 thf(fact_4266_zero__less__eq__of__bool,axiom,
% 0.22/0.48 ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.22/0.48
% 0.22/0.48 % zero_less_eq_of_bool
% 0.22/0.48 thf(fact_4267_zero__less__eq__of__bool,axiom,
% 0.22/0.48 ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).
% 0.22/0.48
% 0.22/0.48 % zero_less_eq_of_bool
% 0.22/0.48 thf(fact_4268_mod__eq__self__iff__div__eq__0,axiom,
% 0.22/0.48 ! [A: nat,B: nat] :
% 0.22/0.48 ( ( ( modulo_modulo_nat @ A @ B )
% 0.22/0.48 = A )
% 0.22/0.48 = ( ( divide_divide_nat @ A @ B )
% 0.22/0.48 = zero_zero_nat ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_eq_self_iff_div_eq_0
% 0.22/0.48 thf(fact_4269_mod__eq__self__iff__div__eq__0,axiom,
% 0.22/0.48 ! [A: int,B: int] :
% 0.22/0.48 ( ( ( modulo_modulo_int @ A @ B )
% 0.22/0.48 = A )
% 0.22/0.48 = ( ( divide_divide_int @ A @ B )
% 0.22/0.48 = zero_zero_int ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_eq_self_iff_div_eq_0
% 0.22/0.48 thf(fact_4270_mod__eq__self__iff__div__eq__0,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer] :
% 0.22/0.48 ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.22/0.48 = A )
% 0.22/0.48 = ( ( divide6298287555418463151nteger @ A @ B )
% 0.22/0.48 = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_eq_self_iff_div_eq_0
% 0.22/0.48 thf(fact_4271_cong__exp__iff__simps_I9_J,axiom,
% 0.22/0.48 ! [M: num,Q2: num,N: num] :
% 0.22/0.48 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.22/0.48 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.22/0.48 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(9)
% 0.22/0.48 thf(fact_4272_cong__exp__iff__simps_I9_J,axiom,
% 0.22/0.48 ! [M: num,Q2: num,N: num] :
% 0.22/0.48 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.22/0.48 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 0.22/0.48 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(9)
% 0.22/0.48 thf(fact_4273_cong__exp__iff__simps_I9_J,axiom,
% 0.22/0.48 ! [M: num,Q2: num,N: num] :
% 0.22/0.48 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.22/0.48 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(9)
% 0.22/0.48 thf(fact_4274_cong__exp__iff__simps_I4_J,axiom,
% 0.22/0.48 ! [M: num,N: num] :
% 0.22/0.48 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
% 0.22/0.48 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(4)
% 0.22/0.48 thf(fact_4275_cong__exp__iff__simps_I4_J,axiom,
% 0.22/0.48 ! [M: num,N: num] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
% 0.22/0.48 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(4)
% 0.22/0.48 thf(fact_4276_cong__exp__iff__simps_I4_J,axiom,
% 0.22/0.48 ! [M: num,N: num] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(4)
% 0.22/0.48 thf(fact_4277_mod__eqE,axiom,
% 0.22/0.48 ! [A: int,C: int,B: int] :
% 0.22/0.48 ( ( ( modulo_modulo_int @ A @ C )
% 0.22/0.48 = ( modulo_modulo_int @ B @ C ) )
% 0.22/0.48 => ~ ! [D2: int] :
% 0.22/0.48 ( B
% 0.22/0.48 != ( plus_plus_int @ A @ ( times_times_int @ C @ D2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_eqE
% 0.22/0.48 thf(fact_4278_mod__eqE,axiom,
% 0.22/0.48 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.22/0.48 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ B @ C ) )
% 0.22/0.48 => ~ ! [D2: code_integer] :
% 0.22/0.48 ( B
% 0.22/0.48 != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_eqE
% 0.22/0.48 thf(fact_4279_mod__eq__0__iff__dvd,axiom,
% 0.22/0.48 ! [A: nat,B: nat] :
% 0.22/0.48 ( ( ( modulo_modulo_nat @ A @ B )
% 0.22/0.48 = zero_zero_nat )
% 0.22/0.48 = ( dvd_dvd_nat @ B @ A ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_eq_0_iff_dvd
% 0.22/0.48 thf(fact_4280_mod__eq__0__iff__dvd,axiom,
% 0.22/0.48 ! [A: int,B: int] :
% 0.22/0.48 ( ( ( modulo_modulo_int @ A @ B )
% 0.22/0.48 = zero_zero_int )
% 0.22/0.48 = ( dvd_dvd_int @ B @ A ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_eq_0_iff_dvd
% 0.22/0.48 thf(fact_4281_mod__eq__0__iff__dvd,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer] :
% 0.22/0.48 ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.22/0.48 = zero_z3403309356797280102nteger )
% 0.22/0.48 = ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_eq_0_iff_dvd
% 0.22/0.48 thf(fact_4282_dvd__eq__mod__eq__0,axiom,
% 0.22/0.48 ( dvd_dvd_nat
% 0.22/0.48 = ( ^ [A3: nat,B2: nat] :
% 0.22/0.48 ( ( modulo_modulo_nat @ B2 @ A3 )
% 0.22/0.48 = zero_zero_nat ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % dvd_eq_mod_eq_0
% 0.22/0.48 thf(fact_4283_dvd__eq__mod__eq__0,axiom,
% 0.22/0.48 ( dvd_dvd_int
% 0.22/0.48 = ( ^ [A3: int,B2: int] :
% 0.22/0.48 ( ( modulo_modulo_int @ B2 @ A3 )
% 0.22/0.48 = zero_zero_int ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % dvd_eq_mod_eq_0
% 0.22/0.48 thf(fact_4284_dvd__eq__mod__eq__0,axiom,
% 0.22/0.48 ( dvd_dvd_Code_integer
% 0.22/0.48 = ( ^ [A3: code_integer,B2: code_integer] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ B2 @ A3 )
% 0.22/0.48 = zero_z3403309356797280102nteger ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % dvd_eq_mod_eq_0
% 0.22/0.48 thf(fact_4285_mod__0__imp__dvd,axiom,
% 0.22/0.48 ! [A: nat,B: nat] :
% 0.22/0.48 ( ( ( modulo_modulo_nat @ A @ B )
% 0.22/0.48 = zero_zero_nat )
% 0.22/0.48 => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_0_imp_dvd
% 0.22/0.48 thf(fact_4286_mod__0__imp__dvd,axiom,
% 0.22/0.48 ! [A: int,B: int] :
% 0.22/0.48 ( ( ( modulo_modulo_int @ A @ B )
% 0.22/0.48 = zero_zero_int )
% 0.22/0.48 => ( dvd_dvd_int @ B @ A ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_0_imp_dvd
% 0.22/0.48 thf(fact_4287_mod__0__imp__dvd,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer] :
% 0.22/0.48 ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.22/0.48 = zero_z3403309356797280102nteger )
% 0.22/0.48 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_0_imp_dvd
% 0.22/0.48 thf(fact_4288_div__add1__eq,axiom,
% 0.22/0.48 ! [A: nat,B: nat,C: nat] :
% 0.22/0.48 ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.22/0.48 = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % div_add1_eq
% 0.22/0.48 thf(fact_4289_div__add1__eq,axiom,
% 0.22/0.48 ! [A: int,B: int,C: int] :
% 0.22/0.48 ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.22/0.48 = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % div_add1_eq
% 0.22/0.48 thf(fact_4290_div__add1__eq,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.22/0.48 ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.22/0.48 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % div_add1_eq
% 0.22/0.48 thf(fact_4291_of__bool__less__eq__one,axiom,
% 0.22/0.48 ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_less_eq_one
% 0.22/0.48 thf(fact_4292_of__bool__less__eq__one,axiom,
% 0.22/0.48 ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_less_eq_one
% 0.22/0.48 thf(fact_4293_of__bool__less__eq__one,axiom,
% 0.22/0.48 ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_less_eq_one
% 0.22/0.48 thf(fact_4294_of__bool__less__eq__one,axiom,
% 0.22/0.48 ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_less_eq_one
% 0.22/0.48 thf(fact_4295_of__bool__less__eq__one,axiom,
% 0.22/0.48 ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_less_eq_one
% 0.22/0.48 thf(fact_4296_split__of__bool__asm,axiom,
% 0.22/0.48 ! [P: complex > $o,P5: $o] :
% 0.22/0.48 ( ( P @ ( zero_n1201886186963655149omplex @ P5 ) )
% 0.22/0.48 = ( ~ ( ( P5
% 0.22/0.48 & ~ ( P @ one_one_complex ) )
% 0.22/0.48 | ( ~ P5
% 0.22/0.48 & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % split_of_bool_asm
% 0.22/0.48 thf(fact_4297_split__of__bool__asm,axiom,
% 0.22/0.48 ! [P: real > $o,P5: $o] :
% 0.22/0.48 ( ( P @ ( zero_n3304061248610475627l_real @ P5 ) )
% 0.22/0.48 = ( ~ ( ( P5
% 0.22/0.48 & ~ ( P @ one_one_real ) )
% 0.22/0.48 | ( ~ P5
% 0.22/0.48 & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % split_of_bool_asm
% 0.22/0.48 thf(fact_4298_split__of__bool__asm,axiom,
% 0.22/0.48 ! [P: rat > $o,P5: $o] :
% 0.22/0.48 ( ( P @ ( zero_n2052037380579107095ol_rat @ P5 ) )
% 0.22/0.48 = ( ~ ( ( P5
% 0.22/0.48 & ~ ( P @ one_one_rat ) )
% 0.22/0.48 | ( ~ P5
% 0.22/0.48 & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % split_of_bool_asm
% 0.22/0.48 thf(fact_4299_split__of__bool__asm,axiom,
% 0.22/0.48 ! [P: nat > $o,P5: $o] :
% 0.22/0.48 ( ( P @ ( zero_n2687167440665602831ol_nat @ P5 ) )
% 0.22/0.48 = ( ~ ( ( P5
% 0.22/0.48 & ~ ( P @ one_one_nat ) )
% 0.22/0.48 | ( ~ P5
% 0.22/0.48 & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % split_of_bool_asm
% 0.22/0.48 thf(fact_4300_split__of__bool__asm,axiom,
% 0.22/0.48 ! [P: int > $o,P5: $o] :
% 0.22/0.48 ( ( P @ ( zero_n2684676970156552555ol_int @ P5 ) )
% 0.22/0.48 = ( ~ ( ( P5
% 0.22/0.48 & ~ ( P @ one_one_int ) )
% 0.22/0.48 | ( ~ P5
% 0.22/0.48 & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % split_of_bool_asm
% 0.22/0.48 thf(fact_4301_split__of__bool__asm,axiom,
% 0.22/0.48 ! [P: code_integer > $o,P5: $o] :
% 0.22/0.48 ( ( P @ ( zero_n356916108424825756nteger @ P5 ) )
% 0.22/0.48 = ( ~ ( ( P5
% 0.22/0.48 & ~ ( P @ one_one_Code_integer ) )
% 0.22/0.48 | ( ~ P5
% 0.22/0.48 & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % split_of_bool_asm
% 0.22/0.48 thf(fact_4302_split__of__bool,axiom,
% 0.22/0.48 ! [P: complex > $o,P5: $o] :
% 0.22/0.48 ( ( P @ ( zero_n1201886186963655149omplex @ P5 ) )
% 0.22/0.48 = ( ( P5
% 0.22/0.48 => ( P @ one_one_complex ) )
% 0.22/0.48 & ( ~ P5
% 0.22/0.48 => ( P @ zero_zero_complex ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % split_of_bool
% 0.22/0.48 thf(fact_4303_split__of__bool,axiom,
% 0.22/0.48 ! [P: real > $o,P5: $o] :
% 0.22/0.48 ( ( P @ ( zero_n3304061248610475627l_real @ P5 ) )
% 0.22/0.48 = ( ( P5
% 0.22/0.48 => ( P @ one_one_real ) )
% 0.22/0.48 & ( ~ P5
% 0.22/0.48 => ( P @ zero_zero_real ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % split_of_bool
% 0.22/0.48 thf(fact_4304_split__of__bool,axiom,
% 0.22/0.48 ! [P: rat > $o,P5: $o] :
% 0.22/0.48 ( ( P @ ( zero_n2052037380579107095ol_rat @ P5 ) )
% 0.22/0.48 = ( ( P5
% 0.22/0.48 => ( P @ one_one_rat ) )
% 0.22/0.48 & ( ~ P5
% 0.22/0.48 => ( P @ zero_zero_rat ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % split_of_bool
% 0.22/0.48 thf(fact_4305_split__of__bool,axiom,
% 0.22/0.48 ! [P: nat > $o,P5: $o] :
% 0.22/0.48 ( ( P @ ( zero_n2687167440665602831ol_nat @ P5 ) )
% 0.22/0.48 = ( ( P5
% 0.22/0.48 => ( P @ one_one_nat ) )
% 0.22/0.48 & ( ~ P5
% 0.22/0.48 => ( P @ zero_zero_nat ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % split_of_bool
% 0.22/0.48 thf(fact_4306_split__of__bool,axiom,
% 0.22/0.48 ! [P: int > $o,P5: $o] :
% 0.22/0.48 ( ( P @ ( zero_n2684676970156552555ol_int @ P5 ) )
% 0.22/0.48 = ( ( P5
% 0.22/0.48 => ( P @ one_one_int ) )
% 0.22/0.48 & ( ~ P5
% 0.22/0.48 => ( P @ zero_zero_int ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % split_of_bool
% 0.22/0.48 thf(fact_4307_split__of__bool,axiom,
% 0.22/0.48 ! [P: code_integer > $o,P5: $o] :
% 0.22/0.48 ( ( P @ ( zero_n356916108424825756nteger @ P5 ) )
% 0.22/0.48 = ( ( P5
% 0.22/0.48 => ( P @ one_one_Code_integer ) )
% 0.22/0.48 & ( ~ P5
% 0.22/0.48 => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % split_of_bool
% 0.22/0.48 thf(fact_4308_of__bool__def,axiom,
% 0.22/0.48 ( zero_n1201886186963655149omplex
% 0.22/0.48 = ( ^ [P6: $o] : ( if_complex @ P6 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_def
% 0.22/0.48 thf(fact_4309_of__bool__def,axiom,
% 0.22/0.48 ( zero_n3304061248610475627l_real
% 0.22/0.48 = ( ^ [P6: $o] : ( if_real @ P6 @ one_one_real @ zero_zero_real ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_def
% 0.22/0.48 thf(fact_4310_of__bool__def,axiom,
% 0.22/0.48 ( zero_n2052037380579107095ol_rat
% 0.22/0.48 = ( ^ [P6: $o] : ( if_rat @ P6 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_def
% 0.22/0.48 thf(fact_4311_of__bool__def,axiom,
% 0.22/0.48 ( zero_n2687167440665602831ol_nat
% 0.22/0.48 = ( ^ [P6: $o] : ( if_nat @ P6 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_def
% 0.22/0.48 thf(fact_4312_of__bool__def,axiom,
% 0.22/0.48 ( zero_n2684676970156552555ol_int
% 0.22/0.48 = ( ^ [P6: $o] : ( if_int @ P6 @ one_one_int @ zero_zero_int ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_def
% 0.22/0.48 thf(fact_4313_of__bool__def,axiom,
% 0.22/0.48 ( zero_n356916108424825756nteger
% 0.22/0.48 = ( ^ [P6: $o] : ( if_Code_integer @ P6 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_bool_def
% 0.22/0.48 thf(fact_4314_mod__eq__dvd__iff,axiom,
% 0.22/0.48 ! [A: int,C: int,B: int] :
% 0.22/0.48 ( ( ( modulo_modulo_int @ A @ C )
% 0.22/0.48 = ( modulo_modulo_int @ B @ C ) )
% 0.22/0.48 = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_eq_dvd_iff
% 0.22/0.48 thf(fact_4315_mod__eq__dvd__iff,axiom,
% 0.22/0.48 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.22/0.48 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.22/0.48 = ( modulo364778990260209775nteger @ B @ C ) )
% 0.22/0.48 = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_eq_dvd_iff
% 0.22/0.48 thf(fact_4316_dvd__minus__mod,axiom,
% 0.22/0.48 ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % dvd_minus_mod
% 0.22/0.48 thf(fact_4317_dvd__minus__mod,axiom,
% 0.22/0.48 ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % dvd_minus_mod
% 0.22/0.48 thf(fact_4318_dvd__minus__mod,axiom,
% 0.22/0.48 ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % dvd_minus_mod
% 0.22/0.48 thf(fact_4319_mod__Suc,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 0.22/0.48 = N )
% 0.22/0.48 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 0.22/0.48 = zero_zero_nat ) )
% 0.22/0.48 & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 0.22/0.48 != N )
% 0.22/0.48 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 0.22/0.48 = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_Suc
% 0.22/0.48 thf(fact_4320_mod__induct,axiom,
% 0.22/0.48 ! [P: nat > $o,N: nat,P5: nat,M: nat] :
% 0.22/0.48 ( ( P @ N )
% 0.22/0.48 => ( ( ord_less_nat @ N @ P5 )
% 0.22/0.48 => ( ( ord_less_nat @ M @ P5 )
% 0.22/0.48 => ( ! [N2: nat] :
% 0.22/0.48 ( ( ord_less_nat @ N2 @ P5 )
% 0.22/0.48 => ( ( P @ N2 )
% 0.22/0.48 => ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P5 ) ) ) )
% 0.22/0.48 => ( P @ M ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_induct
% 0.22/0.48 thf(fact_4321_mod__less__divisor,axiom,
% 0.22/0.48 ! [N: nat,M: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.48 => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_less_divisor
% 0.22/0.48 thf(fact_4322_gcd__nat__induct,axiom,
% 0.22/0.48 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.22/0.48 ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
% 0.22/0.48 => ( ! [M4: nat,N2: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.22/0.48 => ( ( P @ N2 @ ( modulo_modulo_nat @ M4 @ N2 ) )
% 0.22/0.48 => ( P @ M4 @ N2 ) ) )
% 0.22/0.48 => ( P @ M @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % gcd_nat_induct
% 0.22/0.48 thf(fact_4323_mod__Suc__le__divisor,axiom,
% 0.22/0.48 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% 0.22/0.48
% 0.22/0.48 % mod_Suc_le_divisor
% 0.22/0.48 thf(fact_4324_mod__eq__0D,axiom,
% 0.22/0.48 ! [M: nat,D: nat] :
% 0.22/0.48 ( ( ( modulo_modulo_nat @ M @ D )
% 0.22/0.48 = zero_zero_nat )
% 0.22/0.48 => ? [Q4: nat] :
% 0.22/0.48 ( M
% 0.22/0.48 = ( times_times_nat @ D @ Q4 ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_eq_0D
% 0.22/0.48 thf(fact_4325_mod__if,axiom,
% 0.22/0.48 ( modulo_modulo_nat
% 0.22/0.48 = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( ord_less_nat @ M5 @ N4 ) @ M5 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M5 @ N4 ) @ N4 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_if
% 0.22/0.48 thf(fact_4326_mod__geq,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ~ ( ord_less_nat @ M @ N )
% 0.22/0.48 => ( ( modulo_modulo_nat @ M @ N )
% 0.22/0.48 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_geq
% 0.22/0.48 thf(fact_4327_le__mod__geq,axiom,
% 0.22/0.48 ! [N: nat,M: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.48 => ( ( modulo_modulo_nat @ M @ N )
% 0.22/0.48 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % le_mod_geq
% 0.22/0.48 thf(fact_4328_nat__mod__eq__iff,axiom,
% 0.22/0.48 ! [X: nat,N: nat,Y4: nat] :
% 0.22/0.48 ( ( ( modulo_modulo_nat @ X @ N )
% 0.22/0.48 = ( modulo_modulo_nat @ Y4 @ N ) )
% 0.22/0.48 = ( ? [Q1: nat,Q22: nat] :
% 0.22/0.48 ( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
% 0.22/0.48 = ( plus_plus_nat @ Y4 @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % nat_mod_eq_iff
% 0.22/0.48 thf(fact_4329_pochhammer__pos,axiom,
% 0.22/0.48 ! [X: real,N: nat] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_pos
% 0.22/0.48 thf(fact_4330_pochhammer__pos,axiom,
% 0.22/0.48 ! [X: rat,N: nat] :
% 0.22/0.48 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.22/0.48 => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_pos
% 0.22/0.48 thf(fact_4331_pochhammer__pos,axiom,
% 0.22/0.48 ! [X: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.22/0.48 => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_pos
% 0.22/0.48 thf(fact_4332_pochhammer__pos,axiom,
% 0.22/0.48 ! [X: int,N: nat] :
% 0.22/0.48 ( ( ord_less_int @ zero_zero_int @ X )
% 0.22/0.48 => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_pos
% 0.22/0.48 thf(fact_4333_pochhammer__eq__0__mono,axiom,
% 0.22/0.48 ! [A: complex,N: nat,M: nat] :
% 0.22/0.48 ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.22/0.48 = zero_zero_complex )
% 0.22/0.48 => ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.48 => ( ( comm_s2602460028002588243omplex @ A @ M )
% 0.22/0.48 = zero_zero_complex ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_eq_0_mono
% 0.22/0.48 thf(fact_4334_pochhammer__eq__0__mono,axiom,
% 0.22/0.48 ! [A: real,N: nat,M: nat] :
% 0.22/0.48 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.22/0.48 = zero_zero_real )
% 0.22/0.48 => ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.48 => ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.22/0.48 = zero_zero_real ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_eq_0_mono
% 0.22/0.48 thf(fact_4335_pochhammer__eq__0__mono,axiom,
% 0.22/0.48 ! [A: rat,N: nat,M: nat] :
% 0.22/0.48 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.22/0.48 = zero_zero_rat )
% 0.22/0.48 => ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.48 => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.22/0.48 = zero_zero_rat ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_eq_0_mono
% 0.22/0.48 thf(fact_4336_pochhammer__neq__0__mono,axiom,
% 0.22/0.48 ! [A: complex,M: nat,N: nat] :
% 0.22/0.48 ( ( ( comm_s2602460028002588243omplex @ A @ M )
% 0.22/0.48 != zero_zero_complex )
% 0.22/0.48 => ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.48 => ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.22/0.48 != zero_zero_complex ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_neq_0_mono
% 0.22/0.48 thf(fact_4337_pochhammer__neq__0__mono,axiom,
% 0.22/0.48 ! [A: real,M: nat,N: nat] :
% 0.22/0.48 ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.22/0.48 != zero_zero_real )
% 0.22/0.48 => ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.48 => ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.22/0.48 != zero_zero_real ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_neq_0_mono
% 0.22/0.48 thf(fact_4338_pochhammer__neq__0__mono,axiom,
% 0.22/0.48 ! [A: rat,M: nat,N: nat] :
% 0.22/0.48 ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.22/0.48 != zero_zero_rat )
% 0.22/0.48 => ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.48 => ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.22/0.48 != zero_zero_rat ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_neq_0_mono
% 0.22/0.48 thf(fact_4339_exp__mod__exp,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % exp_mod_exp
% 0.22/0.48 thf(fact_4340_exp__mod__exp,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % exp_mod_exp
% 0.22/0.48 thf(fact_4341_exp__mod__exp,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % exp_mod_exp
% 0.22/0.48 thf(fact_4342_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.22/0.48 ! [B: code_integer,A: code_integer] :
% 0.22/0.48 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.22/0.48 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.22/0.48 thf(fact_4343_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.22/0.48 ! [B: nat,A: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.22/0.48 => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.22/0.48 thf(fact_4344_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.22/0.48 ! [B: int,A: int] :
% 0.22/0.48 ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.48 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.22/0.48 thf(fact_4345_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer] :
% 0.22/0.48 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.22/0.48 => ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.22/0.48 => ( ( modulo364778990260209775nteger @ A @ B )
% 0.22/0.48 = A ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % unique_euclidean_semiring_numeral_class.mod_less
% 0.22/0.48 thf(fact_4346_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.22/0.48 ! [A: nat,B: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.22/0.48 => ( ( ord_less_nat @ A @ B )
% 0.22/0.48 => ( ( modulo_modulo_nat @ A @ B )
% 0.22/0.48 = A ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % unique_euclidean_semiring_numeral_class.mod_less
% 0.22/0.48 thf(fact_4347_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.22/0.48 ! [A: int,B: int] :
% 0.22/0.48 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.22/0.48 => ( ( ord_less_int @ A @ B )
% 0.22/0.48 => ( ( modulo_modulo_int @ A @ B )
% 0.22/0.48 = A ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % unique_euclidean_semiring_numeral_class.mod_less
% 0.22/0.48 thf(fact_4348_cong__exp__iff__simps_I2_J,axiom,
% 0.22/0.48 ! [N: num,Q2: num] :
% 0.22/0.48 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 = zero_zero_nat )
% 0.22/0.48 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.22/0.48 = zero_zero_nat ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(2)
% 0.22/0.48 thf(fact_4349_cong__exp__iff__simps_I2_J,axiom,
% 0.22/0.48 ! [N: num,Q2: num] :
% 0.22/0.48 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 = zero_zero_int )
% 0.22/0.48 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 0.22/0.48 = zero_zero_int ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(2)
% 0.22/0.48 thf(fact_4350_cong__exp__iff__simps_I2_J,axiom,
% 0.22/0.48 ! [N: num,Q2: num] :
% 0.22/0.48 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 = zero_z3403309356797280102nteger )
% 0.22/0.48 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.22/0.48 = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(2)
% 0.22/0.48 thf(fact_4351_cong__exp__iff__simps_I1_J,axiom,
% 0.22/0.48 ! [N: num] :
% 0.22/0.48 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
% 0.22/0.48 = zero_zero_nat ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(1)
% 0.22/0.48 thf(fact_4352_cong__exp__iff__simps_I1_J,axiom,
% 0.22/0.48 ! [N: num] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
% 0.22/0.48 = zero_zero_int ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(1)
% 0.22/0.48 thf(fact_4353_cong__exp__iff__simps_I1_J,axiom,
% 0.22/0.48 ! [N: num] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
% 0.22/0.48 = zero_z3403309356797280102nteger ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(1)
% 0.22/0.48 thf(fact_4354_cong__exp__iff__simps_I6_J,axiom,
% 0.22/0.48 ! [Q2: num,N: num] :
% 0.22/0.48 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(6)
% 0.22/0.48 thf(fact_4355_cong__exp__iff__simps_I6_J,axiom,
% 0.22/0.48 ! [Q2: num,N: num] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(6)
% 0.22/0.48 thf(fact_4356_cong__exp__iff__simps_I6_J,axiom,
% 0.22/0.48 ! [Q2: num,N: num] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(6)
% 0.22/0.48 thf(fact_4357_cong__exp__iff__simps_I8_J,axiom,
% 0.22/0.48 ! [M: num,Q2: num] :
% 0.22/0.48 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(8)
% 0.22/0.48 thf(fact_4358_cong__exp__iff__simps_I8_J,axiom,
% 0.22/0.48 ! [M: num,Q2: num] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(8)
% 0.22/0.48 thf(fact_4359_cong__exp__iff__simps_I8_J,axiom,
% 0.22/0.48 ! [M: num,Q2: num] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(8)
% 0.22/0.48 thf(fact_4360_div__mult1__eq,axiom,
% 0.22/0.48 ! [A: nat,B: nat,C: nat] :
% 0.22/0.48 ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.22/0.48 = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % div_mult1_eq
% 0.22/0.48 thf(fact_4361_div__mult1__eq,axiom,
% 0.22/0.48 ! [A: int,B: int,C: int] :
% 0.22/0.48 ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
% 0.22/0.48 = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % div_mult1_eq
% 0.22/0.48 thf(fact_4362_div__mult1__eq,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.22/0.48 ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.22/0.48 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % div_mult1_eq
% 0.22/0.48 thf(fact_4363_mult__div__mod__eq,axiom,
% 0.22/0.48 ! [B: nat,A: nat] :
% 0.22/0.48 ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
% 0.22/0.48 = A ) ).
% 0.22/0.48
% 0.22/0.48 % mult_div_mod_eq
% 0.22/0.48 thf(fact_4364_mult__div__mod__eq,axiom,
% 0.22/0.48 ! [B: int,A: int] :
% 0.22/0.48 ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
% 0.22/0.48 = A ) ).
% 0.22/0.48
% 0.22/0.48 % mult_div_mod_eq
% 0.22/0.48 thf(fact_4365_mult__div__mod__eq,axiom,
% 0.22/0.48 ! [B: code_integer,A: code_integer] :
% 0.22/0.48 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.22/0.48 = A ) ).
% 0.22/0.48
% 0.22/0.48 % mult_div_mod_eq
% 0.22/0.48 thf(fact_4366_mod__mult__div__eq,axiom,
% 0.22/0.48 ! [A: nat,B: nat] :
% 0.22/0.48 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 0.22/0.48 = A ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult_div_eq
% 0.22/0.48 thf(fact_4367_mod__mult__div__eq,axiom,
% 0.22/0.48 ! [A: int,B: int] :
% 0.22/0.48 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 0.22/0.48 = A ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult_div_eq
% 0.22/0.48 thf(fact_4368_mod__mult__div__eq,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer] :
% 0.22/0.48 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 0.22/0.48 = A ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult_div_eq
% 0.22/0.48 thf(fact_4369_mod__div__mult__eq,axiom,
% 0.22/0.48 ! [A: nat,B: nat] :
% 0.22/0.48 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 0.22/0.48 = A ) ).
% 0.22/0.48
% 0.22/0.48 % mod_div_mult_eq
% 0.22/0.48 thf(fact_4370_mod__div__mult__eq,axiom,
% 0.22/0.48 ! [A: int,B: int] :
% 0.22/0.48 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 0.22/0.48 = A ) ).
% 0.22/0.48
% 0.22/0.48 % mod_div_mult_eq
% 0.22/0.48 thf(fact_4371_mod__div__mult__eq,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer] :
% 0.22/0.48 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 0.22/0.48 = A ) ).
% 0.22/0.48
% 0.22/0.48 % mod_div_mult_eq
% 0.22/0.48 thf(fact_4372_div__mult__mod__eq,axiom,
% 0.22/0.48 ! [A: nat,B: nat] :
% 0.22/0.48 ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
% 0.22/0.48 = A ) ).
% 0.22/0.48
% 0.22/0.48 % div_mult_mod_eq
% 0.22/0.48 thf(fact_4373_div__mult__mod__eq,axiom,
% 0.22/0.48 ! [A: int,B: int] :
% 0.22/0.48 ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
% 0.22/0.48 = A ) ).
% 0.22/0.48
% 0.22/0.48 % div_mult_mod_eq
% 0.22/0.48 thf(fact_4374_div__mult__mod__eq,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer] :
% 0.22/0.48 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.22/0.48 = A ) ).
% 0.22/0.48
% 0.22/0.48 % div_mult_mod_eq
% 0.22/0.48 thf(fact_4375_mod__div__decomp,axiom,
% 0.22/0.48 ! [A: nat,B: nat] :
% 0.22/0.48 ( A
% 0.22/0.48 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_div_decomp
% 0.22/0.48 thf(fact_4376_mod__div__decomp,axiom,
% 0.22/0.48 ! [A: int,B: int] :
% 0.22/0.48 ( A
% 0.22/0.48 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_div_decomp
% 0.22/0.48 thf(fact_4377_mod__div__decomp,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer] :
% 0.22/0.48 ( A
% 0.22/0.48 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_div_decomp
% 0.22/0.48 thf(fact_4378_cancel__div__mod__rules_I1_J,axiom,
% 0.22/0.48 ! [A: nat,B: nat,C: nat] :
% 0.22/0.48 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 0.22/0.48 = ( plus_plus_nat @ A @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % cancel_div_mod_rules(1)
% 0.22/0.48 thf(fact_4379_cancel__div__mod__rules_I1_J,axiom,
% 0.22/0.48 ! [A: int,B: int,C: int] :
% 0.22/0.48 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 0.22/0.48 = ( plus_plus_int @ A @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % cancel_div_mod_rules(1)
% 0.22/0.48 thf(fact_4380_cancel__div__mod__rules_I1_J,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.22/0.48 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 0.22/0.48 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % cancel_div_mod_rules(1)
% 0.22/0.48 thf(fact_4381_cancel__div__mod__rules_I2_J,axiom,
% 0.22/0.48 ! [B: nat,A: nat,C: nat] :
% 0.22/0.48 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 0.22/0.48 = ( plus_plus_nat @ A @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % cancel_div_mod_rules(2)
% 0.22/0.48 thf(fact_4382_cancel__div__mod__rules_I2_J,axiom,
% 0.22/0.48 ! [B: int,A: int,C: int] :
% 0.22/0.48 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 0.22/0.48 = ( plus_plus_int @ A @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % cancel_div_mod_rules(2)
% 0.22/0.48 thf(fact_4383_cancel__div__mod__rules_I2_J,axiom,
% 0.22/0.48 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.22/0.48 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 0.22/0.48 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % cancel_div_mod_rules(2)
% 0.22/0.48 thf(fact_4384_unit__imp__mod__eq__0,axiom,
% 0.22/0.48 ! [B: nat,A: nat] :
% 0.22/0.48 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.22/0.48 => ( ( modulo_modulo_nat @ A @ B )
% 0.22/0.48 = zero_zero_nat ) ) ).
% 0.22/0.48
% 0.22/0.48 % unit_imp_mod_eq_0
% 0.22/0.48 thf(fact_4385_unit__imp__mod__eq__0,axiom,
% 0.22/0.48 ! [B: int,A: int] :
% 0.22/0.48 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.22/0.48 => ( ( modulo_modulo_int @ A @ B )
% 0.22/0.48 = zero_zero_int ) ) ).
% 0.22/0.48
% 0.22/0.48 % unit_imp_mod_eq_0
% 0.22/0.48 thf(fact_4386_unit__imp__mod__eq__0,axiom,
% 0.22/0.48 ! [B: code_integer,A: code_integer] :
% 0.22/0.48 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.22/0.48 => ( ( modulo364778990260209775nteger @ A @ B )
% 0.22/0.48 = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.48
% 0.22/0.48 % unit_imp_mod_eq_0
% 0.22/0.48 thf(fact_4387_minus__mult__div__eq__mod,axiom,
% 0.22/0.48 ! [A: nat,B: nat] :
% 0.22/0.48 ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 0.22/0.48 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.22/0.48
% 0.22/0.48 % minus_mult_div_eq_mod
% 0.22/0.48 thf(fact_4388_minus__mult__div__eq__mod,axiom,
% 0.22/0.48 ! [A: int,B: int] :
% 0.22/0.48 ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 0.22/0.48 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.22/0.48
% 0.22/0.48 % minus_mult_div_eq_mod
% 0.22/0.48 thf(fact_4389_minus__mult__div__eq__mod,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer] :
% 0.22/0.48 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 0.22/0.48 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.22/0.48
% 0.22/0.48 % minus_mult_div_eq_mod
% 0.22/0.48 thf(fact_4390_minus__mod__eq__mult__div,axiom,
% 0.22/0.48 ! [A: nat,B: nat] :
% 0.22/0.48 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 0.22/0.48 = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % minus_mod_eq_mult_div
% 0.22/0.48 thf(fact_4391_minus__mod__eq__mult__div,axiom,
% 0.22/0.48 ! [A: int,B: int] :
% 0.22/0.48 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 0.22/0.48 = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % minus_mod_eq_mult_div
% 0.22/0.48 thf(fact_4392_minus__mod__eq__mult__div,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer] :
% 0.22/0.48 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.22/0.48 = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % minus_mod_eq_mult_div
% 0.22/0.48 thf(fact_4393_minus__mod__eq__div__mult,axiom,
% 0.22/0.48 ! [A: nat,B: nat] :
% 0.22/0.48 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 0.22/0.48 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).
% 0.22/0.48
% 0.22/0.48 % minus_mod_eq_div_mult
% 0.22/0.48 thf(fact_4394_minus__mod__eq__div__mult,axiom,
% 0.22/0.48 ! [A: int,B: int] :
% 0.22/0.48 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 0.22/0.48 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).
% 0.22/0.48
% 0.22/0.48 % minus_mod_eq_div_mult
% 0.22/0.48 thf(fact_4395_minus__mod__eq__div__mult,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer] :
% 0.22/0.48 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.22/0.48 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).
% 0.22/0.48
% 0.22/0.48 % minus_mod_eq_div_mult
% 0.22/0.48 thf(fact_4396_minus__div__mult__eq__mod,axiom,
% 0.22/0.48 ! [A: nat,B: nat] :
% 0.22/0.48 ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 0.22/0.48 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.22/0.48
% 0.22/0.48 % minus_div_mult_eq_mod
% 0.22/0.48 thf(fact_4397_minus__div__mult__eq__mod,axiom,
% 0.22/0.48 ! [A: int,B: int] :
% 0.22/0.48 ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 0.22/0.48 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.22/0.48
% 0.22/0.48 % minus_div_mult_eq_mod
% 0.22/0.48 thf(fact_4398_minus__div__mult__eq__mod,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer] :
% 0.22/0.48 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 0.22/0.48 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.22/0.48
% 0.22/0.48 % minus_div_mult_eq_mod
% 0.22/0.48 thf(fact_4399_cong__exp__iff__simps_I10_J,axiom,
% 0.22/0.48 ! [M: num,Q2: num,N: num] :
% 0.22/0.48 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(10)
% 0.22/0.48 thf(fact_4400_cong__exp__iff__simps_I10_J,axiom,
% 0.22/0.48 ! [M: num,Q2: num,N: num] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(10)
% 0.22/0.48 thf(fact_4401_cong__exp__iff__simps_I10_J,axiom,
% 0.22/0.48 ! [M: num,Q2: num,N: num] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(10)
% 0.22/0.48 thf(fact_4402_cong__exp__iff__simps_I12_J,axiom,
% 0.22/0.48 ! [M: num,Q2: num,N: num] :
% 0.22/0.48 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(12)
% 0.22/0.48 thf(fact_4403_cong__exp__iff__simps_I12_J,axiom,
% 0.22/0.48 ! [M: num,Q2: num,N: num] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(12)
% 0.22/0.48 thf(fact_4404_cong__exp__iff__simps_I12_J,axiom,
% 0.22/0.48 ! [M: num,Q2: num,N: num] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(12)
% 0.22/0.48 thf(fact_4405_cong__exp__iff__simps_I13_J,axiom,
% 0.22/0.48 ! [M: num,Q2: num,N: num] :
% 0.22/0.48 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.22/0.48 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.22/0.48 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(13)
% 0.22/0.48 thf(fact_4406_cong__exp__iff__simps_I13_J,axiom,
% 0.22/0.48 ! [M: num,Q2: num,N: num] :
% 0.22/0.48 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.22/0.48 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 0.22/0.48 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(13)
% 0.22/0.48 thf(fact_4407_cong__exp__iff__simps_I13_J,axiom,
% 0.22/0.48 ! [M: num,Q2: num,N: num] :
% 0.22/0.48 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.22/0.48 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(13)
% 0.22/0.48 thf(fact_4408_mod__le__divisor,axiom,
% 0.22/0.48 ! [N: nat,M: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.48 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_le_divisor
% 0.22/0.48 thf(fact_4409_mod__greater__zero__iff__not__dvd,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
% 0.22/0.48 = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_greater_zero_iff_not_dvd
% 0.22/0.48 thf(fact_4410_div__less__mono,axiom,
% 0.22/0.48 ! [A2: nat,B4: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_nat @ A2 @ B4 )
% 0.22/0.48 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.48 => ( ( ( modulo_modulo_nat @ A2 @ N )
% 0.22/0.48 = zero_zero_nat )
% 0.22/0.48 => ( ( ( modulo_modulo_nat @ B4 @ N )
% 0.22/0.48 = zero_zero_nat )
% 0.22/0.48 => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B4 @ N ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % div_less_mono
% 0.22/0.48 thf(fact_4411_mod__eq__nat1E,axiom,
% 0.22/0.48 ! [M: nat,Q2: nat,N: nat] :
% 0.22/0.48 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 0.22/0.48 = ( modulo_modulo_nat @ N @ Q2 ) )
% 0.22/0.48 => ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.48 => ~ ! [S3: nat] :
% 0.22/0.48 ( M
% 0.22/0.48 != ( plus_plus_nat @ N @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_eq_nat1E
% 0.22/0.48 thf(fact_4412_mod__eq__nat2E,axiom,
% 0.22/0.48 ! [M: nat,Q2: nat,N: nat] :
% 0.22/0.48 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 0.22/0.48 = ( modulo_modulo_nat @ N @ Q2 ) )
% 0.22/0.48 => ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.48 => ~ ! [S3: nat] :
% 0.22/0.48 ( N
% 0.22/0.48 != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_eq_nat2E
% 0.22/0.48 thf(fact_4413_nat__mod__eq__lemma,axiom,
% 0.22/0.48 ! [X: nat,N: nat,Y4: nat] :
% 0.22/0.48 ( ( ( modulo_modulo_nat @ X @ N )
% 0.22/0.48 = ( modulo_modulo_nat @ Y4 @ N ) )
% 0.22/0.48 => ( ( ord_less_eq_nat @ Y4 @ X )
% 0.22/0.48 => ? [Q4: nat] :
% 0.22/0.48 ( X
% 0.22/0.48 = ( plus_plus_nat @ Y4 @ ( times_times_nat @ N @ Q4 ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % nat_mod_eq_lemma
% 0.22/0.48 thf(fact_4414_pochhammer__nonneg,axiom,
% 0.22/0.48 ! [X: rat,N: nat] :
% 0.22/0.48 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.22/0.48 => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_nonneg
% 0.22/0.48 thf(fact_4415_pochhammer__nonneg,axiom,
% 0.22/0.48 ! [X: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.22/0.48 => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_nonneg
% 0.22/0.48 thf(fact_4416_pochhammer__nonneg,axiom,
% 0.22/0.48 ! [X: int,N: nat] :
% 0.22/0.48 ( ( ord_less_int @ zero_zero_int @ X )
% 0.22/0.48 => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_nonneg
% 0.22/0.48 thf(fact_4417_pochhammer__nonneg,axiom,
% 0.22/0.48 ! [X: real,N: nat] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_nonneg
% 0.22/0.48 thf(fact_4418_mod__eq__dvd__iff__nat,axiom,
% 0.22/0.48 ! [N: nat,M: nat,Q2: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.48 => ( ( ( modulo_modulo_nat @ M @ Q2 )
% 0.22/0.48 = ( modulo_modulo_nat @ N @ Q2 ) )
% 0.22/0.48 = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_eq_dvd_iff_nat
% 0.22/0.48 thf(fact_4419_mod__mult2__eq,axiom,
% 0.22/0.48 ! [M: nat,N: nat,Q2: nat] :
% 0.22/0.48 ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 0.22/0.48 = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult2_eq
% 0.22/0.48 thf(fact_4420_div__mod__decomp,axiom,
% 0.22/0.48 ! [A2: nat,N: nat] :
% 0.22/0.48 ( A2
% 0.22/0.48 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N ) @ N ) @ ( modulo_modulo_nat @ A2 @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % div_mod_decomp
% 0.22/0.48 thf(fact_4421_modulo__nat__def,axiom,
% 0.22/0.48 ( modulo_modulo_nat
% 0.22/0.48 = ( ^ [M5: nat,N4: nat] : ( minus_minus_nat @ M5 @ ( times_times_nat @ ( divide_divide_nat @ M5 @ N4 ) @ N4 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % modulo_nat_def
% 0.22/0.48 thf(fact_4422_pochhammer__0__left,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( ( N = zero_zero_nat )
% 0.22/0.48 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 0.22/0.48 = one_one_complex ) )
% 0.22/0.48 & ( ( N != zero_zero_nat )
% 0.22/0.48 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 0.22/0.48 = zero_zero_complex ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_0_left
% 0.22/0.48 thf(fact_4423_pochhammer__0__left,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( ( N = zero_zero_nat )
% 0.22/0.48 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 0.22/0.48 = one_one_real ) )
% 0.22/0.48 & ( ( N != zero_zero_nat )
% 0.22/0.48 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 0.22/0.48 = zero_zero_real ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_0_left
% 0.22/0.48 thf(fact_4424_pochhammer__0__left,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( ( N = zero_zero_nat )
% 0.22/0.48 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 0.22/0.48 = one_one_rat ) )
% 0.22/0.48 & ( ( N != zero_zero_nat )
% 0.22/0.48 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 0.22/0.48 = zero_zero_rat ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_0_left
% 0.22/0.48 thf(fact_4425_pochhammer__0__left,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( ( N = zero_zero_nat )
% 0.22/0.48 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 0.22/0.48 = one_one_nat ) )
% 0.22/0.48 & ( ( N != zero_zero_nat )
% 0.22/0.48 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 0.22/0.48 = zero_zero_nat ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_0_left
% 0.22/0.48 thf(fact_4426_pochhammer__0__left,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( ( N = zero_zero_nat )
% 0.22/0.48 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 0.22/0.48 = one_one_int ) )
% 0.22/0.48 & ( ( N != zero_zero_nat )
% 0.22/0.48 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 0.22/0.48 = zero_zero_int ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_0_left
% 0.22/0.48 thf(fact_4427_cong__exp__iff__simps_I3_J,axiom,
% 0.22/0.48 ! [N: num,Q2: num] :
% 0.22/0.48 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 != zero_zero_nat ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(3)
% 0.22/0.48 thf(fact_4428_cong__exp__iff__simps_I3_J,axiom,
% 0.22/0.48 ! [N: num,Q2: num] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 != zero_zero_int ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(3)
% 0.22/0.48 thf(fact_4429_cong__exp__iff__simps_I3_J,axiom,
% 0.22/0.48 ! [N: num,Q2: num] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 != zero_z3403309356797280102nteger ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(3)
% 0.22/0.48 thf(fact_4430_mod__mult2__eq_H,axiom,
% 0.22/0.48 ! [A: code_integer,M: nat,N: nat] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
% 0.22/0.48 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult2_eq'
% 0.22/0.48 thf(fact_4431_mod__mult2__eq_H,axiom,
% 0.22/0.48 ! [A: nat,M: nat,N: nat] :
% 0.22/0.48 ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 0.22/0.48 = ( 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 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult2_eq'
% 0.22/0.48 thf(fact_4432_mod__mult2__eq_H,axiom,
% 0.22/0.48 ! [A: int,M: nat,N: nat] :
% 0.22/0.48 ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.22/0.48 = ( 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 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_mult2_eq'
% 0.22/0.48 thf(fact_4433_even__even__mod__4__iff,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.48 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % even_even_mod_4_iff
% 0.22/0.48 thf(fact_4434_field__char__0__class_Oof__nat__div,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
% 0.22/0.48 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % field_char_0_class.of_nat_div
% 0.22/0.48 thf(fact_4435_field__char__0__class_Oof__nat__div,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
% 0.22/0.48 = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % field_char_0_class.of_nat_div
% 0.22/0.48 thf(fact_4436_field__char__0__class_Oof__nat__div,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
% 0.22/0.48 = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % field_char_0_class.of_nat_div
% 0.22/0.48 thf(fact_4437_split__mod,axiom,
% 0.22/0.48 ! [P: nat > $o,M: nat,N: nat] :
% 0.22/0.48 ( ( P @ ( modulo_modulo_nat @ M @ N ) )
% 0.22/0.48 = ( ( ( N = zero_zero_nat )
% 0.22/0.48 => ( P @ M ) )
% 0.22/0.48 & ( ( N != zero_zero_nat )
% 0.22/0.48 => ! [I2: nat,J3: nat] :
% 0.22/0.48 ( ( ord_less_nat @ J3 @ N )
% 0.22/0.48 => ( ( M
% 0.22/0.48 = ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J3 ) )
% 0.22/0.48 => ( P @ J3 ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % split_mod
% 0.22/0.48 thf(fact_4438_mod__nat__eqI,axiom,
% 0.22/0.48 ! [R2: nat,N: nat,M: nat] :
% 0.22/0.48 ( ( ord_less_nat @ R2 @ N )
% 0.22/0.48 => ( ( ord_less_eq_nat @ R2 @ M )
% 0.22/0.48 => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R2 ) )
% 0.22/0.48 => ( ( modulo_modulo_nat @ M @ N )
% 0.22/0.48 = R2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_nat_eqI
% 0.22/0.48 thf(fact_4439_pochhammer__rec,axiom,
% 0.22/0.48 ! [A: complex,N: nat] :
% 0.22/0.48 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_rec
% 0.22/0.48 thf(fact_4440_pochhammer__rec,axiom,
% 0.22/0.48 ! [A: real,N: nat] :
% 0.22/0.48 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_rec
% 0.22/0.48 thf(fact_4441_pochhammer__rec,axiom,
% 0.22/0.48 ! [A: rat,N: nat] :
% 0.22/0.48 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_rec
% 0.22/0.48 thf(fact_4442_pochhammer__rec,axiom,
% 0.22/0.48 ! [A: nat,N: nat] :
% 0.22/0.48 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_rec
% 0.22/0.48 thf(fact_4443_pochhammer__rec,axiom,
% 0.22/0.48 ! [A: int,N: nat] :
% 0.22/0.48 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_rec
% 0.22/0.48 thf(fact_4444_pochhammer__rec_H,axiom,
% 0.22/0.48 ! [Z: complex,N: nat] :
% 0.22/0.48 ( ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_complex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_rec'
% 0.22/0.48 thf(fact_4445_pochhammer__rec_H,axiom,
% 0.22/0.48 ! [Z: real,N: nat] :
% 0.22/0.48 ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_rec'
% 0.22/0.48 thf(fact_4446_pochhammer__rec_H,axiom,
% 0.22/0.48 ! [Z: rat,N: nat] :
% 0.22/0.48 ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_rec'
% 0.22/0.48 thf(fact_4447_pochhammer__rec_H,axiom,
% 0.22/0.48 ! [Z: nat,N: nat] :
% 0.22/0.48 ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_rec'
% 0.22/0.48 thf(fact_4448_pochhammer__rec_H,axiom,
% 0.22/0.48 ! [Z: int,N: nat] :
% 0.22/0.48 ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_rec'
% 0.22/0.48 thf(fact_4449_pochhammer__Suc,axiom,
% 0.22/0.48 ! [A: complex,N: nat] :
% 0.22/0.48 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_Suc
% 0.22/0.48 thf(fact_4450_pochhammer__Suc,axiom,
% 0.22/0.48 ! [A: real,N: nat] :
% 0.22/0.48 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_Suc
% 0.22/0.48 thf(fact_4451_pochhammer__Suc,axiom,
% 0.22/0.48 ! [A: rat,N: nat] :
% 0.22/0.48 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_Suc
% 0.22/0.48 thf(fact_4452_pochhammer__Suc,axiom,
% 0.22/0.48 ! [A: nat,N: nat] :
% 0.22/0.48 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_Suc
% 0.22/0.48 thf(fact_4453_pochhammer__Suc,axiom,
% 0.22/0.48 ! [A: int,N: nat] :
% 0.22/0.48 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_Suc
% 0.22/0.48 thf(fact_4454_real__of__nat__div__aux,axiom,
% 0.22/0.48 ! [X: nat,D: nat] :
% 0.22/0.48 ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
% 0.22/0.48 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % real_of_nat_div_aux
% 0.22/0.48 thf(fact_4455_pochhammer__product_H,axiom,
% 0.22/0.48 ! [Z: complex,N: nat,M: nat] :
% 0.22/0.48 ( ( comm_s2602460028002588243omplex @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.22/0.48 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ N ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ M ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_product'
% 0.22/0.48 thf(fact_4456_pochhammer__product_H,axiom,
% 0.22/0.48 ! [Z: real,N: nat,M: nat] :
% 0.22/0.48 ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.22/0.48 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ M ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_product'
% 0.22/0.48 thf(fact_4457_pochhammer__product_H,axiom,
% 0.22/0.48 ! [Z: rat,N: nat,M: nat] :
% 0.22/0.48 ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.22/0.48 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_product'
% 0.22/0.48 thf(fact_4458_pochhammer__product_H,axiom,
% 0.22/0.48 ! [Z: nat,N: nat,M: nat] :
% 0.22/0.48 ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.22/0.48 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_product'
% 0.22/0.48 thf(fact_4459_pochhammer__product_H,axiom,
% 0.22/0.48 ! [Z: int,N: nat,M: nat] :
% 0.22/0.48 ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.22/0.48 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_product'
% 0.22/0.48 thf(fact_4460_binomial__maximum_H,axiom,
% 0.22/0.48 ! [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 ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_maximum'
% 0.22/0.48 thf(fact_4461_binomial__mono,axiom,
% 0.22/0.48 ! [K: nat,K6: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ K @ K6 )
% 0.22/0.48 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 0.22/0.48 => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_mono
% 0.22/0.48 thf(fact_4462_binomial__antimono,axiom,
% 0.22/0.48 ! [K: nat,K6: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ K @ K6 )
% 0.22/0.48 => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 0.22/0.48 => ( ( ord_less_eq_nat @ K6 @ N )
% 0.22/0.48 => ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_antimono
% 0.22/0.48 thf(fact_4463_binomial__maximum,axiom,
% 0.22/0.48 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_maximum
% 0.22/0.48 thf(fact_4464_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.22/0.48 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.22/0.48 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.22/0.48 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.22/0.48 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.22/0.48 thf(fact_4465_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.22/0.48 ! [C: nat,A: nat,B: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.22/0.48 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.22/0.48 = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.22/0.48 thf(fact_4466_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.22/0.48 ! [C: int,A: int,B: int] :
% 0.22/0.48 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.22/0.48 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 0.22/0.48 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.22/0.48 thf(fact_4467_even__iff__mod__2__eq__zero,axiom,
% 0.22/0.48 ! [A: nat] :
% 0.22/0.48 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.48 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.48 = zero_zero_nat ) ) ).
% 0.22/0.48
% 0.22/0.48 % even_iff_mod_2_eq_zero
% 0.22/0.48 thf(fact_4468_even__iff__mod__2__eq__zero,axiom,
% 0.22/0.48 ! [A: int] :
% 0.22/0.48 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.48 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.48 = zero_zero_int ) ) ).
% 0.22/0.48
% 0.22/0.48 % even_iff_mod_2_eq_zero
% 0.22/0.48 thf(fact_4469_even__iff__mod__2__eq__zero,axiom,
% 0.22/0.48 ! [A: code_integer] :
% 0.22/0.48 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.48 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.48 = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.48
% 0.22/0.48 % even_iff_mod_2_eq_zero
% 0.22/0.48 thf(fact_4470_cong__exp__iff__simps_I7_J,axiom,
% 0.22/0.48 ! [Q2: num,N: num] :
% 0.22/0.48 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.22/0.48 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.22/0.48 = zero_zero_nat ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(7)
% 0.22/0.48 thf(fact_4471_cong__exp__iff__simps_I7_J,axiom,
% 0.22/0.48 ! [Q2: num,N: num] :
% 0.22/0.48 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.22/0.48 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 0.22/0.48 = zero_zero_int ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(7)
% 0.22/0.48 thf(fact_4472_cong__exp__iff__simps_I7_J,axiom,
% 0.22/0.48 ! [Q2: num,N: num] :
% 0.22/0.48 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.22/0.48 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.22/0.48 = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(7)
% 0.22/0.48 thf(fact_4473_cong__exp__iff__simps_I11_J,axiom,
% 0.22/0.48 ! [M: num,Q2: num] :
% 0.22/0.48 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.22/0.48 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.22/0.48 = zero_zero_nat ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(11)
% 0.22/0.48 thf(fact_4474_cong__exp__iff__simps_I11_J,axiom,
% 0.22/0.48 ! [M: num,Q2: num] :
% 0.22/0.48 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.22/0.48 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 0.22/0.48 = zero_zero_int ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(11)
% 0.22/0.48 thf(fact_4475_cong__exp__iff__simps_I11_J,axiom,
% 0.22/0.48 ! [M: num,Q2: num] :
% 0.22/0.48 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.22/0.48 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.22/0.48 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.22/0.48 = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.48
% 0.22/0.48 % cong_exp_iff_simps(11)
% 0.22/0.48 thf(fact_4476_odd__iff__mod__2__eq__one,axiom,
% 0.22/0.48 ! [A: nat] :
% 0.22/0.48 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.22/0.48 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.48 = one_one_nat ) ) ).
% 0.22/0.48
% 0.22/0.48 % odd_iff_mod_2_eq_one
% 0.22/0.48 thf(fact_4477_odd__iff__mod__2__eq__one,axiom,
% 0.22/0.48 ! [A: int] :
% 0.22/0.48 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.22/0.48 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.48 = one_one_int ) ) ).
% 0.22/0.48
% 0.22/0.48 % odd_iff_mod_2_eq_one
% 0.22/0.48 thf(fact_4478_odd__iff__mod__2__eq__one,axiom,
% 0.22/0.48 ! [A: code_integer] :
% 0.22/0.48 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.22/0.48 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.48 = one_one_Code_integer ) ) ).
% 0.22/0.48
% 0.22/0.48 % odd_iff_mod_2_eq_one
% 0.22/0.48 thf(fact_4479_Suc__mod__eq__add3__mod,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 0.22/0.48 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % Suc_mod_eq_add3_mod
% 0.22/0.48 thf(fact_4480_Suc__times__mod__eq,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.22/0.48 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
% 0.22/0.48 = one_one_nat ) ) ).
% 0.22/0.48
% 0.22/0.48 % Suc_times_mod_eq
% 0.22/0.48 thf(fact_4481_pochhammer__product,axiom,
% 0.22/0.48 ! [M: nat,N: nat,Z: complex] :
% 0.22/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.48 => ( ( comm_s2602460028002588243omplex @ Z @ N )
% 0.22/0.48 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_product
% 0.22/0.48 thf(fact_4482_pochhammer__product,axiom,
% 0.22/0.48 ! [M: nat,N: nat,Z: real] :
% 0.22/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.48 => ( ( comm_s7457072308508201937r_real @ Z @ N )
% 0.22/0.48 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_product
% 0.22/0.48 thf(fact_4483_pochhammer__product,axiom,
% 0.22/0.48 ! [M: nat,N: nat,Z: rat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.48 => ( ( comm_s4028243227959126397er_rat @ Z @ N )
% 0.22/0.48 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_product
% 0.22/0.48 thf(fact_4484_pochhammer__product,axiom,
% 0.22/0.48 ! [M: nat,N: nat,Z: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.48 => ( ( comm_s4663373288045622133er_nat @ Z @ N )
% 0.22/0.48 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_product
% 0.22/0.48 thf(fact_4485_pochhammer__product,axiom,
% 0.22/0.48 ! [M: nat,N: nat,Z: int] :
% 0.22/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.48 => ( ( comm_s4660882817536571857er_int @ Z @ N )
% 0.22/0.48 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pochhammer_product
% 0.22/0.48 thf(fact_4486_binomial__strict__antimono,axiom,
% 0.22/0.48 ! [K: nat,K6: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_nat @ K @ K6 )
% 0.22/0.48 => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 0.22/0.48 => ( ( ord_less_eq_nat @ K6 @ N )
% 0.22/0.48 => ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_strict_antimono
% 0.22/0.48 thf(fact_4487_binomial__strict__mono,axiom,
% 0.22/0.48 ! [K: nat,K6: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_nat @ K @ K6 )
% 0.22/0.48 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 0.22/0.48 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_strict_mono
% 0.22/0.48 thf(fact_4488_binomial__less__binomial__Suc,axiom,
% 0.22/0.48 ! [K: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.48 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_less_binomial_Suc
% 0.22/0.48 thf(fact_4489_central__binomial__odd,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.48 => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.48 = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % central_binomial_odd
% 0.22/0.48 thf(fact_4490_divmod__digit__0_I2_J,axiom,
% 0.22/0.48 ! [B: nat,A: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.22/0.48 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.22/0.48 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
% 0.22/0.48 = ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % divmod_digit_0(2)
% 0.22/0.48 thf(fact_4491_divmod__digit__0_I2_J,axiom,
% 0.22/0.48 ! [B: int,A: int] :
% 0.22/0.48 ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.48 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.22/0.48 => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
% 0.22/0.48 = ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % divmod_digit_0(2)
% 0.22/0.48 thf(fact_4492_divmod__digit__0_I2_J,axiom,
% 0.22/0.48 ! [B: code_integer,A: code_integer] :
% 0.22/0.48 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.22/0.48 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.22/0.48 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
% 0.22/0.48 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % divmod_digit_0(2)
% 0.22/0.48 thf(fact_4493_bits__stable__imp__add__self,axiom,
% 0.22/0.48 ! [A: nat] :
% 0.22/0.48 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.48 = A )
% 0.22/0.48 => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.48 = zero_zero_nat ) ) ).
% 0.22/0.48
% 0.22/0.48 % bits_stable_imp_add_self
% 0.22/0.48 thf(fact_4494_bits__stable__imp__add__self,axiom,
% 0.22/0.48 ! [A: int] :
% 0.22/0.48 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.48 = A )
% 0.22/0.48 => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.22/0.48 = zero_zero_int ) ) ).
% 0.22/0.48
% 0.22/0.48 % bits_stable_imp_add_self
% 0.22/0.48 thf(fact_4495_bits__stable__imp__add__self,axiom,
% 0.22/0.48 ! [A: code_integer] :
% 0.22/0.48 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.48 = A )
% 0.22/0.48 => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.22/0.48 = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.48
% 0.22/0.48 % bits_stable_imp_add_self
% 0.22/0.48 thf(fact_4496_parity__cases,axiom,
% 0.22/0.48 ! [A: nat] :
% 0.22/0.48 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.48 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.48 != zero_zero_nat ) )
% 0.22/0.48 => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.48 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.48 != one_one_nat ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % parity_cases
% 0.22/0.48 thf(fact_4497_parity__cases,axiom,
% 0.22/0.48 ! [A: int] :
% 0.22/0.48 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.48 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.48 != zero_zero_int ) )
% 0.22/0.48 => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.48 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.48 != one_one_int ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % parity_cases
% 0.22/0.48 thf(fact_4498_parity__cases,axiom,
% 0.22/0.48 ! [A: code_integer] :
% 0.22/0.48 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.48 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.48 != zero_z3403309356797280102nteger ) )
% 0.22/0.48 => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.48 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.48 != one_one_Code_integer ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % parity_cases
% 0.22/0.48 thf(fact_4499_mod2__eq__if,axiom,
% 0.22/0.48 ! [A: nat] :
% 0.22/0.48 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.48 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.48 = zero_zero_nat ) )
% 0.22/0.48 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.48 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.48 = one_one_nat ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod2_eq_if
% 0.22/0.48 thf(fact_4500_mod2__eq__if,axiom,
% 0.22/0.48 ! [A: int] :
% 0.22/0.48 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.48 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.48 = zero_zero_int ) )
% 0.22/0.48 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.48 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.48 = one_one_int ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod2_eq_if
% 0.22/0.48 thf(fact_4501_mod2__eq__if,axiom,
% 0.22/0.48 ! [A: code_integer] :
% 0.22/0.48 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.48 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.48 = zero_z3403309356797280102nteger ) )
% 0.22/0.48 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.48 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.48 = one_one_Code_integer ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod2_eq_if
% 0.22/0.48 thf(fact_4502_div__exp__mod__exp__eq,axiom,
% 0.22/0.48 ! [A: nat,N: nat,M: nat] :
% 0.22/0.48 ( ( 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 ) )
% 0.22/0.48 = ( 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 ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % div_exp_mod_exp_eq
% 0.22/0.48 thf(fact_4503_div__exp__mod__exp__eq,axiom,
% 0.22/0.48 ! [A: int,N: nat,M: nat] :
% 0.22/0.48 ( ( 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 ) )
% 0.22/0.48 = ( 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 ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % div_exp_mod_exp_eq
% 0.22/0.48 thf(fact_4504_div__exp__mod__exp__eq,axiom,
% 0.22/0.48 ! [A: code_integer,N: nat,M: nat] :
% 0.22/0.48 ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.22/0.48 = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % div_exp_mod_exp_eq
% 0.22/0.48 thf(fact_4505_bits__induct,axiom,
% 0.22/0.48 ! [P: nat > $o,A: nat] :
% 0.22/0.48 ( ! [A4: nat] :
% 0.22/0.48 ( ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.48 = A4 )
% 0.22/0.48 => ( P @ A4 ) )
% 0.22/0.48 => ( ! [A4: nat,B3: $o] :
% 0.22/0.48 ( ( P @ A4 )
% 0.22/0.48 => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.48 = A4 )
% 0.22/0.48 => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 0.22/0.48 => ( P @ A ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % bits_induct
% 0.22/0.48 thf(fact_4506_bits__induct,axiom,
% 0.22/0.48 ! [P: int > $o,A: int] :
% 0.22/0.48 ( ! [A4: int] :
% 0.22/0.48 ( ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.48 = A4 )
% 0.22/0.48 => ( P @ A4 ) )
% 0.22/0.48 => ( ! [A4: int,B3: $o] :
% 0.22/0.48 ( ( P @ A4 )
% 0.22/0.48 => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.48 = A4 )
% 0.22/0.48 => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 0.22/0.48 => ( P @ A ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % bits_induct
% 0.22/0.48 thf(fact_4507_bits__induct,axiom,
% 0.22/0.48 ! [P: code_integer > $o,A: code_integer] :
% 0.22/0.48 ( ! [A4: code_integer] :
% 0.22/0.48 ( ( ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.48 = A4 )
% 0.22/0.48 => ( P @ A4 ) )
% 0.22/0.48 => ( ! [A4: code_integer,B3: $o] :
% 0.22/0.48 ( ( P @ A4 )
% 0.22/0.48 => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.48 = A4 )
% 0.22/0.48 => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 0.22/0.48 => ( P @ A ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % bits_induct
% 0.22/0.48 thf(fact_4508_verit__le__mono__div,axiom,
% 0.22/0.48 ! [A2: nat,B4: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_nat @ A2 @ B4 )
% 0.22/0.48 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.48 => ( ord_less_eq_nat
% 0.22/0.48 @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N )
% 0.22/0.48 @ ( if_nat
% 0.22/0.48 @ ( ( modulo_modulo_nat @ B4 @ N )
% 0.22/0.48 = zero_zero_nat )
% 0.22/0.48 @ one_one_nat
% 0.22/0.48 @ zero_zero_nat ) )
% 0.22/0.48 @ ( divide_divide_nat @ B4 @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % verit_le_mono_div
% 0.22/0.48 thf(fact_4509_divmod__digit__0_I1_J,axiom,
% 0.22/0.48 ! [B: nat,A: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.22/0.48 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.22/0.48 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.22/0.48 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % divmod_digit_0(1)
% 0.22/0.48 thf(fact_4510_divmod__digit__0_I1_J,axiom,
% 0.22/0.48 ! [B: int,A: int] :
% 0.22/0.48 ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.48 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.22/0.48 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.22/0.48 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % divmod_digit_0(1)
% 0.22/0.48 thf(fact_4511_divmod__digit__0_I1_J,axiom,
% 0.22/0.48 ! [B: code_integer,A: code_integer] :
% 0.22/0.48 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.22/0.48 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.22/0.48 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.22/0.48 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % divmod_digit_0(1)
% 0.22/0.48 thf(fact_4512_mult__exp__mod__exp__eq,axiom,
% 0.22/0.48 ! [M: nat,N: nat,A: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.48 => ( ( 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 ) )
% 0.22/0.48 = ( 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 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mult_exp_mod_exp_eq
% 0.22/0.48 thf(fact_4513_mult__exp__mod__exp__eq,axiom,
% 0.22/0.48 ! [M: nat,N: nat,A: int] :
% 0.22/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.48 => ( ( 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 ) )
% 0.22/0.48 = ( 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 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mult_exp_mod_exp_eq
% 0.22/0.48 thf(fact_4514_mult__exp__mod__exp__eq,axiom,
% 0.22/0.48 ! [M: nat,N: nat,A: code_integer] :
% 0.22/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.48 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mult_exp_mod_exp_eq
% 0.22/0.48 thf(fact_4515_mod__double__modulus,axiom,
% 0.22/0.48 ! [M: code_integer,X: code_integer] :
% 0.22/0.48 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 0.22/0.48 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 0.22/0.48 => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.22/0.48 = ( modulo364778990260209775nteger @ X @ M ) )
% 0.22/0.48 | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.22/0.48 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_double_modulus
% 0.22/0.48 thf(fact_4516_mod__double__modulus,axiom,
% 0.22/0.48 ! [M: nat,X: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.22/0.48 => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.22/0.48 => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.22/0.48 = ( modulo_modulo_nat @ X @ M ) )
% 0.22/0.48 | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.22/0.48 = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_double_modulus
% 0.22/0.48 thf(fact_4517_mod__double__modulus,axiom,
% 0.22/0.48 ! [M: int,X: int] :
% 0.22/0.48 ( ( ord_less_int @ zero_zero_int @ M )
% 0.22/0.48 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.22/0.48 => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.22/0.48 = ( modulo_modulo_int @ X @ M ) )
% 0.22/0.48 | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.22/0.48 = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_double_modulus
% 0.22/0.48 thf(fact_4518_divmod__digit__1_I2_J,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer] :
% 0.22/0.48 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.22/0.48 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.22/0.48 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.22/0.48 => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.22/0.48 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % divmod_digit_1(2)
% 0.22/0.48 thf(fact_4519_divmod__digit__1_I2_J,axiom,
% 0.22/0.48 ! [A: nat,B: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.22/0.48 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.22/0.48 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.22/0.48 => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.22/0.48 = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % divmod_digit_1(2)
% 0.22/0.48 thf(fact_4520_divmod__digit__1_I2_J,axiom,
% 0.22/0.48 ! [A: int,B: int] :
% 0.22/0.48 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.22/0.48 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.48 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.22/0.48 => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.22/0.48 = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % divmod_digit_1(2)
% 0.22/0.48 thf(fact_4521_unset__bit__Suc,axiom,
% 0.22/0.48 ! [N: nat,A: int] :
% 0.22/0.48 ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
% 0.22/0.48 = ( 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 ) ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % unset_bit_Suc
% 0.22/0.48 thf(fact_4522_unset__bit__Suc,axiom,
% 0.22/0.48 ! [N: nat,A: nat] :
% 0.22/0.48 ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
% 0.22/0.48 = ( 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 ) ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % unset_bit_Suc
% 0.22/0.48 thf(fact_4523_unset__bit__Suc,axiom,
% 0.22/0.48 ! [N: nat,A: code_integer] :
% 0.22/0.48 ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
% 0.22/0.48 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % unset_bit_Suc
% 0.22/0.48 thf(fact_4524_set__bit__Suc,axiom,
% 0.22/0.48 ! [N: nat,A: int] :
% 0.22/0.48 ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
% 0.22/0.48 = ( 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 ) ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % set_bit_Suc
% 0.22/0.48 thf(fact_4525_set__bit__Suc,axiom,
% 0.22/0.48 ! [N: nat,A: nat] :
% 0.22/0.48 ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
% 0.22/0.48 = ( 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 ) ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % set_bit_Suc
% 0.22/0.48 thf(fact_4526_set__bit__Suc,axiom,
% 0.22/0.48 ! [N: nat,A: code_integer] :
% 0.22/0.48 ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
% 0.22/0.48 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % set_bit_Suc
% 0.22/0.48 thf(fact_4527_flip__bit__Suc,axiom,
% 0.22/0.48 ! [N: nat,A: int] :
% 0.22/0.48 ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
% 0.22/0.48 = ( 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 ) ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % flip_bit_Suc
% 0.22/0.48 thf(fact_4528_flip__bit__Suc,axiom,
% 0.22/0.48 ! [N: nat,A: nat] :
% 0.22/0.48 ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
% 0.22/0.48 = ( 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 ) ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % flip_bit_Suc
% 0.22/0.48 thf(fact_4529_flip__bit__Suc,axiom,
% 0.22/0.48 ! [N: nat,A: code_integer] :
% 0.22/0.48 ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
% 0.22/0.48 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % flip_bit_Suc
% 0.22/0.48 thf(fact_4530_even__mod__4__div__2,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.22/0.48 = ( suc @ zero_zero_nat ) )
% 0.22/0.48 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % even_mod_4_div_2
% 0.22/0.48 thf(fact_4531_exp__div__exp__eq,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( times_times_nat
% 0.22/0.48 @ ( zero_n2687167440665602831ol_nat
% 0.22/0.48 @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.22/0.48 != zero_zero_nat )
% 0.22/0.48 & ( ord_less_eq_nat @ N @ M ) ) )
% 0.22/0.48 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % exp_div_exp_eq
% 0.22/0.48 thf(fact_4532_exp__div__exp__eq,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( times_times_int
% 0.22/0.48 @ ( zero_n2684676970156552555ol_int
% 0.22/0.48 @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.22/0.48 != zero_zero_int )
% 0.22/0.48 & ( ord_less_eq_nat @ N @ M ) ) )
% 0.22/0.48 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % exp_div_exp_eq
% 0.22/0.48 thf(fact_4533_exp__div__exp__eq,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( times_3573771949741848930nteger
% 0.22/0.48 @ ( zero_n356916108424825756nteger
% 0.22/0.48 @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 0.22/0.48 != zero_z3403309356797280102nteger )
% 0.22/0.48 & ( ord_less_eq_nat @ N @ M ) ) )
% 0.22/0.48 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % exp_div_exp_eq
% 0.22/0.48 thf(fact_4534_odd__mod__4__div__2,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.22/0.48 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.22/0.48 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % odd_mod_4_div_2
% 0.22/0.48 thf(fact_4535_divmod__digit__1_I1_J,axiom,
% 0.22/0.48 ! [A: code_integer,B: code_integer] :
% 0.22/0.48 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.22/0.48 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.22/0.48 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.22/0.48 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
% 0.22/0.48 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % divmod_digit_1(1)
% 0.22/0.48 thf(fact_4536_divmod__digit__1_I1_J,axiom,
% 0.22/0.48 ! [A: nat,B: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.22/0.48 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.22/0.48 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.22/0.48 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
% 0.22/0.48 = ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % divmod_digit_1(1)
% 0.22/0.48 thf(fact_4537_divmod__digit__1_I1_J,axiom,
% 0.22/0.48 ! [A: int,B: int] :
% 0.22/0.48 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.22/0.48 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.48 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.22/0.48 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
% 0.22/0.48 = ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % divmod_digit_1(1)
% 0.22/0.48 thf(fact_4538_zero__less__binomial__iff,axiom,
% 0.22/0.48 ! [N: nat,K: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
% 0.22/0.48 = ( ord_less_eq_nat @ K @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % zero_less_binomial_iff
% 0.22/0.48 thf(fact_4539_choose__two,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.48 = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % choose_two
% 0.22/0.48 thf(fact_4540_binomial__n__0,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( binomial @ N @ zero_zero_nat )
% 0.22/0.48 = one_one_nat ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_n_0
% 0.22/0.48 thf(fact_4541_binomial__Suc__Suc,axiom,
% 0.22/0.48 ! [N: nat,K: nat] :
% 0.22/0.48 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 0.22/0.48 = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_Suc_Suc
% 0.22/0.48 thf(fact_4542_binomial__eq__0__iff,axiom,
% 0.22/0.48 ! [N: nat,K: nat] :
% 0.22/0.48 ( ( ( binomial @ N @ K )
% 0.22/0.48 = zero_zero_nat )
% 0.22/0.48 = ( ord_less_nat @ N @ K ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_eq_0_iff
% 0.22/0.48 thf(fact_4543_binomial__1,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
% 0.22/0.48 = N ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_1
% 0.22/0.48 thf(fact_4544_binomial__0__Suc,axiom,
% 0.22/0.48 ! [K: nat] :
% 0.22/0.48 ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
% 0.22/0.48 = zero_zero_nat ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_0_Suc
% 0.22/0.48 thf(fact_4545_binomial__addition__formula,axiom,
% 0.22/0.48 ! [N: nat,K: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.48 => ( ( binomial @ N @ ( suc @ K ) )
% 0.22/0.48 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_addition_formula
% 0.22/0.48 thf(fact_4546_binomial__Suc__n,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( binomial @ ( suc @ N ) @ N )
% 0.22/0.48 = ( suc @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_Suc_n
% 0.22/0.48 thf(fact_4547_mod__neg__neg__trivial,axiom,
% 0.22/0.48 ! [K: int,L: int] :
% 0.22/0.48 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.22/0.48 => ( ( ord_less_int @ L @ K )
% 0.22/0.48 => ( ( modulo_modulo_int @ K @ L )
% 0.22/0.48 = K ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_neg_neg_trivial
% 0.22/0.48 thf(fact_4548_mod__pos__pos__trivial,axiom,
% 0.22/0.48 ! [K: int,L: int] :
% 0.22/0.48 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.22/0.48 => ( ( ord_less_int @ K @ L )
% 0.22/0.48 => ( ( modulo_modulo_int @ K @ L )
% 0.22/0.48 = K ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_pos_pos_trivial
% 0.22/0.48 thf(fact_4549_zmod__numeral__Bit0,axiom,
% 0.22/0.48 ! [V: num,W: num] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.22/0.48 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % zmod_numeral_Bit0
% 0.22/0.48 thf(fact_4550_zmod__numeral__Bit1,axiom,
% 0.22/0.48 ! [V: num,W: num] :
% 0.22/0.48 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.22/0.48 = ( 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 ) ) ).
% 0.22/0.48
% 0.22/0.48 % zmod_numeral_Bit1
% 0.22/0.48 thf(fact_4551_zmod__le__nonneg__dividend,axiom,
% 0.22/0.48 ! [M: int,K: int] :
% 0.22/0.48 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 0.22/0.48 => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% 0.22/0.48
% 0.22/0.48 % zmod_le_nonneg_dividend
% 0.22/0.48 thf(fact_4552_neg__mod__bound,axiom,
% 0.22/0.48 ! [L: int,K: int] :
% 0.22/0.48 ( ( ord_less_int @ L @ zero_zero_int )
% 0.22/0.48 => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % neg_mod_bound
% 0.22/0.48 thf(fact_4553_Euclidean__Division_Opos__mod__bound,axiom,
% 0.22/0.48 ! [L: int,K: int] :
% 0.22/0.48 ( ( ord_less_int @ zero_zero_int @ L )
% 0.22/0.48 => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% 0.22/0.48
% 0.22/0.48 % Euclidean_Division.pos_mod_bound
% 0.22/0.48 thf(fact_4554_zmod__eq__0D,axiom,
% 0.22/0.48 ! [M: int,D: int] :
% 0.22/0.48 ( ( ( modulo_modulo_int @ M @ D )
% 0.22/0.48 = zero_zero_int )
% 0.22/0.48 => ? [Q4: int] :
% 0.22/0.48 ( M
% 0.22/0.48 = ( times_times_int @ D @ Q4 ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % zmod_eq_0D
% 0.22/0.48 thf(fact_4555_zmod__eq__0__iff,axiom,
% 0.22/0.48 ! [M: int,D: int] :
% 0.22/0.48 ( ( ( modulo_modulo_int @ M @ D )
% 0.22/0.48 = zero_zero_int )
% 0.22/0.48 = ( ? [Q3: int] :
% 0.22/0.48 ( M
% 0.22/0.48 = ( times_times_int @ D @ Q3 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % zmod_eq_0_iff
% 0.22/0.48 thf(fact_4556_zmod__int,axiom,
% 0.22/0.48 ! [A: nat,B: nat] :
% 0.22/0.48 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
% 0.22/0.48 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % zmod_int
% 0.22/0.48 thf(fact_4557_neg__mod__sign,axiom,
% 0.22/0.48 ! [L: int,K: int] :
% 0.22/0.48 ( ( ord_less_int @ L @ zero_zero_int )
% 0.22/0.48 => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% 0.22/0.48
% 0.22/0.48 % neg_mod_sign
% 0.22/0.48 thf(fact_4558_Euclidean__Division_Opos__mod__sign,axiom,
% 0.22/0.48 ! [L: int,K: int] :
% 0.22/0.48 ( ( ord_less_int @ zero_zero_int @ L )
% 0.22/0.48 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % Euclidean_Division.pos_mod_sign
% 0.22/0.48 thf(fact_4559_zmod__trivial__iff,axiom,
% 0.22/0.48 ! [I: int,K: int] :
% 0.22/0.48 ( ( ( modulo_modulo_int @ I @ K )
% 0.22/0.48 = I )
% 0.22/0.48 = ( ( K = zero_zero_int )
% 0.22/0.48 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.22/0.48 & ( ord_less_int @ I @ K ) )
% 0.22/0.48 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 0.22/0.48 & ( ord_less_int @ K @ I ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % zmod_trivial_iff
% 0.22/0.48 thf(fact_4560_pos__mod__conj,axiom,
% 0.22/0.48 ! [B: int,A: int] :
% 0.22/0.48 ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.48 => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
% 0.22/0.48 & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pos_mod_conj
% 0.22/0.48 thf(fact_4561_neg__mod__conj,axiom,
% 0.22/0.48 ! [B: int,A: int] :
% 0.22/0.48 ( ( ord_less_int @ B @ zero_zero_int )
% 0.22/0.48 => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
% 0.22/0.48 & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % neg_mod_conj
% 0.22/0.48 thf(fact_4562_zdiv__mono__strict,axiom,
% 0.22/0.48 ! [A2: int,B4: int,N: int] :
% 0.22/0.48 ( ( ord_less_int @ A2 @ B4 )
% 0.22/0.48 => ( ( ord_less_int @ zero_zero_int @ N )
% 0.22/0.48 => ( ( ( modulo_modulo_int @ A2 @ N )
% 0.22/0.48 = zero_zero_int )
% 0.22/0.48 => ( ( ( modulo_modulo_int @ B4 @ N )
% 0.22/0.48 = zero_zero_int )
% 0.22/0.48 => ( ord_less_int @ ( divide_divide_int @ A2 @ N ) @ ( divide_divide_int @ B4 @ N ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % zdiv_mono_strict
% 0.22/0.48 thf(fact_4563_div__mod__decomp__int,axiom,
% 0.22/0.48 ! [A2: int,N: int] :
% 0.22/0.48 ( A2
% 0.22/0.48 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N ) @ N ) @ ( modulo_modulo_int @ A2 @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % div_mod_decomp_int
% 0.22/0.48 thf(fact_4564_mod__pos__neg__trivial,axiom,
% 0.22/0.48 ! [K: int,L: int] :
% 0.22/0.48 ( ( ord_less_int @ zero_zero_int @ K )
% 0.22/0.48 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 0.22/0.48 => ( ( modulo_modulo_int @ K @ L )
% 0.22/0.48 = ( plus_plus_int @ K @ L ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_pos_neg_trivial
% 0.22/0.48 thf(fact_4565_mod__pos__geq,axiom,
% 0.22/0.48 ! [L: int,K: int] :
% 0.22/0.48 ( ( ord_less_int @ zero_zero_int @ L )
% 0.22/0.48 => ( ( ord_less_eq_int @ L @ K )
% 0.22/0.48 => ( ( modulo_modulo_int @ K @ L )
% 0.22/0.48 = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_pos_geq
% 0.22/0.48 thf(fact_4566_mod__int__pos__iff,axiom,
% 0.22/0.48 ! [K: int,L: int] :
% 0.22/0.48 ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
% 0.22/0.48 = ( ( dvd_dvd_int @ L @ K )
% 0.22/0.48 | ( ( L = zero_zero_int )
% 0.22/0.48 & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 0.22/0.48 | ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_int_pos_iff
% 0.22/0.48 thf(fact_4567_split__zmod,axiom,
% 0.22/0.48 ! [P: int > $o,N: int,K: int] :
% 0.22/0.48 ( ( P @ ( modulo_modulo_int @ N @ K ) )
% 0.22/0.48 = ( ( ( K = zero_zero_int )
% 0.22/0.48 => ( P @ N ) )
% 0.22/0.48 & ( ( ord_less_int @ zero_zero_int @ K )
% 0.22/0.48 => ! [I2: int,J3: int] :
% 0.22/0.48 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.22/0.48 & ( ord_less_int @ J3 @ K )
% 0.22/0.48 & ( N
% 0.22/0.48 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 0.22/0.48 => ( P @ J3 ) ) )
% 0.22/0.48 & ( ( ord_less_int @ K @ zero_zero_int )
% 0.22/0.48 => ! [I2: int,J3: int] :
% 0.22/0.48 ( ( ( ord_less_int @ K @ J3 )
% 0.22/0.48 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.22/0.48 & ( N
% 0.22/0.48 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 0.22/0.48 => ( P @ J3 ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % split_zmod
% 0.22/0.48 thf(fact_4568_int__mod__neg__eq,axiom,
% 0.22/0.48 ! [A: int,B: int,Q2: int,R2: int] :
% 0.22/0.48 ( ( A
% 0.22/0.48 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
% 0.22/0.48 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.22/0.48 => ( ( ord_less_int @ B @ R2 )
% 0.22/0.48 => ( ( modulo_modulo_int @ A @ B )
% 0.22/0.48 = R2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % int_mod_neg_eq
% 0.22/0.48 thf(fact_4569_int__mod__pos__eq,axiom,
% 0.22/0.48 ! [A: int,B: int,Q2: int,R2: int] :
% 0.22/0.48 ( ( A
% 0.22/0.48 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
% 0.22/0.48 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.22/0.48 => ( ( ord_less_int @ R2 @ B )
% 0.22/0.48 => ( ( modulo_modulo_int @ A @ B )
% 0.22/0.48 = R2 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % int_mod_pos_eq
% 0.22/0.48 thf(fact_4570_zmod__zmult2__eq,axiom,
% 0.22/0.48 ! [C: int,A: int,B: int] :
% 0.22/0.48 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.22/0.48 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 0.22/0.48 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % zmod_zmult2_eq
% 0.22/0.48 thf(fact_4571_verit__le__mono__div__int,axiom,
% 0.22/0.48 ! [A2: int,B4: int,N: int] :
% 0.22/0.48 ( ( ord_less_int @ A2 @ B4 )
% 0.22/0.48 => ( ( ord_less_int @ zero_zero_int @ N )
% 0.22/0.48 => ( ord_less_eq_int
% 0.22/0.48 @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N )
% 0.22/0.48 @ ( if_int
% 0.22/0.48 @ ( ( modulo_modulo_int @ B4 @ N )
% 0.22/0.48 = zero_zero_int )
% 0.22/0.48 @ one_one_int
% 0.22/0.48 @ zero_zero_int ) )
% 0.22/0.48 @ ( divide_divide_int @ B4 @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % verit_le_mono_div_int
% 0.22/0.48 thf(fact_4572_split__neg__lemma,axiom,
% 0.22/0.48 ! [K: int,P: int > int > $o,N: int] :
% 0.22/0.48 ( ( ord_less_int @ K @ zero_zero_int )
% 0.22/0.48 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 0.22/0.48 = ( ! [I2: int,J3: int] :
% 0.22/0.48 ( ( ( ord_less_int @ K @ J3 )
% 0.22/0.48 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.22/0.48 & ( N
% 0.22/0.48 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 0.22/0.48 => ( P @ I2 @ J3 ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % split_neg_lemma
% 0.22/0.48 thf(fact_4573_split__pos__lemma,axiom,
% 0.22/0.48 ! [K: int,P: int > int > $o,N: int] :
% 0.22/0.48 ( ( ord_less_int @ zero_zero_int @ K )
% 0.22/0.48 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 0.22/0.48 = ( ! [I2: int,J3: int] :
% 0.22/0.48 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.22/0.48 & ( ord_less_int @ J3 @ K )
% 0.22/0.48 & ( N
% 0.22/0.48 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 0.22/0.48 => ( P @ I2 @ J3 ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % split_pos_lemma
% 0.22/0.48 thf(fact_4574_pos__zmod__mult__2,axiom,
% 0.22/0.48 ! [A: int,B: int] :
% 0.22/0.48 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.22/0.48 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.22/0.48 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % pos_zmod_mult_2
% 0.22/0.48 thf(fact_4575_neg__zmod__mult__2,axiom,
% 0.22/0.48 ! [A: int,B: int] :
% 0.22/0.48 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.22/0.48 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.22/0.48 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % neg_zmod_mult_2
% 0.22/0.48 thf(fact_4576_binomial__eq__0,axiom,
% 0.22/0.48 ! [N: nat,K: nat] :
% 0.22/0.48 ( ( ord_less_nat @ N @ K )
% 0.22/0.48 => ( ( binomial @ N @ K )
% 0.22/0.48 = zero_zero_nat ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_eq_0
% 0.22/0.48 thf(fact_4577_Suc__times__binomial__eq,axiom,
% 0.22/0.48 ! [N: nat,K: nat] :
% 0.22/0.48 ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
% 0.22/0.48 = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % Suc_times_binomial_eq
% 0.22/0.48 thf(fact_4578_Suc__times__binomial,axiom,
% 0.22/0.48 ! [K: nat,N: nat] :
% 0.22/0.48 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
% 0.22/0.48 = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % Suc_times_binomial
% 0.22/0.48 thf(fact_4579_binomial__symmetric,axiom,
% 0.22/0.48 ! [K: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.48 => ( ( binomial @ N @ K )
% 0.22/0.48 = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_symmetric
% 0.22/0.48 thf(fact_4580_choose__mult__lemma,axiom,
% 0.22/0.48 ! [M: nat,R2: nat,K: nat] :
% 0.22/0.48 ( ( 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 ) )
% 0.22/0.48 = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % choose_mult_lemma
% 0.22/0.48 thf(fact_4581_binomial__le__pow,axiom,
% 0.22/0.48 ! [R2: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ R2 @ N )
% 0.22/0.48 => ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_le_pow
% 0.22/0.48 thf(fact_4582_zero__less__binomial,axiom,
% 0.22/0.48 ! [K: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.48 => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % zero_less_binomial
% 0.22/0.48 thf(fact_4583_Suc__times__binomial__add,axiom,
% 0.22/0.48 ! [A: nat,B: nat] :
% 0.22/0.48 ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
% 0.22/0.48 = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % Suc_times_binomial_add
% 0.22/0.48 thf(fact_4584_choose__mult,axiom,
% 0.22/0.48 ! [K: nat,M: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ K @ M )
% 0.22/0.48 => ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.48 => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 0.22/0.48 = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % choose_mult
% 0.22/0.48 thf(fact_4585_binomial__Suc__Suc__eq__times,axiom,
% 0.22/0.48 ! [N: nat,K: nat] :
% 0.22/0.48 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 0.22/0.48 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_Suc_Suc_eq_times
% 0.22/0.48 thf(fact_4586_binomial__absorb__comp,axiom,
% 0.22/0.48 ! [N: nat,K: nat] :
% 0.22/0.48 ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
% 0.22/0.48 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_absorb_comp
% 0.22/0.48 thf(fact_4587_binomial__absorption,axiom,
% 0.22/0.48 ! [K: nat,N: nat] :
% 0.22/0.48 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
% 0.22/0.48 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_absorption
% 0.22/0.48 thf(fact_4588_binomial__ge__n__over__k__pow__k,axiom,
% 0.22/0.48 ! [K: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.48 => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_ge_n_over_k_pow_k
% 0.22/0.48 thf(fact_4589_binomial__ge__n__over__k__pow__k,axiom,
% 0.22/0.48 ! [K: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.48 => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_ge_n_over_k_pow_k
% 0.22/0.48 thf(fact_4590_binomial__le__pow2,axiom,
% 0.22/0.48 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_le_pow2
% 0.22/0.48 thf(fact_4591_choose__reduce__nat,axiom,
% 0.22/0.48 ! [N: nat,K: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.48 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.22/0.48 => ( ( binomial @ N @ K )
% 0.22/0.48 = ( 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 ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % choose_reduce_nat
% 0.22/0.48 thf(fact_4592_times__binomial__minus1__eq,axiom,
% 0.22/0.48 ! [K: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.22/0.48 => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
% 0.22/0.48 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % times_binomial_minus1_eq
% 0.22/0.48 thf(fact_4593_mod__exhaust__less__4,axiom,
% 0.22/0.48 ! [M: nat] :
% 0.22/0.48 ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.22/0.48 = zero_zero_nat )
% 0.22/0.48 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.22/0.48 = one_one_nat )
% 0.22/0.48 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.22/0.48 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.48 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.22/0.48 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % mod_exhaust_less_4
% 0.22/0.48 thf(fact_4594_artanh__def,axiom,
% 0.22/0.48 ( artanh_real
% 0.22/0.48 = ( ^ [X3: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % artanh_def
% 0.22/0.48 thf(fact_4595_signed__take__bit__Suc,axiom,
% 0.22/0.48 ! [N: nat,A: code_integer] :
% 0.22/0.48 ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
% 0.22/0.48 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % signed_take_bit_Suc
% 0.22/0.48 thf(fact_4596_signed__take__bit__Suc,axiom,
% 0.22/0.48 ! [N: nat,A: int] :
% 0.22/0.48 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
% 0.22/0.48 = ( 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 ) ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % signed_take_bit_Suc
% 0.22/0.48 thf(fact_4597__C10_C,axiom,
% 0.22/0.48 ! [L: nat] :
% 0.22/0.48 ( ( foldr_real_real @ plus_plus_real @ ( replicate_real @ L @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ zero_zero_real )
% 0.22/0.48 = ( times_times_real @ ( semiri5074537144036343181t_real @ L ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % "10"
% 0.22/0.48 thf(fact_4598_add__scale__eq__noteq,axiom,
% 0.22/0.48 ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
% 0.22/0.48 ( ( R2 != zero_zero_complex )
% 0.22/0.48 => ( ( ( A = B )
% 0.22/0.48 & ( C != D ) )
% 0.22/0.48 => ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C ) )
% 0.22/0.48 != ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % add_scale_eq_noteq
% 0.22/0.48 thf(fact_4599_add__scale__eq__noteq,axiom,
% 0.22/0.48 ! [R2: real,A: real,B: real,C: real,D: real] :
% 0.22/0.48 ( ( R2 != zero_zero_real )
% 0.22/0.48 => ( ( ( A = B )
% 0.22/0.48 & ( C != D ) )
% 0.22/0.48 => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
% 0.22/0.48 != ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % add_scale_eq_noteq
% 0.22/0.48 thf(fact_4600_add__scale__eq__noteq,axiom,
% 0.22/0.48 ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
% 0.22/0.48 ( ( R2 != zero_zero_rat )
% 0.22/0.48 => ( ( ( A = B )
% 0.22/0.48 & ( C != D ) )
% 0.22/0.48 => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
% 0.22/0.48 != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % add_scale_eq_noteq
% 0.22/0.48 thf(fact_4601_add__scale__eq__noteq,axiom,
% 0.22/0.48 ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
% 0.22/0.48 ( ( R2 != zero_zero_nat )
% 0.22/0.48 => ( ( ( A = B )
% 0.22/0.48 & ( C != D ) )
% 0.22/0.48 => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
% 0.22/0.48 != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % add_scale_eq_noteq
% 0.22/0.48 thf(fact_4602_add__scale__eq__noteq,axiom,
% 0.22/0.48 ! [R2: int,A: int,B: int,C: int,D: int] :
% 0.22/0.48 ( ( R2 != zero_zero_int )
% 0.22/0.48 => ( ( ( A = B )
% 0.22/0.48 & ( C != D ) )
% 0.22/0.48 => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
% 0.22/0.48 != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % add_scale_eq_noteq
% 0.22/0.48 thf(fact_4603_fact__double,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( 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 ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_double
% 0.22/0.48 thf(fact_4604_fact__double,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( 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 ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_double
% 0.22/0.48 thf(fact_4605_fact__double,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.48 = ( 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 ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_double
% 0.22/0.48 thf(fact_4606_num_Osize__gen_I3_J,axiom,
% 0.22/0.48 ! [X33: num] :
% 0.22/0.48 ( ( size_num @ ( bit1 @ X33 ) )
% 0.22/0.48 = ( plus_plus_nat @ ( size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % num.size_gen(3)
% 0.22/0.48 thf(fact_4607_foldr0,axiom,
% 0.22/0.48 ! [Xs: list_real,C: real,D: real] :
% 0.22/0.48 ( ( foldr_real_real @ plus_plus_real @ Xs @ ( plus_plus_real @ C @ D ) )
% 0.22/0.48 = ( plus_plus_real @ ( foldr_real_real @ plus_plus_real @ Xs @ D ) @ C ) ) ).
% 0.22/0.48
% 0.22/0.48 % foldr0
% 0.22/0.48 thf(fact_4608_ln__less__cancel__iff,axiom,
% 0.22/0.48 ! [X: real,Y4: real] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.22/0.48 => ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y4 ) )
% 0.22/0.48 = ( ord_less_real @ X @ Y4 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_less_cancel_iff
% 0.22/0.48 thf(fact_4609_ln__inj__iff,axiom,
% 0.22/0.48 ! [X: real,Y4: real] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.22/0.48 => ( ( ( ln_ln_real @ X )
% 0.22/0.48 = ( ln_ln_real @ Y4 ) )
% 0.22/0.48 = ( X = Y4 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_inj_iff
% 0.22/0.48 thf(fact_4610_of__nat__fact,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( semiri681578069525770553at_rat @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.22/0.48 = ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_nat_fact
% 0.22/0.48 thf(fact_4611_of__nat__fact,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( semiri1314217659103216013at_int @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.22/0.48 = ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_nat_fact
% 0.22/0.48 thf(fact_4612_of__nat__fact,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( semiri1316708129612266289at_nat @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.22/0.48 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_nat_fact
% 0.22/0.48 thf(fact_4613_of__nat__fact,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( semiri5074537144036343181t_real @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.22/0.48 = ( semiri2265585572941072030t_real @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_nat_fact
% 0.22/0.48 thf(fact_4614_of__nat__fact,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( semiri8010041392384452111omplex @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.22/0.48 = ( semiri5044797733671781792omplex @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % of_nat_fact
% 0.22/0.48 thf(fact_4615_signed__take__bit__of__0,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
% 0.22/0.48 = zero_zero_int ) ).
% 0.22/0.48
% 0.22/0.48 % signed_take_bit_of_0
% 0.22/0.48 thf(fact_4616_replicate__eq__replicate,axiom,
% 0.22/0.48 ! [M: nat,X: real,N: nat,Y4: real] :
% 0.22/0.48 ( ( ( replicate_real @ M @ X )
% 0.22/0.48 = ( replicate_real @ N @ Y4 ) )
% 0.22/0.48 = ( ( M = N )
% 0.22/0.48 & ( ( M != zero_zero_nat )
% 0.22/0.48 => ( X = Y4 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % replicate_eq_replicate
% 0.22/0.48 thf(fact_4617_replicate__eq__replicate,axiom,
% 0.22/0.48 ! [M: nat,X: vEBT_VEBT,N: nat,Y4: vEBT_VEBT] :
% 0.22/0.48 ( ( ( replicate_VEBT_VEBT @ M @ X )
% 0.22/0.48 = ( replicate_VEBT_VEBT @ N @ Y4 ) )
% 0.22/0.48 = ( ( M = N )
% 0.22/0.48 & ( ( M != zero_zero_nat )
% 0.22/0.48 => ( X = Y4 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % replicate_eq_replicate
% 0.22/0.48 thf(fact_4618_ln__le__cancel__iff,axiom,
% 0.22/0.48 ! [X: real,Y4: real] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.22/0.48 => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y4 ) )
% 0.22/0.48 = ( ord_less_eq_real @ X @ Y4 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_le_cancel_iff
% 0.22/0.48 thf(fact_4619_ln__less__zero__iff,axiom,
% 0.22/0.48 ! [X: real] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 0.22/0.48 = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_less_zero_iff
% 0.22/0.48 thf(fact_4620_ln__gt__zero__iff,axiom,
% 0.22/0.48 ! [X: real] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.22/0.48 = ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_gt_zero_iff
% 0.22/0.48 thf(fact_4621_ln__eq__zero__iff,axiom,
% 0.22/0.48 ! [X: real] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ( ( ln_ln_real @ X )
% 0.22/0.48 = zero_zero_real )
% 0.22/0.48 = ( X = one_one_real ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_eq_zero_iff
% 0.22/0.48 thf(fact_4622_ln__one,axiom,
% 0.22/0.48 ( ( ln_ln_real @ one_one_real )
% 0.22/0.48 = zero_zero_real ) ).
% 0.22/0.48
% 0.22/0.48 % ln_one
% 0.22/0.48 thf(fact_4623_fact__0,axiom,
% 0.22/0.48 ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
% 0.22/0.48 = one_one_rat ) ).
% 0.22/0.48
% 0.22/0.48 % fact_0
% 0.22/0.48 thf(fact_4624_fact__0,axiom,
% 0.22/0.48 ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
% 0.22/0.48 = one_one_int ) ).
% 0.22/0.48
% 0.22/0.48 % fact_0
% 0.22/0.48 thf(fact_4625_fact__0,axiom,
% 0.22/0.48 ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
% 0.22/0.48 = one_one_nat ) ).
% 0.22/0.48
% 0.22/0.48 % fact_0
% 0.22/0.48 thf(fact_4626_fact__0,axiom,
% 0.22/0.48 ( ( semiri2265585572941072030t_real @ zero_zero_nat )
% 0.22/0.48 = one_one_real ) ).
% 0.22/0.48
% 0.22/0.48 % fact_0
% 0.22/0.48 thf(fact_4627_fact__0,axiom,
% 0.22/0.48 ( ( semiri5044797733671781792omplex @ zero_zero_nat )
% 0.22/0.48 = one_one_complex ) ).
% 0.22/0.48
% 0.22/0.48 % fact_0
% 0.22/0.48 thf(fact_4628_signed__take__bit__Suc__1,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
% 0.22/0.48 = one_one_int ) ).
% 0.22/0.48
% 0.22/0.48 % signed_take_bit_Suc_1
% 0.22/0.48 thf(fact_4629_signed__take__bit__numeral__of__1,axiom,
% 0.22/0.48 ! [K: num] :
% 0.22/0.48 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
% 0.22/0.48 = one_one_int ) ).
% 0.22/0.48
% 0.22/0.48 % signed_take_bit_numeral_of_1
% 0.22/0.48 thf(fact_4630_ln__ge__zero__iff,axiom,
% 0.22/0.48 ! [X: real] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.22/0.48 = ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_ge_zero_iff
% 0.22/0.48 thf(fact_4631_ln__le__zero__iff,axiom,
% 0.22/0.48 ! [X: real] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 0.22/0.48 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_le_zero_iff
% 0.22/0.48 thf(fact_4632_fact__Suc__0,axiom,
% 0.22/0.48 ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
% 0.22/0.48 = one_one_rat ) ).
% 0.22/0.48
% 0.22/0.48 % fact_Suc_0
% 0.22/0.48 thf(fact_4633_fact__Suc__0,axiom,
% 0.22/0.48 ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
% 0.22/0.48 = one_one_int ) ).
% 0.22/0.48
% 0.22/0.48 % fact_Suc_0
% 0.22/0.48 thf(fact_4634_fact__Suc__0,axiom,
% 0.22/0.48 ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
% 0.22/0.48 = one_one_nat ) ).
% 0.22/0.48
% 0.22/0.48 % fact_Suc_0
% 0.22/0.48 thf(fact_4635_fact__Suc__0,axiom,
% 0.22/0.48 ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
% 0.22/0.48 = one_one_real ) ).
% 0.22/0.48
% 0.22/0.48 % fact_Suc_0
% 0.22/0.48 thf(fact_4636_fact__Suc__0,axiom,
% 0.22/0.48 ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
% 0.22/0.48 = one_one_complex ) ).
% 0.22/0.48
% 0.22/0.48 % fact_Suc_0
% 0.22/0.48 thf(fact_4637_fact__Suc,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_Suc
% 0.22/0.48 thf(fact_4638_fact__Suc,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_Suc
% 0.22/0.48 thf(fact_4639_fact__Suc,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_Suc
% 0.22/0.48 thf(fact_4640_fact__Suc,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_Suc
% 0.22/0.48 thf(fact_4641_fact__Suc,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( semiri5044797733671781792omplex @ ( suc @ N ) )
% 0.22/0.48 = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_Suc
% 0.22/0.48 thf(fact_4642_fact__2,axiom,
% 0.22/0.48 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.48 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_2
% 0.22/0.48 thf(fact_4643_fact__2,axiom,
% 0.22/0.48 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.48 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_2
% 0.22/0.48 thf(fact_4644_fact__2,axiom,
% 0.22/0.48 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.48 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_2
% 0.22/0.48 thf(fact_4645_fact__2,axiom,
% 0.22/0.48 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.48 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_2
% 0.22/0.48 thf(fact_4646_fact__2,axiom,
% 0.22/0.48 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.48 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_2
% 0.22/0.48 thf(fact_4647_signed__take__bit__Suc__bit0,axiom,
% 0.22/0.48 ! [N: nat,K: num] :
% 0.22/0.48 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.22/0.48 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % signed_take_bit_Suc_bit0
% 0.22/0.48 thf(fact_4648_signed__take__bit__Suc__bit1,axiom,
% 0.22/0.48 ! [N: nat,K: num] :
% 0.22/0.48 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.22/0.48 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.22/0.48
% 0.22/0.48 % signed_take_bit_Suc_bit1
% 0.22/0.48 thf(fact_4649_fact__mono__nat,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.48 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_mono_nat
% 0.22/0.48 thf(fact_4650_fact__ge__self,axiom,
% 0.22/0.48 ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_ge_self
% 0.22/0.48 thf(fact_4651_fact__nonzero,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( semiri773545260158071498ct_rat @ N )
% 0.22/0.48 != zero_zero_rat ) ).
% 0.22/0.48
% 0.22/0.48 % fact_nonzero
% 0.22/0.48 thf(fact_4652_fact__nonzero,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( semiri1406184849735516958ct_int @ N )
% 0.22/0.48 != zero_zero_int ) ).
% 0.22/0.48
% 0.22/0.48 % fact_nonzero
% 0.22/0.48 thf(fact_4653_fact__nonzero,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( semiri1408675320244567234ct_nat @ N )
% 0.22/0.48 != zero_zero_nat ) ).
% 0.22/0.48
% 0.22/0.48 % fact_nonzero
% 0.22/0.48 thf(fact_4654_fact__nonzero,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( semiri2265585572941072030t_real @ N )
% 0.22/0.48 != zero_zero_real ) ).
% 0.22/0.48
% 0.22/0.48 % fact_nonzero
% 0.22/0.48 thf(fact_4655_fact__nonzero,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ( ( semiri5044797733671781792omplex @ N )
% 0.22/0.48 != zero_zero_complex ) ).
% 0.22/0.48
% 0.22/0.48 % fact_nonzero
% 0.22/0.48 thf(fact_4656_signed__take__bit__mult,axiom,
% 0.22/0.48 ! [N: nat,K: int,L: int] :
% 0.22/0.48 ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 0.22/0.48 = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % signed_take_bit_mult
% 0.22/0.48 thf(fact_4657_signed__take__bit__add,axiom,
% 0.22/0.48 ! [N: nat,K: int,L: int] :
% 0.22/0.48 ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 0.22/0.48 = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % signed_take_bit_add
% 0.22/0.48 thf(fact_4658_signed__take__bit__diff,axiom,
% 0.22/0.48 ! [N: nat,K: int,L: int] :
% 0.22/0.48 ( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 0.22/0.48 = ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % signed_take_bit_diff
% 0.22/0.48 thf(fact_4659_ln__less__self,axiom,
% 0.22/0.48 ! [X: real] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_less_self
% 0.22/0.48 thf(fact_4660_fact__less__mono__nat,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.22/0.48 => ( ( ord_less_nat @ M @ N )
% 0.22/0.48 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_less_mono_nat
% 0.22/0.48 thf(fact_4661_fact__ge__zero,axiom,
% 0.22/0.48 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_ge_zero
% 0.22/0.48 thf(fact_4662_fact__ge__zero,axiom,
% 0.22/0.48 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_ge_zero
% 0.22/0.48 thf(fact_4663_fact__ge__zero,axiom,
% 0.22/0.48 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_ge_zero
% 0.22/0.48 thf(fact_4664_fact__ge__zero,axiom,
% 0.22/0.48 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_ge_zero
% 0.22/0.48 thf(fact_4665_fact__gt__zero,axiom,
% 0.22/0.48 ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_gt_zero
% 0.22/0.48 thf(fact_4666_fact__gt__zero,axiom,
% 0.22/0.48 ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_gt_zero
% 0.22/0.48 thf(fact_4667_fact__gt__zero,axiom,
% 0.22/0.48 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_gt_zero
% 0.22/0.48 thf(fact_4668_fact__gt__zero,axiom,
% 0.22/0.48 ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_gt_zero
% 0.22/0.48 thf(fact_4669_fact__not__neg,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% 0.22/0.48
% 0.22/0.48 % fact_not_neg
% 0.22/0.48 thf(fact_4670_fact__not__neg,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% 0.22/0.48
% 0.22/0.48 % fact_not_neg
% 0.22/0.48 thf(fact_4671_fact__not__neg,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% 0.22/0.48
% 0.22/0.48 % fact_not_neg
% 0.22/0.48 thf(fact_4672_fact__not__neg,axiom,
% 0.22/0.48 ! [N: nat] :
% 0.22/0.48 ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% 0.22/0.48
% 0.22/0.48 % fact_not_neg
% 0.22/0.48 thf(fact_4673_fact__ge__1,axiom,
% 0.22/0.48 ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_ge_1
% 0.22/0.48 thf(fact_4674_fact__ge__1,axiom,
% 0.22/0.48 ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_ge_1
% 0.22/0.48 thf(fact_4675_fact__ge__1,axiom,
% 0.22/0.48 ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_ge_1
% 0.22/0.48 thf(fact_4676_fact__ge__1,axiom,
% 0.22/0.48 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_ge_1
% 0.22/0.48 thf(fact_4677_fact__mono,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.48 => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_mono
% 0.22/0.48 thf(fact_4678_fact__mono,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.48 => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_mono
% 0.22/0.48 thf(fact_4679_fact__mono,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.48 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_mono
% 0.22/0.48 thf(fact_4680_fact__mono,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.48 => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_mono
% 0.22/0.48 thf(fact_4681_fact__dvd,axiom,
% 0.22/0.48 ! [N: nat,M: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.48 => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_dvd
% 0.22/0.48 thf(fact_4682_fact__dvd,axiom,
% 0.22/0.48 ! [N: nat,M: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.48 => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_dvd
% 0.22/0.48 thf(fact_4683_fact__dvd,axiom,
% 0.22/0.48 ! [N: nat,M: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.48 => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_dvd
% 0.22/0.48 thf(fact_4684_fact__dvd,axiom,
% 0.22/0.48 ! [N: nat,M: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.48 => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_dvd
% 0.22/0.48 thf(fact_4685_fact__diff__Suc,axiom,
% 0.22/0.48 ! [N: nat,M: nat] :
% 0.22/0.48 ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.22/0.48 => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
% 0.22/0.48 = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_diff_Suc
% 0.22/0.48 thf(fact_4686_ln__bound,axiom,
% 0.22/0.48 ! [X: real] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_bound
% 0.22/0.48 thf(fact_4687_ln__gt__zero,axiom,
% 0.22/0.48 ! [X: real] :
% 0.22/0.48 ( ( ord_less_real @ one_one_real @ X )
% 0.22/0.48 => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_gt_zero
% 0.22/0.48 thf(fact_4688_ln__less__zero,axiom,
% 0.22/0.48 ! [X: real] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ( ord_less_real @ X @ one_one_real )
% 0.22/0.48 => ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_less_zero
% 0.22/0.48 thf(fact_4689_ln__gt__zero__imp__gt__one,axiom,
% 0.22/0.48 ! [X: real] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.22/0.48 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_gt_zero_imp_gt_one
% 0.22/0.48 thf(fact_4690_ln__ge__zero,axiom,
% 0.22/0.48 ! [X: real] :
% 0.22/0.48 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.22/0.48 => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_ge_zero
% 0.22/0.48 thf(fact_4691_fact__ge__Suc__0__nat,axiom,
% 0.22/0.48 ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_ge_Suc_0_nat
% 0.22/0.48 thf(fact_4692_dvd__fact,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.22/0.48 => ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.48 => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % dvd_fact
% 0.22/0.48 thf(fact_4693_fact__less__mono,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.22/0.48 => ( ( ord_less_nat @ M @ N )
% 0.22/0.48 => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_less_mono
% 0.22/0.48 thf(fact_4694_fact__less__mono,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.22/0.48 => ( ( ord_less_nat @ M @ N )
% 0.22/0.48 => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_less_mono
% 0.22/0.48 thf(fact_4695_fact__less__mono,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.22/0.48 => ( ( ord_less_nat @ M @ N )
% 0.22/0.48 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_less_mono
% 0.22/0.48 thf(fact_4696_fact__less__mono,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.22/0.48 => ( ( ord_less_nat @ M @ N )
% 0.22/0.48 => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_less_mono
% 0.22/0.48 thf(fact_4697_fact__fact__dvd__fact,axiom,
% 0.22/0.48 ! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_fact_dvd_fact
% 0.22/0.48 thf(fact_4698_fact__fact__dvd__fact,axiom,
% 0.22/0.48 ! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_fact_dvd_fact
% 0.22/0.48 thf(fact_4699_fact__fact__dvd__fact,axiom,
% 0.22/0.48 ! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_fact_dvd_fact
% 0.22/0.48 thf(fact_4700_fact__fact__dvd__fact,axiom,
% 0.22/0.48 ! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_fact_dvd_fact
% 0.22/0.48 thf(fact_4701_fact__fact__dvd__fact,axiom,
% 0.22/0.48 ! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_fact_dvd_fact
% 0.22/0.48 thf(fact_4702_fact__mod,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.48 => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
% 0.22/0.48 = zero_zero_int ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_mod
% 0.22/0.48 thf(fact_4703_fact__mod,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.48 => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
% 0.22/0.48 = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_mod
% 0.22/0.48 thf(fact_4704_fact__mod,axiom,
% 0.22/0.48 ! [M: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.48 => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
% 0.22/0.48 = zero_zero_nat ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_mod
% 0.22/0.48 thf(fact_4705_fact__le__power,axiom,
% 0.22/0.48 ! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_le_power
% 0.22/0.48 thf(fact_4706_fact__le__power,axiom,
% 0.22/0.48 ! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_le_power
% 0.22/0.48 thf(fact_4707_fact__le__power,axiom,
% 0.22/0.48 ! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_le_power
% 0.22/0.48 thf(fact_4708_fact__le__power,axiom,
% 0.22/0.48 ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_le_power
% 0.22/0.48 thf(fact_4709_ln__2__less__1,axiom,
% 0.22/0.48 ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 0.22/0.48
% 0.22/0.48 % ln_2_less_1
% 0.22/0.48 thf(fact_4710_ln__ge__zero__imp__ge__one,axiom,
% 0.22/0.48 ! [X: real] :
% 0.22/0.48 ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.22/0.48 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_ge_zero_imp_ge_one
% 0.22/0.48 thf(fact_4711_ln__add__one__self__le__self,axiom,
% 0.22/0.48 ! [X: real] :
% 0.22/0.48 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_add_one_self_le_self
% 0.22/0.48 thf(fact_4712_ln__mult,axiom,
% 0.22/0.48 ! [X: real,Y4: real] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.22/0.48 => ( ( ln_ln_real @ ( times_times_real @ X @ Y4 ) )
% 0.22/0.48 = ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y4 ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_mult
% 0.22/0.48 thf(fact_4713_ln__eq__minus__one,axiom,
% 0.22/0.48 ! [X: real] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ( ( ln_ln_real @ X )
% 0.22/0.48 = ( minus_minus_real @ X @ one_one_real ) )
% 0.22/0.48 => ( X = one_one_real ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_eq_minus_one
% 0.22/0.48 thf(fact_4714_ln__div,axiom,
% 0.22/0.48 ! [X: real,Y4: real] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.22/0.48 => ( ( ln_ln_real @ ( divide_divide_real @ X @ Y4 ) )
% 0.22/0.48 = ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y4 ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_div
% 0.22/0.48 thf(fact_4715_fact__div__fact__le__pow,axiom,
% 0.22/0.48 ! [R2: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ R2 @ N )
% 0.22/0.48 => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_div_fact_le_pow
% 0.22/0.48 thf(fact_4716_binomial__fact__lemma,axiom,
% 0.22/0.48 ! [K: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.48 => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 0.22/0.48 = ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_fact_lemma
% 0.22/0.48 thf(fact_4717_choose__dvd,axiom,
% 0.22/0.48 ! [K: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.48 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % choose_dvd
% 0.22/0.48 thf(fact_4718_choose__dvd,axiom,
% 0.22/0.48 ! [K: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.48 => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % choose_dvd
% 0.22/0.48 thf(fact_4719_choose__dvd,axiom,
% 0.22/0.48 ! [K: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.48 => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % choose_dvd
% 0.22/0.48 thf(fact_4720_choose__dvd,axiom,
% 0.22/0.48 ! [K: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.48 => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % choose_dvd
% 0.22/0.48 thf(fact_4721_choose__dvd,axiom,
% 0.22/0.48 ! [K: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.48 => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % choose_dvd
% 0.22/0.48 thf(fact_4722_ln__le__minus__one,axiom,
% 0.22/0.48 ! [X: real] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_le_minus_one
% 0.22/0.48 thf(fact_4723_ln__diff__le,axiom,
% 0.22/0.48 ! [X: real,Y4: real] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.22/0.48 => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y4 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y4 ) @ Y4 ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_diff_le
% 0.22/0.48 thf(fact_4724_ln__realpow,axiom,
% 0.22/0.48 ! [X: real,N: nat] :
% 0.22/0.48 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.48 => ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
% 0.22/0.48 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % ln_realpow
% 0.22/0.48 thf(fact_4725_binomial__altdef__nat,axiom,
% 0.22/0.48 ! [K: nat,N: nat] :
% 0.22/0.48 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.48 => ( ( binomial @ N @ K )
% 0.22/0.48 = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % binomial_altdef_nat
% 0.22/0.48 thf(fact_4726_signed__take__bit__int__less__exp,axiom,
% 0.22/0.48 ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.22/0.48
% 0.22/0.48 % signed_take_bit_int_less_exp
% 0.22/0.48 thf(fact_4727_num_Osize__gen_I1_J,axiom,
% 0.22/0.48 ( ( size_num @ one )
% 0.22/0.48 = zero_zero_nat ) ).
% 0.22/0.48
% 0.22/0.48 % num.size_gen(1)
% 0.22/0.48 thf(fact_4728_even__signed__take__bit__iff,axiom,
% 0.22/0.48 ! [M: nat,A: code_integer] :
% 0.22/0.48 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
% 0.22/0.48 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.22/0.48
% 0.22/0.48 % even_signed_take_bit_iff
% 0.22/0.48 thf(fact_4729_even__signed__take__bit__iff,axiom,
% 0.22/0.48 ! [M: nat,A: int] :
% 0.22/0.48 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
% 0.22/0.48 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.22/0.48
% 0.22/0.48 % even_signed_take_bit_iff
% 0.22/0.48 thf(fact_4730_square__fact__le__2__fact,axiom,
% 0.22/0.48 ! [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 ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % square_fact_le_2_fact
% 0.22/0.48 thf(fact_4731_fact__num__eq__if,axiom,
% 0.22/0.48 ( semiri773545260158071498ct_rat
% 0.22/0.48 = ( ^ [M5: nat] : ( if_rat @ ( M5 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M5 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 0.22/0.48
% 0.22/0.48 % fact_num_eq_if
% 0.22/0.48 thf(fact_4732_fact__num__eq__if,axiom,
% 0.22/0.48 ( semiri1406184849735516958ct_int
% 0.22/0.48 = ( ^ [M5: nat] : ( if_int @ ( M5 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % fact_num_eq_if
% 0.22/0.49 thf(fact_4733_fact__num__eq__if,axiom,
% 0.22/0.49 ( semiri1408675320244567234ct_nat
% 0.22/0.49 = ( ^ [M5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M5 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % fact_num_eq_if
% 0.22/0.49 thf(fact_4734_fact__num__eq__if,axiom,
% 0.22/0.49 ( semiri2265585572941072030t_real
% 0.22/0.49 = ( ^ [M5: nat] : ( if_real @ ( M5 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % fact_num_eq_if
% 0.22/0.49 thf(fact_4735_fact__num__eq__if,axiom,
% 0.22/0.49 ( semiri5044797733671781792omplex
% 0.22/0.49 = ( ^ [M5: nat] : ( if_complex @ ( M5 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M5 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % fact_num_eq_if
% 0.22/0.49 thf(fact_4736_fact__reduce,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.49 => ( ( semiri773545260158071498ct_rat @ N )
% 0.22/0.49 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % fact_reduce
% 0.22/0.49 thf(fact_4737_fact__reduce,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.49 => ( ( semiri1406184849735516958ct_int @ N )
% 0.22/0.49 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % fact_reduce
% 0.22/0.49 thf(fact_4738_fact__reduce,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.49 => ( ( semiri1408675320244567234ct_nat @ N )
% 0.22/0.49 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % fact_reduce
% 0.22/0.49 thf(fact_4739_fact__reduce,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.49 => ( ( semiri2265585572941072030t_real @ N )
% 0.22/0.49 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % fact_reduce
% 0.22/0.49 thf(fact_4740_fact__reduce,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.49 => ( ( semiri5044797733671781792omplex @ N )
% 0.22/0.49 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % fact_reduce
% 0.22/0.49 thf(fact_4741_signed__take__bit__int__less__self__iff,axiom,
% 0.22/0.49 ! [N: nat,K: int] :
% 0.22/0.49 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 0.22/0.49 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 0.22/0.49
% 0.22/0.49 % signed_take_bit_int_less_self_iff
% 0.22/0.49 thf(fact_4742_signed__take__bit__int__greater__eq__self__iff,axiom,
% 0.22/0.49 ! [K: int,N: nat] :
% 0.22/0.49 ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.22/0.49 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % signed_take_bit_int_greater_eq_self_iff
% 0.22/0.49 thf(fact_4743_fact__binomial,axiom,
% 0.22/0.49 ! [K: nat,N: nat] :
% 0.22/0.49 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.49 => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
% 0.22/0.49 = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % fact_binomial
% 0.22/0.49 thf(fact_4744_fact__binomial,axiom,
% 0.22/0.49 ! [K: nat,N: nat] :
% 0.22/0.49 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.49 => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
% 0.22/0.49 = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % fact_binomial
% 0.22/0.49 thf(fact_4745_fact__binomial,axiom,
% 0.22/0.49 ! [K: nat,N: nat] :
% 0.22/0.49 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.49 => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
% 0.22/0.49 = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % fact_binomial
% 0.22/0.49 thf(fact_4746_binomial__fact,axiom,
% 0.22/0.49 ! [K: nat,N: nat] :
% 0.22/0.49 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.49 => ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 0.22/0.49 = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % binomial_fact
% 0.22/0.49 thf(fact_4747_binomial__fact,axiom,
% 0.22/0.49 ! [K: nat,N: nat] :
% 0.22/0.49 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.49 => ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 0.22/0.49 = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % binomial_fact
% 0.22/0.49 thf(fact_4748_binomial__fact,axiom,
% 0.22/0.49 ! [K: nat,N: nat] :
% 0.22/0.49 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.49 => ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 0.22/0.49 = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % binomial_fact
% 0.22/0.49 thf(fact_4749_add__0__iff,axiom,
% 0.22/0.49 ! [B: complex,A: complex] :
% 0.22/0.49 ( ( B
% 0.22/0.49 = ( plus_plus_complex @ B @ A ) )
% 0.22/0.49 = ( A = zero_zero_complex ) ) ).
% 0.22/0.49
% 0.22/0.49 % add_0_iff
% 0.22/0.49 thf(fact_4750_add__0__iff,axiom,
% 0.22/0.49 ! [B: real,A: real] :
% 0.22/0.49 ( ( B
% 0.22/0.49 = ( plus_plus_real @ B @ A ) )
% 0.22/0.49 = ( A = zero_zero_real ) ) ).
% 0.22/0.49
% 0.22/0.49 % add_0_iff
% 0.22/0.49 thf(fact_4751_add__0__iff,axiom,
% 0.22/0.49 ! [B: rat,A: rat] :
% 0.22/0.49 ( ( B
% 0.22/0.49 = ( plus_plus_rat @ B @ A ) )
% 0.22/0.49 = ( A = zero_zero_rat ) ) ).
% 0.22/0.49
% 0.22/0.49 % add_0_iff
% 0.22/0.49 thf(fact_4752_add__0__iff,axiom,
% 0.22/0.49 ! [B: nat,A: nat] :
% 0.22/0.49 ( ( B
% 0.22/0.49 = ( plus_plus_nat @ B @ A ) )
% 0.22/0.49 = ( A = zero_zero_nat ) ) ).
% 0.22/0.49
% 0.22/0.49 % add_0_iff
% 0.22/0.49 thf(fact_4753_add__0__iff,axiom,
% 0.22/0.49 ! [B: int,A: int] :
% 0.22/0.49 ( ( B
% 0.22/0.49 = ( plus_plus_int @ B @ A ) )
% 0.22/0.49 = ( A = zero_zero_int ) ) ).
% 0.22/0.49
% 0.22/0.49 % add_0_iff
% 0.22/0.49 thf(fact_4754_crossproduct__eq,axiom,
% 0.22/0.49 ! [W: real,Y4: real,X: real,Z: real] :
% 0.22/0.49 ( ( ( plus_plus_real @ ( times_times_real @ W @ Y4 ) @ ( times_times_real @ X @ Z ) )
% 0.22/0.49 = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X @ Y4 ) ) )
% 0.22/0.49 = ( ( W = X )
% 0.22/0.49 | ( Y4 = Z ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % crossproduct_eq
% 0.22/0.49 thf(fact_4755_crossproduct__eq,axiom,
% 0.22/0.49 ! [W: rat,Y4: rat,X: rat,Z: rat] :
% 0.22/0.49 ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y4 ) @ ( times_times_rat @ X @ Z ) )
% 0.22/0.49 = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y4 ) ) )
% 0.22/0.49 = ( ( W = X )
% 0.22/0.49 | ( Y4 = Z ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % crossproduct_eq
% 0.22/0.49 thf(fact_4756_crossproduct__eq,axiom,
% 0.22/0.49 ! [W: nat,Y4: nat,X: nat,Z: nat] :
% 0.22/0.49 ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y4 ) @ ( times_times_nat @ X @ Z ) )
% 0.22/0.49 = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y4 ) ) )
% 0.22/0.49 = ( ( W = X )
% 0.22/0.49 | ( Y4 = Z ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % crossproduct_eq
% 0.22/0.49 thf(fact_4757_crossproduct__eq,axiom,
% 0.22/0.49 ! [W: int,Y4: int,X: int,Z: int] :
% 0.22/0.49 ( ( ( plus_plus_int @ ( times_times_int @ W @ Y4 ) @ ( times_times_int @ X @ Z ) )
% 0.22/0.49 = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y4 ) ) )
% 0.22/0.49 = ( ( W = X )
% 0.22/0.49 | ( Y4 = Z ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % crossproduct_eq
% 0.22/0.49 thf(fact_4758_crossproduct__noteq,axiom,
% 0.22/0.49 ! [A: real,B: real,C: real,D: real] :
% 0.22/0.49 ( ( ( A != B )
% 0.22/0.49 & ( C != D ) )
% 0.22/0.49 = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
% 0.22/0.49 != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % crossproduct_noteq
% 0.22/0.49 thf(fact_4759_crossproduct__noteq,axiom,
% 0.22/0.49 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.22/0.49 ( ( ( A != B )
% 0.22/0.49 & ( C != D ) )
% 0.22/0.49 = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
% 0.22/0.49 != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % crossproduct_noteq
% 0.22/0.49 thf(fact_4760_crossproduct__noteq,axiom,
% 0.22/0.49 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.22/0.49 ( ( ( A != B )
% 0.22/0.49 & ( C != D ) )
% 0.22/0.49 = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
% 0.22/0.49 != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % crossproduct_noteq
% 0.22/0.49 thf(fact_4761_crossproduct__noteq,axiom,
% 0.22/0.49 ! [A: int,B: int,C: int,D: int] :
% 0.22/0.49 ( ( ( A != B )
% 0.22/0.49 & ( C != D ) )
% 0.22/0.49 = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
% 0.22/0.49 != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % crossproduct_noteq
% 0.22/0.49 thf(fact_4762_signed__take__bit__int__less__eq,axiom,
% 0.22/0.49 ! [N: nat,K: int] :
% 0.22/0.49 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 0.22/0.49 => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % signed_take_bit_int_less_eq
% 0.22/0.49 thf(fact_4763_eq__diff__eq_H,axiom,
% 0.22/0.49 ! [X: real,Y4: real,Z: real] :
% 0.22/0.49 ( ( X
% 0.22/0.49 = ( minus_minus_real @ Y4 @ Z ) )
% 0.22/0.49 = ( Y4
% 0.22/0.49 = ( plus_plus_real @ X @ Z ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % eq_diff_eq'
% 0.22/0.49 thf(fact_4764_ln__one__plus__pos__lower__bound,axiom,
% 0.22/0.49 ! [X: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.49 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.22/0.49 => ( ord_less_eq_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % ln_one_plus_pos_lower_bound
% 0.22/0.49 thf(fact_4765_num_Osize__gen_I2_J,axiom,
% 0.22/0.49 ! [X2: num] :
% 0.22/0.49 ( ( size_num @ ( bit0 @ X2 ) )
% 0.22/0.49 = ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % num.size_gen(2)
% 0.22/0.49 thf(fact_4766_binomial__code,axiom,
% 0.22/0.49 ( binomial
% 0.22/0.49 = ( ^ [N4: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N4 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N4 @ ( minus_minus_nat @ N4 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N4 @ K3 ) @ one_one_nat ) @ N4 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % binomial_code
% 0.22/0.49 thf(fact_4767_tanh__ln__real,axiom,
% 0.22/0.49 ! [X: real] :
% 0.22/0.49 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.49 => ( ( tanh_real @ ( ln_ln_real @ X ) )
% 0.22/0.49 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % tanh_ln_real
% 0.22/0.49 thf(fact_4768_ln__one__minus__pos__lower__bound,axiom,
% 0.22/0.49 ! [X: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.49 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.49 => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % ln_one_minus_pos_lower_bound
% 0.22/0.49 thf(fact_4769_signed__take__bit__Suc__minus__bit1,axiom,
% 0.22/0.49 ! [N: nat,K: num] :
% 0.22/0.49 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.22/0.49 = ( 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 ) ) ).
% 0.22/0.49
% 0.22/0.49 % signed_take_bit_Suc_minus_bit1
% 0.22/0.49 thf(fact_4770_signed__take__bit__rec,axiom,
% 0.22/0.49 ( bit_ri6519982836138164636nteger
% 0.22/0.49 = ( ^ [N4: nat,A3: code_integer] : ( if_Code_integer @ ( N4 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % signed_take_bit_rec
% 0.22/0.49 thf(fact_4771_signed__take__bit__rec,axiom,
% 0.22/0.49 ( bit_ri631733984087533419it_int
% 0.22/0.49 = ( ^ [N4: nat,A3: int] : ( if_int @ ( N4 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % signed_take_bit_rec
% 0.22/0.49 thf(fact_4772_map__cnt,axiom,
% 0.22/0.49 ( ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ zero_zero_real )
% 0.22/0.49 = ( times_times_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % map_cnt
% 0.22/0.49 thf(fact_4773_abs__ln__one__plus__x__minus__x__bound,axiom,
% 0.22/0.49 ! [X: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.49 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_ln_one_plus_x_minus_x_bound
% 0.22/0.49 thf(fact_4774_add_Oinverse__inverse,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 0.22/0.49 = A ) ).
% 0.22/0.49
% 0.22/0.49 % add.inverse_inverse
% 0.22/0.49 thf(fact_4775_add_Oinverse__inverse,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 0.22/0.49 = A ) ).
% 0.22/0.49
% 0.22/0.49 % add.inverse_inverse
% 0.22/0.49 thf(fact_4776_add_Oinverse__inverse,axiom,
% 0.22/0.49 ! [A: complex] :
% 0.22/0.49 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
% 0.22/0.49 = A ) ).
% 0.22/0.49
% 0.22/0.49 % add.inverse_inverse
% 0.22/0.49 thf(fact_4777_add_Oinverse__inverse,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.22/0.49 = A ) ).
% 0.22/0.49
% 0.22/0.49 % add.inverse_inverse
% 0.22/0.49 thf(fact_4778_add_Oinverse__inverse,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
% 0.22/0.49 = A ) ).
% 0.22/0.49
% 0.22/0.49 % add.inverse_inverse
% 0.22/0.49 thf(fact_4779_neg__equal__iff__equal,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( ( uminus_uminus_real @ A )
% 0.22/0.49 = ( uminus_uminus_real @ B ) )
% 0.22/0.49 = ( A = B ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_equal_iff_equal
% 0.22/0.49 thf(fact_4780_neg__equal__iff__equal,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( ( uminus_uminus_int @ A )
% 0.22/0.49 = ( uminus_uminus_int @ B ) )
% 0.22/0.49 = ( A = B ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_equal_iff_equal
% 0.22/0.49 thf(fact_4781_neg__equal__iff__equal,axiom,
% 0.22/0.49 ! [A: complex,B: complex] :
% 0.22/0.49 ( ( ( uminus1482373934393186551omplex @ A )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ B ) )
% 0.22/0.49 = ( A = B ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_equal_iff_equal
% 0.22/0.49 thf(fact_4782_neg__equal__iff__equal,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( ( uminus1351360451143612070nteger @ A )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ B ) )
% 0.22/0.49 = ( A = B ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_equal_iff_equal
% 0.22/0.49 thf(fact_4783_neg__equal__iff__equal,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( ( uminus_uminus_rat @ A )
% 0.22/0.49 = ( uminus_uminus_rat @ B ) )
% 0.22/0.49 = ( A = B ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_equal_iff_equal
% 0.22/0.49 thf(fact_4784_verit__minus__simplify_I4_J,axiom,
% 0.22/0.49 ! [B: real] :
% 0.22/0.49 ( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
% 0.22/0.49 = B ) ).
% 0.22/0.49
% 0.22/0.49 % verit_minus_simplify(4)
% 0.22/0.49 thf(fact_4785_verit__minus__simplify_I4_J,axiom,
% 0.22/0.49 ! [B: int] :
% 0.22/0.49 ( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
% 0.22/0.49 = B ) ).
% 0.22/0.49
% 0.22/0.49 % verit_minus_simplify(4)
% 0.22/0.49 thf(fact_4786_verit__minus__simplify_I4_J,axiom,
% 0.22/0.49 ! [B: complex] :
% 0.22/0.49 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B ) )
% 0.22/0.49 = B ) ).
% 0.22/0.49
% 0.22/0.49 % verit_minus_simplify(4)
% 0.22/0.49 thf(fact_4787_verit__minus__simplify_I4_J,axiom,
% 0.22/0.49 ! [B: code_integer] :
% 0.22/0.49 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B ) )
% 0.22/0.49 = B ) ).
% 0.22/0.49
% 0.22/0.49 % verit_minus_simplify(4)
% 0.22/0.49 thf(fact_4788_verit__minus__simplify_I4_J,axiom,
% 0.22/0.49 ! [B: rat] :
% 0.22/0.49 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B ) )
% 0.22/0.49 = B ) ).
% 0.22/0.49
% 0.22/0.49 % verit_minus_simplify(4)
% 0.22/0.49 thf(fact_4789_Compl__subset__Compl__iff,axiom,
% 0.22/0.49 ! [A2: set_int,B4: set_int] :
% 0.22/0.49 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B4 ) )
% 0.22/0.49 = ( ord_less_eq_set_int @ B4 @ A2 ) ) ).
% 0.22/0.49
% 0.22/0.49 % Compl_subset_Compl_iff
% 0.22/0.49 thf(fact_4790_Compl__anti__mono,axiom,
% 0.22/0.49 ! [A2: set_int,B4: set_int] :
% 0.22/0.49 ( ( ord_less_eq_set_int @ A2 @ B4 )
% 0.22/0.49 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B4 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % Compl_anti_mono
% 0.22/0.49 thf(fact_4791_abs__idempotent,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.22/0.49 = ( abs_abs_real @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_idempotent
% 0.22/0.49 thf(fact_4792_abs__idempotent,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 0.22/0.49 = ( abs_abs_int @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_idempotent
% 0.22/0.49 thf(fact_4793_abs__idempotent,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.22/0.49 = ( abs_abs_Code_integer @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_idempotent
% 0.22/0.49 thf(fact_4794_abs__idempotent,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 0.22/0.49 = ( abs_abs_rat @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_idempotent
% 0.22/0.49 thf(fact_4795_abs__abs,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.22/0.49 = ( abs_abs_real @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_abs
% 0.22/0.49 thf(fact_4796_abs__abs,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 0.22/0.49 = ( abs_abs_int @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_abs
% 0.22/0.49 thf(fact_4797_abs__abs,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.22/0.49 = ( abs_abs_Code_integer @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_abs
% 0.22/0.49 thf(fact_4798_abs__abs,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 0.22/0.49 = ( abs_abs_rat @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_abs
% 0.22/0.49 thf(fact_4799_neg__le__iff__le,axiom,
% 0.22/0.49 ! [B: code_integer,A: code_integer] :
% 0.22/0.49 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.22/0.49 = ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_le_iff_le
% 0.22/0.49 thf(fact_4800_neg__le__iff__le,axiom,
% 0.22/0.49 ! [B: rat,A: rat] :
% 0.22/0.49 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 0.22/0.49 = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_le_iff_le
% 0.22/0.49 thf(fact_4801_neg__le__iff__le,axiom,
% 0.22/0.49 ! [B: int,A: int] :
% 0.22/0.49 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 0.22/0.49 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_le_iff_le
% 0.22/0.49 thf(fact_4802_neg__le__iff__le,axiom,
% 0.22/0.49 ! [B: real,A: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.22/0.49 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_le_iff_le
% 0.22/0.49 thf(fact_4803_neg__equal__zero,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( ( uminus_uminus_real @ A )
% 0.22/0.49 = A )
% 0.22/0.49 = ( A = zero_zero_real ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_equal_zero
% 0.22/0.49 thf(fact_4804_neg__equal__zero,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( ( uminus_uminus_int @ A )
% 0.22/0.49 = A )
% 0.22/0.49 = ( A = zero_zero_int ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_equal_zero
% 0.22/0.49 thf(fact_4805_neg__equal__zero,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( ( uminus1351360451143612070nteger @ A )
% 0.22/0.49 = A )
% 0.22/0.49 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_equal_zero
% 0.22/0.49 thf(fact_4806_neg__equal__zero,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( ( uminus_uminus_rat @ A )
% 0.22/0.49 = A )
% 0.22/0.49 = ( A = zero_zero_rat ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_equal_zero
% 0.22/0.49 thf(fact_4807_equal__neg__zero,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( A
% 0.22/0.49 = ( uminus_uminus_real @ A ) )
% 0.22/0.49 = ( A = zero_zero_real ) ) ).
% 0.22/0.49
% 0.22/0.49 % equal_neg_zero
% 0.22/0.49 thf(fact_4808_equal__neg__zero,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( A
% 0.22/0.49 = ( uminus_uminus_int @ A ) )
% 0.22/0.49 = ( A = zero_zero_int ) ) ).
% 0.22/0.49
% 0.22/0.49 % equal_neg_zero
% 0.22/0.49 thf(fact_4809_equal__neg__zero,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( A
% 0.22/0.49 = ( uminus1351360451143612070nteger @ A ) )
% 0.22/0.49 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.49
% 0.22/0.49 % equal_neg_zero
% 0.22/0.49 thf(fact_4810_equal__neg__zero,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( A
% 0.22/0.49 = ( uminus_uminus_rat @ A ) )
% 0.22/0.49 = ( A = zero_zero_rat ) ) ).
% 0.22/0.49
% 0.22/0.49 % equal_neg_zero
% 0.22/0.49 thf(fact_4811_neg__equal__0__iff__equal,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( ( uminus_uminus_real @ A )
% 0.22/0.49 = zero_zero_real )
% 0.22/0.49 = ( A = zero_zero_real ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_equal_0_iff_equal
% 0.22/0.49 thf(fact_4812_neg__equal__0__iff__equal,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( ( uminus_uminus_int @ A )
% 0.22/0.49 = zero_zero_int )
% 0.22/0.49 = ( A = zero_zero_int ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_equal_0_iff_equal
% 0.22/0.49 thf(fact_4813_neg__equal__0__iff__equal,axiom,
% 0.22/0.49 ! [A: complex] :
% 0.22/0.49 ( ( ( uminus1482373934393186551omplex @ A )
% 0.22/0.49 = zero_zero_complex )
% 0.22/0.49 = ( A = zero_zero_complex ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_equal_0_iff_equal
% 0.22/0.49 thf(fact_4814_neg__equal__0__iff__equal,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( ( uminus1351360451143612070nteger @ A )
% 0.22/0.49 = zero_z3403309356797280102nteger )
% 0.22/0.49 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_equal_0_iff_equal
% 0.22/0.49 thf(fact_4815_neg__equal__0__iff__equal,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( ( uminus_uminus_rat @ A )
% 0.22/0.49 = zero_zero_rat )
% 0.22/0.49 = ( A = zero_zero_rat ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_equal_0_iff_equal
% 0.22/0.49 thf(fact_4816_neg__0__equal__iff__equal,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( zero_zero_real
% 0.22/0.49 = ( uminus_uminus_real @ A ) )
% 0.22/0.49 = ( zero_zero_real = A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_0_equal_iff_equal
% 0.22/0.49 thf(fact_4817_neg__0__equal__iff__equal,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( zero_zero_int
% 0.22/0.49 = ( uminus_uminus_int @ A ) )
% 0.22/0.49 = ( zero_zero_int = A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_0_equal_iff_equal
% 0.22/0.49 thf(fact_4818_neg__0__equal__iff__equal,axiom,
% 0.22/0.49 ! [A: complex] :
% 0.22/0.49 ( ( zero_zero_complex
% 0.22/0.49 = ( uminus1482373934393186551omplex @ A ) )
% 0.22/0.49 = ( zero_zero_complex = A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_0_equal_iff_equal
% 0.22/0.49 thf(fact_4819_neg__0__equal__iff__equal,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( zero_z3403309356797280102nteger
% 0.22/0.49 = ( uminus1351360451143612070nteger @ A ) )
% 0.22/0.49 = ( zero_z3403309356797280102nteger = A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_0_equal_iff_equal
% 0.22/0.49 thf(fact_4820_neg__0__equal__iff__equal,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( zero_zero_rat
% 0.22/0.49 = ( uminus_uminus_rat @ A ) )
% 0.22/0.49 = ( zero_zero_rat = A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_0_equal_iff_equal
% 0.22/0.49 thf(fact_4821_add_Oinverse__neutral,axiom,
% 0.22/0.49 ( ( uminus_uminus_real @ zero_zero_real )
% 0.22/0.49 = zero_zero_real ) ).
% 0.22/0.49
% 0.22/0.49 % add.inverse_neutral
% 0.22/0.49 thf(fact_4822_add_Oinverse__neutral,axiom,
% 0.22/0.49 ( ( uminus_uminus_int @ zero_zero_int )
% 0.22/0.49 = zero_zero_int ) ).
% 0.22/0.49
% 0.22/0.49 % add.inverse_neutral
% 0.22/0.49 thf(fact_4823_add_Oinverse__neutral,axiom,
% 0.22/0.49 ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 0.22/0.49 = zero_zero_complex ) ).
% 0.22/0.49
% 0.22/0.49 % add.inverse_neutral
% 0.22/0.49 thf(fact_4824_add_Oinverse__neutral,axiom,
% 0.22/0.49 ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 0.22/0.49 = zero_z3403309356797280102nteger ) ).
% 0.22/0.49
% 0.22/0.49 % add.inverse_neutral
% 0.22/0.49 thf(fact_4825_add_Oinverse__neutral,axiom,
% 0.22/0.49 ( ( uminus_uminus_rat @ zero_zero_rat )
% 0.22/0.49 = zero_zero_rat ) ).
% 0.22/0.49
% 0.22/0.49 % add.inverse_neutral
% 0.22/0.49 thf(fact_4826_neg__less__iff__less,axiom,
% 0.22/0.49 ! [B: real,A: real] :
% 0.22/0.49 ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.22/0.49 = ( ord_less_real @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_less_iff_less
% 0.22/0.49 thf(fact_4827_neg__less__iff__less,axiom,
% 0.22/0.49 ! [B: int,A: int] :
% 0.22/0.49 ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 0.22/0.49 = ( ord_less_int @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_less_iff_less
% 0.22/0.49 thf(fact_4828_neg__less__iff__less,axiom,
% 0.22/0.49 ! [B: code_integer,A: code_integer] :
% 0.22/0.49 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.22/0.49 = ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_less_iff_less
% 0.22/0.49 thf(fact_4829_neg__less__iff__less,axiom,
% 0.22/0.49 ! [B: rat,A: rat] :
% 0.22/0.49 ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 0.22/0.49 = ( ord_less_rat @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_less_iff_less
% 0.22/0.49 thf(fact_4830_neg__numeral__eq__iff,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 0.22/0.49 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.22/0.49 = ( M = N ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_numeral_eq_iff
% 0.22/0.49 thf(fact_4831_neg__numeral__eq__iff,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 0.22/0.49 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.49 = ( M = N ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_numeral_eq_iff
% 0.22/0.49 thf(fact_4832_neg__numeral__eq__iff,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.22/0.49 = ( M = N ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_numeral_eq_iff
% 0.22/0.49 thf(fact_4833_neg__numeral__eq__iff,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.22/0.49 = ( M = N ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_numeral_eq_iff
% 0.22/0.49 thf(fact_4834_neg__numeral__eq__iff,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 0.22/0.49 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.22/0.49 = ( M = N ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_numeral_eq_iff
% 0.22/0.49 thf(fact_4835_mult__minus__left,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 0.22/0.49 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus_left
% 0.22/0.49 thf(fact_4836_mult__minus__left,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 0.22/0.49 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus_left
% 0.22/0.49 thf(fact_4837_mult__minus__left,axiom,
% 0.22/0.49 ! [A: complex,B: complex] :
% 0.22/0.49 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus_left
% 0.22/0.49 thf(fact_4838_mult__minus__left,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus_left
% 0.22/0.49 thf(fact_4839_mult__minus__left,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.22/0.49 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus_left
% 0.22/0.49 thf(fact_4840_minus__mult__minus,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.22/0.49 = ( times_times_real @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_mult_minus
% 0.22/0.49 thf(fact_4841_minus__mult__minus,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.22/0.49 = ( times_times_int @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_mult_minus
% 0.22/0.49 thf(fact_4842_minus__mult__minus,axiom,
% 0.22/0.49 ! [A: complex,B: complex] :
% 0.22/0.49 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.22/0.49 = ( times_times_complex @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_mult_minus
% 0.22/0.49 thf(fact_4843_minus__mult__minus,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.22/0.49 = ( times_3573771949741848930nteger @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_mult_minus
% 0.22/0.49 thf(fact_4844_minus__mult__minus,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.22/0.49 = ( times_times_rat @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_mult_minus
% 0.22/0.49 thf(fact_4845_mult__minus__right,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
% 0.22/0.49 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus_right
% 0.22/0.49 thf(fact_4846_mult__minus__right,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
% 0.22/0.49 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus_right
% 0.22/0.49 thf(fact_4847_mult__minus__right,axiom,
% 0.22/0.49 ! [A: complex,B: complex] :
% 0.22/0.49 ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus_right
% 0.22/0.49 thf(fact_4848_mult__minus__right,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus_right
% 0.22/0.49 thf(fact_4849_mult__minus__right,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.22/0.49 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus_right
% 0.22/0.49 thf(fact_4850_add__minus__cancel,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
% 0.22/0.49 = B ) ).
% 0.22/0.49
% 0.22/0.49 % add_minus_cancel
% 0.22/0.49 thf(fact_4851_add__minus__cancel,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 0.22/0.49 = B ) ).
% 0.22/0.49
% 0.22/0.49 % add_minus_cancel
% 0.22/0.49 thf(fact_4852_add__minus__cancel,axiom,
% 0.22/0.49 ! [A: complex,B: complex] :
% 0.22/0.49 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
% 0.22/0.49 = B ) ).
% 0.22/0.49
% 0.22/0.49 % add_minus_cancel
% 0.22/0.49 thf(fact_4853_add__minus__cancel,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
% 0.22/0.49 = B ) ).
% 0.22/0.49
% 0.22/0.49 % add_minus_cancel
% 0.22/0.49 thf(fact_4854_add__minus__cancel,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
% 0.22/0.49 = B ) ).
% 0.22/0.49
% 0.22/0.49 % add_minus_cancel
% 0.22/0.49 thf(fact_4855_minus__add__cancel,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
% 0.22/0.49 = B ) ).
% 0.22/0.49
% 0.22/0.49 % minus_add_cancel
% 0.22/0.49 thf(fact_4856_minus__add__cancel,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 0.22/0.49 = B ) ).
% 0.22/0.49
% 0.22/0.49 % minus_add_cancel
% 0.22/0.49 thf(fact_4857_minus__add__cancel,axiom,
% 0.22/0.49 ! [A: complex,B: complex] :
% 0.22/0.49 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
% 0.22/0.49 = B ) ).
% 0.22/0.49
% 0.22/0.49 % minus_add_cancel
% 0.22/0.49 thf(fact_4858_minus__add__cancel,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.22/0.49 = B ) ).
% 0.22/0.49
% 0.22/0.49 % minus_add_cancel
% 0.22/0.49 thf(fact_4859_minus__add__cancel,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
% 0.22/0.49 = B ) ).
% 0.22/0.49
% 0.22/0.49 % minus_add_cancel
% 0.22/0.49 thf(fact_4860_minus__add__distrib,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.22/0.49 = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_add_distrib
% 0.22/0.49 thf(fact_4861_minus__add__distrib,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.22/0.49 = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_add_distrib
% 0.22/0.49 thf(fact_4862_minus__add__distrib,axiom,
% 0.22/0.49 ! [A: complex,B: complex] :
% 0.22/0.49 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 0.22/0.49 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_add_distrib
% 0.22/0.49 thf(fact_4863_minus__add__distrib,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.22/0.49 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_add_distrib
% 0.22/0.49 thf(fact_4864_minus__add__distrib,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.22/0.49 = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_add_distrib
% 0.22/0.49 thf(fact_4865_minus__diff__eq,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
% 0.22/0.49 = ( minus_minus_real @ B @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_diff_eq
% 0.22/0.49 thf(fact_4866_minus__diff__eq,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 0.22/0.49 = ( minus_minus_int @ B @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_diff_eq
% 0.22/0.49 thf(fact_4867_minus__diff__eq,axiom,
% 0.22/0.49 ! [A: complex,B: complex] :
% 0.22/0.49 ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
% 0.22/0.49 = ( minus_minus_complex @ B @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_diff_eq
% 0.22/0.49 thf(fact_4868_minus__diff__eq,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.22/0.49 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_diff_eq
% 0.22/0.49 thf(fact_4869_minus__diff__eq,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
% 0.22/0.49 = ( minus_minus_rat @ B @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_diff_eq
% 0.22/0.49 thf(fact_4870_div__minus__minus,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.22/0.49 = ( divide_divide_int @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % div_minus_minus
% 0.22/0.49 thf(fact_4871_div__minus__minus,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.22/0.49 = ( divide6298287555418463151nteger @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % div_minus_minus
% 0.22/0.49 thf(fact_4872_abs__0,axiom,
% 0.22/0.49 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.22/0.49 = zero_z3403309356797280102nteger ) ).
% 0.22/0.49
% 0.22/0.49 % abs_0
% 0.22/0.49 thf(fact_4873_abs__0,axiom,
% 0.22/0.49 ( ( abs_abs_complex @ zero_zero_complex )
% 0.22/0.49 = zero_zero_complex ) ).
% 0.22/0.49
% 0.22/0.49 % abs_0
% 0.22/0.49 thf(fact_4874_abs__0,axiom,
% 0.22/0.49 ( ( abs_abs_real @ zero_zero_real )
% 0.22/0.49 = zero_zero_real ) ).
% 0.22/0.49
% 0.22/0.49 % abs_0
% 0.22/0.49 thf(fact_4875_abs__0,axiom,
% 0.22/0.49 ( ( abs_abs_rat @ zero_zero_rat )
% 0.22/0.49 = zero_zero_rat ) ).
% 0.22/0.49
% 0.22/0.49 % abs_0
% 0.22/0.49 thf(fact_4876_abs__0,axiom,
% 0.22/0.49 ( ( abs_abs_int @ zero_zero_int )
% 0.22/0.49 = zero_zero_int ) ).
% 0.22/0.49
% 0.22/0.49 % abs_0
% 0.22/0.49 thf(fact_4877_abs__zero,axiom,
% 0.22/0.49 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.22/0.49 = zero_z3403309356797280102nteger ) ).
% 0.22/0.49
% 0.22/0.49 % abs_zero
% 0.22/0.49 thf(fact_4878_abs__zero,axiom,
% 0.22/0.49 ( ( abs_abs_real @ zero_zero_real )
% 0.22/0.49 = zero_zero_real ) ).
% 0.22/0.49
% 0.22/0.49 % abs_zero
% 0.22/0.49 thf(fact_4879_abs__zero,axiom,
% 0.22/0.49 ( ( abs_abs_rat @ zero_zero_rat )
% 0.22/0.49 = zero_zero_rat ) ).
% 0.22/0.49
% 0.22/0.49 % abs_zero
% 0.22/0.49 thf(fact_4880_abs__zero,axiom,
% 0.22/0.49 ( ( abs_abs_int @ zero_zero_int )
% 0.22/0.49 = zero_zero_int ) ).
% 0.22/0.49
% 0.22/0.49 % abs_zero
% 0.22/0.49 thf(fact_4881_abs__eq__0,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( ( abs_abs_Code_integer @ A )
% 0.22/0.49 = zero_z3403309356797280102nteger )
% 0.22/0.49 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_eq_0
% 0.22/0.49 thf(fact_4882_abs__eq__0,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( ( abs_abs_real @ A )
% 0.22/0.49 = zero_zero_real )
% 0.22/0.49 = ( A = zero_zero_real ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_eq_0
% 0.22/0.49 thf(fact_4883_abs__eq__0,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( ( abs_abs_rat @ A )
% 0.22/0.49 = zero_zero_rat )
% 0.22/0.49 = ( A = zero_zero_rat ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_eq_0
% 0.22/0.49 thf(fact_4884_abs__eq__0,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( ( abs_abs_int @ A )
% 0.22/0.49 = zero_zero_int )
% 0.22/0.49 = ( A = zero_zero_int ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_eq_0
% 0.22/0.49 thf(fact_4885_abs__0__eq,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( zero_z3403309356797280102nteger
% 0.22/0.49 = ( abs_abs_Code_integer @ A ) )
% 0.22/0.49 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_0_eq
% 0.22/0.49 thf(fact_4886_abs__0__eq,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( zero_zero_real
% 0.22/0.49 = ( abs_abs_real @ A ) )
% 0.22/0.49 = ( A = zero_zero_real ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_0_eq
% 0.22/0.49 thf(fact_4887_abs__0__eq,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( zero_zero_rat
% 0.22/0.49 = ( abs_abs_rat @ A ) )
% 0.22/0.49 = ( A = zero_zero_rat ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_0_eq
% 0.22/0.49 thf(fact_4888_abs__0__eq,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( zero_zero_int
% 0.22/0.49 = ( abs_abs_int @ A ) )
% 0.22/0.49 = ( A = zero_zero_int ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_0_eq
% 0.22/0.49 thf(fact_4889_minus__dvd__iff,axiom,
% 0.22/0.49 ! [X: real,Y4: real] :
% 0.22/0.49 ( ( dvd_dvd_real @ ( uminus_uminus_real @ X ) @ Y4 )
% 0.22/0.49 = ( dvd_dvd_real @ X @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_dvd_iff
% 0.22/0.49 thf(fact_4890_minus__dvd__iff,axiom,
% 0.22/0.49 ! [X: int,Y4: int] :
% 0.22/0.49 ( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y4 )
% 0.22/0.49 = ( dvd_dvd_int @ X @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_dvd_iff
% 0.22/0.49 thf(fact_4891_minus__dvd__iff,axiom,
% 0.22/0.49 ! [X: complex,Y4: complex] :
% 0.22/0.49 ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X ) @ Y4 )
% 0.22/0.49 = ( dvd_dvd_complex @ X @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_dvd_iff
% 0.22/0.49 thf(fact_4892_minus__dvd__iff,axiom,
% 0.22/0.49 ! [X: code_integer,Y4: code_integer] :
% 0.22/0.49 ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ Y4 )
% 0.22/0.49 = ( dvd_dvd_Code_integer @ X @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_dvd_iff
% 0.22/0.49 thf(fact_4893_minus__dvd__iff,axiom,
% 0.22/0.49 ! [X: rat,Y4: rat] :
% 0.22/0.49 ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X ) @ Y4 )
% 0.22/0.49 = ( dvd_dvd_rat @ X @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_dvd_iff
% 0.22/0.49 thf(fact_4894_dvd__minus__iff,axiom,
% 0.22/0.49 ! [X: real,Y4: real] :
% 0.22/0.49 ( ( dvd_dvd_real @ X @ ( uminus_uminus_real @ Y4 ) )
% 0.22/0.49 = ( dvd_dvd_real @ X @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % dvd_minus_iff
% 0.22/0.49 thf(fact_4895_dvd__minus__iff,axiom,
% 0.22/0.49 ! [X: int,Y4: int] :
% 0.22/0.49 ( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y4 ) )
% 0.22/0.49 = ( dvd_dvd_int @ X @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % dvd_minus_iff
% 0.22/0.49 thf(fact_4896_dvd__minus__iff,axiom,
% 0.22/0.49 ! [X: complex,Y4: complex] :
% 0.22/0.49 ( ( dvd_dvd_complex @ X @ ( uminus1482373934393186551omplex @ Y4 ) )
% 0.22/0.49 = ( dvd_dvd_complex @ X @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % dvd_minus_iff
% 0.22/0.49 thf(fact_4897_dvd__minus__iff,axiom,
% 0.22/0.49 ! [X: code_integer,Y4: code_integer] :
% 0.22/0.49 ( ( dvd_dvd_Code_integer @ X @ ( uminus1351360451143612070nteger @ Y4 ) )
% 0.22/0.49 = ( dvd_dvd_Code_integer @ X @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % dvd_minus_iff
% 0.22/0.49 thf(fact_4898_dvd__minus__iff,axiom,
% 0.22/0.49 ! [X: rat,Y4: rat] :
% 0.22/0.49 ( ( dvd_dvd_rat @ X @ ( uminus_uminus_rat @ Y4 ) )
% 0.22/0.49 = ( dvd_dvd_rat @ X @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % dvd_minus_iff
% 0.22/0.49 thf(fact_4899_abs__numeral,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
% 0.22/0.49 = ( numera6620942414471956472nteger @ N ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_numeral
% 0.22/0.49 thf(fact_4900_abs__numeral,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
% 0.22/0.49 = ( numeral_numeral_real @ N ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_numeral
% 0.22/0.49 thf(fact_4901_abs__numeral,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
% 0.22/0.49 = ( numeral_numeral_rat @ N ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_numeral
% 0.22/0.49 thf(fact_4902_abs__numeral,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
% 0.22/0.49 = ( numeral_numeral_int @ N ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_numeral
% 0.22/0.49 thf(fact_4903_abs__mult__self__eq,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 0.22/0.49 = ( times_3573771949741848930nteger @ A @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_mult_self_eq
% 0.22/0.49 thf(fact_4904_abs__mult__self__eq,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
% 0.22/0.49 = ( times_times_real @ A @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_mult_self_eq
% 0.22/0.49 thf(fact_4905_abs__mult__self__eq,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
% 0.22/0.49 = ( times_times_rat @ A @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_mult_self_eq
% 0.22/0.49 thf(fact_4906_abs__mult__self__eq,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
% 0.22/0.49 = ( times_times_int @ A @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_mult_self_eq
% 0.22/0.49 thf(fact_4907_abs__add__abs,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
% 0.22/0.49 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_add_abs
% 0.22/0.49 thf(fact_4908_abs__add__abs,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 0.22/0.49 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_add_abs
% 0.22/0.49 thf(fact_4909_abs__add__abs,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
% 0.22/0.49 = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_add_abs
% 0.22/0.49 thf(fact_4910_abs__add__abs,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
% 0.22/0.49 = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_add_abs
% 0.22/0.49 thf(fact_4911_abs__1,axiom,
% 0.22/0.49 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 0.22/0.49 = one_one_Code_integer ) ).
% 0.22/0.49
% 0.22/0.49 % abs_1
% 0.22/0.49 thf(fact_4912_abs__1,axiom,
% 0.22/0.49 ( ( abs_abs_complex @ one_one_complex )
% 0.22/0.49 = one_one_complex ) ).
% 0.22/0.49
% 0.22/0.49 % abs_1
% 0.22/0.49 thf(fact_4913_abs__1,axiom,
% 0.22/0.49 ( ( abs_abs_real @ one_one_real )
% 0.22/0.49 = one_one_real ) ).
% 0.22/0.49
% 0.22/0.49 % abs_1
% 0.22/0.49 thf(fact_4914_abs__1,axiom,
% 0.22/0.49 ( ( abs_abs_rat @ one_one_rat )
% 0.22/0.49 = one_one_rat ) ).
% 0.22/0.49
% 0.22/0.49 % abs_1
% 0.22/0.49 thf(fact_4915_abs__1,axiom,
% 0.22/0.49 ( ( abs_abs_int @ one_one_int )
% 0.22/0.49 = one_one_int ) ).
% 0.22/0.49
% 0.22/0.49 % abs_1
% 0.22/0.49 thf(fact_4916_abs__divide,axiom,
% 0.22/0.49 ! [A: complex,B: complex] :
% 0.22/0.49 ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.22/0.49 = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_divide
% 0.22/0.49 thf(fact_4917_abs__divide,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 0.22/0.49 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_divide
% 0.22/0.49 thf(fact_4918_abs__divide,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 0.22/0.49 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_divide
% 0.22/0.49 thf(fact_4919_abs__minus__cancel,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.22/0.49 = ( abs_abs_real @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_minus_cancel
% 0.22/0.49 thf(fact_4920_abs__minus__cancel,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 0.22/0.49 = ( abs_abs_int @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_minus_cancel
% 0.22/0.49 thf(fact_4921_abs__minus__cancel,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 0.22/0.49 = ( abs_abs_Code_integer @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_minus_cancel
% 0.22/0.49 thf(fact_4922_abs__minus__cancel,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 0.22/0.49 = ( abs_abs_rat @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_minus_cancel
% 0.22/0.49 thf(fact_4923_abs__minus,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.22/0.49 = ( abs_abs_real @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_minus
% 0.22/0.49 thf(fact_4924_abs__minus,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 0.22/0.49 = ( abs_abs_int @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_minus
% 0.22/0.49 thf(fact_4925_abs__minus,axiom,
% 0.22/0.49 ! [A: complex] :
% 0.22/0.49 ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
% 0.22/0.49 = ( abs_abs_complex @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_minus
% 0.22/0.49 thf(fact_4926_abs__minus,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 0.22/0.49 = ( abs_abs_Code_integer @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_minus
% 0.22/0.49 thf(fact_4927_abs__minus,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 0.22/0.49 = ( abs_abs_rat @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_minus
% 0.22/0.49 thf(fact_4928_mod__minus__minus,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.22/0.49 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mod_minus_minus
% 0.22/0.49 thf(fact_4929_mod__minus__minus,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mod_minus_minus
% 0.22/0.49 thf(fact_4930_dvd__abs__iff,axiom,
% 0.22/0.49 ! [M: real,K: real] :
% 0.22/0.49 ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
% 0.22/0.49 = ( dvd_dvd_real @ M @ K ) ) ).
% 0.22/0.49
% 0.22/0.49 % dvd_abs_iff
% 0.22/0.49 thf(fact_4931_dvd__abs__iff,axiom,
% 0.22/0.49 ! [M: int,K: int] :
% 0.22/0.49 ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
% 0.22/0.49 = ( dvd_dvd_int @ M @ K ) ) ).
% 0.22/0.49
% 0.22/0.49 % dvd_abs_iff
% 0.22/0.49 thf(fact_4932_dvd__abs__iff,axiom,
% 0.22/0.49 ! [M: code_integer,K: code_integer] :
% 0.22/0.49 ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
% 0.22/0.49 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 0.22/0.49
% 0.22/0.49 % dvd_abs_iff
% 0.22/0.49 thf(fact_4933_dvd__abs__iff,axiom,
% 0.22/0.49 ! [M: rat,K: rat] :
% 0.22/0.49 ( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
% 0.22/0.49 = ( dvd_dvd_rat @ M @ K ) ) ).
% 0.22/0.49
% 0.22/0.49 % dvd_abs_iff
% 0.22/0.49 thf(fact_4934_abs__dvd__iff,axiom,
% 0.22/0.49 ! [M: real,K: real] :
% 0.22/0.49 ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
% 0.22/0.49 = ( dvd_dvd_real @ M @ K ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_dvd_iff
% 0.22/0.49 thf(fact_4935_abs__dvd__iff,axiom,
% 0.22/0.49 ! [M: int,K: int] :
% 0.22/0.49 ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
% 0.22/0.49 = ( dvd_dvd_int @ M @ K ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_dvd_iff
% 0.22/0.49 thf(fact_4936_abs__dvd__iff,axiom,
% 0.22/0.49 ! [M: code_integer,K: code_integer] :
% 0.22/0.49 ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
% 0.22/0.49 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_dvd_iff
% 0.22/0.49 thf(fact_4937_abs__dvd__iff,axiom,
% 0.22/0.49 ! [M: rat,K: rat] :
% 0.22/0.49 ( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
% 0.22/0.49 = ( dvd_dvd_rat @ M @ K ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_dvd_iff
% 0.22/0.49 thf(fact_4938_abs__of__nat,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 0.22/0.49 = ( semiri4939895301339042750nteger @ N ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_of_nat
% 0.22/0.49 thf(fact_4939_abs__of__nat,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.22/0.49 = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_of_nat
% 0.22/0.49 thf(fact_4940_abs__of__nat,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.22/0.49 = ( semiri681578069525770553at_rat @ N ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_of_nat
% 0.22/0.49 thf(fact_4941_abs__of__nat,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.22/0.49 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_of_nat
% 0.22/0.49 thf(fact_4942_abs__bool__eq,axiom,
% 0.22/0.49 ! [P: $o] :
% 0.22/0.49 ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
% 0.22/0.49 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_bool_eq
% 0.22/0.49 thf(fact_4943_abs__bool__eq,axiom,
% 0.22/0.49 ! [P: $o] :
% 0.22/0.49 ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 0.22/0.49 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_bool_eq
% 0.22/0.49 thf(fact_4944_abs__bool__eq,axiom,
% 0.22/0.49 ! [P: $o] :
% 0.22/0.49 ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.22/0.49 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_bool_eq
% 0.22/0.49 thf(fact_4945_abs__bool__eq,axiom,
% 0.22/0.49 ! [P: $o] :
% 0.22/0.49 ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
% 0.22/0.49 = ( zero_n356916108424825756nteger @ P ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_bool_eq
% 0.22/0.49 thf(fact_4946_tanh__0,axiom,
% 0.22/0.49 ( ( tanh_complex @ zero_zero_complex )
% 0.22/0.49 = zero_zero_complex ) ).
% 0.22/0.49
% 0.22/0.49 % tanh_0
% 0.22/0.49 thf(fact_4947_tanh__0,axiom,
% 0.22/0.49 ( ( tanh_real @ zero_zero_real )
% 0.22/0.49 = zero_zero_real ) ).
% 0.22/0.49
% 0.22/0.49 % tanh_0
% 0.22/0.49 thf(fact_4948_tanh__real__zero__iff,axiom,
% 0.22/0.49 ! [X: real] :
% 0.22/0.49 ( ( ( tanh_real @ X )
% 0.22/0.49 = zero_zero_real )
% 0.22/0.49 = ( X = zero_zero_real ) ) ).
% 0.22/0.49
% 0.22/0.49 % tanh_real_zero_iff
% 0.22/0.49 thf(fact_4949_tanh__real__le__iff,axiom,
% 0.22/0.49 ! [X: real,Y4: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ ( tanh_real @ X ) @ ( tanh_real @ Y4 ) )
% 0.22/0.49 = ( ord_less_eq_real @ X @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % tanh_real_le_iff
% 0.22/0.49 thf(fact_4950_neg__less__eq__nonneg,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.22/0.49 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_less_eq_nonneg
% 0.22/0.49 thf(fact_4951_neg__less__eq__nonneg,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.22/0.49 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_less_eq_nonneg
% 0.22/0.49 thf(fact_4952_neg__less__eq__nonneg,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 0.22/0.49 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_less_eq_nonneg
% 0.22/0.49 thf(fact_4953_neg__less__eq__nonneg,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 0.22/0.49 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_less_eq_nonneg
% 0.22/0.49 thf(fact_4954_less__eq__neg__nonpos,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.22/0.49 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.22/0.49
% 0.22/0.49 % less_eq_neg_nonpos
% 0.22/0.49 thf(fact_4955_less__eq__neg__nonpos,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.22/0.49 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.22/0.49
% 0.22/0.49 % less_eq_neg_nonpos
% 0.22/0.49 thf(fact_4956_less__eq__neg__nonpos,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 0.22/0.49 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.22/0.49
% 0.22/0.49 % less_eq_neg_nonpos
% 0.22/0.49 thf(fact_4957_less__eq__neg__nonpos,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 0.22/0.49 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.22/0.49
% 0.22/0.49 % less_eq_neg_nonpos
% 0.22/0.49 thf(fact_4958_neg__le__0__iff__le,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.22/0.49 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_le_0_iff_le
% 0.22/0.49 thf(fact_4959_neg__le__0__iff__le,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.22/0.49 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_le_0_iff_le
% 0.22/0.49 thf(fact_4960_neg__le__0__iff__le,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.22/0.49 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_le_0_iff_le
% 0.22/0.49 thf(fact_4961_neg__le__0__iff__le,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.22/0.49 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_le_0_iff_le
% 0.22/0.49 thf(fact_4962_neg__0__le__iff__le,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.22/0.49 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_0_le_iff_le
% 0.22/0.49 thf(fact_4963_neg__0__le__iff__le,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.22/0.49 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_0_le_iff_le
% 0.22/0.49 thf(fact_4964_neg__0__le__iff__le,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.22/0.49 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_0_le_iff_le
% 0.22/0.49 thf(fact_4965_neg__0__le__iff__le,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.22/0.49 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_0_le_iff_le
% 0.22/0.49 thf(fact_4966_less__neg__neg,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 0.22/0.49 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.22/0.49
% 0.22/0.49 % less_neg_neg
% 0.22/0.49 thf(fact_4967_less__neg__neg,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 0.22/0.49 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.22/0.49
% 0.22/0.49 % less_neg_neg
% 0.22/0.49 thf(fact_4968_less__neg__neg,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.22/0.49 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.22/0.49
% 0.22/0.49 % less_neg_neg
% 0.22/0.49 thf(fact_4969_less__neg__neg,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.22/0.49 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.22/0.49
% 0.22/0.49 % less_neg_neg
% 0.22/0.49 thf(fact_4970_neg__less__pos,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 0.22/0.49 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_less_pos
% 0.22/0.49 thf(fact_4971_neg__less__pos,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 0.22/0.49 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_less_pos
% 0.22/0.49 thf(fact_4972_neg__less__pos,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.22/0.49 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_less_pos
% 0.22/0.49 thf(fact_4973_neg__less__pos,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.22/0.49 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_less_pos
% 0.22/0.49 thf(fact_4974_neg__0__less__iff__less,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.22/0.49 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_0_less_iff_less
% 0.22/0.49 thf(fact_4975_neg__0__less__iff__less,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.22/0.49 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_0_less_iff_less
% 0.22/0.49 thf(fact_4976_neg__0__less__iff__less,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.22/0.49 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_0_less_iff_less
% 0.22/0.49 thf(fact_4977_neg__0__less__iff__less,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.22/0.49 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_0_less_iff_less
% 0.22/0.49 thf(fact_4978_neg__less__0__iff__less,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.22/0.49 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_less_0_iff_less
% 0.22/0.49 thf(fact_4979_neg__less__0__iff__less,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.22/0.49 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_less_0_iff_less
% 0.22/0.49 thf(fact_4980_neg__less__0__iff__less,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.22/0.49 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_less_0_iff_less
% 0.22/0.49 thf(fact_4981_neg__less__0__iff__less,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.22/0.49 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_less_0_iff_less
% 0.22/0.49 thf(fact_4982_ab__left__minus,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.22/0.49 = zero_zero_real ) ).
% 0.22/0.49
% 0.22/0.49 % ab_left_minus
% 0.22/0.49 thf(fact_4983_ab__left__minus,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.22/0.49 = zero_zero_int ) ).
% 0.22/0.49
% 0.22/0.49 % ab_left_minus
% 0.22/0.49 thf(fact_4984_ab__left__minus,axiom,
% 0.22/0.49 ! [A: complex] :
% 0.22/0.49 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.22/0.49 = zero_zero_complex ) ).
% 0.22/0.49
% 0.22/0.49 % ab_left_minus
% 0.22/0.49 thf(fact_4985_ab__left__minus,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.22/0.49 = zero_z3403309356797280102nteger ) ).
% 0.22/0.49
% 0.22/0.49 % ab_left_minus
% 0.22/0.49 thf(fact_4986_ab__left__minus,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.22/0.49 = zero_zero_rat ) ).
% 0.22/0.49
% 0.22/0.49 % ab_left_minus
% 0.22/0.49 thf(fact_4987_add_Oright__inverse,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 0.22/0.49 = zero_zero_real ) ).
% 0.22/0.49
% 0.22/0.49 % add.right_inverse
% 0.22/0.49 thf(fact_4988_add_Oright__inverse,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 0.22/0.49 = zero_zero_int ) ).
% 0.22/0.49
% 0.22/0.49 % add.right_inverse
% 0.22/0.49 thf(fact_4989_add_Oright__inverse,axiom,
% 0.22/0.49 ! [A: complex] :
% 0.22/0.49 ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 0.22/0.49 = zero_zero_complex ) ).
% 0.22/0.49
% 0.22/0.49 % add.right_inverse
% 0.22/0.49 thf(fact_4990_add_Oright__inverse,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.22/0.49 = zero_z3403309356797280102nteger ) ).
% 0.22/0.49
% 0.22/0.49 % add.right_inverse
% 0.22/0.49 thf(fact_4991_add_Oright__inverse,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.22/0.49 = zero_zero_rat ) ).
% 0.22/0.49
% 0.22/0.49 % add.right_inverse
% 0.22/0.49 thf(fact_4992_verit__minus__simplify_I3_J,axiom,
% 0.22/0.49 ! [B: real] :
% 0.22/0.49 ( ( minus_minus_real @ zero_zero_real @ B )
% 0.22/0.49 = ( uminus_uminus_real @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % verit_minus_simplify(3)
% 0.22/0.49 thf(fact_4993_verit__minus__simplify_I3_J,axiom,
% 0.22/0.49 ! [B: int] :
% 0.22/0.49 ( ( minus_minus_int @ zero_zero_int @ B )
% 0.22/0.49 = ( uminus_uminus_int @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % verit_minus_simplify(3)
% 0.22/0.49 thf(fact_4994_verit__minus__simplify_I3_J,axiom,
% 0.22/0.49 ! [B: complex] :
% 0.22/0.49 ( ( minus_minus_complex @ zero_zero_complex @ B )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % verit_minus_simplify(3)
% 0.22/0.49 thf(fact_4995_verit__minus__simplify_I3_J,axiom,
% 0.22/0.49 ! [B: code_integer] :
% 0.22/0.49 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % verit_minus_simplify(3)
% 0.22/0.49 thf(fact_4996_verit__minus__simplify_I3_J,axiom,
% 0.22/0.49 ! [B: rat] :
% 0.22/0.49 ( ( minus_minus_rat @ zero_zero_rat @ B )
% 0.22/0.49 = ( uminus_uminus_rat @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % verit_minus_simplify(3)
% 0.22/0.49 thf(fact_4997_diff__0,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( minus_minus_real @ zero_zero_real @ A )
% 0.22/0.49 = ( uminus_uminus_real @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_0
% 0.22/0.49 thf(fact_4998_diff__0,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( minus_minus_int @ zero_zero_int @ A )
% 0.22/0.49 = ( uminus_uminus_int @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_0
% 0.22/0.49 thf(fact_4999_diff__0,axiom,
% 0.22/0.49 ! [A: complex] :
% 0.22/0.49 ( ( minus_minus_complex @ zero_zero_complex @ A )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_0
% 0.22/0.49 thf(fact_5000_diff__0,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_0
% 0.22/0.49 thf(fact_5001_diff__0,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( minus_minus_rat @ zero_zero_rat @ A )
% 0.22/0.49 = ( uminus_uminus_rat @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_0
% 0.22/0.49 thf(fact_5002_add__neg__numeral__simps_I3_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.22/0.49 = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_simps(3)
% 0.22/0.49 thf(fact_5003_add__neg__numeral__simps_I3_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.49 = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_simps(3)
% 0.22/0.49 thf(fact_5004_add__neg__numeral__simps_I3_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_simps(3)
% 0.22/0.49 thf(fact_5005_add__neg__numeral__simps_I3_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_simps(3)
% 0.22/0.49 thf(fact_5006_add__neg__numeral__simps_I3_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.22/0.49 = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_simps(3)
% 0.22/0.49 thf(fact_5007_mult__minus1__right,axiom,
% 0.22/0.49 ! [Z: real] :
% 0.22/0.49 ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.49 = ( uminus_uminus_real @ Z ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus1_right
% 0.22/0.49 thf(fact_5008_mult__minus1__right,axiom,
% 0.22/0.49 ! [Z: int] :
% 0.22/0.49 ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.49 = ( uminus_uminus_int @ Z ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus1_right
% 0.22/0.49 thf(fact_5009_mult__minus1__right,axiom,
% 0.22/0.49 ! [Z: complex] :
% 0.22/0.49 ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus1_right
% 0.22/0.49 thf(fact_5010_mult__minus1__right,axiom,
% 0.22/0.49 ! [Z: code_integer] :
% 0.22/0.49 ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus1_right
% 0.22/0.49 thf(fact_5011_mult__minus1__right,axiom,
% 0.22/0.49 ! [Z: rat] :
% 0.22/0.49 ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.22/0.49 = ( uminus_uminus_rat @ Z ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus1_right
% 0.22/0.49 thf(fact_5012_mult__minus1,axiom,
% 0.22/0.49 ! [Z: real] :
% 0.22/0.49 ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
% 0.22/0.49 = ( uminus_uminus_real @ Z ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus1
% 0.22/0.49 thf(fact_5013_mult__minus1,axiom,
% 0.22/0.49 ! [Z: int] :
% 0.22/0.49 ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
% 0.22/0.49 = ( uminus_uminus_int @ Z ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus1
% 0.22/0.49 thf(fact_5014_mult__minus1,axiom,
% 0.22/0.49 ! [Z: complex] :
% 0.22/0.49 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus1
% 0.22/0.49 thf(fact_5015_mult__minus1,axiom,
% 0.22/0.49 ! [Z: code_integer] :
% 0.22/0.49 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus1
% 0.22/0.49 thf(fact_5016_mult__minus1,axiom,
% 0.22/0.49 ! [Z: rat] :
% 0.22/0.49 ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
% 0.22/0.49 = ( uminus_uminus_rat @ Z ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_minus1
% 0.22/0.49 thf(fact_5017_abs__le__zero__iff,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 0.22/0.49 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_le_zero_iff
% 0.22/0.49 thf(fact_5018_abs__le__zero__iff,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 0.22/0.49 = ( A = zero_zero_rat ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_le_zero_iff
% 0.22/0.49 thf(fact_5019_abs__le__zero__iff,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 0.22/0.49 = ( A = zero_zero_int ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_le_zero_iff
% 0.22/0.49 thf(fact_5020_abs__le__zero__iff,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 0.22/0.49 = ( A = zero_zero_real ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_le_zero_iff
% 0.22/0.49 thf(fact_5021_abs__le__self__iff,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 0.22/0.49 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_le_self_iff
% 0.22/0.49 thf(fact_5022_abs__le__self__iff,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 0.22/0.49 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_le_self_iff
% 0.22/0.49 thf(fact_5023_abs__le__self__iff,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 0.22/0.49 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_le_self_iff
% 0.22/0.49 thf(fact_5024_abs__le__self__iff,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 0.22/0.49 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_le_self_iff
% 0.22/0.49 thf(fact_5025_abs__of__nonneg,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.22/0.49 => ( ( abs_abs_Code_integer @ A )
% 0.22/0.49 = A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_of_nonneg
% 0.22/0.49 thf(fact_5026_abs__of__nonneg,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.22/0.49 => ( ( abs_abs_rat @ A )
% 0.22/0.49 = A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_of_nonneg
% 0.22/0.49 thf(fact_5027_abs__of__nonneg,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.22/0.49 => ( ( abs_abs_int @ A )
% 0.22/0.49 = A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_of_nonneg
% 0.22/0.49 thf(fact_5028_abs__of__nonneg,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.49 => ( ( abs_abs_real @ A )
% 0.22/0.49 = A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_of_nonneg
% 0.22/0.49 thf(fact_5029_diff__minus__eq__add,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
% 0.22/0.49 = ( plus_plus_real @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_minus_eq_add
% 0.22/0.49 thf(fact_5030_diff__minus__eq__add,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 0.22/0.49 = ( plus_plus_int @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_minus_eq_add
% 0.22/0.49 thf(fact_5031_diff__minus__eq__add,axiom,
% 0.22/0.49 ! [A: complex,B: complex] :
% 0.22/0.49 ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 0.22/0.49 = ( plus_plus_complex @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_minus_eq_add
% 0.22/0.49 thf(fact_5032_diff__minus__eq__add,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.22/0.49 = ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_minus_eq_add
% 0.22/0.49 thf(fact_5033_diff__minus__eq__add,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.22/0.49 = ( plus_plus_rat @ A @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_minus_eq_add
% 0.22/0.49 thf(fact_5034_uminus__add__conv__diff,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
% 0.22/0.49 = ( minus_minus_real @ B @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % uminus_add_conv_diff
% 0.22/0.49 thf(fact_5035_uminus__add__conv__diff,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 0.22/0.49 = ( minus_minus_int @ B @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % uminus_add_conv_diff
% 0.22/0.49 thf(fact_5036_uminus__add__conv__diff,axiom,
% 0.22/0.49 ! [A: complex,B: complex] :
% 0.22/0.49 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.22/0.49 = ( minus_minus_complex @ B @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % uminus_add_conv_diff
% 0.22/0.49 thf(fact_5037_uminus__add__conv__diff,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.22/0.49 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % uminus_add_conv_diff
% 0.22/0.49 thf(fact_5038_uminus__add__conv__diff,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.22/0.49 = ( minus_minus_rat @ B @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % uminus_add_conv_diff
% 0.22/0.49 thf(fact_5039_zero__less__abs__iff,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 0.22/0.49 = ( A != zero_z3403309356797280102nteger ) ) ).
% 0.22/0.49
% 0.22/0.49 % zero_less_abs_iff
% 0.22/0.49 thf(fact_5040_zero__less__abs__iff,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 0.22/0.49 = ( A != zero_zero_real ) ) ).
% 0.22/0.49
% 0.22/0.49 % zero_less_abs_iff
% 0.22/0.49 thf(fact_5041_zero__less__abs__iff,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 0.22/0.49 = ( A != zero_zero_rat ) ) ).
% 0.22/0.49
% 0.22/0.49 % zero_less_abs_iff
% 0.22/0.49 thf(fact_5042_zero__less__abs__iff,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 0.22/0.49 = ( A != zero_zero_int ) ) ).
% 0.22/0.49
% 0.22/0.49 % zero_less_abs_iff
% 0.22/0.49 thf(fact_5043_div__minus1__right,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.49 = ( uminus_uminus_int @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % div_minus1_right
% 0.22/0.49 thf(fact_5044_div__minus1__right,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % div_minus1_right
% 0.22/0.49 thf(fact_5045_divide__minus1,axiom,
% 0.22/0.49 ! [X: real] :
% 0.22/0.49 ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.49 = ( uminus_uminus_real @ X ) ) ).
% 0.22/0.49
% 0.22/0.49 % divide_minus1
% 0.22/0.49 thf(fact_5046_divide__minus1,axiom,
% 0.22/0.49 ! [X: complex] :
% 0.22/0.49 ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ X ) ) ).
% 0.22/0.49
% 0.22/0.49 % divide_minus1
% 0.22/0.49 thf(fact_5047_divide__minus1,axiom,
% 0.22/0.49 ! [X: rat] :
% 0.22/0.49 ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.22/0.49 = ( uminus_uminus_rat @ X ) ) ).
% 0.22/0.49
% 0.22/0.49 % divide_minus1
% 0.22/0.49 thf(fact_5048_abs__neg__numeral,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.22/0.49 = ( numeral_numeral_real @ N ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_neg_numeral
% 0.22/0.49 thf(fact_5049_abs__neg__numeral,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.49 = ( numeral_numeral_int @ N ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_neg_numeral
% 0.22/0.49 thf(fact_5050_abs__neg__numeral,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.22/0.49 = ( numera6620942414471956472nteger @ N ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_neg_numeral
% 0.22/0.49 thf(fact_5051_abs__neg__numeral,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.22/0.49 = ( numeral_numeral_rat @ N ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_neg_numeral
% 0.22/0.49 thf(fact_5052_abs__neg__one,axiom,
% 0.22/0.49 ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.49 = one_one_real ) ).
% 0.22/0.49
% 0.22/0.49 % abs_neg_one
% 0.22/0.49 thf(fact_5053_abs__neg__one,axiom,
% 0.22/0.49 ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.49 = one_one_int ) ).
% 0.22/0.49
% 0.22/0.49 % abs_neg_one
% 0.22/0.49 thf(fact_5054_abs__neg__one,axiom,
% 0.22/0.49 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.49 = one_one_Code_integer ) ).
% 0.22/0.49
% 0.22/0.49 % abs_neg_one
% 0.22/0.49 thf(fact_5055_abs__neg__one,axiom,
% 0.22/0.49 ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.22/0.49 = one_one_rat ) ).
% 0.22/0.49
% 0.22/0.49 % abs_neg_one
% 0.22/0.49 thf(fact_5056_minus__mod__self1,axiom,
% 0.22/0.49 ! [B: int,A: int] :
% 0.22/0.49 ( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
% 0.22/0.49 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_mod_self1
% 0.22/0.49 thf(fact_5057_minus__mod__self1,axiom,
% 0.22/0.49 ! [B: code_integer,A: code_integer] :
% 0.22/0.49 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ B )
% 0.22/0.49 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_mod_self1
% 0.22/0.49 thf(fact_5058_negative__eq__positive,axiom,
% 0.22/0.49 ! [N: nat,M: nat] :
% 0.22/0.49 ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.22/0.49 = ( semiri1314217659103216013at_int @ M ) )
% 0.22/0.49 = ( ( N = zero_zero_nat )
% 0.22/0.49 & ( M = zero_zero_nat ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % negative_eq_positive
% 0.22/0.49 thf(fact_5059_real__add__minus__iff,axiom,
% 0.22/0.49 ! [X: real,A: real] :
% 0.22/0.49 ( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
% 0.22/0.49 = zero_zero_real )
% 0.22/0.49 = ( X = A ) ) ).
% 0.22/0.49
% 0.22/0.49 % real_add_minus_iff
% 0.22/0.49 thf(fact_5060_signed__take__bit__of__minus__1,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.49
% 0.22/0.49 % signed_take_bit_of_minus_1
% 0.22/0.49 thf(fact_5061_signed__take__bit__of__minus__1,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.49 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.49
% 0.22/0.49 % signed_take_bit_of_minus_1
% 0.22/0.49 thf(fact_5062_negative__zle,axiom,
% 0.22/0.49 ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 0.22/0.49
% 0.22/0.49 % negative_zle
% 0.22/0.49 thf(fact_5063_tanh__real__neg__iff,axiom,
% 0.22/0.49 ! [X: real] :
% 0.22/0.49 ( ( ord_less_real @ ( tanh_real @ X ) @ zero_zero_real )
% 0.22/0.49 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.22/0.49
% 0.22/0.49 % tanh_real_neg_iff
% 0.22/0.49 thf(fact_5064_tanh__real__pos__iff,axiom,
% 0.22/0.49 ! [X: real] :
% 0.22/0.49 ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X ) )
% 0.22/0.49 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.22/0.49
% 0.22/0.49 % tanh_real_pos_iff
% 0.22/0.49 thf(fact_5065_tanh__real__nonpos__iff,axiom,
% 0.22/0.49 ! [X: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ ( tanh_real @ X ) @ zero_zero_real )
% 0.22/0.49 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.22/0.49
% 0.22/0.49 % tanh_real_nonpos_iff
% 0.22/0.49 thf(fact_5066_tanh__real__nonneg__iff,axiom,
% 0.22/0.49 ! [X: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X ) )
% 0.22/0.49 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.22/0.49
% 0.22/0.49 % tanh_real_nonneg_iff
% 0.22/0.49 thf(fact_5067_dbl__simps_I1_J,axiom,
% 0.22/0.49 ! [K: num] :
% 0.22/0.49 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.22/0.49 = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % dbl_simps(1)
% 0.22/0.49 thf(fact_5068_dbl__simps_I1_J,axiom,
% 0.22/0.49 ! [K: num] :
% 0.22/0.49 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.22/0.49 = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % dbl_simps(1)
% 0.22/0.49 thf(fact_5069_dbl__simps_I1_J,axiom,
% 0.22/0.49 ! [K: num] :
% 0.22/0.49 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % dbl_simps(1)
% 0.22/0.49 thf(fact_5070_dbl__simps_I1_J,axiom,
% 0.22/0.49 ! [K: num] :
% 0.22/0.49 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % dbl_simps(1)
% 0.22/0.49 thf(fact_5071_dbl__simps_I1_J,axiom,
% 0.22/0.49 ! [K: num] :
% 0.22/0.49 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.22/0.49 = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % dbl_simps(1)
% 0.22/0.49 thf(fact_5072_dbl__inc__simps_I4_J,axiom,
% 0.22/0.49 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.49 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.22/0.49
% 0.22/0.49 % dbl_inc_simps(4)
% 0.22/0.49 thf(fact_5073_dbl__inc__simps_I4_J,axiom,
% 0.22/0.49 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.49 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.49
% 0.22/0.49 % dbl_inc_simps(4)
% 0.22/0.49 thf(fact_5074_dbl__inc__simps_I4_J,axiom,
% 0.22/0.49 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.22/0.49
% 0.22/0.49 % dbl_inc_simps(4)
% 0.22/0.49 thf(fact_5075_dbl__inc__simps_I4_J,axiom,
% 0.22/0.49 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.49
% 0.22/0.49 % dbl_inc_simps(4)
% 0.22/0.49 thf(fact_5076_dbl__inc__simps_I4_J,axiom,
% 0.22/0.49 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.22/0.49 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.22/0.49
% 0.22/0.49 % dbl_inc_simps(4)
% 0.22/0.49 thf(fact_5077_add__neg__numeral__special_I8_J,axiom,
% 0.22/0.49 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.22/0.49 = zero_zero_real ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_special(8)
% 0.22/0.49 thf(fact_5078_add__neg__numeral__special_I8_J,axiom,
% 0.22/0.49 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.22/0.49 = zero_zero_int ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_special(8)
% 0.22/0.49 thf(fact_5079_add__neg__numeral__special_I8_J,axiom,
% 0.22/0.49 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 0.22/0.49 = zero_zero_complex ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_special(8)
% 0.22/0.49 thf(fact_5080_add__neg__numeral__special_I8_J,axiom,
% 0.22/0.49 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.22/0.49 = zero_z3403309356797280102nteger ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_special(8)
% 0.22/0.49 thf(fact_5081_add__neg__numeral__special_I8_J,axiom,
% 0.22/0.49 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.22/0.49 = zero_zero_rat ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_special(8)
% 0.22/0.49 thf(fact_5082_add__neg__numeral__special_I7_J,axiom,
% 0.22/0.49 ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.49 = zero_zero_real ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_special(7)
% 0.22/0.49 thf(fact_5083_add__neg__numeral__special_I7_J,axiom,
% 0.22/0.49 ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.49 = zero_zero_int ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_special(7)
% 0.22/0.49 thf(fact_5084_add__neg__numeral__special_I7_J,axiom,
% 0.22/0.49 ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.22/0.49 = zero_zero_complex ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_special(7)
% 0.22/0.49 thf(fact_5085_add__neg__numeral__special_I7_J,axiom,
% 0.22/0.49 ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.49 = zero_z3403309356797280102nteger ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_special(7)
% 0.22/0.49 thf(fact_5086_add__neg__numeral__special_I7_J,axiom,
% 0.22/0.49 ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.22/0.49 = zero_zero_rat ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_special(7)
% 0.22/0.49 thf(fact_5087_diff__numeral__special_I12_J,axiom,
% 0.22/0.49 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.49 = zero_zero_real ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(12)
% 0.22/0.49 thf(fact_5088_diff__numeral__special_I12_J,axiom,
% 0.22/0.49 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.49 = zero_zero_int ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(12)
% 0.22/0.49 thf(fact_5089_diff__numeral__special_I12_J,axiom,
% 0.22/0.49 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.22/0.49 = zero_zero_complex ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(12)
% 0.22/0.49 thf(fact_5090_diff__numeral__special_I12_J,axiom,
% 0.22/0.49 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.49 = zero_z3403309356797280102nteger ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(12)
% 0.22/0.49 thf(fact_5091_diff__numeral__special_I12_J,axiom,
% 0.22/0.49 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.22/0.49 = zero_zero_rat ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(12)
% 0.22/0.49 thf(fact_5092_neg__one__eq__numeral__iff,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( ( uminus_uminus_real @ one_one_real )
% 0.22/0.49 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.22/0.49 = ( N = one ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_one_eq_numeral_iff
% 0.22/0.49 thf(fact_5093_neg__one__eq__numeral__iff,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( ( uminus_uminus_int @ one_one_int )
% 0.22/0.49 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.49 = ( N = one ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_one_eq_numeral_iff
% 0.22/0.49 thf(fact_5094_neg__one__eq__numeral__iff,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( ( uminus1482373934393186551omplex @ one_one_complex )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.22/0.49 = ( N = one ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_one_eq_numeral_iff
% 0.22/0.49 thf(fact_5095_neg__one__eq__numeral__iff,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.22/0.49 = ( N = one ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_one_eq_numeral_iff
% 0.22/0.49 thf(fact_5096_neg__one__eq__numeral__iff,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( ( uminus_uminus_rat @ one_one_rat )
% 0.22/0.49 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.22/0.49 = ( N = one ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_one_eq_numeral_iff
% 0.22/0.49 thf(fact_5097_numeral__eq__neg__one__iff,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
% 0.22/0.49 = ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.49 = ( N = one ) ) ).
% 0.22/0.49
% 0.22/0.49 % numeral_eq_neg_one_iff
% 0.22/0.49 thf(fact_5098_numeral__eq__neg__one__iff,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
% 0.22/0.49 = ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.49 = ( N = one ) ) ).
% 0.22/0.49
% 0.22/0.49 % numeral_eq_neg_one_iff
% 0.22/0.49 thf(fact_5099_numeral__eq__neg__one__iff,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.22/0.49 = ( N = one ) ) ).
% 0.22/0.49
% 0.22/0.49 % numeral_eq_neg_one_iff
% 0.22/0.49 thf(fact_5100_numeral__eq__neg__one__iff,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.49 = ( N = one ) ) ).
% 0.22/0.49
% 0.22/0.49 % numeral_eq_neg_one_iff
% 0.22/0.49 thf(fact_5101_numeral__eq__neg__one__iff,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
% 0.22/0.49 = ( uminus_uminus_rat @ one_one_rat ) )
% 0.22/0.49 = ( N = one ) ) ).
% 0.22/0.49
% 0.22/0.49 % numeral_eq_neg_one_iff
% 0.22/0.49 thf(fact_5102_divide__le__0__abs__iff,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
% 0.22/0.49 = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.22/0.49 | ( B = zero_zero_rat ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % divide_le_0_abs_iff
% 0.22/0.49 thf(fact_5103_divide__le__0__abs__iff,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
% 0.22/0.49 = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.22/0.49 | ( B = zero_zero_real ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % divide_le_0_abs_iff
% 0.22/0.49 thf(fact_5104_zero__le__divide__abs__iff,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
% 0.22/0.49 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.22/0.49 | ( B = zero_zero_rat ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % zero_le_divide_abs_iff
% 0.22/0.49 thf(fact_5105_zero__le__divide__abs__iff,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
% 0.22/0.49 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.49 | ( B = zero_zero_real ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % zero_le_divide_abs_iff
% 0.22/0.49 thf(fact_5106_abs__of__nonpos,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
% 0.22/0.49 => ( ( abs_abs_Code_integer @ A )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_of_nonpos
% 0.22/0.49 thf(fact_5107_abs__of__nonpos,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.22/0.49 => ( ( abs_abs_rat @ A )
% 0.22/0.49 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_of_nonpos
% 0.22/0.49 thf(fact_5108_abs__of__nonpos,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.22/0.49 => ( ( abs_abs_int @ A )
% 0.22/0.49 = ( uminus_uminus_int @ A ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_of_nonpos
% 0.22/0.49 thf(fact_5109_abs__of__nonpos,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.22/0.49 => ( ( abs_abs_real @ A )
% 0.22/0.49 = ( uminus_uminus_real @ A ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_of_nonpos
% 0.22/0.49 thf(fact_5110_minus__one__mult__self,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
% 0.22/0.49 = one_one_real ) ).
% 0.22/0.49
% 0.22/0.49 % minus_one_mult_self
% 0.22/0.49 thf(fact_5111_minus__one__mult__self,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
% 0.22/0.49 = one_one_int ) ).
% 0.22/0.49
% 0.22/0.49 % minus_one_mult_self
% 0.22/0.49 thf(fact_5112_minus__one__mult__self,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
% 0.22/0.49 = one_one_complex ) ).
% 0.22/0.49
% 0.22/0.49 % minus_one_mult_self
% 0.22/0.49 thf(fact_5113_minus__one__mult__self,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
% 0.22/0.49 = one_one_Code_integer ) ).
% 0.22/0.49
% 0.22/0.49 % minus_one_mult_self
% 0.22/0.49 thf(fact_5114_minus__one__mult__self,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
% 0.22/0.49 = one_one_rat ) ).
% 0.22/0.49
% 0.22/0.49 % minus_one_mult_self
% 0.22/0.49 thf(fact_5115_left__minus__one__mult__self,axiom,
% 0.22/0.49 ! [N: nat,A: real] :
% 0.22/0.49 ( ( 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 ) )
% 0.22/0.49 = A ) ).
% 0.22/0.49
% 0.22/0.49 % left_minus_one_mult_self
% 0.22/0.49 thf(fact_5116_left__minus__one__mult__self,axiom,
% 0.22/0.49 ! [N: nat,A: int] :
% 0.22/0.49 ( ( 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 ) )
% 0.22/0.49 = A ) ).
% 0.22/0.49
% 0.22/0.49 % left_minus_one_mult_self
% 0.22/0.49 thf(fact_5117_left__minus__one__mult__self,axiom,
% 0.22/0.49 ! [N: nat,A: complex] :
% 0.22/0.49 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
% 0.22/0.49 = A ) ).
% 0.22/0.49
% 0.22/0.49 % left_minus_one_mult_self
% 0.22/0.49 thf(fact_5118_left__minus__one__mult__self,axiom,
% 0.22/0.49 ! [N: nat,A: code_integer] :
% 0.22/0.49 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
% 0.22/0.49 = A ) ).
% 0.22/0.49
% 0.22/0.49 % left_minus_one_mult_self
% 0.22/0.49 thf(fact_5119_left__minus__one__mult__self,axiom,
% 0.22/0.49 ! [N: nat,A: rat] :
% 0.22/0.49 ( ( 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 ) )
% 0.22/0.49 = A ) ).
% 0.22/0.49
% 0.22/0.49 % left_minus_one_mult_self
% 0.22/0.49 thf(fact_5120_mod__minus1__right,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.49 = zero_zero_int ) ).
% 0.22/0.49
% 0.22/0.49 % mod_minus1_right
% 0.22/0.49 thf(fact_5121_mod__minus1__right,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.49 = zero_z3403309356797280102nteger ) ).
% 0.22/0.49
% 0.22/0.49 % mod_minus1_right
% 0.22/0.49 thf(fact_5122_norm__neg__numeral,axiom,
% 0.22/0.49 ! [W: num] :
% 0.22/0.49 ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.22/0.49 = ( numeral_numeral_real @ W ) ) ).
% 0.22/0.49
% 0.22/0.49 % norm_neg_numeral
% 0.22/0.49 thf(fact_5123_norm__neg__numeral,axiom,
% 0.22/0.49 ! [W: num] :
% 0.22/0.49 ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.22/0.49 = ( numeral_numeral_real @ W ) ) ).
% 0.22/0.49
% 0.22/0.49 % norm_neg_numeral
% 0.22/0.49 thf(fact_5124_semiring__norm_I168_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: real] :
% 0.22/0.49 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y4 ) )
% 0.22/0.49 = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(168)
% 0.22/0.49 thf(fact_5125_semiring__norm_I168_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: int] :
% 0.22/0.49 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y4 ) )
% 0.22/0.49 = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(168)
% 0.22/0.49 thf(fact_5126_semiring__norm_I168_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: complex] :
% 0.22/0.49 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y4 ) )
% 0.22/0.49 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(168)
% 0.22/0.49 thf(fact_5127_semiring__norm_I168_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: code_integer] :
% 0.22/0.49 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y4 ) )
% 0.22/0.49 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(168)
% 0.22/0.49 thf(fact_5128_semiring__norm_I168_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: rat] :
% 0.22/0.49 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y4 ) )
% 0.22/0.49 = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(168)
% 0.22/0.49 thf(fact_5129_diff__numeral__simps_I3_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 0.22/0.49 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_simps(3)
% 0.22/0.49 thf(fact_5130_diff__numeral__simps_I3_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.22/0.49 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_simps(3)
% 0.22/0.49 thf(fact_5131_diff__numeral__simps_I3_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_simps(3)
% 0.22/0.49 thf(fact_5132_diff__numeral__simps_I3_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_simps(3)
% 0.22/0.49 thf(fact_5133_diff__numeral__simps_I3_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 0.22/0.49 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_simps(3)
% 0.22/0.49 thf(fact_5134_diff__numeral__simps_I2_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.22/0.49 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_simps(2)
% 0.22/0.49 thf(fact_5135_diff__numeral__simps_I2_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.49 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_simps(2)
% 0.22/0.49 thf(fact_5136_diff__numeral__simps_I2_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.22/0.49 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_simps(2)
% 0.22/0.49 thf(fact_5137_diff__numeral__simps_I2_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.22/0.49 = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_simps(2)
% 0.22/0.49 thf(fact_5138_diff__numeral__simps_I2_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.22/0.49 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_simps(2)
% 0.22/0.49 thf(fact_5139_semiring__norm_I172_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: real] :
% 0.22/0.49 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y4 ) )
% 0.22/0.49 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(172)
% 0.22/0.49 thf(fact_5140_semiring__norm_I172_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: int] :
% 0.22/0.49 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y4 ) )
% 0.22/0.49 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(172)
% 0.22/0.49 thf(fact_5141_semiring__norm_I172_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: complex] :
% 0.22/0.49 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y4 ) )
% 0.22/0.49 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(172)
% 0.22/0.49 thf(fact_5142_semiring__norm_I172_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: code_integer] :
% 0.22/0.49 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y4 ) )
% 0.22/0.49 = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(172)
% 0.22/0.49 thf(fact_5143_semiring__norm_I172_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: rat] :
% 0.22/0.49 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y4 ) )
% 0.22/0.49 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(172)
% 0.22/0.49 thf(fact_5144_semiring__norm_I171_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: real] :
% 0.22/0.49 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y4 ) )
% 0.22/0.49 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(171)
% 0.22/0.49 thf(fact_5145_semiring__norm_I171_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: int] :
% 0.22/0.49 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y4 ) )
% 0.22/0.49 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(171)
% 0.22/0.49 thf(fact_5146_semiring__norm_I171_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: complex] :
% 0.22/0.49 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y4 ) )
% 0.22/0.49 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(171)
% 0.22/0.49 thf(fact_5147_semiring__norm_I171_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: code_integer] :
% 0.22/0.49 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y4 ) )
% 0.22/0.49 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(171)
% 0.22/0.49 thf(fact_5148_semiring__norm_I171_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: rat] :
% 0.22/0.49 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y4 ) )
% 0.22/0.49 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(171)
% 0.22/0.49 thf(fact_5149_semiring__norm_I170_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: real] :
% 0.22/0.49 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y4 ) )
% 0.22/0.49 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(170)
% 0.22/0.49 thf(fact_5150_semiring__norm_I170_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: int] :
% 0.22/0.49 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y4 ) )
% 0.22/0.49 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(170)
% 0.22/0.49 thf(fact_5151_semiring__norm_I170_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: complex] :
% 0.22/0.49 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y4 ) )
% 0.22/0.49 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(170)
% 0.22/0.49 thf(fact_5152_semiring__norm_I170_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: code_integer] :
% 0.22/0.49 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y4 ) )
% 0.22/0.49 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(170)
% 0.22/0.49 thf(fact_5153_semiring__norm_I170_J,axiom,
% 0.22/0.49 ! [V: num,W: num,Y4: rat] :
% 0.22/0.49 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y4 ) )
% 0.22/0.49 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 0.22/0.49
% 0.22/0.49 % semiring_norm(170)
% 0.22/0.49 thf(fact_5154_mult__neg__numeral__simps_I3_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.22/0.49 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_neg_numeral_simps(3)
% 0.22/0.49 thf(fact_5155_mult__neg__numeral__simps_I3_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.49 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_neg_numeral_simps(3)
% 0.22/0.49 thf(fact_5156_mult__neg__numeral__simps_I3_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_neg_numeral_simps(3)
% 0.22/0.49 thf(fact_5157_mult__neg__numeral__simps_I3_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_neg_numeral_simps(3)
% 0.22/0.49 thf(fact_5158_mult__neg__numeral__simps_I3_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.22/0.49 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_neg_numeral_simps(3)
% 0.22/0.49 thf(fact_5159_mult__neg__numeral__simps_I2_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 0.22/0.49 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_neg_numeral_simps(2)
% 0.22/0.49 thf(fact_5160_mult__neg__numeral__simps_I2_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.22/0.49 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_neg_numeral_simps(2)
% 0.22/0.49 thf(fact_5161_mult__neg__numeral__simps_I2_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_neg_numeral_simps(2)
% 0.22/0.49 thf(fact_5162_mult__neg__numeral__simps_I2_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_neg_numeral_simps(2)
% 0.22/0.49 thf(fact_5163_mult__neg__numeral__simps_I2_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 0.22/0.49 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_neg_numeral_simps(2)
% 0.22/0.49 thf(fact_5164_mult__neg__numeral__simps_I1_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.22/0.49 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_neg_numeral_simps(1)
% 0.22/0.49 thf(fact_5165_mult__neg__numeral__simps_I1_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.49 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_neg_numeral_simps(1)
% 0.22/0.49 thf(fact_5166_mult__neg__numeral__simps_I1_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.22/0.49 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_neg_numeral_simps(1)
% 0.22/0.49 thf(fact_5167_mult__neg__numeral__simps_I1_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.22/0.49 = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_neg_numeral_simps(1)
% 0.22/0.49 thf(fact_5168_mult__neg__numeral__simps_I1_J,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.22/0.49 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % mult_neg_numeral_simps(1)
% 0.22/0.49 thf(fact_5169_negative__zless,axiom,
% 0.22/0.49 ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 0.22/0.49
% 0.22/0.49 % negative_zless
% 0.22/0.49 thf(fact_5170_neg__numeral__le__iff,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.22/0.49 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_numeral_le_iff
% 0.22/0.49 thf(fact_5171_neg__numeral__le__iff,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.22/0.49 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_numeral_le_iff
% 0.22/0.49 thf(fact_5172_neg__numeral__le__iff,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.49 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_numeral_le_iff
% 0.22/0.49 thf(fact_5173_neg__numeral__le__iff,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.22/0.49 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_numeral_le_iff
% 0.22/0.49 thf(fact_5174_neg__numeral__less__iff,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.22/0.49 = ( ord_less_num @ N @ M ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_numeral_less_iff
% 0.22/0.49 thf(fact_5175_neg__numeral__less__iff,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.49 = ( ord_less_num @ N @ M ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_numeral_less_iff
% 0.22/0.49 thf(fact_5176_neg__numeral__less__iff,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.22/0.49 = ( ord_less_num @ N @ M ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_numeral_less_iff
% 0.22/0.49 thf(fact_5177_neg__numeral__less__iff,axiom,
% 0.22/0.49 ! [M: num,N: num] :
% 0.22/0.49 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.22/0.49 = ( ord_less_num @ N @ M ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_numeral_less_iff
% 0.22/0.49 thf(fact_5178_not__neg__one__le__neg__numeral__iff,axiom,
% 0.22/0.49 ! [M: num] :
% 0.22/0.49 ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 0.22/0.49 = ( M != one ) ) ).
% 0.22/0.49
% 0.22/0.49 % not_neg_one_le_neg_numeral_iff
% 0.22/0.49 thf(fact_5179_not__neg__one__le__neg__numeral__iff,axiom,
% 0.22/0.49 ! [M: num] :
% 0.22/0.49 ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 0.22/0.49 = ( M != one ) ) ).
% 0.22/0.49
% 0.22/0.49 % not_neg_one_le_neg_numeral_iff
% 0.22/0.49 thf(fact_5180_not__neg__one__le__neg__numeral__iff,axiom,
% 0.22/0.49 ! [M: num] :
% 0.22/0.49 ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 0.22/0.49 = ( M != one ) ) ).
% 0.22/0.49
% 0.22/0.49 % not_neg_one_le_neg_numeral_iff
% 0.22/0.49 thf(fact_5181_not__neg__one__le__neg__numeral__iff,axiom,
% 0.22/0.49 ! [M: num] :
% 0.22/0.49 ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 0.22/0.49 = ( M != one ) ) ).
% 0.22/0.49
% 0.22/0.49 % not_neg_one_le_neg_numeral_iff
% 0.22/0.49 thf(fact_5182_divide__le__eq__numeral1_I2_J,axiom,
% 0.22/0.49 ! [B: rat,W: num,A: rat] :
% 0.22/0.49 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 0.22/0.49 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % divide_le_eq_numeral1(2)
% 0.22/0.49 thf(fact_5183_divide__le__eq__numeral1_I2_J,axiom,
% 0.22/0.49 ! [B: real,W: num,A: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 0.22/0.49 = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % divide_le_eq_numeral1(2)
% 0.22/0.49 thf(fact_5184_le__divide__eq__numeral1_I2_J,axiom,
% 0.22/0.49 ! [A: rat,B: rat,W: num] :
% 0.22/0.49 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.22/0.49 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % le_divide_eq_numeral1(2)
% 0.22/0.49 thf(fact_5185_le__divide__eq__numeral1_I2_J,axiom,
% 0.22/0.49 ! [A: real,B: real,W: num] :
% 0.22/0.49 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.22/0.49 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % le_divide_eq_numeral1(2)
% 0.22/0.49 thf(fact_5186_divide__eq__eq__numeral1_I2_J,axiom,
% 0.22/0.49 ! [B: real,W: num,A: real] :
% 0.22/0.49 ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.22/0.49 = A )
% 0.22/0.49 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.22/0.49 != zero_zero_real )
% 0.22/0.49 => ( B
% 0.22/0.49 = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
% 0.22/0.49 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.22/0.49 = zero_zero_real )
% 0.22/0.49 => ( A = zero_zero_real ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % divide_eq_eq_numeral1(2)
% 0.22/0.49 thf(fact_5187_divide__eq__eq__numeral1_I2_J,axiom,
% 0.22/0.49 ! [B: complex,W: num,A: complex] :
% 0.22/0.49 ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.22/0.49 = A )
% 0.22/0.49 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.22/0.49 != zero_zero_complex )
% 0.22/0.49 => ( B
% 0.22/0.49 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
% 0.22/0.49 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.22/0.49 = zero_zero_complex )
% 0.22/0.49 => ( A = zero_zero_complex ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % divide_eq_eq_numeral1(2)
% 0.22/0.49 thf(fact_5188_divide__eq__eq__numeral1_I2_J,axiom,
% 0.22/0.49 ! [B: rat,W: num,A: rat] :
% 0.22/0.49 ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.22/0.49 = A )
% 0.22/0.49 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.22/0.49 != zero_zero_rat )
% 0.22/0.49 => ( B
% 0.22/0.49 = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
% 0.22/0.49 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.22/0.49 = zero_zero_rat )
% 0.22/0.49 => ( A = zero_zero_rat ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % divide_eq_eq_numeral1(2)
% 0.22/0.49 thf(fact_5189_eq__divide__eq__numeral1_I2_J,axiom,
% 0.22/0.49 ! [A: real,B: real,W: num] :
% 0.22/0.49 ( ( A
% 0.22/0.49 = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.22/0.49 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.22/0.49 != zero_zero_real )
% 0.22/0.49 => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.22/0.49 = B ) )
% 0.22/0.49 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.22/0.49 = zero_zero_real )
% 0.22/0.49 => ( A = zero_zero_real ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % eq_divide_eq_numeral1(2)
% 0.22/0.49 thf(fact_5190_eq__divide__eq__numeral1_I2_J,axiom,
% 0.22/0.49 ! [A: complex,B: complex,W: num] :
% 0.22/0.49 ( ( A
% 0.22/0.49 = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
% 0.22/0.49 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.22/0.49 != zero_zero_complex )
% 0.22/0.49 => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.22/0.49 = B ) )
% 0.22/0.49 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.22/0.49 = zero_zero_complex )
% 0.22/0.49 => ( A = zero_zero_complex ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % eq_divide_eq_numeral1(2)
% 0.22/0.49 thf(fact_5191_eq__divide__eq__numeral1_I2_J,axiom,
% 0.22/0.49 ! [A: rat,B: rat,W: num] :
% 0.22/0.49 ( ( A
% 0.22/0.49 = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.22/0.49 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.22/0.49 != zero_zero_rat )
% 0.22/0.49 => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.22/0.49 = B ) )
% 0.22/0.49 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.22/0.49 = zero_zero_rat )
% 0.22/0.49 => ( A = zero_zero_rat ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % eq_divide_eq_numeral1(2)
% 0.22/0.49 thf(fact_5192_neg__numeral__less__neg__one__iff,axiom,
% 0.22/0.49 ! [M: num] :
% 0.22/0.49 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.49 = ( M != one ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_numeral_less_neg_one_iff
% 0.22/0.49 thf(fact_5193_neg__numeral__less__neg__one__iff,axiom,
% 0.22/0.49 ! [M: num] :
% 0.22/0.49 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.49 = ( M != one ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_numeral_less_neg_one_iff
% 0.22/0.49 thf(fact_5194_neg__numeral__less__neg__one__iff,axiom,
% 0.22/0.49 ! [M: num] :
% 0.22/0.49 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.49 = ( M != one ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_numeral_less_neg_one_iff
% 0.22/0.49 thf(fact_5195_neg__numeral__less__neg__one__iff,axiom,
% 0.22/0.49 ! [M: num] :
% 0.22/0.49 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.22/0.49 = ( M != one ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_numeral_less_neg_one_iff
% 0.22/0.49 thf(fact_5196_divide__less__eq__numeral1_I2_J,axiom,
% 0.22/0.49 ! [B: real,W: num,A: real] :
% 0.22/0.49 ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 0.22/0.49 = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % divide_less_eq_numeral1(2)
% 0.22/0.49 thf(fact_5197_divide__less__eq__numeral1_I2_J,axiom,
% 0.22/0.49 ! [B: rat,W: num,A: rat] :
% 0.22/0.49 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 0.22/0.49 = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % divide_less_eq_numeral1(2)
% 0.22/0.49 thf(fact_5198_less__divide__eq__numeral1_I2_J,axiom,
% 0.22/0.49 ! [A: real,B: real,W: num] :
% 0.22/0.49 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.22/0.49 = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % less_divide_eq_numeral1(2)
% 0.22/0.49 thf(fact_5199_less__divide__eq__numeral1_I2_J,axiom,
% 0.22/0.49 ! [A: rat,B: rat,W: num] :
% 0.22/0.49 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.22/0.49 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % less_divide_eq_numeral1(2)
% 0.22/0.49 thf(fact_5200_power2__minus,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.49 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % power2_minus
% 0.22/0.49 thf(fact_5201_power2__minus,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.49 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % power2_minus
% 0.22/0.49 thf(fact_5202_power2__minus,axiom,
% 0.22/0.49 ! [A: complex] :
% 0.22/0.49 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.49 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % power2_minus
% 0.22/0.49 thf(fact_5203_power2__minus,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.49 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % power2_minus
% 0.22/0.49 thf(fact_5204_power2__minus,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.49 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % power2_minus
% 0.22/0.49 thf(fact_5205_zero__less__power__abs__iff,axiom,
% 0.22/0.49 ! [A: code_integer,N: nat] :
% 0.22/0.49 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
% 0.22/0.49 = ( ( A != zero_z3403309356797280102nteger )
% 0.22/0.49 | ( N = zero_zero_nat ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % zero_less_power_abs_iff
% 0.22/0.49 thf(fact_5206_zero__less__power__abs__iff,axiom,
% 0.22/0.49 ! [A: real,N: nat] :
% 0.22/0.49 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 0.22/0.49 = ( ( A != zero_zero_real )
% 0.22/0.49 | ( N = zero_zero_nat ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % zero_less_power_abs_iff
% 0.22/0.49 thf(fact_5207_zero__less__power__abs__iff,axiom,
% 0.22/0.49 ! [A: rat,N: nat] :
% 0.22/0.49 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
% 0.22/0.49 = ( ( A != zero_zero_rat )
% 0.22/0.49 | ( N = zero_zero_nat ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % zero_less_power_abs_iff
% 0.22/0.49 thf(fact_5208_zero__less__power__abs__iff,axiom,
% 0.22/0.49 ! [A: int,N: nat] :
% 0.22/0.49 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
% 0.22/0.49 = ( ( A != zero_zero_int )
% 0.22/0.49 | ( N = zero_zero_nat ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % zero_less_power_abs_iff
% 0.22/0.49 thf(fact_5209_abs__power2,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.49 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_power2
% 0.22/0.49 thf(fact_5210_abs__power2,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.49 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_power2
% 0.22/0.49 thf(fact_5211_abs__power2,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.49 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_power2
% 0.22/0.49 thf(fact_5212_abs__power2,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.49 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_power2
% 0.22/0.49 thf(fact_5213_power2__abs,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.49 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % power2_abs
% 0.22/0.49 thf(fact_5214_power2__abs,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.49 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % power2_abs
% 0.22/0.49 thf(fact_5215_power2__abs,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.49 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % power2_abs
% 0.22/0.49 thf(fact_5216_power2__abs,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.49 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % power2_abs
% 0.22/0.49 thf(fact_5217_add__neg__numeral__special_I9_J,axiom,
% 0.22/0.49 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.49 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_special(9)
% 0.22/0.49 thf(fact_5218_add__neg__numeral__special_I9_J,axiom,
% 0.22/0.49 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.49 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_special(9)
% 0.22/0.49 thf(fact_5219_add__neg__numeral__special_I9_J,axiom,
% 0.22/0.49 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_special(9)
% 0.22/0.49 thf(fact_5220_add__neg__numeral__special_I9_J,axiom,
% 0.22/0.49 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_special(9)
% 0.22/0.49 thf(fact_5221_add__neg__numeral__special_I9_J,axiom,
% 0.22/0.49 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.22/0.49 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % add_neg_numeral_special(9)
% 0.22/0.49 thf(fact_5222_diff__numeral__special_I11_J,axiom,
% 0.22/0.49 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.49 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(11)
% 0.22/0.49 thf(fact_5223_diff__numeral__special_I11_J,axiom,
% 0.22/0.49 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.49 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(11)
% 0.22/0.49 thf(fact_5224_diff__numeral__special_I11_J,axiom,
% 0.22/0.49 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.22/0.49 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(11)
% 0.22/0.49 thf(fact_5225_diff__numeral__special_I11_J,axiom,
% 0.22/0.49 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.49 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(11)
% 0.22/0.49 thf(fact_5226_diff__numeral__special_I11_J,axiom,
% 0.22/0.49 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.22/0.49 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(11)
% 0.22/0.49 thf(fact_5227_diff__numeral__special_I10_J,axiom,
% 0.22/0.49 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.22/0.49 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(10)
% 0.22/0.49 thf(fact_5228_diff__numeral__special_I10_J,axiom,
% 0.22/0.49 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.22/0.49 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(10)
% 0.22/0.49 thf(fact_5229_diff__numeral__special_I10_J,axiom,
% 0.22/0.49 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(10)
% 0.22/0.49 thf(fact_5230_diff__numeral__special_I10_J,axiom,
% 0.22/0.49 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(10)
% 0.22/0.49 thf(fact_5231_diff__numeral__special_I10_J,axiom,
% 0.22/0.49 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.22/0.49 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(10)
% 0.22/0.49 thf(fact_5232_minus__1__div__2__eq,axiom,
% 0.22/0.49 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.49 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_1_div_2_eq
% 0.22/0.49 thf(fact_5233_minus__1__div__2__eq,axiom,
% 0.22/0.49 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_1_div_2_eq
% 0.22/0.49 thf(fact_5234_minus__1__mod__2__eq,axiom,
% 0.22/0.49 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.49 = one_one_int ) ).
% 0.22/0.49
% 0.22/0.49 % minus_1_mod_2_eq
% 0.22/0.49 thf(fact_5235_minus__1__mod__2__eq,axiom,
% 0.22/0.49 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.49 = one_one_Code_integer ) ).
% 0.22/0.49
% 0.22/0.49 % minus_1_mod_2_eq
% 0.22/0.49 thf(fact_5236_bits__minus__1__mod__2__eq,axiom,
% 0.22/0.49 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.49 = one_one_int ) ).
% 0.22/0.49
% 0.22/0.49 % bits_minus_1_mod_2_eq
% 0.22/0.49 thf(fact_5237_bits__minus__1__mod__2__eq,axiom,
% 0.22/0.49 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.49 = one_one_Code_integer ) ).
% 0.22/0.49
% 0.22/0.49 % bits_minus_1_mod_2_eq
% 0.22/0.49 thf(fact_5238_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.22/0.49 ! [A: real,N: nat] :
% 0.22/0.49 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.49 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % Power.ring_1_class.power_minus_even
% 0.22/0.49 thf(fact_5239_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.22/0.49 ! [A: int,N: nat] :
% 0.22/0.49 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.49 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % Power.ring_1_class.power_minus_even
% 0.22/0.49 thf(fact_5240_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.22/0.49 ! [A: complex,N: nat] :
% 0.22/0.49 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.49 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % Power.ring_1_class.power_minus_even
% 0.22/0.49 thf(fact_5241_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.22/0.49 ! [A: code_integer,N: nat] :
% 0.22/0.49 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.49 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % Power.ring_1_class.power_minus_even
% 0.22/0.49 thf(fact_5242_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.22/0.49 ! [A: rat,N: nat] :
% 0.22/0.49 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.49 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % Power.ring_1_class.power_minus_even
% 0.22/0.49 thf(fact_5243_power__minus__odd,axiom,
% 0.22/0.49 ! [N: nat,A: real] :
% 0.22/0.49 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.22/0.49 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % power_minus_odd
% 0.22/0.49 thf(fact_5244_power__minus__odd,axiom,
% 0.22/0.49 ! [N: nat,A: int] :
% 0.22/0.49 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.22/0.49 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % power_minus_odd
% 0.22/0.49 thf(fact_5245_power__minus__odd,axiom,
% 0.22/0.49 ! [N: nat,A: complex] :
% 0.22/0.49 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % power_minus_odd
% 0.22/0.49 thf(fact_5246_power__minus__odd,axiom,
% 0.22/0.49 ! [N: nat,A: code_integer] :
% 0.22/0.49 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % power_minus_odd
% 0.22/0.49 thf(fact_5247_power__minus__odd,axiom,
% 0.22/0.49 ! [N: nat,A: rat] :
% 0.22/0.49 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.22/0.49 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % power_minus_odd
% 0.22/0.49 thf(fact_5248_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.22/0.49 ! [N: nat,A: real] :
% 0.22/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.22/0.49 = ( power_power_real @ A @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % Parity.ring_1_class.power_minus_even
% 0.22/0.49 thf(fact_5249_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.22/0.49 ! [N: nat,A: int] :
% 0.22/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.22/0.49 = ( power_power_int @ A @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % Parity.ring_1_class.power_minus_even
% 0.22/0.49 thf(fact_5250_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.22/0.49 ! [N: nat,A: complex] :
% 0.22/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.22/0.49 = ( power_power_complex @ A @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % Parity.ring_1_class.power_minus_even
% 0.22/0.49 thf(fact_5251_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.22/0.49 ! [N: nat,A: code_integer] :
% 0.22/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.22/0.49 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % Parity.ring_1_class.power_minus_even
% 0.22/0.49 thf(fact_5252_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.22/0.49 ! [N: nat,A: rat] :
% 0.22/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.22/0.49 = ( power_power_rat @ A @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % Parity.ring_1_class.power_minus_even
% 0.22/0.49 thf(fact_5253_power__even__abs__numeral,axiom,
% 0.22/0.49 ! [W: num,A: code_integer] :
% 0.22/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.22/0.49 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.22/0.49 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % power_even_abs_numeral
% 0.22/0.49 thf(fact_5254_power__even__abs__numeral,axiom,
% 0.22/0.49 ! [W: num,A: rat] :
% 0.22/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.22/0.49 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.22/0.49 = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % power_even_abs_numeral
% 0.22/0.49 thf(fact_5255_power__even__abs__numeral,axiom,
% 0.22/0.49 ! [W: num,A: int] :
% 0.22/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.22/0.49 => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.22/0.49 = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % power_even_abs_numeral
% 0.22/0.49 thf(fact_5256_power__even__abs__numeral,axiom,
% 0.22/0.49 ! [W: num,A: real] :
% 0.22/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.22/0.49 => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.22/0.49 = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % power_even_abs_numeral
% 0.22/0.49 thf(fact_5257_diff__numeral__special_I3_J,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.22/0.49 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(3)
% 0.22/0.49 thf(fact_5258_diff__numeral__special_I3_J,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.49 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(3)
% 0.22/0.49 thf(fact_5259_diff__numeral__special_I3_J,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.22/0.49 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(3)
% 0.22/0.49 thf(fact_5260_diff__numeral__special_I3_J,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.22/0.49 = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(3)
% 0.22/0.49 thf(fact_5261_diff__numeral__special_I3_J,axiom,
% 0.22/0.49 ! [N: num] :
% 0.22/0.49 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.22/0.49 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(3)
% 0.22/0.49 thf(fact_5262_diff__numeral__special_I4_J,axiom,
% 0.22/0.49 ! [M: num] :
% 0.22/0.49 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
% 0.22/0.49 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(4)
% 0.22/0.49 thf(fact_5263_diff__numeral__special_I4_J,axiom,
% 0.22/0.49 ! [M: num] :
% 0.22/0.49 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
% 0.22/0.49 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(4)
% 0.22/0.49 thf(fact_5264_diff__numeral__special_I4_J,axiom,
% 0.22/0.49 ! [M: num] :
% 0.22/0.49 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(4)
% 0.22/0.49 thf(fact_5265_diff__numeral__special_I4_J,axiom,
% 0.22/0.49 ! [M: num] :
% 0.22/0.49 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(4)
% 0.22/0.49 thf(fact_5266_diff__numeral__special_I4_J,axiom,
% 0.22/0.49 ! [M: num] :
% 0.22/0.49 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
% 0.22/0.49 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % diff_numeral_special(4)
% 0.22/0.49 thf(fact_5267_dbl__simps_I4_J,axiom,
% 0.22/0.49 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.49 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % dbl_simps(4)
% 0.22/0.49 thf(fact_5268_dbl__simps_I4_J,axiom,
% 0.22/0.49 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.49 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % dbl_simps(4)
% 0.22/0.49 thf(fact_5269_dbl__simps_I4_J,axiom,
% 0.22/0.49 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % dbl_simps(4)
% 0.22/0.49 thf(fact_5270_dbl__simps_I4_J,axiom,
% 0.22/0.49 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % dbl_simps(4)
% 0.22/0.49 thf(fact_5271_dbl__simps_I4_J,axiom,
% 0.22/0.49 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.22/0.49 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % dbl_simps(4)
% 0.22/0.49 thf(fact_5272_power__minus1__even,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.49 = one_one_real ) ).
% 0.22/0.49
% 0.22/0.49 % power_minus1_even
% 0.22/0.49 thf(fact_5273_power__minus1__even,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.49 = one_one_int ) ).
% 0.22/0.49
% 0.22/0.49 % power_minus1_even
% 0.22/0.49 thf(fact_5274_power__minus1__even,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.49 = one_one_complex ) ).
% 0.22/0.49
% 0.22/0.49 % power_minus1_even
% 0.22/0.49 thf(fact_5275_power__minus1__even,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.49 = one_one_Code_integer ) ).
% 0.22/0.49
% 0.22/0.49 % power_minus1_even
% 0.22/0.49 thf(fact_5276_power__minus1__even,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.49 = one_one_rat ) ).
% 0.22/0.49
% 0.22/0.49 % power_minus1_even
% 0.22/0.49 thf(fact_5277_neg__one__odd__power,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.22/0.49 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_one_odd_power
% 0.22/0.49 thf(fact_5278_neg__one__odd__power,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.22/0.49 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_one_odd_power
% 0.22/0.49 thf(fact_5279_neg__one__odd__power,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.22/0.49 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_one_odd_power
% 0.22/0.49 thf(fact_5280_neg__one__odd__power,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_one_odd_power
% 0.22/0.49 thf(fact_5281_neg__one__odd__power,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.22/0.49 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_one_odd_power
% 0.22/0.49 thf(fact_5282_neg__one__even__power,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.22/0.49 = one_one_real ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_one_even_power
% 0.22/0.49 thf(fact_5283_neg__one__even__power,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.22/0.49 = one_one_int ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_one_even_power
% 0.22/0.49 thf(fact_5284_neg__one__even__power,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.22/0.49 = one_one_complex ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_one_even_power
% 0.22/0.49 thf(fact_5285_neg__one__even__power,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.22/0.49 = one_one_Code_integer ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_one_even_power
% 0.22/0.49 thf(fact_5286_neg__one__even__power,axiom,
% 0.22/0.49 ! [N: nat] :
% 0.22/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.49 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.22/0.49 = one_one_rat ) ) ).
% 0.22/0.49
% 0.22/0.49 % neg_one_even_power
% 0.22/0.49 thf(fact_5287_signed__take__bit__0,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % signed_take_bit_0
% 0.22/0.49 thf(fact_5288_signed__take__bit__0,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 0.22/0.49 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % signed_take_bit_0
% 0.22/0.49 thf(fact_5289_signed__take__bit__Suc__minus__bit0,axiom,
% 0.22/0.49 ! [N: nat,K: num] :
% 0.22/0.49 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.22/0.49 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % signed_take_bit_Suc_minus_bit0
% 0.22/0.49 thf(fact_5290_signed__take__bit__minus,axiom,
% 0.22/0.49 ! [N: nat,K: int] :
% 0.22/0.49 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
% 0.22/0.49 = ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % signed_take_bit_minus
% 0.22/0.49 thf(fact_5291_equation__minus__iff,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( A
% 0.22/0.49 = ( uminus_uminus_real @ B ) )
% 0.22/0.49 = ( B
% 0.22/0.49 = ( uminus_uminus_real @ A ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % equation_minus_iff
% 0.22/0.49 thf(fact_5292_equation__minus__iff,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( A
% 0.22/0.49 = ( uminus_uminus_int @ B ) )
% 0.22/0.49 = ( B
% 0.22/0.49 = ( uminus_uminus_int @ A ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % equation_minus_iff
% 0.22/0.49 thf(fact_5293_equation__minus__iff,axiom,
% 0.22/0.49 ! [A: complex,B: complex] :
% 0.22/0.49 ( ( A
% 0.22/0.49 = ( uminus1482373934393186551omplex @ B ) )
% 0.22/0.49 = ( B
% 0.22/0.49 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % equation_minus_iff
% 0.22/0.49 thf(fact_5294_equation__minus__iff,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( A
% 0.22/0.49 = ( uminus1351360451143612070nteger @ B ) )
% 0.22/0.49 = ( B
% 0.22/0.49 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % equation_minus_iff
% 0.22/0.49 thf(fact_5295_equation__minus__iff,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( A
% 0.22/0.49 = ( uminus_uminus_rat @ B ) )
% 0.22/0.49 = ( B
% 0.22/0.49 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % equation_minus_iff
% 0.22/0.49 thf(fact_5296_minus__equation__iff,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( ( uminus_uminus_real @ A )
% 0.22/0.49 = B )
% 0.22/0.49 = ( ( uminus_uminus_real @ B )
% 0.22/0.49 = A ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_equation_iff
% 0.22/0.49 thf(fact_5297_minus__equation__iff,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( ( uminus_uminus_int @ A )
% 0.22/0.49 = B )
% 0.22/0.49 = ( ( uminus_uminus_int @ B )
% 0.22/0.49 = A ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_equation_iff
% 0.22/0.49 thf(fact_5298_minus__equation__iff,axiom,
% 0.22/0.49 ! [A: complex,B: complex] :
% 0.22/0.49 ( ( ( uminus1482373934393186551omplex @ A )
% 0.22/0.49 = B )
% 0.22/0.49 = ( ( uminus1482373934393186551omplex @ B )
% 0.22/0.49 = A ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_equation_iff
% 0.22/0.49 thf(fact_5299_minus__equation__iff,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( ( uminus1351360451143612070nteger @ A )
% 0.22/0.49 = B )
% 0.22/0.49 = ( ( uminus1351360451143612070nteger @ B )
% 0.22/0.49 = A ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_equation_iff
% 0.22/0.49 thf(fact_5300_minus__equation__iff,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( ( uminus_uminus_rat @ A )
% 0.22/0.49 = B )
% 0.22/0.49 = ( ( uminus_uminus_rat @ B )
% 0.22/0.49 = A ) ) ).
% 0.22/0.49
% 0.22/0.49 % minus_equation_iff
% 0.22/0.49 thf(fact_5301_abs__eq__iff,axiom,
% 0.22/0.49 ! [X: real,Y4: real] :
% 0.22/0.49 ( ( ( abs_abs_real @ X )
% 0.22/0.49 = ( abs_abs_real @ Y4 ) )
% 0.22/0.49 = ( ( X = Y4 )
% 0.22/0.49 | ( X
% 0.22/0.49 = ( uminus_uminus_real @ Y4 ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_eq_iff
% 0.22/0.49 thf(fact_5302_abs__eq__iff,axiom,
% 0.22/0.49 ! [X: int,Y4: int] :
% 0.22/0.49 ( ( ( abs_abs_int @ X )
% 0.22/0.49 = ( abs_abs_int @ Y4 ) )
% 0.22/0.49 = ( ( X = Y4 )
% 0.22/0.49 | ( X
% 0.22/0.49 = ( uminus_uminus_int @ Y4 ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_eq_iff
% 0.22/0.49 thf(fact_5303_abs__eq__iff,axiom,
% 0.22/0.49 ! [X: code_integer,Y4: code_integer] :
% 0.22/0.49 ( ( ( abs_abs_Code_integer @ X )
% 0.22/0.49 = ( abs_abs_Code_integer @ Y4 ) )
% 0.22/0.49 = ( ( X = Y4 )
% 0.22/0.49 | ( X
% 0.22/0.49 = ( uminus1351360451143612070nteger @ Y4 ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_eq_iff
% 0.22/0.49 thf(fact_5304_abs__eq__iff,axiom,
% 0.22/0.49 ! [X: rat,Y4: rat] :
% 0.22/0.49 ( ( ( abs_abs_rat @ X )
% 0.22/0.49 = ( abs_abs_rat @ Y4 ) )
% 0.22/0.49 = ( ( X = Y4 )
% 0.22/0.49 | ( X
% 0.22/0.49 = ( uminus_uminus_rat @ Y4 ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_eq_iff
% 0.22/0.49 thf(fact_5305_verit__negate__coefficient_I3_J,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( A = B )
% 0.22/0.49 => ( ( uminus_uminus_real @ A )
% 0.22/0.49 = ( uminus_uminus_real @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % verit_negate_coefficient(3)
% 0.22/0.49 thf(fact_5306_verit__negate__coefficient_I3_J,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( A = B )
% 0.22/0.49 => ( ( uminus_uminus_int @ A )
% 0.22/0.49 = ( uminus_uminus_int @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % verit_negate_coefficient(3)
% 0.22/0.49 thf(fact_5307_verit__negate__coefficient_I3_J,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( A = B )
% 0.22/0.49 => ( ( uminus1351360451143612070nteger @ A )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % verit_negate_coefficient(3)
% 0.22/0.49 thf(fact_5308_verit__negate__coefficient_I3_J,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( A = B )
% 0.22/0.49 => ( ( uminus_uminus_rat @ A )
% 0.22/0.49 = ( uminus_uminus_rat @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % verit_negate_coefficient(3)
% 0.22/0.49 thf(fact_5309_abs__ge__minus__self,axiom,
% 0.22/0.49 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_ge_minus_self
% 0.22/0.49 thf(fact_5310_abs__ge__minus__self,axiom,
% 0.22/0.49 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_ge_minus_self
% 0.22/0.49 thf(fact_5311_abs__ge__minus__self,axiom,
% 0.22/0.49 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_ge_minus_self
% 0.22/0.49 thf(fact_5312_abs__ge__minus__self,axiom,
% 0.22/0.49 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_ge_minus_self
% 0.22/0.49 thf(fact_5313_abs__le__iff,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.22/0.49 = ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.22/0.49 & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_le_iff
% 0.22/0.49 thf(fact_5314_abs__le__iff,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.22/0.49 = ( ( ord_less_eq_rat @ A @ B )
% 0.22/0.49 & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_le_iff
% 0.22/0.49 thf(fact_5315_abs__le__iff,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.22/0.49 = ( ( ord_less_eq_int @ A @ B )
% 0.22/0.49 & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_le_iff
% 0.22/0.49 thf(fact_5316_abs__le__iff,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.22/0.49 = ( ( ord_less_eq_real @ A @ B )
% 0.22/0.49 & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_le_iff
% 0.22/0.49 thf(fact_5317_abs__le__D2,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.22/0.49 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_le_D2
% 0.22/0.49 thf(fact_5318_abs__le__D2,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.22/0.49 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_le_D2
% 0.22/0.49 thf(fact_5319_abs__le__D2,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.22/0.49 => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_le_D2
% 0.22/0.49 thf(fact_5320_abs__le__D2,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.22/0.49 => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_le_D2
% 0.22/0.49 thf(fact_5321_abs__leI,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.22/0.49 => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.22/0.49 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_leI
% 0.22/0.49 thf(fact_5322_abs__leI,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( ord_less_eq_rat @ A @ B )
% 0.22/0.49 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.22/0.49 => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_leI
% 0.22/0.49 thf(fact_5323_abs__leI,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( ord_less_eq_int @ A @ B )
% 0.22/0.49 => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 0.22/0.49 => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_leI
% 0.22/0.49 thf(fact_5324_abs__leI,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( ord_less_eq_real @ A @ B )
% 0.22/0.49 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.22/0.49 => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_leI
% 0.22/0.49 thf(fact_5325_abs__less__iff,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
% 0.22/0.49 = ( ( ord_less_real @ A @ B )
% 0.22/0.49 & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_less_iff
% 0.22/0.49 thf(fact_5326_abs__less__iff,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
% 0.22/0.49 = ( ( ord_less_int @ A @ B )
% 0.22/0.49 & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_less_iff
% 0.22/0.49 thf(fact_5327_abs__less__iff,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.22/0.49 = ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.22/0.49 & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_less_iff
% 0.22/0.49 thf(fact_5328_abs__less__iff,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
% 0.22/0.49 = ( ( ord_less_rat @ A @ B )
% 0.22/0.49 & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_less_iff
% 0.22/0.49 thf(fact_5329_eq__abs__iff_H,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( A
% 0.22/0.49 = ( abs_abs_Code_integer @ B ) )
% 0.22/0.49 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.22/0.49 & ( ( B = A )
% 0.22/0.49 | ( B
% 0.22/0.49 = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % eq_abs_iff'
% 0.22/0.49 thf(fact_5330_eq__abs__iff_H,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( A
% 0.22/0.49 = ( abs_abs_rat @ B ) )
% 0.22/0.49 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.22/0.49 & ( ( B = A )
% 0.22/0.49 | ( B
% 0.22/0.49 = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % eq_abs_iff'
% 0.22/0.49 thf(fact_5331_eq__abs__iff_H,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( A
% 0.22/0.49 = ( abs_abs_int @ B ) )
% 0.22/0.49 = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.22/0.49 & ( ( B = A )
% 0.22/0.49 | ( B
% 0.22/0.49 = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % eq_abs_iff'
% 0.22/0.49 thf(fact_5332_eq__abs__iff_H,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( A
% 0.22/0.49 = ( abs_abs_real @ B ) )
% 0.22/0.49 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.49 & ( ( B = A )
% 0.22/0.49 | ( B
% 0.22/0.49 = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % eq_abs_iff'
% 0.22/0.49 thf(fact_5333_abs__eq__iff_H,axiom,
% 0.22/0.49 ! [A: code_integer,B: code_integer] :
% 0.22/0.49 ( ( ( abs_abs_Code_integer @ A )
% 0.22/0.49 = B )
% 0.22/0.49 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.22/0.49 & ( ( A = B )
% 0.22/0.49 | ( A
% 0.22/0.49 = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_eq_iff'
% 0.22/0.49 thf(fact_5334_abs__eq__iff_H,axiom,
% 0.22/0.49 ! [A: rat,B: rat] :
% 0.22/0.49 ( ( ( abs_abs_rat @ A )
% 0.22/0.49 = B )
% 0.22/0.49 = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.22/0.49 & ( ( A = B )
% 0.22/0.49 | ( A
% 0.22/0.49 = ( uminus_uminus_rat @ B ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_eq_iff'
% 0.22/0.49 thf(fact_5335_abs__eq__iff_H,axiom,
% 0.22/0.49 ! [A: int,B: int] :
% 0.22/0.49 ( ( ( abs_abs_int @ A )
% 0.22/0.49 = B )
% 0.22/0.49 = ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.22/0.49 & ( ( A = B )
% 0.22/0.49 | ( A
% 0.22/0.49 = ( uminus_uminus_int @ B ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_eq_iff'
% 0.22/0.49 thf(fact_5336_abs__eq__iff_H,axiom,
% 0.22/0.49 ! [A: real,B: real] :
% 0.22/0.49 ( ( ( abs_abs_real @ A )
% 0.22/0.49 = B )
% 0.22/0.49 = ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.22/0.49 & ( ( A = B )
% 0.22/0.49 | ( A
% 0.22/0.49 = ( uminus_uminus_real @ B ) ) ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_eq_iff'
% 0.22/0.49 thf(fact_5337_abs__minus__le__zero,axiom,
% 0.22/0.49 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 0.22/0.49
% 0.22/0.49 % abs_minus_le_zero
% 0.22/0.49 thf(fact_5338_abs__minus__le__zero,axiom,
% 0.22/0.49 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 0.22/0.49
% 0.22/0.49 % abs_minus_le_zero
% 0.22/0.49 thf(fact_5339_abs__minus__le__zero,axiom,
% 0.22/0.49 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 0.22/0.49
% 0.22/0.49 % abs_minus_le_zero
% 0.22/0.49 thf(fact_5340_abs__minus__le__zero,axiom,
% 0.22/0.49 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 0.22/0.49
% 0.22/0.49 % abs_minus_le_zero
% 0.22/0.49 thf(fact_5341_abs__if__raw,axiom,
% 0.22/0.49 ( abs_abs_real
% 0.22/0.49 = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_if_raw
% 0.22/0.49 thf(fact_5342_abs__if__raw,axiom,
% 0.22/0.49 ( abs_abs_int
% 0.22/0.49 = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_if_raw
% 0.22/0.49 thf(fact_5343_abs__if__raw,axiom,
% 0.22/0.49 ( abs_abs_Code_integer
% 0.22/0.49 = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_if_raw
% 0.22/0.49 thf(fact_5344_abs__if__raw,axiom,
% 0.22/0.49 ( abs_abs_rat
% 0.22/0.49 = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_if_raw
% 0.22/0.49 thf(fact_5345_abs__of__neg,axiom,
% 0.22/0.49 ! [A: real] :
% 0.22/0.49 ( ( ord_less_real @ A @ zero_zero_real )
% 0.22/0.49 => ( ( abs_abs_real @ A )
% 0.22/0.49 = ( uminus_uminus_real @ A ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_of_neg
% 0.22/0.49 thf(fact_5346_abs__of__neg,axiom,
% 0.22/0.49 ! [A: int] :
% 0.22/0.49 ( ( ord_less_int @ A @ zero_zero_int )
% 0.22/0.49 => ( ( abs_abs_int @ A )
% 0.22/0.49 = ( uminus_uminus_int @ A ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_of_neg
% 0.22/0.49 thf(fact_5347_abs__of__neg,axiom,
% 0.22/0.49 ! [A: code_integer] :
% 0.22/0.49 ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 0.22/0.49 => ( ( abs_abs_Code_integer @ A )
% 0.22/0.49 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_of_neg
% 0.22/0.49 thf(fact_5348_abs__of__neg,axiom,
% 0.22/0.49 ! [A: rat] :
% 0.22/0.49 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.22/0.49 => ( ( abs_abs_rat @ A )
% 0.22/0.49 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_of_neg
% 0.22/0.49 thf(fact_5349_abs__if,axiom,
% 0.22/0.49 ( abs_abs_real
% 0.22/0.49 = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 0.22/0.49
% 0.22/0.49 % abs_if
% 0.22/0.50 thf(fact_5350_abs__if,axiom,
% 0.22/0.50 ( abs_abs_int
% 0.22/0.50 = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_if
% 0.22/0.50 thf(fact_5351_abs__if,axiom,
% 0.22/0.50 ( abs_abs_Code_integer
% 0.22/0.50 = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_if
% 0.22/0.50 thf(fact_5352_abs__if,axiom,
% 0.22/0.50 ( abs_abs_rat
% 0.22/0.50 = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_if
% 0.22/0.50 thf(fact_5353_abs__real__def,axiom,
% 0.22/0.50 ( abs_abs_real
% 0.22/0.50 = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_real_def
% 0.22/0.50 thf(fact_5354_abs__le__D1,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.22/0.50 => ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_le_D1
% 0.22/0.50 thf(fact_5355_abs__le__D1,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.22/0.50 => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_le_D1
% 0.22/0.50 thf(fact_5356_abs__le__D1,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.22/0.50 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_le_D1
% 0.22/0.50 thf(fact_5357_abs__le__D1,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.22/0.50 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_le_D1
% 0.22/0.50 thf(fact_5358_abs__ge__self,axiom,
% 0.22/0.50 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_ge_self
% 0.22/0.50 thf(fact_5359_abs__ge__self,axiom,
% 0.22/0.50 ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_ge_self
% 0.22/0.50 thf(fact_5360_abs__ge__self,axiom,
% 0.22/0.50 ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_ge_self
% 0.22/0.50 thf(fact_5361_abs__ge__self,axiom,
% 0.22/0.50 ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_ge_self
% 0.22/0.50 thf(fact_5362_abs__eq__0__iff,axiom,
% 0.22/0.50 ! [A: code_integer] :
% 0.22/0.50 ( ( ( abs_abs_Code_integer @ A )
% 0.22/0.50 = zero_z3403309356797280102nteger )
% 0.22/0.50 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_eq_0_iff
% 0.22/0.50 thf(fact_5363_abs__eq__0__iff,axiom,
% 0.22/0.50 ! [A: complex] :
% 0.22/0.50 ( ( ( abs_abs_complex @ A )
% 0.22/0.50 = zero_zero_complex )
% 0.22/0.50 = ( A = zero_zero_complex ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_eq_0_iff
% 0.22/0.50 thf(fact_5364_abs__eq__0__iff,axiom,
% 0.22/0.50 ! [A: real] :
% 0.22/0.50 ( ( ( abs_abs_real @ A )
% 0.22/0.50 = zero_zero_real )
% 0.22/0.50 = ( A = zero_zero_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_eq_0_iff
% 0.22/0.50 thf(fact_5365_abs__eq__0__iff,axiom,
% 0.22/0.50 ! [A: rat] :
% 0.22/0.50 ( ( ( abs_abs_rat @ A )
% 0.22/0.50 = zero_zero_rat )
% 0.22/0.50 = ( A = zero_zero_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_eq_0_iff
% 0.22/0.50 thf(fact_5366_abs__eq__0__iff,axiom,
% 0.22/0.50 ! [A: int] :
% 0.22/0.50 ( ( ( abs_abs_int @ A )
% 0.22/0.50 = zero_zero_int )
% 0.22/0.50 = ( A = zero_zero_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_eq_0_iff
% 0.22/0.50 thf(fact_5367_abs__mult,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.22/0.50 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_mult
% 0.22/0.50 thf(fact_5368_abs__mult,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 0.22/0.50 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_mult
% 0.22/0.50 thf(fact_5369_abs__mult,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 0.22/0.50 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_mult
% 0.22/0.50 thf(fact_5370_abs__mult,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 0.22/0.50 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_mult
% 0.22/0.50 thf(fact_5371_abs__one,axiom,
% 0.22/0.50 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 0.22/0.50 = one_one_Code_integer ) ).
% 0.22/0.50
% 0.22/0.50 % abs_one
% 0.22/0.50 thf(fact_5372_abs__one,axiom,
% 0.22/0.50 ( ( abs_abs_real @ one_one_real )
% 0.22/0.50 = one_one_real ) ).
% 0.22/0.50
% 0.22/0.50 % abs_one
% 0.22/0.50 thf(fact_5373_abs__one,axiom,
% 0.22/0.50 ( ( abs_abs_rat @ one_one_rat )
% 0.22/0.50 = one_one_rat ) ).
% 0.22/0.50
% 0.22/0.50 % abs_one
% 0.22/0.50 thf(fact_5374_abs__one,axiom,
% 0.22/0.50 ( ( abs_abs_int @ one_one_int )
% 0.22/0.50 = one_one_int ) ).
% 0.22/0.50
% 0.22/0.50 % abs_one
% 0.22/0.50 thf(fact_5375_abs__minus__commute,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.22/0.50 = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_minus_commute
% 0.22/0.50 thf(fact_5376_abs__minus__commute,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 0.22/0.50 = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_minus_commute
% 0.22/0.50 thf(fact_5377_abs__minus__commute,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
% 0.22/0.50 = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_minus_commute
% 0.22/0.50 thf(fact_5378_abs__minus__commute,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
% 0.22/0.50 = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_minus_commute
% 0.22/0.50 thf(fact_5379_dvd__if__abs__eq,axiom,
% 0.22/0.50 ! [L: real,K: real] :
% 0.22/0.50 ( ( ( abs_abs_real @ L )
% 0.22/0.50 = ( abs_abs_real @ K ) )
% 0.22/0.50 => ( dvd_dvd_real @ L @ K ) ) ).
% 0.22/0.50
% 0.22/0.50 % dvd_if_abs_eq
% 0.22/0.50 thf(fact_5380_dvd__if__abs__eq,axiom,
% 0.22/0.50 ! [L: int,K: int] :
% 0.22/0.50 ( ( ( abs_abs_int @ L )
% 0.22/0.50 = ( abs_abs_int @ K ) )
% 0.22/0.50 => ( dvd_dvd_int @ L @ K ) ) ).
% 0.22/0.50
% 0.22/0.50 % dvd_if_abs_eq
% 0.22/0.50 thf(fact_5381_dvd__if__abs__eq,axiom,
% 0.22/0.50 ! [L: code_integer,K: code_integer] :
% 0.22/0.50 ( ( ( abs_abs_Code_integer @ L )
% 0.22/0.50 = ( abs_abs_Code_integer @ K ) )
% 0.22/0.50 => ( dvd_dvd_Code_integer @ L @ K ) ) ).
% 0.22/0.50
% 0.22/0.50 % dvd_if_abs_eq
% 0.22/0.50 thf(fact_5382_dvd__if__abs__eq,axiom,
% 0.22/0.50 ! [L: rat,K: rat] :
% 0.22/0.50 ( ( ( abs_abs_rat @ L )
% 0.22/0.50 = ( abs_abs_rat @ K ) )
% 0.22/0.50 => ( dvd_dvd_rat @ L @ K ) ) ).
% 0.22/0.50
% 0.22/0.50 % dvd_if_abs_eq
% 0.22/0.50 thf(fact_5383_le__minus__iff,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.22/0.50 = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % le_minus_iff
% 0.22/0.50 thf(fact_5384_le__minus__iff,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.22/0.50 = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % le_minus_iff
% 0.22/0.50 thf(fact_5385_le__minus__iff,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 0.22/0.50 = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % le_minus_iff
% 0.22/0.50 thf(fact_5386_le__minus__iff,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 0.22/0.50 = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % le_minus_iff
% 0.22/0.50 thf(fact_5387_minus__le__iff,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.22/0.50 = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_le_iff
% 0.22/0.50 thf(fact_5388_minus__le__iff,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.22/0.50 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_le_iff
% 0.22/0.50 thf(fact_5389_minus__le__iff,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 0.22/0.50 = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_le_iff
% 0.22/0.50 thf(fact_5390_minus__le__iff,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.22/0.50 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_le_iff
% 0.22/0.50 thf(fact_5391_le__imp__neg__le,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.22/0.50 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % le_imp_neg_le
% 0.22/0.50 thf(fact_5392_le__imp__neg__le,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( ord_less_eq_rat @ A @ B )
% 0.22/0.50 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % le_imp_neg_le
% 0.22/0.50 thf(fact_5393_le__imp__neg__le,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( ord_less_eq_int @ A @ B )
% 0.22/0.50 => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % le_imp_neg_le
% 0.22/0.50 thf(fact_5394_le__imp__neg__le,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( ord_less_eq_real @ A @ B )
% 0.22/0.50 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % le_imp_neg_le
% 0.22/0.50 thf(fact_5395_less__minus__iff,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 0.22/0.50 = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % less_minus_iff
% 0.22/0.50 thf(fact_5396_less__minus__iff,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
% 0.22/0.50 = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % less_minus_iff
% 0.22/0.50 thf(fact_5397_less__minus__iff,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.22/0.50 = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % less_minus_iff
% 0.22/0.50 thf(fact_5398_less__minus__iff,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.22/0.50 = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % less_minus_iff
% 0.22/0.50 thf(fact_5399_minus__less__iff,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 0.22/0.50 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_less_iff
% 0.22/0.50 thf(fact_5400_minus__less__iff,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
% 0.22/0.50 = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_less_iff
% 0.22/0.50 thf(fact_5401_minus__less__iff,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.22/0.50 = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_less_iff
% 0.22/0.50 thf(fact_5402_minus__less__iff,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.22/0.50 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_less_iff
% 0.22/0.50 thf(fact_5403_verit__negate__coefficient_I2_J,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( ord_less_real @ A @ B )
% 0.22/0.50 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % verit_negate_coefficient(2)
% 0.22/0.50 thf(fact_5404_verit__negate__coefficient_I2_J,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( ord_less_int @ A @ B )
% 0.22/0.50 => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % verit_negate_coefficient(2)
% 0.22/0.50 thf(fact_5405_verit__negate__coefficient_I2_J,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.22/0.50 => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % verit_negate_coefficient(2)
% 0.22/0.50 thf(fact_5406_verit__negate__coefficient_I2_J,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( ord_less_rat @ A @ B )
% 0.22/0.50 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % verit_negate_coefficient(2)
% 0.22/0.50 thf(fact_5407_neg__numeral__neq__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] :
% 0.22/0.50 ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 0.22/0.50 != ( numeral_numeral_real @ N ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_neq_numeral
% 0.22/0.50 thf(fact_5408_neg__numeral__neq__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] :
% 0.22/0.50 ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 0.22/0.50 != ( numeral_numeral_int @ N ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_neq_numeral
% 0.22/0.50 thf(fact_5409_neg__numeral__neq__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] :
% 0.22/0.50 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 0.22/0.50 != ( numera6690914467698888265omplex @ N ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_neq_numeral
% 0.22/0.50 thf(fact_5410_neg__numeral__neq__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] :
% 0.22/0.50 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 0.22/0.50 != ( numera6620942414471956472nteger @ N ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_neq_numeral
% 0.22/0.50 thf(fact_5411_neg__numeral__neq__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] :
% 0.22/0.50 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 0.22/0.50 != ( numeral_numeral_rat @ N ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_neq_numeral
% 0.22/0.50 thf(fact_5412_numeral__neq__neg__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] :
% 0.22/0.50 ( ( numeral_numeral_real @ M )
% 0.22/0.50 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % numeral_neq_neg_numeral
% 0.22/0.50 thf(fact_5413_numeral__neq__neg__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] :
% 0.22/0.50 ( ( numeral_numeral_int @ M )
% 0.22/0.50 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % numeral_neq_neg_numeral
% 0.22/0.50 thf(fact_5414_numeral__neq__neg__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] :
% 0.22/0.50 ( ( numera6690914467698888265omplex @ M )
% 0.22/0.50 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % numeral_neq_neg_numeral
% 0.22/0.50 thf(fact_5415_numeral__neq__neg__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] :
% 0.22/0.50 ( ( numera6620942414471956472nteger @ M )
% 0.22/0.50 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % numeral_neq_neg_numeral
% 0.22/0.50 thf(fact_5416_numeral__neq__neg__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] :
% 0.22/0.50 ( ( numeral_numeral_rat @ M )
% 0.22/0.50 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % numeral_neq_neg_numeral
% 0.22/0.50 thf(fact_5417_square__eq__iff,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( ( times_times_real @ A @ A )
% 0.22/0.50 = ( times_times_real @ B @ B ) )
% 0.22/0.50 = ( ( A = B )
% 0.22/0.50 | ( A
% 0.22/0.50 = ( uminus_uminus_real @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % square_eq_iff
% 0.22/0.50 thf(fact_5418_square__eq__iff,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( ( times_times_int @ A @ A )
% 0.22/0.50 = ( times_times_int @ B @ B ) )
% 0.22/0.50 = ( ( A = B )
% 0.22/0.50 | ( A
% 0.22/0.50 = ( uminus_uminus_int @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % square_eq_iff
% 0.22/0.50 thf(fact_5419_square__eq__iff,axiom,
% 0.22/0.50 ! [A: complex,B: complex] :
% 0.22/0.50 ( ( ( times_times_complex @ A @ A )
% 0.22/0.50 = ( times_times_complex @ B @ B ) )
% 0.22/0.50 = ( ( A = B )
% 0.22/0.50 | ( A
% 0.22/0.50 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % square_eq_iff
% 0.22/0.50 thf(fact_5420_square__eq__iff,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( ( times_3573771949741848930nteger @ A @ A )
% 0.22/0.50 = ( times_3573771949741848930nteger @ B @ B ) )
% 0.22/0.50 = ( ( A = B )
% 0.22/0.50 | ( A
% 0.22/0.50 = ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % square_eq_iff
% 0.22/0.50 thf(fact_5421_square__eq__iff,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( ( times_times_rat @ A @ A )
% 0.22/0.50 = ( times_times_rat @ B @ B ) )
% 0.22/0.50 = ( ( A = B )
% 0.22/0.50 | ( A
% 0.22/0.50 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % square_eq_iff
% 0.22/0.50 thf(fact_5422_minus__mult__commute,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 0.22/0.50 = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_mult_commute
% 0.22/0.50 thf(fact_5423_minus__mult__commute,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 0.22/0.50 = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_mult_commute
% 0.22/0.50 thf(fact_5424_minus__mult__commute,axiom,
% 0.22/0.50 ! [A: complex,B: complex] :
% 0.22/0.50 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.22/0.50 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_mult_commute
% 0.22/0.50 thf(fact_5425_minus__mult__commute,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.22/0.50 = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_mult_commute
% 0.22/0.50 thf(fact_5426_minus__mult__commute,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.22/0.50 = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_mult_commute
% 0.22/0.50 thf(fact_5427_group__cancel_Oneg1,axiom,
% 0.22/0.50 ! [A2: real,K: real,A: real] :
% 0.22/0.50 ( ( A2
% 0.22/0.50 = ( plus_plus_real @ K @ A ) )
% 0.22/0.50 => ( ( uminus_uminus_real @ A2 )
% 0.22/0.50 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % group_cancel.neg1
% 0.22/0.50 thf(fact_5428_group__cancel_Oneg1,axiom,
% 0.22/0.50 ! [A2: int,K: int,A: int] :
% 0.22/0.50 ( ( A2
% 0.22/0.50 = ( plus_plus_int @ K @ A ) )
% 0.22/0.50 => ( ( uminus_uminus_int @ A2 )
% 0.22/0.50 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % group_cancel.neg1
% 0.22/0.50 thf(fact_5429_group__cancel_Oneg1,axiom,
% 0.22/0.50 ! [A2: complex,K: complex,A: complex] :
% 0.22/0.50 ( ( A2
% 0.22/0.50 = ( plus_plus_complex @ K @ A ) )
% 0.22/0.50 => ( ( uminus1482373934393186551omplex @ A2 )
% 0.22/0.50 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % group_cancel.neg1
% 0.22/0.50 thf(fact_5430_group__cancel_Oneg1,axiom,
% 0.22/0.50 ! [A2: code_integer,K: code_integer,A: code_integer] :
% 0.22/0.50 ( ( A2
% 0.22/0.50 = ( plus_p5714425477246183910nteger @ K @ A ) )
% 0.22/0.50 => ( ( uminus1351360451143612070nteger @ A2 )
% 0.22/0.50 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % group_cancel.neg1
% 0.22/0.50 thf(fact_5431_group__cancel_Oneg1,axiom,
% 0.22/0.50 ! [A2: rat,K: rat,A: rat] :
% 0.22/0.50 ( ( A2
% 0.22/0.50 = ( plus_plus_rat @ K @ A ) )
% 0.22/0.50 => ( ( uminus_uminus_rat @ A2 )
% 0.22/0.50 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % group_cancel.neg1
% 0.22/0.50 thf(fact_5432_add_Oinverse__distrib__swap,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.22/0.50 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % add.inverse_distrib_swap
% 0.22/0.50 thf(fact_5433_add_Oinverse__distrib__swap,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.22/0.50 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % add.inverse_distrib_swap
% 0.22/0.50 thf(fact_5434_add_Oinverse__distrib__swap,axiom,
% 0.22/0.50 ! [A: complex,B: complex] :
% 0.22/0.50 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 0.22/0.50 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % add.inverse_distrib_swap
% 0.22/0.50 thf(fact_5435_add_Oinverse__distrib__swap,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.22/0.50 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % add.inverse_distrib_swap
% 0.22/0.50 thf(fact_5436_add_Oinverse__distrib__swap,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.22/0.50 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % add.inverse_distrib_swap
% 0.22/0.50 thf(fact_5437_is__num__normalize_I8_J,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.22/0.50 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % is_num_normalize(8)
% 0.22/0.50 thf(fact_5438_is__num__normalize_I8_J,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.22/0.50 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % is_num_normalize(8)
% 0.22/0.50 thf(fact_5439_is__num__normalize_I8_J,axiom,
% 0.22/0.50 ! [A: complex,B: complex] :
% 0.22/0.50 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 0.22/0.50 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % is_num_normalize(8)
% 0.22/0.50 thf(fact_5440_is__num__normalize_I8_J,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.22/0.50 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % is_num_normalize(8)
% 0.22/0.50 thf(fact_5441_is__num__normalize_I8_J,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.22/0.50 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % is_num_normalize(8)
% 0.22/0.50 thf(fact_5442_one__neq__neg__one,axiom,
% 0.22/0.50 ( one_one_real
% 0.22/0.50 != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % one_neq_neg_one
% 0.22/0.50 thf(fact_5443_one__neq__neg__one,axiom,
% 0.22/0.50 ( one_one_int
% 0.22/0.50 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % one_neq_neg_one
% 0.22/0.50 thf(fact_5444_one__neq__neg__one,axiom,
% 0.22/0.50 ( one_one_complex
% 0.22/0.50 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.22/0.50
% 0.22/0.50 % one_neq_neg_one
% 0.22/0.50 thf(fact_5445_one__neq__neg__one,axiom,
% 0.22/0.50 ( one_one_Code_integer
% 0.22/0.50 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.50
% 0.22/0.50 % one_neq_neg_one
% 0.22/0.50 thf(fact_5446_one__neq__neg__one,axiom,
% 0.22/0.50 ( one_one_rat
% 0.22/0.50 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % one_neq_neg_one
% 0.22/0.50 thf(fact_5447_minus__diff__commute,axiom,
% 0.22/0.50 ! [B: real,A: real] :
% 0.22/0.50 ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
% 0.22/0.50 = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_diff_commute
% 0.22/0.50 thf(fact_5448_minus__diff__commute,axiom,
% 0.22/0.50 ! [B: int,A: int] :
% 0.22/0.50 ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 0.22/0.50 = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_diff_commute
% 0.22/0.50 thf(fact_5449_minus__diff__commute,axiom,
% 0.22/0.50 ! [B: complex,A: complex] :
% 0.22/0.50 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
% 0.22/0.50 = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_diff_commute
% 0.22/0.50 thf(fact_5450_minus__diff__commute,axiom,
% 0.22/0.50 ! [B: code_integer,A: code_integer] :
% 0.22/0.50 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
% 0.22/0.50 = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_diff_commute
% 0.22/0.50 thf(fact_5451_minus__diff__commute,axiom,
% 0.22/0.50 ! [B: rat,A: rat] :
% 0.22/0.50 ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
% 0.22/0.50 = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_diff_commute
% 0.22/0.50 thf(fact_5452_minus__diff__minus,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.22/0.50 = ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_diff_minus
% 0.22/0.50 thf(fact_5453_minus__diff__minus,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.22/0.50 = ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_diff_minus
% 0.22/0.50 thf(fact_5454_minus__diff__minus,axiom,
% 0.22/0.50 ! [A: complex,B: complex] :
% 0.22/0.50 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.22/0.50 = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_diff_minus
% 0.22/0.50 thf(fact_5455_minus__diff__minus,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.22/0.50 = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_diff_minus
% 0.22/0.50 thf(fact_5456_minus__diff__minus,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.22/0.50 = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_diff_minus
% 0.22/0.50 thf(fact_5457_div__minus__right,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.22/0.50 = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % div_minus_right
% 0.22/0.50 thf(fact_5458_div__minus__right,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.22/0.50 = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % div_minus_right
% 0.22/0.50 thf(fact_5459_minus__divide__right,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.22/0.50 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_right
% 0.22/0.50 thf(fact_5460_minus__divide__right,axiom,
% 0.22/0.50 ! [A: complex,B: complex] :
% 0.22/0.50 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.22/0.50 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_right
% 0.22/0.50 thf(fact_5461_minus__divide__right,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.22/0.50 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_right
% 0.22/0.50 thf(fact_5462_minus__divide__divide,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.22/0.50 = ( divide_divide_real @ A @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_divide
% 0.22/0.50 thf(fact_5463_minus__divide__divide,axiom,
% 0.22/0.50 ! [A: complex,B: complex] :
% 0.22/0.50 ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.22/0.50 = ( divide1717551699836669952omplex @ A @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_divide
% 0.22/0.50 thf(fact_5464_minus__divide__divide,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.22/0.50 = ( divide_divide_rat @ A @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_divide
% 0.22/0.50 thf(fact_5465_minus__divide__left,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.22/0.50 = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_left
% 0.22/0.50 thf(fact_5466_minus__divide__left,axiom,
% 0.22/0.50 ! [A: complex,B: complex] :
% 0.22/0.50 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.22/0.50 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_left
% 0.22/0.50 thf(fact_5467_minus__divide__left,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.22/0.50 = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_left
% 0.22/0.50 thf(fact_5468_mod__minus__eq,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
% 0.22/0.50 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % mod_minus_eq
% 0.22/0.50 thf(fact_5469_mod__minus__eq,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) @ B )
% 0.22/0.50 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % mod_minus_eq
% 0.22/0.50 thf(fact_5470_mod__minus__cong,axiom,
% 0.22/0.50 ! [A: int,B: int,A6: int] :
% 0.22/0.50 ( ( ( modulo_modulo_int @ A @ B )
% 0.22/0.50 = ( modulo_modulo_int @ A6 @ B ) )
% 0.22/0.50 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 0.22/0.50 = ( modulo_modulo_int @ ( uminus_uminus_int @ A6 ) @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % mod_minus_cong
% 0.22/0.50 thf(fact_5471_mod__minus__cong,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer,A6: code_integer] :
% 0.22/0.50 ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.22/0.50 = ( modulo364778990260209775nteger @ A6 @ B ) )
% 0.22/0.50 => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.22/0.50 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A6 ) @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % mod_minus_cong
% 0.22/0.50 thf(fact_5472_mod__minus__right,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 0.22/0.50 = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % mod_minus_right
% 0.22/0.50 thf(fact_5473_mod__minus__right,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.22/0.50 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % mod_minus_right
% 0.22/0.50 thf(fact_5474_uminus__int__code_I1_J,axiom,
% 0.22/0.50 ( ( uminus_uminus_int @ zero_zero_int )
% 0.22/0.50 = zero_zero_int ) ).
% 0.22/0.50
% 0.22/0.50 % uminus_int_code(1)
% 0.22/0.50 thf(fact_5475_int__cases2,axiom,
% 0.22/0.50 ! [Z: int] :
% 0.22/0.50 ( ! [N2: nat] :
% 0.22/0.50 ( Z
% 0.22/0.50 != ( semiri1314217659103216013at_int @ N2 ) )
% 0.22/0.50 => ~ ! [N2: nat] :
% 0.22/0.50 ( Z
% 0.22/0.50 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % int_cases2
% 0.22/0.50 thf(fact_5476_uminus__dvd__conv_I1_J,axiom,
% 0.22/0.50 ( dvd_dvd_int
% 0.22/0.50 = ( ^ [D4: int] : ( dvd_dvd_int @ ( uminus_uminus_int @ D4 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % uminus_dvd_conv(1)
% 0.22/0.50 thf(fact_5477_uminus__dvd__conv_I2_J,axiom,
% 0.22/0.50 ( dvd_dvd_int
% 0.22/0.50 = ( ^ [D4: int,T2: int] : ( dvd_dvd_int @ D4 @ ( uminus_uminus_int @ T2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % uminus_dvd_conv(2)
% 0.22/0.50 thf(fact_5478_abs__ge__zero,axiom,
% 0.22/0.50 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_ge_zero
% 0.22/0.50 thf(fact_5479_abs__ge__zero,axiom,
% 0.22/0.50 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_ge_zero
% 0.22/0.50 thf(fact_5480_abs__ge__zero,axiom,
% 0.22/0.50 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_ge_zero
% 0.22/0.50 thf(fact_5481_abs__ge__zero,axiom,
% 0.22/0.50 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_ge_zero
% 0.22/0.50 thf(fact_5482_abs__of__pos,axiom,
% 0.22/0.50 ! [A: code_integer] :
% 0.22/0.50 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 0.22/0.50 => ( ( abs_abs_Code_integer @ A )
% 0.22/0.50 = A ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_of_pos
% 0.22/0.50 thf(fact_5483_abs__of__pos,axiom,
% 0.22/0.50 ! [A: real] :
% 0.22/0.50 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.50 => ( ( abs_abs_real @ A )
% 0.22/0.50 = A ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_of_pos
% 0.22/0.50 thf(fact_5484_abs__of__pos,axiom,
% 0.22/0.50 ! [A: rat] :
% 0.22/0.50 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.22/0.50 => ( ( abs_abs_rat @ A )
% 0.22/0.50 = A ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_of_pos
% 0.22/0.50 thf(fact_5485_abs__of__pos,axiom,
% 0.22/0.50 ! [A: int] :
% 0.22/0.50 ( ( ord_less_int @ zero_zero_int @ A )
% 0.22/0.50 => ( ( abs_abs_int @ A )
% 0.22/0.50 = A ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_of_pos
% 0.22/0.50 thf(fact_5486_abs__not__less__zero,axiom,
% 0.22/0.50 ! [A: code_integer] :
% 0.22/0.50 ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 0.22/0.50
% 0.22/0.50 % abs_not_less_zero
% 0.22/0.50 thf(fact_5487_abs__not__less__zero,axiom,
% 0.22/0.50 ! [A: real] :
% 0.22/0.50 ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 0.22/0.50
% 0.22/0.50 % abs_not_less_zero
% 0.22/0.50 thf(fact_5488_abs__not__less__zero,axiom,
% 0.22/0.50 ! [A: rat] :
% 0.22/0.50 ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 0.22/0.50
% 0.22/0.50 % abs_not_less_zero
% 0.22/0.50 thf(fact_5489_abs__not__less__zero,axiom,
% 0.22/0.50 ! [A: int] :
% 0.22/0.50 ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 0.22/0.50
% 0.22/0.50 % abs_not_less_zero
% 0.22/0.50 thf(fact_5490_abs__triangle__ineq,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq
% 0.22/0.50 thf(fact_5491_abs__triangle__ineq,axiom,
% 0.22/0.50 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq
% 0.22/0.50 thf(fact_5492_abs__triangle__ineq,axiom,
% 0.22/0.50 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq
% 0.22/0.50 thf(fact_5493_abs__triangle__ineq,axiom,
% 0.22/0.50 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq
% 0.22/0.50 thf(fact_5494_abs__mult__less,axiom,
% 0.22/0.50 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 0.22/0.50 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 0.22/0.50 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
% 0.22/0.50 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_mult_less
% 0.22/0.50 thf(fact_5495_abs__mult__less,axiom,
% 0.22/0.50 ! [A: real,C: real,B: real,D: real] :
% 0.22/0.50 ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 0.22/0.50 => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
% 0.22/0.50 => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_mult_less
% 0.22/0.50 thf(fact_5496_abs__mult__less,axiom,
% 0.22/0.50 ! [A: rat,C: rat,B: rat,D: rat] :
% 0.22/0.50 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 0.22/0.50 => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
% 0.22/0.50 => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_mult_less
% 0.22/0.50 thf(fact_5497_abs__mult__less,axiom,
% 0.22/0.50 ! [A: int,C: int,B: int,D: int] :
% 0.22/0.50 ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 0.22/0.50 => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
% 0.22/0.50 => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_mult_less
% 0.22/0.50 thf(fact_5498_abs__triangle__ineq2,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq2
% 0.22/0.50 thf(fact_5499_abs__triangle__ineq2,axiom,
% 0.22/0.50 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq2
% 0.22/0.50 thf(fact_5500_abs__triangle__ineq2,axiom,
% 0.22/0.50 ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq2
% 0.22/0.50 thf(fact_5501_abs__triangle__ineq2,axiom,
% 0.22/0.50 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq2
% 0.22/0.50 thf(fact_5502_abs__triangle__ineq3,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq3
% 0.22/0.50 thf(fact_5503_abs__triangle__ineq3,axiom,
% 0.22/0.50 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq3
% 0.22/0.50 thf(fact_5504_abs__triangle__ineq3,axiom,
% 0.22/0.50 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq3
% 0.22/0.50 thf(fact_5505_abs__triangle__ineq3,axiom,
% 0.22/0.50 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq3
% 0.22/0.50 thf(fact_5506_abs__triangle__ineq2__sym,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq2_sym
% 0.22/0.50 thf(fact_5507_abs__triangle__ineq2__sym,axiom,
% 0.22/0.50 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq2_sym
% 0.22/0.50 thf(fact_5508_abs__triangle__ineq2__sym,axiom,
% 0.22/0.50 ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq2_sym
% 0.22/0.50 thf(fact_5509_abs__triangle__ineq2__sym,axiom,
% 0.22/0.50 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq2_sym
% 0.22/0.50 thf(fact_5510_nonzero__abs__divide,axiom,
% 0.22/0.50 ! [B: real,A: real] :
% 0.22/0.50 ( ( B != zero_zero_real )
% 0.22/0.50 => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 0.22/0.50 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % nonzero_abs_divide
% 0.22/0.50 thf(fact_5511_nonzero__abs__divide,axiom,
% 0.22/0.50 ! [B: rat,A: rat] :
% 0.22/0.50 ( ( B != zero_zero_rat )
% 0.22/0.50 => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 0.22/0.50 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % nonzero_abs_divide
% 0.22/0.50 thf(fact_5512_neg__numeral__le__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_le_numeral
% 0.22/0.50 thf(fact_5513_neg__numeral__le__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_le_numeral
% 0.22/0.50 thf(fact_5514_neg__numeral__le__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_le_numeral
% 0.22/0.50 thf(fact_5515_neg__numeral__le__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_le_numeral
% 0.22/0.50 thf(fact_5516_not__numeral__le__neg__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] :
% 0.22/0.50 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_numeral_le_neg_numeral
% 0.22/0.50 thf(fact_5517_not__numeral__le__neg__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] :
% 0.22/0.50 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_numeral_le_neg_numeral
% 0.22/0.50 thf(fact_5518_not__numeral__le__neg__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] :
% 0.22/0.50 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_numeral_le_neg_numeral
% 0.22/0.50 thf(fact_5519_not__numeral__le__neg__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] :
% 0.22/0.50 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_numeral_le_neg_numeral
% 0.22/0.50 thf(fact_5520_zero__neq__neg__numeral,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ( zero_zero_real
% 0.22/0.50 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % zero_neq_neg_numeral
% 0.22/0.50 thf(fact_5521_zero__neq__neg__numeral,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ( zero_zero_int
% 0.22/0.50 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % zero_neq_neg_numeral
% 0.22/0.50 thf(fact_5522_zero__neq__neg__numeral,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ( zero_zero_complex
% 0.22/0.50 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % zero_neq_neg_numeral
% 0.22/0.50 thf(fact_5523_zero__neq__neg__numeral,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ( zero_z3403309356797280102nteger
% 0.22/0.50 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % zero_neq_neg_numeral
% 0.22/0.50 thf(fact_5524_zero__neq__neg__numeral,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ( zero_zero_rat
% 0.22/0.50 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % zero_neq_neg_numeral
% 0.22/0.50 thf(fact_5525_not__numeral__less__neg__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] :
% 0.22/0.50 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_numeral_less_neg_numeral
% 0.22/0.50 thf(fact_5526_not__numeral__less__neg__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] :
% 0.22/0.50 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_numeral_less_neg_numeral
% 0.22/0.50 thf(fact_5527_not__numeral__less__neg__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] :
% 0.22/0.50 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_numeral_less_neg_numeral
% 0.22/0.50 thf(fact_5528_not__numeral__less__neg__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] :
% 0.22/0.50 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_numeral_less_neg_numeral
% 0.22/0.50 thf(fact_5529_neg__numeral__less__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_less_numeral
% 0.22/0.50 thf(fact_5530_neg__numeral__less__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_less_numeral
% 0.22/0.50 thf(fact_5531_neg__numeral__less__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_less_numeral
% 0.22/0.50 thf(fact_5532_neg__numeral__less__numeral,axiom,
% 0.22/0.50 ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_less_numeral
% 0.22/0.50 thf(fact_5533_le__minus__one__simps_I2_J,axiom,
% 0.22/0.50 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.22/0.50
% 0.22/0.50 % le_minus_one_simps(2)
% 0.22/0.50 thf(fact_5534_le__minus__one__simps_I2_J,axiom,
% 0.22/0.50 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.22/0.50
% 0.22/0.50 % le_minus_one_simps(2)
% 0.22/0.50 thf(fact_5535_le__minus__one__simps_I2_J,axiom,
% 0.22/0.50 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.22/0.50
% 0.22/0.50 % le_minus_one_simps(2)
% 0.22/0.50 thf(fact_5536_le__minus__one__simps_I2_J,axiom,
% 0.22/0.50 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.22/0.50
% 0.22/0.50 % le_minus_one_simps(2)
% 0.22/0.50 thf(fact_5537_le__minus__one__simps_I4_J,axiom,
% 0.22/0.50 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.50
% 0.22/0.50 % le_minus_one_simps(4)
% 0.22/0.50 thf(fact_5538_le__minus__one__simps_I4_J,axiom,
% 0.22/0.50 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % le_minus_one_simps(4)
% 0.22/0.50 thf(fact_5539_le__minus__one__simps_I4_J,axiom,
% 0.22/0.50 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % le_minus_one_simps(4)
% 0.22/0.50 thf(fact_5540_le__minus__one__simps_I4_J,axiom,
% 0.22/0.50 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % le_minus_one_simps(4)
% 0.22/0.50 thf(fact_5541_neg__eq__iff__add__eq__0,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( ( uminus_uminus_real @ A )
% 0.22/0.50 = B )
% 0.22/0.50 = ( ( plus_plus_real @ A @ B )
% 0.22/0.50 = zero_zero_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_eq_iff_add_eq_0
% 0.22/0.50 thf(fact_5542_neg__eq__iff__add__eq__0,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( ( uminus_uminus_int @ A )
% 0.22/0.50 = B )
% 0.22/0.50 = ( ( plus_plus_int @ A @ B )
% 0.22/0.50 = zero_zero_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_eq_iff_add_eq_0
% 0.22/0.50 thf(fact_5543_neg__eq__iff__add__eq__0,axiom,
% 0.22/0.50 ! [A: complex,B: complex] :
% 0.22/0.50 ( ( ( uminus1482373934393186551omplex @ A )
% 0.22/0.50 = B )
% 0.22/0.50 = ( ( plus_plus_complex @ A @ B )
% 0.22/0.50 = zero_zero_complex ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_eq_iff_add_eq_0
% 0.22/0.50 thf(fact_5544_neg__eq__iff__add__eq__0,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( ( uminus1351360451143612070nteger @ A )
% 0.22/0.50 = B )
% 0.22/0.50 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.22/0.50 = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_eq_iff_add_eq_0
% 0.22/0.50 thf(fact_5545_neg__eq__iff__add__eq__0,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( ( uminus_uminus_rat @ A )
% 0.22/0.50 = B )
% 0.22/0.50 = ( ( plus_plus_rat @ A @ B )
% 0.22/0.50 = zero_zero_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_eq_iff_add_eq_0
% 0.22/0.50 thf(fact_5546_eq__neg__iff__add__eq__0,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( A
% 0.22/0.50 = ( uminus_uminus_real @ B ) )
% 0.22/0.50 = ( ( plus_plus_real @ A @ B )
% 0.22/0.50 = zero_zero_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % eq_neg_iff_add_eq_0
% 0.22/0.50 thf(fact_5547_eq__neg__iff__add__eq__0,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( A
% 0.22/0.50 = ( uminus_uminus_int @ B ) )
% 0.22/0.50 = ( ( plus_plus_int @ A @ B )
% 0.22/0.50 = zero_zero_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % eq_neg_iff_add_eq_0
% 0.22/0.50 thf(fact_5548_eq__neg__iff__add__eq__0,axiom,
% 0.22/0.50 ! [A: complex,B: complex] :
% 0.22/0.50 ( ( A
% 0.22/0.50 = ( uminus1482373934393186551omplex @ B ) )
% 0.22/0.50 = ( ( plus_plus_complex @ A @ B )
% 0.22/0.50 = zero_zero_complex ) ) ).
% 0.22/0.50
% 0.22/0.50 % eq_neg_iff_add_eq_0
% 0.22/0.50 thf(fact_5549_eq__neg__iff__add__eq__0,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( A
% 0.22/0.50 = ( uminus1351360451143612070nteger @ B ) )
% 0.22/0.50 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.22/0.50 = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.50
% 0.22/0.50 % eq_neg_iff_add_eq_0
% 0.22/0.50 thf(fact_5550_eq__neg__iff__add__eq__0,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( A
% 0.22/0.50 = ( uminus_uminus_rat @ B ) )
% 0.22/0.50 = ( ( plus_plus_rat @ A @ B )
% 0.22/0.50 = zero_zero_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % eq_neg_iff_add_eq_0
% 0.22/0.50 thf(fact_5551_add_Oinverse__unique,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( ( plus_plus_real @ A @ B )
% 0.22/0.50 = zero_zero_real )
% 0.22/0.50 => ( ( uminus_uminus_real @ A )
% 0.22/0.50 = B ) ) ).
% 0.22/0.50
% 0.22/0.50 % add.inverse_unique
% 0.22/0.50 thf(fact_5552_add_Oinverse__unique,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( ( plus_plus_int @ A @ B )
% 0.22/0.50 = zero_zero_int )
% 0.22/0.50 => ( ( uminus_uminus_int @ A )
% 0.22/0.50 = B ) ) ).
% 0.22/0.50
% 0.22/0.50 % add.inverse_unique
% 0.22/0.50 thf(fact_5553_add_Oinverse__unique,axiom,
% 0.22/0.50 ! [A: complex,B: complex] :
% 0.22/0.50 ( ( ( plus_plus_complex @ A @ B )
% 0.22/0.50 = zero_zero_complex )
% 0.22/0.50 => ( ( uminus1482373934393186551omplex @ A )
% 0.22/0.50 = B ) ) ).
% 0.22/0.50
% 0.22/0.50 % add.inverse_unique
% 0.22/0.50 thf(fact_5554_add_Oinverse__unique,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.22/0.50 = zero_z3403309356797280102nteger )
% 0.22/0.50 => ( ( uminus1351360451143612070nteger @ A )
% 0.22/0.50 = B ) ) ).
% 0.22/0.50
% 0.22/0.50 % add.inverse_unique
% 0.22/0.50 thf(fact_5555_add_Oinverse__unique,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( ( plus_plus_rat @ A @ B )
% 0.22/0.50 = zero_zero_rat )
% 0.22/0.50 => ( ( uminus_uminus_rat @ A )
% 0.22/0.50 = B ) ) ).
% 0.22/0.50
% 0.22/0.50 % add.inverse_unique
% 0.22/0.50 thf(fact_5556_ab__group__add__class_Oab__left__minus,axiom,
% 0.22/0.50 ! [A: real] :
% 0.22/0.50 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.22/0.50 = zero_zero_real ) ).
% 0.22/0.50
% 0.22/0.50 % ab_group_add_class.ab_left_minus
% 0.22/0.50 thf(fact_5557_ab__group__add__class_Oab__left__minus,axiom,
% 0.22/0.50 ! [A: int] :
% 0.22/0.50 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.22/0.50 = zero_zero_int ) ).
% 0.22/0.50
% 0.22/0.50 % ab_group_add_class.ab_left_minus
% 0.22/0.50 thf(fact_5558_ab__group__add__class_Oab__left__minus,axiom,
% 0.22/0.50 ! [A: complex] :
% 0.22/0.50 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.22/0.50 = zero_zero_complex ) ).
% 0.22/0.50
% 0.22/0.50 % ab_group_add_class.ab_left_minus
% 0.22/0.50 thf(fact_5559_ab__group__add__class_Oab__left__minus,axiom,
% 0.22/0.50 ! [A: code_integer] :
% 0.22/0.50 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.22/0.50 = zero_z3403309356797280102nteger ) ).
% 0.22/0.50
% 0.22/0.50 % ab_group_add_class.ab_left_minus
% 0.22/0.50 thf(fact_5560_ab__group__add__class_Oab__left__minus,axiom,
% 0.22/0.50 ! [A: rat] :
% 0.22/0.50 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.22/0.50 = zero_zero_rat ) ).
% 0.22/0.50
% 0.22/0.50 % ab_group_add_class.ab_left_minus
% 0.22/0.50 thf(fact_5561_add__eq__0__iff,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( ( plus_plus_real @ A @ B )
% 0.22/0.50 = zero_zero_real )
% 0.22/0.50 = ( B
% 0.22/0.50 = ( uminus_uminus_real @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % add_eq_0_iff
% 0.22/0.50 thf(fact_5562_add__eq__0__iff,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( ( plus_plus_int @ A @ B )
% 0.22/0.50 = zero_zero_int )
% 0.22/0.50 = ( B
% 0.22/0.50 = ( uminus_uminus_int @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % add_eq_0_iff
% 0.22/0.50 thf(fact_5563_add__eq__0__iff,axiom,
% 0.22/0.50 ! [A: complex,B: complex] :
% 0.22/0.50 ( ( ( plus_plus_complex @ A @ B )
% 0.22/0.50 = zero_zero_complex )
% 0.22/0.50 = ( B
% 0.22/0.50 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % add_eq_0_iff
% 0.22/0.50 thf(fact_5564_add__eq__0__iff,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.22/0.50 = zero_z3403309356797280102nteger )
% 0.22/0.50 = ( B
% 0.22/0.50 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % add_eq_0_iff
% 0.22/0.50 thf(fact_5565_add__eq__0__iff,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( ( plus_plus_rat @ A @ B )
% 0.22/0.50 = zero_zero_rat )
% 0.22/0.50 = ( B
% 0.22/0.50 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % add_eq_0_iff
% 0.22/0.50 thf(fact_5566_zero__neq__neg__one,axiom,
% 0.22/0.50 ( zero_zero_real
% 0.22/0.50 != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % zero_neq_neg_one
% 0.22/0.50 thf(fact_5567_zero__neq__neg__one,axiom,
% 0.22/0.50 ( zero_zero_int
% 0.22/0.50 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % zero_neq_neg_one
% 0.22/0.50 thf(fact_5568_zero__neq__neg__one,axiom,
% 0.22/0.50 ( zero_zero_complex
% 0.22/0.50 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.22/0.50
% 0.22/0.50 % zero_neq_neg_one
% 0.22/0.50 thf(fact_5569_zero__neq__neg__one,axiom,
% 0.22/0.50 ( zero_z3403309356797280102nteger
% 0.22/0.50 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.50
% 0.22/0.50 % zero_neq_neg_one
% 0.22/0.50 thf(fact_5570_zero__neq__neg__one,axiom,
% 0.22/0.50 ( zero_zero_rat
% 0.22/0.50 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % zero_neq_neg_one
% 0.22/0.50 thf(fact_5571_less__minus__one__simps_I4_J,axiom,
% 0.22/0.50 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % less_minus_one_simps(4)
% 0.22/0.50 thf(fact_5572_less__minus__one__simps_I4_J,axiom,
% 0.22/0.50 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % less_minus_one_simps(4)
% 0.22/0.50 thf(fact_5573_less__minus__one__simps_I4_J,axiom,
% 0.22/0.50 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.50
% 0.22/0.50 % less_minus_one_simps(4)
% 0.22/0.50 thf(fact_5574_less__minus__one__simps_I4_J,axiom,
% 0.22/0.50 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % less_minus_one_simps(4)
% 0.22/0.50 thf(fact_5575_less__minus__one__simps_I2_J,axiom,
% 0.22/0.50 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.22/0.50
% 0.22/0.50 % less_minus_one_simps(2)
% 0.22/0.50 thf(fact_5576_less__minus__one__simps_I2_J,axiom,
% 0.22/0.50 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.22/0.50
% 0.22/0.50 % less_minus_one_simps(2)
% 0.22/0.50 thf(fact_5577_less__minus__one__simps_I2_J,axiom,
% 0.22/0.50 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.22/0.50
% 0.22/0.50 % less_minus_one_simps(2)
% 0.22/0.50 thf(fact_5578_less__minus__one__simps_I2_J,axiom,
% 0.22/0.50 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.22/0.50
% 0.22/0.50 % less_minus_one_simps(2)
% 0.22/0.50 thf(fact_5579_numeral__times__minus__swap,axiom,
% 0.22/0.50 ! [W: num,X: real] :
% 0.22/0.50 ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X ) )
% 0.22/0.50 = ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % numeral_times_minus_swap
% 0.22/0.50 thf(fact_5580_numeral__times__minus__swap,axiom,
% 0.22/0.50 ! [W: num,X: int] :
% 0.22/0.50 ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
% 0.22/0.50 = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % numeral_times_minus_swap
% 0.22/0.50 thf(fact_5581_numeral__times__minus__swap,axiom,
% 0.22/0.50 ! [W: num,X: complex] :
% 0.22/0.50 ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X ) )
% 0.22/0.50 = ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % numeral_times_minus_swap
% 0.22/0.50 thf(fact_5582_numeral__times__minus__swap,axiom,
% 0.22/0.50 ! [W: num,X: code_integer] :
% 0.22/0.50 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X ) )
% 0.22/0.50 = ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % numeral_times_minus_swap
% 0.22/0.50 thf(fact_5583_numeral__times__minus__swap,axiom,
% 0.22/0.50 ! [W: num,X: rat] :
% 0.22/0.50 ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
% 0.22/0.50 = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % numeral_times_minus_swap
% 0.22/0.50 thf(fact_5584_nonzero__minus__divide__right,axiom,
% 0.22/0.50 ! [B: real,A: real] :
% 0.22/0.50 ( ( B != zero_zero_real )
% 0.22/0.50 => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.22/0.50 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % nonzero_minus_divide_right
% 0.22/0.50 thf(fact_5585_nonzero__minus__divide__right,axiom,
% 0.22/0.50 ! [B: complex,A: complex] :
% 0.22/0.50 ( ( B != zero_zero_complex )
% 0.22/0.50 => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.22/0.50 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % nonzero_minus_divide_right
% 0.22/0.50 thf(fact_5586_nonzero__minus__divide__right,axiom,
% 0.22/0.50 ! [B: rat,A: rat] :
% 0.22/0.50 ( ( B != zero_zero_rat )
% 0.22/0.50 => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.22/0.50 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % nonzero_minus_divide_right
% 0.22/0.50 thf(fact_5587_nonzero__minus__divide__divide,axiom,
% 0.22/0.50 ! [B: real,A: real] :
% 0.22/0.50 ( ( B != zero_zero_real )
% 0.22/0.50 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.22/0.50 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % nonzero_minus_divide_divide
% 0.22/0.50 thf(fact_5588_nonzero__minus__divide__divide,axiom,
% 0.22/0.50 ! [B: complex,A: complex] :
% 0.22/0.50 ( ( B != zero_zero_complex )
% 0.22/0.50 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.22/0.50 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % nonzero_minus_divide_divide
% 0.22/0.50 thf(fact_5589_nonzero__minus__divide__divide,axiom,
% 0.22/0.50 ! [B: rat,A: rat] :
% 0.22/0.50 ( ( B != zero_zero_rat )
% 0.22/0.50 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.22/0.50 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % nonzero_minus_divide_divide
% 0.22/0.50 thf(fact_5590_one__neq__neg__numeral,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ( one_one_real
% 0.22/0.50 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % one_neq_neg_numeral
% 0.22/0.50 thf(fact_5591_one__neq__neg__numeral,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ( one_one_int
% 0.22/0.50 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % one_neq_neg_numeral
% 0.22/0.50 thf(fact_5592_one__neq__neg__numeral,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ( one_one_complex
% 0.22/0.50 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % one_neq_neg_numeral
% 0.22/0.50 thf(fact_5593_one__neq__neg__numeral,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ( one_one_Code_integer
% 0.22/0.50 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % one_neq_neg_numeral
% 0.22/0.50 thf(fact_5594_one__neq__neg__numeral,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ( one_one_rat
% 0.22/0.50 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % one_neq_neg_numeral
% 0.22/0.50 thf(fact_5595_numeral__neq__neg__one,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ( ( numeral_numeral_real @ N )
% 0.22/0.50 != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % numeral_neq_neg_one
% 0.22/0.50 thf(fact_5596_numeral__neq__neg__one,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ( ( numeral_numeral_int @ N )
% 0.22/0.50 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % numeral_neq_neg_one
% 0.22/0.50 thf(fact_5597_numeral__neq__neg__one,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ( ( numera6690914467698888265omplex @ N )
% 0.22/0.50 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.22/0.50
% 0.22/0.50 % numeral_neq_neg_one
% 0.22/0.50 thf(fact_5598_numeral__neq__neg__one,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ( ( numera6620942414471956472nteger @ N )
% 0.22/0.50 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.50
% 0.22/0.50 % numeral_neq_neg_one
% 0.22/0.50 thf(fact_5599_numeral__neq__neg__one,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ( ( numeral_numeral_rat @ N )
% 0.22/0.50 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % numeral_neq_neg_one
% 0.22/0.50 thf(fact_5600_square__eq__1__iff,axiom,
% 0.22/0.50 ! [X: real] :
% 0.22/0.50 ( ( ( times_times_real @ X @ X )
% 0.22/0.50 = one_one_real )
% 0.22/0.50 = ( ( X = one_one_real )
% 0.22/0.50 | ( X
% 0.22/0.50 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % square_eq_1_iff
% 0.22/0.50 thf(fact_5601_square__eq__1__iff,axiom,
% 0.22/0.50 ! [X: int] :
% 0.22/0.50 ( ( ( times_times_int @ X @ X )
% 0.22/0.50 = one_one_int )
% 0.22/0.50 = ( ( X = one_one_int )
% 0.22/0.50 | ( X
% 0.22/0.50 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % square_eq_1_iff
% 0.22/0.50 thf(fact_5602_square__eq__1__iff,axiom,
% 0.22/0.50 ! [X: complex] :
% 0.22/0.50 ( ( ( times_times_complex @ X @ X )
% 0.22/0.50 = one_one_complex )
% 0.22/0.50 = ( ( X = one_one_complex )
% 0.22/0.50 | ( X
% 0.22/0.50 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % square_eq_1_iff
% 0.22/0.50 thf(fact_5603_square__eq__1__iff,axiom,
% 0.22/0.50 ! [X: code_integer] :
% 0.22/0.50 ( ( ( times_3573771949741848930nteger @ X @ X )
% 0.22/0.50 = one_one_Code_integer )
% 0.22/0.50 = ( ( X = one_one_Code_integer )
% 0.22/0.50 | ( X
% 0.22/0.50 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % square_eq_1_iff
% 0.22/0.50 thf(fact_5604_square__eq__1__iff,axiom,
% 0.22/0.50 ! [X: rat] :
% 0.22/0.50 ( ( ( times_times_rat @ X @ X )
% 0.22/0.50 = one_one_rat )
% 0.22/0.50 = ( ( X = one_one_rat )
% 0.22/0.50 | ( X
% 0.22/0.50 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % square_eq_1_iff
% 0.22/0.50 thf(fact_5605_group__cancel_Osub2,axiom,
% 0.22/0.50 ! [B4: real,K: real,B: real,A: real] :
% 0.22/0.50 ( ( B4
% 0.22/0.50 = ( plus_plus_real @ K @ B ) )
% 0.22/0.50 => ( ( minus_minus_real @ A @ B4 )
% 0.22/0.50 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % group_cancel.sub2
% 0.22/0.50 thf(fact_5606_group__cancel_Osub2,axiom,
% 0.22/0.50 ! [B4: int,K: int,B: int,A: int] :
% 0.22/0.50 ( ( B4
% 0.22/0.50 = ( plus_plus_int @ K @ B ) )
% 0.22/0.50 => ( ( minus_minus_int @ A @ B4 )
% 0.22/0.50 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % group_cancel.sub2
% 0.22/0.50 thf(fact_5607_group__cancel_Osub2,axiom,
% 0.22/0.50 ! [B4: complex,K: complex,B: complex,A: complex] :
% 0.22/0.50 ( ( B4
% 0.22/0.50 = ( plus_plus_complex @ K @ B ) )
% 0.22/0.50 => ( ( minus_minus_complex @ A @ B4 )
% 0.22/0.50 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % group_cancel.sub2
% 0.22/0.50 thf(fact_5608_group__cancel_Osub2,axiom,
% 0.22/0.50 ! [B4: code_integer,K: code_integer,B: code_integer,A: code_integer] :
% 0.22/0.50 ( ( B4
% 0.22/0.50 = ( plus_p5714425477246183910nteger @ K @ B ) )
% 0.22/0.50 => ( ( minus_8373710615458151222nteger @ A @ B4 )
% 0.22/0.50 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % group_cancel.sub2
% 0.22/0.50 thf(fact_5609_group__cancel_Osub2,axiom,
% 0.22/0.50 ! [B4: rat,K: rat,B: rat,A: rat] :
% 0.22/0.50 ( ( B4
% 0.22/0.50 = ( plus_plus_rat @ K @ B ) )
% 0.22/0.50 => ( ( minus_minus_rat @ A @ B4 )
% 0.22/0.50 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % group_cancel.sub2
% 0.22/0.50 thf(fact_5610_diff__conv__add__uminus,axiom,
% 0.22/0.50 ( minus_minus_real
% 0.22/0.50 = ( ^ [A3: real,B2: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % diff_conv_add_uminus
% 0.22/0.50 thf(fact_5611_diff__conv__add__uminus,axiom,
% 0.22/0.50 ( minus_minus_int
% 0.22/0.50 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % diff_conv_add_uminus
% 0.22/0.50 thf(fact_5612_diff__conv__add__uminus,axiom,
% 0.22/0.50 ( minus_minus_complex
% 0.22/0.50 = ( ^ [A3: complex,B2: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % diff_conv_add_uminus
% 0.22/0.50 thf(fact_5613_diff__conv__add__uminus,axiom,
% 0.22/0.50 ( minus_8373710615458151222nteger
% 0.22/0.50 = ( ^ [A3: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % diff_conv_add_uminus
% 0.22/0.50 thf(fact_5614_diff__conv__add__uminus,axiom,
% 0.22/0.50 ( minus_minus_rat
% 0.22/0.50 = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % diff_conv_add_uminus
% 0.22/0.50 thf(fact_5615_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.22/0.50 ( minus_minus_real
% 0.22/0.50 = ( ^ [A3: real,B2: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.22/0.50 thf(fact_5616_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.22/0.50 ( minus_minus_int
% 0.22/0.50 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.22/0.50 thf(fact_5617_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.22/0.50 ( minus_minus_complex
% 0.22/0.50 = ( ^ [A3: complex,B2: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.22/0.50 thf(fact_5618_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.22/0.50 ( minus_8373710615458151222nteger
% 0.22/0.50 = ( ^ [A3: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.22/0.50 thf(fact_5619_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.22/0.50 ( minus_minus_rat
% 0.22/0.50 = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.22/0.50 thf(fact_5620_dvd__div__neg,axiom,
% 0.22/0.50 ! [B: real,A: real] :
% 0.22/0.50 ( ( dvd_dvd_real @ B @ A )
% 0.22/0.50 => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
% 0.22/0.50 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % dvd_div_neg
% 0.22/0.50 thf(fact_5621_dvd__div__neg,axiom,
% 0.22/0.50 ! [B: int,A: int] :
% 0.22/0.50 ( ( dvd_dvd_int @ B @ A )
% 0.22/0.50 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.22/0.50 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % dvd_div_neg
% 0.22/0.50 thf(fact_5622_dvd__div__neg,axiom,
% 0.22/0.50 ! [B: complex,A: complex] :
% 0.22/0.50 ( ( dvd_dvd_complex @ B @ A )
% 0.22/0.50 => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 0.22/0.50 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % dvd_div_neg
% 0.22/0.50 thf(fact_5623_dvd__div__neg,axiom,
% 0.22/0.50 ! [B: code_integer,A: code_integer] :
% 0.22/0.50 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.22/0.50 => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.22/0.50 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % dvd_div_neg
% 0.22/0.50 thf(fact_5624_dvd__div__neg,axiom,
% 0.22/0.50 ! [B: rat,A: rat] :
% 0.22/0.50 ( ( dvd_dvd_rat @ B @ A )
% 0.22/0.50 => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.22/0.50 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % dvd_div_neg
% 0.22/0.50 thf(fact_5625_dvd__neg__div,axiom,
% 0.22/0.50 ! [B: real,A: real] :
% 0.22/0.50 ( ( dvd_dvd_real @ B @ A )
% 0.22/0.50 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
% 0.22/0.50 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % dvd_neg_div
% 0.22/0.50 thf(fact_5626_dvd__neg__div,axiom,
% 0.22/0.50 ! [B: int,A: int] :
% 0.22/0.50 ( ( dvd_dvd_int @ B @ A )
% 0.22/0.50 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 0.22/0.50 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % dvd_neg_div
% 0.22/0.50 thf(fact_5627_dvd__neg__div,axiom,
% 0.22/0.50 ! [B: complex,A: complex] :
% 0.22/0.50 ( ( dvd_dvd_complex @ B @ A )
% 0.22/0.50 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.22/0.50 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % dvd_neg_div
% 0.22/0.50 thf(fact_5628_dvd__neg__div,axiom,
% 0.22/0.50 ! [B: code_integer,A: code_integer] :
% 0.22/0.50 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.22/0.50 => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.22/0.50 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % dvd_neg_div
% 0.22/0.50 thf(fact_5629_dvd__neg__div,axiom,
% 0.22/0.50 ! [B: rat,A: rat] :
% 0.22/0.50 ( ( dvd_dvd_rat @ B @ A )
% 0.22/0.50 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.22/0.50 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % dvd_neg_div
% 0.22/0.50 thf(fact_5630_real__minus__mult__self__le,axiom,
% 0.22/0.50 ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% 0.22/0.50
% 0.22/0.50 % real_minus_mult_self_le
% 0.22/0.50 thf(fact_5631_int__of__nat__induct,axiom,
% 0.22/0.50 ! [P: int > $o,Z: int] :
% 0.22/0.50 ( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.22/0.50 => ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
% 0.22/0.50 => ( P @ Z ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % int_of_nat_induct
% 0.22/0.50 thf(fact_5632_int__cases,axiom,
% 0.22/0.50 ! [Z: int] :
% 0.22/0.50 ( ! [N2: nat] :
% 0.22/0.50 ( Z
% 0.22/0.50 != ( semiri1314217659103216013at_int @ N2 ) )
% 0.22/0.50 => ~ ! [N2: nat] :
% 0.22/0.50 ( Z
% 0.22/0.50 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % int_cases
% 0.22/0.50 thf(fact_5633_zmult__eq__1__iff,axiom,
% 0.22/0.50 ! [M: int,N: int] :
% 0.22/0.50 ( ( ( times_times_int @ M @ N )
% 0.22/0.50 = one_one_int )
% 0.22/0.50 = ( ( ( M = one_one_int )
% 0.22/0.50 & ( N = one_one_int ) )
% 0.22/0.50 | ( ( M
% 0.22/0.50 = ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.50 & ( N
% 0.22/0.50 = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % zmult_eq_1_iff
% 0.22/0.50 thf(fact_5634_pos__zmult__eq__1__iff__lemma,axiom,
% 0.22/0.50 ! [M: int,N: int] :
% 0.22/0.50 ( ( ( times_times_int @ M @ N )
% 0.22/0.50 = one_one_int )
% 0.22/0.50 => ( ( M = one_one_int )
% 0.22/0.50 | ( M
% 0.22/0.50 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % pos_zmult_eq_1_iff_lemma
% 0.22/0.50 thf(fact_5635_minus__int__code_I2_J,axiom,
% 0.22/0.50 ! [L: int] :
% 0.22/0.50 ( ( minus_minus_int @ zero_zero_int @ L )
% 0.22/0.50 = ( uminus_uminus_int @ L ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_int_code(2)
% 0.22/0.50 thf(fact_5636_not__int__zless__negative,axiom,
% 0.22/0.50 ! [N: nat,M: nat] :
% 0.22/0.50 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_int_zless_negative
% 0.22/0.50 thf(fact_5637_zmod__zminus1__not__zero,axiom,
% 0.22/0.50 ! [K: int,L: int] :
% 0.22/0.50 ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
% 0.22/0.50 != zero_zero_int )
% 0.22/0.50 => ( ( modulo_modulo_int @ K @ L )
% 0.22/0.50 != zero_zero_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % zmod_zminus1_not_zero
% 0.22/0.50 thf(fact_5638_zmod__zminus2__not__zero,axiom,
% 0.22/0.50 ! [K: int,L: int] :
% 0.22/0.50 ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L ) )
% 0.22/0.50 != zero_zero_int )
% 0.22/0.50 => ( ( modulo_modulo_int @ K @ L )
% 0.22/0.50 != zero_zero_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % zmod_zminus2_not_zero
% 0.22/0.50 thf(fact_5639_minus__real__def,axiom,
% 0.22/0.50 ( minus_minus_real
% 0.22/0.50 = ( ^ [X3: real,Y6: real] : ( plus_plus_real @ X3 @ ( uminus_uminus_real @ Y6 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_real_def
% 0.22/0.50 thf(fact_5640_dense__eq0__I,axiom,
% 0.22/0.50 ! [X: rat] :
% 0.22/0.50 ( ! [E: rat] :
% 0.22/0.50 ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.22/0.50 => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E ) )
% 0.22/0.50 => ( X = zero_zero_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % dense_eq0_I
% 0.22/0.50 thf(fact_5641_dense__eq0__I,axiom,
% 0.22/0.50 ! [X: real] :
% 0.22/0.50 ( ! [E: real] :
% 0.22/0.50 ( ( ord_less_real @ zero_zero_real @ E )
% 0.22/0.50 => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E ) )
% 0.22/0.50 => ( X = zero_zero_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % dense_eq0_I
% 0.22/0.50 thf(fact_5642_abs__mult__pos,axiom,
% 0.22/0.50 ! [X: code_integer,Y4: code_integer] :
% 0.22/0.50 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 0.22/0.50 => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y4 ) @ X )
% 0.22/0.50 = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y4 @ X ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_mult_pos
% 0.22/0.50 thf(fact_5643_abs__mult__pos,axiom,
% 0.22/0.50 ! [X: rat,Y4: rat] :
% 0.22/0.50 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.22/0.50 => ( ( times_times_rat @ ( abs_abs_rat @ Y4 ) @ X )
% 0.22/0.50 = ( abs_abs_rat @ ( times_times_rat @ Y4 @ X ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_mult_pos
% 0.22/0.50 thf(fact_5644_abs__mult__pos,axiom,
% 0.22/0.50 ! [X: int,Y4: int] :
% 0.22/0.50 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.22/0.50 => ( ( times_times_int @ ( abs_abs_int @ Y4 ) @ X )
% 0.22/0.50 = ( abs_abs_int @ ( times_times_int @ Y4 @ X ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_mult_pos
% 0.22/0.50 thf(fact_5645_abs__mult__pos,axiom,
% 0.22/0.50 ! [X: real,Y4: real] :
% 0.22/0.50 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.50 => ( ( times_times_real @ ( abs_abs_real @ Y4 ) @ X )
% 0.22/0.50 = ( abs_abs_real @ ( times_times_real @ Y4 @ X ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_mult_pos
% 0.22/0.50 thf(fact_5646_abs__eq__mult,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.22/0.50 | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 0.22/0.50 & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.22/0.50 | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
% 0.22/0.50 => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.22/0.50 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_eq_mult
% 0.22/0.50 thf(fact_5647_abs__eq__mult,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.22/0.50 | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.22/0.50 & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.22/0.50 | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 0.22/0.50 => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 0.22/0.50 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_eq_mult
% 0.22/0.50 thf(fact_5648_abs__eq__mult,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.22/0.50 | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.22/0.50 & ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.22/0.50 | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 0.22/0.50 => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 0.22/0.50 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_eq_mult
% 0.22/0.50 thf(fact_5649_abs__eq__mult,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.50 | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.22/0.50 & ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.22/0.50 | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.22/0.50 => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 0.22/0.50 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_eq_mult
% 0.22/0.50 thf(fact_5650_zero__le__power__abs,axiom,
% 0.22/0.50 ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 0.22/0.50
% 0.22/0.50 % zero_le_power_abs
% 0.22/0.50 thf(fact_5651_zero__le__power__abs,axiom,
% 0.22/0.50 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 0.22/0.50
% 0.22/0.50 % zero_le_power_abs
% 0.22/0.50 thf(fact_5652_zero__le__power__abs,axiom,
% 0.22/0.50 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 0.22/0.50
% 0.22/0.50 % zero_le_power_abs
% 0.22/0.50 thf(fact_5653_zero__le__power__abs,axiom,
% 0.22/0.50 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 0.22/0.50
% 0.22/0.50 % zero_le_power_abs
% 0.22/0.50 thf(fact_5654_abs__div__pos,axiom,
% 0.22/0.50 ! [Y4: real,X: real] :
% 0.22/0.50 ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.22/0.50 => ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y4 )
% 0.22/0.50 = ( abs_abs_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_div_pos
% 0.22/0.50 thf(fact_5655_abs__div__pos,axiom,
% 0.22/0.50 ! [Y4: rat,X: rat] :
% 0.22/0.50 ( ( ord_less_rat @ zero_zero_rat @ Y4 )
% 0.22/0.50 => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y4 )
% 0.22/0.50 = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_div_pos
% 0.22/0.50 thf(fact_5656_abs__diff__le__iff,axiom,
% 0.22/0.50 ! [X: code_integer,A: code_integer,R2: code_integer] :
% 0.22/0.50 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
% 0.22/0.50 = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
% 0.22/0.50 & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_diff_le_iff
% 0.22/0.50 thf(fact_5657_abs__diff__le__iff,axiom,
% 0.22/0.50 ! [X: rat,A: rat,R2: rat] :
% 0.22/0.50 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
% 0.22/0.50 = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
% 0.22/0.50 & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_diff_le_iff
% 0.22/0.50 thf(fact_5658_abs__diff__le__iff,axiom,
% 0.22/0.50 ! [X: int,A: int,R2: int] :
% 0.22/0.50 ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
% 0.22/0.50 = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X )
% 0.22/0.50 & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_diff_le_iff
% 0.22/0.50 thf(fact_5659_abs__diff__le__iff,axiom,
% 0.22/0.50 ! [X: real,A: real,R2: real] :
% 0.22/0.50 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
% 0.22/0.50 = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X )
% 0.22/0.50 & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_diff_le_iff
% 0.22/0.50 thf(fact_5660_abs__triangle__ineq4,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq4
% 0.22/0.50 thf(fact_5661_abs__triangle__ineq4,axiom,
% 0.22/0.50 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq4
% 0.22/0.50 thf(fact_5662_abs__triangle__ineq4,axiom,
% 0.22/0.50 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq4
% 0.22/0.50 thf(fact_5663_abs__triangle__ineq4,axiom,
% 0.22/0.50 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_triangle_ineq4
% 0.22/0.50 thf(fact_5664_abs__diff__triangle__ineq,axiom,
% 0.22/0.50 ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_diff_triangle_ineq
% 0.22/0.50 thf(fact_5665_abs__diff__triangle__ineq,axiom,
% 0.22/0.50 ! [A: rat,B: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_diff_triangle_ineq
% 0.22/0.50 thf(fact_5666_abs__diff__triangle__ineq,axiom,
% 0.22/0.50 ! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_diff_triangle_ineq
% 0.22/0.50 thf(fact_5667_abs__diff__triangle__ineq,axiom,
% 0.22/0.50 ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_diff_triangle_ineq
% 0.22/0.50 thf(fact_5668_abs__diff__less__iff,axiom,
% 0.22/0.50 ! [X: code_integer,A: code_integer,R2: code_integer] :
% 0.22/0.50 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
% 0.22/0.50 = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
% 0.22/0.50 & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_diff_less_iff
% 0.22/0.50 thf(fact_5669_abs__diff__less__iff,axiom,
% 0.22/0.50 ! [X: real,A: real,R2: real] :
% 0.22/0.50 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
% 0.22/0.50 = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X )
% 0.22/0.50 & ( ord_less_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_diff_less_iff
% 0.22/0.50 thf(fact_5670_abs__diff__less__iff,axiom,
% 0.22/0.50 ! [X: rat,A: rat,R2: rat] :
% 0.22/0.50 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
% 0.22/0.50 = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
% 0.22/0.50 & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_diff_less_iff
% 0.22/0.50 thf(fact_5671_abs__diff__less__iff,axiom,
% 0.22/0.50 ! [X: int,A: int,R2: int] :
% 0.22/0.50 ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
% 0.22/0.50 = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X )
% 0.22/0.50 & ( ord_less_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_diff_less_iff
% 0.22/0.50 thf(fact_5672_neg__numeral__le__zero,axiom,
% 0.22/0.50 ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_le_zero
% 0.22/0.50 thf(fact_5673_neg__numeral__le__zero,axiom,
% 0.22/0.50 ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_le_zero
% 0.22/0.50 thf(fact_5674_neg__numeral__le__zero,axiom,
% 0.22/0.50 ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_le_zero
% 0.22/0.50 thf(fact_5675_neg__numeral__le__zero,axiom,
% 0.22/0.50 ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_le_zero
% 0.22/0.50 thf(fact_5676_not__zero__le__neg__numeral,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_zero_le_neg_numeral
% 0.22/0.50 thf(fact_5677_not__zero__le__neg__numeral,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_zero_le_neg_numeral
% 0.22/0.50 thf(fact_5678_not__zero__le__neg__numeral,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_zero_le_neg_numeral
% 0.22/0.50 thf(fact_5679_not__zero__le__neg__numeral,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_zero_le_neg_numeral
% 0.22/0.50 thf(fact_5680_neg__numeral__less__zero,axiom,
% 0.22/0.50 ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_less_zero
% 0.22/0.50 thf(fact_5681_neg__numeral__less__zero,axiom,
% 0.22/0.50 ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_less_zero
% 0.22/0.50 thf(fact_5682_neg__numeral__less__zero,axiom,
% 0.22/0.50 ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_less_zero
% 0.22/0.50 thf(fact_5683_neg__numeral__less__zero,axiom,
% 0.22/0.50 ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_less_zero
% 0.22/0.50 thf(fact_5684_not__zero__less__neg__numeral,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_zero_less_neg_numeral
% 0.22/0.50 thf(fact_5685_not__zero__less__neg__numeral,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_zero_less_neg_numeral
% 0.22/0.50 thf(fact_5686_not__zero__less__neg__numeral,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_zero_less_neg_numeral
% 0.22/0.50 thf(fact_5687_not__zero__less__neg__numeral,axiom,
% 0.22/0.50 ! [N: num] :
% 0.22/0.50 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_zero_less_neg_numeral
% 0.22/0.50 thf(fact_5688_le__minus__one__simps_I1_J,axiom,
% 0.22/0.50 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.22/0.50
% 0.22/0.50 % le_minus_one_simps(1)
% 0.22/0.50 thf(fact_5689_le__minus__one__simps_I1_J,axiom,
% 0.22/0.50 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.22/0.50
% 0.22/0.50 % le_minus_one_simps(1)
% 0.22/0.50 thf(fact_5690_le__minus__one__simps_I1_J,axiom,
% 0.22/0.50 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.22/0.50
% 0.22/0.50 % le_minus_one_simps(1)
% 0.22/0.50 thf(fact_5691_le__minus__one__simps_I1_J,axiom,
% 0.22/0.50 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.22/0.50
% 0.22/0.50 % le_minus_one_simps(1)
% 0.22/0.50 thf(fact_5692_le__minus__one__simps_I3_J,axiom,
% 0.22/0.50 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.50
% 0.22/0.50 % le_minus_one_simps(3)
% 0.22/0.50 thf(fact_5693_le__minus__one__simps_I3_J,axiom,
% 0.22/0.50 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % le_minus_one_simps(3)
% 0.22/0.50 thf(fact_5694_le__minus__one__simps_I3_J,axiom,
% 0.22/0.50 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % le_minus_one_simps(3)
% 0.22/0.50 thf(fact_5695_le__minus__one__simps_I3_J,axiom,
% 0.22/0.50 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % le_minus_one_simps(3)
% 0.22/0.50 thf(fact_5696_less__minus__one__simps_I3_J,axiom,
% 0.22/0.50 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % less_minus_one_simps(3)
% 0.22/0.50 thf(fact_5697_less__minus__one__simps_I3_J,axiom,
% 0.22/0.50 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % less_minus_one_simps(3)
% 0.22/0.50 thf(fact_5698_less__minus__one__simps_I3_J,axiom,
% 0.22/0.50 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.50
% 0.22/0.50 % less_minus_one_simps(3)
% 0.22/0.50 thf(fact_5699_less__minus__one__simps_I3_J,axiom,
% 0.22/0.50 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % less_minus_one_simps(3)
% 0.22/0.50 thf(fact_5700_less__minus__one__simps_I1_J,axiom,
% 0.22/0.50 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.22/0.50
% 0.22/0.50 % less_minus_one_simps(1)
% 0.22/0.50 thf(fact_5701_less__minus__one__simps_I1_J,axiom,
% 0.22/0.50 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.22/0.50
% 0.22/0.50 % less_minus_one_simps(1)
% 0.22/0.50 thf(fact_5702_less__minus__one__simps_I1_J,axiom,
% 0.22/0.50 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.22/0.50
% 0.22/0.50 % less_minus_one_simps(1)
% 0.22/0.50 thf(fact_5703_less__minus__one__simps_I1_J,axiom,
% 0.22/0.50 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.22/0.50
% 0.22/0.50 % less_minus_one_simps(1)
% 0.22/0.50 thf(fact_5704_neg__numeral__le__one,axiom,
% 0.22/0.50 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_le_one
% 0.22/0.50 thf(fact_5705_neg__numeral__le__one,axiom,
% 0.22/0.50 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_le_one
% 0.22/0.50 thf(fact_5706_neg__numeral__le__one,axiom,
% 0.22/0.50 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_le_one
% 0.22/0.50 thf(fact_5707_neg__numeral__le__one,axiom,
% 0.22/0.50 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_le_one
% 0.22/0.50 thf(fact_5708_neg__one__le__numeral,axiom,
% 0.22/0.50 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_one_le_numeral
% 0.22/0.50 thf(fact_5709_neg__one__le__numeral,axiom,
% 0.22/0.50 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_one_le_numeral
% 0.22/0.50 thf(fact_5710_neg__one__le__numeral,axiom,
% 0.22/0.50 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_one_le_numeral
% 0.22/0.50 thf(fact_5711_neg__one__le__numeral,axiom,
% 0.22/0.50 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_one_le_numeral
% 0.22/0.50 thf(fact_5712_neg__numeral__le__neg__one,axiom,
% 0.22/0.50 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_le_neg_one
% 0.22/0.50 thf(fact_5713_neg__numeral__le__neg__one,axiom,
% 0.22/0.50 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_le_neg_one
% 0.22/0.50 thf(fact_5714_neg__numeral__le__neg__one,axiom,
% 0.22/0.50 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_le_neg_one
% 0.22/0.50 thf(fact_5715_neg__numeral__le__neg__one,axiom,
% 0.22/0.50 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_le_neg_one
% 0.22/0.50 thf(fact_5716_not__numeral__le__neg__one,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_numeral_le_neg_one
% 0.22/0.50 thf(fact_5717_not__numeral__le__neg__one,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_numeral_le_neg_one
% 0.22/0.50 thf(fact_5718_not__numeral__le__neg__one,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_numeral_le_neg_one
% 0.22/0.50 thf(fact_5719_not__numeral__le__neg__one,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_numeral_le_neg_one
% 0.22/0.50 thf(fact_5720_not__one__le__neg__numeral,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_one_le_neg_numeral
% 0.22/0.50 thf(fact_5721_not__one__le__neg__numeral,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_one_le_neg_numeral
% 0.22/0.50 thf(fact_5722_not__one__le__neg__numeral,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_one_le_neg_numeral
% 0.22/0.50 thf(fact_5723_not__one__le__neg__numeral,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_one_le_neg_numeral
% 0.22/0.50 thf(fact_5724_neg__numeral__less__one,axiom,
% 0.22/0.50 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_less_one
% 0.22/0.50 thf(fact_5725_neg__numeral__less__one,axiom,
% 0.22/0.50 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_less_one
% 0.22/0.50 thf(fact_5726_neg__numeral__less__one,axiom,
% 0.22/0.50 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_less_one
% 0.22/0.50 thf(fact_5727_neg__numeral__less__one,axiom,
% 0.22/0.50 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.22/0.50
% 0.22/0.50 % neg_numeral_less_one
% 0.22/0.50 thf(fact_5728_neg__one__less__numeral,axiom,
% 0.22/0.50 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_one_less_numeral
% 0.22/0.50 thf(fact_5729_neg__one__less__numeral,axiom,
% 0.22/0.50 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_one_less_numeral
% 0.22/0.50 thf(fact_5730_neg__one__less__numeral,axiom,
% 0.22/0.50 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_one_less_numeral
% 0.22/0.50 thf(fact_5731_neg__one__less__numeral,axiom,
% 0.22/0.50 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_one_less_numeral
% 0.22/0.50 thf(fact_5732_not__numeral__less__neg__one,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_numeral_less_neg_one
% 0.22/0.50 thf(fact_5733_not__numeral__less__neg__one,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_numeral_less_neg_one
% 0.22/0.50 thf(fact_5734_not__numeral__less__neg__one,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_numeral_less_neg_one
% 0.22/0.50 thf(fact_5735_not__numeral__less__neg__one,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_numeral_less_neg_one
% 0.22/0.50 thf(fact_5736_not__one__less__neg__numeral,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_one_less_neg_numeral
% 0.22/0.50 thf(fact_5737_not__one__less__neg__numeral,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_one_less_neg_numeral
% 0.22/0.50 thf(fact_5738_not__one__less__neg__numeral,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_one_less_neg_numeral
% 0.22/0.50 thf(fact_5739_not__one__less__neg__numeral,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_one_less_neg_numeral
% 0.22/0.50 thf(fact_5740_not__neg__one__less__neg__numeral,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_neg_one_less_neg_numeral
% 0.22/0.50 thf(fact_5741_not__neg__one__less__neg__numeral,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_neg_one_less_neg_numeral
% 0.22/0.50 thf(fact_5742_not__neg__one__less__neg__numeral,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_neg_one_less_neg_numeral
% 0.22/0.50 thf(fact_5743_not__neg__one__less__neg__numeral,axiom,
% 0.22/0.50 ! [M: num] :
% 0.22/0.50 ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_neg_one_less_neg_numeral
% 0.22/0.50 thf(fact_5744_eq__minus__divide__eq,axiom,
% 0.22/0.50 ! [A: real,B: real,C: real] :
% 0.22/0.50 ( ( A
% 0.22/0.50 = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.22/0.50 = ( ( ( C != zero_zero_real )
% 0.22/0.50 => ( ( times_times_real @ A @ C )
% 0.22/0.50 = ( uminus_uminus_real @ B ) ) )
% 0.22/0.50 & ( ( C = zero_zero_real )
% 0.22/0.50 => ( A = zero_zero_real ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % eq_minus_divide_eq
% 0.22/0.50 thf(fact_5745_eq__minus__divide__eq,axiom,
% 0.22/0.50 ! [A: complex,B: complex,C: complex] :
% 0.22/0.50 ( ( A
% 0.22/0.50 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
% 0.22/0.50 = ( ( ( C != zero_zero_complex )
% 0.22/0.50 => ( ( times_times_complex @ A @ C )
% 0.22/0.50 = ( uminus1482373934393186551omplex @ B ) ) )
% 0.22/0.50 & ( ( C = zero_zero_complex )
% 0.22/0.50 => ( A = zero_zero_complex ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % eq_minus_divide_eq
% 0.22/0.50 thf(fact_5746_eq__minus__divide__eq,axiom,
% 0.22/0.50 ! [A: rat,B: rat,C: rat] :
% 0.22/0.50 ( ( A
% 0.22/0.50 = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.22/0.50 = ( ( ( C != zero_zero_rat )
% 0.22/0.50 => ( ( times_times_rat @ A @ C )
% 0.22/0.50 = ( uminus_uminus_rat @ B ) ) )
% 0.22/0.50 & ( ( C = zero_zero_rat )
% 0.22/0.50 => ( A = zero_zero_rat ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % eq_minus_divide_eq
% 0.22/0.50 thf(fact_5747_minus__divide__eq__eq,axiom,
% 0.22/0.50 ! [B: real,C: real,A: real] :
% 0.22/0.50 ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
% 0.22/0.50 = A )
% 0.22/0.50 = ( ( ( C != zero_zero_real )
% 0.22/0.50 => ( ( uminus_uminus_real @ B )
% 0.22/0.50 = ( times_times_real @ A @ C ) ) )
% 0.22/0.50 & ( ( C = zero_zero_real )
% 0.22/0.50 => ( A = zero_zero_real ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_eq_eq
% 0.22/0.50 thf(fact_5748_minus__divide__eq__eq,axiom,
% 0.22/0.50 ! [B: complex,C: complex,A: complex] :
% 0.22/0.50 ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.22/0.50 = A )
% 0.22/0.50 = ( ( ( C != zero_zero_complex )
% 0.22/0.50 => ( ( uminus1482373934393186551omplex @ B )
% 0.22/0.50 = ( times_times_complex @ A @ C ) ) )
% 0.22/0.50 & ( ( C = zero_zero_complex )
% 0.22/0.50 => ( A = zero_zero_complex ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_eq_eq
% 0.22/0.50 thf(fact_5749_minus__divide__eq__eq,axiom,
% 0.22/0.50 ! [B: rat,C: rat,A: rat] :
% 0.22/0.50 ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
% 0.22/0.50 = A )
% 0.22/0.50 = ( ( ( C != zero_zero_rat )
% 0.22/0.50 => ( ( uminus_uminus_rat @ B )
% 0.22/0.50 = ( times_times_rat @ A @ C ) ) )
% 0.22/0.50 & ( ( C = zero_zero_rat )
% 0.22/0.50 => ( A = zero_zero_rat ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_eq_eq
% 0.22/0.50 thf(fact_5750_nonzero__neg__divide__eq__eq,axiom,
% 0.22/0.50 ! [B: real,A: real,C: real] :
% 0.22/0.50 ( ( B != zero_zero_real )
% 0.22/0.50 => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.22/0.50 = C )
% 0.22/0.50 = ( ( uminus_uminus_real @ A )
% 0.22/0.50 = ( times_times_real @ C @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % nonzero_neg_divide_eq_eq
% 0.22/0.50 thf(fact_5751_nonzero__neg__divide__eq__eq,axiom,
% 0.22/0.50 ! [B: complex,A: complex,C: complex] :
% 0.22/0.50 ( ( B != zero_zero_complex )
% 0.22/0.50 => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.22/0.50 = C )
% 0.22/0.50 = ( ( uminus1482373934393186551omplex @ A )
% 0.22/0.50 = ( times_times_complex @ C @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % nonzero_neg_divide_eq_eq
% 0.22/0.50 thf(fact_5752_nonzero__neg__divide__eq__eq,axiom,
% 0.22/0.50 ! [B: rat,A: rat,C: rat] :
% 0.22/0.50 ( ( B != zero_zero_rat )
% 0.22/0.50 => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.22/0.50 = C )
% 0.22/0.50 = ( ( uminus_uminus_rat @ A )
% 0.22/0.50 = ( times_times_rat @ C @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % nonzero_neg_divide_eq_eq
% 0.22/0.50 thf(fact_5753_nonzero__neg__divide__eq__eq2,axiom,
% 0.22/0.50 ! [B: real,C: real,A: real] :
% 0.22/0.50 ( ( B != zero_zero_real )
% 0.22/0.50 => ( ( C
% 0.22/0.50 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
% 0.22/0.50 = ( ( times_times_real @ C @ B )
% 0.22/0.50 = ( uminus_uminus_real @ A ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % nonzero_neg_divide_eq_eq2
% 0.22/0.50 thf(fact_5754_nonzero__neg__divide__eq__eq2,axiom,
% 0.22/0.50 ! [B: complex,C: complex,A: complex] :
% 0.22/0.50 ( ( B != zero_zero_complex )
% 0.22/0.50 => ( ( C
% 0.22/0.50 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.22/0.50 = ( ( times_times_complex @ C @ B )
% 0.22/0.50 = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % nonzero_neg_divide_eq_eq2
% 0.22/0.50 thf(fact_5755_nonzero__neg__divide__eq__eq2,axiom,
% 0.22/0.50 ! [B: rat,C: rat,A: rat] :
% 0.22/0.50 ( ( B != zero_zero_rat )
% 0.22/0.50 => ( ( C
% 0.22/0.50 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
% 0.22/0.50 = ( ( times_times_rat @ C @ B )
% 0.22/0.50 = ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % nonzero_neg_divide_eq_eq2
% 0.22/0.50 thf(fact_5756_mult__1s__ring__1_I2_J,axiom,
% 0.22/0.50 ! [B: real] :
% 0.22/0.50 ( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
% 0.22/0.50 = ( uminus_uminus_real @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % mult_1s_ring_1(2)
% 0.22/0.50 thf(fact_5757_mult__1s__ring__1_I2_J,axiom,
% 0.22/0.50 ! [B: int] :
% 0.22/0.50 ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
% 0.22/0.50 = ( uminus_uminus_int @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % mult_1s_ring_1(2)
% 0.22/0.50 thf(fact_5758_mult__1s__ring__1_I2_J,axiom,
% 0.22/0.50 ! [B: complex] :
% 0.22/0.50 ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
% 0.22/0.50 = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % mult_1s_ring_1(2)
% 0.22/0.50 thf(fact_5759_mult__1s__ring__1_I2_J,axiom,
% 0.22/0.50 ! [B: code_integer] :
% 0.22/0.50 ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
% 0.22/0.50 = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % mult_1s_ring_1(2)
% 0.22/0.50 thf(fact_5760_mult__1s__ring__1_I2_J,axiom,
% 0.22/0.50 ! [B: rat] :
% 0.22/0.50 ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
% 0.22/0.50 = ( uminus_uminus_rat @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % mult_1s_ring_1(2)
% 0.22/0.50 thf(fact_5761_mult__1s__ring__1_I1_J,axiom,
% 0.22/0.50 ! [B: real] :
% 0.22/0.50 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
% 0.22/0.50 = ( uminus_uminus_real @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % mult_1s_ring_1(1)
% 0.22/0.50 thf(fact_5762_mult__1s__ring__1_I1_J,axiom,
% 0.22/0.50 ! [B: int] :
% 0.22/0.50 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
% 0.22/0.50 = ( uminus_uminus_int @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % mult_1s_ring_1(1)
% 0.22/0.50 thf(fact_5763_mult__1s__ring__1_I1_J,axiom,
% 0.22/0.50 ! [B: complex] :
% 0.22/0.50 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
% 0.22/0.50 = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % mult_1s_ring_1(1)
% 0.22/0.50 thf(fact_5764_mult__1s__ring__1_I1_J,axiom,
% 0.22/0.50 ! [B: code_integer] :
% 0.22/0.50 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
% 0.22/0.50 = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % mult_1s_ring_1(1)
% 0.22/0.50 thf(fact_5765_mult__1s__ring__1_I1_J,axiom,
% 0.22/0.50 ! [B: rat] :
% 0.22/0.50 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
% 0.22/0.50 = ( uminus_uminus_rat @ B ) ) ).
% 0.22/0.50
% 0.22/0.50 % mult_1s_ring_1(1)
% 0.22/0.50 thf(fact_5766_divide__eq__minus__1__iff,axiom,
% 0.22/0.50 ! [A: real,B: real] :
% 0.22/0.50 ( ( ( divide_divide_real @ A @ B )
% 0.22/0.50 = ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.50 = ( ( B != zero_zero_real )
% 0.22/0.50 & ( A
% 0.22/0.50 = ( uminus_uminus_real @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % divide_eq_minus_1_iff
% 0.22/0.50 thf(fact_5767_divide__eq__minus__1__iff,axiom,
% 0.22/0.50 ! [A: complex,B: complex] :
% 0.22/0.50 ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.22/0.50 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.22/0.50 = ( ( B != zero_zero_complex )
% 0.22/0.50 & ( A
% 0.22/0.50 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % divide_eq_minus_1_iff
% 0.22/0.50 thf(fact_5768_divide__eq__minus__1__iff,axiom,
% 0.22/0.50 ! [A: rat,B: rat] :
% 0.22/0.50 ( ( ( divide_divide_rat @ A @ B )
% 0.22/0.50 = ( uminus_uminus_rat @ one_one_rat ) )
% 0.22/0.50 = ( ( B != zero_zero_rat )
% 0.22/0.50 & ( A
% 0.22/0.50 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % divide_eq_minus_1_iff
% 0.22/0.50 thf(fact_5769_uminus__numeral__One,axiom,
% 0.22/0.50 ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
% 0.22/0.50 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % uminus_numeral_One
% 0.22/0.50 thf(fact_5770_uminus__numeral__One,axiom,
% 0.22/0.50 ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
% 0.22/0.50 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % uminus_numeral_One
% 0.22/0.50 thf(fact_5771_uminus__numeral__One,axiom,
% 0.22/0.50 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
% 0.22/0.50 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.22/0.50
% 0.22/0.50 % uminus_numeral_One
% 0.22/0.50 thf(fact_5772_uminus__numeral__One,axiom,
% 0.22/0.50 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
% 0.22/0.50 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.50
% 0.22/0.50 % uminus_numeral_One
% 0.22/0.50 thf(fact_5773_uminus__numeral__One,axiom,
% 0.22/0.50 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
% 0.22/0.50 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % uminus_numeral_One
% 0.22/0.50 thf(fact_5774_power__minus,axiom,
% 0.22/0.50 ! [A: real,N: nat] :
% 0.22/0.50 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.22/0.50 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_minus
% 0.22/0.50 thf(fact_5775_power__minus,axiom,
% 0.22/0.50 ! [A: int,N: nat] :
% 0.22/0.50 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.22/0.50 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_minus
% 0.22/0.50 thf(fact_5776_power__minus,axiom,
% 0.22/0.50 ! [A: complex,N: nat] :
% 0.22/0.50 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.22/0.50 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_minus
% 0.22/0.50 thf(fact_5777_power__minus,axiom,
% 0.22/0.50 ! [A: code_integer,N: nat] :
% 0.22/0.50 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.22/0.50 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_minus
% 0.22/0.50 thf(fact_5778_power__minus,axiom,
% 0.22/0.50 ! [A: rat,N: nat] :
% 0.22/0.50 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.22/0.50 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_minus
% 0.22/0.50 thf(fact_5779_power__minus__Bit0,axiom,
% 0.22/0.50 ! [X: real,K: num] :
% 0.22/0.50 ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.22/0.50 = ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_minus_Bit0
% 0.22/0.50 thf(fact_5780_power__minus__Bit0,axiom,
% 0.22/0.50 ! [X: int,K: num] :
% 0.22/0.50 ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.22/0.50 = ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_minus_Bit0
% 0.22/0.50 thf(fact_5781_power__minus__Bit0,axiom,
% 0.22/0.50 ! [X: complex,K: num] :
% 0.22/0.50 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.22/0.50 = ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_minus_Bit0
% 0.22/0.50 thf(fact_5782_power__minus__Bit0,axiom,
% 0.22/0.50 ! [X: code_integer,K: num] :
% 0.22/0.50 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.22/0.50 = ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_minus_Bit0
% 0.22/0.50 thf(fact_5783_power__minus__Bit0,axiom,
% 0.22/0.50 ! [X: rat,K: num] :
% 0.22/0.50 ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.22/0.50 = ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_minus_Bit0
% 0.22/0.50 thf(fact_5784_lemma__interval__lt,axiom,
% 0.22/0.50 ! [A: real,X: real,B: real] :
% 0.22/0.50 ( ( ord_less_real @ A @ X )
% 0.22/0.50 => ( ( ord_less_real @ X @ B )
% 0.22/0.50 => ? [D2: real] :
% 0.22/0.50 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.22/0.50 & ! [Y3: real] :
% 0.22/0.50 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D2 )
% 0.22/0.50 => ( ( ord_less_real @ A @ Y3 )
% 0.22/0.50 & ( ord_less_real @ Y3 @ B ) ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % lemma_interval_lt
% 0.22/0.50 thf(fact_5785_norm__uminus__minus,axiom,
% 0.22/0.50 ! [X: real,Y4: real] :
% 0.22/0.50 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y4 ) )
% 0.22/0.50 = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y4 ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % norm_uminus_minus
% 0.22/0.50 thf(fact_5786_norm__uminus__minus,axiom,
% 0.22/0.50 ! [X: complex,Y4: complex] :
% 0.22/0.50 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y4 ) )
% 0.22/0.50 = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y4 ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % norm_uminus_minus
% 0.22/0.50 thf(fact_5787_power__minus__Bit1,axiom,
% 0.22/0.50 ! [X: real,K: num] :
% 0.22/0.50 ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.22/0.50 = ( uminus_uminus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_minus_Bit1
% 0.22/0.50 thf(fact_5788_power__minus__Bit1,axiom,
% 0.22/0.50 ! [X: int,K: num] :
% 0.22/0.50 ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.22/0.50 = ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_minus_Bit1
% 0.22/0.50 thf(fact_5789_power__minus__Bit1,axiom,
% 0.22/0.50 ! [X: complex,K: num] :
% 0.22/0.50 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.22/0.50 = ( uminus1482373934393186551omplex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_minus_Bit1
% 0.22/0.50 thf(fact_5790_power__minus__Bit1,axiom,
% 0.22/0.50 ! [X: code_integer,K: num] :
% 0.22/0.50 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.22/0.50 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_minus_Bit1
% 0.22/0.50 thf(fact_5791_power__minus__Bit1,axiom,
% 0.22/0.50 ! [X: rat,K: num] :
% 0.22/0.50 ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.22/0.50 = ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_minus_Bit1
% 0.22/0.50 thf(fact_5792_sin__bound__lemma,axiom,
% 0.22/0.50 ! [X: real,Y4: real,U: real,V: real] :
% 0.22/0.50 ( ( X = Y4 )
% 0.22/0.50 => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 0.22/0.50 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y4 ) ) @ V ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % sin_bound_lemma
% 0.22/0.50 thf(fact_5793_int__cases4,axiom,
% 0.22/0.50 ! [M: int] :
% 0.22/0.50 ( ! [N2: nat] :
% 0.22/0.50 ( M
% 0.22/0.50 != ( semiri1314217659103216013at_int @ N2 ) )
% 0.22/0.50 => ~ ! [N2: nat] :
% 0.22/0.50 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.22/0.50 => ( M
% 0.22/0.50 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % int_cases4
% 0.22/0.50 thf(fact_5794_real__0__less__add__iff,axiom,
% 0.22/0.50 ! [X: real,Y4: real] :
% 0.22/0.50 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y4 ) )
% 0.22/0.50 = ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y4 ) ) ).
% 0.22/0.50
% 0.22/0.50 % real_0_less_add_iff
% 0.22/0.50 thf(fact_5795_real__add__less__0__iff,axiom,
% 0.22/0.50 ! [X: real,Y4: real] :
% 0.22/0.50 ( ( ord_less_real @ ( plus_plus_real @ X @ Y4 ) @ zero_zero_real )
% 0.22/0.50 = ( ord_less_real @ Y4 @ ( uminus_uminus_real @ X ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % real_add_less_0_iff
% 0.22/0.50 thf(fact_5796_real__add__le__0__iff,axiom,
% 0.22/0.50 ! [X: real,Y4: real] :
% 0.22/0.50 ( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y4 ) @ zero_zero_real )
% 0.22/0.50 = ( ord_less_eq_real @ Y4 @ ( uminus_uminus_real @ X ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % real_add_le_0_iff
% 0.22/0.50 thf(fact_5797_real__0__le__add__iff,axiom,
% 0.22/0.50 ! [X: real,Y4: real] :
% 0.22/0.50 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y4 ) )
% 0.22/0.50 = ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y4 ) ) ).
% 0.22/0.50
% 0.22/0.50 % real_0_le_add_iff
% 0.22/0.50 thf(fact_5798_int__zle__neg,axiom,
% 0.22/0.50 ! [N: nat,M: nat] :
% 0.22/0.50 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
% 0.22/0.50 = ( ( N = zero_zero_nat )
% 0.22/0.50 & ( M = zero_zero_nat ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % int_zle_neg
% 0.22/0.50 thf(fact_5799_nonpos__int__cases,axiom,
% 0.22/0.50 ! [K: int] :
% 0.22/0.50 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.22/0.50 => ~ ! [N2: nat] :
% 0.22/0.50 ( K
% 0.22/0.50 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % nonpos_int_cases
% 0.22/0.50 thf(fact_5800_negative__zle__0,axiom,
% 0.22/0.50 ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% 0.22/0.50
% 0.22/0.50 % negative_zle_0
% 0.22/0.50 thf(fact_5801_zmod__zminus1__eq__if,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( ( ( modulo_modulo_int @ A @ B )
% 0.22/0.50 = zero_zero_int )
% 0.22/0.50 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 0.22/0.50 = zero_zero_int ) )
% 0.22/0.50 & ( ( ( modulo_modulo_int @ A @ B )
% 0.22/0.50 != zero_zero_int )
% 0.22/0.50 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 0.22/0.50 = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % zmod_zminus1_eq_if
% 0.22/0.50 thf(fact_5802_zmod__zminus2__eq__if,axiom,
% 0.22/0.50 ! [A: int,B: int] :
% 0.22/0.50 ( ( ( ( modulo_modulo_int @ A @ B )
% 0.22/0.50 = zero_zero_int )
% 0.22/0.50 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 0.22/0.50 = zero_zero_int ) )
% 0.22/0.50 & ( ( ( modulo_modulo_int @ A @ B )
% 0.22/0.50 != zero_zero_int )
% 0.22/0.50 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 0.22/0.50 = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % zmod_zminus2_eq_if
% 0.22/0.50 thf(fact_5803_abs__add__one__gt__zero,axiom,
% 0.22/0.50 ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_add_one_gt_zero
% 0.22/0.50 thf(fact_5804_abs__add__one__gt__zero,axiom,
% 0.22/0.50 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_add_one_gt_zero
% 0.22/0.50 thf(fact_5805_abs__add__one__gt__zero,axiom,
% 0.22/0.50 ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_add_one_gt_zero
% 0.22/0.50 thf(fact_5806_abs__add__one__gt__zero,axiom,
% 0.22/0.50 ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_add_one_gt_zero
% 0.22/0.50 thf(fact_5807_less__minus__divide__eq,axiom,
% 0.22/0.50 ! [A: real,B: real,C: real] :
% 0.22/0.50 ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.22/0.50 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.50 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.22/0.50 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.50 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.50 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.22/0.50 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.50 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % less_minus_divide_eq
% 0.22/0.50 thf(fact_5808_less__minus__divide__eq,axiom,
% 0.22/0.50 ! [A: rat,B: rat,C: rat] :
% 0.22/0.50 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.22/0.50 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.50 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.22/0.50 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.50 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.50 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.22/0.50 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.50 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % less_minus_divide_eq
% 0.22/0.50 thf(fact_5809_minus__divide__less__eq,axiom,
% 0.22/0.50 ! [B: real,C: real,A: real] :
% 0.22/0.50 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.22/0.50 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.50 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.22/0.50 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.50 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.50 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.22/0.50 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.50 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_less_eq
% 0.22/0.50 thf(fact_5810_minus__divide__less__eq,axiom,
% 0.22/0.50 ! [B: rat,C: rat,A: rat] :
% 0.22/0.50 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.22/0.50 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.50 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.22/0.50 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.50 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.50 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.22/0.50 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.50 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_less_eq
% 0.22/0.50 thf(fact_5811_neg__less__minus__divide__eq,axiom,
% 0.22/0.50 ! [C: real,A: real,B: real] :
% 0.22/0.50 ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.50 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.22/0.50 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_less_minus_divide_eq
% 0.22/0.50 thf(fact_5812_neg__less__minus__divide__eq,axiom,
% 0.22/0.50 ! [C: rat,A: rat,B: rat] :
% 0.22/0.50 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.50 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.22/0.50 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_less_minus_divide_eq
% 0.22/0.50 thf(fact_5813_neg__minus__divide__less__eq,axiom,
% 0.22/0.50 ! [C: real,B: real,A: real] :
% 0.22/0.50 ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.50 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.22/0.50 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_minus_divide_less_eq
% 0.22/0.50 thf(fact_5814_neg__minus__divide__less__eq,axiom,
% 0.22/0.50 ! [C: rat,B: rat,A: rat] :
% 0.22/0.50 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.50 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.22/0.50 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_minus_divide_less_eq
% 0.22/0.50 thf(fact_5815_pos__less__minus__divide__eq,axiom,
% 0.22/0.50 ! [C: real,A: real,B: real] :
% 0.22/0.50 ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.50 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.22/0.50 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % pos_less_minus_divide_eq
% 0.22/0.50 thf(fact_5816_pos__less__minus__divide__eq,axiom,
% 0.22/0.50 ! [C: rat,A: rat,B: rat] :
% 0.22/0.50 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.50 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.22/0.50 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % pos_less_minus_divide_eq
% 0.22/0.50 thf(fact_5817_pos__minus__divide__less__eq,axiom,
% 0.22/0.50 ! [C: real,B: real,A: real] :
% 0.22/0.50 ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.50 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.22/0.50 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % pos_minus_divide_less_eq
% 0.22/0.50 thf(fact_5818_pos__minus__divide__less__eq,axiom,
% 0.22/0.50 ! [C: rat,B: rat,A: rat] :
% 0.22/0.50 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.50 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.22/0.50 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % pos_minus_divide_less_eq
% 0.22/0.50 thf(fact_5819_divide__eq__eq__numeral_I2_J,axiom,
% 0.22/0.50 ! [B: real,C: real,W: num] :
% 0.22/0.50 ( ( ( divide_divide_real @ B @ C )
% 0.22/0.50 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.22/0.50 = ( ( ( C != zero_zero_real )
% 0.22/0.50 => ( B
% 0.22/0.50 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.22/0.50 & ( ( C = zero_zero_real )
% 0.22/0.50 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.22/0.50 = zero_zero_real ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % divide_eq_eq_numeral(2)
% 0.22/0.50 thf(fact_5820_divide__eq__eq__numeral_I2_J,axiom,
% 0.22/0.50 ! [B: complex,C: complex,W: num] :
% 0.22/0.50 ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.22/0.50 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.22/0.50 = ( ( ( C != zero_zero_complex )
% 0.22/0.50 => ( B
% 0.22/0.50 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
% 0.22/0.50 & ( ( C = zero_zero_complex )
% 0.22/0.50 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.22/0.50 = zero_zero_complex ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % divide_eq_eq_numeral(2)
% 0.22/0.50 thf(fact_5821_divide__eq__eq__numeral_I2_J,axiom,
% 0.22/0.50 ! [B: rat,C: rat,W: num] :
% 0.22/0.50 ( ( ( divide_divide_rat @ B @ C )
% 0.22/0.50 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.22/0.50 = ( ( ( C != zero_zero_rat )
% 0.22/0.50 => ( B
% 0.22/0.50 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.22/0.50 & ( ( C = zero_zero_rat )
% 0.22/0.50 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.22/0.50 = zero_zero_rat ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % divide_eq_eq_numeral(2)
% 0.22/0.50 thf(fact_5822_eq__divide__eq__numeral_I2_J,axiom,
% 0.22/0.50 ! [W: num,B: real,C: real] :
% 0.22/0.50 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.22/0.50 = ( divide_divide_real @ B @ C ) )
% 0.22/0.50 = ( ( ( C != zero_zero_real )
% 0.22/0.50 => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
% 0.22/0.50 = B ) )
% 0.22/0.50 & ( ( C = zero_zero_real )
% 0.22/0.50 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.22/0.50 = zero_zero_real ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % eq_divide_eq_numeral(2)
% 0.22/0.50 thf(fact_5823_eq__divide__eq__numeral_I2_J,axiom,
% 0.22/0.50 ! [W: num,B: complex,C: complex] :
% 0.22/0.50 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.22/0.50 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.22/0.50 = ( ( ( C != zero_zero_complex )
% 0.22/0.50 => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
% 0.22/0.50 = B ) )
% 0.22/0.50 & ( ( C = zero_zero_complex )
% 0.22/0.50 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.22/0.50 = zero_zero_complex ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % eq_divide_eq_numeral(2)
% 0.22/0.50 thf(fact_5824_eq__divide__eq__numeral_I2_J,axiom,
% 0.22/0.50 ! [W: num,B: rat,C: rat] :
% 0.22/0.50 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.22/0.50 = ( divide_divide_rat @ B @ C ) )
% 0.22/0.50 = ( ( ( C != zero_zero_rat )
% 0.22/0.50 => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
% 0.22/0.50 = B ) )
% 0.22/0.50 & ( ( C = zero_zero_rat )
% 0.22/0.50 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.22/0.50 = zero_zero_rat ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % eq_divide_eq_numeral(2)
% 0.22/0.50 thf(fact_5825_add__divide__eq__if__simps_I3_J,axiom,
% 0.22/0.50 ! [Z: real,A: real,B: real] :
% 0.22/0.50 ( ( ( Z = zero_zero_real )
% 0.22/0.50 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.22/0.50 = B ) )
% 0.22/0.50 & ( ( Z != zero_zero_real )
% 0.22/0.50 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.22/0.50 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % add_divide_eq_if_simps(3)
% 0.22/0.50 thf(fact_5826_add__divide__eq__if__simps_I3_J,axiom,
% 0.22/0.50 ! [Z: complex,A: complex,B: complex] :
% 0.22/0.50 ( ( ( Z = zero_zero_complex )
% 0.22/0.50 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.22/0.50 = B ) )
% 0.22/0.50 & ( ( Z != zero_zero_complex )
% 0.22/0.50 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.22/0.50 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % add_divide_eq_if_simps(3)
% 0.22/0.50 thf(fact_5827_add__divide__eq__if__simps_I3_J,axiom,
% 0.22/0.50 ! [Z: rat,A: rat,B: rat] :
% 0.22/0.50 ( ( ( Z = zero_zero_rat )
% 0.22/0.50 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.22/0.50 = B ) )
% 0.22/0.50 & ( ( Z != zero_zero_rat )
% 0.22/0.50 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.22/0.50 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % add_divide_eq_if_simps(3)
% 0.22/0.50 thf(fact_5828_minus__divide__add__eq__iff,axiom,
% 0.22/0.50 ! [Z: real,X: real,Y4: real] :
% 0.22/0.50 ( ( Z != zero_zero_real )
% 0.22/0.50 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y4 )
% 0.22/0.50 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y4 @ Z ) ) @ Z ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_add_eq_iff
% 0.22/0.50 thf(fact_5829_minus__divide__add__eq__iff,axiom,
% 0.22/0.50 ! [Z: complex,X: complex,Y4: complex] :
% 0.22/0.50 ( ( Z != zero_zero_complex )
% 0.22/0.50 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y4 )
% 0.22/0.50 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y4 @ Z ) ) @ Z ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_add_eq_iff
% 0.22/0.50 thf(fact_5830_minus__divide__add__eq__iff,axiom,
% 0.22/0.50 ! [Z: rat,X: rat,Y4: rat] :
% 0.22/0.50 ( ( Z != zero_zero_rat )
% 0.22/0.50 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y4 )
% 0.22/0.50 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y4 @ Z ) ) @ Z ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_add_eq_iff
% 0.22/0.50 thf(fact_5831_add__divide__eq__if__simps_I6_J,axiom,
% 0.22/0.50 ! [Z: real,A: real,B: real] :
% 0.22/0.50 ( ( ( Z = zero_zero_real )
% 0.22/0.50 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.22/0.50 = ( uminus_uminus_real @ B ) ) )
% 0.22/0.50 & ( ( Z != zero_zero_real )
% 0.22/0.50 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.22/0.50 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % add_divide_eq_if_simps(6)
% 0.22/0.50 thf(fact_5832_add__divide__eq__if__simps_I6_J,axiom,
% 0.22/0.50 ! [Z: complex,A: complex,B: complex] :
% 0.22/0.50 ( ( ( Z = zero_zero_complex )
% 0.22/0.50 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.22/0.50 = ( uminus1482373934393186551omplex @ B ) ) )
% 0.22/0.50 & ( ( Z != zero_zero_complex )
% 0.22/0.50 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.22/0.50 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % add_divide_eq_if_simps(6)
% 0.22/0.50 thf(fact_5833_add__divide__eq__if__simps_I6_J,axiom,
% 0.22/0.50 ! [Z: rat,A: rat,B: rat] :
% 0.22/0.50 ( ( ( Z = zero_zero_rat )
% 0.22/0.50 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.22/0.50 = ( uminus_uminus_rat @ B ) ) )
% 0.22/0.50 & ( ( Z != zero_zero_rat )
% 0.22/0.50 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.22/0.50 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % add_divide_eq_if_simps(6)
% 0.22/0.50 thf(fact_5834_add__divide__eq__if__simps_I5_J,axiom,
% 0.22/0.50 ! [Z: real,A: real,B: real] :
% 0.22/0.50 ( ( ( Z = zero_zero_real )
% 0.22/0.50 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.22/0.50 = ( uminus_uminus_real @ B ) ) )
% 0.22/0.50 & ( ( Z != zero_zero_real )
% 0.22/0.50 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.22/0.50 = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % add_divide_eq_if_simps(5)
% 0.22/0.50 thf(fact_5835_add__divide__eq__if__simps_I5_J,axiom,
% 0.22/0.50 ! [Z: complex,A: complex,B: complex] :
% 0.22/0.50 ( ( ( Z = zero_zero_complex )
% 0.22/0.50 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.22/0.50 = ( uminus1482373934393186551omplex @ B ) ) )
% 0.22/0.50 & ( ( Z != zero_zero_complex )
% 0.22/0.50 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.22/0.50 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % add_divide_eq_if_simps(5)
% 0.22/0.50 thf(fact_5836_add__divide__eq__if__simps_I5_J,axiom,
% 0.22/0.50 ! [Z: rat,A: rat,B: rat] :
% 0.22/0.50 ( ( ( Z = zero_zero_rat )
% 0.22/0.50 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.22/0.50 = ( uminus_uminus_rat @ B ) ) )
% 0.22/0.50 & ( ( Z != zero_zero_rat )
% 0.22/0.50 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.22/0.50 = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % add_divide_eq_if_simps(5)
% 0.22/0.50 thf(fact_5837_minus__divide__diff__eq__iff,axiom,
% 0.22/0.50 ! [Z: real,X: real,Y4: real] :
% 0.22/0.50 ( ( Z != zero_zero_real )
% 0.22/0.50 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y4 )
% 0.22/0.50 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y4 @ Z ) ) @ Z ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_diff_eq_iff
% 0.22/0.50 thf(fact_5838_minus__divide__diff__eq__iff,axiom,
% 0.22/0.50 ! [Z: complex,X: complex,Y4: complex] :
% 0.22/0.50 ( ( Z != zero_zero_complex )
% 0.22/0.50 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y4 )
% 0.22/0.50 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y4 @ Z ) ) @ Z ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_diff_eq_iff
% 0.22/0.50 thf(fact_5839_minus__divide__diff__eq__iff,axiom,
% 0.22/0.50 ! [Z: rat,X: rat,Y4: rat] :
% 0.22/0.50 ( ( Z != zero_zero_rat )
% 0.22/0.50 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y4 )
% 0.22/0.50 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y4 @ Z ) ) @ Z ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_diff_eq_iff
% 0.22/0.50 thf(fact_5840_even__minus,axiom,
% 0.22/0.50 ! [A: int] :
% 0.22/0.50 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
% 0.22/0.50 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.22/0.50
% 0.22/0.50 % even_minus
% 0.22/0.50 thf(fact_5841_even__minus,axiom,
% 0.22/0.50 ! [A: code_integer] :
% 0.22/0.50 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.22/0.50 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.22/0.50
% 0.22/0.50 % even_minus
% 0.22/0.50 thf(fact_5842_lemma__interval,axiom,
% 0.22/0.50 ! [A: real,X: real,B: real] :
% 0.22/0.50 ( ( ord_less_real @ A @ X )
% 0.22/0.50 => ( ( ord_less_real @ X @ B )
% 0.22/0.50 => ? [D2: real] :
% 0.22/0.50 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.22/0.50 & ! [Y3: real] :
% 0.22/0.50 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D2 )
% 0.22/0.50 => ( ( ord_less_eq_real @ A @ Y3 )
% 0.22/0.50 & ( ord_less_eq_real @ Y3 @ B ) ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % lemma_interval
% 0.22/0.50 thf(fact_5843_power2__eq__iff,axiom,
% 0.22/0.50 ! [X: real,Y4: real] :
% 0.22/0.50 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.50 = ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.50 = ( ( X = Y4 )
% 0.22/0.50 | ( X
% 0.22/0.50 = ( uminus_uminus_real @ Y4 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power2_eq_iff
% 0.22/0.50 thf(fact_5844_power2__eq__iff,axiom,
% 0.22/0.50 ! [X: int,Y4: int] :
% 0.22/0.50 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.50 = ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.50 = ( ( X = Y4 )
% 0.22/0.50 | ( X
% 0.22/0.50 = ( uminus_uminus_int @ Y4 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power2_eq_iff
% 0.22/0.50 thf(fact_5845_power2__eq__iff,axiom,
% 0.22/0.50 ! [X: complex,Y4: complex] :
% 0.22/0.50 ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.50 = ( power_power_complex @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.50 = ( ( X = Y4 )
% 0.22/0.50 | ( X
% 0.22/0.50 = ( uminus1482373934393186551omplex @ Y4 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power2_eq_iff
% 0.22/0.50 thf(fact_5846_power2__eq__iff,axiom,
% 0.22/0.50 ! [X: code_integer,Y4: code_integer] :
% 0.22/0.50 ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.50 = ( power_8256067586552552935nteger @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.50 = ( ( X = Y4 )
% 0.22/0.50 | ( X
% 0.22/0.50 = ( uminus1351360451143612070nteger @ Y4 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power2_eq_iff
% 0.22/0.50 thf(fact_5847_power2__eq__iff,axiom,
% 0.22/0.50 ! [X: rat,Y4: rat] :
% 0.22/0.50 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.50 = ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.50 = ( ( X = Y4 )
% 0.22/0.50 | ( X
% 0.22/0.50 = ( uminus_uminus_rat @ Y4 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power2_eq_iff
% 0.22/0.50 thf(fact_5848_norm__triangle__ineq3,axiom,
% 0.22/0.50 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % norm_triangle_ineq3
% 0.22/0.50 thf(fact_5849_norm__triangle__ineq3,axiom,
% 0.22/0.50 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % norm_triangle_ineq3
% 0.22/0.50 thf(fact_5850_int__cases3,axiom,
% 0.22/0.50 ! [K: int] :
% 0.22/0.50 ( ( K != zero_zero_int )
% 0.22/0.50 => ( ! [N2: nat] :
% 0.22/0.50 ( ( K
% 0.22/0.50 = ( semiri1314217659103216013at_int @ N2 ) )
% 0.22/0.50 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 0.22/0.50 => ~ ! [N2: nat] :
% 0.22/0.50 ( ( K
% 0.22/0.50 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.22/0.50 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % int_cases3
% 0.22/0.50 thf(fact_5851_pochhammer__eq__0__iff,axiom,
% 0.22/0.50 ! [A: complex,N: nat] :
% 0.22/0.50 ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.22/0.50 = zero_zero_complex )
% 0.22/0.50 = ( ? [K3: nat] :
% 0.22/0.50 ( ( ord_less_nat @ K3 @ N )
% 0.22/0.50 & ( A
% 0.22/0.50 = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % pochhammer_eq_0_iff
% 0.22/0.50 thf(fact_5852_pochhammer__eq__0__iff,axiom,
% 0.22/0.50 ! [A: real,N: nat] :
% 0.22/0.50 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.22/0.50 = zero_zero_real )
% 0.22/0.50 = ( ? [K3: nat] :
% 0.22/0.50 ( ( ord_less_nat @ K3 @ N )
% 0.22/0.50 & ( A
% 0.22/0.50 = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % pochhammer_eq_0_iff
% 0.22/0.50 thf(fact_5853_pochhammer__eq__0__iff,axiom,
% 0.22/0.50 ! [A: rat,N: nat] :
% 0.22/0.50 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.22/0.50 = zero_zero_rat )
% 0.22/0.50 = ( ? [K3: nat] :
% 0.22/0.50 ( ( ord_less_nat @ K3 @ N )
% 0.22/0.50 & ( A
% 0.22/0.50 = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % pochhammer_eq_0_iff
% 0.22/0.50 thf(fact_5854_pochhammer__of__nat__eq__0__iff,axiom,
% 0.22/0.50 ! [N: nat,K: nat] :
% 0.22/0.50 ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.22/0.50 = zero_z3403309356797280102nteger )
% 0.22/0.50 = ( ord_less_nat @ N @ K ) ) ).
% 0.22/0.50
% 0.22/0.50 % pochhammer_of_nat_eq_0_iff
% 0.22/0.50 thf(fact_5855_pochhammer__of__nat__eq__0__iff,axiom,
% 0.22/0.50 ! [N: nat,K: nat] :
% 0.22/0.50 ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.22/0.50 = zero_zero_complex )
% 0.22/0.50 = ( ord_less_nat @ N @ K ) ) ).
% 0.22/0.50
% 0.22/0.50 % pochhammer_of_nat_eq_0_iff
% 0.22/0.50 thf(fact_5856_pochhammer__of__nat__eq__0__iff,axiom,
% 0.22/0.50 ! [N: nat,K: nat] :
% 0.22/0.50 ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.22/0.50 = zero_zero_real )
% 0.22/0.50 = ( ord_less_nat @ N @ K ) ) ).
% 0.22/0.50
% 0.22/0.50 % pochhammer_of_nat_eq_0_iff
% 0.22/0.50 thf(fact_5857_pochhammer__of__nat__eq__0__iff,axiom,
% 0.22/0.50 ! [N: nat,K: nat] :
% 0.22/0.50 ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.22/0.50 = zero_zero_rat )
% 0.22/0.50 = ( ord_less_nat @ N @ K ) ) ).
% 0.22/0.50
% 0.22/0.50 % pochhammer_of_nat_eq_0_iff
% 0.22/0.50 thf(fact_5858_pochhammer__of__nat__eq__0__iff,axiom,
% 0.22/0.50 ! [N: nat,K: nat] :
% 0.22/0.50 ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.22/0.50 = zero_zero_int )
% 0.22/0.50 = ( ord_less_nat @ N @ K ) ) ).
% 0.22/0.50
% 0.22/0.50 % pochhammer_of_nat_eq_0_iff
% 0.22/0.50 thf(fact_5859_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.22/0.50 ! [N: nat,K: nat] :
% 0.22/0.50 ( ( ord_less_nat @ N @ K )
% 0.22/0.50 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.22/0.50 = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.50
% 0.22/0.50 % pochhammer_of_nat_eq_0_lemma
% 0.22/0.50 thf(fact_5860_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.22/0.50 ! [N: nat,K: nat] :
% 0.22/0.50 ( ( ord_less_nat @ N @ K )
% 0.22/0.50 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.22/0.50 = zero_zero_complex ) ) ).
% 0.22/0.50
% 0.22/0.50 % pochhammer_of_nat_eq_0_lemma
% 0.22/0.50 thf(fact_5861_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.22/0.50 ! [N: nat,K: nat] :
% 0.22/0.50 ( ( ord_less_nat @ N @ K )
% 0.22/0.50 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.22/0.50 = zero_zero_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % pochhammer_of_nat_eq_0_lemma
% 0.22/0.50 thf(fact_5862_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.22/0.50 ! [N: nat,K: nat] :
% 0.22/0.50 ( ( ord_less_nat @ N @ K )
% 0.22/0.50 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.22/0.50 = zero_zero_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % pochhammer_of_nat_eq_0_lemma
% 0.22/0.50 thf(fact_5863_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.22/0.50 ! [N: nat,K: nat] :
% 0.22/0.50 ( ( ord_less_nat @ N @ K )
% 0.22/0.50 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.22/0.50 = zero_zero_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % pochhammer_of_nat_eq_0_lemma
% 0.22/0.50 thf(fact_5864_ln__add__one__self__le__self2,axiom,
% 0.22/0.50 ! [X: real] :
% 0.22/0.50 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.22/0.50 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 0.22/0.50
% 0.22/0.50 % ln_add_one_self_le_self2
% 0.22/0.50 thf(fact_5865_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.22/0.50 ! [K: nat,N: nat] :
% 0.22/0.50 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.50 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.22/0.50 != zero_z3403309356797280102nteger ) ) ).
% 0.22/0.50
% 0.22/0.50 % pochhammer_of_nat_eq_0_lemma'
% 0.22/0.50 thf(fact_5866_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.22/0.50 ! [K: nat,N: nat] :
% 0.22/0.50 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.50 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.22/0.50 != zero_zero_complex ) ) ).
% 0.22/0.50
% 0.22/0.50 % pochhammer_of_nat_eq_0_lemma'
% 0.22/0.50 thf(fact_5867_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.22/0.50 ! [K: nat,N: nat] :
% 0.22/0.50 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.50 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.22/0.50 != zero_zero_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % pochhammer_of_nat_eq_0_lemma'
% 0.22/0.50 thf(fact_5868_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.22/0.50 ! [K: nat,N: nat] :
% 0.22/0.50 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.50 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.22/0.50 != zero_zero_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % pochhammer_of_nat_eq_0_lemma'
% 0.22/0.50 thf(fact_5869_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.22/0.50 ! [K: nat,N: nat] :
% 0.22/0.50 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.50 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.22/0.50 != zero_zero_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % pochhammer_of_nat_eq_0_lemma'
% 0.22/0.50 thf(fact_5870_not__zle__0__negative,axiom,
% 0.22/0.50 ! [N: nat] :
% 0.22/0.50 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % not_zle_0_negative
% 0.22/0.50 thf(fact_5871_negD,axiom,
% 0.22/0.50 ! [X: int] :
% 0.22/0.50 ( ( ord_less_int @ X @ zero_zero_int )
% 0.22/0.50 => ? [N2: nat] :
% 0.22/0.50 ( X
% 0.22/0.50 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % negD
% 0.22/0.50 thf(fact_5872_negative__zless__0,axiom,
% 0.22/0.50 ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% 0.22/0.50
% 0.22/0.50 % negative_zless_0
% 0.22/0.50 thf(fact_5873_verit__less__mono__div__int2,axiom,
% 0.22/0.50 ! [A2: int,B4: int,N: int] :
% 0.22/0.50 ( ( ord_less_eq_int @ A2 @ B4 )
% 0.22/0.50 => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
% 0.22/0.50 => ( ord_less_eq_int @ ( divide_divide_int @ B4 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % verit_less_mono_div_int2
% 0.22/0.50 thf(fact_5874_div__eq__minus1,axiom,
% 0.22/0.50 ! [B: int] :
% 0.22/0.50 ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.50 => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 0.22/0.50 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % div_eq_minus1
% 0.22/0.50 thf(fact_5875_abs__le__square__iff,axiom,
% 0.22/0.50 ! [X: code_integer,Y4: code_integer] :
% 0.22/0.50 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y4 ) )
% 0.22/0.50 = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_le_square_iff
% 0.22/0.50 thf(fact_5876_abs__le__square__iff,axiom,
% 0.22/0.50 ! [X: rat,Y4: rat] :
% 0.22/0.50 ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y4 ) )
% 0.22/0.50 = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_le_square_iff
% 0.22/0.50 thf(fact_5877_abs__le__square__iff,axiom,
% 0.22/0.50 ! [X: int,Y4: int] :
% 0.22/0.50 ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y4 ) )
% 0.22/0.50 = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_le_square_iff
% 0.22/0.50 thf(fact_5878_abs__le__square__iff,axiom,
% 0.22/0.50 ! [X: real,Y4: real] :
% 0.22/0.50 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y4 ) )
% 0.22/0.50 = ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_le_square_iff
% 0.22/0.50 thf(fact_5879_abs__square__eq__1,axiom,
% 0.22/0.50 ! [X: code_integer] :
% 0.22/0.50 ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.50 = one_one_Code_integer )
% 0.22/0.50 = ( ( abs_abs_Code_integer @ X )
% 0.22/0.50 = one_one_Code_integer ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_square_eq_1
% 0.22/0.50 thf(fact_5880_abs__square__eq__1,axiom,
% 0.22/0.50 ! [X: rat] :
% 0.22/0.50 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.50 = one_one_rat )
% 0.22/0.50 = ( ( abs_abs_rat @ X )
% 0.22/0.50 = one_one_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_square_eq_1
% 0.22/0.50 thf(fact_5881_abs__square__eq__1,axiom,
% 0.22/0.50 ! [X: int] :
% 0.22/0.50 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.50 = one_one_int )
% 0.22/0.50 = ( ( abs_abs_int @ X )
% 0.22/0.50 = one_one_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_square_eq_1
% 0.22/0.50 thf(fact_5882_abs__square__eq__1,axiom,
% 0.22/0.50 ! [X: real] :
% 0.22/0.50 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.50 = one_one_real )
% 0.22/0.50 = ( ( abs_abs_real @ X )
% 0.22/0.50 = one_one_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_square_eq_1
% 0.22/0.50 thf(fact_5883_power__even__abs,axiom,
% 0.22/0.50 ! [N: nat,A: code_integer] :
% 0.22/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.50 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
% 0.22/0.50 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_even_abs
% 0.22/0.50 thf(fact_5884_power__even__abs,axiom,
% 0.22/0.50 ! [N: nat,A: rat] :
% 0.22/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.50 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N )
% 0.22/0.50 = ( power_power_rat @ A @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_even_abs
% 0.22/0.50 thf(fact_5885_power__even__abs,axiom,
% 0.22/0.50 ! [N: nat,A: int] :
% 0.22/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.50 => ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
% 0.22/0.50 = ( power_power_int @ A @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_even_abs
% 0.22/0.50 thf(fact_5886_power__even__abs,axiom,
% 0.22/0.50 ! [N: nat,A: real] :
% 0.22/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.50 => ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
% 0.22/0.50 = ( power_power_real @ A @ N ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_even_abs
% 0.22/0.50 thf(fact_5887_le__minus__divide__eq,axiom,
% 0.22/0.50 ! [A: rat,B: rat,C: rat] :
% 0.22/0.50 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.22/0.50 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.50 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.22/0.50 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.50 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.50 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.22/0.50 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.50 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % le_minus_divide_eq
% 0.22/0.50 thf(fact_5888_le__minus__divide__eq,axiom,
% 0.22/0.50 ! [A: real,B: real,C: real] :
% 0.22/0.50 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.22/0.50 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.50 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.22/0.50 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.50 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.50 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.22/0.50 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.50 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % le_minus_divide_eq
% 0.22/0.50 thf(fact_5889_minus__divide__le__eq,axiom,
% 0.22/0.50 ! [B: rat,C: rat,A: rat] :
% 0.22/0.50 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.22/0.50 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.50 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.22/0.50 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.50 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.50 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.22/0.50 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.50 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_le_eq
% 0.22/0.50 thf(fact_5890_minus__divide__le__eq,axiom,
% 0.22/0.50 ! [B: real,C: real,A: real] :
% 0.22/0.50 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.22/0.50 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.50 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.22/0.50 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.50 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.50 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.22/0.50 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.50 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_divide_le_eq
% 0.22/0.50 thf(fact_5891_neg__le__minus__divide__eq,axiom,
% 0.22/0.50 ! [C: rat,A: rat,B: rat] :
% 0.22/0.50 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.50 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.22/0.50 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_le_minus_divide_eq
% 0.22/0.50 thf(fact_5892_neg__le__minus__divide__eq,axiom,
% 0.22/0.50 ! [C: real,A: real,B: real] :
% 0.22/0.50 ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.50 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.22/0.50 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_le_minus_divide_eq
% 0.22/0.50 thf(fact_5893_neg__minus__divide__le__eq,axiom,
% 0.22/0.50 ! [C: rat,B: rat,A: rat] :
% 0.22/0.50 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.50 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.22/0.50 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_minus_divide_le_eq
% 0.22/0.50 thf(fact_5894_neg__minus__divide__le__eq,axiom,
% 0.22/0.50 ! [C: real,B: real,A: real] :
% 0.22/0.50 ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.50 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.22/0.50 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_minus_divide_le_eq
% 0.22/0.50 thf(fact_5895_pos__le__minus__divide__eq,axiom,
% 0.22/0.50 ! [C: rat,A: rat,B: rat] :
% 0.22/0.50 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.50 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.22/0.50 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % pos_le_minus_divide_eq
% 0.22/0.50 thf(fact_5896_pos__le__minus__divide__eq,axiom,
% 0.22/0.50 ! [C: real,A: real,B: real] :
% 0.22/0.50 ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.50 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.22/0.50 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % pos_le_minus_divide_eq
% 0.22/0.50 thf(fact_5897_pos__minus__divide__le__eq,axiom,
% 0.22/0.50 ! [C: rat,B: rat,A: rat] :
% 0.22/0.50 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.50 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.22/0.50 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % pos_minus_divide_le_eq
% 0.22/0.50 thf(fact_5898_pos__minus__divide__le__eq,axiom,
% 0.22/0.50 ! [C: real,B: real,A: real] :
% 0.22/0.50 ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.50 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.22/0.50 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % pos_minus_divide_le_eq
% 0.22/0.50 thf(fact_5899_divide__less__eq__numeral_I2_J,axiom,
% 0.22/0.50 ! [B: real,C: real,W: num] :
% 0.22/0.50 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.22/0.50 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.50 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.22/0.50 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.50 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.50 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.22/0.50 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.50 => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % divide_less_eq_numeral(2)
% 0.22/0.50 thf(fact_5900_divide__less__eq__numeral_I2_J,axiom,
% 0.22/0.50 ! [B: rat,C: rat,W: num] :
% 0.22/0.50 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.22/0.50 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.50 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.22/0.50 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.50 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.50 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.22/0.50 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.50 => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % divide_less_eq_numeral(2)
% 0.22/0.50 thf(fact_5901_less__divide__eq__numeral_I2_J,axiom,
% 0.22/0.50 ! [W: num,B: real,C: real] :
% 0.22/0.50 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 0.22/0.50 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.50 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.22/0.50 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.50 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.50 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.22/0.50 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.50 => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % less_divide_eq_numeral(2)
% 0.22/0.50 thf(fact_5902_less__divide__eq__numeral_I2_J,axiom,
% 0.22/0.50 ! [W: num,B: rat,C: rat] :
% 0.22/0.50 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 0.22/0.50 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.50 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.22/0.50 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.50 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.50 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.22/0.50 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.50 => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % less_divide_eq_numeral(2)
% 0.22/0.50 thf(fact_5903_power2__eq__1__iff,axiom,
% 0.22/0.50 ! [A: real] :
% 0.22/0.50 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.50 = one_one_real )
% 0.22/0.50 = ( ( A = one_one_real )
% 0.22/0.50 | ( A
% 0.22/0.50 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power2_eq_1_iff
% 0.22/0.50 thf(fact_5904_power2__eq__1__iff,axiom,
% 0.22/0.50 ! [A: int] :
% 0.22/0.50 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.50 = one_one_int )
% 0.22/0.50 = ( ( A = one_one_int )
% 0.22/0.50 | ( A
% 0.22/0.50 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power2_eq_1_iff
% 0.22/0.50 thf(fact_5905_power2__eq__1__iff,axiom,
% 0.22/0.50 ! [A: complex] :
% 0.22/0.50 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.50 = one_one_complex )
% 0.22/0.50 = ( ( A = one_one_complex )
% 0.22/0.50 | ( A
% 0.22/0.50 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power2_eq_1_iff
% 0.22/0.50 thf(fact_5906_power2__eq__1__iff,axiom,
% 0.22/0.50 ! [A: code_integer] :
% 0.22/0.50 ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.50 = one_one_Code_integer )
% 0.22/0.50 = ( ( A = one_one_Code_integer )
% 0.22/0.50 | ( A
% 0.22/0.50 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power2_eq_1_iff
% 0.22/0.50 thf(fact_5907_power2__eq__1__iff,axiom,
% 0.22/0.50 ! [A: rat] :
% 0.22/0.50 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.50 = one_one_rat )
% 0.22/0.50 = ( ( A = one_one_rat )
% 0.22/0.50 | ( A
% 0.22/0.50 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power2_eq_1_iff
% 0.22/0.50 thf(fact_5908_uminus__power__if,axiom,
% 0.22/0.50 ! [N: nat,A: real] :
% 0.22/0.50 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.50 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.22/0.50 = ( power_power_real @ A @ N ) ) )
% 0.22/0.50 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.50 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.22/0.50 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % uminus_power_if
% 0.22/0.50 thf(fact_5909_uminus__power__if,axiom,
% 0.22/0.50 ! [N: nat,A: int] :
% 0.22/0.50 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.50 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.22/0.50 = ( power_power_int @ A @ N ) ) )
% 0.22/0.50 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.50 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.22/0.50 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % uminus_power_if
% 0.22/0.50 thf(fact_5910_uminus__power__if,axiom,
% 0.22/0.50 ! [N: nat,A: complex] :
% 0.22/0.50 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.50 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.22/0.50 = ( power_power_complex @ A @ N ) ) )
% 0.22/0.50 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.50 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.22/0.50 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % uminus_power_if
% 0.22/0.50 thf(fact_5911_uminus__power__if,axiom,
% 0.22/0.50 ! [N: nat,A: code_integer] :
% 0.22/0.50 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.50 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.22/0.50 = ( power_8256067586552552935nteger @ A @ N ) ) )
% 0.22/0.50 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.50 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.22/0.50 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % uminus_power_if
% 0.22/0.50 thf(fact_5912_uminus__power__if,axiom,
% 0.22/0.50 ! [N: nat,A: rat] :
% 0.22/0.50 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.50 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.22/0.50 = ( power_power_rat @ A @ N ) ) )
% 0.22/0.50 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.50 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.22/0.50 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % uminus_power_if
% 0.22/0.50 thf(fact_5913_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.22/0.50 ! [K: nat,N: nat] :
% 0.22/0.50 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.50 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
% 0.22/0.50 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_one_power_add_eq_neg_one_power_diff
% 0.22/0.50 thf(fact_5914_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.22/0.50 ! [K: nat,N: nat] :
% 0.22/0.50 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.50 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
% 0.22/0.50 = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_one_power_add_eq_neg_one_power_diff
% 0.22/0.50 thf(fact_5915_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.22/0.50 ! [K: nat,N: nat] :
% 0.22/0.50 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.50 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
% 0.22/0.50 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_one_power_add_eq_neg_one_power_diff
% 0.22/0.50 thf(fact_5916_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.22/0.50 ! [K: nat,N: nat] :
% 0.22/0.50 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.50 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
% 0.22/0.50 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_one_power_add_eq_neg_one_power_diff
% 0.22/0.50 thf(fact_5917_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.22/0.50 ! [K: nat,N: nat] :
% 0.22/0.50 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.50 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
% 0.22/0.50 = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_one_power_add_eq_neg_one_power_diff
% 0.22/0.50 thf(fact_5918_realpow__square__minus__le,axiom,
% 0.22/0.50 ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % realpow_square_minus_le
% 0.22/0.50 thf(fact_5919_neg__int__cases,axiom,
% 0.22/0.50 ! [K: int] :
% 0.22/0.50 ( ( ord_less_int @ K @ zero_zero_int )
% 0.22/0.50 => ~ ! [N2: nat] :
% 0.22/0.50 ( ( K
% 0.22/0.50 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.22/0.50 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % neg_int_cases
% 0.22/0.50 thf(fact_5920_ln__one__minus__pos__upper__bound,axiom,
% 0.22/0.50 ! [X: real] :
% 0.22/0.50 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.50 => ( ( ord_less_real @ X @ one_one_real )
% 0.22/0.50 => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % ln_one_minus_pos_upper_bound
% 0.22/0.50 thf(fact_5921_signed__take__bit__int__less__eq__self__iff,axiom,
% 0.22/0.50 ! [N: nat,K: int] :
% 0.22/0.50 ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 0.22/0.50 = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% 0.22/0.50
% 0.22/0.50 % signed_take_bit_int_less_eq_self_iff
% 0.22/0.50 thf(fact_5922_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 0.22/0.50 ! [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 ) ) ).
% 0.22/0.50
% 0.22/0.50 % signed_take_bit_int_greater_eq_minus_exp
% 0.22/0.50 thf(fact_5923_signed__take__bit__int__greater__self__iff,axiom,
% 0.22/0.50 ! [K: int,N: nat] :
% 0.22/0.50 ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.22/0.50 = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % signed_take_bit_int_greater_self_iff
% 0.22/0.50 thf(fact_5924_minus__mod__int__eq,axiom,
% 0.22/0.50 ! [L: int,K: int] :
% 0.22/0.50 ( ( ord_less_eq_int @ zero_zero_int @ L )
% 0.22/0.50 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
% 0.22/0.50 = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % minus_mod_int_eq
% 0.22/0.50 thf(fact_5925_zmod__minus1,axiom,
% 0.22/0.50 ! [B: int] :
% 0.22/0.50 ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.50 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 0.22/0.50 = ( minus_minus_int @ B @ one_one_int ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % zmod_minus1
% 0.22/0.50 thf(fact_5926_zdiv__zminus2__eq__if,axiom,
% 0.22/0.50 ! [B: int,A: int] :
% 0.22/0.50 ( ( B != zero_zero_int )
% 0.22/0.50 => ( ( ( ( modulo_modulo_int @ A @ B )
% 0.22/0.50 = zero_zero_int )
% 0.22/0.50 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.22/0.50 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 0.22/0.50 & ( ( ( modulo_modulo_int @ A @ B )
% 0.22/0.50 != zero_zero_int )
% 0.22/0.50 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.22/0.50 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % zdiv_zminus2_eq_if
% 0.22/0.50 thf(fact_5927_zdiv__zminus1__eq__if,axiom,
% 0.22/0.50 ! [B: int,A: int] :
% 0.22/0.50 ( ( B != zero_zero_int )
% 0.22/0.50 => ( ( ( ( modulo_modulo_int @ A @ B )
% 0.22/0.50 = zero_zero_int )
% 0.22/0.50 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 0.22/0.50 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 0.22/0.50 & ( ( ( modulo_modulo_int @ A @ B )
% 0.22/0.50 != zero_zero_int )
% 0.22/0.50 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 0.22/0.50 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % zdiv_zminus1_eq_if
% 0.22/0.50 thf(fact_5928_power2__le__iff__abs__le,axiom,
% 0.22/0.50 ! [Y4: code_integer,X: code_integer] :
% 0.22/0.50 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y4 )
% 0.22/0.50 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.50 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y4 ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power2_le_iff_abs_le
% 0.22/0.50 thf(fact_5929_power2__le__iff__abs__le,axiom,
% 0.22/0.50 ! [Y4: rat,X: rat] :
% 0.22/0.50 ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 0.22/0.50 => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.50 = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y4 ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power2_le_iff_abs_le
% 0.22/0.50 thf(fact_5930_power2__le__iff__abs__le,axiom,
% 0.22/0.50 ! [Y4: int,X: int] :
% 0.22/0.50 ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.22/0.50 => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.50 = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y4 ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power2_le_iff_abs_le
% 0.22/0.50 thf(fact_5931_power2__le__iff__abs__le,axiom,
% 0.22/0.50 ! [Y4: real,X: real] :
% 0.22/0.50 ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.50 => ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.50 = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y4 ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power2_le_iff_abs_le
% 0.22/0.50 thf(fact_5932_abs__square__le__1,axiom,
% 0.22/0.50 ! [X: code_integer] :
% 0.22/0.50 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.22/0.50 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_square_le_1
% 0.22/0.50 thf(fact_5933_abs__square__le__1,axiom,
% 0.22/0.50 ! [X: rat] :
% 0.22/0.50 ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.22/0.50 = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_square_le_1
% 0.22/0.50 thf(fact_5934_abs__square__le__1,axiom,
% 0.22/0.50 ! [X: int] :
% 0.22/0.50 ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.22/0.50 = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_square_le_1
% 0.22/0.50 thf(fact_5935_abs__square__le__1,axiom,
% 0.22/0.50 ! [X: real] :
% 0.22/0.50 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.22/0.50 = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_square_le_1
% 0.22/0.50 thf(fact_5936_abs__square__less__1,axiom,
% 0.22/0.50 ! [X: code_integer] :
% 0.22/0.50 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.22/0.50 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_square_less_1
% 0.22/0.50 thf(fact_5937_abs__square__less__1,axiom,
% 0.22/0.50 ! [X: real] :
% 0.22/0.50 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.22/0.50 = ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_square_less_1
% 0.22/0.50 thf(fact_5938_abs__square__less__1,axiom,
% 0.22/0.50 ! [X: rat] :
% 0.22/0.50 ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.22/0.50 = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_square_less_1
% 0.22/0.50 thf(fact_5939_abs__square__less__1,axiom,
% 0.22/0.50 ! [X: int] :
% 0.22/0.50 ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.22/0.50 = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 0.22/0.50
% 0.22/0.50 % abs_square_less_1
% 0.22/0.50 thf(fact_5940_power__mono__even,axiom,
% 0.22/0.50 ! [N: nat,A: code_integer,B: code_integer] :
% 0.22/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.50 => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
% 0.22/0.50 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_mono_even
% 0.22/0.50 thf(fact_5941_power__mono__even,axiom,
% 0.22/0.50 ! [N: nat,A: rat,B: rat] :
% 0.22/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.50 => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
% 0.22/0.50 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_mono_even
% 0.22/0.50 thf(fact_5942_power__mono__even,axiom,
% 0.22/0.50 ! [N: nat,A: int,B: int] :
% 0.22/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.50 => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
% 0.22/0.50 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_mono_even
% 0.22/0.50 thf(fact_5943_power__mono__even,axiom,
% 0.22/0.50 ! [N: nat,A: real,B: real] :
% 0.22/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.50 => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
% 0.22/0.50 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.22/0.50
% 0.22/0.50 % power_mono_even
% 0.22/0.50 thf(fact_5944_divide__le__eq__numeral_I2_J,axiom,
% 0.22/0.50 ! [B: rat,C: rat,W: num] :
% 0.22/0.50 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.22/0.50 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.50 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.22/0.50 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.51 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.51 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.22/0.51 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.51 => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % divide_le_eq_numeral(2)
% 0.22/0.51 thf(fact_5945_divide__le__eq__numeral_I2_J,axiom,
% 0.22/0.51 ! [B: real,C: real,W: num] :
% 0.22/0.51 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.22/0.51 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.51 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.22/0.51 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.51 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.51 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.22/0.51 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.51 => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % divide_le_eq_numeral(2)
% 0.22/0.51 thf(fact_5946_le__divide__eq__numeral_I2_J,axiom,
% 0.22/0.51 ! [W: num,B: rat,C: rat] :
% 0.22/0.51 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 0.22/0.51 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.51 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.22/0.51 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.22/0.51 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.51 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.22/0.51 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.22/0.51 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % le_divide_eq_numeral(2)
% 0.22/0.51 thf(fact_5947_le__divide__eq__numeral_I2_J,axiom,
% 0.22/0.51 ! [W: num,B: real,C: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 0.22/0.51 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.51 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.22/0.51 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.51 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.51 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.22/0.51 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.51 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % le_divide_eq_numeral(2)
% 0.22/0.51 thf(fact_5948_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.22/0.51 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.22/0.51 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % abs_ln_one_plus_x_minus_x_bound_nonpos
% 0.22/0.51 thf(fact_5949_square__le__1,axiom,
% 0.22/0.51 ! [X: code_integer] :
% 0.22/0.51 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 0.22/0.51 => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
% 0.22/0.51 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % square_le_1
% 0.22/0.51 thf(fact_5950_square__le__1,axiom,
% 0.22/0.51 ! [X: rat] :
% 0.22/0.51 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
% 0.22/0.51 => ( ( ord_less_eq_rat @ X @ one_one_rat )
% 0.22/0.51 => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % square_le_1
% 0.22/0.51 thf(fact_5951_square__le__1,axiom,
% 0.22/0.51 ! [X: int] :
% 0.22/0.51 ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 0.22/0.51 => ( ( ord_less_eq_int @ X @ one_one_int )
% 0.22/0.51 => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % square_le_1
% 0.22/0.51 thf(fact_5952_square__le__1,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.22/0.51 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.22/0.51 => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % square_le_1
% 0.22/0.51 thf(fact_5953_minus__power__mult__self,axiom,
% 0.22/0.51 ! [A: real,N: nat] :
% 0.22/0.51 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 0.22/0.51 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % minus_power_mult_self
% 0.22/0.51 thf(fact_5954_minus__power__mult__self,axiom,
% 0.22/0.51 ! [A: int,N: nat] :
% 0.22/0.51 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 0.22/0.51 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % minus_power_mult_self
% 0.22/0.51 thf(fact_5955_minus__power__mult__self,axiom,
% 0.22/0.51 ! [A: complex,N: nat] :
% 0.22/0.51 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
% 0.22/0.51 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % minus_power_mult_self
% 0.22/0.51 thf(fact_5956_minus__power__mult__self,axiom,
% 0.22/0.51 ! [A: code_integer,N: nat] :
% 0.22/0.51 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 0.22/0.51 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % minus_power_mult_self
% 0.22/0.51 thf(fact_5957_minus__power__mult__self,axiom,
% 0.22/0.51 ! [A: rat,N: nat] :
% 0.22/0.51 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 0.22/0.51 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % minus_power_mult_self
% 0.22/0.51 thf(fact_5958_minus__one__power__iff,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.51 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.22/0.51 = one_one_real ) )
% 0.22/0.51 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.51 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.22/0.51 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % minus_one_power_iff
% 0.22/0.51 thf(fact_5959_minus__one__power__iff,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.51 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.22/0.51 = one_one_int ) )
% 0.22/0.51 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.51 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.22/0.51 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % minus_one_power_iff
% 0.22/0.51 thf(fact_5960_minus__one__power__iff,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.51 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.22/0.51 = one_one_complex ) )
% 0.22/0.51 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.51 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.22/0.51 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % minus_one_power_iff
% 0.22/0.51 thf(fact_5961_minus__one__power__iff,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.51 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.22/0.51 = one_one_Code_integer ) )
% 0.22/0.51 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.51 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.22/0.51 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % minus_one_power_iff
% 0.22/0.51 thf(fact_5962_minus__one__power__iff,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.51 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.22/0.51 = one_one_rat ) )
% 0.22/0.51 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.51 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.22/0.51 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % minus_one_power_iff
% 0.22/0.51 thf(fact_5963_pochhammer__absorb__comp,axiom,
% 0.22/0.51 ! [R2: code_integer,K: nat] :
% 0.22/0.51 ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K ) )
% 0.22/0.51 = ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_absorb_comp
% 0.22/0.51 thf(fact_5964_pochhammer__absorb__comp,axiom,
% 0.22/0.51 ! [R2: complex,K: nat] :
% 0.22/0.51 ( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K ) )
% 0.22/0.51 = ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_absorb_comp
% 0.22/0.51 thf(fact_5965_pochhammer__absorb__comp,axiom,
% 0.22/0.51 ! [R2: real,K: nat] :
% 0.22/0.51 ( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K ) )
% 0.22/0.51 = ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_absorb_comp
% 0.22/0.51 thf(fact_5966_pochhammer__absorb__comp,axiom,
% 0.22/0.51 ! [R2: rat,K: nat] :
% 0.22/0.51 ( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K ) )
% 0.22/0.51 = ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_absorb_comp
% 0.22/0.51 thf(fact_5967_pochhammer__absorb__comp,axiom,
% 0.22/0.51 ! [R2: int,K: nat] :
% 0.22/0.51 ( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K ) )
% 0.22/0.51 = ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_absorb_comp
% 0.22/0.51 thf(fact_5968_pochhammer__same,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ N )
% 0.22/0.51 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_same
% 0.22/0.51 thf(fact_5969_pochhammer__same,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ N )
% 0.22/0.51 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_same
% 0.22/0.51 thf(fact_5970_pochhammer__same,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ N )
% 0.22/0.51 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_same
% 0.22/0.51 thf(fact_5971_pochhammer__same,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ N )
% 0.22/0.51 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_same
% 0.22/0.51 thf(fact_5972_pochhammer__same,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ N )
% 0.22/0.51 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_same
% 0.22/0.51 thf(fact_5973_signed__take__bit__int__eq__self__iff,axiom,
% 0.22/0.51 ! [N: nat,K: int] :
% 0.22/0.51 ( ( ( bit_ri631733984087533419it_int @ N @ K )
% 0.22/0.51 = K )
% 0.22/0.51 = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 0.22/0.51 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % signed_take_bit_int_eq_self_iff
% 0.22/0.51 thf(fact_5974_signed__take__bit__int__eq__self,axiom,
% 0.22/0.51 ! [N: nat,K: int] :
% 0.22/0.51 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 0.22/0.51 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.51 => ( ( bit_ri631733984087533419it_int @ N @ K )
% 0.22/0.51 = K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % signed_take_bit_int_eq_self
% 0.22/0.51 thf(fact_5975_Bernoulli__inequality,axiom,
% 0.22/0.51 ! [X: real,N: nat] :
% 0.22/0.51 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.22/0.51 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % Bernoulli_inequality
% 0.22/0.51 thf(fact_5976_minus__1__div__exp__eq__int,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.51 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.51
% 0.22/0.51 % minus_1_div_exp_eq_int
% 0.22/0.51 thf(fact_5977_div__pos__neg__trivial,axiom,
% 0.22/0.51 ! [K: int,L: int] :
% 0.22/0.51 ( ( ord_less_int @ zero_zero_int @ K )
% 0.22/0.51 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 0.22/0.51 => ( ( divide_divide_int @ K @ L )
% 0.22/0.51 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % div_pos_neg_trivial
% 0.22/0.51 thf(fact_5978_power__minus1__odd,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.22/0.51 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % power_minus1_odd
% 0.22/0.51 thf(fact_5979_power__minus1__odd,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.22/0.51 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.51
% 0.22/0.51 % power_minus1_odd
% 0.22/0.51 thf(fact_5980_power__minus1__odd,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.22/0.51 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.22/0.51
% 0.22/0.51 % power_minus1_odd
% 0.22/0.51 thf(fact_5981_power__minus1__odd,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.22/0.51 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.51
% 0.22/0.51 % power_minus1_odd
% 0.22/0.51 thf(fact_5982_power__minus1__odd,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.22/0.51 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.22/0.51
% 0.22/0.51 % power_minus1_odd
% 0.22/0.51 thf(fact_5983_pochhammer__minus,axiom,
% 0.22/0.51 ! [B: code_integer,K: nat] :
% 0.22/0.51 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
% 0.22/0.51 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_minus
% 0.22/0.51 thf(fact_5984_pochhammer__minus,axiom,
% 0.22/0.51 ! [B: complex,K: nat] :
% 0.22/0.51 ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
% 0.22/0.51 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_minus
% 0.22/0.51 thf(fact_5985_pochhammer__minus,axiom,
% 0.22/0.51 ! [B: real,K: nat] :
% 0.22/0.51 ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
% 0.22/0.51 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_minus
% 0.22/0.51 thf(fact_5986_pochhammer__minus,axiom,
% 0.22/0.51 ! [B: rat,K: nat] :
% 0.22/0.51 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
% 0.22/0.51 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_minus
% 0.22/0.51 thf(fact_5987_pochhammer__minus,axiom,
% 0.22/0.51 ! [B: int,K: nat] :
% 0.22/0.51 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
% 0.22/0.51 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_minus
% 0.22/0.51 thf(fact_5988_pochhammer__minus_H,axiom,
% 0.22/0.51 ! [B: code_integer,K: nat] :
% 0.22/0.51 ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
% 0.22/0.51 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_minus'
% 0.22/0.51 thf(fact_5989_pochhammer__minus_H,axiom,
% 0.22/0.51 ! [B: complex,K: nat] :
% 0.22/0.51 ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
% 0.22/0.51 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_minus'
% 0.22/0.51 thf(fact_5990_pochhammer__minus_H,axiom,
% 0.22/0.51 ! [B: real,K: nat] :
% 0.22/0.51 ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
% 0.22/0.51 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_minus'
% 0.22/0.51 thf(fact_5991_pochhammer__minus_H,axiom,
% 0.22/0.51 ! [B: rat,K: nat] :
% 0.22/0.51 ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
% 0.22/0.51 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_minus'
% 0.22/0.51 thf(fact_5992_pochhammer__minus_H,axiom,
% 0.22/0.51 ! [B: int,K: nat] :
% 0.22/0.51 ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
% 0.22/0.51 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pochhammer_minus'
% 0.22/0.51 thf(fact_5993_int__bit__induct,axiom,
% 0.22/0.51 ! [P: int > $o,K: int] :
% 0.22/0.51 ( ( P @ zero_zero_int )
% 0.22/0.51 => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.51 => ( ! [K2: int] :
% 0.22/0.51 ( ( P @ K2 )
% 0.22/0.51 => ( ( K2 != zero_zero_int )
% 0.22/0.51 => ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
% 0.22/0.51 => ( ! [K2: int] :
% 0.22/0.51 ( ( P @ K2 )
% 0.22/0.51 => ( ( K2
% 0.22/0.51 != ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.51 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
% 0.22/0.51 => ( P @ K ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % int_bit_induct
% 0.22/0.51 thf(fact_5994_signed__take__bit__int__greater__eq,axiom,
% 0.22/0.51 ! [K: int,N: nat] :
% 0.22/0.51 ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.22/0.51 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % signed_take_bit_int_greater_eq
% 0.22/0.51 thf(fact_5995_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.51 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.22/0.51 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % abs_ln_one_plus_x_minus_x_bound_nonneg
% 0.22/0.51 thf(fact_5996_fact__code,axiom,
% 0.22/0.51 ( semiri773545260158071498ct_rat
% 0.22/0.51 = ( ^ [N4: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % fact_code
% 0.22/0.51 thf(fact_5997_fact__code,axiom,
% 0.22/0.51 ( semiri1406184849735516958ct_int
% 0.22/0.51 = ( ^ [N4: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % fact_code
% 0.22/0.51 thf(fact_5998_fact__code,axiom,
% 0.22/0.51 ( semiri1408675320244567234ct_nat
% 0.22/0.51 = ( ^ [N4: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % fact_code
% 0.22/0.51 thf(fact_5999_fact__code,axiom,
% 0.22/0.51 ( semiri2265585572941072030t_real
% 0.22/0.51 = ( ^ [N4: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % fact_code
% 0.22/0.51 thf(fact_6000_fact__code,axiom,
% 0.22/0.51 ( semiri5044797733671781792omplex
% 0.22/0.51 = ( ^ [N4: nat] : ( semiri8010041392384452111omplex @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % fact_code
% 0.22/0.51 thf(fact_6001_abs__sqrt__wlog,axiom,
% 0.22/0.51 ! [P: code_integer > code_integer > $o,X: code_integer] :
% 0.22/0.51 ( ! [X4: code_integer] :
% 0.22/0.51 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
% 0.22/0.51 => ( P @ X4 @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.51 => ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % abs_sqrt_wlog
% 0.22/0.51 thf(fact_6002_abs__sqrt__wlog,axiom,
% 0.22/0.51 ! [P: rat > rat > $o,X: rat] :
% 0.22/0.51 ( ! [X4: rat] :
% 0.22/0.51 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 0.22/0.51 => ( P @ X4 @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.51 => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % abs_sqrt_wlog
% 0.22/0.51 thf(fact_6003_abs__sqrt__wlog,axiom,
% 0.22/0.51 ! [P: int > int > $o,X: int] :
% 0.22/0.51 ( ! [X4: int] :
% 0.22/0.51 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.22/0.51 => ( P @ X4 @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.51 => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % abs_sqrt_wlog
% 0.22/0.51 thf(fact_6004_abs__sqrt__wlog,axiom,
% 0.22/0.51 ! [P: real > real > $o,X: real] :
% 0.22/0.51 ( ! [X4: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.22/0.51 => ( P @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.51 => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % abs_sqrt_wlog
% 0.22/0.51 thf(fact_6005_sin__coeff__def,axiom,
% 0.22/0.51 ( sin_coeff
% 0.22/0.51 = ( ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N4 ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sin_coeff_def
% 0.22/0.51 thf(fact_6006_cos__coeff__def,axiom,
% 0.22/0.51 ( cos_coeff
% 0.22/0.51 = ( ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N4 ) ) @ zero_zero_real ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % cos_coeff_def
% 0.22/0.51 thf(fact_6007_compl__le__compl__iff,axiom,
% 0.22/0.51 ! [X: set_int,Y4: set_int] :
% 0.22/0.51 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( uminus1532241313380277803et_int @ Y4 ) )
% 0.22/0.51 = ( ord_less_eq_set_int @ Y4 @ X ) ) ).
% 0.22/0.51
% 0.22/0.51 % compl_le_compl_iff
% 0.22/0.51 thf(fact_6008_listsum__bound,axiom,
% 0.22/0.51 ! [Xs: list_complex,F: complex > real,Y4: real] :
% 0.22/0.51 ( ! [X4: complex] :
% 0.22/0.51 ( ( member_complex @ X4 @ ( set_complex2 @ Xs ) )
% 0.22/0.51 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.22/0.51 => ( ord_less_eq_real @ Y4 @ ( foldr_real_real @ plus_plus_real @ ( map_complex_real @ F @ Xs ) @ Y4 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % listsum_bound
% 0.22/0.51 thf(fact_6009_listsum__bound,axiom,
% 0.22/0.51 ! [Xs: list_set_nat,F: set_nat > real,Y4: real] :
% 0.22/0.51 ( ! [X4: set_nat] :
% 0.22/0.51 ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
% 0.22/0.51 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.22/0.51 => ( ord_less_eq_real @ Y4 @ ( foldr_real_real @ plus_plus_real @ ( map_set_nat_real @ F @ Xs ) @ Y4 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % listsum_bound
% 0.22/0.51 thf(fact_6010_listsum__bound,axiom,
% 0.22/0.51 ! [Xs: list_VEBT_VEBT,F: vEBT_VEBT > real,Y4: real] :
% 0.22/0.51 ( ! [X4: vEBT_VEBT] :
% 0.22/0.51 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.22/0.51 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.22/0.51 => ( ord_less_eq_real @ Y4 @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ F @ Xs ) @ Y4 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % listsum_bound
% 0.22/0.51 thf(fact_6011_listsum__bound,axiom,
% 0.22/0.51 ! [Xs: list_nat,F: nat > real,Y4: real] :
% 0.22/0.51 ( ! [X4: nat] :
% 0.22/0.51 ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 0.22/0.51 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.22/0.51 => ( ord_less_eq_real @ Y4 @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ F @ Xs ) @ Y4 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % listsum_bound
% 0.22/0.51 thf(fact_6012_listsum__bound,axiom,
% 0.22/0.51 ! [Xs: list_real,F: real > real,Y4: real] :
% 0.22/0.51 ( ! [X4: real] :
% 0.22/0.51 ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
% 0.22/0.51 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.22/0.51 => ( ord_less_eq_real @ Y4 @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ F @ Xs ) @ Y4 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % listsum_bound
% 0.22/0.51 thf(fact_6013_listsum__bound,axiom,
% 0.22/0.51 ! [Xs: list_int,F: int > real,Y4: real] :
% 0.22/0.51 ( ! [X4: int] :
% 0.22/0.51 ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 0.22/0.51 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.22/0.51 => ( ord_less_eq_real @ Y4 @ ( foldr_real_real @ plus_plus_real @ ( map_int_real @ F @ Xs ) @ Y4 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % listsum_bound
% 0.22/0.51 thf(fact_6014_signed__take__bit__numeral__minus__bit1,axiom,
% 0.22/0.51 ! [L: num,K: num] :
% 0.22/0.51 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.22/0.51 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.22/0.51
% 0.22/0.51 % signed_take_bit_numeral_minus_bit1
% 0.22/0.51 thf(fact_6015_dbl__dec__simps_I4_J,axiom,
% 0.22/0.51 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.51 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_simps(4)
% 0.22/0.51 thf(fact_6016_dbl__dec__simps_I4_J,axiom,
% 0.22/0.51 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.51 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_simps(4)
% 0.22/0.51 thf(fact_6017_dbl__dec__simps_I4_J,axiom,
% 0.22/0.51 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.22/0.51 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_simps(4)
% 0.22/0.51 thf(fact_6018_dbl__dec__simps_I4_J,axiom,
% 0.22/0.51 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.51 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_simps(4)
% 0.22/0.51 thf(fact_6019_dbl__dec__simps_I4_J,axiom,
% 0.22/0.51 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.22/0.51 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_simps(4)
% 0.22/0.51 thf(fact_6020_ComplI,axiom,
% 0.22/0.51 ! [C: complex,A2: set_complex] :
% 0.22/0.51 ( ~ ( member_complex @ C @ A2 )
% 0.22/0.51 => ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % ComplI
% 0.22/0.51 thf(fact_6021_ComplI,axiom,
% 0.22/0.51 ! [C: real,A2: set_real] :
% 0.22/0.51 ( ~ ( member_real @ C @ A2 )
% 0.22/0.51 => ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % ComplI
% 0.22/0.51 thf(fact_6022_ComplI,axiom,
% 0.22/0.51 ! [C: set_nat,A2: set_set_nat] :
% 0.22/0.51 ( ~ ( member_set_nat @ C @ A2 )
% 0.22/0.51 => ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % ComplI
% 0.22/0.51 thf(fact_6023_ComplI,axiom,
% 0.22/0.51 ! [C: nat,A2: set_nat] :
% 0.22/0.51 ( ~ ( member_nat @ C @ A2 )
% 0.22/0.51 => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % ComplI
% 0.22/0.51 thf(fact_6024_ComplI,axiom,
% 0.22/0.51 ! [C: int,A2: set_int] :
% 0.22/0.51 ( ~ ( member_int @ C @ A2 )
% 0.22/0.51 => ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % ComplI
% 0.22/0.51 thf(fact_6025_Compl__iff,axiom,
% 0.22/0.51 ! [C: complex,A2: set_complex] :
% 0.22/0.51 ( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) )
% 0.22/0.51 = ( ~ ( member_complex @ C @ A2 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % Compl_iff
% 0.22/0.51 thf(fact_6026_Compl__iff,axiom,
% 0.22/0.51 ! [C: real,A2: set_real] :
% 0.22/0.51 ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
% 0.22/0.51 = ( ~ ( member_real @ C @ A2 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % Compl_iff
% 0.22/0.51 thf(fact_6027_Compl__iff,axiom,
% 0.22/0.51 ! [C: set_nat,A2: set_set_nat] :
% 0.22/0.51 ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
% 0.22/0.51 = ( ~ ( member_set_nat @ C @ A2 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % Compl_iff
% 0.22/0.51 thf(fact_6028_Compl__iff,axiom,
% 0.22/0.51 ! [C: nat,A2: set_nat] :
% 0.22/0.51 ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
% 0.22/0.51 = ( ~ ( member_nat @ C @ A2 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % Compl_iff
% 0.22/0.51 thf(fact_6029_Compl__iff,axiom,
% 0.22/0.51 ! [C: int,A2: set_int] :
% 0.22/0.51 ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
% 0.22/0.51 = ( ~ ( member_int @ C @ A2 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % Compl_iff
% 0.22/0.51 thf(fact_6030_zdvd1__eq,axiom,
% 0.22/0.51 ! [X: int] :
% 0.22/0.51 ( ( dvd_dvd_int @ X @ one_one_int )
% 0.22/0.51 = ( ( abs_abs_int @ X )
% 0.22/0.51 = one_one_int ) ) ).
% 0.22/0.51
% 0.22/0.51 % zdvd1_eq
% 0.22/0.51 thf(fact_6031_dbl__dec__simps_I3_J,axiom,
% 0.22/0.51 ( ( neg_nu6511756317524482435omplex @ one_one_complex )
% 0.22/0.51 = one_one_complex ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_simps(3)
% 0.22/0.51 thf(fact_6032_dbl__dec__simps_I3_J,axiom,
% 0.22/0.51 ( ( neg_nu6075765906172075777c_real @ one_one_real )
% 0.22/0.51 = one_one_real ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_simps(3)
% 0.22/0.51 thf(fact_6033_dbl__dec__simps_I3_J,axiom,
% 0.22/0.51 ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
% 0.22/0.51 = one_one_rat ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_simps(3)
% 0.22/0.51 thf(fact_6034_dbl__dec__simps_I3_J,axiom,
% 0.22/0.51 ( ( neg_nu3811975205180677377ec_int @ one_one_int )
% 0.22/0.51 = one_one_int ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_simps(3)
% 0.22/0.51 thf(fact_6035_zabs__less__one__iff,axiom,
% 0.22/0.51 ! [Z: int] :
% 0.22/0.51 ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 0.22/0.51 = ( Z = zero_zero_int ) ) ).
% 0.22/0.51
% 0.22/0.51 % zabs_less_one_iff
% 0.22/0.51 thf(fact_6036_in__set__replicate,axiom,
% 0.22/0.51 ! [X: complex,N: nat,Y4: complex] :
% 0.22/0.51 ( ( member_complex @ X @ ( set_complex2 @ ( replicate_complex @ N @ Y4 ) ) )
% 0.22/0.51 = ( ( X = Y4 )
% 0.22/0.51 & ( N != zero_zero_nat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % in_set_replicate
% 0.22/0.51 thf(fact_6037_in__set__replicate,axiom,
% 0.22/0.51 ! [X: set_nat,N: nat,Y4: set_nat] :
% 0.22/0.51 ( ( member_set_nat @ X @ ( set_set_nat2 @ ( replicate_set_nat @ N @ Y4 ) ) )
% 0.22/0.51 = ( ( X = Y4 )
% 0.22/0.51 & ( N != zero_zero_nat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % in_set_replicate
% 0.22/0.51 thf(fact_6038_in__set__replicate,axiom,
% 0.22/0.51 ! [X: nat,N: nat,Y4: nat] :
% 0.22/0.51 ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ Y4 ) ) )
% 0.22/0.51 = ( ( X = Y4 )
% 0.22/0.51 & ( N != zero_zero_nat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % in_set_replicate
% 0.22/0.51 thf(fact_6039_in__set__replicate,axiom,
% 0.22/0.51 ! [X: int,N: nat,Y4: int] :
% 0.22/0.51 ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ Y4 ) ) )
% 0.22/0.51 = ( ( X = Y4 )
% 0.22/0.51 & ( N != zero_zero_nat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % in_set_replicate
% 0.22/0.51 thf(fact_6040_in__set__replicate,axiom,
% 0.22/0.51 ! [X: real,N: nat,Y4: real] :
% 0.22/0.51 ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N @ Y4 ) ) )
% 0.22/0.51 = ( ( X = Y4 )
% 0.22/0.51 & ( N != zero_zero_nat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % in_set_replicate
% 0.22/0.51 thf(fact_6041_in__set__replicate,axiom,
% 0.22/0.51 ! [X: vEBT_VEBT,N: nat,Y4: vEBT_VEBT] :
% 0.22/0.51 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y4 ) ) )
% 0.22/0.51 = ( ( X = Y4 )
% 0.22/0.51 & ( N != zero_zero_nat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % in_set_replicate
% 0.22/0.51 thf(fact_6042_Bex__set__replicate,axiom,
% 0.22/0.51 ! [N: nat,A: nat,P: nat > $o] :
% 0.22/0.51 ( ( ? [X3: nat] :
% 0.22/0.51 ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
% 0.22/0.51 & ( P @ X3 ) ) )
% 0.22/0.51 = ( ( P @ A )
% 0.22/0.51 & ( N != zero_zero_nat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % Bex_set_replicate
% 0.22/0.51 thf(fact_6043_Bex__set__replicate,axiom,
% 0.22/0.51 ! [N: nat,A: int,P: int > $o] :
% 0.22/0.51 ( ( ? [X3: int] :
% 0.22/0.51 ( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
% 0.22/0.51 & ( P @ X3 ) ) )
% 0.22/0.51 = ( ( P @ A )
% 0.22/0.51 & ( N != zero_zero_nat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % Bex_set_replicate
% 0.22/0.51 thf(fact_6044_Bex__set__replicate,axiom,
% 0.22/0.51 ! [N: nat,A: real,P: real > $o] :
% 0.22/0.51 ( ( ? [X3: real] :
% 0.22/0.51 ( ( member_real @ X3 @ ( set_real2 @ ( replicate_real @ N @ A ) ) )
% 0.22/0.51 & ( P @ X3 ) ) )
% 0.22/0.51 = ( ( P @ A )
% 0.22/0.51 & ( N != zero_zero_nat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % Bex_set_replicate
% 0.22/0.51 thf(fact_6045_Bex__set__replicate,axiom,
% 0.22/0.51 ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.22/0.51 ( ( ? [X3: vEBT_VEBT] :
% 0.22/0.51 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
% 0.22/0.51 & ( P @ X3 ) ) )
% 0.22/0.51 = ( ( P @ A )
% 0.22/0.51 & ( N != zero_zero_nat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % Bex_set_replicate
% 0.22/0.51 thf(fact_6046_Ball__set__replicate,axiom,
% 0.22/0.51 ! [N: nat,A: nat,P: nat > $o] :
% 0.22/0.51 ( ( ! [X3: nat] :
% 0.22/0.51 ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
% 0.22/0.51 => ( P @ X3 ) ) )
% 0.22/0.51 = ( ( P @ A )
% 0.22/0.51 | ( N = zero_zero_nat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % Ball_set_replicate
% 0.22/0.51 thf(fact_6047_Ball__set__replicate,axiom,
% 0.22/0.51 ! [N: nat,A: int,P: int > $o] :
% 0.22/0.51 ( ( ! [X3: int] :
% 0.22/0.51 ( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
% 0.22/0.51 => ( P @ X3 ) ) )
% 0.22/0.51 = ( ( P @ A )
% 0.22/0.51 | ( N = zero_zero_nat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % Ball_set_replicate
% 0.22/0.51 thf(fact_6048_Ball__set__replicate,axiom,
% 0.22/0.51 ! [N: nat,A: real,P: real > $o] :
% 0.22/0.51 ( ( ! [X3: real] :
% 0.22/0.51 ( ( member_real @ X3 @ ( set_real2 @ ( replicate_real @ N @ A ) ) )
% 0.22/0.51 => ( P @ X3 ) ) )
% 0.22/0.51 = ( ( P @ A )
% 0.22/0.51 | ( N = zero_zero_nat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % Ball_set_replicate
% 0.22/0.51 thf(fact_6049_Ball__set__replicate,axiom,
% 0.22/0.51 ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.22/0.51 ( ( ! [X3: vEBT_VEBT] :
% 0.22/0.51 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
% 0.22/0.51 => ( P @ X3 ) ) )
% 0.22/0.51 = ( ( P @ A )
% 0.22/0.51 | ( N = zero_zero_nat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % Ball_set_replicate
% 0.22/0.51 thf(fact_6050_pred__numeral__simps_I1_J,axiom,
% 0.22/0.51 ( ( pred_numeral @ one )
% 0.22/0.51 = zero_zero_nat ) ).
% 0.22/0.51
% 0.22/0.51 % pred_numeral_simps(1)
% 0.22/0.51 thf(fact_6051_eq__numeral__Suc,axiom,
% 0.22/0.51 ! [K: num,N: nat] :
% 0.22/0.51 ( ( ( numeral_numeral_nat @ K )
% 0.22/0.51 = ( suc @ N ) )
% 0.22/0.51 = ( ( pred_numeral @ K )
% 0.22/0.51 = N ) ) ).
% 0.22/0.51
% 0.22/0.51 % eq_numeral_Suc
% 0.22/0.51 thf(fact_6052_Suc__eq__numeral,axiom,
% 0.22/0.51 ! [N: nat,K: num] :
% 0.22/0.51 ( ( ( suc @ N )
% 0.22/0.51 = ( numeral_numeral_nat @ K ) )
% 0.22/0.51 = ( N
% 0.22/0.51 = ( pred_numeral @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % Suc_eq_numeral
% 0.22/0.51 thf(fact_6053_sin__coeff__0,axiom,
% 0.22/0.51 ( ( sin_coeff @ zero_zero_nat )
% 0.22/0.51 = zero_zero_real ) ).
% 0.22/0.51
% 0.22/0.51 % sin_coeff_0
% 0.22/0.51 thf(fact_6054_cos__coeff__0,axiom,
% 0.22/0.51 ( ( cos_coeff @ zero_zero_nat )
% 0.22/0.51 = one_one_real ) ).
% 0.22/0.51
% 0.22/0.51 % cos_coeff_0
% 0.22/0.51 thf(fact_6055_pred__numeral__simps_I3_J,axiom,
% 0.22/0.51 ! [K: num] :
% 0.22/0.51 ( ( pred_numeral @ ( bit1 @ K ) )
% 0.22/0.51 = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pred_numeral_simps(3)
% 0.22/0.51 thf(fact_6056_less__numeral__Suc,axiom,
% 0.22/0.51 ! [K: num,N: nat] :
% 0.22/0.51 ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.22/0.51 = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.22/0.51
% 0.22/0.51 % less_numeral_Suc
% 0.22/0.51 thf(fact_6057_less__Suc__numeral,axiom,
% 0.22/0.51 ! [N: nat,K: num] :
% 0.22/0.51 ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.22/0.51 = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % less_Suc_numeral
% 0.22/0.51 thf(fact_6058_le__numeral__Suc,axiom,
% 0.22/0.51 ! [K: num,N: nat] :
% 0.22/0.51 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.22/0.51 = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.22/0.51
% 0.22/0.51 % le_numeral_Suc
% 0.22/0.51 thf(fact_6059_le__Suc__numeral,axiom,
% 0.22/0.51 ! [N: nat,K: num] :
% 0.22/0.51 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.22/0.51 = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % le_Suc_numeral
% 0.22/0.51 thf(fact_6060_diff__numeral__Suc,axiom,
% 0.22/0.51 ! [K: num,N: nat] :
% 0.22/0.51 ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.22/0.51 = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.22/0.51
% 0.22/0.51 % diff_numeral_Suc
% 0.22/0.51 thf(fact_6061_diff__Suc__numeral,axiom,
% 0.22/0.51 ! [N: nat,K: num] :
% 0.22/0.51 ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.22/0.51 = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % diff_Suc_numeral
% 0.22/0.51 thf(fact_6062_dbl__dec__simps_I2_J,axiom,
% 0.22/0.51 ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 0.22/0.51 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_simps(2)
% 0.22/0.51 thf(fact_6063_dbl__dec__simps_I2_J,axiom,
% 0.22/0.51 ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 0.22/0.51 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_simps(2)
% 0.22/0.51 thf(fact_6064_dbl__dec__simps_I2_J,axiom,
% 0.22/0.51 ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 0.22/0.51 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_simps(2)
% 0.22/0.51 thf(fact_6065_dbl__dec__simps_I2_J,axiom,
% 0.22/0.51 ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 0.22/0.51 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_simps(2)
% 0.22/0.51 thf(fact_6066_dbl__dec__simps_I2_J,axiom,
% 0.22/0.51 ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 0.22/0.51 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_simps(2)
% 0.22/0.51 thf(fact_6067_dbl__inc__simps_I1_J,axiom,
% 0.22/0.51 ! [K: num] :
% 0.22/0.51 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.22/0.51 = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_inc_simps(1)
% 0.22/0.51 thf(fact_6068_dbl__inc__simps_I1_J,axiom,
% 0.22/0.51 ! [K: num] :
% 0.22/0.51 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.22/0.51 = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_inc_simps(1)
% 0.22/0.51 thf(fact_6069_dbl__inc__simps_I1_J,axiom,
% 0.22/0.51 ! [K: num] :
% 0.22/0.51 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.22/0.51 = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_inc_simps(1)
% 0.22/0.51 thf(fact_6070_dbl__inc__simps_I1_J,axiom,
% 0.22/0.51 ! [K: num] :
% 0.22/0.51 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.22/0.51 = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_inc_simps(1)
% 0.22/0.51 thf(fact_6071_dbl__inc__simps_I1_J,axiom,
% 0.22/0.51 ! [K: num] :
% 0.22/0.51 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.22/0.51 = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_inc_simps(1)
% 0.22/0.51 thf(fact_6072_dbl__dec__simps_I1_J,axiom,
% 0.22/0.51 ! [K: num] :
% 0.22/0.51 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.22/0.51 = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_simps(1)
% 0.22/0.51 thf(fact_6073_dbl__dec__simps_I1_J,axiom,
% 0.22/0.51 ! [K: num] :
% 0.22/0.51 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.22/0.51 = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_simps(1)
% 0.22/0.51 thf(fact_6074_dbl__dec__simps_I1_J,axiom,
% 0.22/0.51 ! [K: num] :
% 0.22/0.51 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.22/0.51 = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_simps(1)
% 0.22/0.51 thf(fact_6075_dbl__dec__simps_I1_J,axiom,
% 0.22/0.51 ! [K: num] :
% 0.22/0.51 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.22/0.51 = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_simps(1)
% 0.22/0.51 thf(fact_6076_dbl__dec__simps_I1_J,axiom,
% 0.22/0.51 ! [K: num] :
% 0.22/0.51 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.22/0.51 = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_simps(1)
% 0.22/0.51 thf(fact_6077_signed__take__bit__numeral__bit0,axiom,
% 0.22/0.51 ! [L: num,K: num] :
% 0.22/0.51 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.22/0.51 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % signed_take_bit_numeral_bit0
% 0.22/0.51 thf(fact_6078_signed__take__bit__numeral__minus__bit0,axiom,
% 0.22/0.51 ! [L: num,K: num] :
% 0.22/0.51 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.22/0.51 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % signed_take_bit_numeral_minus_bit0
% 0.22/0.51 thf(fact_6079_signed__take__bit__numeral__bit1,axiom,
% 0.22/0.51 ! [L: num,K: num] :
% 0.22/0.51 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.22/0.51 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.22/0.51
% 0.22/0.51 % signed_take_bit_numeral_bit1
% 0.22/0.51 thf(fact_6080_ComplD,axiom,
% 0.22/0.51 ! [C: complex,A2: set_complex] :
% 0.22/0.51 ( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) )
% 0.22/0.51 => ~ ( member_complex @ C @ A2 ) ) ).
% 0.22/0.51
% 0.22/0.51 % ComplD
% 0.22/0.51 thf(fact_6081_ComplD,axiom,
% 0.22/0.51 ! [C: real,A2: set_real] :
% 0.22/0.51 ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
% 0.22/0.51 => ~ ( member_real @ C @ A2 ) ) ).
% 0.22/0.51
% 0.22/0.51 % ComplD
% 0.22/0.51 thf(fact_6082_ComplD,axiom,
% 0.22/0.51 ! [C: set_nat,A2: set_set_nat] :
% 0.22/0.51 ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
% 0.22/0.51 => ~ ( member_set_nat @ C @ A2 ) ) ).
% 0.22/0.51
% 0.22/0.51 % ComplD
% 0.22/0.51 thf(fact_6083_ComplD,axiom,
% 0.22/0.51 ! [C: nat,A2: set_nat] :
% 0.22/0.51 ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
% 0.22/0.51 => ~ ( member_nat @ C @ A2 ) ) ).
% 0.22/0.51
% 0.22/0.51 % ComplD
% 0.22/0.51 thf(fact_6084_ComplD,axiom,
% 0.22/0.51 ! [C: int,A2: set_int] :
% 0.22/0.51 ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
% 0.22/0.51 => ~ ( member_int @ C @ A2 ) ) ).
% 0.22/0.51
% 0.22/0.51 % ComplD
% 0.22/0.51 thf(fact_6085_subset__code_I1_J,axiom,
% 0.22/0.51 ! [Xs: list_complex,B4: set_complex] :
% 0.22/0.51 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ B4 )
% 0.22/0.51 = ( ! [X3: complex] :
% 0.22/0.51 ( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
% 0.22/0.51 => ( member_complex @ X3 @ B4 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % subset_code(1)
% 0.22/0.51 thf(fact_6086_subset__code_I1_J,axiom,
% 0.22/0.51 ! [Xs: list_set_nat,B4: set_set_nat] :
% 0.22/0.51 ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B4 )
% 0.22/0.51 = ( ! [X3: set_nat] :
% 0.22/0.51 ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
% 0.22/0.51 => ( member_set_nat @ X3 @ B4 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % subset_code(1)
% 0.22/0.51 thf(fact_6087_subset__code_I1_J,axiom,
% 0.22/0.51 ! [Xs: list_VEBT_VEBT,B4: set_VEBT_VEBT] :
% 0.22/0.51 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B4 )
% 0.22/0.51 = ( ! [X3: vEBT_VEBT] :
% 0.22/0.51 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.22/0.51 => ( member_VEBT_VEBT @ X3 @ B4 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % subset_code(1)
% 0.22/0.51 thf(fact_6088_subset__code_I1_J,axiom,
% 0.22/0.51 ! [Xs: list_nat,B4: set_nat] :
% 0.22/0.51 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B4 )
% 0.22/0.51 = ( ! [X3: nat] :
% 0.22/0.51 ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 0.22/0.51 => ( member_nat @ X3 @ B4 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % subset_code(1)
% 0.22/0.51 thf(fact_6089_subset__code_I1_J,axiom,
% 0.22/0.51 ! [Xs: list_real,B4: set_real] :
% 0.22/0.51 ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B4 )
% 0.22/0.51 = ( ! [X3: real] :
% 0.22/0.51 ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 0.22/0.51 => ( member_real @ X3 @ B4 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % subset_code(1)
% 0.22/0.51 thf(fact_6090_subset__code_I1_J,axiom,
% 0.22/0.51 ! [Xs: list_int,B4: set_int] :
% 0.22/0.51 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B4 )
% 0.22/0.51 = ( ! [X3: int] :
% 0.22/0.51 ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 0.22/0.51 => ( member_int @ X3 @ B4 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % subset_code(1)
% 0.22/0.51 thf(fact_6091_zdvd__antisym__abs,axiom,
% 0.22/0.51 ! [A: int,B: int] :
% 0.22/0.51 ( ( dvd_dvd_int @ A @ B )
% 0.22/0.51 => ( ( dvd_dvd_int @ B @ A )
% 0.22/0.51 => ( ( abs_abs_int @ A )
% 0.22/0.51 = ( abs_abs_int @ B ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % zdvd_antisym_abs
% 0.22/0.51 thf(fact_6092_abs__zmult__eq__1,axiom,
% 0.22/0.51 ! [M: int,N: int] :
% 0.22/0.51 ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
% 0.22/0.51 = one_one_int )
% 0.22/0.51 => ( ( abs_abs_int @ M )
% 0.22/0.51 = one_one_int ) ) ).
% 0.22/0.51
% 0.22/0.51 % abs_zmult_eq_1
% 0.22/0.51 thf(fact_6093_abs__div,axiom,
% 0.22/0.51 ! [Y4: int,X: int] :
% 0.22/0.51 ( ( dvd_dvd_int @ Y4 @ X )
% 0.22/0.51 => ( ( abs_abs_int @ ( divide_divide_int @ X @ Y4 ) )
% 0.22/0.51 = ( divide_divide_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y4 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % abs_div
% 0.22/0.51 thf(fact_6094_numeral__eq__Suc,axiom,
% 0.22/0.51 ( numeral_numeral_nat
% 0.22/0.51 = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % numeral_eq_Suc
% 0.22/0.51 thf(fact_6095_sin__coeff__Suc,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( sin_coeff @ ( suc @ N ) )
% 0.22/0.51 = ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sin_coeff_Suc
% 0.22/0.51 thf(fact_6096_zabs__def,axiom,
% 0.22/0.51 ( abs_abs_int
% 0.22/0.51 = ( ^ [I2: int] : ( if_int @ ( ord_less_int @ I2 @ zero_zero_int ) @ ( uminus_uminus_int @ I2 ) @ I2 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % zabs_def
% 0.22/0.51 thf(fact_6097_dvd__imp__le__int,axiom,
% 0.22/0.51 ! [I: int,D: int] :
% 0.22/0.51 ( ( I != zero_zero_int )
% 0.22/0.51 => ( ( dvd_dvd_int @ D @ I )
% 0.22/0.51 => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dvd_imp_le_int
% 0.22/0.51 thf(fact_6098_abs__mod__less,axiom,
% 0.22/0.51 ! [L: int,K: int] :
% 0.22/0.51 ( ( L != zero_zero_int )
% 0.22/0.51 => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % abs_mod_less
% 0.22/0.51 thf(fact_6099_cos__coeff__Suc,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( cos_coeff @ ( suc @ N ) )
% 0.22/0.51 = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % cos_coeff_Suc
% 0.22/0.51 thf(fact_6100_pred__numeral__def,axiom,
% 0.22/0.51 ( pred_numeral
% 0.22/0.51 = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % pred_numeral_def
% 0.22/0.51 thf(fact_6101_zdvd__mult__cancel1,axiom,
% 0.22/0.51 ! [M: int,N: int] :
% 0.22/0.51 ( ( M != zero_zero_int )
% 0.22/0.51 => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
% 0.22/0.51 = ( ( abs_abs_int @ N )
% 0.22/0.51 = one_one_int ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % zdvd_mult_cancel1
% 0.22/0.51 thf(fact_6102_fact__numeral,axiom,
% 0.22/0.51 ! [K: num] :
% 0.22/0.51 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
% 0.22/0.51 = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % fact_numeral
% 0.22/0.51 thf(fact_6103_fact__numeral,axiom,
% 0.22/0.51 ! [K: num] :
% 0.22/0.51 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
% 0.22/0.51 = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % fact_numeral
% 0.22/0.51 thf(fact_6104_fact__numeral,axiom,
% 0.22/0.51 ! [K: num] :
% 0.22/0.51 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
% 0.22/0.51 = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % fact_numeral
% 0.22/0.51 thf(fact_6105_fact__numeral,axiom,
% 0.22/0.51 ! [K: num] :
% 0.22/0.51 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
% 0.22/0.51 = ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % fact_numeral
% 0.22/0.51 thf(fact_6106_fact__numeral,axiom,
% 0.22/0.51 ! [K: num] :
% 0.22/0.51 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
% 0.22/0.51 = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % fact_numeral
% 0.22/0.51 thf(fact_6107_even__abs__add__iff,axiom,
% 0.22/0.51 ! [K: int,L: int] :
% 0.22/0.51 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L ) )
% 0.22/0.51 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % even_abs_add_iff
% 0.22/0.51 thf(fact_6108_even__add__abs__iff,axiom,
% 0.22/0.51 ! [K: int,L: int] :
% 0.22/0.51 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L ) ) )
% 0.22/0.51 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % even_add_abs_iff
% 0.22/0.51 thf(fact_6109_nat__intermed__int__val,axiom,
% 0.22/0.51 ! [M: nat,N: nat,F: nat > int,K: int] :
% 0.22/0.51 ( ! [I3: nat] :
% 0.22/0.51 ( ( ( ord_less_eq_nat @ M @ I3 )
% 0.22/0.51 & ( ord_less_nat @ I3 @ N ) )
% 0.22/0.51 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 0.22/0.51 => ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.51 => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 0.22/0.51 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.22/0.51 => ? [I3: nat] :
% 0.22/0.51 ( ( ord_less_eq_nat @ M @ I3 )
% 0.22/0.51 & ( ord_less_eq_nat @ I3 @ N )
% 0.22/0.51 & ( ( F @ I3 )
% 0.22/0.51 = K ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % nat_intermed_int_val
% 0.22/0.51 thf(fact_6110_dbl__dec__def,axiom,
% 0.22/0.51 ( neg_nu6511756317524482435omplex
% 0.22/0.51 = ( ^ [X3: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X3 @ X3 ) @ one_one_complex ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_def
% 0.22/0.51 thf(fact_6111_dbl__dec__def,axiom,
% 0.22/0.51 ( neg_nu6075765906172075777c_real
% 0.22/0.51 = ( ^ [X3: real] : ( minus_minus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_def
% 0.22/0.51 thf(fact_6112_dbl__dec__def,axiom,
% 0.22/0.51 ( neg_nu3179335615603231917ec_rat
% 0.22/0.51 = ( ^ [X3: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_def
% 0.22/0.51 thf(fact_6113_dbl__dec__def,axiom,
% 0.22/0.51 ( neg_nu3811975205180677377ec_int
% 0.22/0.51 = ( ^ [X3: int] : ( minus_minus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dbl_dec_def
% 0.22/0.51 thf(fact_6114_incr__lemma,axiom,
% 0.22/0.51 ! [D: int,Z: int,X: int] :
% 0.22/0.51 ( ( ord_less_int @ zero_zero_int @ D )
% 0.22/0.51 => ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % incr_lemma
% 0.22/0.51 thf(fact_6115_decr__lemma,axiom,
% 0.22/0.51 ! [D: int,X: int,Z: int] :
% 0.22/0.51 ( ( ord_less_int @ zero_zero_int @ D )
% 0.22/0.51 => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% 0.22/0.51
% 0.22/0.51 % decr_lemma
% 0.22/0.51 thf(fact_6116_compl__mono,axiom,
% 0.22/0.51 ! [X: set_int,Y4: set_int] :
% 0.22/0.51 ( ( ord_less_eq_set_int @ X @ Y4 )
% 0.22/0.51 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y4 ) @ ( uminus1532241313380277803et_int @ X ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % compl_mono
% 0.22/0.51 thf(fact_6117_compl__le__swap1,axiom,
% 0.22/0.51 ! [Y4: set_int,X: set_int] :
% 0.22/0.51 ( ( ord_less_eq_set_int @ Y4 @ ( uminus1532241313380277803et_int @ X ) )
% 0.22/0.51 => ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y4 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % compl_le_swap1
% 0.22/0.51 thf(fact_6118_compl__le__swap2,axiom,
% 0.22/0.51 ! [Y4: set_int,X: set_int] :
% 0.22/0.51 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y4 ) @ X )
% 0.22/0.51 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y4 ) ) ).
% 0.22/0.51
% 0.22/0.51 % compl_le_swap2
% 0.22/0.51 thf(fact_6119_nat__ivt__aux,axiom,
% 0.22/0.51 ! [N: nat,F: nat > int,K: int] :
% 0.22/0.51 ( ! [I3: nat] :
% 0.22/0.51 ( ( ord_less_nat @ I3 @ N )
% 0.22/0.51 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 0.22/0.51 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.22/0.51 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.22/0.51 => ? [I3: nat] :
% 0.22/0.51 ( ( ord_less_eq_nat @ I3 @ N )
% 0.22/0.51 & ( ( F @ I3 )
% 0.22/0.51 = K ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % nat_ivt_aux
% 0.22/0.51 thf(fact_6120_complex__mod__minus__le__complex__mod,axiom,
% 0.22/0.51 ! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.22/0.51
% 0.22/0.51 % complex_mod_minus_le_complex_mod
% 0.22/0.51 thf(fact_6121_complex__mod__triangle__ineq2,axiom,
% 0.22/0.51 ! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% 0.22/0.51
% 0.22/0.51 % complex_mod_triangle_ineq2
% 0.22/0.51 thf(fact_6122_nat0__intermed__int__val,axiom,
% 0.22/0.51 ! [N: nat,F: nat > int,K: int] :
% 0.22/0.51 ( ! [I3: nat] :
% 0.22/0.51 ( ( ord_less_nat @ I3 @ N )
% 0.22/0.51 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 0.22/0.51 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.22/0.51 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.22/0.51 => ? [I3: nat] :
% 0.22/0.51 ( ( ord_less_eq_nat @ I3 @ N )
% 0.22/0.51 & ( ( F @ I3 )
% 0.22/0.51 = K ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % nat0_intermed_int_val
% 0.22/0.51 thf(fact_6123_set__n__deg__not__0,axiom,
% 0.22/0.51 ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
% 0.22/0.51 ( ! [X4: vEBT_VEBT] :
% 0.22/0.51 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.22/0.51 => ( vEBT_invar_vebt @ X4 @ N ) )
% 0.22/0.51 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.22/0.51 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.22/0.51 => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % set_n_deg_not_0
% 0.22/0.51 thf(fact_6124_arctan__double,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.22/0.51 => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
% 0.22/0.51 = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % arctan_double
% 0.22/0.51 thf(fact_6125_concat__bit__Suc,axiom,
% 0.22/0.51 ! [N: nat,K: int,L: int] :
% 0.22/0.51 ( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
% 0.22/0.51 = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % concat_bit_Suc
% 0.22/0.51 thf(fact_6126_modulo__int__unfold,axiom,
% 0.22/0.51 ! [L: int,K: int,N: nat,M: nat] :
% 0.22/0.51 ( ( ( ( ( sgn_sgn_int @ L )
% 0.22/0.51 = zero_zero_int )
% 0.22/0.51 | ( ( sgn_sgn_int @ K )
% 0.22/0.51 = zero_zero_int )
% 0.22/0.51 | ( N = zero_zero_nat ) )
% 0.22/0.51 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.22/0.51 = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 0.22/0.51 & ( ~ ( ( ( sgn_sgn_int @ L )
% 0.22/0.51 = zero_zero_int )
% 0.22/0.51 | ( ( sgn_sgn_int @ K )
% 0.22/0.51 = zero_zero_int )
% 0.22/0.51 | ( N = zero_zero_nat ) )
% 0.22/0.51 => ( ( ( ( sgn_sgn_int @ K )
% 0.22/0.51 = ( sgn_sgn_int @ L ) )
% 0.22/0.51 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.22/0.51 = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
% 0.22/0.51 & ( ( ( sgn_sgn_int @ K )
% 0.22/0.51 != ( sgn_sgn_int @ L ) )
% 0.22/0.51 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.22/0.51 = ( times_times_int @ ( sgn_sgn_int @ L )
% 0.22/0.51 @ ( minus_minus_int
% 0.22/0.51 @ ( semiri1314217659103216013at_int
% 0.22/0.51 @ ( times_times_nat @ N
% 0.22/0.51 @ ( zero_n2687167440665602831ol_nat
% 0.22/0.51 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
% 0.22/0.51 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % modulo_int_unfold
% 0.22/0.51 thf(fact_6127_mask__numeral,axiom,
% 0.22/0.51 ! [N: num] :
% 0.22/0.51 ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N ) )
% 0.22/0.51 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % mask_numeral
% 0.22/0.51 thf(fact_6128_mask__numeral,axiom,
% 0.22/0.51 ! [N: num] :
% 0.22/0.51 ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N ) )
% 0.22/0.51 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % mask_numeral
% 0.22/0.51 thf(fact_6129_sgn__sgn,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( sgn_sgn_int @ ( sgn_sgn_int @ A ) )
% 0.22/0.51 = ( sgn_sgn_int @ A ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_sgn
% 0.22/0.51 thf(fact_6130_sgn__sgn,axiom,
% 0.22/0.51 ! [A: real] :
% 0.22/0.51 ( ( sgn_sgn_real @ ( sgn_sgn_real @ A ) )
% 0.22/0.51 = ( sgn_sgn_real @ A ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_sgn
% 0.22/0.51 thf(fact_6131_sgn__sgn,axiom,
% 0.22/0.51 ! [A: complex] :
% 0.22/0.51 ( ( sgn_sgn_complex @ ( sgn_sgn_complex @ A ) )
% 0.22/0.51 = ( sgn_sgn_complex @ A ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_sgn
% 0.22/0.51 thf(fact_6132_sgn__sgn,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( sgn_sgn_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.22/0.51 = ( sgn_sgn_Code_integer @ A ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_sgn
% 0.22/0.51 thf(fact_6133_sgn__sgn,axiom,
% 0.22/0.51 ! [A: rat] :
% 0.22/0.51 ( ( sgn_sgn_rat @ ( sgn_sgn_rat @ A ) )
% 0.22/0.51 = ( sgn_sgn_rat @ A ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_sgn
% 0.22/0.51 thf(fact_6134_mask__nat__positive__iff,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.22/0.51 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.22/0.51
% 0.22/0.51 % mask_nat_positive_iff
% 0.22/0.51 thf(fact_6135_sgn__zero,axiom,
% 0.22/0.51 ( ( sgn_sgn_complex @ zero_zero_complex )
% 0.22/0.51 = zero_zero_complex ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_zero
% 0.22/0.51 thf(fact_6136_sgn__zero,axiom,
% 0.22/0.51 ( ( sgn_sgn_real @ zero_zero_real )
% 0.22/0.51 = zero_zero_real ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_zero
% 0.22/0.51 thf(fact_6137_sgn__0,axiom,
% 0.22/0.51 ( ( sgn_sgn_Code_integer @ zero_z3403309356797280102nteger )
% 0.22/0.51 = zero_z3403309356797280102nteger ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_0
% 0.22/0.51 thf(fact_6138_sgn__0,axiom,
% 0.22/0.51 ( ( sgn_sgn_complex @ zero_zero_complex )
% 0.22/0.51 = zero_zero_complex ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_0
% 0.22/0.51 thf(fact_6139_sgn__0,axiom,
% 0.22/0.51 ( ( sgn_sgn_real @ zero_zero_real )
% 0.22/0.51 = zero_zero_real ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_0
% 0.22/0.51 thf(fact_6140_sgn__0,axiom,
% 0.22/0.51 ( ( sgn_sgn_rat @ zero_zero_rat )
% 0.22/0.51 = zero_zero_rat ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_0
% 0.22/0.51 thf(fact_6141_sgn__0,axiom,
% 0.22/0.51 ( ( sgn_sgn_int @ zero_zero_int )
% 0.22/0.51 = zero_zero_int ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_0
% 0.22/0.51 thf(fact_6142_sgn__1,axiom,
% 0.22/0.51 ( ( sgn_sgn_int @ one_one_int )
% 0.22/0.51 = one_one_int ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_1
% 0.22/0.51 thf(fact_6143_sgn__1,axiom,
% 0.22/0.51 ( ( sgn_sgn_real @ one_one_real )
% 0.22/0.51 = one_one_real ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_1
% 0.22/0.51 thf(fact_6144_sgn__1,axiom,
% 0.22/0.51 ( ( sgn_sgn_complex @ one_one_complex )
% 0.22/0.51 = one_one_complex ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_1
% 0.22/0.51 thf(fact_6145_sgn__1,axiom,
% 0.22/0.51 ( ( sgn_sgn_Code_integer @ one_one_Code_integer )
% 0.22/0.51 = one_one_Code_integer ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_1
% 0.22/0.51 thf(fact_6146_sgn__1,axiom,
% 0.22/0.51 ( ( sgn_sgn_rat @ one_one_rat )
% 0.22/0.51 = one_one_rat ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_1
% 0.22/0.51 thf(fact_6147_sgn__divide,axiom,
% 0.22/0.51 ! [A: complex,B: complex] :
% 0.22/0.51 ( ( sgn_sgn_complex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.22/0.51 = ( divide1717551699836669952omplex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_divide
% 0.22/0.51 thf(fact_6148_sgn__divide,axiom,
% 0.22/0.51 ! [A: real,B: real] :
% 0.22/0.51 ( ( sgn_sgn_real @ ( divide_divide_real @ A @ B ) )
% 0.22/0.51 = ( divide_divide_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_divide
% 0.22/0.51 thf(fact_6149_sgn__divide,axiom,
% 0.22/0.51 ! [A: rat,B: rat] :
% 0.22/0.51 ( ( sgn_sgn_rat @ ( divide_divide_rat @ A @ B ) )
% 0.22/0.51 = ( divide_divide_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_divide
% 0.22/0.51 thf(fact_6150_idom__abs__sgn__class_Osgn__minus,axiom,
% 0.22/0.51 ! [A: real] :
% 0.22/0.51 ( ( sgn_sgn_real @ ( uminus_uminus_real @ A ) )
% 0.22/0.51 = ( uminus_uminus_real @ ( sgn_sgn_real @ A ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % idom_abs_sgn_class.sgn_minus
% 0.22/0.51 thf(fact_6151_idom__abs__sgn__class_Osgn__minus,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( sgn_sgn_int @ ( uminus_uminus_int @ A ) )
% 0.22/0.51 = ( uminus_uminus_int @ ( sgn_sgn_int @ A ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % idom_abs_sgn_class.sgn_minus
% 0.22/0.51 thf(fact_6152_idom__abs__sgn__class_Osgn__minus,axiom,
% 0.22/0.51 ! [A: complex] :
% 0.22/0.51 ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ A ) )
% 0.22/0.51 = ( uminus1482373934393186551omplex @ ( sgn_sgn_complex @ A ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % idom_abs_sgn_class.sgn_minus
% 0.22/0.51 thf(fact_6153_idom__abs__sgn__class_Osgn__minus,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 0.22/0.51 = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ A ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % idom_abs_sgn_class.sgn_minus
% 0.22/0.51 thf(fact_6154_idom__abs__sgn__class_Osgn__minus,axiom,
% 0.22/0.51 ! [A: rat] :
% 0.22/0.51 ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ A ) )
% 0.22/0.51 = ( uminus_uminus_rat @ ( sgn_sgn_rat @ A ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % idom_abs_sgn_class.sgn_minus
% 0.22/0.51 thf(fact_6155_arctan__eq__zero__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ( arctan @ X )
% 0.22/0.51 = zero_zero_real )
% 0.22/0.51 = ( X = zero_zero_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % arctan_eq_zero_iff
% 0.22/0.51 thf(fact_6156_arctan__zero__zero,axiom,
% 0.22/0.51 ( ( arctan @ zero_zero_real )
% 0.22/0.51 = zero_zero_real ) ).
% 0.22/0.51
% 0.22/0.51 % arctan_zero_zero
% 0.22/0.51 thf(fact_6157_concat__bit__0,axiom,
% 0.22/0.51 ! [K: int,L: int] :
% 0.22/0.51 ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
% 0.22/0.51 = L ) ).
% 0.22/0.51
% 0.22/0.51 % concat_bit_0
% 0.22/0.51 thf(fact_6158_sgn__less,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 0.22/0.51 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_less
% 0.22/0.51 thf(fact_6159_sgn__less,axiom,
% 0.22/0.51 ! [A: real] :
% 0.22/0.51 ( ( ord_less_real @ ( sgn_sgn_real @ A ) @ zero_zero_real )
% 0.22/0.51 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_less
% 0.22/0.51 thf(fact_6160_sgn__less,axiom,
% 0.22/0.51 ! [A: rat] :
% 0.22/0.51 ( ( ord_less_rat @ ( sgn_sgn_rat @ A ) @ zero_zero_rat )
% 0.22/0.51 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_less
% 0.22/0.51 thf(fact_6161_sgn__less,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( ord_less_int @ ( sgn_sgn_int @ A ) @ zero_zero_int )
% 0.22/0.51 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_less
% 0.22/0.51 thf(fact_6162_sgn__greater,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A ) )
% 0.22/0.51 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_greater
% 0.22/0.51 thf(fact_6163_sgn__greater,axiom,
% 0.22/0.51 ! [A: real] :
% 0.22/0.51 ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A ) )
% 0.22/0.51 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_greater
% 0.22/0.51 thf(fact_6164_sgn__greater,axiom,
% 0.22/0.51 ! [A: rat] :
% 0.22/0.51 ( ( ord_less_rat @ zero_zero_rat @ ( sgn_sgn_rat @ A ) )
% 0.22/0.51 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_greater
% 0.22/0.51 thf(fact_6165_sgn__greater,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A ) )
% 0.22/0.51 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_greater
% 0.22/0.51 thf(fact_6166_divide__sgn,axiom,
% 0.22/0.51 ! [A: real,B: real] :
% 0.22/0.51 ( ( divide_divide_real @ A @ ( sgn_sgn_real @ B ) )
% 0.22/0.51 = ( times_times_real @ A @ ( sgn_sgn_real @ B ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % divide_sgn
% 0.22/0.51 thf(fact_6167_divide__sgn,axiom,
% 0.22/0.51 ! [A: rat,B: rat] :
% 0.22/0.51 ( ( divide_divide_rat @ A @ ( sgn_sgn_rat @ B ) )
% 0.22/0.51 = ( times_times_rat @ A @ ( sgn_sgn_rat @ B ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % divide_sgn
% 0.22/0.51 thf(fact_6168_mask__0,axiom,
% 0.22/0.51 ( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
% 0.22/0.51 = zero_zero_nat ) ).
% 0.22/0.51
% 0.22/0.51 % mask_0
% 0.22/0.51 thf(fact_6169_mask__0,axiom,
% 0.22/0.51 ( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
% 0.22/0.51 = zero_zero_int ) ).
% 0.22/0.51
% 0.22/0.51 % mask_0
% 0.22/0.51 thf(fact_6170_mask__eq__0__iff,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( ( bit_se2002935070580805687sk_nat @ N )
% 0.22/0.51 = zero_zero_nat )
% 0.22/0.51 = ( N = zero_zero_nat ) ) ).
% 0.22/0.51
% 0.22/0.51 % mask_eq_0_iff
% 0.22/0.51 thf(fact_6171_mask__eq__0__iff,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( ( bit_se2000444600071755411sk_int @ N )
% 0.22/0.51 = zero_zero_int )
% 0.22/0.51 = ( N = zero_zero_nat ) ) ).
% 0.22/0.51
% 0.22/0.51 % mask_eq_0_iff
% 0.22/0.51 thf(fact_6172_arctan__less__zero__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_real @ ( arctan @ X ) @ zero_zero_real )
% 0.22/0.51 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % arctan_less_zero_iff
% 0.22/0.51 thf(fact_6173_zero__less__arctan__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_real @ zero_zero_real @ ( arctan @ X ) )
% 0.22/0.51 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.22/0.51
% 0.22/0.51 % zero_less_arctan_iff
% 0.22/0.51 thf(fact_6174_zero__le__arctan__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X ) )
% 0.22/0.51 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.22/0.51
% 0.22/0.51 % zero_le_arctan_iff
% 0.22/0.51 thf(fact_6175_arctan__le__zero__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ ( arctan @ X ) @ zero_zero_real )
% 0.22/0.51 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % arctan_le_zero_iff
% 0.22/0.51 thf(fact_6176_concat__bit__nonnegative__iff,axiom,
% 0.22/0.51 ! [N: nat,K: int,L: int] :
% 0.22/0.51 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
% 0.22/0.51 = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).
% 0.22/0.51
% 0.22/0.51 % concat_bit_nonnegative_iff
% 0.22/0.51 thf(fact_6177_concat__bit__negative__iff,axiom,
% 0.22/0.51 ! [N: nat,K: int,L: int] :
% 0.22/0.51 ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
% 0.22/0.51 = ( ord_less_int @ L @ zero_zero_int ) ) ).
% 0.22/0.51
% 0.22/0.51 % concat_bit_negative_iff
% 0.22/0.51 thf(fact_6178_sgn__pos,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 0.22/0.51 => ( ( sgn_sgn_Code_integer @ A )
% 0.22/0.51 = one_one_Code_integer ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_pos
% 0.22/0.51 thf(fact_6179_sgn__pos,axiom,
% 0.22/0.51 ! [A: real] :
% 0.22/0.51 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.51 => ( ( sgn_sgn_real @ A )
% 0.22/0.51 = one_one_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_pos
% 0.22/0.51 thf(fact_6180_sgn__pos,axiom,
% 0.22/0.51 ! [A: rat] :
% 0.22/0.51 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.22/0.51 => ( ( sgn_sgn_rat @ A )
% 0.22/0.51 = one_one_rat ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_pos
% 0.22/0.51 thf(fact_6181_sgn__pos,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( ord_less_int @ zero_zero_int @ A )
% 0.22/0.51 => ( ( sgn_sgn_int @ A )
% 0.22/0.51 = one_one_int ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_pos
% 0.22/0.51 thf(fact_6182_abs__sgn__eq__1,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( A != zero_z3403309356797280102nteger )
% 0.22/0.51 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.22/0.51 = one_one_Code_integer ) ) ).
% 0.22/0.51
% 0.22/0.51 % abs_sgn_eq_1
% 0.22/0.51 thf(fact_6183_abs__sgn__eq__1,axiom,
% 0.22/0.51 ! [A: real] :
% 0.22/0.51 ( ( A != zero_zero_real )
% 0.22/0.51 => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.22/0.51 = one_one_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % abs_sgn_eq_1
% 0.22/0.51 thf(fact_6184_abs__sgn__eq__1,axiom,
% 0.22/0.51 ! [A: rat] :
% 0.22/0.51 ( ( A != zero_zero_rat )
% 0.22/0.51 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.22/0.51 = one_one_rat ) ) ).
% 0.22/0.51
% 0.22/0.51 % abs_sgn_eq_1
% 0.22/0.51 thf(fact_6185_abs__sgn__eq__1,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( A != zero_zero_int )
% 0.22/0.51 => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.22/0.51 = one_one_int ) ) ).
% 0.22/0.51
% 0.22/0.51 % abs_sgn_eq_1
% 0.22/0.51 thf(fact_6186_sgn__mult__self__eq,axiom,
% 0.22/0.51 ! [A: real] :
% 0.22/0.51 ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ A ) )
% 0.22/0.51 = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_mult_self_eq
% 0.22/0.51 thf(fact_6187_sgn__mult__self__eq,axiom,
% 0.22/0.51 ! [A: rat] :
% 0.22/0.51 ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ A ) )
% 0.22/0.51 = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_mult_self_eq
% 0.22/0.51 thf(fact_6188_sgn__mult__self__eq,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ A ) )
% 0.22/0.51 = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_mult_self_eq
% 0.22/0.51 thf(fact_6189_sgn__mult__self__eq,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
% 0.22/0.51 = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_mult_self_eq
% 0.22/0.51 thf(fact_6190_mask__Suc__0,axiom,
% 0.22/0.51 ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
% 0.22/0.51 = one_one_nat ) ).
% 0.22/0.51
% 0.22/0.51 % mask_Suc_0
% 0.22/0.51 thf(fact_6191_mask__Suc__0,axiom,
% 0.22/0.51 ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
% 0.22/0.51 = one_one_int ) ).
% 0.22/0.51
% 0.22/0.51 % mask_Suc_0
% 0.22/0.51 thf(fact_6192_sgn__abs,axiom,
% 0.22/0.51 ! [A: complex] :
% 0.22/0.51 ( ( abs_abs_complex @ ( sgn_sgn_complex @ A ) )
% 0.22/0.51 = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_abs
% 0.22/0.51 thf(fact_6193_sgn__abs,axiom,
% 0.22/0.51 ! [A: real] :
% 0.22/0.51 ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.22/0.51 = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_abs
% 0.22/0.51 thf(fact_6194_sgn__abs,axiom,
% 0.22/0.51 ! [A: rat] :
% 0.22/0.51 ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.22/0.51 = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_abs
% 0.22/0.51 thf(fact_6195_sgn__abs,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.22/0.51 = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_abs
% 0.22/0.51 thf(fact_6196_sgn__abs,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.22/0.51 = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_abs
% 0.22/0.51 thf(fact_6197_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.22/0.51 ! [A: complex] :
% 0.22/0.51 ( ( sgn_sgn_complex @ ( abs_abs_complex @ A ) )
% 0.22/0.51 = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % idom_abs_sgn_class.abs_sgn
% 0.22/0.51 thf(fact_6198_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.22/0.51 ! [A: real] :
% 0.22/0.51 ( ( sgn_sgn_real @ ( abs_abs_real @ A ) )
% 0.22/0.51 = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % idom_abs_sgn_class.abs_sgn
% 0.22/0.51 thf(fact_6199_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.22/0.51 ! [A: rat] :
% 0.22/0.51 ( ( sgn_sgn_rat @ ( abs_abs_rat @ A ) )
% 0.22/0.51 = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % idom_abs_sgn_class.abs_sgn
% 0.22/0.51 thf(fact_6200_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( sgn_sgn_int @ ( abs_abs_int @ A ) )
% 0.22/0.51 = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % idom_abs_sgn_class.abs_sgn
% 0.22/0.51 thf(fact_6201_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( sgn_sgn_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.22/0.51 = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % idom_abs_sgn_class.abs_sgn
% 0.22/0.51 thf(fact_6202_sgn__mult__dvd__iff,axiom,
% 0.22/0.51 ! [R2: int,L: int,K: int] :
% 0.22/0.51 ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K )
% 0.22/0.51 = ( ( dvd_dvd_int @ L @ K )
% 0.22/0.51 & ( ( R2 = zero_zero_int )
% 0.22/0.51 => ( K = zero_zero_int ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_mult_dvd_iff
% 0.22/0.51 thf(fact_6203_mult__sgn__dvd__iff,axiom,
% 0.22/0.51 ! [L: int,R2: int,K: int] :
% 0.22/0.51 ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K )
% 0.22/0.51 = ( ( dvd_dvd_int @ L @ K )
% 0.22/0.51 & ( ( R2 = zero_zero_int )
% 0.22/0.51 => ( K = zero_zero_int ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % mult_sgn_dvd_iff
% 0.22/0.51 thf(fact_6204_dvd__sgn__mult__iff,axiom,
% 0.22/0.51 ! [L: int,R2: int,K: int] :
% 0.22/0.51 ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
% 0.22/0.51 = ( ( dvd_dvd_int @ L @ K )
% 0.22/0.51 | ( R2 = zero_zero_int ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dvd_sgn_mult_iff
% 0.22/0.51 thf(fact_6205_dvd__mult__sgn__iff,axiom,
% 0.22/0.51 ! [L: int,K: int,R2: int] :
% 0.22/0.51 ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
% 0.22/0.51 = ( ( dvd_dvd_int @ L @ K )
% 0.22/0.51 | ( R2 = zero_zero_int ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % dvd_mult_sgn_iff
% 0.22/0.51 thf(fact_6206_sgn__neg,axiom,
% 0.22/0.51 ! [A: real] :
% 0.22/0.51 ( ( ord_less_real @ A @ zero_zero_real )
% 0.22/0.51 => ( ( sgn_sgn_real @ A )
% 0.22/0.51 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_neg
% 0.22/0.51 thf(fact_6207_sgn__neg,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( ord_less_int @ A @ zero_zero_int )
% 0.22/0.51 => ( ( sgn_sgn_int @ A )
% 0.22/0.51 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_neg
% 0.22/0.51 thf(fact_6208_sgn__neg,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 0.22/0.51 => ( ( sgn_sgn_Code_integer @ A )
% 0.22/0.51 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_neg
% 0.22/0.51 thf(fact_6209_sgn__neg,axiom,
% 0.22/0.51 ! [A: rat] :
% 0.22/0.51 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.22/0.51 => ( ( sgn_sgn_rat @ A )
% 0.22/0.51 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_neg
% 0.22/0.51 thf(fact_6210_sgn__of__nat,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( sgn_sgn_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.22/0.51 = ( zero_n3304061248610475627l_real @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_of_nat
% 0.22/0.51 thf(fact_6211_sgn__of__nat,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( sgn_sgn_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.22/0.51 = ( zero_n2052037380579107095ol_rat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_of_nat
% 0.22/0.51 thf(fact_6212_sgn__of__nat,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( sgn_sgn_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.22/0.51 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_of_nat
% 0.22/0.51 thf(fact_6213_sgn__of__nat,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( sgn_sgn_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 0.22/0.51 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_of_nat
% 0.22/0.51 thf(fact_6214_of__nat__mask__eq,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.22/0.51 = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 0.22/0.51
% 0.22/0.51 % of_nat_mask_eq
% 0.22/0.51 thf(fact_6215_of__nat__mask__eq,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.22/0.51 = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.22/0.51
% 0.22/0.51 % of_nat_mask_eq
% 0.22/0.51 thf(fact_6216_size__neq__size__imp__neq,axiom,
% 0.22/0.51 ! [X: list_real,Y4: list_real] :
% 0.22/0.51 ( ( ( size_size_list_real @ X )
% 0.22/0.51 != ( size_size_list_real @ Y4 ) )
% 0.22/0.51 => ( X != Y4 ) ) ).
% 0.22/0.51
% 0.22/0.51 % size_neq_size_imp_neq
% 0.22/0.51 thf(fact_6217_size__neq__size__imp__neq,axiom,
% 0.22/0.51 ! [X: list_o,Y4: list_o] :
% 0.22/0.51 ( ( ( size_size_list_o @ X )
% 0.22/0.51 != ( size_size_list_o @ Y4 ) )
% 0.22/0.51 => ( X != Y4 ) ) ).
% 0.22/0.51
% 0.22/0.51 % size_neq_size_imp_neq
% 0.22/0.51 thf(fact_6218_size__neq__size__imp__neq,axiom,
% 0.22/0.51 ! [X: list_nat,Y4: list_nat] :
% 0.22/0.51 ( ( ( size_size_list_nat @ X )
% 0.22/0.51 != ( size_size_list_nat @ Y4 ) )
% 0.22/0.51 => ( X != Y4 ) ) ).
% 0.22/0.51
% 0.22/0.51 % size_neq_size_imp_neq
% 0.22/0.51 thf(fact_6219_size__neq__size__imp__neq,axiom,
% 0.22/0.51 ! [X: list_int,Y4: list_int] :
% 0.22/0.51 ( ( ( size_size_list_int @ X )
% 0.22/0.51 != ( size_size_list_int @ Y4 ) )
% 0.22/0.51 => ( X != Y4 ) ) ).
% 0.22/0.51
% 0.22/0.51 % size_neq_size_imp_neq
% 0.22/0.51 thf(fact_6220_size__neq__size__imp__neq,axiom,
% 0.22/0.51 ! [X: num,Y4: num] :
% 0.22/0.51 ( ( ( size_size_num @ X )
% 0.22/0.51 != ( size_size_num @ Y4 ) )
% 0.22/0.51 => ( X != Y4 ) ) ).
% 0.22/0.51
% 0.22/0.51 % size_neq_size_imp_neq
% 0.22/0.51 thf(fact_6221_sgn__zero__iff,axiom,
% 0.22/0.51 ! [X: complex] :
% 0.22/0.51 ( ( ( sgn_sgn_complex @ X )
% 0.22/0.51 = zero_zero_complex )
% 0.22/0.51 = ( X = zero_zero_complex ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_zero_iff
% 0.22/0.51 thf(fact_6222_sgn__zero__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ( sgn_sgn_real @ X )
% 0.22/0.51 = zero_zero_real )
% 0.22/0.51 = ( X = zero_zero_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_zero_iff
% 0.22/0.51 thf(fact_6223_sgn__0__0,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( ( sgn_sgn_Code_integer @ A )
% 0.22/0.51 = zero_z3403309356797280102nteger )
% 0.22/0.51 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_0_0
% 0.22/0.51 thf(fact_6224_sgn__0__0,axiom,
% 0.22/0.51 ! [A: real] :
% 0.22/0.51 ( ( ( sgn_sgn_real @ A )
% 0.22/0.51 = zero_zero_real )
% 0.22/0.51 = ( A = zero_zero_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_0_0
% 0.22/0.51 thf(fact_6225_sgn__0__0,axiom,
% 0.22/0.51 ! [A: rat] :
% 0.22/0.51 ( ( ( sgn_sgn_rat @ A )
% 0.22/0.51 = zero_zero_rat )
% 0.22/0.51 = ( A = zero_zero_rat ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_0_0
% 0.22/0.51 thf(fact_6226_sgn__0__0,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( ( sgn_sgn_int @ A )
% 0.22/0.51 = zero_zero_int )
% 0.22/0.51 = ( A = zero_zero_int ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_0_0
% 0.22/0.51 thf(fact_6227_sgn__eq__0__iff,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( ( sgn_sgn_Code_integer @ A )
% 0.22/0.51 = zero_z3403309356797280102nteger )
% 0.22/0.51 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_eq_0_iff
% 0.22/0.51 thf(fact_6228_sgn__eq__0__iff,axiom,
% 0.22/0.51 ! [A: complex] :
% 0.22/0.51 ( ( ( sgn_sgn_complex @ A )
% 0.22/0.51 = zero_zero_complex )
% 0.22/0.51 = ( A = zero_zero_complex ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_eq_0_iff
% 0.22/0.51 thf(fact_6229_sgn__eq__0__iff,axiom,
% 0.22/0.51 ! [A: real] :
% 0.22/0.51 ( ( ( sgn_sgn_real @ A )
% 0.22/0.51 = zero_zero_real )
% 0.22/0.51 = ( A = zero_zero_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_eq_0_iff
% 0.22/0.51 thf(fact_6230_sgn__eq__0__iff,axiom,
% 0.22/0.51 ! [A: rat] :
% 0.22/0.51 ( ( ( sgn_sgn_rat @ A )
% 0.22/0.51 = zero_zero_rat )
% 0.22/0.51 = ( A = zero_zero_rat ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_eq_0_iff
% 0.22/0.51 thf(fact_6231_sgn__eq__0__iff,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( ( sgn_sgn_int @ A )
% 0.22/0.51 = zero_zero_int )
% 0.22/0.51 = ( A = zero_zero_int ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_eq_0_iff
% 0.22/0.51 thf(fact_6232_Real__Vector__Spaces_Osgn__mult,axiom,
% 0.22/0.51 ! [X: complex,Y4: complex] :
% 0.22/0.51 ( ( sgn_sgn_complex @ ( times_times_complex @ X @ Y4 ) )
% 0.22/0.51 = ( times_times_complex @ ( sgn_sgn_complex @ X ) @ ( sgn_sgn_complex @ Y4 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % Real_Vector_Spaces.sgn_mult
% 0.22/0.51 thf(fact_6233_Real__Vector__Spaces_Osgn__mult,axiom,
% 0.22/0.51 ! [X: real,Y4: real] :
% 0.22/0.51 ( ( sgn_sgn_real @ ( times_times_real @ X @ Y4 ) )
% 0.22/0.51 = ( times_times_real @ ( sgn_sgn_real @ X ) @ ( sgn_sgn_real @ Y4 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % Real_Vector_Spaces.sgn_mult
% 0.22/0.51 thf(fact_6234_sgn__mult,axiom,
% 0.22/0.51 ! [A: complex,B: complex] :
% 0.22/0.51 ( ( sgn_sgn_complex @ ( times_times_complex @ A @ B ) )
% 0.22/0.51 = ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_mult
% 0.22/0.51 thf(fact_6235_sgn__mult,axiom,
% 0.22/0.51 ! [A: code_integer,B: code_integer] :
% 0.22/0.51 ( ( sgn_sgn_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.22/0.51 = ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ B ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_mult
% 0.22/0.51 thf(fact_6236_sgn__mult,axiom,
% 0.22/0.51 ! [A: real,B: real] :
% 0.22/0.51 ( ( sgn_sgn_real @ ( times_times_real @ A @ B ) )
% 0.22/0.51 = ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_mult
% 0.22/0.51 thf(fact_6237_sgn__mult,axiom,
% 0.22/0.51 ! [A: rat,B: rat] :
% 0.22/0.51 ( ( sgn_sgn_rat @ ( times_times_rat @ A @ B ) )
% 0.22/0.51 = ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_mult
% 0.22/0.51 thf(fact_6238_sgn__mult,axiom,
% 0.22/0.51 ! [A: int,B: int] :
% 0.22/0.51 ( ( sgn_sgn_int @ ( times_times_int @ A @ B ) )
% 0.22/0.51 = ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_mult
% 0.22/0.51 thf(fact_6239_same__sgn__sgn__add,axiom,
% 0.22/0.51 ! [B: code_integer,A: code_integer] :
% 0.22/0.51 ( ( ( sgn_sgn_Code_integer @ B )
% 0.22/0.51 = ( sgn_sgn_Code_integer @ A ) )
% 0.22/0.51 => ( ( sgn_sgn_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.22/0.51 = ( sgn_sgn_Code_integer @ A ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % same_sgn_sgn_add
% 0.22/0.51 thf(fact_6240_same__sgn__sgn__add,axiom,
% 0.22/0.51 ! [B: real,A: real] :
% 0.22/0.51 ( ( ( sgn_sgn_real @ B )
% 0.22/0.51 = ( sgn_sgn_real @ A ) )
% 0.22/0.51 => ( ( sgn_sgn_real @ ( plus_plus_real @ A @ B ) )
% 0.22/0.51 = ( sgn_sgn_real @ A ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % same_sgn_sgn_add
% 0.22/0.51 thf(fact_6241_same__sgn__sgn__add,axiom,
% 0.22/0.51 ! [B: rat,A: rat] :
% 0.22/0.51 ( ( ( sgn_sgn_rat @ B )
% 0.22/0.51 = ( sgn_sgn_rat @ A ) )
% 0.22/0.51 => ( ( sgn_sgn_rat @ ( plus_plus_rat @ A @ B ) )
% 0.22/0.51 = ( sgn_sgn_rat @ A ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % same_sgn_sgn_add
% 0.22/0.51 thf(fact_6242_same__sgn__sgn__add,axiom,
% 0.22/0.51 ! [B: int,A: int] :
% 0.22/0.51 ( ( ( sgn_sgn_int @ B )
% 0.22/0.51 = ( sgn_sgn_int @ A ) )
% 0.22/0.51 => ( ( sgn_sgn_int @ ( plus_plus_int @ A @ B ) )
% 0.22/0.51 = ( sgn_sgn_int @ A ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % same_sgn_sgn_add
% 0.22/0.51 thf(fact_6243_div__eq__sgn__abs,axiom,
% 0.22/0.51 ! [K: int,L: int] :
% 0.22/0.51 ( ( ( sgn_sgn_int @ K )
% 0.22/0.51 = ( sgn_sgn_int @ L ) )
% 0.22/0.51 => ( ( divide_divide_int @ K @ L )
% 0.22/0.51 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % div_eq_sgn_abs
% 0.22/0.51 thf(fact_6244_arctan__monotone_H,axiom,
% 0.22/0.51 ! [X: real,Y4: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ X @ Y4 )
% 0.22/0.51 => ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y4 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % arctan_monotone'
% 0.22/0.51 thf(fact_6245_arctan__le__iff,axiom,
% 0.22/0.51 ! [X: real,Y4: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y4 ) )
% 0.22/0.51 = ( ord_less_eq_real @ X @ Y4 ) ) ).
% 0.22/0.51
% 0.22/0.51 % arctan_le_iff
% 0.22/0.51 thf(fact_6246_less__eq__mask,axiom,
% 0.22/0.51 ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 0.22/0.51
% 0.22/0.51 % less_eq_mask
% 0.22/0.51 thf(fact_6247_concat__bit__assoc,axiom,
% 0.22/0.51 ! [N: nat,K: int,M: nat,L: int,R2: int] :
% 0.22/0.51 ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L @ R2 ) )
% 0.22/0.51 = ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R2 ) ) ).
% 0.22/0.51
% 0.22/0.51 % concat_bit_assoc
% 0.22/0.51 thf(fact_6248_sgn__not__eq__imp,axiom,
% 0.22/0.51 ! [B: real,A: real] :
% 0.22/0.51 ( ( ( sgn_sgn_real @ B )
% 0.22/0.51 != ( sgn_sgn_real @ A ) )
% 0.22/0.51 => ( ( ( sgn_sgn_real @ A )
% 0.22/0.51 != zero_zero_real )
% 0.22/0.51 => ( ( ( sgn_sgn_real @ B )
% 0.22/0.51 != zero_zero_real )
% 0.22/0.51 => ( ( sgn_sgn_real @ A )
% 0.22/0.51 = ( uminus_uminus_real @ ( sgn_sgn_real @ B ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_not_eq_imp
% 0.22/0.51 thf(fact_6249_sgn__not__eq__imp,axiom,
% 0.22/0.51 ! [B: int,A: int] :
% 0.22/0.51 ( ( ( sgn_sgn_int @ B )
% 0.22/0.51 != ( sgn_sgn_int @ A ) )
% 0.22/0.51 => ( ( ( sgn_sgn_int @ A )
% 0.22/0.51 != zero_zero_int )
% 0.22/0.51 => ( ( ( sgn_sgn_int @ B )
% 0.22/0.51 != zero_zero_int )
% 0.22/0.51 => ( ( sgn_sgn_int @ A )
% 0.22/0.51 = ( uminus_uminus_int @ ( sgn_sgn_int @ B ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_not_eq_imp
% 0.22/0.51 thf(fact_6250_sgn__not__eq__imp,axiom,
% 0.22/0.51 ! [B: code_integer,A: code_integer] :
% 0.22/0.51 ( ( ( sgn_sgn_Code_integer @ B )
% 0.22/0.51 != ( sgn_sgn_Code_integer @ A ) )
% 0.22/0.51 => ( ( ( sgn_sgn_Code_integer @ A )
% 0.22/0.51 != zero_z3403309356797280102nteger )
% 0.22/0.51 => ( ( ( sgn_sgn_Code_integer @ B )
% 0.22/0.51 != zero_z3403309356797280102nteger )
% 0.22/0.51 => ( ( sgn_sgn_Code_integer @ A )
% 0.22/0.51 = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ B ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_not_eq_imp
% 0.22/0.51 thf(fact_6251_sgn__not__eq__imp,axiom,
% 0.22/0.51 ! [B: rat,A: rat] :
% 0.22/0.51 ( ( ( sgn_sgn_rat @ B )
% 0.22/0.51 != ( sgn_sgn_rat @ A ) )
% 0.22/0.51 => ( ( ( sgn_sgn_rat @ A )
% 0.22/0.51 != zero_zero_rat )
% 0.22/0.51 => ( ( ( sgn_sgn_rat @ B )
% 0.22/0.51 != zero_zero_rat )
% 0.22/0.51 => ( ( sgn_sgn_rat @ A )
% 0.22/0.51 = ( uminus_uminus_rat @ ( sgn_sgn_rat @ B ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_not_eq_imp
% 0.22/0.51 thf(fact_6252_sgn__minus__1,axiom,
% 0.22/0.51 ( ( sgn_sgn_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.51 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_minus_1
% 0.22/0.51 thf(fact_6253_sgn__minus__1,axiom,
% 0.22/0.51 ( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.51 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_minus_1
% 0.22/0.51 thf(fact_6254_sgn__minus__1,axiom,
% 0.22/0.51 ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.22/0.51 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_minus_1
% 0.22/0.51 thf(fact_6255_sgn__minus__1,axiom,
% 0.22/0.51 ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.51 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_minus_1
% 0.22/0.51 thf(fact_6256_sgn__minus__1,axiom,
% 0.22/0.51 ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.22/0.51 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_minus_1
% 0.22/0.51 thf(fact_6257_linordered__idom__class_Oabs__sgn,axiom,
% 0.22/0.51 ( abs_abs_Code_integer
% 0.22/0.51 = ( ^ [K3: code_integer] : ( times_3573771949741848930nteger @ K3 @ ( sgn_sgn_Code_integer @ K3 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % linordered_idom_class.abs_sgn
% 0.22/0.51 thf(fact_6258_linordered__idom__class_Oabs__sgn,axiom,
% 0.22/0.51 ( abs_abs_real
% 0.22/0.51 = ( ^ [K3: real] : ( times_times_real @ K3 @ ( sgn_sgn_real @ K3 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % linordered_idom_class.abs_sgn
% 0.22/0.51 thf(fact_6259_linordered__idom__class_Oabs__sgn,axiom,
% 0.22/0.51 ( abs_abs_rat
% 0.22/0.51 = ( ^ [K3: rat] : ( times_times_rat @ K3 @ ( sgn_sgn_rat @ K3 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % linordered_idom_class.abs_sgn
% 0.22/0.51 thf(fact_6260_linordered__idom__class_Oabs__sgn,axiom,
% 0.22/0.51 ( abs_abs_int
% 0.22/0.51 = ( ^ [K3: int] : ( times_times_int @ K3 @ ( sgn_sgn_int @ K3 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % linordered_idom_class.abs_sgn
% 0.22/0.51 thf(fact_6261_abs__mult__sgn,axiom,
% 0.22/0.51 ! [A: complex] :
% 0.22/0.51 ( ( times_times_complex @ ( abs_abs_complex @ A ) @ ( sgn_sgn_complex @ A ) )
% 0.22/0.51 = A ) ).
% 0.22/0.51
% 0.22/0.51 % abs_mult_sgn
% 0.22/0.51 thf(fact_6262_abs__mult__sgn,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
% 0.22/0.51 = A ) ).
% 0.22/0.51
% 0.22/0.51 % abs_mult_sgn
% 0.22/0.51 thf(fact_6263_abs__mult__sgn,axiom,
% 0.22/0.51 ! [A: real] :
% 0.22/0.51 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( sgn_sgn_real @ A ) )
% 0.22/0.51 = A ) ).
% 0.22/0.51
% 0.22/0.51 % abs_mult_sgn
% 0.22/0.51 thf(fact_6264_abs__mult__sgn,axiom,
% 0.22/0.51 ! [A: rat] :
% 0.22/0.51 ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( sgn_sgn_rat @ A ) )
% 0.22/0.51 = A ) ).
% 0.22/0.51
% 0.22/0.51 % abs_mult_sgn
% 0.22/0.51 thf(fact_6265_abs__mult__sgn,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( times_times_int @ ( abs_abs_int @ A ) @ ( sgn_sgn_int @ A ) )
% 0.22/0.51 = A ) ).
% 0.22/0.51
% 0.22/0.51 % abs_mult_sgn
% 0.22/0.51 thf(fact_6266_sgn__mult__abs,axiom,
% 0.22/0.51 ! [A: complex] :
% 0.22/0.51 ( ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( abs_abs_complex @ A ) )
% 0.22/0.51 = A ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_mult_abs
% 0.22/0.51 thf(fact_6267_sgn__mult__abs,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 0.22/0.51 = A ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_mult_abs
% 0.22/0.51 thf(fact_6268_sgn__mult__abs,axiom,
% 0.22/0.51 ! [A: real] :
% 0.22/0.51 ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( abs_abs_real @ A ) )
% 0.22/0.51 = A ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_mult_abs
% 0.22/0.51 thf(fact_6269_sgn__mult__abs,axiom,
% 0.22/0.51 ! [A: rat] :
% 0.22/0.51 ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( abs_abs_rat @ A ) )
% 0.22/0.51 = A ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_mult_abs
% 0.22/0.51 thf(fact_6270_sgn__mult__abs,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( abs_abs_int @ A ) )
% 0.22/0.51 = A ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_mult_abs
% 0.22/0.51 thf(fact_6271_mult__sgn__abs,axiom,
% 0.22/0.51 ! [X: code_integer] :
% 0.22/0.51 ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ X ) @ ( abs_abs_Code_integer @ X ) )
% 0.22/0.51 = X ) ).
% 0.22/0.51
% 0.22/0.51 % mult_sgn_abs
% 0.22/0.51 thf(fact_6272_mult__sgn__abs,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( times_times_real @ ( sgn_sgn_real @ X ) @ ( abs_abs_real @ X ) )
% 0.22/0.51 = X ) ).
% 0.22/0.51
% 0.22/0.51 % mult_sgn_abs
% 0.22/0.51 thf(fact_6273_mult__sgn__abs,axiom,
% 0.22/0.51 ! [X: rat] :
% 0.22/0.51 ( ( times_times_rat @ ( sgn_sgn_rat @ X ) @ ( abs_abs_rat @ X ) )
% 0.22/0.51 = X ) ).
% 0.22/0.51
% 0.22/0.51 % mult_sgn_abs
% 0.22/0.51 thf(fact_6274_mult__sgn__abs,axiom,
% 0.22/0.51 ! [X: int] :
% 0.22/0.51 ( ( times_times_int @ ( sgn_sgn_int @ X ) @ ( abs_abs_int @ X ) )
% 0.22/0.51 = X ) ).
% 0.22/0.51
% 0.22/0.51 % mult_sgn_abs
% 0.22/0.51 thf(fact_6275_int__sgnE,axiom,
% 0.22/0.51 ! [K: int] :
% 0.22/0.51 ~ ! [N2: nat,L2: int] :
% 0.22/0.51 ( K
% 0.22/0.51 != ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % int_sgnE
% 0.22/0.51 thf(fact_6276_same__sgn__abs__add,axiom,
% 0.22/0.51 ! [B: code_integer,A: code_integer] :
% 0.22/0.51 ( ( ( sgn_sgn_Code_integer @ B )
% 0.22/0.51 = ( sgn_sgn_Code_integer @ A ) )
% 0.22/0.51 => ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.22/0.51 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % same_sgn_abs_add
% 0.22/0.51 thf(fact_6277_same__sgn__abs__add,axiom,
% 0.22/0.51 ! [B: real,A: real] :
% 0.22/0.51 ( ( ( sgn_sgn_real @ B )
% 0.22/0.51 = ( sgn_sgn_real @ A ) )
% 0.22/0.51 => ( ( abs_abs_real @ ( plus_plus_real @ A @ B ) )
% 0.22/0.51 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % same_sgn_abs_add
% 0.22/0.51 thf(fact_6278_same__sgn__abs__add,axiom,
% 0.22/0.51 ! [B: rat,A: rat] :
% 0.22/0.51 ( ( ( sgn_sgn_rat @ B )
% 0.22/0.51 = ( sgn_sgn_rat @ A ) )
% 0.22/0.51 => ( ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) )
% 0.22/0.51 = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % same_sgn_abs_add
% 0.22/0.51 thf(fact_6279_same__sgn__abs__add,axiom,
% 0.22/0.51 ! [B: int,A: int] :
% 0.22/0.51 ( ( ( sgn_sgn_int @ B )
% 0.22/0.51 = ( sgn_sgn_int @ A ) )
% 0.22/0.51 => ( ( abs_abs_int @ ( plus_plus_int @ A @ B ) )
% 0.22/0.51 = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % same_sgn_abs_add
% 0.22/0.51 thf(fact_6280_mask__nonnegative__int,axiom,
% 0.22/0.51 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.22/0.51
% 0.22/0.51 % mask_nonnegative_int
% 0.22/0.51 thf(fact_6281_not__mask__negative__int,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% 0.22/0.51
% 0.22/0.51 % not_mask_negative_int
% 0.22/0.51 thf(fact_6282_sgn__1__pos,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( ( sgn_sgn_Code_integer @ A )
% 0.22/0.51 = one_one_Code_integer )
% 0.22/0.51 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_1_pos
% 0.22/0.51 thf(fact_6283_sgn__1__pos,axiom,
% 0.22/0.51 ! [A: real] :
% 0.22/0.51 ( ( ( sgn_sgn_real @ A )
% 0.22/0.51 = one_one_real )
% 0.22/0.51 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_1_pos
% 0.22/0.51 thf(fact_6284_sgn__1__pos,axiom,
% 0.22/0.51 ! [A: rat] :
% 0.22/0.51 ( ( ( sgn_sgn_rat @ A )
% 0.22/0.51 = one_one_rat )
% 0.22/0.51 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_1_pos
% 0.22/0.51 thf(fact_6285_sgn__1__pos,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( ( sgn_sgn_int @ A )
% 0.22/0.51 = one_one_int )
% 0.22/0.51 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_1_pos
% 0.22/0.51 thf(fact_6286_abs__sgn__eq,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( ( A = zero_z3403309356797280102nteger )
% 0.22/0.51 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.22/0.51 = zero_z3403309356797280102nteger ) )
% 0.22/0.51 & ( ( A != zero_z3403309356797280102nteger )
% 0.22/0.51 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.22/0.51 = one_one_Code_integer ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % abs_sgn_eq
% 0.22/0.51 thf(fact_6287_abs__sgn__eq,axiom,
% 0.22/0.51 ! [A: real] :
% 0.22/0.51 ( ( ( A = zero_zero_real )
% 0.22/0.51 => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.22/0.51 = zero_zero_real ) )
% 0.22/0.51 & ( ( A != zero_zero_real )
% 0.22/0.51 => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.22/0.51 = one_one_real ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % abs_sgn_eq
% 0.22/0.51 thf(fact_6288_abs__sgn__eq,axiom,
% 0.22/0.51 ! [A: rat] :
% 0.22/0.51 ( ( ( A = zero_zero_rat )
% 0.22/0.51 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.22/0.51 = zero_zero_rat ) )
% 0.22/0.51 & ( ( A != zero_zero_rat )
% 0.22/0.51 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.22/0.51 = one_one_rat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % abs_sgn_eq
% 0.22/0.51 thf(fact_6289_abs__sgn__eq,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( ( A = zero_zero_int )
% 0.22/0.51 => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.22/0.51 = zero_zero_int ) )
% 0.22/0.51 & ( ( A != zero_zero_int )
% 0.22/0.51 => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.22/0.51 = one_one_int ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % abs_sgn_eq
% 0.22/0.51 thf(fact_6290_div__sgn__abs__cancel,axiom,
% 0.22/0.51 ! [V: int,K: int,L: int] :
% 0.22/0.51 ( ( V != zero_zero_int )
% 0.22/0.51 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
% 0.22/0.51 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % div_sgn_abs_cancel
% 0.22/0.51 thf(fact_6291_div__dvd__sgn__abs,axiom,
% 0.22/0.51 ! [L: int,K: int] :
% 0.22/0.51 ( ( dvd_dvd_int @ L @ K )
% 0.22/0.51 => ( ( divide_divide_int @ K @ L )
% 0.22/0.51 = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % div_dvd_sgn_abs
% 0.22/0.51 thf(fact_6292_sgn__mod,axiom,
% 0.22/0.51 ! [L: int,K: int] :
% 0.22/0.51 ( ( L != zero_zero_int )
% 0.22/0.51 => ( ~ ( dvd_dvd_int @ L @ K )
% 0.22/0.51 => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
% 0.22/0.51 = ( sgn_sgn_int @ L ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_mod
% 0.22/0.51 thf(fact_6293_length__pos__if__in__set,axiom,
% 0.22/0.51 ! [X: complex,Xs: list_complex] :
% 0.22/0.51 ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
% 0.22/0.51 => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % length_pos_if_in_set
% 0.22/0.51 thf(fact_6294_length__pos__if__in__set,axiom,
% 0.22/0.51 ! [X: set_nat,Xs: list_set_nat] :
% 0.22/0.51 ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 0.22/0.51 => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % length_pos_if_in_set
% 0.22/0.51 thf(fact_6295_length__pos__if__in__set,axiom,
% 0.22/0.51 ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.22/0.51 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.22/0.51 => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % length_pos_if_in_set
% 0.22/0.51 thf(fact_6296_length__pos__if__in__set,axiom,
% 0.22/0.51 ! [X: real,Xs: list_real] :
% 0.22/0.51 ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.22/0.51 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % length_pos_if_in_set
% 0.22/0.51 thf(fact_6297_length__pos__if__in__set,axiom,
% 0.22/0.51 ! [X: $o,Xs: list_o] :
% 0.22/0.51 ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 0.22/0.51 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % length_pos_if_in_set
% 0.22/0.51 thf(fact_6298_length__pos__if__in__set,axiom,
% 0.22/0.51 ! [X: nat,Xs: list_nat] :
% 0.22/0.51 ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.22/0.51 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % length_pos_if_in_set
% 0.22/0.51 thf(fact_6299_length__pos__if__in__set,axiom,
% 0.22/0.51 ! [X: int,Xs: list_int] :
% 0.22/0.51 ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.22/0.51 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % length_pos_if_in_set
% 0.22/0.51 thf(fact_6300_less__mask,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.22/0.51 => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % less_mask
% 0.22/0.51 thf(fact_6301_sgn__if,axiom,
% 0.22/0.51 ( sgn_sgn_real
% 0.22/0.51 = ( ^ [X3: real] : ( if_real @ ( X3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_if
% 0.22/0.51 thf(fact_6302_sgn__if,axiom,
% 0.22/0.51 ( sgn_sgn_int
% 0.22/0.51 = ( ^ [X3: int] : ( if_int @ ( X3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_if
% 0.22/0.51 thf(fact_6303_sgn__if,axiom,
% 0.22/0.51 ( sgn_sgn_Code_integer
% 0.22/0.51 = ( ^ [X3: code_integer] : ( if_Code_integer @ ( X3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X3 ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_if
% 0.22/0.51 thf(fact_6304_sgn__if,axiom,
% 0.22/0.51 ( sgn_sgn_rat
% 0.22/0.51 = ( ^ [X3: rat] : ( if_rat @ ( X3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_if
% 0.22/0.51 thf(fact_6305_sgn__1__neg,axiom,
% 0.22/0.51 ! [A: real] :
% 0.22/0.51 ( ( ( sgn_sgn_real @ A )
% 0.22/0.51 = ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.51 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_1_neg
% 0.22/0.51 thf(fact_6306_sgn__1__neg,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( ( sgn_sgn_int @ A )
% 0.22/0.51 = ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.51 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_1_neg
% 0.22/0.51 thf(fact_6307_sgn__1__neg,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( ( sgn_sgn_Code_integer @ A )
% 0.22/0.51 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.51 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_1_neg
% 0.22/0.51 thf(fact_6308_sgn__1__neg,axiom,
% 0.22/0.51 ! [A: rat] :
% 0.22/0.51 ( ( ( sgn_sgn_rat @ A )
% 0.22/0.51 = ( uminus_uminus_rat @ one_one_rat ) )
% 0.22/0.51 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_1_neg
% 0.22/0.51 thf(fact_6309_zsgn__def,axiom,
% 0.22/0.51 ( sgn_sgn_int
% 0.22/0.51 = ( ^ [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 ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % zsgn_def
% 0.22/0.51 thf(fact_6310_norm__sgn,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ( X = zero_zero_real )
% 0.22/0.51 => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
% 0.22/0.51 = zero_zero_real ) )
% 0.22/0.51 & ( ( X != zero_zero_real )
% 0.22/0.51 => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
% 0.22/0.51 = one_one_real ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % norm_sgn
% 0.22/0.51 thf(fact_6311_norm__sgn,axiom,
% 0.22/0.51 ! [X: complex] :
% 0.22/0.51 ( ( ( X = zero_zero_complex )
% 0.22/0.51 => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
% 0.22/0.51 = zero_zero_real ) )
% 0.22/0.51 & ( ( X != zero_zero_complex )
% 0.22/0.51 => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
% 0.22/0.51 = one_one_real ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % norm_sgn
% 0.22/0.51 thf(fact_6312_div__noneq__sgn__abs,axiom,
% 0.22/0.51 ! [L: int,K: int] :
% 0.22/0.51 ( ( L != zero_zero_int )
% 0.22/0.51 => ( ( ( sgn_sgn_int @ K )
% 0.22/0.51 != ( sgn_sgn_int @ L ) )
% 0.22/0.51 => ( ( divide_divide_int @ K @ L )
% 0.22/0.51 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) )
% 0.22/0.51 @ ( zero_n2684676970156552555ol_int
% 0.22/0.51 @ ~ ( dvd_dvd_int @ L @ K ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % div_noneq_sgn_abs
% 0.22/0.51 thf(fact_6313_Suc__mask__eq__exp,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.22/0.51 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.22/0.51
% 0.22/0.51 % Suc_mask_eq_exp
% 0.22/0.51 thf(fact_6314_mask__nat__less__exp,axiom,
% 0.22/0.51 ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.22/0.51
% 0.22/0.51 % mask_nat_less_exp
% 0.22/0.51 thf(fact_6315_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N ) )
% 0.22/0.51 = ( N = zero_zero_nat ) ) ).
% 0.22/0.51
% 0.22/0.51 % semiring_bit_operations_class.even_mask_iff
% 0.22/0.51 thf(fact_6316_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.22/0.51 = ( N = zero_zero_nat ) ) ).
% 0.22/0.51
% 0.22/0.51 % semiring_bit_operations_class.even_mask_iff
% 0.22/0.51 thf(fact_6317_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) )
% 0.22/0.51 = ( N = zero_zero_nat ) ) ).
% 0.22/0.51
% 0.22/0.51 % semiring_bit_operations_class.even_mask_iff
% 0.22/0.51 thf(fact_6318_mask__nat__def,axiom,
% 0.22/0.51 ( bit_se2002935070580805687sk_nat
% 0.22/0.51 = ( ^ [N4: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % mask_nat_def
% 0.22/0.51 thf(fact_6319_mask__half__int,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.51 = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % mask_half_int
% 0.22/0.51 thf(fact_6320_mask__int__def,axiom,
% 0.22/0.51 ( bit_se2000444600071755411sk_int
% 0.22/0.51 = ( ^ [N4: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) @ one_one_int ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % mask_int_def
% 0.22/0.51 thf(fact_6321_mask__eq__exp__minus__1,axiom,
% 0.22/0.51 ( bit_se2002935070580805687sk_nat
% 0.22/0.51 = ( ^ [N4: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % mask_eq_exp_minus_1
% 0.22/0.51 thf(fact_6322_mask__eq__exp__minus__1,axiom,
% 0.22/0.51 ( bit_se2000444600071755411sk_int
% 0.22/0.51 = ( ^ [N4: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) @ one_one_int ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % mask_eq_exp_minus_1
% 0.22/0.51 thf(fact_6323_arctan__add,axiom,
% 0.22/0.51 ! [X: real,Y4: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.22/0.51 => ( ( ord_less_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
% 0.22/0.51 => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y4 ) )
% 0.22/0.51 = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y4 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y4 ) ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % arctan_add
% 0.22/0.51 thf(fact_6324_divide__int__unfold,axiom,
% 0.22/0.51 ! [L: int,K: int,N: nat,M: nat] :
% 0.22/0.51 ( ( ( ( ( sgn_sgn_int @ L )
% 0.22/0.51 = zero_zero_int )
% 0.22/0.51 | ( ( sgn_sgn_int @ K )
% 0.22/0.51 = zero_zero_int )
% 0.22/0.51 | ( N = zero_zero_nat ) )
% 0.22/0.51 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.22/0.51 = zero_zero_int ) )
% 0.22/0.51 & ( ~ ( ( ( sgn_sgn_int @ L )
% 0.22/0.51 = zero_zero_int )
% 0.22/0.51 | ( ( sgn_sgn_int @ K )
% 0.22/0.51 = zero_zero_int )
% 0.22/0.51 | ( N = zero_zero_nat ) )
% 0.22/0.51 => ( ( ( ( sgn_sgn_int @ K )
% 0.22/0.51 = ( sgn_sgn_int @ L ) )
% 0.22/0.51 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.22/0.51 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
% 0.22/0.51 & ( ( ( sgn_sgn_int @ K )
% 0.22/0.51 != ( sgn_sgn_int @ L ) )
% 0.22/0.51 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.22/0.51 = ( uminus_uminus_int
% 0.22/0.51 @ ( semiri1314217659103216013at_int
% 0.22/0.51 @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
% 0.22/0.51 @ ( zero_n2687167440665602831ol_nat
% 0.22/0.51 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % divide_int_unfold
% 0.22/0.51 thf(fact_6325_divide__int__def,axiom,
% 0.22/0.51 ( divide_divide_int
% 0.22/0.51 = ( ^ [K3: int,L3: int] :
% 0.22/0.51 ( if_int @ ( L3 = zero_zero_int ) @ zero_zero_int
% 0.22/0.51 @ ( if_int
% 0.22/0.51 @ ( ( sgn_sgn_int @ K3 )
% 0.22/0.51 = ( sgn_sgn_int @ L3 ) )
% 0.22/0.51 @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) )
% 0.22/0.51 @ ( uminus_uminus_int
% 0.22/0.51 @ ( semiri1314217659103216013at_int
% 0.22/0.51 @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) )
% 0.22/0.51 @ ( zero_n2687167440665602831ol_nat
% 0.22/0.51 @ ~ ( dvd_dvd_int @ L3 @ K3 ) ) ) ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % divide_int_def
% 0.22/0.51 thf(fact_6326_take__bit__rec,axiom,
% 0.22/0.51 ( bit_se1745604003318907178nteger
% 0.22/0.51 = ( ^ [N4: nat,A3: code_integer] : ( if_Code_integer @ ( N4 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_rec
% 0.22/0.51 thf(fact_6327_take__bit__rec,axiom,
% 0.22/0.51 ( bit_se2923211474154528505it_int
% 0.22/0.51 = ( ^ [N4: nat,A3: int] : ( if_int @ ( N4 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_rec
% 0.22/0.51 thf(fact_6328_take__bit__rec,axiom,
% 0.22/0.51 ( bit_se2925701944663578781it_nat
% 0.22/0.51 = ( ^ [N4: nat,A3: nat] : ( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_rec
% 0.22/0.51 thf(fact_6329_tanh__real__altdef,axiom,
% 0.22/0.51 ( tanh_real
% 0.22/0.51 = ( ^ [X3: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % tanh_real_altdef
% 0.22/0.51 thf(fact_6330_power__numeral,axiom,
% 0.22/0.51 ! [K: num,L: num] :
% 0.22/0.51 ( ( power_power_complex @ ( numera6690914467698888265omplex @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.22/0.51 = ( numera6690914467698888265omplex @ ( pow @ K @ L ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % power_numeral
% 0.22/0.51 thf(fact_6331_power__numeral,axiom,
% 0.22/0.51 ! [K: num,L: num] :
% 0.22/0.51 ( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.22/0.51 = ( numeral_numeral_real @ ( pow @ K @ L ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % power_numeral
% 0.22/0.51 thf(fact_6332_power__numeral,axiom,
% 0.22/0.51 ! [K: num,L: num] :
% 0.22/0.51 ( ( power_power_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.22/0.51 = ( numeral_numeral_rat @ ( pow @ K @ L ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % power_numeral
% 0.22/0.51 thf(fact_6333_power__numeral,axiom,
% 0.22/0.51 ! [K: num,L: num] :
% 0.22/0.51 ( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.22/0.51 = ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % power_numeral
% 0.22/0.51 thf(fact_6334_power__numeral,axiom,
% 0.22/0.51 ! [K: num,L: num] :
% 0.22/0.51 ( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.22/0.51 = ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % power_numeral
% 0.22/0.51 thf(fact_6335_and__int__unfold,axiom,
% 0.22/0.51 ( bit_se725231765392027082nd_int
% 0.22/0.51 = ( ^ [K3: int,L3: int] :
% 0.22/0.51 ( if_int
% 0.22/0.51 @ ( ( K3 = zero_zero_int )
% 0.22/0.51 | ( L3 = zero_zero_int ) )
% 0.22/0.51 @ zero_zero_int
% 0.22/0.51 @ ( if_int
% 0.22/0.51 @ ( K3
% 0.22/0.51 = ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.51 @ L3
% 0.22/0.51 @ ( if_int
% 0.22/0.51 @ ( L3
% 0.22/0.51 = ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.51 @ K3
% 0.22/0.51 @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % and_int_unfold
% 0.22/0.51 thf(fact_6336_arctan__half,axiom,
% 0.22/0.51 ( arctan
% 0.22/0.51 = ( ^ [X3: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X3 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % arctan_half
% 0.22/0.51 thf(fact_6337_and_Oright__idem,axiom,
% 0.22/0.51 ! [A: int,B: int] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ B )
% 0.22/0.51 = ( bit_se725231765392027082nd_int @ A @ B ) ) ).
% 0.22/0.51
% 0.22/0.51 % and.right_idem
% 0.22/0.51 thf(fact_6338_and_Oright__idem,axiom,
% 0.22/0.51 ! [A: nat,B: nat] :
% 0.22/0.51 ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ B )
% 0.22/0.51 = ( bit_se727722235901077358nd_nat @ A @ B ) ) ).
% 0.22/0.51
% 0.22/0.51 % and.right_idem
% 0.22/0.51 thf(fact_6339_and_Oleft__idem,axiom,
% 0.22/0.51 ! [A: int,B: int] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 0.22/0.51 = ( bit_se725231765392027082nd_int @ A @ B ) ) ).
% 0.22/0.51
% 0.22/0.51 % and.left_idem
% 0.22/0.51 thf(fact_6340_and_Oleft__idem,axiom,
% 0.22/0.51 ! [A: nat,B: nat] :
% 0.22/0.51 ( ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 0.22/0.51 = ( bit_se727722235901077358nd_nat @ A @ B ) ) ).
% 0.22/0.51
% 0.22/0.51 % and.left_idem
% 0.22/0.51 thf(fact_6341_and_Oidem,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ A @ A )
% 0.22/0.51 = A ) ).
% 0.22/0.51
% 0.22/0.51 % and.idem
% 0.22/0.51 thf(fact_6342_and_Oidem,axiom,
% 0.22/0.51 ! [A: nat] :
% 0.22/0.51 ( ( bit_se727722235901077358nd_nat @ A @ A )
% 0.22/0.51 = A ) ).
% 0.22/0.51
% 0.22/0.51 % and.idem
% 0.22/0.51 thf(fact_6343_real__sqrt__eq__iff,axiom,
% 0.22/0.51 ! [X: real,Y4: real] :
% 0.22/0.51 ( ( ( sqrt @ X )
% 0.22/0.51 = ( sqrt @ Y4 ) )
% 0.22/0.51 = ( X = Y4 ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_eq_iff
% 0.22/0.51 thf(fact_6344_sgn__le__0__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ ( sgn_sgn_real @ X ) @ zero_zero_real )
% 0.22/0.51 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % sgn_le_0_iff
% 0.22/0.51 thf(fact_6345_zero__le__sgn__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X ) )
% 0.22/0.51 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.22/0.51
% 0.22/0.51 % zero_le_sgn_iff
% 0.22/0.51 thf(fact_6346_take__bit__of__0,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
% 0.22/0.51 = zero_zero_int ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_of_0
% 0.22/0.51 thf(fact_6347_take__bit__of__0,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
% 0.22/0.51 = zero_zero_nat ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_of_0
% 0.22/0.51 thf(fact_6348_bit_Oconj__zero__right,axiom,
% 0.22/0.51 ! [X: int] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
% 0.22/0.51 = zero_zero_int ) ).
% 0.22/0.51
% 0.22/0.51 % bit.conj_zero_right
% 0.22/0.51 thf(fact_6349_bit_Oconj__zero__left,axiom,
% 0.22/0.51 ! [X: int] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
% 0.22/0.51 = zero_zero_int ) ).
% 0.22/0.51
% 0.22/0.51 % bit.conj_zero_left
% 0.22/0.51 thf(fact_6350_zero__and__eq,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
% 0.22/0.51 = zero_zero_int ) ).
% 0.22/0.51
% 0.22/0.51 % zero_and_eq
% 0.22/0.51 thf(fact_6351_zero__and__eq,axiom,
% 0.22/0.51 ! [A: nat] :
% 0.22/0.51 ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
% 0.22/0.51 = zero_zero_nat ) ).
% 0.22/0.51
% 0.22/0.51 % zero_and_eq
% 0.22/0.51 thf(fact_6352_and__zero__eq,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
% 0.22/0.51 = zero_zero_int ) ).
% 0.22/0.51
% 0.22/0.51 % and_zero_eq
% 0.22/0.51 thf(fact_6353_and__zero__eq,axiom,
% 0.22/0.51 ! [A: nat] :
% 0.22/0.51 ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
% 0.22/0.51 = zero_zero_nat ) ).
% 0.22/0.51
% 0.22/0.51 % and_zero_eq
% 0.22/0.51 thf(fact_6354_real__sqrt__eq__zero__cancel__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ( sqrt @ X )
% 0.22/0.51 = zero_zero_real )
% 0.22/0.51 = ( X = zero_zero_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_eq_zero_cancel_iff
% 0.22/0.51 thf(fact_6355_real__sqrt__zero,axiom,
% 0.22/0.51 ( ( sqrt @ zero_zero_real )
% 0.22/0.51 = zero_zero_real ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_zero
% 0.22/0.51 thf(fact_6356_real__sqrt__less__iff,axiom,
% 0.22/0.51 ! [X: real,Y4: real] :
% 0.22/0.51 ( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y4 ) )
% 0.22/0.51 = ( ord_less_real @ X @ Y4 ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_less_iff
% 0.22/0.51 thf(fact_6357_real__sqrt__le__iff,axiom,
% 0.22/0.51 ! [X: real,Y4: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y4 ) )
% 0.22/0.51 = ( ord_less_eq_real @ X @ Y4 ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_le_iff
% 0.22/0.51 thf(fact_6358_real__sqrt__eq__1__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ( sqrt @ X )
% 0.22/0.51 = one_one_real )
% 0.22/0.51 = ( X = one_one_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_eq_1_iff
% 0.22/0.51 thf(fact_6359_real__sqrt__one,axiom,
% 0.22/0.51 ( ( sqrt @ one_one_real )
% 0.22/0.51 = one_one_real ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_one
% 0.22/0.51 thf(fact_6360_nat__int,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
% 0.22/0.51 = N ) ).
% 0.22/0.51
% 0.22/0.51 % nat_int
% 0.22/0.51 thf(fact_6361_take__bit__and,axiom,
% 0.22/0.51 ! [N: nat,A: int,B: int] :
% 0.22/0.51 ( ( bit_se2923211474154528505it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 0.22/0.51 = ( bit_se725231765392027082nd_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_and
% 0.22/0.51 thf(fact_6362_take__bit__and,axiom,
% 0.22/0.51 ! [N: nat,A: nat,B: nat] :
% 0.22/0.51 ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 0.22/0.51 = ( bit_se727722235901077358nd_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_and
% 0.22/0.51 thf(fact_6363_exp__le__cancel__iff,axiom,
% 0.22/0.51 ! [X: real,Y4: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ ( exp_real @ X ) @ ( exp_real @ Y4 ) )
% 0.22/0.51 = ( ord_less_eq_real @ X @ Y4 ) ) ).
% 0.22/0.51
% 0.22/0.51 % exp_le_cancel_iff
% 0.22/0.51 thf(fact_6364_concat__bit__of__zero__2,axiom,
% 0.22/0.51 ! [N: nat,K: int] :
% 0.22/0.51 ( ( bit_concat_bit @ N @ K @ zero_zero_int )
% 0.22/0.51 = ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 0.22/0.51
% 0.22/0.51 % concat_bit_of_zero_2
% 0.22/0.51 thf(fact_6365_exp__zero,axiom,
% 0.22/0.51 ( ( exp_complex @ zero_zero_complex )
% 0.22/0.51 = one_one_complex ) ).
% 0.22/0.51
% 0.22/0.51 % exp_zero
% 0.22/0.51 thf(fact_6366_exp__zero,axiom,
% 0.22/0.51 ( ( exp_real @ zero_zero_real )
% 0.22/0.51 = one_one_real ) ).
% 0.22/0.51
% 0.22/0.51 % exp_zero
% 0.22/0.51 thf(fact_6367_take__bit__0,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
% 0.22/0.51 = zero_zero_int ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_0
% 0.22/0.51 thf(fact_6368_take__bit__0,axiom,
% 0.22/0.51 ! [A: nat] :
% 0.22/0.51 ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
% 0.22/0.51 = zero_zero_nat ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_0
% 0.22/0.51 thf(fact_6369_take__bit__Suc__1,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
% 0.22/0.51 = one_one_int ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_Suc_1
% 0.22/0.51 thf(fact_6370_take__bit__Suc__1,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
% 0.22/0.51 = one_one_nat ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_Suc_1
% 0.22/0.51 thf(fact_6371_bit_Oconj__one__right,axiom,
% 0.22/0.51 ! [X: code_integer] :
% 0.22/0.51 ( ( bit_se3949692690581998587nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.51 = X ) ).
% 0.22/0.51
% 0.22/0.51 % bit.conj_one_right
% 0.22/0.51 thf(fact_6372_bit_Oconj__one__right,axiom,
% 0.22/0.51 ! [X: int] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ X @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.51 = X ) ).
% 0.22/0.51
% 0.22/0.51 % bit.conj_one_right
% 0.22/0.51 thf(fact_6373_and_Oright__neutral,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.51 = A ) ).
% 0.22/0.51
% 0.22/0.51 % and.right_neutral
% 0.22/0.51 thf(fact_6374_and_Oright__neutral,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.51 = A ) ).
% 0.22/0.51
% 0.22/0.51 % and.right_neutral
% 0.22/0.51 thf(fact_6375_and_Oleft__neutral,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
% 0.22/0.51 = A ) ).
% 0.22/0.51
% 0.22/0.51 % and.left_neutral
% 0.22/0.51 thf(fact_6376_and_Oleft__neutral,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
% 0.22/0.51 = A ) ).
% 0.22/0.51
% 0.22/0.51 % and.left_neutral
% 0.22/0.51 thf(fact_6377_take__bit__numeral__1,axiom,
% 0.22/0.51 ! [L: num] :
% 0.22/0.51 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ one_one_int )
% 0.22/0.51 = one_one_int ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_numeral_1
% 0.22/0.51 thf(fact_6378_take__bit__numeral__1,axiom,
% 0.22/0.51 ! [L: num] :
% 0.22/0.51 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ one_one_nat )
% 0.22/0.51 = one_one_nat ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_numeral_1
% 0.22/0.51 thf(fact_6379_nat__numeral,axiom,
% 0.22/0.51 ! [K: num] :
% 0.22/0.51 ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 0.22/0.51 = ( numeral_numeral_nat @ K ) ) ).
% 0.22/0.51
% 0.22/0.51 % nat_numeral
% 0.22/0.51 thf(fact_6380_real__sqrt__lt__0__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
% 0.22/0.51 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_lt_0_iff
% 0.22/0.51 thf(fact_6381_real__sqrt__gt__0__iff,axiom,
% 0.22/0.51 ! [Y4: real] :
% 0.22/0.51 ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y4 ) )
% 0.22/0.51 = ( ord_less_real @ zero_zero_real @ Y4 ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_gt_0_iff
% 0.22/0.51 thf(fact_6382_real__sqrt__ge__0__iff,axiom,
% 0.22/0.51 ! [Y4: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y4 ) )
% 0.22/0.51 = ( ord_less_eq_real @ zero_zero_real @ Y4 ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_ge_0_iff
% 0.22/0.51 thf(fact_6383_real__sqrt__le__0__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
% 0.22/0.51 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_le_0_iff
% 0.22/0.51 thf(fact_6384_real__sqrt__lt__1__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
% 0.22/0.51 = ( ord_less_real @ X @ one_one_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_lt_1_iff
% 0.22/0.51 thf(fact_6385_real__sqrt__gt__1__iff,axiom,
% 0.22/0.51 ! [Y4: real] :
% 0.22/0.51 ( ( ord_less_real @ one_one_real @ ( sqrt @ Y4 ) )
% 0.22/0.51 = ( ord_less_real @ one_one_real @ Y4 ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_gt_1_iff
% 0.22/0.51 thf(fact_6386_real__sqrt__ge__1__iff,axiom,
% 0.22/0.51 ! [Y4: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y4 ) )
% 0.22/0.51 = ( ord_less_eq_real @ one_one_real @ Y4 ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_ge_1_iff
% 0.22/0.51 thf(fact_6387_real__sqrt__le__1__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
% 0.22/0.51 = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_le_1_iff
% 0.22/0.51 thf(fact_6388_exp__eq__one__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ( exp_real @ X )
% 0.22/0.51 = one_one_real )
% 0.22/0.51 = ( X = zero_zero_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % exp_eq_one_iff
% 0.22/0.51 thf(fact_6389_and__nonnegative__int__iff,axiom,
% 0.22/0.51 ! [K: int,L: int] :
% 0.22/0.51 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 0.22/0.51 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.22/0.51 | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % and_nonnegative_int_iff
% 0.22/0.51 thf(fact_6390_and__negative__int__iff,axiom,
% 0.22/0.51 ! [K: int,L: int] :
% 0.22/0.51 ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
% 0.22/0.51 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.22/0.51 & ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % and_negative_int_iff
% 0.22/0.51 thf(fact_6391_real__sqrt__mult__self,axiom,
% 0.22/0.51 ! [A: real] :
% 0.22/0.51 ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
% 0.22/0.51 = ( abs_abs_real @ A ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_mult_self
% 0.22/0.51 thf(fact_6392_real__sqrt__abs2,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( sqrt @ ( times_times_real @ X @ X ) )
% 0.22/0.51 = ( abs_abs_real @ X ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_abs2
% 0.22/0.51 thf(fact_6393_nat__of__bool,axiom,
% 0.22/0.51 ! [P: $o] :
% 0.22/0.51 ( ( nat2 @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.22/0.51 = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.22/0.51
% 0.22/0.51 % nat_of_bool
% 0.22/0.51 thf(fact_6394_take__bit__of__1__eq__0__iff,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 0.22/0.51 = zero_zero_int )
% 0.22/0.51 = ( N = zero_zero_nat ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_of_1_eq_0_iff
% 0.22/0.51 thf(fact_6395_take__bit__of__1__eq__0__iff,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 0.22/0.51 = zero_zero_nat )
% 0.22/0.51 = ( N = zero_zero_nat ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_of_1_eq_0_iff
% 0.22/0.51 thf(fact_6396_and__numerals_I2_J,axiom,
% 0.22/0.51 ! [Y4: num] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y4 ) ) )
% 0.22/0.51 = one_one_int ) ).
% 0.22/0.51
% 0.22/0.51 % and_numerals(2)
% 0.22/0.51 thf(fact_6397_and__numerals_I2_J,axiom,
% 0.22/0.51 ! [Y4: num] :
% 0.22/0.51 ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 0.22/0.51 = one_one_nat ) ).
% 0.22/0.51
% 0.22/0.51 % and_numerals(2)
% 0.22/0.51 thf(fact_6398_and__numerals_I8_J,axiom,
% 0.22/0.51 ! [X: num] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
% 0.22/0.51 = one_one_int ) ).
% 0.22/0.51
% 0.22/0.51 % and_numerals(8)
% 0.22/0.51 thf(fact_6399_and__numerals_I8_J,axiom,
% 0.22/0.51 ! [X: num] :
% 0.22/0.51 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
% 0.22/0.51 = one_one_nat ) ).
% 0.22/0.51
% 0.22/0.51 % and_numerals(8)
% 0.22/0.51 thf(fact_6400_real__sqrt__four,axiom,
% 0.22/0.51 ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.22/0.51 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_four
% 0.22/0.51 thf(fact_6401_nat__1,axiom,
% 0.22/0.51 ( ( nat2 @ one_one_int )
% 0.22/0.51 = ( suc @ zero_zero_nat ) ) ).
% 0.22/0.51
% 0.22/0.51 % nat_1
% 0.22/0.51 thf(fact_6402_nat__0__iff,axiom,
% 0.22/0.51 ! [I: int] :
% 0.22/0.51 ( ( ( nat2 @ I )
% 0.22/0.51 = zero_zero_nat )
% 0.22/0.51 = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% 0.22/0.51
% 0.22/0.51 % nat_0_iff
% 0.22/0.51 thf(fact_6403_nat__le__0,axiom,
% 0.22/0.51 ! [Z: int] :
% 0.22/0.51 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 0.22/0.51 => ( ( nat2 @ Z )
% 0.22/0.51 = zero_zero_nat ) ) ).
% 0.22/0.51
% 0.22/0.51 % nat_le_0
% 0.22/0.51 thf(fact_6404_zless__nat__conj,axiom,
% 0.22/0.51 ! [W: int,Z: int] :
% 0.22/0.51 ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.22/0.51 = ( ( ord_less_int @ zero_zero_int @ Z )
% 0.22/0.51 & ( ord_less_int @ W @ Z ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % zless_nat_conj
% 0.22/0.51 thf(fact_6405_nat__neg__numeral,axiom,
% 0.22/0.51 ! [K: num] :
% 0.22/0.51 ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.22/0.51 = zero_zero_nat ) ).
% 0.22/0.51
% 0.22/0.51 % nat_neg_numeral
% 0.22/0.51 thf(fact_6406_one__less__exp__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
% 0.22/0.51 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.22/0.51
% 0.22/0.51 % one_less_exp_iff
% 0.22/0.51 thf(fact_6407_exp__less__one__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
% 0.22/0.51 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % exp_less_one_iff
% 0.22/0.51 thf(fact_6408_exp__le__one__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
% 0.22/0.51 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.22/0.51
% 0.22/0.51 % exp_le_one_iff
% 0.22/0.51 thf(fact_6409_one__le__exp__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
% 0.22/0.51 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.22/0.51
% 0.22/0.51 % one_le_exp_iff
% 0.22/0.51 thf(fact_6410_nat__zminus__int,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.22/0.51 = zero_zero_nat ) ).
% 0.22/0.51
% 0.22/0.51 % nat_zminus_int
% 0.22/0.51 thf(fact_6411_int__nat__eq,axiom,
% 0.22/0.51 ! [Z: int] :
% 0.22/0.51 ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.22/0.51 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.22/0.51 = Z ) )
% 0.22/0.51 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.22/0.51 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.22/0.51 = zero_zero_int ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % int_nat_eq
% 0.22/0.51 thf(fact_6412_take__bit__minus__one__eq__mask,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( bit_se1745604003318907178nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.51 = ( bit_se2119862282449309892nteger @ N ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_minus_one_eq_mask
% 0.22/0.51 thf(fact_6413_take__bit__minus__one__eq__mask,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.51 = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_minus_one_eq_mask
% 0.22/0.51 thf(fact_6414_exp__ln,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.51 => ( ( exp_real @ ( ln_ln_real @ X ) )
% 0.22/0.51 = X ) ) ).
% 0.22/0.51
% 0.22/0.51 % exp_ln
% 0.22/0.51 thf(fact_6415_exp__ln__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ( exp_real @ ( ln_ln_real @ X ) )
% 0.22/0.51 = X )
% 0.22/0.51 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.22/0.51
% 0.22/0.51 % exp_ln_iff
% 0.22/0.51 thf(fact_6416_take__bit__of__Suc__0,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.22/0.51 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_of_Suc_0
% 0.22/0.51 thf(fact_6417_and__numerals_I1_J,axiom,
% 0.22/0.51 ! [Y4: num] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y4 ) ) )
% 0.22/0.51 = zero_zero_int ) ).
% 0.22/0.51
% 0.22/0.51 % and_numerals(1)
% 0.22/0.51 thf(fact_6418_and__numerals_I1_J,axiom,
% 0.22/0.51 ! [Y4: num] :
% 0.22/0.51 ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 0.22/0.51 = zero_zero_nat ) ).
% 0.22/0.51
% 0.22/0.51 % and_numerals(1)
% 0.22/0.51 thf(fact_6419_and__numerals_I5_J,axiom,
% 0.22/0.51 ! [X: num] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
% 0.22/0.51 = zero_zero_int ) ).
% 0.22/0.51
% 0.22/0.51 % and_numerals(5)
% 0.22/0.51 thf(fact_6420_and__numerals_I5_J,axiom,
% 0.22/0.51 ! [X: num] :
% 0.22/0.51 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
% 0.22/0.51 = zero_zero_nat ) ).
% 0.22/0.51
% 0.22/0.51 % and_numerals(5)
% 0.22/0.51 thf(fact_6421_and__numerals_I3_J,axiom,
% 0.22/0.51 ! [X: num,Y4: num] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y4 ) ) )
% 0.22/0.51 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % and_numerals(3)
% 0.22/0.51 thf(fact_6422_and__numerals_I3_J,axiom,
% 0.22/0.51 ! [X: num,Y4: num] :
% 0.22/0.51 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 0.22/0.51 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % and_numerals(3)
% 0.22/0.51 thf(fact_6423_zero__less__nat__eq,axiom,
% 0.22/0.51 ! [Z: int] :
% 0.22/0.51 ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
% 0.22/0.51 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.22/0.51
% 0.22/0.51 % zero_less_nat_eq
% 0.22/0.51 thf(fact_6424_take__bit__of__1,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
% 0.22/0.51 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_of_1
% 0.22/0.51 thf(fact_6425_take__bit__of__1,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 0.22/0.51 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_of_1
% 0.22/0.51 thf(fact_6426_take__bit__of__1,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 0.22/0.51 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_of_1
% 0.22/0.51 thf(fact_6427_diff__nat__numeral,axiom,
% 0.22/0.51 ! [V: num,V2: num] :
% 0.22/0.51 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V2 ) )
% 0.22/0.51 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V2 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % diff_nat_numeral
% 0.22/0.51 thf(fact_6428_and__minus__numerals_I6_J,axiom,
% 0.22/0.51 ! [N: num] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 0.22/0.51 = one_one_int ) ).
% 0.22/0.51
% 0.22/0.51 % and_minus_numerals(6)
% 0.22/0.51 thf(fact_6429_and__minus__numerals_I2_J,axiom,
% 0.22/0.51 ! [N: num] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.22/0.51 = one_one_int ) ).
% 0.22/0.51
% 0.22/0.51 % and_minus_numerals(2)
% 0.22/0.51 thf(fact_6430_nat__eq__numeral__power__cancel__iff,axiom,
% 0.22/0.51 ! [Y4: int,X: num,N: nat] :
% 0.22/0.51 ( ( ( nat2 @ Y4 )
% 0.22/0.51 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.22/0.51 = ( Y4
% 0.22/0.51 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % nat_eq_numeral_power_cancel_iff
% 0.22/0.51 thf(fact_6431_numeral__power__eq__nat__cancel__iff,axiom,
% 0.22/0.51 ! [X: num,N: nat,Y4: int] :
% 0.22/0.51 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.22/0.51 = ( nat2 @ Y4 ) )
% 0.22/0.51 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.22/0.51 = Y4 ) ) ).
% 0.22/0.51
% 0.22/0.51 % numeral_power_eq_nat_cancel_iff
% 0.22/0.51 thf(fact_6432_dvd__nat__abs__iff,axiom,
% 0.22/0.51 ! [N: nat,K: int] :
% 0.22/0.51 ( ( dvd_dvd_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 0.22/0.51 = ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).
% 0.22/0.51
% 0.22/0.51 % dvd_nat_abs_iff
% 0.22/0.51 thf(fact_6433_nat__abs__dvd__iff,axiom,
% 0.22/0.51 ! [K: int,N: nat] :
% 0.22/0.51 ( ( dvd_dvd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 0.22/0.51 = ( dvd_dvd_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % nat_abs_dvd_iff
% 0.22/0.51 thf(fact_6434_and__numerals_I4_J,axiom,
% 0.22/0.51 ! [X: num,Y4: num] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y4 ) ) )
% 0.22/0.51 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % and_numerals(4)
% 0.22/0.51 thf(fact_6435_and__numerals_I4_J,axiom,
% 0.22/0.51 ! [X: num,Y4: num] :
% 0.22/0.51 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 0.22/0.51 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % and_numerals(4)
% 0.22/0.51 thf(fact_6436_and__numerals_I6_J,axiom,
% 0.22/0.51 ! [X: num,Y4: num] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y4 ) ) )
% 0.22/0.51 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % and_numerals(6)
% 0.22/0.51 thf(fact_6437_and__numerals_I6_J,axiom,
% 0.22/0.51 ! [X: num,Y4: num] :
% 0.22/0.51 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 0.22/0.51 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % and_numerals(6)
% 0.22/0.51 thf(fact_6438_even__take__bit__eq,axiom,
% 0.22/0.51 ! [N: nat,A: code_integer] :
% 0.22/0.51 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N @ A ) )
% 0.22/0.51 = ( ( N = zero_zero_nat )
% 0.22/0.51 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % even_take_bit_eq
% 0.22/0.51 thf(fact_6439_even__take__bit__eq,axiom,
% 0.22/0.51 ! [N: nat,A: int] :
% 0.22/0.51 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.22/0.51 = ( ( N = zero_zero_nat )
% 0.22/0.51 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % even_take_bit_eq
% 0.22/0.51 thf(fact_6440_even__take__bit__eq,axiom,
% 0.22/0.51 ! [N: nat,A: nat] :
% 0.22/0.51 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 0.22/0.51 = ( ( N = zero_zero_nat )
% 0.22/0.51 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % even_take_bit_eq
% 0.22/0.51 thf(fact_6441_one__less__nat__eq,axiom,
% 0.22/0.51 ! [Z: int] :
% 0.22/0.51 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
% 0.22/0.51 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.22/0.51
% 0.22/0.51 % one_less_nat_eq
% 0.22/0.51 thf(fact_6442_real__sqrt__abs,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( sqrt @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.51 = ( abs_abs_real @ X ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_abs
% 0.22/0.51 thf(fact_6443_and__minus__numerals_I5_J,axiom,
% 0.22/0.51 ! [N: num] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 0.22/0.51 = zero_zero_int ) ).
% 0.22/0.51
% 0.22/0.51 % and_minus_numerals(5)
% 0.22/0.51 thf(fact_6444_and__minus__numerals_I1_J,axiom,
% 0.22/0.51 ! [N: num] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.22/0.51 = zero_zero_int ) ).
% 0.22/0.51
% 0.22/0.51 % and_minus_numerals(1)
% 0.22/0.51 thf(fact_6445_take__bit__Suc__0,axiom,
% 0.22/0.51 ! [A: code_integer] :
% 0.22/0.51 ( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
% 0.22/0.51 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_Suc_0
% 0.22/0.51 thf(fact_6446_take__bit__Suc__0,axiom,
% 0.22/0.51 ! [A: int] :
% 0.22/0.51 ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
% 0.22/0.51 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_Suc_0
% 0.22/0.51 thf(fact_6447_take__bit__Suc__0,axiom,
% 0.22/0.51 ! [A: nat] :
% 0.22/0.51 ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
% 0.22/0.51 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_Suc_0
% 0.22/0.51 thf(fact_6448_real__sqrt__pow2__iff,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.51 = X )
% 0.22/0.51 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_pow2_iff
% 0.22/0.51 thf(fact_6449_real__sqrt__pow2,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.51 => ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.51 = X ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_pow2
% 0.22/0.51 thf(fact_6450_real__sqrt__sum__squares__mult__squared__eq,axiom,
% 0.22/0.51 ! [X: real,Y4: real,Xa: real,Ya: real] :
% 0.22/0.51 ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( 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 ) ) )
% 0.22/0.51 = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( 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 ) ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_sum_squares_mult_squared_eq
% 0.22/0.51 thf(fact_6451_nat__numeral__diff__1,axiom,
% 0.22/0.51 ! [V: num] :
% 0.22/0.51 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
% 0.22/0.51 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % nat_numeral_diff_1
% 0.22/0.51 thf(fact_6452_numeral__power__less__nat__cancel__iff,axiom,
% 0.22/0.51 ! [X: num,N: nat,A: int] :
% 0.22/0.51 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 0.22/0.51 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.22/0.51
% 0.22/0.51 % numeral_power_less_nat_cancel_iff
% 0.22/0.51 thf(fact_6453_nat__less__numeral__power__cancel__iff,axiom,
% 0.22/0.51 ! [A: int,X: num,N: nat] :
% 0.22/0.51 ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.22/0.51 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % nat_less_numeral_power_cancel_iff
% 0.22/0.51 thf(fact_6454_nat__le__numeral__power__cancel__iff,axiom,
% 0.22/0.51 ! [A: int,X: num,N: nat] :
% 0.22/0.51 ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.22/0.51 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % nat_le_numeral_power_cancel_iff
% 0.22/0.51 thf(fact_6455_numeral__power__le__nat__cancel__iff,axiom,
% 0.22/0.51 ! [X: num,N: nat,A: int] :
% 0.22/0.51 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 0.22/0.51 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.22/0.51
% 0.22/0.51 % numeral_power_le_nat_cancel_iff
% 0.22/0.51 thf(fact_6456_and__numerals_I7_J,axiom,
% 0.22/0.51 ! [X: num,Y4: num] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y4 ) ) )
% 0.22/0.51 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % and_numerals(7)
% 0.22/0.51 thf(fact_6457_and__numerals_I7_J,axiom,
% 0.22/0.51 ! [X: num,Y4: num] :
% 0.22/0.51 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 0.22/0.51 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % and_numerals(7)
% 0.22/0.51 thf(fact_6458_take__bit__of__exp,axiom,
% 0.22/0.51 ! [M: nat,N: nat] :
% 0.22/0.51 ( ( bit_se1745604003318907178nteger @ M @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.51 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_of_exp
% 0.22/0.51 thf(fact_6459_take__bit__of__exp,axiom,
% 0.22/0.51 ! [M: nat,N: nat] :
% 0.22/0.51 ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.51 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_of_exp
% 0.22/0.51 thf(fact_6460_take__bit__of__exp,axiom,
% 0.22/0.51 ! [M: nat,N: nat] :
% 0.22/0.51 ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.51 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_of_exp
% 0.22/0.51 thf(fact_6461_take__bit__of__2,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.51 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_of_2
% 0.22/0.51 thf(fact_6462_take__bit__of__2,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.51 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_of_2
% 0.22/0.51 thf(fact_6463_take__bit__of__2,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.51 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_of_2
% 0.22/0.51 thf(fact_6464_take__bit__eq__mask,axiom,
% 0.22/0.51 ( bit_se2923211474154528505it_int
% 0.22/0.51 = ( ^ [N4: nat,A3: int] : ( bit_se725231765392027082nd_int @ A3 @ ( bit_se2000444600071755411sk_int @ N4 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_eq_mask
% 0.22/0.51 thf(fact_6465_take__bit__eq__mask,axiom,
% 0.22/0.51 ( bit_se2925701944663578781it_nat
% 0.22/0.51 = ( ^ [N4: nat,A3: nat] : ( bit_se727722235901077358nd_nat @ A3 @ ( bit_se2002935070580805687sk_nat @ N4 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_eq_mask
% 0.22/0.51 thf(fact_6466_nat__mask__eq,axiom,
% 0.22/0.51 ! [N: nat] :
% 0.22/0.51 ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
% 0.22/0.51 = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 0.22/0.51
% 0.22/0.51 % nat_mask_eq
% 0.22/0.51 thf(fact_6467_and_Oleft__commute,axiom,
% 0.22/0.51 ! [B: int,A: int,C: int] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ B @ ( bit_se725231765392027082nd_int @ A @ C ) )
% 0.22/0.51 = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % and.left_commute
% 0.22/0.51 thf(fact_6468_and_Oleft__commute,axiom,
% 0.22/0.51 ! [B: nat,A: nat,C: nat] :
% 0.22/0.51 ( ( bit_se727722235901077358nd_nat @ B @ ( bit_se727722235901077358nd_nat @ A @ C ) )
% 0.22/0.51 = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % and.left_commute
% 0.22/0.51 thf(fact_6469_and_Ocommute,axiom,
% 0.22/0.51 ( bit_se725231765392027082nd_int
% 0.22/0.51 = ( ^ [A3: int,B2: int] : ( bit_se725231765392027082nd_int @ B2 @ A3 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % and.commute
% 0.22/0.51 thf(fact_6470_and_Ocommute,axiom,
% 0.22/0.51 ( bit_se727722235901077358nd_nat
% 0.22/0.51 = ( ^ [A3: nat,B2: nat] : ( bit_se727722235901077358nd_nat @ B2 @ A3 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % and.commute
% 0.22/0.51 thf(fact_6471_and_Oassoc,axiom,
% 0.22/0.51 ! [A: int,B: int,C: int] :
% 0.22/0.51 ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ C )
% 0.22/0.51 = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % and.assoc
% 0.22/0.51 thf(fact_6472_and_Oassoc,axiom,
% 0.22/0.51 ! [A: nat,B: nat,C: nat] :
% 0.22/0.51 ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ C )
% 0.22/0.51 = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % and.assoc
% 0.22/0.51 thf(fact_6473_of__nat__and__eq,axiom,
% 0.22/0.51 ! [M: nat,N: nat] :
% 0.22/0.51 ( ( semiri1314217659103216013at_int @ ( bit_se727722235901077358nd_nat @ M @ N ) )
% 0.22/0.51 = ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % of_nat_and_eq
% 0.22/0.51 thf(fact_6474_of__nat__and__eq,axiom,
% 0.22/0.51 ! [M: nat,N: nat] :
% 0.22/0.51 ( ( semiri1316708129612266289at_nat @ ( bit_se727722235901077358nd_nat @ M @ N ) )
% 0.22/0.51 = ( bit_se727722235901077358nd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % of_nat_and_eq
% 0.22/0.51 thf(fact_6475_take__bit__of__nat,axiom,
% 0.22/0.51 ! [N: nat,M: nat] :
% 0.22/0.51 ( ( bit_se2923211474154528505it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 0.22/0.51 = ( semiri1314217659103216013at_int @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_of_nat
% 0.22/0.51 thf(fact_6476_take__bit__of__nat,axiom,
% 0.22/0.51 ! [N: nat,M: nat] :
% 0.22/0.51 ( ( bit_se2925701944663578781it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 0.22/0.51 = ( semiri1316708129612266289at_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_of_nat
% 0.22/0.51 thf(fact_6477_nat__take__bit__eq,axiom,
% 0.22/0.51 ! [K: int,N: nat] :
% 0.22/0.51 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.22/0.51 => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.22/0.51 = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % nat_take_bit_eq
% 0.22/0.51 thf(fact_6478_take__bit__nat__eq,axiom,
% 0.22/0.51 ! [K: int,N: nat] :
% 0.22/0.51 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.22/0.51 => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
% 0.22/0.51 = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_nat_eq
% 0.22/0.51 thf(fact_6479_norm__exp,axiom,
% 0.22/0.51 ! [X: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % norm_exp
% 0.22/0.51 thf(fact_6480_norm__exp,axiom,
% 0.22/0.51 ! [X: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % norm_exp
% 0.22/0.51 thf(fact_6481_take__bit__add,axiom,
% 0.22/0.51 ! [N: nat,A: int,B: int] :
% 0.22/0.51 ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) )
% 0.22/0.51 = ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ A @ B ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_add
% 0.22/0.51 thf(fact_6482_take__bit__add,axiom,
% 0.22/0.51 ! [N: nat,A: nat,B: nat] :
% 0.22/0.51 ( ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) )
% 0.22/0.51 = ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ A @ B ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_add
% 0.22/0.51 thf(fact_6483_take__bit__tightened,axiom,
% 0.22/0.51 ! [N: nat,A: int,B: int,M: nat] :
% 0.22/0.51 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.22/0.51 = ( bit_se2923211474154528505it_int @ N @ B ) )
% 0.22/0.51 => ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.51 => ( ( bit_se2923211474154528505it_int @ M @ A )
% 0.22/0.51 = ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_tightened
% 0.22/0.51 thf(fact_6484_take__bit__tightened,axiom,
% 0.22/0.51 ! [N: nat,A: nat,B: nat,M: nat] :
% 0.22/0.51 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.22/0.51 = ( bit_se2925701944663578781it_nat @ N @ B ) )
% 0.22/0.51 => ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.51 => ( ( bit_se2925701944663578781it_nat @ M @ A )
% 0.22/0.51 = ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_tightened
% 0.22/0.51 thf(fact_6485_take__bit__nat__less__eq__self,axiom,
% 0.22/0.51 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_nat_less_eq_self
% 0.22/0.51 thf(fact_6486_take__bit__tightened__less__eq__nat,axiom,
% 0.22/0.51 ! [M: nat,N: nat,Q2: nat] :
% 0.22/0.51 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.51 => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N @ Q2 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_tightened_less_eq_nat
% 0.22/0.51 thf(fact_6487_real__sqrt__less__mono,axiom,
% 0.22/0.51 ! [X: real,Y4: real] :
% 0.22/0.51 ( ( ord_less_real @ X @ Y4 )
% 0.22/0.51 => ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y4 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_less_mono
% 0.22/0.51 thf(fact_6488_real__sqrt__le__mono,axiom,
% 0.22/0.51 ! [X: real,Y4: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ X @ Y4 )
% 0.22/0.51 => ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y4 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_le_mono
% 0.22/0.51 thf(fact_6489_real__sqrt__divide,axiom,
% 0.22/0.51 ! [X: real,Y4: real] :
% 0.22/0.51 ( ( sqrt @ ( divide_divide_real @ X @ Y4 ) )
% 0.22/0.51 = ( divide_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y4 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_divide
% 0.22/0.51 thf(fact_6490_real__sqrt__mult,axiom,
% 0.22/0.51 ! [X: real,Y4: real] :
% 0.22/0.51 ( ( sqrt @ ( times_times_real @ X @ Y4 ) )
% 0.22/0.51 = ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y4 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_mult
% 0.22/0.51 thf(fact_6491_real__sqrt__power,axiom,
% 0.22/0.51 ! [X: real,K: nat] :
% 0.22/0.51 ( ( sqrt @ ( power_power_real @ X @ K ) )
% 0.22/0.51 = ( power_power_real @ ( sqrt @ X ) @ K ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_power
% 0.22/0.51 thf(fact_6492_real__sqrt__minus,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( sqrt @ ( uminus_uminus_real @ X ) )
% 0.22/0.51 = ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_minus
% 0.22/0.51 thf(fact_6493_exp__not__eq__zero,axiom,
% 0.22/0.51 ! [X: complex] :
% 0.22/0.51 ( ( exp_complex @ X )
% 0.22/0.51 != zero_zero_complex ) ).
% 0.22/0.51
% 0.22/0.51 % exp_not_eq_zero
% 0.22/0.51 thf(fact_6494_exp__not__eq__zero,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( exp_real @ X )
% 0.22/0.51 != zero_zero_real ) ).
% 0.22/0.51
% 0.22/0.51 % exp_not_eq_zero
% 0.22/0.51 thf(fact_6495_take__bit__minus,axiom,
% 0.22/0.51 ! [N: nat,K: int] :
% 0.22/0.51 ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.22/0.51 = ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_minus
% 0.22/0.51 thf(fact_6496_exp__times__arg__commute,axiom,
% 0.22/0.51 ! [A2: complex] :
% 0.22/0.51 ( ( times_times_complex @ ( exp_complex @ A2 ) @ A2 )
% 0.22/0.51 = ( times_times_complex @ A2 @ ( exp_complex @ A2 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % exp_times_arg_commute
% 0.22/0.51 thf(fact_6497_exp__times__arg__commute,axiom,
% 0.22/0.51 ! [A2: real] :
% 0.22/0.51 ( ( times_times_real @ ( exp_real @ A2 ) @ A2 )
% 0.22/0.51 = ( times_times_real @ A2 @ ( exp_real @ A2 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % exp_times_arg_commute
% 0.22/0.51 thf(fact_6498_take__bit__mult,axiom,
% 0.22/0.51 ! [N: nat,K: int,L: int] :
% 0.22/0.51 ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
% 0.22/0.51 = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_mult
% 0.22/0.51 thf(fact_6499_take__bit__diff,axiom,
% 0.22/0.51 ! [N: nat,K: int,L: int] :
% 0.22/0.51 ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
% 0.22/0.51 = ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_diff
% 0.22/0.51 thf(fact_6500_concat__bit__take__bit__eq,axiom,
% 0.22/0.51 ! [N: nat,B: int] :
% 0.22/0.51 ( ( bit_concat_bit @ N @ ( bit_se2923211474154528505it_int @ N @ B ) )
% 0.22/0.51 = ( bit_concat_bit @ N @ B ) ) ).
% 0.22/0.51
% 0.22/0.51 % concat_bit_take_bit_eq
% 0.22/0.51 thf(fact_6501_concat__bit__eq__iff,axiom,
% 0.22/0.51 ! [N: nat,K: int,L: int,R2: int,S2: int] :
% 0.22/0.51 ( ( ( bit_concat_bit @ N @ K @ L )
% 0.22/0.51 = ( bit_concat_bit @ N @ R2 @ S2 ) )
% 0.22/0.51 = ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.22/0.51 = ( bit_se2923211474154528505it_int @ N @ R2 ) )
% 0.22/0.51 & ( L = S2 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % concat_bit_eq_iff
% 0.22/0.51 thf(fact_6502_and__eq__minus__1__iff,axiom,
% 0.22/0.51 ! [A: code_integer,B: code_integer] :
% 0.22/0.51 ( ( ( bit_se3949692690581998587nteger @ A @ B )
% 0.22/0.51 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.51 = ( ( A
% 0.22/0.51 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.51 & ( B
% 0.22/0.51 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % and_eq_minus_1_iff
% 0.22/0.51 thf(fact_6503_and__eq__minus__1__iff,axiom,
% 0.22/0.51 ! [A: int,B: int] :
% 0.22/0.51 ( ( ( bit_se725231765392027082nd_int @ A @ B )
% 0.22/0.51 = ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.51 = ( ( A
% 0.22/0.51 = ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.51 & ( B
% 0.22/0.51 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % and_eq_minus_1_iff
% 0.22/0.51 thf(fact_6504_real__sqrt__gt__zero,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.51 => ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_gt_zero
% 0.22/0.51 thf(fact_6505_exp__total,axiom,
% 0.22/0.51 ! [Y4: real] :
% 0.22/0.51 ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.22/0.51 => ? [X4: real] :
% 0.22/0.51 ( ( exp_real @ X4 )
% 0.22/0.51 = Y4 ) ) ).
% 0.22/0.51
% 0.22/0.51 % exp_total
% 0.22/0.51 thf(fact_6506_exp__gt__zero,axiom,
% 0.22/0.51 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% 0.22/0.51
% 0.22/0.51 % exp_gt_zero
% 0.22/0.51 thf(fact_6507_not__exp__less__zero,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% 0.22/0.51
% 0.22/0.51 % not_exp_less_zero
% 0.22/0.51 thf(fact_6508_real__sqrt__ge__zero,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.51 => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_ge_zero
% 0.22/0.51 thf(fact_6509_real__sqrt__eq__zero__cancel,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.51 => ( ( ( sqrt @ X )
% 0.22/0.51 = zero_zero_real )
% 0.22/0.51 => ( X = zero_zero_real ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_eq_zero_cancel
% 0.22/0.51 thf(fact_6510_exp__ge__zero,axiom,
% 0.22/0.51 ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% 0.22/0.51
% 0.22/0.51 % exp_ge_zero
% 0.22/0.51 thf(fact_6511_not__exp__le__zero,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% 0.22/0.51
% 0.22/0.51 % not_exp_le_zero
% 0.22/0.51 thf(fact_6512_real__sqrt__ge__one,axiom,
% 0.22/0.51 ! [X: real] :
% 0.22/0.51 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.22/0.51 => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sqrt_ge_one
% 0.22/0.51 thf(fact_6513_nat__zero__as__int,axiom,
% 0.22/0.51 ( zero_zero_nat
% 0.22/0.51 = ( nat2 @ zero_zero_int ) ) ).
% 0.22/0.51
% 0.22/0.51 % nat_zero_as_int
% 0.22/0.51 thf(fact_6514_AND__upper2_H,axiom,
% 0.22/0.51 ! [Y4: int,Z: int,X: int] :
% 0.22/0.51 ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.22/0.51 => ( ( ord_less_eq_int @ Y4 @ Z )
% 0.22/0.51 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y4 ) @ Z ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % AND_upper2'
% 0.22/0.51 thf(fact_6515_AND__upper1_H,axiom,
% 0.22/0.51 ! [Y4: int,Z: int,Ya: int] :
% 0.22/0.51 ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.22/0.51 => ( ( ord_less_eq_int @ Y4 @ Z )
% 0.22/0.51 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y4 @ Ya ) @ Z ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % AND_upper1'
% 0.22/0.51 thf(fact_6516_AND__upper2,axiom,
% 0.22/0.51 ! [Y4: int,X: int] :
% 0.22/0.51 ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.22/0.51 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y4 ) @ Y4 ) ) ).
% 0.22/0.51
% 0.22/0.51 % AND_upper2
% 0.22/0.51 thf(fact_6517_AND__upper1,axiom,
% 0.22/0.51 ! [X: int,Y4: int] :
% 0.22/0.51 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.22/0.51 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y4 ) @ X ) ) ).
% 0.22/0.51
% 0.22/0.51 % AND_upper1
% 0.22/0.51 thf(fact_6518_AND__lower,axiom,
% 0.22/0.51 ! [X: int,Y4: int] :
% 0.22/0.51 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.22/0.51 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y4 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % AND_lower
% 0.22/0.51 thf(fact_6519_take__bit__tightened__less__eq__int,axiom,
% 0.22/0.51 ! [M: nat,N: nat,K: int] :
% 0.22/0.51 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.51 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_tightened_less_eq_int
% 0.22/0.51 thf(fact_6520_nat__numeral__as__int,axiom,
% 0.22/0.51 ( numeral_numeral_nat
% 0.22/0.51 = ( ^ [I2: num] : ( nat2 @ ( numeral_numeral_int @ I2 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % nat_numeral_as_int
% 0.22/0.51 thf(fact_6521_take__bit__int__less__eq__self__iff,axiom,
% 0.22/0.51 ! [N: nat,K: int] :
% 0.22/0.51 ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 0.22/0.51 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_int_less_eq_self_iff
% 0.22/0.51 thf(fact_6522_take__bit__nonnegative,axiom,
% 0.22/0.51 ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_nonnegative
% 0.22/0.51 thf(fact_6523_take__bit__int__greater__self__iff,axiom,
% 0.22/0.51 ! [K: int,N: nat] :
% 0.22/0.51 ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.22/0.51 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.22/0.51
% 0.22/0.51 % take_bit_int_greater_self_iff
% 0.22/0.51 thf(fact_6524_not__take__bit__negative,axiom,
% 0.22/0.51 ! [N: nat,K: int] :
% 0.22/0.51 ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% 0.22/0.51
% 0.22/0.51 % not_take_bit_negative
% 0.22/0.51 thf(fact_6525_signed__take__bit__eq__iff__take__bit__eq,axiom,
% 0.22/0.51 ! [N: nat,A: int,B: int] :
% 0.22/0.51 ( ( ( bit_ri631733984087533419it_int @ N @ A )
% 0.22/0.51 = ( bit_ri631733984087533419it_int @ N @ B ) )
% 0.22/0.51 = ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 0.22/0.51 = ( bit_se2923211474154528505it_int @ ( suc @ N ) @ B ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % signed_take_bit_eq_iff_take_bit_eq
% 0.22/0.51 thf(fact_6526_real__sgn__eq,axiom,
% 0.22/0.51 ( sgn_sgn_real
% 0.22/0.51 = ( ^ [X3: real] : ( divide_divide_real @ X3 @ ( abs_abs_real @ X3 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % real_sgn_eq
% 0.22/0.51 thf(fact_6527_nat__mono,axiom,
% 0.22/0.51 ! [X: int,Y4: int] :
% 0.22/0.51 ( ( ord_less_eq_int @ X @ Y4 )
% 0.22/0.51 => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y4 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % nat_mono
% 0.22/0.51 thf(fact_6528_signed__take__bit__take__bit,axiom,
% 0.22/0.51 ! [M: nat,N: nat,A: int] :
% 0.22/0.51 ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.22/0.51 = ( if_int_int @ ( ord_less_eq_nat @ N @ M ) @ ( bit_se2923211474154528505it_int @ N ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).
% 0.22/0.51
% 0.22/0.51 % signed_take_bit_take_bit
% 0.22/0.51 thf(fact_6529_eq__nat__nat__iff,axiom,
% 0.22/0.51 ! [Z: int,Z6: int] :
% 0.22/0.51 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.22/0.51 => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 0.22/0.51 => ( ( ( nat2 @ Z )
% 0.22/0.51 = ( nat2 @ Z6 ) )
% 0.22/0.51 = ( Z = Z6 ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % eq_nat_nat_iff
% 0.22/0.51 thf(fact_6530_all__nat,axiom,
% 0.22/0.51 ( ( ^ [P2: nat > $o] :
% 0.22/0.51 ! [X6: nat] : ( P2 @ X6 ) )
% 0.22/0.51 = ( ^ [P3: nat > $o] :
% 0.22/0.51 ! [X3: int] :
% 0.22/0.51 ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.22/0.51 => ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % all_nat
% 0.22/0.51 thf(fact_6531_ex__nat,axiom,
% 0.22/0.51 ( ( ^ [P2: nat > $o] :
% 0.22/0.51 ? [X6: nat] : ( P2 @ X6 ) )
% 0.22/0.51 = ( ^ [P3: nat > $o] :
% 0.22/0.51 ? [X3: int] :
% 0.22/0.51 ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.22/0.51 & ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % ex_nat
% 0.22/0.51 thf(fact_6532_mult__exp__exp,axiom,
% 0.22/0.51 ! [X: complex,Y4: complex] :
% 0.22/0.51 ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y4 ) )
% 0.22/0.51 = ( exp_complex @ ( plus_plus_complex @ X @ Y4 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % mult_exp_exp
% 0.22/0.51 thf(fact_6533_mult__exp__exp,axiom,
% 0.22/0.51 ! [X: real,Y4: real] :
% 0.22/0.51 ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y4 ) )
% 0.22/0.51 = ( exp_real @ ( plus_plus_real @ X @ Y4 ) ) ) ).
% 0.22/0.51
% 0.22/0.51 % mult_exp_exp
% 0.22/0.51 thf(fact_6534_exp__add__commuting,axiom,
% 0.22/0.51 ! [X: complex,Y4: complex] :
% 0.22/0.51 ( ( ( times_times_complex @ X @ Y4 )
% 0.22/0.51 = ( times_times_complex @ Y4 @ X ) )
% 0.22/0.52 => ( ( exp_complex @ ( plus_plus_complex @ X @ Y4 ) )
% 0.22/0.52 = ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_add_commuting
% 0.22/0.52 thf(fact_6535_exp__add__commuting,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ( times_times_real @ X @ Y4 )
% 0.22/0.52 = ( times_times_real @ Y4 @ X ) )
% 0.22/0.52 => ( ( exp_real @ ( plus_plus_real @ X @ Y4 ) )
% 0.22/0.52 = ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_add_commuting
% 0.22/0.52 thf(fact_6536_exp__diff,axiom,
% 0.22/0.52 ! [X: complex,Y4: complex] :
% 0.22/0.52 ( ( exp_complex @ ( minus_minus_complex @ X @ Y4 ) )
% 0.22/0.52 = ( divide1717551699836669952omplex @ ( exp_complex @ X ) @ ( exp_complex @ Y4 ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_diff
% 0.22/0.52 thf(fact_6537_exp__diff,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( exp_real @ ( minus_minus_real @ X @ Y4 ) )
% 0.22/0.52 = ( divide_divide_real @ ( exp_real @ X ) @ ( exp_real @ Y4 ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_diff
% 0.22/0.52 thf(fact_6538_nat__one__as__int,axiom,
% 0.22/0.52 ( one_one_nat
% 0.22/0.52 = ( nat2 @ one_one_int ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_one_as_int
% 0.22/0.52 thf(fact_6539_take__bit__unset__bit__eq,axiom,
% 0.22/0.52 ! [N: nat,M: nat,A: int] :
% 0.22/0.52 ( ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.52 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.22/0.52 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.22/0.52 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.22/0.52 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.22/0.52 = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_unset_bit_eq
% 0.22/0.52 thf(fact_6540_take__bit__unset__bit__eq,axiom,
% 0.22/0.52 ! [N: nat,M: nat,A: nat] :
% 0.22/0.52 ( ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.52 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.22/0.52 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 0.22/0.52 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.22/0.52 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.22/0.52 = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_unset_bit_eq
% 0.22/0.52 thf(fact_6541_take__bit__unset__bit__eq,axiom,
% 0.22/0.52 ! [N: nat,M: nat,A: code_integer] :
% 0.22/0.52 ( ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.52 => ( ( bit_se1745604003318907178nteger @ N @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 0.22/0.52 = ( bit_se1745604003318907178nteger @ N @ A ) ) )
% 0.22/0.52 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.22/0.52 => ( ( bit_se1745604003318907178nteger @ N @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 0.22/0.52 = ( bit_se8260200283734997820nteger @ M @ ( bit_se1745604003318907178nteger @ N @ A ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_unset_bit_eq
% 0.22/0.52 thf(fact_6542_take__bit__set__bit__eq,axiom,
% 0.22/0.52 ! [N: nat,M: nat,A: code_integer] :
% 0.22/0.52 ( ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.52 => ( ( bit_se1745604003318907178nteger @ N @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 0.22/0.52 = ( bit_se1745604003318907178nteger @ N @ A ) ) )
% 0.22/0.52 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.22/0.52 => ( ( bit_se1745604003318907178nteger @ N @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 0.22/0.52 = ( bit_se2793503036327961859nteger @ M @ ( bit_se1745604003318907178nteger @ N @ A ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_set_bit_eq
% 0.22/0.52 thf(fact_6543_take__bit__set__bit__eq,axiom,
% 0.22/0.52 ! [N: nat,M: nat,A: int] :
% 0.22/0.52 ( ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.52 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.22/0.52 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.22/0.52 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.22/0.52 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.22/0.52 = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_set_bit_eq
% 0.22/0.52 thf(fact_6544_take__bit__set__bit__eq,axiom,
% 0.22/0.52 ! [N: nat,M: nat,A: nat] :
% 0.22/0.52 ( ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.52 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.22/0.52 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 0.22/0.52 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.22/0.52 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.22/0.52 = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_set_bit_eq
% 0.22/0.52 thf(fact_6545_take__bit__flip__bit__eq,axiom,
% 0.22/0.52 ! [N: nat,M: nat,A: code_integer] :
% 0.22/0.52 ( ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.52 => ( ( bit_se1745604003318907178nteger @ N @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 0.22/0.52 = ( bit_se1745604003318907178nteger @ N @ A ) ) )
% 0.22/0.52 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.22/0.52 => ( ( bit_se1745604003318907178nteger @ N @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 0.22/0.52 = ( bit_se1345352211410354436nteger @ M @ ( bit_se1745604003318907178nteger @ N @ A ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_flip_bit_eq
% 0.22/0.52 thf(fact_6546_take__bit__flip__bit__eq,axiom,
% 0.22/0.52 ! [N: nat,M: nat,A: int] :
% 0.22/0.52 ( ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.52 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.22/0.52 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.22/0.52 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.22/0.52 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.22/0.52 = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_flip_bit_eq
% 0.22/0.52 thf(fact_6547_take__bit__flip__bit__eq,axiom,
% 0.22/0.52 ! [N: nat,M: nat,A: nat] :
% 0.22/0.52 ( ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.52 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.22/0.52 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 0.22/0.52 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.22/0.52 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.22/0.52 = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_flip_bit_eq
% 0.22/0.52 thf(fact_6548_unset__bit__nat__def,axiom,
% 0.22/0.52 ( bit_se4205575877204974255it_nat
% 0.22/0.52 = ( ^ [M5: nat,N4: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M5 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % unset_bit_nat_def
% 0.22/0.52 thf(fact_6549_pow_Osimps_I1_J,axiom,
% 0.22/0.52 ! [X: num] :
% 0.22/0.52 ( ( pow @ X @ one )
% 0.22/0.52 = X ) ).
% 0.22/0.52
% 0.22/0.52 % pow.simps(1)
% 0.22/0.52 thf(fact_6550_exp__gt__one,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_gt_one
% 0.22/0.52 thf(fact_6551_real__div__sqrt,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( divide_divide_real @ X @ ( sqrt @ X ) )
% 0.22/0.52 = ( sqrt @ X ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % real_div_sqrt
% 0.22/0.52 thf(fact_6552_sqrt__add__le__add__sqrt,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.52 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y4 ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y4 ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sqrt_add_le_add_sqrt
% 0.22/0.52 thf(fact_6553_take__bit__signed__take__bit,axiom,
% 0.22/0.52 ! [M: nat,N: nat,A: int] :
% 0.22/0.52 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.22/0.52 => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N @ A ) )
% 0.22/0.52 = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_signed_take_bit
% 0.22/0.52 thf(fact_6554_exp__ge__add__one__self,axiom,
% 0.22/0.52 ! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_ge_add_one_self
% 0.22/0.52 thf(fact_6555_le__real__sqrt__sumsq,axiom,
% 0.22/0.52 ! [X: real,Y4: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % le_real_sqrt_sumsq
% 0.22/0.52 thf(fact_6556_and__less__eq,axiom,
% 0.22/0.52 ! [L: int,K: int] :
% 0.22/0.52 ( ( ord_less_int @ L @ zero_zero_int )
% 0.22/0.52 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).
% 0.22/0.52
% 0.22/0.52 % and_less_eq
% 0.22/0.52 thf(fact_6557_AND__upper1_H_H,axiom,
% 0.22/0.52 ! [Y4: int,Z: int,Ya: int] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.22/0.52 => ( ( ord_less_int @ Y4 @ Z )
% 0.22/0.52 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y4 @ Ya ) @ Z ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % AND_upper1''
% 0.22/0.52 thf(fact_6558_AND__upper2_H_H,axiom,
% 0.22/0.52 ! [Y4: int,Z: int,X: int] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.22/0.52 => ( ( ord_less_int @ Y4 @ Z )
% 0.22/0.52 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y4 ) @ Z ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % AND_upper2''
% 0.22/0.52 thf(fact_6559_nat__mono__iff,axiom,
% 0.22/0.52 ! [Z: int,W: int] :
% 0.22/0.52 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.22/0.52 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.22/0.52 = ( ord_less_int @ W @ Z ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_mono_iff
% 0.22/0.52 thf(fact_6560_take__bit__eq__mask__iff,axiom,
% 0.22/0.52 ! [N: nat,K: int] :
% 0.22/0.52 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.22/0.52 = ( bit_se2000444600071755411sk_int @ N ) )
% 0.22/0.52 = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 0.22/0.52 = zero_zero_int ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_eq_mask_iff
% 0.22/0.52 thf(fact_6561_exp__minus__inverse,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) )
% 0.22/0.52 = one_one_real ) ).
% 0.22/0.52
% 0.22/0.52 % exp_minus_inverse
% 0.22/0.52 thf(fact_6562_exp__minus__inverse,axiom,
% 0.22/0.52 ! [X: complex] :
% 0.22/0.52 ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) )
% 0.22/0.52 = one_one_complex ) ).
% 0.22/0.52
% 0.22/0.52 % exp_minus_inverse
% 0.22/0.52 thf(fact_6563_take__bit__decr__eq,axiom,
% 0.22/0.52 ! [N: nat,K: int] :
% 0.22/0.52 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.22/0.52 != zero_zero_int )
% 0.22/0.52 => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
% 0.22/0.52 = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_decr_eq
% 0.22/0.52 thf(fact_6564_zless__nat__eq__int__zless,axiom,
% 0.22/0.52 ! [M: nat,Z: int] :
% 0.22/0.52 ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
% 0.22/0.52 = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% 0.22/0.52
% 0.22/0.52 % zless_nat_eq_int_zless
% 0.22/0.52 thf(fact_6565_nat__le__iff,axiom,
% 0.22/0.52 ! [X: int,N: nat] :
% 0.22/0.52 ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
% 0.22/0.52 = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_le_iff
% 0.22/0.52 thf(fact_6566_nat__0__le,axiom,
% 0.22/0.52 ! [Z: int] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.22/0.52 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.22/0.52 = Z ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_0_le
% 0.22/0.52 thf(fact_6567_int__eq__iff,axiom,
% 0.22/0.52 ! [M: nat,Z: int] :
% 0.22/0.52 ( ( ( semiri1314217659103216013at_int @ M )
% 0.22/0.52 = Z )
% 0.22/0.52 = ( ( M
% 0.22/0.52 = ( nat2 @ Z ) )
% 0.22/0.52 & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % int_eq_iff
% 0.22/0.52 thf(fact_6568_nat__int__add,axiom,
% 0.22/0.52 ! [A: nat,B: nat] :
% 0.22/0.52 ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
% 0.22/0.52 = ( plus_plus_nat @ A @ B ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_int_add
% 0.22/0.52 thf(fact_6569_exp__of__nat2__mult,axiom,
% 0.22/0.52 ! [X: complex,N: nat] :
% 0.22/0.52 ( ( exp_complex @ ( times_times_complex @ X @ ( semiri8010041392384452111omplex @ N ) ) )
% 0.22/0.52 = ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_of_nat2_mult
% 0.22/0.52 thf(fact_6570_exp__of__nat2__mult,axiom,
% 0.22/0.52 ! [X: real,N: nat] :
% 0.22/0.52 ( ( exp_real @ ( times_times_real @ X @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.22/0.52 = ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_of_nat2_mult
% 0.22/0.52 thf(fact_6571_exp__of__nat__mult,axiom,
% 0.22/0.52 ! [N: nat,X: complex] :
% 0.22/0.52 ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ X ) )
% 0.22/0.52 = ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_of_nat_mult
% 0.22/0.52 thf(fact_6572_exp__of__nat__mult,axiom,
% 0.22/0.52 ! [N: nat,X: real] :
% 0.22/0.52 ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) )
% 0.22/0.52 = ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_of_nat_mult
% 0.22/0.52 thf(fact_6573_int__minus,axiom,
% 0.22/0.52 ! [N: nat,M: nat] :
% 0.22/0.52 ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
% 0.22/0.52 = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % int_minus
% 0.22/0.52 thf(fact_6574_nat__abs__mult__distrib,axiom,
% 0.22/0.52 ! [W: int,Z: int] :
% 0.22/0.52 ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
% 0.22/0.52 = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_abs_mult_distrib
% 0.22/0.52 thf(fact_6575_even__and__iff,axiom,
% 0.22/0.52 ! [A: code_integer,B: code_integer] :
% 0.22/0.52 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ A @ B ) )
% 0.22/0.52 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.52 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % even_and_iff
% 0.22/0.52 thf(fact_6576_even__and__iff,axiom,
% 0.22/0.52 ! [A: int,B: int] :
% 0.22/0.52 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 0.22/0.52 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.52 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % even_and_iff
% 0.22/0.52 thf(fact_6577_even__and__iff,axiom,
% 0.22/0.52 ! [A: nat,B: nat] :
% 0.22/0.52 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 0.22/0.52 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.52 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % even_and_iff
% 0.22/0.52 thf(fact_6578_sqrt2__less__2,axiom,
% 0.22/0.52 ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.22/0.52
% 0.22/0.52 % sqrt2_less_2
% 0.22/0.52 thf(fact_6579_sgn__real__def,axiom,
% 0.22/0.52 ( sgn_sgn_real
% 0.22/0.52 = ( ^ [A3: real] : ( if_real @ ( A3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sgn_real_def
% 0.22/0.52 thf(fact_6580_exp__ge__add__one__self__aux,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_ge_add_one_self_aux
% 0.22/0.52 thf(fact_6581_even__and__iff__int,axiom,
% 0.22/0.52 ! [K: int,L: int] :
% 0.22/0.52 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 0.22/0.52 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.22/0.52 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % even_and_iff_int
% 0.22/0.52 thf(fact_6582_lemma__exp__total,axiom,
% 0.22/0.52 ! [Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ one_one_real @ Y4 )
% 0.22/0.52 => ? [X4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.22/0.52 & ( ord_less_eq_real @ X4 @ ( minus_minus_real @ Y4 @ one_one_real ) )
% 0.22/0.52 & ( ( exp_real @ X4 )
% 0.22/0.52 = Y4 ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % lemma_exp_total
% 0.22/0.52 thf(fact_6583_ln__ge__iff,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ Y4 @ ( ln_ln_real @ X ) )
% 0.22/0.52 = ( ord_less_eq_real @ ( exp_real @ Y4 ) @ X ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ln_ge_iff
% 0.22/0.52 thf(fact_6584_ln__x__over__x__mono,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ X @ Y4 )
% 0.22/0.52 => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y4 ) @ Y4 ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ln_x_over_x_mono
% 0.22/0.52 thf(fact_6585_nat__less__eq__zless,axiom,
% 0.22/0.52 ! [W: int,Z: int] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.22/0.52 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.22/0.52 = ( ord_less_int @ W @ Z ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_less_eq_zless
% 0.22/0.52 thf(fact_6586_nat__le__eq__zle,axiom,
% 0.22/0.52 ! [W: int,Z: int] :
% 0.22/0.52 ( ( ( ord_less_int @ zero_zero_int @ W )
% 0.22/0.52 | ( ord_less_eq_int @ zero_zero_int @ Z ) )
% 0.22/0.52 => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.22/0.52 = ( ord_less_eq_int @ W @ Z ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_le_eq_zle
% 0.22/0.52 thf(fact_6587_nat__eq__iff2,axiom,
% 0.22/0.52 ! [M: nat,W: int] :
% 0.22/0.52 ( ( M
% 0.22/0.52 = ( nat2 @ W ) )
% 0.22/0.52 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.22/0.52 => ( W
% 0.22/0.52 = ( semiri1314217659103216013at_int @ M ) ) )
% 0.22/0.52 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 0.22/0.52 => ( M = zero_zero_nat ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_eq_iff2
% 0.22/0.52 thf(fact_6588_nat__eq__iff,axiom,
% 0.22/0.52 ! [W: int,M: nat] :
% 0.22/0.52 ( ( ( nat2 @ W )
% 0.22/0.52 = M )
% 0.22/0.52 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.22/0.52 => ( W
% 0.22/0.52 = ( semiri1314217659103216013at_int @ M ) ) )
% 0.22/0.52 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 0.22/0.52 => ( M = zero_zero_nat ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_eq_iff
% 0.22/0.52 thf(fact_6589_split__nat,axiom,
% 0.22/0.52 ! [P: nat > $o,I: int] :
% 0.22/0.52 ( ( P @ ( nat2 @ I ) )
% 0.22/0.52 = ( ! [N4: nat] :
% 0.22/0.52 ( ( I
% 0.22/0.52 = ( semiri1314217659103216013at_int @ N4 ) )
% 0.22/0.52 => ( P @ N4 ) )
% 0.22/0.52 & ( ( ord_less_int @ I @ zero_zero_int )
% 0.22/0.52 => ( P @ zero_zero_nat ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % split_nat
% 0.22/0.52 thf(fact_6590_le__nat__iff,axiom,
% 0.22/0.52 ! [K: int,N: nat] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.22/0.52 => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 0.22/0.52 = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % le_nat_iff
% 0.22/0.52 thf(fact_6591_nat__add__distrib,axiom,
% 0.22/0.52 ! [Z: int,Z6: int] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.22/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 0.22/0.52 => ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
% 0.22/0.52 = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_add_distrib
% 0.22/0.52 thf(fact_6592_nat__mult__distrib,axiom,
% 0.22/0.52 ! [Z: int,Z6: int] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.22/0.52 => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
% 0.22/0.52 = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_mult_distrib
% 0.22/0.52 thf(fact_6593_Suc__as__int,axiom,
% 0.22/0.52 ( suc
% 0.22/0.52 = ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % Suc_as_int
% 0.22/0.52 thf(fact_6594_nat__diff__distrib,axiom,
% 0.22/0.52 ! [Z6: int,Z: int] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 0.22/0.52 => ( ( ord_less_eq_int @ Z6 @ Z )
% 0.22/0.52 => ( ( nat2 @ ( minus_minus_int @ Z @ Z6 ) )
% 0.22/0.52 = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_diff_distrib
% 0.22/0.52 thf(fact_6595_nat__diff__distrib_H,axiom,
% 0.22/0.52 ! [X: int,Y4: int] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.22/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.22/0.52 => ( ( nat2 @ ( minus_minus_int @ X @ Y4 ) )
% 0.22/0.52 = ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y4 ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_diff_distrib'
% 0.22/0.52 thf(fact_6596_nat__abs__triangle__ineq,axiom,
% 0.22/0.52 ! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_abs_triangle_ineq
% 0.22/0.52 thf(fact_6597_nat__div__distrib,axiom,
% 0.22/0.52 ! [X: int,Y4: int] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.22/0.52 => ( ( nat2 @ ( divide_divide_int @ X @ Y4 ) )
% 0.22/0.52 = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_div_distrib
% 0.22/0.52 thf(fact_6598_nat__div__distrib_H,axiom,
% 0.22/0.52 ! [Y4: int,X: int] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.22/0.52 => ( ( nat2 @ ( divide_divide_int @ X @ Y4 ) )
% 0.22/0.52 = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_div_distrib'
% 0.22/0.52 thf(fact_6599_nat__power__eq,axiom,
% 0.22/0.52 ! [Z: int,N: nat] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.22/0.52 => ( ( nat2 @ ( power_power_int @ Z @ N ) )
% 0.22/0.52 = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_power_eq
% 0.22/0.52 thf(fact_6600_nat__mod__distrib,axiom,
% 0.22/0.52 ! [X: int,Y4: int] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.22/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.22/0.52 => ( ( nat2 @ ( modulo_modulo_int @ X @ Y4 ) )
% 0.22/0.52 = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y4 ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_mod_distrib
% 0.22/0.52 thf(fact_6601_div__abs__eq__div__nat,axiom,
% 0.22/0.52 ! [K: int,L: int] :
% 0.22/0.52 ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
% 0.22/0.52 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % div_abs_eq_div_nat
% 0.22/0.52 thf(fact_6602_num_Osize_I4_J,axiom,
% 0.22/0.52 ( ( size_size_num @ one )
% 0.22/0.52 = zero_zero_nat ) ).
% 0.22/0.52
% 0.22/0.52 % num.size(4)
% 0.22/0.52 thf(fact_6603_mod__abs__eq__div__nat,axiom,
% 0.22/0.52 ! [K: int,L: int] :
% 0.22/0.52 ( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
% 0.22/0.52 = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % mod_abs_eq_div_nat
% 0.22/0.52 thf(fact_6604_take__bit__Suc__bit0,axiom,
% 0.22/0.52 ! [N: nat,K: num] :
% 0.22/0.52 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.22/0.52 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_Suc_bit0
% 0.22/0.52 thf(fact_6605_take__bit__Suc__bit0,axiom,
% 0.22/0.52 ! [N: nat,K: num] :
% 0.22/0.52 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.22/0.52 = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_Suc_bit0
% 0.22/0.52 thf(fact_6606_take__bit__eq__mod,axiom,
% 0.22/0.52 ( bit_se1745604003318907178nteger
% 0.22/0.52 = ( ^ [N4: nat,A3: code_integer] : ( modulo364778990260209775nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_eq_mod
% 0.22/0.52 thf(fact_6607_take__bit__eq__mod,axiom,
% 0.22/0.52 ( bit_se2923211474154528505it_int
% 0.22/0.52 = ( ^ [N4: nat,A3: int] : ( modulo_modulo_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_eq_mod
% 0.22/0.52 thf(fact_6608_take__bit__eq__mod,axiom,
% 0.22/0.52 ( bit_se2925701944663578781it_nat
% 0.22/0.52 = ( ^ [N4: nat,A3: nat] : ( modulo_modulo_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_eq_mod
% 0.22/0.52 thf(fact_6609_one__and__eq,axiom,
% 0.22/0.52 ! [A: code_integer] :
% 0.22/0.52 ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
% 0.22/0.52 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % one_and_eq
% 0.22/0.52 thf(fact_6610_one__and__eq,axiom,
% 0.22/0.52 ! [A: int] :
% 0.22/0.52 ( ( bit_se725231765392027082nd_int @ one_one_int @ A )
% 0.22/0.52 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % one_and_eq
% 0.22/0.52 thf(fact_6611_one__and__eq,axiom,
% 0.22/0.52 ! [A: nat] :
% 0.22/0.52 ( ( bit_se727722235901077358nd_nat @ one_one_nat @ A )
% 0.22/0.52 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % one_and_eq
% 0.22/0.52 thf(fact_6612_and__one__eq,axiom,
% 0.22/0.52 ! [A: code_integer] :
% 0.22/0.52 ( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
% 0.22/0.52 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % and_one_eq
% 0.22/0.52 thf(fact_6613_and__one__eq,axiom,
% 0.22/0.52 ! [A: int] :
% 0.22/0.52 ( ( bit_se725231765392027082nd_int @ A @ one_one_int )
% 0.22/0.52 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % and_one_eq
% 0.22/0.52 thf(fact_6614_and__one__eq,axiom,
% 0.22/0.52 ! [A: nat] :
% 0.22/0.52 ( ( bit_se727722235901077358nd_nat @ A @ one_one_nat )
% 0.22/0.52 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % and_one_eq
% 0.22/0.52 thf(fact_6615_take__bit__nat__eq__self__iff,axiom,
% 0.22/0.52 ! [N: nat,M: nat] :
% 0.22/0.52 ( ( ( bit_se2925701944663578781it_nat @ N @ M )
% 0.22/0.52 = M )
% 0.22/0.52 = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_nat_eq_self_iff
% 0.22/0.52 thf(fact_6616_take__bit__nat__less__exp,axiom,
% 0.22/0.52 ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_nat_less_exp
% 0.22/0.52 thf(fact_6617_take__bit__nat__eq__self,axiom,
% 0.22/0.52 ! [M: nat,N: nat] :
% 0.22/0.52 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.52 => ( ( bit_se2925701944663578781it_nat @ N @ M )
% 0.22/0.52 = M ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_nat_eq_self
% 0.22/0.52 thf(fact_6618_real__less__rsqrt,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y4 )
% 0.22/0.52 => ( ord_less_real @ X @ ( sqrt @ Y4 ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % real_less_rsqrt
% 0.22/0.52 thf(fact_6619_real__le__rsqrt,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y4 )
% 0.22/0.52 => ( ord_less_eq_real @ X @ ( sqrt @ Y4 ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % real_le_rsqrt
% 0.22/0.52 thf(fact_6620_sqrt__le__D,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( sqrt @ X ) @ Y4 )
% 0.22/0.52 => ( ord_less_eq_real @ X @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sqrt_le_D
% 0.22/0.52 thf(fact_6621_exp__le,axiom,
% 0.22/0.52 ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_le
% 0.22/0.52 thf(fact_6622_take__bit__nat__def,axiom,
% 0.22/0.52 ( bit_se2925701944663578781it_nat
% 0.22/0.52 = ( ^ [N4: nat,M5: nat] : ( modulo_modulo_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_nat_def
% 0.22/0.52 thf(fact_6623_nat__2,axiom,
% 0.22/0.52 ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.52 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_2
% 0.22/0.52 thf(fact_6624_sgn__power__injE,axiom,
% 0.22/0.52 ! [A: real,N: nat,X: real,B: real] :
% 0.22/0.52 ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 0.22/0.52 = X )
% 0.22/0.52 => ( ( X
% 0.22/0.52 = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
% 0.22/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.52 => ( A = B ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sgn_power_injE
% 0.22/0.52 thf(fact_6625_take__bit__int__less__exp,axiom,
% 0.22/0.52 ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_int_less_exp
% 0.22/0.52 thf(fact_6626_Suc__nat__eq__nat__zadd1,axiom,
% 0.22/0.52 ! [Z: int] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.22/0.52 => ( ( suc @ ( nat2 @ Z ) )
% 0.22/0.52 = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % Suc_nat_eq_nat_zadd1
% 0.22/0.52 thf(fact_6627_nat__less__iff,axiom,
% 0.22/0.52 ! [W: int,M: nat] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.22/0.52 => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
% 0.22/0.52 = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_less_iff
% 0.22/0.52 thf(fact_6628_take__bit__int__def,axiom,
% 0.22/0.52 ( bit_se2923211474154528505it_int
% 0.22/0.52 = ( ^ [N4: nat,K3: int] : ( modulo_modulo_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_int_def
% 0.22/0.52 thf(fact_6629_nat__mult__distrib__neg,axiom,
% 0.22/0.52 ! [Z: int,Z6: int] :
% 0.22/0.52 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 0.22/0.52 => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
% 0.22/0.52 = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_mult_distrib_neg
% 0.22/0.52 thf(fact_6630_exp__divide__power__eq,axiom,
% 0.22/0.52 ! [N: nat,X: complex] :
% 0.22/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.52 => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
% 0.22/0.52 = ( exp_complex @ X ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_divide_power_eq
% 0.22/0.52 thf(fact_6631_exp__divide__power__eq,axiom,
% 0.22/0.52 ! [N: nat,X: real] :
% 0.22/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.52 => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
% 0.22/0.52 = ( exp_real @ X ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_divide_power_eq
% 0.22/0.52 thf(fact_6632_nat__abs__int__diff,axiom,
% 0.22/0.52 ! [A: nat,B: nat] :
% 0.22/0.52 ( ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.52 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 0.22/0.52 = ( minus_minus_nat @ B @ A ) ) )
% 0.22/0.52 & ( ~ ( ord_less_eq_nat @ A @ B )
% 0.22/0.52 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 0.22/0.52 = ( minus_minus_nat @ A @ B ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_abs_int_diff
% 0.22/0.52 thf(fact_6633_tanh__altdef,axiom,
% 0.22/0.52 ( tanh_real
% 0.22/0.52 = ( ^ [X3: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) @ ( plus_plus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % tanh_altdef
% 0.22/0.52 thf(fact_6634_tanh__altdef,axiom,
% 0.22/0.52 ( tanh_complex
% 0.22/0.52 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % tanh_altdef
% 0.22/0.52 thf(fact_6635_take__bit__eq__0__iff,axiom,
% 0.22/0.52 ! [N: nat,A: code_integer] :
% 0.22/0.52 ( ( ( bit_se1745604003318907178nteger @ N @ A )
% 0.22/0.52 = zero_z3403309356797280102nteger )
% 0.22/0.52 = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_eq_0_iff
% 0.22/0.52 thf(fact_6636_take__bit__eq__0__iff,axiom,
% 0.22/0.52 ! [N: nat,A: int] :
% 0.22/0.52 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.22/0.52 = zero_zero_int )
% 0.22/0.52 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_eq_0_iff
% 0.22/0.52 thf(fact_6637_take__bit__eq__0__iff,axiom,
% 0.22/0.52 ! [N: nat,A: nat] :
% 0.22/0.52 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.22/0.52 = zero_zero_nat )
% 0.22/0.52 = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_eq_0_iff
% 0.22/0.52 thf(fact_6638_take__bit__numeral__bit0,axiom,
% 0.22/0.52 ! [L: num,K: num] :
% 0.22/0.52 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.22/0.52 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_numeral_bit0
% 0.22/0.52 thf(fact_6639_take__bit__numeral__bit0,axiom,
% 0.22/0.52 ! [L: num,K: num] :
% 0.22/0.52 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.22/0.52 = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_numeral_bit0
% 0.22/0.52 thf(fact_6640_take__bit__nat__less__self__iff,axiom,
% 0.22/0.52 ! [N: nat,M: nat] :
% 0.22/0.52 ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
% 0.22/0.52 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_nat_less_self_iff
% 0.22/0.52 thf(fact_6641_real__sqrt__unique,axiom,
% 0.22/0.52 ! [Y4: real,X: real] :
% 0.22/0.52 ( ( ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.52 = X )
% 0.22/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.52 => ( ( sqrt @ X )
% 0.22/0.52 = Y4 ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % real_sqrt_unique
% 0.22/0.52 thf(fact_6642_real__le__lsqrt,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.52 => ( ( ord_less_eq_real @ X @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ord_less_eq_real @ ( sqrt @ X ) @ Y4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % real_le_lsqrt
% 0.22/0.52 thf(fact_6643_lemma__real__divide__sqrt__less,axiom,
% 0.22/0.52 ! [U: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ U )
% 0.22/0.52 => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 0.22/0.52
% 0.22/0.52 % lemma_real_divide_sqrt_less
% 0.22/0.52 thf(fact_6644_real__sqrt__sum__squares__eq__cancel2,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.52 = Y4 )
% 0.22/0.52 => ( X = zero_zero_real ) ) ).
% 0.22/0.52
% 0.22/0.52 % real_sqrt_sum_squares_eq_cancel2
% 0.22/0.52 thf(fact_6645_real__sqrt__sum__squares__eq__cancel,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.52 = X )
% 0.22/0.52 => ( Y4 = zero_zero_real ) ) ).
% 0.22/0.52
% 0.22/0.52 % real_sqrt_sum_squares_eq_cancel
% 0.22/0.52 thf(fact_6646_real__sqrt__sum__squares__triangle__ineq,axiom,
% 0.22/0.52 ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % real_sqrt_sum_squares_triangle_ineq
% 0.22/0.52 thf(fact_6647_real__sqrt__sum__squares__ge2,axiom,
% 0.22/0.52 ! [Y4: real,X: real] : ( ord_less_eq_real @ Y4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % real_sqrt_sum_squares_ge2
% 0.22/0.52 thf(fact_6648_real__sqrt__sum__squares__ge1,axiom,
% 0.22/0.52 ! [X: real,Y4: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % real_sqrt_sum_squares_ge1
% 0.22/0.52 thf(fact_6649_exp__half__le2,axiom,
% 0.22/0.52 ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_half_le2
% 0.22/0.52 thf(fact_6650_sqrt__ge__absD,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y4 ) )
% 0.22/0.52 => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y4 ) ) ).
% 0.22/0.52
% 0.22/0.52 % sqrt_ge_absD
% 0.22/0.52 thf(fact_6651_take__bit__Suc__minus__bit0,axiom,
% 0.22/0.52 ! [N: nat,K: num] :
% 0.22/0.52 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.22/0.52 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_Suc_minus_bit0
% 0.22/0.52 thf(fact_6652_take__bit__int__greater__eq__self__iff,axiom,
% 0.22/0.52 ! [K: int,N: nat] :
% 0.22/0.52 ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.22/0.52 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_int_greater_eq_self_iff
% 0.22/0.52 thf(fact_6653_take__bit__int__less__self__iff,axiom,
% 0.22/0.52 ! [N: nat,K: int] :
% 0.22/0.52 ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 0.22/0.52 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_int_less_self_iff
% 0.22/0.52 thf(fact_6654_exp__double,axiom,
% 0.22/0.52 ! [Z: complex] :
% 0.22/0.52 ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) )
% 0.22/0.52 = ( power_power_complex @ ( exp_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_double
% 0.22/0.52 thf(fact_6655_exp__double,axiom,
% 0.22/0.52 ! [Z: real] :
% 0.22/0.52 ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) )
% 0.22/0.52 = ( power_power_real @ ( exp_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_double
% 0.22/0.52 thf(fact_6656_nat__dvd__iff,axiom,
% 0.22/0.52 ! [Z: int,M: nat] :
% 0.22/0.52 ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
% 0.22/0.52 = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.22/0.52 => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
% 0.22/0.52 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.22/0.52 => ( M = zero_zero_nat ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_dvd_iff
% 0.22/0.52 thf(fact_6657_real__less__lsqrt,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.52 => ( ( ord_less_real @ X @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ord_less_real @ ( sqrt @ X ) @ Y4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % real_less_lsqrt
% 0.22/0.52 thf(fact_6658_sqrt__sum__squares__le__sum,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.52 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sqrt_sum_squares_le_sum
% 0.22/0.52 thf(fact_6659_sqrt__sum__squares__le__sum__abs,axiom,
% 0.22/0.52 ! [X: real,Y4: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y4 ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sqrt_sum_squares_le_sum_abs
% 0.22/0.52 thf(fact_6660_real__sqrt__ge__abs2,axiom,
% 0.22/0.52 ! [Y4: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % real_sqrt_ge_abs2
% 0.22/0.52 thf(fact_6661_real__sqrt__ge__abs1,axiom,
% 0.22/0.52 ! [X: real,Y4: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % real_sqrt_ge_abs1
% 0.22/0.52 thf(fact_6662_sqrt__even__pow2,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.52 => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.52 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sqrt_even_pow2
% 0.22/0.52 thf(fact_6663_ln__sqrt,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ln_ln_real @ ( sqrt @ X ) )
% 0.22/0.52 = ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ln_sqrt
% 0.22/0.52 thf(fact_6664_arsinh__real__def,axiom,
% 0.22/0.52 ( arsinh_real
% 0.22/0.52 = ( ^ [X3: real] : ( ln_ln_real @ ( plus_plus_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % arsinh_real_def
% 0.22/0.52 thf(fact_6665_take__bit__int__eq__self,axiom,
% 0.22/0.52 ! [K: int,N: nat] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.22/0.52 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.52 => ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.22/0.52 = K ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_int_eq_self
% 0.22/0.52 thf(fact_6666_take__bit__int__eq__self__iff,axiom,
% 0.22/0.52 ! [N: nat,K: int] :
% 0.22/0.52 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.22/0.52 = K )
% 0.22/0.52 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.22/0.52 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_int_eq_self_iff
% 0.22/0.52 thf(fact_6667_and__int__rec,axiom,
% 0.22/0.52 ( bit_se725231765392027082nd_int
% 0.22/0.52 = ( ^ [K3: int,L3: int] :
% 0.22/0.52 ( plus_plus_int
% 0.22/0.52 @ ( zero_n2684676970156552555ol_int
% 0.22/0.52 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.22/0.52 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
% 0.22/0.52 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % and_int_rec
% 0.22/0.52 thf(fact_6668_take__bit__numeral__minus__bit0,axiom,
% 0.22/0.52 ! [L: num,K: num] :
% 0.22/0.52 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.22/0.52 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_numeral_minus_bit0
% 0.22/0.52 thf(fact_6669_take__bit__incr__eq,axiom,
% 0.22/0.52 ! [N: nat,K: int] :
% 0.22/0.52 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.22/0.52 != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 0.22/0.52 => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 0.22/0.52 = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_incr_eq
% 0.22/0.52 thf(fact_6670_take__bit__eq__mask__iff__exp__dvd,axiom,
% 0.22/0.52 ! [N: nat,K: int] :
% 0.22/0.52 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.22/0.52 = ( bit_se2000444600071755411sk_int @ N ) )
% 0.22/0.52 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_eq_mask_iff_exp_dvd
% 0.22/0.52 thf(fact_6671_exp__bound__half,axiom,
% 0.22/0.52 ! [Z: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_bound_half
% 0.22/0.52 thf(fact_6672_exp__bound__half,axiom,
% 0.22/0.52 ! [Z: complex] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_bound_half
% 0.22/0.52 thf(fact_6673_take__bit__Suc__minus__1__eq,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.52 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_Code_integer ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_Suc_minus_1_eq
% 0.22/0.52 thf(fact_6674_take__bit__Suc__minus__1__eq,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.52 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_Suc_minus_1_eq
% 0.22/0.52 thf(fact_6675_take__bit__Suc__bit1,axiom,
% 0.22/0.52 ! [N: nat,K: num] :
% 0.22/0.52 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.22/0.52 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_Suc_bit1
% 0.22/0.52 thf(fact_6676_take__bit__Suc__bit1,axiom,
% 0.22/0.52 ! [N: nat,K: num] :
% 0.22/0.52 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.22/0.52 = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_Suc_bit1
% 0.22/0.52 thf(fact_6677_take__bit__numeral__minus__1__eq,axiom,
% 0.22/0.52 ! [K: num] :
% 0.22/0.52 ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.52 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_Code_integer ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_numeral_minus_1_eq
% 0.22/0.52 thf(fact_6678_take__bit__numeral__minus__1__eq,axiom,
% 0.22/0.52 ! [K: num] :
% 0.22/0.52 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.52 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_numeral_minus_1_eq
% 0.22/0.52 thf(fact_6679_take__bit__Suc,axiom,
% 0.22/0.52 ! [N: nat,A: code_integer] :
% 0.22/0.52 ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A )
% 0.22/0.52 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_Suc
% 0.22/0.52 thf(fact_6680_take__bit__Suc,axiom,
% 0.22/0.52 ! [N: nat,A: int] :
% 0.22/0.52 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 0.22/0.52 = ( 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 ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_Suc
% 0.22/0.52 thf(fact_6681_take__bit__Suc,axiom,
% 0.22/0.52 ! [N: nat,A: nat] :
% 0.22/0.52 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A )
% 0.22/0.52 = ( 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 ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_Suc
% 0.22/0.52 thf(fact_6682_arsinh__real__aux,axiom,
% 0.22/0.52 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % arsinh_real_aux
% 0.22/0.52 thf(fact_6683_exp__bound,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.22/0.52 => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_bound
% 0.22/0.52 thf(fact_6684_real__sqrt__sum__squares__mult__ge__zero,axiom,
% 0.22/0.52 ! [X: real,Y4: real,Xa: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( 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 ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % real_sqrt_sum_squares_mult_ge_zero
% 0.22/0.52 thf(fact_6685_real__sqrt__power__even,axiom,
% 0.22/0.52 ! [N: nat,X: real] :
% 0.22/0.52 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( power_power_real @ ( sqrt @ X ) @ N )
% 0.22/0.52 = ( power_power_real @ X @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % real_sqrt_power_even
% 0.22/0.52 thf(fact_6686_arith__geo__mean__sqrt,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.52 => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y4 ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % arith_geo_mean_sqrt
% 0.22/0.52 thf(fact_6687_take__bit__int__less__eq,axiom,
% 0.22/0.52 ! [N: nat,K: int] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 0.22/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.52 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_int_less_eq
% 0.22/0.52 thf(fact_6688_take__bit__int__greater__eq,axiom,
% 0.22/0.52 ! [K: int,N: nat] :
% 0.22/0.52 ( ( ord_less_int @ K @ zero_zero_int )
% 0.22/0.52 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_int_greater_eq
% 0.22/0.52 thf(fact_6689_even__nat__iff,axiom,
% 0.22/0.52 ! [K: int] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.22/0.52 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
% 0.22/0.52 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % even_nat_iff
% 0.22/0.52 thf(fact_6690_signed__take__bit__eq__take__bit__shift,axiom,
% 0.22/0.52 ( bit_ri631733984087533419it_int
% 0.22/0.52 = ( ^ [N4: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % signed_take_bit_eq_take_bit_shift
% 0.22/0.52 thf(fact_6691_num_Osize_I5_J,axiom,
% 0.22/0.52 ! [X2: num] :
% 0.22/0.52 ( ( size_size_num @ ( bit0 @ X2 ) )
% 0.22/0.52 = ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % num.size(5)
% 0.22/0.52 thf(fact_6692_stable__imp__take__bit__eq,axiom,
% 0.22/0.52 ! [A: code_integer,N: nat] :
% 0.22/0.52 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.52 = A )
% 0.22/0.52 => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.52 => ( ( bit_se1745604003318907178nteger @ N @ A )
% 0.22/0.52 = zero_z3403309356797280102nteger ) )
% 0.22/0.52 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.52 => ( ( bit_se1745604003318907178nteger @ N @ A )
% 0.22/0.52 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % stable_imp_take_bit_eq
% 0.22/0.52 thf(fact_6693_stable__imp__take__bit__eq,axiom,
% 0.22/0.52 ! [A: int,N: nat] :
% 0.22/0.52 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.52 = A )
% 0.22/0.52 => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.52 => ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.22/0.52 = zero_zero_int ) )
% 0.22/0.52 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.52 => ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.22/0.52 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % stable_imp_take_bit_eq
% 0.22/0.52 thf(fact_6694_stable__imp__take__bit__eq,axiom,
% 0.22/0.52 ! [A: nat,N: nat] :
% 0.22/0.52 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.52 = A )
% 0.22/0.52 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.52 => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.22/0.52 = zero_zero_nat ) )
% 0.22/0.52 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.52 => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.22/0.52 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % stable_imp_take_bit_eq
% 0.22/0.52 thf(fact_6695_num_Osize_I6_J,axiom,
% 0.22/0.52 ! [X33: num] :
% 0.22/0.52 ( ( size_size_num @ ( bit1 @ X33 ) )
% 0.22/0.52 = ( plus_plus_nat @ ( size_size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % num.size(6)
% 0.22/0.52 thf(fact_6696_take__bit__numeral__bit1,axiom,
% 0.22/0.52 ! [L: num,K: num] :
% 0.22/0.52 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.22/0.52 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_numeral_bit1
% 0.22/0.52 thf(fact_6697_take__bit__numeral__bit1,axiom,
% 0.22/0.52 ! [L: num,K: num] :
% 0.22/0.52 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.22/0.52 = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_numeral_bit1
% 0.22/0.52 thf(fact_6698_real__exp__bound__lemma,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % real_exp_bound_lemma
% 0.22/0.52 thf(fact_6699_cos__x__y__le__one,axiom,
% 0.22/0.52 ! [X: real,Y4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% 0.22/0.52
% 0.22/0.52 % cos_x_y_le_one
% 0.22/0.52 thf(fact_6700_real__sqrt__sum__squares__less,axiom,
% 0.22/0.52 ! [X: real,U: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.22/0.52 => ( ( ord_less_real @ ( abs_abs_real @ Y4 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.22/0.52 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % real_sqrt_sum_squares_less
% 0.22/0.52 thf(fact_6701_arcosh__real__def,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.22/0.52 => ( ( arcosh_real @ X )
% 0.22/0.52 = ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % arcosh_real_def
% 0.22/0.52 thf(fact_6702_exp__ge__one__plus__x__over__n__power__n,axiom,
% 0.22/0.52 ! [N: nat,X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
% 0.22/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.52 => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_ge_one_plus_x_over_n_power_n
% 0.22/0.52 thf(fact_6703_exp__ge__one__minus__x__over__n__power__n,axiom,
% 0.22/0.52 ! [X: real,N: nat] :
% 0.22/0.52 ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 0.22/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.52 => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_ge_one_minus_x_over_n_power_n
% 0.22/0.52 thf(fact_6704_take__bit__minus__small__eq,axiom,
% 0.22/0.52 ! [K: int,N: nat] :
% 0.22/0.52 ( ( ord_less_int @ zero_zero_int @ K )
% 0.22/0.52 => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.52 => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
% 0.22/0.52 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_minus_small_eq
% 0.22/0.52 thf(fact_6705_exp__bound__lemma,axiom,
% 0.22/0.52 ! [Z: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( 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 ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_bound_lemma
% 0.22/0.52 thf(fact_6706_exp__bound__lemma,axiom,
% 0.22/0.52 ! [Z: complex] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( 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 ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_bound_lemma
% 0.22/0.52 thf(fact_6707_modulo__int__def,axiom,
% 0.22/0.52 ( modulo_modulo_int
% 0.22/0.52 = ( ^ [K3: int,L3: int] :
% 0.22/0.52 ( if_int @ ( L3 = zero_zero_int ) @ K3
% 0.22/0.52 @ ( if_int
% 0.22/0.52 @ ( ( sgn_sgn_int @ K3 )
% 0.22/0.52 = ( sgn_sgn_int @ L3 ) )
% 0.22/0.52 @ ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) ) )
% 0.22/0.52 @ ( times_times_int @ ( sgn_sgn_int @ L3 )
% 0.22/0.52 @ ( minus_minus_int
% 0.22/0.52 @ ( times_times_int @ ( abs_abs_int @ L3 )
% 0.22/0.52 @ ( zero_n2684676970156552555ol_int
% 0.22/0.52 @ ~ ( dvd_dvd_int @ L3 @ K3 ) ) )
% 0.22/0.52 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % modulo_int_def
% 0.22/0.52 thf(fact_6708_sqrt__sum__squares__half__less,axiom,
% 0.22/0.52 ! [X: real,U: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ( ord_less_real @ Y4 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.52 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sqrt_sum_squares_half_less
% 0.22/0.52 thf(fact_6709_exp__lower__Taylor__quadratic,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( divide_divide_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % exp_lower_Taylor_quadratic
% 0.22/0.52 thf(fact_6710_take__bit__numeral__minus__bit1,axiom,
% 0.22/0.52 ! [L: num,K: num] :
% 0.22/0.52 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.22/0.52 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_numeral_minus_bit1
% 0.22/0.52 thf(fact_6711_take__bit__Suc__minus__bit1,axiom,
% 0.22/0.52 ! [N: nat,K: num] :
% 0.22/0.52 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.22/0.52 = ( 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 ) ) ).
% 0.22/0.52
% 0.22/0.52 % take_bit_Suc_minus_bit1
% 0.22/0.52 thf(fact_6712_foldr__same__int,axiom,
% 0.22/0.52 ! [Xs: list_nat,Y4: nat] :
% 0.22/0.52 ( ! [X4: nat,Y: nat] :
% 0.22/0.52 ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 0.22/0.52 => ( ( member_nat @ Y @ ( set_nat2 @ Xs ) )
% 0.22/0.52 => ( X4 = Y ) ) )
% 0.22/0.52 => ( ! [X4: nat] :
% 0.22/0.52 ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 0.22/0.52 => ( X4 = Y4 ) )
% 0.22/0.52 => ( ( foldr_nat_nat @ plus_plus_nat @ Xs @ zero_zero_nat )
% 0.22/0.52 = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ Y4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % foldr_same_int
% 0.22/0.52 thf(fact_6713_log__base__10__eq1,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 0.22/0.52 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_base_10_eq1
% 0.22/0.52 thf(fact_6714_log__base__10__eq2,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 0.22/0.52 = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_base_10_eq2
% 0.22/0.52 thf(fact_6715_machin,axiom,
% 0.22/0.52 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.22/0.52 = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % machin
% 0.22/0.52 thf(fact_6716_foldr__one,axiom,
% 0.22/0.52 ! [D: nat,Ys: list_nat] : ( ord_less_eq_nat @ D @ ( foldr_nat_nat @ plus_plus_nat @ Ys @ D ) ) ).
% 0.22/0.52
% 0.22/0.52 % foldr_one
% 0.22/0.52 thf(fact_6717_log__one,axiom,
% 0.22/0.52 ! [A: real] :
% 0.22/0.52 ( ( log @ A @ one_one_real )
% 0.22/0.52 = zero_zero_real ) ).
% 0.22/0.52
% 0.22/0.52 % log_one
% 0.22/0.52 thf(fact_6718_pred__numeral__inc,axiom,
% 0.22/0.52 ! [K: num] :
% 0.22/0.52 ( ( pred_numeral @ ( inc @ K ) )
% 0.22/0.52 = ( numeral_numeral_nat @ K ) ) ).
% 0.22/0.52
% 0.22/0.52 % pred_numeral_inc
% 0.22/0.52 thf(fact_6719_log__eq__one,axiom,
% 0.22/0.52 ! [A: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.52 => ( ( A != one_one_real )
% 0.22/0.52 => ( ( log @ A @ A )
% 0.22/0.52 = one_one_real ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_eq_one
% 0.22/0.52 thf(fact_6720_log__less__cancel__iff,axiom,
% 0.22/0.52 ! [A: real,X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_real @ one_one_real @ A )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.22/0.52 => ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y4 ) )
% 0.22/0.52 = ( ord_less_real @ X @ Y4 ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_less_cancel_iff
% 0.22/0.52 thf(fact_6721_log__less__one__cancel__iff,axiom,
% 0.22/0.52 ! [A: real,X: real] :
% 0.22/0.52 ( ( ord_less_real @ one_one_real @ A )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_real @ ( log @ A @ X ) @ one_one_real )
% 0.22/0.52 = ( ord_less_real @ X @ A ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_less_one_cancel_iff
% 0.22/0.52 thf(fact_6722_one__less__log__cancel__iff,axiom,
% 0.22/0.52 ! [A: real,X: real] :
% 0.22/0.52 ( ( ord_less_real @ one_one_real @ A )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_real @ one_one_real @ ( log @ A @ X ) )
% 0.22/0.52 = ( ord_less_real @ A @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % one_less_log_cancel_iff
% 0.22/0.52 thf(fact_6723_log__less__zero__cancel__iff,axiom,
% 0.22/0.52 ! [A: real,X: real] :
% 0.22/0.52 ( ( ord_less_real @ one_one_real @ A )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_real @ ( log @ A @ X ) @ zero_zero_real )
% 0.22/0.52 = ( ord_less_real @ X @ one_one_real ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_less_zero_cancel_iff
% 0.22/0.52 thf(fact_6724_zero__less__log__cancel__iff,axiom,
% 0.22/0.52 ! [A: real,X: real] :
% 0.22/0.52 ( ( ord_less_real @ one_one_real @ A )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X ) )
% 0.22/0.52 = ( ord_less_real @ one_one_real @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % zero_less_log_cancel_iff
% 0.22/0.52 thf(fact_6725_log__le__cancel__iff,axiom,
% 0.22/0.52 ! [A: real,X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_real @ one_one_real @ A )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.22/0.52 => ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y4 ) )
% 0.22/0.52 = ( ord_less_eq_real @ X @ Y4 ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_le_cancel_iff
% 0.22/0.52 thf(fact_6726_log__le__one__cancel__iff,axiom,
% 0.22/0.52 ! [A: real,X: real] :
% 0.22/0.52 ( ( ord_less_real @ one_one_real @ A )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
% 0.22/0.52 = ( ord_less_eq_real @ X @ A ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_le_one_cancel_iff
% 0.22/0.52 thf(fact_6727_one__le__log__cancel__iff,axiom,
% 0.22/0.52 ! [A: real,X: real] :
% 0.22/0.52 ( ( ord_less_real @ one_one_real @ A )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
% 0.22/0.52 = ( ord_less_eq_real @ A @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % one_le_log_cancel_iff
% 0.22/0.52 thf(fact_6728_log__le__zero__cancel__iff,axiom,
% 0.22/0.52 ! [A: real,X: real] :
% 0.22/0.52 ( ( ord_less_real @ one_one_real @ A )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
% 0.22/0.52 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_le_zero_cancel_iff
% 0.22/0.52 thf(fact_6729_zero__le__log__cancel__iff,axiom,
% 0.22/0.52 ! [A: real,X: real] :
% 0.22/0.52 ( ( ord_less_real @ one_one_real @ A )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
% 0.22/0.52 = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % zero_le_log_cancel_iff
% 0.22/0.52 thf(fact_6730_add__neg__numeral__special_I6_J,axiom,
% 0.22/0.52 ! [M: num] :
% 0.22/0.52 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.52 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % add_neg_numeral_special(6)
% 0.22/0.52 thf(fact_6731_add__neg__numeral__special_I6_J,axiom,
% 0.22/0.52 ! [M: num] :
% 0.22/0.52 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.52 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % add_neg_numeral_special(6)
% 0.22/0.52 thf(fact_6732_add__neg__numeral__special_I6_J,axiom,
% 0.22/0.52 ! [M: num] :
% 0.22/0.52 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.22/0.52 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % add_neg_numeral_special(6)
% 0.22/0.52 thf(fact_6733_add__neg__numeral__special_I6_J,axiom,
% 0.22/0.52 ! [M: num] :
% 0.22/0.52 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.52 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % add_neg_numeral_special(6)
% 0.22/0.52 thf(fact_6734_add__neg__numeral__special_I6_J,axiom,
% 0.22/0.52 ! [M: num] :
% 0.22/0.52 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.22/0.52 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % add_neg_numeral_special(6)
% 0.22/0.52 thf(fact_6735_add__neg__numeral__special_I5_J,axiom,
% 0.22/0.52 ! [N: num] :
% 0.22/0.52 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.22/0.52 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % add_neg_numeral_special(5)
% 0.22/0.52 thf(fact_6736_add__neg__numeral__special_I5_J,axiom,
% 0.22/0.52 ! [N: num] :
% 0.22/0.52 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.52 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % add_neg_numeral_special(5)
% 0.22/0.52 thf(fact_6737_add__neg__numeral__special_I5_J,axiom,
% 0.22/0.52 ! [N: num] :
% 0.22/0.52 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.22/0.52 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % add_neg_numeral_special(5)
% 0.22/0.52 thf(fact_6738_add__neg__numeral__special_I5_J,axiom,
% 0.22/0.52 ! [N: num] :
% 0.22/0.52 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.22/0.52 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % add_neg_numeral_special(5)
% 0.22/0.52 thf(fact_6739_add__neg__numeral__special_I5_J,axiom,
% 0.22/0.52 ! [N: num] :
% 0.22/0.52 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.22/0.52 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % add_neg_numeral_special(5)
% 0.22/0.52 thf(fact_6740_diff__numeral__special_I6_J,axiom,
% 0.22/0.52 ! [M: num] :
% 0.22/0.52 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.52 = ( numeral_numeral_real @ ( inc @ M ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % diff_numeral_special(6)
% 0.22/0.52 thf(fact_6741_diff__numeral__special_I6_J,axiom,
% 0.22/0.52 ! [M: num] :
% 0.22/0.52 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.52 = ( numeral_numeral_int @ ( inc @ M ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % diff_numeral_special(6)
% 0.22/0.52 thf(fact_6742_diff__numeral__special_I6_J,axiom,
% 0.22/0.52 ! [M: num] :
% 0.22/0.52 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.22/0.52 = ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % diff_numeral_special(6)
% 0.22/0.52 thf(fact_6743_diff__numeral__special_I6_J,axiom,
% 0.22/0.52 ! [M: num] :
% 0.22/0.52 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.52 = ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % diff_numeral_special(6)
% 0.22/0.52 thf(fact_6744_diff__numeral__special_I6_J,axiom,
% 0.22/0.52 ! [M: num] :
% 0.22/0.52 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.22/0.52 = ( numeral_numeral_rat @ ( inc @ M ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % diff_numeral_special(6)
% 0.22/0.52 thf(fact_6745_diff__numeral__special_I5_J,axiom,
% 0.22/0.52 ! [N: num] :
% 0.22/0.52 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N ) )
% 0.22/0.52 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % diff_numeral_special(5)
% 0.22/0.52 thf(fact_6746_diff__numeral__special_I5_J,axiom,
% 0.22/0.52 ! [N: num] :
% 0.22/0.52 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.22/0.52 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % diff_numeral_special(5)
% 0.22/0.52 thf(fact_6747_diff__numeral__special_I5_J,axiom,
% 0.22/0.52 ! [N: num] :
% 0.22/0.52 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N ) )
% 0.22/0.52 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % diff_numeral_special(5)
% 0.22/0.52 thf(fact_6748_diff__numeral__special_I5_J,axiom,
% 0.22/0.52 ! [N: num] :
% 0.22/0.52 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
% 0.22/0.52 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % diff_numeral_special(5)
% 0.22/0.52 thf(fact_6749_diff__numeral__special_I5_J,axiom,
% 0.22/0.52 ! [N: num] :
% 0.22/0.52 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
% 0.22/0.52 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % diff_numeral_special(5)
% 0.22/0.52 thf(fact_6750_and__nat__numerals_I3_J,axiom,
% 0.22/0.52 ! [X: num] :
% 0.22/0.52 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.22/0.52 = zero_zero_nat ) ).
% 0.22/0.52
% 0.22/0.52 % and_nat_numerals(3)
% 0.22/0.52 thf(fact_6751_and__nat__numerals_I1_J,axiom,
% 0.22/0.52 ! [Y4: num] :
% 0.22/0.52 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 0.22/0.52 = zero_zero_nat ) ).
% 0.22/0.52
% 0.22/0.52 % and_nat_numerals(1)
% 0.22/0.52 thf(fact_6752_log__pow__cancel,axiom,
% 0.22/0.52 ! [A: real,B: nat] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.52 => ( ( A != one_one_real )
% 0.22/0.52 => ( ( log @ A @ ( power_power_real @ A @ B ) )
% 0.22/0.52 = ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_pow_cancel
% 0.22/0.52 thf(fact_6753_and__nat__numerals_I4_J,axiom,
% 0.22/0.52 ! [X: num] :
% 0.22/0.52 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.22/0.52 = one_one_nat ) ).
% 0.22/0.52
% 0.22/0.52 % and_nat_numerals(4)
% 0.22/0.52 thf(fact_6754_and__nat__numerals_I2_J,axiom,
% 0.22/0.52 ! [Y4: num] :
% 0.22/0.52 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 0.22/0.52 = one_one_nat ) ).
% 0.22/0.52
% 0.22/0.52 % and_nat_numerals(2)
% 0.22/0.52 thf(fact_6755_Suc__0__and__eq,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.22/0.52 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % Suc_0_and_eq
% 0.22/0.52 thf(fact_6756_and__Suc__0__eq,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.22/0.52 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % and_Suc_0_eq
% 0.22/0.52 thf(fact_6757_pi__neq__zero,axiom,
% 0.22/0.52 pi != zero_zero_real ).
% 0.22/0.52
% 0.22/0.52 % pi_neq_zero
% 0.22/0.52 thf(fact_6758_num__induct,axiom,
% 0.22/0.52 ! [P: num > $o,X: num] :
% 0.22/0.52 ( ( P @ one )
% 0.22/0.52 => ( ! [X4: num] :
% 0.22/0.52 ( ( P @ X4 )
% 0.22/0.52 => ( P @ ( inc @ X4 ) ) )
% 0.22/0.52 => ( P @ X ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % num_induct
% 0.22/0.52 thf(fact_6759_add__inc,axiom,
% 0.22/0.52 ! [X: num,Y4: num] :
% 0.22/0.52 ( ( plus_plus_num @ X @ ( inc @ Y4 ) )
% 0.22/0.52 = ( inc @ ( plus_plus_num @ X @ Y4 ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % add_inc
% 0.22/0.52 thf(fact_6760_log__def,axiom,
% 0.22/0.52 ( log
% 0.22/0.52 = ( ^ [A3: real,X3: real] : ( divide_divide_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ A3 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_def
% 0.22/0.52 thf(fact_6761_pi__gt__zero,axiom,
% 0.22/0.52 ord_less_real @ zero_zero_real @ pi ).
% 0.22/0.52
% 0.22/0.52 % pi_gt_zero
% 0.22/0.52 thf(fact_6762_pi__not__less__zero,axiom,
% 0.22/0.52 ~ ( ord_less_real @ pi @ zero_zero_real ) ).
% 0.22/0.52
% 0.22/0.52 % pi_not_less_zero
% 0.22/0.52 thf(fact_6763_pi__ge__zero,axiom,
% 0.22/0.52 ord_less_eq_real @ zero_zero_real @ pi ).
% 0.22/0.52
% 0.22/0.52 % pi_ge_zero
% 0.22/0.52 thf(fact_6764_inc_Osimps_I1_J,axiom,
% 0.22/0.52 ( ( inc @ one )
% 0.22/0.52 = ( bit0 @ one ) ) ).
% 0.22/0.52
% 0.22/0.52 % inc.simps(1)
% 0.22/0.52 thf(fact_6765_inc_Osimps_I3_J,axiom,
% 0.22/0.52 ! [X: num] :
% 0.22/0.52 ( ( inc @ ( bit1 @ X ) )
% 0.22/0.52 = ( bit0 @ ( inc @ X ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % inc.simps(3)
% 0.22/0.52 thf(fact_6766_inc_Osimps_I2_J,axiom,
% 0.22/0.52 ! [X: num] :
% 0.22/0.52 ( ( inc @ ( bit0 @ X ) )
% 0.22/0.52 = ( bit1 @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % inc.simps(2)
% 0.22/0.52 thf(fact_6767_add__One,axiom,
% 0.22/0.52 ! [X: num] :
% 0.22/0.52 ( ( plus_plus_num @ X @ one )
% 0.22/0.52 = ( inc @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % add_One
% 0.22/0.52 thf(fact_6768_and__nat__def,axiom,
% 0.22/0.52 ( bit_se727722235901077358nd_nat
% 0.22/0.52 = ( ^ [M5: nat,N4: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % and_nat_def
% 0.22/0.52 thf(fact_6769_mult__inc,axiom,
% 0.22/0.52 ! [X: num,Y4: num] :
% 0.22/0.52 ( ( times_times_num @ X @ ( inc @ Y4 ) )
% 0.22/0.52 = ( plus_plus_num @ ( times_times_num @ X @ Y4 ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % mult_inc
% 0.22/0.52 thf(fact_6770_log__base__change,axiom,
% 0.22/0.52 ! [A: real,B: real,X: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.52 => ( ( A != one_one_real )
% 0.22/0.52 => ( ( log @ B @ X )
% 0.22/0.52 = ( divide_divide_real @ ( log @ A @ X ) @ ( log @ A @ B ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_base_change
% 0.22/0.52 thf(fact_6771_log__of__power__eq,axiom,
% 0.22/0.52 ! [M: nat,B: real,N: nat] :
% 0.22/0.52 ( ( ( semiri5074537144036343181t_real @ M )
% 0.22/0.52 = ( power_power_real @ B @ N ) )
% 0.22/0.52 => ( ( ord_less_real @ one_one_real @ B )
% 0.22/0.52 => ( ( semiri5074537144036343181t_real @ N )
% 0.22/0.52 = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_of_power_eq
% 0.22/0.52 thf(fact_6772_less__log__of__power,axiom,
% 0.22/0.52 ! [B: real,N: nat,M: real] :
% 0.22/0.52 ( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
% 0.22/0.52 => ( ( ord_less_real @ one_one_real @ B )
% 0.22/0.52 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % less_log_of_power
% 0.22/0.52 thf(fact_6773_numeral__inc,axiom,
% 0.22/0.52 ! [X: num] :
% 0.22/0.52 ( ( numera6690914467698888265omplex @ ( inc @ X ) )
% 0.22/0.52 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% 0.22/0.52
% 0.22/0.52 % numeral_inc
% 0.22/0.52 thf(fact_6774_numeral__inc,axiom,
% 0.22/0.52 ! [X: num] :
% 0.22/0.52 ( ( numeral_numeral_real @ ( inc @ X ) )
% 0.22/0.52 = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% 0.22/0.52
% 0.22/0.52 % numeral_inc
% 0.22/0.52 thf(fact_6775_numeral__inc,axiom,
% 0.22/0.52 ! [X: num] :
% 0.22/0.52 ( ( numeral_numeral_rat @ ( inc @ X ) )
% 0.22/0.52 = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% 0.22/0.52
% 0.22/0.52 % numeral_inc
% 0.22/0.52 thf(fact_6776_numeral__inc,axiom,
% 0.22/0.52 ! [X: num] :
% 0.22/0.52 ( ( numeral_numeral_nat @ ( inc @ X ) )
% 0.22/0.52 = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% 0.22/0.52
% 0.22/0.52 % numeral_inc
% 0.22/0.52 thf(fact_6777_numeral__inc,axiom,
% 0.22/0.52 ! [X: num] :
% 0.22/0.52 ( ( numeral_numeral_int @ ( inc @ X ) )
% 0.22/0.52 = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% 0.22/0.52
% 0.22/0.52 % numeral_inc
% 0.22/0.52 thf(fact_6778_pi__less__4,axiom,
% 0.22/0.52 ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % pi_less_4
% 0.22/0.52 thf(fact_6779_pi__ge__two,axiom,
% 0.22/0.52 ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% 0.22/0.52
% 0.22/0.52 % pi_ge_two
% 0.22/0.52 thf(fact_6780_pi__half__neq__two,axiom,
% 0.22/0.52 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.22/0.52 != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % pi_half_neq_two
% 0.22/0.52 thf(fact_6781_log__mult,axiom,
% 0.22/0.52 ! [A: real,X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.52 => ( ( A != one_one_real )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.22/0.52 => ( ( log @ A @ ( times_times_real @ X @ Y4 ) )
% 0.22/0.52 = ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y4 ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_mult
% 0.22/0.52 thf(fact_6782_log__divide,axiom,
% 0.22/0.52 ! [A: real,X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.52 => ( ( A != one_one_real )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.22/0.52 => ( ( log @ A @ ( divide_divide_real @ X @ Y4 ) )
% 0.22/0.52 = ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y4 ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_divide
% 0.22/0.52 thf(fact_6783_le__log__of__power,axiom,
% 0.22/0.52 ! [B: real,N: nat,M: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
% 0.22/0.52 => ( ( ord_less_real @ one_one_real @ B )
% 0.22/0.52 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % le_log_of_power
% 0.22/0.52 thf(fact_6784_log__base__pow,axiom,
% 0.22/0.52 ! [A: real,N: nat,X: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.52 => ( ( log @ ( power_power_real @ A @ N ) @ X )
% 0.22/0.52 = ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_base_pow
% 0.22/0.52 thf(fact_6785_log__nat__power,axiom,
% 0.22/0.52 ! [X: real,B: real,N: nat] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( log @ B @ ( power_power_real @ X @ N ) )
% 0.22/0.52 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_nat_power
% 0.22/0.52 thf(fact_6786_pi__half__neq__zero,axiom,
% 0.22/0.52 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.22/0.52 != zero_zero_real ) ).
% 0.22/0.52
% 0.22/0.52 % pi_half_neq_zero
% 0.22/0.52 thf(fact_6787_pi__half__less__two,axiom,
% 0.22/0.52 ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.22/0.52
% 0.22/0.52 % pi_half_less_two
% 0.22/0.52 thf(fact_6788_pi__half__le__two,axiom,
% 0.22/0.52 ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.22/0.52
% 0.22/0.52 % pi_half_le_two
% 0.22/0.52 thf(fact_6789_log2__of__power__eq,axiom,
% 0.22/0.52 ! [M: nat,N: nat] :
% 0.22/0.52 ( ( M
% 0.22/0.52 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.52 => ( ( semiri5074537144036343181t_real @ N )
% 0.22/0.52 = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log2_of_power_eq
% 0.22/0.52 thf(fact_6790_log__of__power__less,axiom,
% 0.22/0.52 ! [M: nat,B: real,N: nat] :
% 0.22/0.52 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 0.22/0.52 => ( ( ord_less_real @ one_one_real @ B )
% 0.22/0.52 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.22/0.52 => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_of_power_less
% 0.22/0.52 thf(fact_6791_log__eq__div__ln__mult__log,axiom,
% 0.22/0.52 ! [A: real,B: real,X: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.52 => ( ( A != one_one_real )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.22/0.52 => ( ( B != one_one_real )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( log @ A @ X )
% 0.22/0.52 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_eq_div_ln_mult_log
% 0.22/0.52 thf(fact_6792_pi__half__gt__zero,axiom,
% 0.22/0.52 ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % pi_half_gt_zero
% 0.22/0.52 thf(fact_6793_pi__half__ge__zero,axiom,
% 0.22/0.52 ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % pi_half_ge_zero
% 0.22/0.52 thf(fact_6794_m2pi__less__pi,axiom,
% 0.22/0.52 ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 0.22/0.52
% 0.22/0.52 % m2pi_less_pi
% 0.22/0.52 thf(fact_6795_arctan__ubound,axiom,
% 0.22/0.52 ! [Y4: real] : ( ord_less_real @ ( arctan @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % arctan_ubound
% 0.22/0.52 thf(fact_6796_arctan__one,axiom,
% 0.22/0.52 ( ( arctan @ one_one_real )
% 0.22/0.52 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % arctan_one
% 0.22/0.52 thf(fact_6797_log__of__power__le,axiom,
% 0.22/0.52 ! [M: nat,B: real,N: nat] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 0.22/0.52 => ( ( ord_less_real @ one_one_real @ B )
% 0.22/0.52 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.22/0.52 => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_of_power_le
% 0.22/0.52 thf(fact_6798_minus__pi__half__less__zero,axiom,
% 0.22/0.52 ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 0.22/0.52
% 0.22/0.52 % minus_pi_half_less_zero
% 0.22/0.52 thf(fact_6799_arctan__lbound,axiom,
% 0.22/0.52 ! [Y4: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y4 ) ) ).
% 0.22/0.52
% 0.22/0.52 % arctan_lbound
% 0.22/0.52 thf(fact_6800_arctan__bounded,axiom,
% 0.22/0.52 ! [Y4: real] :
% 0.22/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y4 ) )
% 0.22/0.52 & ( ord_less_real @ ( arctan @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % arctan_bounded
% 0.22/0.52 thf(fact_6801_and__nat__unfold,axiom,
% 0.22/0.52 ( bit_se727722235901077358nd_nat
% 0.22/0.52 = ( ^ [M5: nat,N4: nat] :
% 0.22/0.52 ( if_nat
% 0.22/0.52 @ ( ( M5 = zero_zero_nat )
% 0.22/0.52 | ( N4 = zero_zero_nat ) )
% 0.22/0.52 @ zero_zero_nat
% 0.22/0.52 @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % and_nat_unfold
% 0.22/0.52 thf(fact_6802_less__log2__of__power,axiom,
% 0.22/0.52 ! [N: nat,M: nat] :
% 0.22/0.52 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.22/0.52 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % less_log2_of_power
% 0.22/0.52 thf(fact_6803_le__log2__of__power,axiom,
% 0.22/0.52 ! [N: nat,M: nat] :
% 0.22/0.52 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.22/0.52 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % le_log2_of_power
% 0.22/0.52 thf(fact_6804_and__nat__rec,axiom,
% 0.22/0.52 ( bit_se727722235901077358nd_nat
% 0.22/0.52 = ( ^ [M5: nat,N4: nat] :
% 0.22/0.52 ( plus_plus_nat
% 0.22/0.52 @ ( zero_n2687167440665602831ol_nat
% 0.22/0.52 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 )
% 0.22/0.52 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
% 0.22/0.52 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % and_nat_rec
% 0.22/0.52 thf(fact_6805_log2__of__power__less,axiom,
% 0.22/0.52 ! [M: nat,N: nat] :
% 0.22/0.52 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.52 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.22/0.52 => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log2_of_power_less
% 0.22/0.52 thf(fact_6806_log2__of__power__le,axiom,
% 0.22/0.52 ! [M: nat,N: nat] :
% 0.22/0.52 ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.52 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.22/0.52 => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log2_of_power_le
% 0.22/0.52 thf(fact_6807_machin__Euler,axiom,
% 0.22/0.52 ( ( 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 ) ) ) ) ) ) ) ) ) ) )
% 0.22/0.52 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % machin_Euler
% 0.22/0.52 thf(fact_6808_arctan__inverse,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( X != zero_zero_real )
% 0.22/0.52 => ( ( arctan @ ( divide_divide_real @ one_one_real @ X ) )
% 0.22/0.52 = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % arctan_inverse
% 0.22/0.52 thf(fact_6809_sin__cos__npi,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( 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 ) ) ) )
% 0.22/0.52 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_cos_npi
% 0.22/0.52 thf(fact_6810_ceiling__log__nat__eq__powr__iff,axiom,
% 0.22/0.52 ! [B: nat,K: nat,N: nat] :
% 0.22/0.52 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.22/0.52 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.22/0.52 => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.22/0.52 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
% 0.22/0.52 = ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.22/0.52 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_log_nat_eq_powr_iff
% 0.22/0.52 thf(fact_6811_cos__pi__eq__zero,axiom,
% 0.22/0.52 ! [M: nat] :
% 0.22/0.52 ( ( 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 ) ) ) )
% 0.22/0.52 = zero_zero_real ) ).
% 0.22/0.52
% 0.22/0.52 % cos_pi_eq_zero
% 0.22/0.52 thf(fact_6812_ceiling__log__nat__eq__if,axiom,
% 0.22/0.52 ! [B: nat,N: nat,K: nat] :
% 0.22/0.52 ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.22/0.52 => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 0.22/0.52 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.22/0.52 => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.22/0.52 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_log_nat_eq_if
% 0.22/0.52 thf(fact_6813_ceiling__log2__div2,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.52 => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.22/0.52 = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_log2_div2
% 0.22/0.52 thf(fact_6814_floor__log__nat__eq__powr__iff,axiom,
% 0.22/0.52 ! [B: nat,K: nat,N: nat] :
% 0.22/0.52 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.22/0.52 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.22/0.52 => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.22/0.52 = ( semiri1314217659103216013at_int @ N ) )
% 0.22/0.52 = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.22/0.52 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_log_nat_eq_powr_iff
% 0.22/0.52 thf(fact_6815_sin__zero,axiom,
% 0.22/0.52 ( ( sin_complex @ zero_zero_complex )
% 0.22/0.52 = zero_zero_complex ) ).
% 0.22/0.52
% 0.22/0.52 % sin_zero
% 0.22/0.52 thf(fact_6816_sin__zero,axiom,
% 0.22/0.52 ( ( sin_real @ zero_zero_real )
% 0.22/0.52 = zero_zero_real ) ).
% 0.22/0.52
% 0.22/0.52 % sin_zero
% 0.22/0.52 thf(fact_6817_cos__zero,axiom,
% 0.22/0.52 ( ( cos_complex @ zero_zero_complex )
% 0.22/0.52 = one_one_complex ) ).
% 0.22/0.52
% 0.22/0.52 % cos_zero
% 0.22/0.52 thf(fact_6818_cos__zero,axiom,
% 0.22/0.52 ( ( cos_real @ zero_zero_real )
% 0.22/0.52 = one_one_real ) ).
% 0.22/0.52
% 0.22/0.52 % cos_zero
% 0.22/0.52 thf(fact_6819_floor__zero,axiom,
% 0.22/0.52 ( ( archim6058952711729229775r_real @ zero_zero_real )
% 0.22/0.52 = zero_zero_int ) ).
% 0.22/0.52
% 0.22/0.52 % floor_zero
% 0.22/0.52 thf(fact_6820_floor__zero,axiom,
% 0.22/0.52 ( ( archim3151403230148437115or_rat @ zero_zero_rat )
% 0.22/0.52 = zero_zero_int ) ).
% 0.22/0.52
% 0.22/0.52 % floor_zero
% 0.22/0.52 thf(fact_6821_floor__numeral,axiom,
% 0.22/0.52 ! [V: num] :
% 0.22/0.52 ( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V ) )
% 0.22/0.52 = ( numeral_numeral_int @ V ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_numeral
% 0.22/0.52 thf(fact_6822_floor__numeral,axiom,
% 0.22/0.52 ! [V: num] :
% 0.22/0.52 ( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
% 0.22/0.52 = ( numeral_numeral_int @ V ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_numeral
% 0.22/0.52 thf(fact_6823_ceiling__zero,axiom,
% 0.22/0.52 ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
% 0.22/0.52 = zero_zero_int ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_zero
% 0.22/0.52 thf(fact_6824_ceiling__zero,axiom,
% 0.22/0.52 ( ( archim7802044766580827645g_real @ zero_zero_real )
% 0.22/0.52 = zero_zero_int ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_zero
% 0.22/0.52 thf(fact_6825_ceiling__numeral,axiom,
% 0.22/0.52 ! [V: num] :
% 0.22/0.52 ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
% 0.22/0.52 = ( numeral_numeral_int @ V ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_numeral
% 0.22/0.52 thf(fact_6826_ceiling__numeral,axiom,
% 0.22/0.52 ! [V: num] :
% 0.22/0.52 ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
% 0.22/0.52 = ( numeral_numeral_int @ V ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_numeral
% 0.22/0.52 thf(fact_6827_sin__pi,axiom,
% 0.22/0.52 ( ( sin_real @ pi )
% 0.22/0.52 = zero_zero_real ) ).
% 0.22/0.52
% 0.22/0.52 % sin_pi
% 0.22/0.52 thf(fact_6828_floor__of__nat,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( archim6058952711729229775r_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.22/0.52 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_of_nat
% 0.22/0.52 thf(fact_6829_floor__of__nat,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( archim3151403230148437115or_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.22/0.52 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_of_nat
% 0.22/0.52 thf(fact_6830_ceiling__of__nat,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( archim7802044766580827645g_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.22/0.52 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_of_nat
% 0.22/0.52 thf(fact_6831_ceiling__of__nat,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( archim2889992004027027881ng_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.22/0.52 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_of_nat
% 0.22/0.52 thf(fact_6832_sin__pi__minus,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( sin_real @ ( minus_minus_real @ pi @ X ) )
% 0.22/0.52 = ( sin_real @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_pi_minus
% 0.22/0.52 thf(fact_6833_cos__minus__pi,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( cos_real @ ( minus_minus_real @ X @ pi ) )
% 0.22/0.52 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_minus_pi
% 0.22/0.52 thf(fact_6834_cos__pi__minus,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( cos_real @ ( minus_minus_real @ pi @ X ) )
% 0.22/0.52 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_pi_minus
% 0.22/0.52 thf(fact_6835_sin__minus__pi,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( sin_real @ ( minus_minus_real @ X @ pi ) )
% 0.22/0.52 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_minus_pi
% 0.22/0.52 thf(fact_6836_zero__le__floor,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
% 0.22/0.52 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % zero_le_floor
% 0.22/0.52 thf(fact_6837_zero__le__floor,axiom,
% 0.22/0.52 ! [X: rat] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
% 0.22/0.52 = ( ord_less_eq_rat @ zero_zero_rat @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % zero_le_floor
% 0.22/0.52 thf(fact_6838_sin__cos__squared__add3,axiom,
% 0.22/0.52 ! [X: complex] :
% 0.22/0.52 ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ X ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ X ) ) )
% 0.22/0.52 = one_one_complex ) ).
% 0.22/0.52
% 0.22/0.52 % sin_cos_squared_add3
% 0.22/0.52 thf(fact_6839_sin__cos__squared__add3,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ X ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ X ) ) )
% 0.22/0.52 = one_one_real ) ).
% 0.22/0.52
% 0.22/0.52 % sin_cos_squared_add3
% 0.22/0.52 thf(fact_6840_floor__less__zero,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
% 0.22/0.52 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_less_zero
% 0.22/0.52 thf(fact_6841_floor__less__zero,axiom,
% 0.22/0.52 ! [X: rat] :
% 0.22/0.52 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
% 0.22/0.52 = ( ord_less_rat @ X @ zero_zero_rat ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_less_zero
% 0.22/0.52 thf(fact_6842_numeral__le__floor,axiom,
% 0.22/0.52 ! [V: num,X: real] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
% 0.22/0.52 = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % numeral_le_floor
% 0.22/0.52 thf(fact_6843_numeral__le__floor,axiom,
% 0.22/0.52 ! [V: num,X: rat] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
% 0.22/0.52 = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % numeral_le_floor
% 0.22/0.52 thf(fact_6844_zero__less__floor,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
% 0.22/0.52 = ( ord_less_eq_real @ one_one_real @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % zero_less_floor
% 0.22/0.52 thf(fact_6845_zero__less__floor,axiom,
% 0.22/0.52 ! [X: rat] :
% 0.22/0.52 ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
% 0.22/0.52 = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % zero_less_floor
% 0.22/0.52 thf(fact_6846_floor__le__zero,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
% 0.22/0.52 = ( ord_less_real @ X @ one_one_real ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_le_zero
% 0.22/0.52 thf(fact_6847_floor__le__zero,axiom,
% 0.22/0.52 ! [X: rat] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
% 0.22/0.52 = ( ord_less_rat @ X @ one_one_rat ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_le_zero
% 0.22/0.52 thf(fact_6848_ceiling__le__zero,axiom,
% 0.22/0.52 ! [X: rat] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
% 0.22/0.52 = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_le_zero
% 0.22/0.52 thf(fact_6849_ceiling__le__zero,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
% 0.22/0.52 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_le_zero
% 0.22/0.52 thf(fact_6850_floor__less__numeral,axiom,
% 0.22/0.52 ! [X: real,V: num] :
% 0.22/0.52 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
% 0.22/0.52 = ( ord_less_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_less_numeral
% 0.22/0.52 thf(fact_6851_floor__less__numeral,axiom,
% 0.22/0.52 ! [X: rat,V: num] :
% 0.22/0.52 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 0.22/0.52 = ( ord_less_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_less_numeral
% 0.22/0.52 thf(fact_6852_zero__less__ceiling,axiom,
% 0.22/0.52 ! [X: rat] :
% 0.22/0.52 ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
% 0.22/0.52 = ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % zero_less_ceiling
% 0.22/0.52 thf(fact_6853_zero__less__ceiling,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
% 0.22/0.52 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % zero_less_ceiling
% 0.22/0.52 thf(fact_6854_one__le__floor,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
% 0.22/0.52 = ( ord_less_eq_real @ one_one_real @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % one_le_floor
% 0.22/0.52 thf(fact_6855_one__le__floor,axiom,
% 0.22/0.52 ! [X: rat] :
% 0.22/0.52 ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
% 0.22/0.52 = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % one_le_floor
% 0.22/0.52 thf(fact_6856_ceiling__le__numeral,axiom,
% 0.22/0.52 ! [X: rat,V: num] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 0.22/0.52 = ( ord_less_eq_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_le_numeral
% 0.22/0.52 thf(fact_6857_ceiling__le__numeral,axiom,
% 0.22/0.52 ! [X: real,V: num] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
% 0.22/0.52 = ( ord_less_eq_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_le_numeral
% 0.22/0.52 thf(fact_6858_ceiling__less__one,axiom,
% 0.22/0.52 ! [X: rat] :
% 0.22/0.52 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
% 0.22/0.52 = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_less_one
% 0.22/0.52 thf(fact_6859_ceiling__less__one,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
% 0.22/0.52 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_less_one
% 0.22/0.52 thf(fact_6860_one__le__ceiling,axiom,
% 0.22/0.52 ! [X: rat] :
% 0.22/0.52 ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
% 0.22/0.52 = ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % one_le_ceiling
% 0.22/0.52 thf(fact_6861_one__le__ceiling,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
% 0.22/0.52 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % one_le_ceiling
% 0.22/0.52 thf(fact_6862_numeral__less__ceiling,axiom,
% 0.22/0.52 ! [V: num,X: real] :
% 0.22/0.52 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
% 0.22/0.52 = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % numeral_less_ceiling
% 0.22/0.52 thf(fact_6863_numeral__less__ceiling,axiom,
% 0.22/0.52 ! [V: num,X: rat] :
% 0.22/0.52 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
% 0.22/0.52 = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % numeral_less_ceiling
% 0.22/0.52 thf(fact_6864_floor__neg__numeral,axiom,
% 0.22/0.52 ! [V: num] :
% 0.22/0.52 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.22/0.52 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_neg_numeral
% 0.22/0.52 thf(fact_6865_floor__neg__numeral,axiom,
% 0.22/0.52 ! [V: num] :
% 0.22/0.52 ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.22/0.52 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_neg_numeral
% 0.22/0.52 thf(fact_6866_ceiling__le__one,axiom,
% 0.22/0.52 ! [X: rat] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
% 0.22/0.52 = ( ord_less_eq_rat @ X @ one_one_rat ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_le_one
% 0.22/0.52 thf(fact_6867_ceiling__le__one,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
% 0.22/0.52 = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_le_one
% 0.22/0.52 thf(fact_6868_ceiling__add__numeral,axiom,
% 0.22/0.52 ! [X: real,V: num] :
% 0.22/0.52 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ V ) ) )
% 0.22/0.52 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_add_numeral
% 0.22/0.52 thf(fact_6869_ceiling__add__numeral,axiom,
% 0.22/0.52 ! [X: rat,V: num] :
% 0.22/0.52 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
% 0.22/0.52 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_add_numeral
% 0.22/0.52 thf(fact_6870_floor__diff__numeral,axiom,
% 0.22/0.52 ! [X: real,V: num] :
% 0.22/0.52 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
% 0.22/0.52 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_diff_numeral
% 0.22/0.52 thf(fact_6871_floor__diff__numeral,axiom,
% 0.22/0.52 ! [X: rat,V: num] :
% 0.22/0.52 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
% 0.22/0.52 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_diff_numeral
% 0.22/0.52 thf(fact_6872_ceiling__neg__numeral,axiom,
% 0.22/0.52 ! [V: num] :
% 0.22/0.52 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.22/0.52 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_neg_numeral
% 0.22/0.52 thf(fact_6873_ceiling__neg__numeral,axiom,
% 0.22/0.52 ! [V: num] :
% 0.22/0.52 ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.22/0.52 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_neg_numeral
% 0.22/0.52 thf(fact_6874_floor__diff__one,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ one_one_real ) )
% 0.22/0.52 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_diff_one
% 0.22/0.52 thf(fact_6875_floor__diff__one,axiom,
% 0.22/0.52 ! [X: rat] :
% 0.22/0.52 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
% 0.22/0.52 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_diff_one
% 0.22/0.52 thf(fact_6876_ceiling__diff__numeral,axiom,
% 0.22/0.52 ! [X: real,V: num] :
% 0.22/0.52 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
% 0.22/0.52 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_diff_numeral
% 0.22/0.52 thf(fact_6877_ceiling__diff__numeral,axiom,
% 0.22/0.52 ! [X: rat,V: num] :
% 0.22/0.52 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
% 0.22/0.52 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_diff_numeral
% 0.22/0.52 thf(fact_6878_ceiling__diff__one,axiom,
% 0.22/0.52 ! [X: rat] :
% 0.22/0.52 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
% 0.22/0.52 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_diff_one
% 0.22/0.52 thf(fact_6879_ceiling__diff__one,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ one_one_real ) )
% 0.22/0.52 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_diff_one
% 0.22/0.52 thf(fact_6880_floor__numeral__power,axiom,
% 0.22/0.52 ! [X: num,N: nat] :
% 0.22/0.52 ( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.22/0.52 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_numeral_power
% 0.22/0.52 thf(fact_6881_floor__numeral__power,axiom,
% 0.22/0.52 ! [X: num,N: nat] :
% 0.22/0.52 ( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.22/0.52 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_numeral_power
% 0.22/0.52 thf(fact_6882_ceiling__numeral__power,axiom,
% 0.22/0.52 ! [X: num,N: nat] :
% 0.22/0.52 ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.22/0.52 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_numeral_power
% 0.22/0.52 thf(fact_6883_ceiling__numeral__power,axiom,
% 0.22/0.52 ! [X: num,N: nat] :
% 0.22/0.52 ( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.22/0.52 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_numeral_power
% 0.22/0.52 thf(fact_6884_sin__npi2,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.22/0.52 = zero_zero_real ) ).
% 0.22/0.52
% 0.22/0.52 % sin_npi2
% 0.22/0.52 thf(fact_6885_sin__npi,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.22/0.52 = zero_zero_real ) ).
% 0.22/0.52
% 0.22/0.52 % sin_npi
% 0.22/0.52 thf(fact_6886_floor__divide__eq__div__numeral,axiom,
% 0.22/0.52 ! [A: num,B: num] :
% 0.22/0.52 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 0.22/0.52 = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_divide_eq_div_numeral
% 0.22/0.52 thf(fact_6887_nat__ceiling__le__eq,axiom,
% 0.22/0.52 ! [X: real,A: nat] :
% 0.22/0.52 ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
% 0.22/0.52 = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_ceiling_le_eq
% 0.22/0.52 thf(fact_6888_ceiling__less__zero,axiom,
% 0.22/0.52 ! [X: rat] :
% 0.22/0.52 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
% 0.22/0.52 = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_less_zero
% 0.22/0.52 thf(fact_6889_ceiling__less__zero,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
% 0.22/0.52 = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_less_zero
% 0.22/0.52 thf(fact_6890_zero__le__ceiling,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
% 0.22/0.52 = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % zero_le_ceiling
% 0.22/0.52 thf(fact_6891_zero__le__ceiling,axiom,
% 0.22/0.52 ! [X: rat] :
% 0.22/0.52 ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
% 0.22/0.52 = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % zero_le_ceiling
% 0.22/0.52 thf(fact_6892_ceiling__divide__eq__div__numeral,axiom,
% 0.22/0.52 ! [A: num,B: num] :
% 0.22/0.52 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 0.22/0.52 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_divide_eq_div_numeral
% 0.22/0.52 thf(fact_6893_numeral__less__floor,axiom,
% 0.22/0.52 ! [V: num,X: real] :
% 0.22/0.52 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
% 0.22/0.52 = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % numeral_less_floor
% 0.22/0.52 thf(fact_6894_numeral__less__floor,axiom,
% 0.22/0.52 ! [V: num,X: rat] :
% 0.22/0.52 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
% 0.22/0.52 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % numeral_less_floor
% 0.22/0.52 thf(fact_6895_floor__le__numeral,axiom,
% 0.22/0.52 ! [X: real,V: num] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
% 0.22/0.52 = ( ord_less_real @ X @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_le_numeral
% 0.22/0.52 thf(fact_6896_floor__le__numeral,axiom,
% 0.22/0.52 ! [X: rat,V: num] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 0.22/0.52 = ( ord_less_rat @ X @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_le_numeral
% 0.22/0.52 thf(fact_6897_one__less__floor,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
% 0.22/0.52 = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % one_less_floor
% 0.22/0.52 thf(fact_6898_one__less__floor,axiom,
% 0.22/0.52 ! [X: rat] :
% 0.22/0.52 ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
% 0.22/0.52 = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % one_less_floor
% 0.22/0.52 thf(fact_6899_floor__le__one,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
% 0.22/0.52 = ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_le_one
% 0.22/0.52 thf(fact_6900_floor__le__one,axiom,
% 0.22/0.52 ! [X: rat] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
% 0.22/0.52 = ( ord_less_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_le_one
% 0.22/0.52 thf(fact_6901_ceiling__less__numeral,axiom,
% 0.22/0.52 ! [X: rat,V: num] :
% 0.22/0.52 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 0.22/0.52 = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_less_numeral
% 0.22/0.52 thf(fact_6902_ceiling__less__numeral,axiom,
% 0.22/0.52 ! [X: real,V: num] :
% 0.22/0.52 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
% 0.22/0.52 = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_less_numeral
% 0.22/0.52 thf(fact_6903_numeral__le__ceiling,axiom,
% 0.22/0.52 ! [V: num,X: real] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
% 0.22/0.52 = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % numeral_le_ceiling
% 0.22/0.52 thf(fact_6904_numeral__le__ceiling,axiom,
% 0.22/0.52 ! [V: num,X: rat] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
% 0.22/0.52 = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % numeral_le_ceiling
% 0.22/0.52 thf(fact_6905_neg__numeral__le__floor,axiom,
% 0.22/0.52 ! [V: num,X: real] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
% 0.22/0.52 = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % neg_numeral_le_floor
% 0.22/0.52 thf(fact_6906_neg__numeral__le__floor,axiom,
% 0.22/0.52 ! [V: num,X: rat] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
% 0.22/0.52 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % neg_numeral_le_floor
% 0.22/0.52 thf(fact_6907_floor__less__neg__numeral,axiom,
% 0.22/0.52 ! [X: real,V: num] :
% 0.22/0.52 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.22/0.52 = ( ord_less_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_less_neg_numeral
% 0.22/0.52 thf(fact_6908_floor__less__neg__numeral,axiom,
% 0.22/0.52 ! [X: rat,V: num] :
% 0.22/0.52 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.22/0.52 = ( ord_less_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_less_neg_numeral
% 0.22/0.52 thf(fact_6909_ceiling__le__neg__numeral,axiom,
% 0.22/0.52 ! [X: rat,V: num] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.22/0.52 = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_le_neg_numeral
% 0.22/0.52 thf(fact_6910_ceiling__le__neg__numeral,axiom,
% 0.22/0.52 ! [X: real,V: num] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.22/0.52 = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_le_neg_numeral
% 0.22/0.52 thf(fact_6911_neg__numeral__less__ceiling,axiom,
% 0.22/0.52 ! [V: num,X: real] :
% 0.22/0.52 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
% 0.22/0.52 = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % neg_numeral_less_ceiling
% 0.22/0.52 thf(fact_6912_neg__numeral__less__ceiling,axiom,
% 0.22/0.52 ! [V: num,X: rat] :
% 0.22/0.52 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
% 0.22/0.52 = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % neg_numeral_less_ceiling
% 0.22/0.52 thf(fact_6913_cos__pi__half,axiom,
% 0.22/0.52 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 = zero_zero_real ) ).
% 0.22/0.52
% 0.22/0.52 % cos_pi_half
% 0.22/0.52 thf(fact_6914_sin__two__pi,axiom,
% 0.22/0.52 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.22/0.52 = zero_zero_real ) ).
% 0.22/0.52
% 0.22/0.52 % sin_two_pi
% 0.22/0.52 thf(fact_6915_sin__pi__half,axiom,
% 0.22/0.52 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 = one_one_real ) ).
% 0.22/0.52
% 0.22/0.52 % sin_pi_half
% 0.22/0.52 thf(fact_6916_cos__two__pi,axiom,
% 0.22/0.52 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.22/0.52 = one_one_real ) ).
% 0.22/0.52
% 0.22/0.52 % cos_two_pi
% 0.22/0.52 thf(fact_6917_cos__periodic,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( cos_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.22/0.52 = ( cos_real @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_periodic
% 0.22/0.52 thf(fact_6918_sin__periodic,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( sin_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.22/0.52 = ( sin_real @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_periodic
% 0.22/0.52 thf(fact_6919_cos__2pi__minus,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 0.22/0.52 = ( cos_real @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_2pi_minus
% 0.22/0.52 thf(fact_6920_cos__npi2,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.22/0.52 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_npi2
% 0.22/0.52 thf(fact_6921_cos__npi,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.22/0.52 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_npi
% 0.22/0.52 thf(fact_6922_floor__one__divide__eq__div__numeral,axiom,
% 0.22/0.52 ! [B: num] :
% 0.22/0.52 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
% 0.22/0.52 = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_one_divide_eq_div_numeral
% 0.22/0.52 thf(fact_6923_floor__minus__divide__eq__div__numeral,axiom,
% 0.22/0.52 ! [A: num,B: num] :
% 0.22/0.52 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 0.22/0.52 = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_minus_divide_eq_div_numeral
% 0.22/0.52 thf(fact_6924_ceiling__minus__divide__eq__div__numeral,axiom,
% 0.22/0.52 ! [A: num,B: num] :
% 0.22/0.52 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 0.22/0.52 = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_minus_divide_eq_div_numeral
% 0.22/0.52 thf(fact_6925_sin__cos__squared__add2,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.52 = one_one_real ) ).
% 0.22/0.52
% 0.22/0.52 % sin_cos_squared_add2
% 0.22/0.52 thf(fact_6926_sin__cos__squared__add2,axiom,
% 0.22/0.52 ! [X: complex] :
% 0.22/0.52 ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.52 = one_one_complex ) ).
% 0.22/0.52
% 0.22/0.52 % sin_cos_squared_add2
% 0.22/0.52 thf(fact_6927_sin__cos__squared__add,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.52 = one_one_real ) ).
% 0.22/0.52
% 0.22/0.52 % sin_cos_squared_add
% 0.22/0.52 thf(fact_6928_sin__cos__squared__add,axiom,
% 0.22/0.52 ! [X: complex] :
% 0.22/0.52 ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.52 = one_one_complex ) ).
% 0.22/0.52
% 0.22/0.52 % sin_cos_squared_add
% 0.22/0.52 thf(fact_6929_sin__2npi,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 0.22/0.52 = zero_zero_real ) ).
% 0.22/0.52
% 0.22/0.52 % sin_2npi
% 0.22/0.52 thf(fact_6930_cos__2npi,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 0.22/0.52 = one_one_real ) ).
% 0.22/0.52
% 0.22/0.52 % cos_2npi
% 0.22/0.52 thf(fact_6931_sin__2pi__minus,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 0.22/0.52 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_2pi_minus
% 0.22/0.52 thf(fact_6932_neg__numeral__less__floor,axiom,
% 0.22/0.52 ! [V: num,X: real] :
% 0.22/0.52 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
% 0.22/0.52 = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % neg_numeral_less_floor
% 0.22/0.52 thf(fact_6933_neg__numeral__less__floor,axiom,
% 0.22/0.52 ! [V: num,X: rat] :
% 0.22/0.52 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
% 0.22/0.52 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % neg_numeral_less_floor
% 0.22/0.52 thf(fact_6934_floor__le__neg__numeral,axiom,
% 0.22/0.52 ! [X: real,V: num] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.22/0.52 = ( ord_less_real @ X @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_le_neg_numeral
% 0.22/0.52 thf(fact_6935_floor__le__neg__numeral,axiom,
% 0.22/0.52 ! [X: rat,V: num] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.22/0.52 = ( ord_less_rat @ X @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_le_neg_numeral
% 0.22/0.52 thf(fact_6936_ceiling__less__neg__numeral,axiom,
% 0.22/0.52 ! [X: rat,V: num] :
% 0.22/0.52 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.22/0.52 = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_less_neg_numeral
% 0.22/0.52 thf(fact_6937_ceiling__less__neg__numeral,axiom,
% 0.22/0.52 ! [X: real,V: num] :
% 0.22/0.52 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.22/0.52 = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_less_neg_numeral
% 0.22/0.52 thf(fact_6938_neg__numeral__le__ceiling,axiom,
% 0.22/0.52 ! [V: num,X: real] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
% 0.22/0.52 = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % neg_numeral_le_ceiling
% 0.22/0.52 thf(fact_6939_neg__numeral__le__ceiling,axiom,
% 0.22/0.52 ! [V: num,X: rat] :
% 0.22/0.52 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
% 0.22/0.52 = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % neg_numeral_le_ceiling
% 0.22/0.52 thf(fact_6940_cos__3over2__pi,axiom,
% 0.22/0.52 ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.22/0.52 = zero_zero_real ) ).
% 0.22/0.52
% 0.22/0.52 % cos_3over2_pi
% 0.22/0.52 thf(fact_6941_floor__minus__one__divide__eq__div__numeral,axiom,
% 0.22/0.52 ! [B: num] :
% 0.22/0.52 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
% 0.22/0.52 = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_minus_one_divide_eq_div_numeral
% 0.22/0.52 thf(fact_6942_sin__3over2__pi,axiom,
% 0.22/0.52 ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.22/0.52 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_3over2_pi
% 0.22/0.52 thf(fact_6943_sin__add,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( sin_real @ ( plus_plus_real @ X @ Y4 ) )
% 0.22/0.52 = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y4 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_add
% 0.22/0.52 thf(fact_6944_floor__le__ceiling,axiom,
% 0.22/0.52 ! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_le_ceiling
% 0.22/0.52 thf(fact_6945_floor__le__ceiling,axiom,
% 0.22/0.52 ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim2889992004027027881ng_rat @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_le_ceiling
% 0.22/0.52 thf(fact_6946_polar__Ex,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ? [R3: real,A4: real] :
% 0.22/0.52 ( ( X
% 0.22/0.52 = ( times_times_real @ R3 @ ( cos_real @ A4 ) ) )
% 0.22/0.52 & ( Y4
% 0.22/0.52 = ( times_times_real @ R3 @ ( sin_real @ A4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % polar_Ex
% 0.22/0.52 thf(fact_6947_sin__diff,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( sin_real @ ( minus_minus_real @ X @ Y4 ) )
% 0.22/0.52 = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y4 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_diff
% 0.22/0.52 thf(fact_6948_cos__one__sin__zero,axiom,
% 0.22/0.52 ! [X: complex] :
% 0.22/0.52 ( ( ( cos_complex @ X )
% 0.22/0.52 = one_one_complex )
% 0.22/0.52 => ( ( sin_complex @ X )
% 0.22/0.52 = zero_zero_complex ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_one_sin_zero
% 0.22/0.52 thf(fact_6949_cos__one__sin__zero,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ( cos_real @ X )
% 0.22/0.52 = one_one_real )
% 0.22/0.52 => ( ( sin_real @ X )
% 0.22/0.52 = zero_zero_real ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_one_sin_zero
% 0.22/0.52 thf(fact_6950_cos__add,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( cos_real @ ( plus_plus_real @ X @ Y4 ) )
% 0.22/0.52 = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_add
% 0.22/0.52 thf(fact_6951_cos__diff,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( cos_real @ ( minus_minus_real @ X @ Y4 ) )
% 0.22/0.52 = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y4 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_diff
% 0.22/0.52 thf(fact_6952_sin__zero__norm__cos__one,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ( sin_real @ X )
% 0.22/0.52 = zero_zero_real )
% 0.22/0.52 => ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
% 0.22/0.52 = one_one_real ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_zero_norm_cos_one
% 0.22/0.52 thf(fact_6953_sin__zero__norm__cos__one,axiom,
% 0.22/0.52 ! [X: complex] :
% 0.22/0.52 ( ( ( sin_complex @ X )
% 0.22/0.52 = zero_zero_complex )
% 0.22/0.52 => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
% 0.22/0.52 = one_one_real ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_zero_norm_cos_one
% 0.22/0.52 thf(fact_6954_ceiling__diff__floor__le__1,axiom,
% 0.22/0.52 ! [X: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim6058952711729229775r_real @ X ) ) @ one_one_int ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_diff_floor_le_1
% 0.22/0.52 thf(fact_6955_ceiling__diff__floor__le__1,axiom,
% 0.22/0.52 ! [X: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim3151403230148437115or_rat @ X ) ) @ one_one_int ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_diff_floor_le_1
% 0.22/0.52 thf(fact_6956_sin__zero__abs__cos__one,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ( sin_real @ X )
% 0.22/0.52 = zero_zero_real )
% 0.22/0.52 => ( ( abs_abs_real @ ( cos_real @ X ) )
% 0.22/0.52 = one_one_real ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_zero_abs_cos_one
% 0.22/0.52 thf(fact_6957_sin__double,axiom,
% 0.22/0.52 ! [X: complex] :
% 0.22/0.52 ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.22/0.52 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X ) ) @ ( cos_complex @ X ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_double
% 0.22/0.52 thf(fact_6958_sin__double,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.22/0.52 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X ) ) @ ( cos_real @ X ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_double
% 0.22/0.52 thf(fact_6959_sincos__principal__value,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ? [Y: real] :
% 0.22/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y )
% 0.22/0.52 & ( ord_less_eq_real @ Y @ pi )
% 0.22/0.52 & ( ( sin_real @ Y )
% 0.22/0.52 = ( sin_real @ X ) )
% 0.22/0.52 & ( ( cos_real @ Y )
% 0.22/0.52 = ( cos_real @ X ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sincos_principal_value
% 0.22/0.52 thf(fact_6960_floor__mono,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ X @ Y4 )
% 0.22/0.52 => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y4 ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_mono
% 0.22/0.52 thf(fact_6961_floor__mono,axiom,
% 0.22/0.52 ! [X: rat,Y4: rat] :
% 0.22/0.52 ( ( ord_less_eq_rat @ X @ Y4 )
% 0.22/0.52 => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y4 ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_mono
% 0.22/0.52 thf(fact_6962_ceiling__mono,axiom,
% 0.22/0.52 ! [Y4: rat,X: rat] :
% 0.22/0.52 ( ( ord_less_eq_rat @ Y4 @ X )
% 0.22/0.52 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y4 ) @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_mono
% 0.22/0.52 thf(fact_6963_ceiling__mono,axiom,
% 0.22/0.52 ! [Y4: real,X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ Y4 @ X )
% 0.22/0.52 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y4 ) @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_mono
% 0.22/0.52 thf(fact_6964_sin__x__le__x,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ord_less_eq_real @ ( sin_real @ X ) @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_x_le_x
% 0.22/0.52 thf(fact_6965_sin__le__one,axiom,
% 0.22/0.52 ! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).
% 0.22/0.52
% 0.22/0.52 % sin_le_one
% 0.22/0.52 thf(fact_6966_cos__le__one,axiom,
% 0.22/0.52 ! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).
% 0.22/0.52
% 0.22/0.52 % cos_le_one
% 0.22/0.52 thf(fact_6967_abs__sin__x__le__abs__x,axiom,
% 0.22/0.52 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ ( abs_abs_real @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % abs_sin_x_le_abs_x
% 0.22/0.52 thf(fact_6968_cos__arctan__not__zero,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( cos_real @ ( arctan @ X ) )
% 0.22/0.52 != zero_zero_real ) ).
% 0.22/0.52
% 0.22/0.52 % cos_arctan_not_zero
% 0.22/0.52 thf(fact_6969_sin__cos__le1,axiom,
% 0.22/0.52 ! [X: real,Y4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y4 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) ) ) @ one_one_real ) ).
% 0.22/0.52
% 0.22/0.52 % sin_cos_le1
% 0.22/0.52 thf(fact_6970_sin__squared__eq,axiom,
% 0.22/0.52 ! [X: complex] :
% 0.22/0.52 ( ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.52 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_squared_eq
% 0.22/0.52 thf(fact_6971_sin__squared__eq,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.52 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_squared_eq
% 0.22/0.52 thf(fact_6972_cos__squared__eq,axiom,
% 0.22/0.52 ! [X: complex] :
% 0.22/0.52 ( ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.52 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_squared_eq
% 0.22/0.52 thf(fact_6973_cos__squared__eq,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.52 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_squared_eq
% 0.22/0.52 thf(fact_6974_le__floor__add,axiom,
% 0.22/0.52 ! [X: real,Y4: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y4 ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y4 ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % le_floor_add
% 0.22/0.52 thf(fact_6975_le__floor__add,axiom,
% 0.22/0.52 ! [X: rat,Y4: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y4 ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y4 ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % le_floor_add
% 0.22/0.52 thf(fact_6976_of__nat__ceiling,axiom,
% 0.22/0.52 ! [R2: rat] : ( ord_less_eq_rat @ R2 @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R2 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % of_nat_ceiling
% 0.22/0.52 thf(fact_6977_of__nat__ceiling,axiom,
% 0.22/0.52 ! [R2: real] : ( ord_less_eq_real @ R2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R2 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % of_nat_ceiling
% 0.22/0.52 thf(fact_6978_sin__gt__zero,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_real @ X @ pi )
% 0.22/0.52 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_gt_zero
% 0.22/0.52 thf(fact_6979_sin__x__ge__neg__x,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ ( sin_real @ X ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_x_ge_neg_x
% 0.22/0.52 thf(fact_6980_sin__ge__zero,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ X @ pi )
% 0.22/0.52 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_ge_zero
% 0.22/0.52 thf(fact_6981_real__nat__ceiling__ge,axiom,
% 0.22/0.52 ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % real_nat_ceiling_ge
% 0.22/0.52 thf(fact_6982_ceiling__add__le,axiom,
% 0.22/0.52 ! [X: rat,Y4: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ Y4 ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y4 ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_add_le
% 0.22/0.52 thf(fact_6983_ceiling__add__le,axiom,
% 0.22/0.52 ! [X: real,Y4: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ Y4 ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y4 ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_add_le
% 0.22/0.52 thf(fact_6984_sin__ge__minus__one,axiom,
% 0.22/0.52 ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_ge_minus_one
% 0.22/0.52 thf(fact_6985_cos__monotone__0__pi__le,axiom,
% 0.22/0.52 ! [Y4: real,X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.52 => ( ( ord_less_eq_real @ Y4 @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ X @ pi )
% 0.22/0.52 => ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_monotone_0_pi_le
% 0.22/0.52 thf(fact_6986_cos__mono__le__eq,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ X @ pi )
% 0.22/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.52 => ( ( ord_less_eq_real @ Y4 @ pi )
% 0.22/0.52 => ( ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) )
% 0.22/0.52 = ( ord_less_eq_real @ Y4 @ X ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_mono_le_eq
% 0.22/0.52 thf(fact_6987_cos__inj__pi,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ X @ pi )
% 0.22/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.52 => ( ( ord_less_eq_real @ Y4 @ pi )
% 0.22/0.52 => ( ( ( cos_real @ X )
% 0.22/0.52 = ( cos_real @ Y4 ) )
% 0.22/0.52 => ( X = Y4 ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_inj_pi
% 0.22/0.52 thf(fact_6988_cos__ge__minus__one,axiom,
% 0.22/0.52 ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_ge_minus_one
% 0.22/0.52 thf(fact_6989_abs__sin__le__one,axiom,
% 0.22/0.52 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).
% 0.22/0.52
% 0.22/0.52 % abs_sin_le_one
% 0.22/0.52 thf(fact_6990_abs__cos__le__one,axiom,
% 0.22/0.52 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).
% 0.22/0.52
% 0.22/0.52 % abs_cos_le_one
% 0.22/0.52 thf(fact_6991_sin__times__sin,axiom,
% 0.22/0.52 ! [W: complex,Z: complex] :
% 0.22/0.52 ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.22/0.52 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_times_sin
% 0.22/0.52 thf(fact_6992_sin__times__sin,axiom,
% 0.22/0.52 ! [W: real,Z: real] :
% 0.22/0.52 ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.22/0.52 = ( 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 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_times_sin
% 0.22/0.52 thf(fact_6993_sin__times__cos,axiom,
% 0.22/0.52 ! [W: complex,Z: complex] :
% 0.22/0.52 ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
% 0.22/0.52 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_times_cos
% 0.22/0.52 thf(fact_6994_sin__times__cos,axiom,
% 0.22/0.52 ! [W: real,Z: real] :
% 0.22/0.52 ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
% 0.22/0.52 = ( 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 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_times_cos
% 0.22/0.52 thf(fact_6995_cos__times__sin,axiom,
% 0.22/0.52 ! [W: complex,Z: complex] :
% 0.22/0.52 ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
% 0.22/0.52 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_times_sin
% 0.22/0.52 thf(fact_6996_cos__times__sin,axiom,
% 0.22/0.52 ! [W: real,Z: real] :
% 0.22/0.52 ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
% 0.22/0.52 = ( 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 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_times_sin
% 0.22/0.52 thf(fact_6997_sin__plus__sin,axiom,
% 0.22/0.52 ! [W: complex,Z: complex] :
% 0.22/0.52 ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.22/0.52 = ( 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 ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_plus_sin
% 0.22/0.52 thf(fact_6998_sin__plus__sin,axiom,
% 0.22/0.52 ! [W: real,Z: real] :
% 0.22/0.52 ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.22/0.52 = ( 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 ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_plus_sin
% 0.22/0.52 thf(fact_6999_sin__diff__sin,axiom,
% 0.22/0.52 ! [W: complex,Z: complex] :
% 0.22/0.52 ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.22/0.52 = ( 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 ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_diff_sin
% 0.22/0.52 thf(fact_7000_sin__diff__sin,axiom,
% 0.22/0.52 ! [W: real,Z: real] :
% 0.22/0.52 ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.22/0.52 = ( 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 ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_diff_sin
% 0.22/0.52 thf(fact_7001_cos__diff__cos,axiom,
% 0.22/0.52 ! [W: complex,Z: complex] :
% 0.22/0.52 ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.22/0.52 = ( 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 ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_diff_cos
% 0.22/0.52 thf(fact_7002_cos__diff__cos,axiom,
% 0.22/0.52 ! [W: real,Z: real] :
% 0.22/0.52 ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.22/0.52 = ( 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 ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_diff_cos
% 0.22/0.52 thf(fact_7003_cos__double,axiom,
% 0.22/0.52 ! [X: complex] :
% 0.22/0.52 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.22/0.52 = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_double
% 0.22/0.52 thf(fact_7004_cos__double,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.22/0.52 = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_double
% 0.22/0.52 thf(fact_7005_cos__double__sin,axiom,
% 0.22/0.52 ! [W: complex] :
% 0.22/0.52 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 0.22/0.52 = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_double_sin
% 0.22/0.52 thf(fact_7006_cos__double__sin,axiom,
% 0.22/0.52 ! [W: real] :
% 0.22/0.52 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 0.22/0.52 = ( 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 ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_double_sin
% 0.22/0.52 thf(fact_7007_of__nat__floor,axiom,
% 0.22/0.52 ! [R2: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 0.22/0.52 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R2 ) ) ) @ R2 ) ) ).
% 0.22/0.52
% 0.22/0.52 % of_nat_floor
% 0.22/0.52 thf(fact_7008_of__nat__floor,axiom,
% 0.22/0.52 ! [R2: rat] :
% 0.22/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 0.22/0.52 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R2 ) ) ) @ R2 ) ) ).
% 0.22/0.52
% 0.22/0.52 % of_nat_floor
% 0.22/0.52 thf(fact_7009_le__mult__nat__floor,axiom,
% 0.22/0.52 ! [A: real,B: real] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ B ) ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % le_mult_nat_floor
% 0.22/0.52 thf(fact_7010_le__mult__nat__floor,axiom,
% 0.22/0.52 ! [A: rat,B: rat] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim3151403230148437115or_rat @ A ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % le_mult_nat_floor
% 0.22/0.52 thf(fact_7011_floor__divide__of__nat__eq,axiom,
% 0.22/0.52 ! [M: nat,N: nat] :
% 0.22/0.52 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.22/0.52 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_divide_of_nat_eq
% 0.22/0.52 thf(fact_7012_floor__divide__of__nat__eq,axiom,
% 0.22/0.52 ! [M: nat,N: nat] :
% 0.22/0.52 ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) )
% 0.22/0.52 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_divide_of_nat_eq
% 0.22/0.52 thf(fact_7013_nat__floor__neg,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.22/0.52 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 0.22/0.52 = zero_zero_nat ) ) ).
% 0.22/0.52
% 0.22/0.52 % nat_floor_neg
% 0.22/0.52 thf(fact_7014_cos__two__neq__zero,axiom,
% 0.22/0.52 ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.22/0.52 != zero_zero_real ) ).
% 0.22/0.52
% 0.22/0.52 % cos_two_neq_zero
% 0.22/0.52 thf(fact_7015_floor__eq3,axiom,
% 0.22/0.52 ! [N: nat,X: real] :
% 0.22/0.52 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 0.22/0.52 => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.22/0.52 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 0.22/0.52 = N ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_eq3
% 0.22/0.52 thf(fact_7016_le__nat__floor,axiom,
% 0.22/0.52 ! [X: nat,A: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
% 0.22/0.52 => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % le_nat_floor
% 0.22/0.52 thf(fact_7017_cos__monotone__0__pi,axiom,
% 0.22/0.52 ! [Y4: real,X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.52 => ( ( ord_less_real @ Y4 @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ X @ pi )
% 0.22/0.52 => ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_monotone_0_pi
% 0.22/0.52 thf(fact_7018_cos__mono__less__eq,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ X @ pi )
% 0.22/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.52 => ( ( ord_less_eq_real @ Y4 @ pi )
% 0.22/0.52 => ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) )
% 0.22/0.52 = ( ord_less_real @ Y4 @ X ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_mono_less_eq
% 0.22/0.52 thf(fact_7019_sin__eq__0__pi,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 0.22/0.52 => ( ( ord_less_real @ X @ pi )
% 0.22/0.52 => ( ( ( sin_real @ X )
% 0.22/0.52 = zero_zero_real )
% 0.22/0.52 => ( X = zero_zero_real ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_eq_0_pi
% 0.22/0.52 thf(fact_7020_sin__zero__pi__iff,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_real @ ( abs_abs_real @ X ) @ pi )
% 0.22/0.52 => ( ( ( sin_real @ X )
% 0.22/0.52 = zero_zero_real )
% 0.22/0.52 = ( X = zero_zero_real ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_zero_pi_iff
% 0.22/0.52 thf(fact_7021_cos__monotone__minus__pi__0_H,axiom,
% 0.22/0.52 ! [Y4: real,X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y4 )
% 0.22/0.52 => ( ( ord_less_eq_real @ Y4 @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.22/0.52 => ( ord_less_eq_real @ ( cos_real @ Y4 ) @ ( cos_real @ X ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_monotone_minus_pi_0'
% 0.22/0.52 thf(fact_7022_sincos__total__pi,axiom,
% 0.22/0.52 ! [Y4: real,X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.52 => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.52 = one_one_real )
% 0.22/0.52 => ? [T3: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.22/0.52 & ( ord_less_eq_real @ T3 @ pi )
% 0.22/0.52 & ( X
% 0.22/0.52 = ( cos_real @ T3 ) )
% 0.22/0.52 & ( Y4
% 0.22/0.52 = ( sin_real @ T3 ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sincos_total_pi
% 0.22/0.52 thf(fact_7023_sin__cos__sqrt,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
% 0.22/0.52 => ( ( sin_real @ X )
% 0.22/0.52 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_cos_sqrt
% 0.22/0.52 thf(fact_7024_sin__expansion__lemma,axiom,
% 0.22/0.52 ! [X: real,M: nat] :
% 0.22/0.52 ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.22/0.52 = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_expansion_lemma
% 0.22/0.52 thf(fact_7025_le__mult__floor,axiom,
% 0.22/0.52 ! [A: real,B: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.22/0.52 => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % le_mult_floor
% 0.22/0.52 thf(fact_7026_le__mult__floor,axiom,
% 0.22/0.52 ! [A: rat,B: rat] :
% 0.22/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.22/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.22/0.52 => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % le_mult_floor
% 0.22/0.52 thf(fact_7027_cos__expansion__lemma,axiom,
% 0.22/0.52 ! [X: real,M: nat] :
% 0.22/0.52 ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.22/0.52 = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_expansion_lemma
% 0.22/0.52 thf(fact_7028_sin__gt__zero__02,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.22/0.52 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_gt_zero_02
% 0.22/0.52 thf(fact_7029_mult__ceiling__le,axiom,
% 0.22/0.52 ! [A: rat,B: rat] :
% 0.22/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.22/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.22/0.52 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % mult_ceiling_le
% 0.22/0.52 thf(fact_7030_mult__ceiling__le,axiom,
% 0.22/0.52 ! [A: real,B: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.22/0.52 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % mult_ceiling_le
% 0.22/0.52 thf(fact_7031_floor__eq4,axiom,
% 0.22/0.52 ! [N: nat,X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 0.22/0.52 => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.22/0.52 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 0.22/0.52 = N ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_eq4
% 0.22/0.52 thf(fact_7032_cos__two__less__zero,axiom,
% 0.22/0.52 ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 0.22/0.52
% 0.22/0.52 % cos_two_less_zero
% 0.22/0.52 thf(fact_7033_cos__two__le__zero,axiom,
% 0.22/0.52 ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 0.22/0.52
% 0.22/0.52 % cos_two_le_zero
% 0.22/0.52 thf(fact_7034_cos__is__zero,axiom,
% 0.22/0.52 ? [X4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.22/0.52 & ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.22/0.52 & ( ( cos_real @ X4 )
% 0.22/0.52 = zero_zero_real )
% 0.22/0.52 & ! [Y3: real] :
% 0.22/0.52 ( ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.22/0.52 & ( ord_less_eq_real @ Y3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.22/0.52 & ( ( cos_real @ Y3 )
% 0.22/0.52 = zero_zero_real ) )
% 0.22/0.52 => ( Y3 = X4 ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_is_zero
% 0.22/0.52 thf(fact_7035_cos__monotone__minus__pi__0,axiom,
% 0.22/0.52 ! [Y4: real,X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y4 )
% 0.22/0.52 => ( ( ord_less_real @ Y4 @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.22/0.52 => ( ord_less_real @ ( cos_real @ Y4 ) @ ( cos_real @ X ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_monotone_minus_pi_0
% 0.22/0.52 thf(fact_7036_cos__total,axiom,
% 0.22/0.52 ! [Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 0.22/0.52 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.22/0.52 => ? [X4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.22/0.52 & ( ord_less_eq_real @ X4 @ pi )
% 0.22/0.52 & ( ( cos_real @ X4 )
% 0.22/0.52 = Y4 )
% 0.22/0.52 & ! [Y3: real] :
% 0.22/0.52 ( ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.22/0.52 & ( ord_less_eq_real @ Y3 @ pi )
% 0.22/0.52 & ( ( cos_real @ Y3 )
% 0.22/0.52 = Y4 ) )
% 0.22/0.52 => ( Y3 = X4 ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_total
% 0.22/0.52 thf(fact_7037_sincos__total__pi__half,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.52 => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.52 = one_one_real )
% 0.22/0.52 => ? [T3: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.22/0.52 & ( ord_less_eq_real @ T3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 & ( X
% 0.22/0.52 = ( cos_real @ T3 ) )
% 0.22/0.52 & ( Y4
% 0.22/0.52 = ( sin_real @ T3 ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sincos_total_pi_half
% 0.22/0.52 thf(fact_7038_sincos__total__2pi__le,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.52 = one_one_real )
% 0.22/0.52 => ? [T3: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.22/0.52 & ( ord_less_eq_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.22/0.52 & ( X
% 0.22/0.52 = ( cos_real @ T3 ) )
% 0.22/0.52 & ( Y4
% 0.22/0.52 = ( sin_real @ T3 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sincos_total_2pi_le
% 0.22/0.52 thf(fact_7039_sincos__total__2pi,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.52 = one_one_real )
% 0.22/0.52 => ~ ! [T3: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.22/0.52 => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.22/0.52 => ( ( X
% 0.22/0.52 = ( cos_real @ T3 ) )
% 0.22/0.52 => ( Y4
% 0.22/0.52 != ( sin_real @ T3 ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sincos_total_2pi
% 0.22/0.52 thf(fact_7040_sin__pi__divide__n__ge__0,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( N != zero_zero_nat )
% 0.22/0.52 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_pi_divide_n_ge_0
% 0.22/0.52 thf(fact_7041_sin__45,axiom,
% 0.22/0.52 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.22/0.52 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_45
% 0.22/0.52 thf(fact_7042_cos__45,axiom,
% 0.22/0.52 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.22/0.52 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_45
% 0.22/0.52 thf(fact_7043_cos__times__cos,axiom,
% 0.22/0.52 ! [W: complex,Z: complex] :
% 0.22/0.52 ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.22/0.52 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_times_cos
% 0.22/0.52 thf(fact_7044_cos__times__cos,axiom,
% 0.22/0.52 ! [W: real,Z: real] :
% 0.22/0.52 ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.22/0.52 = ( 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 ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_times_cos
% 0.22/0.52 thf(fact_7045_cos__plus__cos,axiom,
% 0.22/0.52 ! [W: complex,Z: complex] :
% 0.22/0.52 ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.22/0.52 = ( 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 ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_plus_cos
% 0.22/0.52 thf(fact_7046_cos__plus__cos,axiom,
% 0.22/0.52 ! [W: real,Z: real] :
% 0.22/0.52 ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.22/0.52 = ( 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 ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_plus_cos
% 0.22/0.52 thf(fact_7047_sin__gt__zero2,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_gt_zero2
% 0.22/0.52 thf(fact_7048_sin__lt__zero,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_real @ pi @ X )
% 0.22/0.52 => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.22/0.52 => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_lt_zero
% 0.22/0.52 thf(fact_7049_cos__double__less__one,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.22/0.52 => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_double_less_one
% 0.22/0.52 thf(fact_7050_sin__30,axiom,
% 0.22/0.52 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.22/0.52 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_30
% 0.22/0.52 thf(fact_7051_cos__gt__zero,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_gt_zero
% 0.22/0.52 thf(fact_7052_sin__monotone__2pi__le,axiom,
% 0.22/0.52 ! [Y4: real,X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 0.22/0.52 => ( ( ord_less_eq_real @ Y4 @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ord_less_eq_real @ ( sin_real @ Y4 ) @ ( sin_real @ X ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_monotone_2pi_le
% 0.22/0.52 thf(fact_7053_sin__mono__le__eq,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 0.22/0.52 => ( ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y4 ) )
% 0.22/0.52 = ( ord_less_eq_real @ X @ Y4 ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_mono_le_eq
% 0.22/0.52 thf(fact_7054_sin__inj__pi,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 0.22/0.52 => ( ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ( ( sin_real @ X )
% 0.22/0.52 = ( sin_real @ Y4 ) )
% 0.22/0.52 => ( X = Y4 ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_inj_pi
% 0.22/0.52 thf(fact_7055_cos__60,axiom,
% 0.22/0.52 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.22/0.52 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_60
% 0.22/0.52 thf(fact_7056_sin__60,axiom,
% 0.22/0.52 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.22/0.52 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_60
% 0.22/0.52 thf(fact_7057_cos__30,axiom,
% 0.22/0.52 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.22/0.52 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_30
% 0.22/0.52 thf(fact_7058_cos__double__cos,axiom,
% 0.22/0.52 ! [W: complex] :
% 0.22/0.52 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 0.22/0.52 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_double_cos
% 0.22/0.52 thf(fact_7059_cos__double__cos,axiom,
% 0.22/0.52 ! [W: real] :
% 0.22/0.52 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 0.22/0.52 = ( 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 ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_double_cos
% 0.22/0.52 thf(fact_7060_cos__treble__cos,axiom,
% 0.22/0.52 ! [X: complex] :
% 0.22/0.52 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X ) )
% 0.22/0.52 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_treble_cos
% 0.22/0.52 thf(fact_7061_cos__treble__cos,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X ) )
% 0.22/0.52 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_treble_cos
% 0.22/0.52 thf(fact_7062_sin__le__zero,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ pi @ X )
% 0.22/0.52 => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.22/0.52 => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_le_zero
% 0.22/0.52 thf(fact_7063_sin__less__zero,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 0.22/0.52 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.22/0.52 => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_less_zero
% 0.22/0.52 thf(fact_7064_sin__monotone__2pi,axiom,
% 0.22/0.52 ! [Y4: real,X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 0.22/0.52 => ( ( ord_less_real @ Y4 @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ord_less_real @ ( sin_real @ Y4 ) @ ( sin_real @ X ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_monotone_2pi
% 0.22/0.52 thf(fact_7065_sin__mono__less__eq,axiom,
% 0.22/0.52 ! [X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 0.22/0.52 => ( ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y4 ) )
% 0.22/0.52 = ( ord_less_real @ X @ Y4 ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_mono_less_eq
% 0.22/0.52 thf(fact_7066_sin__total,axiom,
% 0.22/0.52 ! [Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 0.22/0.52 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.22/0.52 => ? [X4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.22/0.52 & ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 & ( ( sin_real @ X4 )
% 0.22/0.52 = Y4 )
% 0.22/0.52 & ! [Y3: real] :
% 0.22/0.52 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.22/0.52 & ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 & ( ( sin_real @ Y3 )
% 0.22/0.52 = Y4 ) )
% 0.22/0.52 => ( Y3 = X4 ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_total
% 0.22/0.52 thf(fact_7067_cos__gt__zero__pi,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.22/0.52 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_gt_zero_pi
% 0.22/0.52 thf(fact_7068_cos__ge__zero,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_ge_zero
% 0.22/0.52 thf(fact_7069_cos__one__2pi,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ( cos_real @ X )
% 0.22/0.52 = one_one_real )
% 0.22/0.52 = ( ? [X3: nat] :
% 0.22/0.52 ( X
% 0.22/0.52 = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.22/0.52 | ? [X3: nat] :
% 0.22/0.52 ( X
% 0.22/0.52 = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_one_2pi
% 0.22/0.52 thf(fact_7070_sin__pi__divide__n__gt__0,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.52 => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_pi_divide_n_gt_0
% 0.22/0.52 thf(fact_7071_sin__arctan,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( sin_real @ ( arctan @ X ) )
% 0.22/0.52 = ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_arctan
% 0.22/0.52 thf(fact_7072_cos__arctan,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( cos_real @ ( arctan @ X ) )
% 0.22/0.52 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_arctan
% 0.22/0.52 thf(fact_7073_floor__log2__div2,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.52 => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.22/0.52 = ( 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 ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_log2_div2
% 0.22/0.52 thf(fact_7074_floor__log__nat__eq__if,axiom,
% 0.22/0.52 ! [B: nat,N: nat,K: nat] :
% 0.22/0.52 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.22/0.52 => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 0.22/0.52 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.22/0.52 => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.22/0.52 = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % floor_log_nat_eq_if
% 0.22/0.52 thf(fact_7075_sin__zero__lemma,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ( sin_real @ X )
% 0.22/0.52 = zero_zero_real )
% 0.22/0.52 => ? [N2: nat] :
% 0.22/0.52 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.22/0.52 & ( X
% 0.22/0.52 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_zero_lemma
% 0.22/0.52 thf(fact_7076_sin__zero__iff,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ( sin_real @ X )
% 0.22/0.52 = zero_zero_real )
% 0.22/0.52 = ( ? [N4: nat] :
% 0.22/0.52 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 0.22/0.52 & ( X
% 0.22/0.52 = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 0.22/0.52 | ? [N4: nat] :
% 0.22/0.52 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 0.22/0.52 & ( X
% 0.22/0.52 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_zero_iff
% 0.22/0.52 thf(fact_7077_cos__zero__lemma,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ( cos_real @ X )
% 0.22/0.52 = zero_zero_real )
% 0.22/0.52 => ? [N2: nat] :
% 0.22/0.52 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.22/0.52 & ( X
% 0.22/0.52 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_zero_lemma
% 0.22/0.52 thf(fact_7078_cos__zero__iff,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ( cos_real @ X )
% 0.22/0.52 = zero_zero_real )
% 0.22/0.52 = ( ? [N4: nat] :
% 0.22/0.52 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 0.22/0.52 & ( X
% 0.22/0.52 = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 0.22/0.52 | ? [N4: nat] :
% 0.22/0.52 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 0.22/0.52 & ( X
% 0.22/0.52 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_zero_iff
% 0.22/0.52 thf(fact_7079_tan__double,axiom,
% 0.22/0.52 ! [X: complex] :
% 0.22/0.52 ( ( ( cos_complex @ X )
% 0.22/0.52 != zero_zero_complex )
% 0.22/0.52 => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.22/0.52 != zero_zero_complex )
% 0.22/0.52 => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.22/0.52 = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % tan_double
% 0.22/0.52 thf(fact_7080_tan__double,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ( cos_real @ X )
% 0.22/0.52 != zero_zero_real )
% 0.22/0.52 => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.22/0.52 != zero_zero_real )
% 0.22/0.52 => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.22/0.52 = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % tan_double
% 0.22/0.52 thf(fact_7081_sin__tan,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ( sin_real @ X )
% 0.22/0.52 = ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_tan
% 0.22/0.52 thf(fact_7082_cos__tan,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 => ( ( cos_real @ X )
% 0.22/0.52 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_tan
% 0.22/0.52 thf(fact_7083_complex__unimodular__polar,axiom,
% 0.22/0.52 ! [Z: complex] :
% 0.22/0.52 ( ( ( real_V1022390504157884413omplex @ Z )
% 0.22/0.52 = one_one_real )
% 0.22/0.52 => ~ ! [T3: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.22/0.52 => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.22/0.52 => ( Z
% 0.22/0.52 != ( complex2 @ ( cos_real @ T3 ) @ ( sin_real @ T3 ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % complex_unimodular_polar
% 0.22/0.52 thf(fact_7084_ceiling__log__eq__powr__iff,axiom,
% 0.22/0.52 ! [X: real,B: real,K: nat] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_real @ one_one_real @ B )
% 0.22/0.52 => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
% 0.22/0.52 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 0.22/0.52 = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
% 0.22/0.52 & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % ceiling_log_eq_powr_iff
% 0.22/0.52 thf(fact_7085_cos__arcsin,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.22/0.52 => ( ( cos_real @ ( arcsin @ X ) )
% 0.22/0.52 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % cos_arcsin
% 0.22/0.52 thf(fact_7086_powr__eq__0__iff,axiom,
% 0.22/0.52 ! [W: real,Z: real] :
% 0.22/0.52 ( ( ( powr_real @ W @ Z )
% 0.22/0.52 = zero_zero_real )
% 0.22/0.52 = ( W = zero_zero_real ) ) ).
% 0.22/0.52
% 0.22/0.52 % powr_eq_0_iff
% 0.22/0.52 thf(fact_7087_powr__0,axiom,
% 0.22/0.52 ! [Z: real] :
% 0.22/0.52 ( ( powr_real @ zero_zero_real @ Z )
% 0.22/0.52 = zero_zero_real ) ).
% 0.22/0.52
% 0.22/0.52 % powr_0
% 0.22/0.52 thf(fact_7088_tan__zero,axiom,
% 0.22/0.52 ( ( tan_complex @ zero_zero_complex )
% 0.22/0.52 = zero_zero_complex ) ).
% 0.22/0.52
% 0.22/0.52 % tan_zero
% 0.22/0.52 thf(fact_7089_tan__zero,axiom,
% 0.22/0.52 ( ( tan_real @ zero_zero_real )
% 0.22/0.52 = zero_zero_real ) ).
% 0.22/0.52
% 0.22/0.52 % tan_zero
% 0.22/0.52 thf(fact_7090_arcsin__0,axiom,
% 0.22/0.52 ( ( arcsin @ zero_zero_real )
% 0.22/0.52 = zero_zero_real ) ).
% 0.22/0.52
% 0.22/0.52 % arcsin_0
% 0.22/0.52 thf(fact_7091_powr__zero__eq__one,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ( X = zero_zero_real )
% 0.22/0.52 => ( ( powr_real @ X @ zero_zero_real )
% 0.22/0.52 = zero_zero_real ) )
% 0.22/0.52 & ( ( X != zero_zero_real )
% 0.22/0.52 => ( ( powr_real @ X @ zero_zero_real )
% 0.22/0.52 = one_one_real ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % powr_zero_eq_one
% 0.22/0.52 thf(fact_7092_powr__gt__zero,axiom,
% 0.22/0.52 ! [X: real,A: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
% 0.22/0.52 = ( X != zero_zero_real ) ) ).
% 0.22/0.52
% 0.22/0.52 % powr_gt_zero
% 0.22/0.52 thf(fact_7093_powr__nonneg__iff,axiom,
% 0.22/0.52 ! [A: real,X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
% 0.22/0.52 = ( A = zero_zero_real ) ) ).
% 0.22/0.52
% 0.22/0.52 % powr_nonneg_iff
% 0.22/0.52 thf(fact_7094_tan__pi,axiom,
% 0.22/0.52 ( ( tan_real @ pi )
% 0.22/0.52 = zero_zero_real ) ).
% 0.22/0.52
% 0.22/0.52 % tan_pi
% 0.22/0.52 thf(fact_7095_powr__eq__one__iff,axiom,
% 0.22/0.52 ! [A: real,X: real] :
% 0.22/0.52 ( ( ord_less_real @ one_one_real @ A )
% 0.22/0.52 => ( ( ( powr_real @ A @ X )
% 0.22/0.52 = one_one_real )
% 0.22/0.52 = ( X = zero_zero_real ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % powr_eq_one_iff
% 0.22/0.52 thf(fact_7096_powr__one__gt__zero__iff,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ( powr_real @ X @ one_one_real )
% 0.22/0.52 = X )
% 0.22/0.52 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % powr_one_gt_zero_iff
% 0.22/0.52 thf(fact_7097_powr__one,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( powr_real @ X @ one_one_real )
% 0.22/0.52 = X ) ) ).
% 0.22/0.52
% 0.22/0.52 % powr_one
% 0.22/0.52 thf(fact_7098_powr__le__cancel__iff,axiom,
% 0.22/0.52 ! [X: real,A: real,B: real] :
% 0.22/0.52 ( ( ord_less_real @ one_one_real @ X )
% 0.22/0.52 => ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 0.22/0.52 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % powr_le_cancel_iff
% 0.22/0.52 thf(fact_7099_numeral__powr__numeral__real,axiom,
% 0.22/0.52 ! [M: num,N: num] :
% 0.22/0.52 ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.22/0.52 = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % numeral_powr_numeral_real
% 0.22/0.52 thf(fact_7100_powr__log__cancel,axiom,
% 0.22/0.52 ! [A: real,X: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.52 => ( ( A != one_one_real )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( powr_real @ A @ ( log @ A @ X ) )
% 0.22/0.52 = X ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % powr_log_cancel
% 0.22/0.52 thf(fact_7101_log__powr__cancel,axiom,
% 0.22/0.52 ! [A: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.52 => ( ( A != one_one_real )
% 0.22/0.52 => ( ( log @ A @ ( powr_real @ A @ Y4 ) )
% 0.22/0.52 = Y4 ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % log_powr_cancel
% 0.22/0.52 thf(fact_7102_tan__npi,axiom,
% 0.22/0.52 ! [N: nat] :
% 0.22/0.52 ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.22/0.52 = zero_zero_real ) ).
% 0.22/0.52
% 0.22/0.52 % tan_npi
% 0.22/0.52 thf(fact_7103_tan__periodic__n,axiom,
% 0.22/0.52 ! [X: real,N: num] :
% 0.22/0.52 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
% 0.22/0.52 = ( tan_real @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % tan_periodic_n
% 0.22/0.52 thf(fact_7104_tan__periodic__nat,axiom,
% 0.22/0.52 ! [X: real,N: nat] :
% 0.22/0.52 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
% 0.22/0.52 = ( tan_real @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % tan_periodic_nat
% 0.22/0.52 thf(fact_7105_sin__arcsin,axiom,
% 0.22/0.52 ! [Y4: real] :
% 0.22/0.52 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 0.22/0.52 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.22/0.52 => ( ( sin_real @ ( arcsin @ Y4 ) )
% 0.22/0.52 = Y4 ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % sin_arcsin
% 0.22/0.52 thf(fact_7106_powr__numeral,axiom,
% 0.22/0.52 ! [X: real,N: num] :
% 0.22/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( powr_real @ X @ ( numeral_numeral_real @ N ) )
% 0.22/0.52 = ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % powr_numeral
% 0.22/0.52 thf(fact_7107_arcsin__1,axiom,
% 0.22/0.52 ( ( arcsin @ one_one_real )
% 0.22/0.52 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % arcsin_1
% 0.22/0.52 thf(fact_7108_tan__periodic,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.22/0.52 = ( tan_real @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % tan_periodic
% 0.22/0.52 thf(fact_7109_arcsin__minus__1,axiom,
% 0.22/0.52 ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.52 = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % arcsin_minus_1
% 0.22/0.52 thf(fact_7110_square__powr__half,axiom,
% 0.22/0.52 ! [X: real] :
% 0.22/0.52 ( ( powr_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.52 = ( abs_abs_real @ X ) ) ).
% 0.22/0.52
% 0.22/0.52 % square_powr_half
% 0.22/0.52 thf(fact_7111_powr__powr,axiom,
% 0.22/0.52 ! [X: real,A: real,B: real] :
% 0.22/0.52 ( ( powr_real @ ( powr_real @ X @ A ) @ B )
% 0.22/0.52 = ( powr_real @ X @ ( times_times_real @ A @ B ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % powr_powr
% 0.22/0.52 thf(fact_7112_Complex__eq__0,axiom,
% 0.22/0.52 ! [A: real,B: real] :
% 0.22/0.52 ( ( ( complex2 @ A @ B )
% 0.22/0.52 = zero_zero_complex )
% 0.22/0.52 = ( ( A = zero_zero_real )
% 0.22/0.52 & ( B = zero_zero_real ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % Complex_eq_0
% 0.22/0.52 thf(fact_7113_zero__complex_Ocode,axiom,
% 0.22/0.52 ( zero_zero_complex
% 0.22/0.52 = ( complex2 @ zero_zero_real @ zero_zero_real ) ) ).
% 0.22/0.52
% 0.22/0.52 % zero_complex.code
% 0.22/0.52 thf(fact_7114_complex__diff,axiom,
% 0.22/0.52 ! [A: real,B: real,C: real,D: real] :
% 0.22/0.52 ( ( minus_minus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 0.22/0.52 = ( complex2 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % complex_diff
% 0.22/0.52 thf(fact_7115_powr__less__mono2__neg,axiom,
% 0.22/0.52 ! [A: real,X: real,Y4: real] :
% 0.22/0.52 ( ( ord_less_real @ A @ zero_zero_real )
% 0.22/0.52 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.52 => ( ( ord_less_real @ X @ Y4 )
% 0.22/0.52 => ( ord_less_real @ ( powr_real @ Y4 @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 0.22/0.52
% 0.22/0.52 % powr_less_mono2_neg
% 0.22/0.52 thf(fact_7116_powr__non__neg,axiom,
% 0.22/0.53 ! [A: real,X: real] :
% 0.22/0.53 ~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).
% 0.22/0.53
% 0.22/0.53 % powr_non_neg
% 0.22/0.53 thf(fact_7117_powr__ge__pzero,axiom,
% 0.22/0.53 ! [X: real,Y4: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y4 ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_ge_pzero
% 0.22/0.53 thf(fact_7118_powr__mono2,axiom,
% 0.22/0.53 ! [A: real,X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ X @ Y4 )
% 0.22/0.53 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y4 @ A ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_mono2
% 0.22/0.53 thf(fact_7119_powr__mono,axiom,
% 0.22/0.53 ! [A: real,B: real,X: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ A @ B )
% 0.22/0.53 => ( ( ord_less_eq_real @ one_one_real @ X )
% 0.22/0.53 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_mono
% 0.22/0.53 thf(fact_7120_Complex__eq__1,axiom,
% 0.22/0.53 ! [A: real,B: real] :
% 0.22/0.53 ( ( ( complex2 @ A @ B )
% 0.22/0.53 = one_one_complex )
% 0.22/0.53 = ( ( A = one_one_real )
% 0.22/0.53 & ( B = zero_zero_real ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % Complex_eq_1
% 0.22/0.53 thf(fact_7121_one__complex_Ocode,axiom,
% 0.22/0.53 ( one_one_complex
% 0.22/0.53 = ( complex2 @ one_one_real @ zero_zero_real ) ) ).
% 0.22/0.53
% 0.22/0.53 % one_complex.code
% 0.22/0.53 thf(fact_7122_Complex__eq__numeral,axiom,
% 0.22/0.53 ! [A: real,B: real,W: num] :
% 0.22/0.53 ( ( ( complex2 @ A @ B )
% 0.22/0.53 = ( numera6690914467698888265omplex @ W ) )
% 0.22/0.53 = ( ( A
% 0.22/0.53 = ( numeral_numeral_real @ W ) )
% 0.22/0.53 & ( B = zero_zero_real ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % Complex_eq_numeral
% 0.22/0.53 thf(fact_7123_powr__less__mono2,axiom,
% 0.22/0.53 ! [A: real,X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( ord_less_real @ X @ Y4 )
% 0.22/0.53 => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y4 @ A ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_less_mono2
% 0.22/0.53 thf(fact_7124_powr__mono2_H,axiom,
% 0.22/0.53 ! [A: real,X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.22/0.53 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ X @ Y4 )
% 0.22/0.53 => ( ord_less_eq_real @ ( powr_real @ Y4 @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_mono2'
% 0.22/0.53 thf(fact_7125_gr__one__powr,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ one_one_real @ X )
% 0.22/0.53 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.22/0.53 => ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % gr_one_powr
% 0.22/0.53 thf(fact_7126_powr__inj,axiom,
% 0.22/0.53 ! [A: real,X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.53 => ( ( A != one_one_real )
% 0.22/0.53 => ( ( ( powr_real @ A @ X )
% 0.22/0.53 = ( powr_real @ A @ Y4 ) )
% 0.22/0.53 = ( X = Y4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_inj
% 0.22/0.53 thf(fact_7127_ge__one__powr__ge__zero,axiom,
% 0.22/0.53 ! [X: real,A: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.53 => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ge_one_powr_ge_zero
% 0.22/0.53 thf(fact_7128_powr__mono__both,axiom,
% 0.22/0.53 ! [A: real,B: real,X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.53 => ( ( ord_less_eq_real @ A @ B )
% 0.22/0.53 => ( ( ord_less_eq_real @ one_one_real @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ X @ Y4 )
% 0.22/0.53 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y4 @ B ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_mono_both
% 0.22/0.53 thf(fact_7129_powr__le1,axiom,
% 0.22/0.53 ! [A: real,X: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.22/0.53 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_le1
% 0.22/0.53 thf(fact_7130_powr__divide,axiom,
% 0.22/0.53 ! [X: real,Y4: real,A: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.53 => ( ( powr_real @ ( divide_divide_real @ X @ Y4 ) @ A )
% 0.22/0.53 = ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y4 @ A ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_divide
% 0.22/0.53 thf(fact_7131_powr__mult,axiom,
% 0.22/0.53 ! [X: real,Y4: real,A: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.53 => ( ( powr_real @ ( times_times_real @ X @ Y4 ) @ A )
% 0.22/0.53 = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y4 @ A ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_mult
% 0.22/0.53 thf(fact_7132_divide__powr__uminus,axiom,
% 0.22/0.53 ! [A: real,B: real,C: real] :
% 0.22/0.53 ( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
% 0.22/0.53 = ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % divide_powr_uminus
% 0.22/0.53 thf(fact_7133_ln__powr,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( X != zero_zero_real )
% 0.22/0.53 => ( ( ln_ln_real @ ( powr_real @ X @ Y4 ) )
% 0.22/0.53 = ( times_times_real @ Y4 @ ( ln_ln_real @ X ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ln_powr
% 0.22/0.53 thf(fact_7134_log__base__powr,axiom,
% 0.22/0.53 ! [A: real,B: real,X: real] :
% 0.22/0.53 ( ( A != zero_zero_real )
% 0.22/0.53 => ( ( log @ ( powr_real @ A @ B ) @ X )
% 0.22/0.53 = ( divide_divide_real @ ( log @ A @ X ) @ B ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % log_base_powr
% 0.22/0.53 thf(fact_7135_log__powr,axiom,
% 0.22/0.53 ! [X: real,B: real,Y4: real] :
% 0.22/0.53 ( ( X != zero_zero_real )
% 0.22/0.53 => ( ( log @ B @ ( powr_real @ X @ Y4 ) )
% 0.22/0.53 = ( times_times_real @ Y4 @ ( log @ B @ X ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % log_powr
% 0.22/0.53 thf(fact_7136_Complex__eq__neg__1,axiom,
% 0.22/0.53 ! [A: real,B: real] :
% 0.22/0.53 ( ( ( complex2 @ A @ B )
% 0.22/0.53 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.22/0.53 = ( ( A
% 0.22/0.53 = ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.53 & ( B = zero_zero_real ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % Complex_eq_neg_1
% 0.22/0.53 thf(fact_7137_Complex__eq__neg__numeral,axiom,
% 0.22/0.53 ! [A: real,B: real,W: num] :
% 0.22/0.53 ( ( ( complex2 @ A @ B )
% 0.22/0.53 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.22/0.53 = ( ( A
% 0.22/0.53 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.22/0.53 & ( B = zero_zero_real ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % Complex_eq_neg_numeral
% 0.22/0.53 thf(fact_7138_powr__add,axiom,
% 0.22/0.53 ! [X: real,A: real,B: real] :
% 0.22/0.53 ( ( powr_real @ X @ ( plus_plus_real @ A @ B ) )
% 0.22/0.53 = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_add
% 0.22/0.53 thf(fact_7139_powr__diff,axiom,
% 0.22/0.53 ! [W: real,Z1: real,Z22: real] :
% 0.22/0.53 ( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z22 ) )
% 0.22/0.53 = ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z22 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_diff
% 0.22/0.53 thf(fact_7140_complex__mult,axiom,
% 0.22/0.53 ! [A: real,B: real,C: real,D: real] :
% 0.22/0.53 ( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 0.22/0.53 = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % complex_mult
% 0.22/0.53 thf(fact_7141_arcsin__le__arcsin,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ X @ Y4 )
% 0.22/0.53 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.22/0.53 => ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y4 ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arcsin_le_arcsin
% 0.22/0.53 thf(fact_7142_arcsin__minus,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.22/0.53 => ( ( arcsin @ ( uminus_uminus_real @ X ) )
% 0.22/0.53 = ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arcsin_minus
% 0.22/0.53 thf(fact_7143_arcsin__le__mono,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.22/0.53 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
% 0.22/0.53 => ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y4 ) )
% 0.22/0.53 = ( ord_less_eq_real @ X @ Y4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arcsin_le_mono
% 0.22/0.53 thf(fact_7144_arcsin__eq__iff,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.22/0.53 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
% 0.22/0.53 => ( ( ( arcsin @ X )
% 0.22/0.53 = ( arcsin @ Y4 ) )
% 0.22/0.53 = ( X = Y4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arcsin_eq_iff
% 0.22/0.53 thf(fact_7145_tan__def,axiom,
% 0.22/0.53 ( tan_complex
% 0.22/0.53 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X3 ) @ ( cos_complex @ X3 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_def
% 0.22/0.53 thf(fact_7146_tan__def,axiom,
% 0.22/0.53 ( tan_real
% 0.22/0.53 = ( ^ [X3: real] : ( divide_divide_real @ ( sin_real @ X3 ) @ ( cos_real @ X3 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_def
% 0.22/0.53 thf(fact_7147_powr__realpow,axiom,
% 0.22/0.53 ! [X: real,N: nat] :
% 0.22/0.53 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 0.22/0.53 = ( power_power_real @ X @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_realpow
% 0.22/0.53 thf(fact_7148_powr__less__iff,axiom,
% 0.22/0.53 ! [B: real,X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ one_one_real @ B )
% 0.22/0.53 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( ord_less_real @ ( powr_real @ B @ Y4 ) @ X )
% 0.22/0.53 = ( ord_less_real @ Y4 @ ( log @ B @ X ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_less_iff
% 0.22/0.53 thf(fact_7149_less__powr__iff,axiom,
% 0.22/0.53 ! [B: real,X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ one_one_real @ B )
% 0.22/0.53 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( ord_less_real @ X @ ( powr_real @ B @ Y4 ) )
% 0.22/0.53 = ( ord_less_real @ ( log @ B @ X ) @ Y4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % less_powr_iff
% 0.22/0.53 thf(fact_7150_log__less__iff,axiom,
% 0.22/0.53 ! [B: real,X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ one_one_real @ B )
% 0.22/0.53 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( ord_less_real @ ( log @ B @ X ) @ Y4 )
% 0.22/0.53 = ( ord_less_real @ X @ ( powr_real @ B @ Y4 ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % log_less_iff
% 0.22/0.53 thf(fact_7151_less__log__iff,axiom,
% 0.22/0.53 ! [B: real,X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ one_one_real @ B )
% 0.22/0.53 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( ord_less_real @ Y4 @ ( log @ B @ X ) )
% 0.22/0.53 = ( ord_less_real @ ( powr_real @ B @ Y4 ) @ X ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % less_log_iff
% 0.22/0.53 thf(fact_7152_powr__minus__divide,axiom,
% 0.22/0.53 ! [X: real,A: real] :
% 0.22/0.53 ( ( powr_real @ X @ ( uminus_uminus_real @ A ) )
% 0.22/0.53 = ( divide_divide_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_minus_divide
% 0.22/0.53 thf(fact_7153_powr__neg__one,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 0.22/0.53 = ( divide_divide_real @ one_one_real @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_neg_one
% 0.22/0.53 thf(fact_7154_arcsin__less__arcsin,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.22/0.53 => ( ( ord_less_real @ X @ Y4 )
% 0.22/0.53 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.22/0.53 => ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y4 ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arcsin_less_arcsin
% 0.22/0.53 thf(fact_7155_powr__mult__base,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( times_times_real @ X @ ( powr_real @ X @ Y4 ) )
% 0.22/0.53 = ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_mult_base
% 0.22/0.53 thf(fact_7156_arcsin__less__mono,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.22/0.53 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
% 0.22/0.53 => ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y4 ) )
% 0.22/0.53 = ( ord_less_real @ X @ Y4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arcsin_less_mono
% 0.22/0.53 thf(fact_7157_le__log__iff,axiom,
% 0.22/0.53 ! [B: real,X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ one_one_real @ B )
% 0.22/0.53 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ Y4 @ ( log @ B @ X ) )
% 0.22/0.53 = ( ord_less_eq_real @ ( powr_real @ B @ Y4 ) @ X ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % le_log_iff
% 0.22/0.53 thf(fact_7158_log__le__iff,axiom,
% 0.22/0.53 ! [B: real,X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ one_one_real @ B )
% 0.22/0.53 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y4 )
% 0.22/0.53 = ( ord_less_eq_real @ X @ ( powr_real @ B @ Y4 ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % log_le_iff
% 0.22/0.53 thf(fact_7159_le__powr__iff,axiom,
% 0.22/0.53 ! [B: real,X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ one_one_real @ B )
% 0.22/0.53 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y4 ) )
% 0.22/0.53 = ( ord_less_eq_real @ ( log @ B @ X ) @ Y4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % le_powr_iff
% 0.22/0.53 thf(fact_7160_powr__le__iff,axiom,
% 0.22/0.53 ! [B: real,X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ one_one_real @ B )
% 0.22/0.53 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ ( powr_real @ B @ Y4 ) @ X )
% 0.22/0.53 = ( ord_less_eq_real @ Y4 @ ( log @ B @ X ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_le_iff
% 0.22/0.53 thf(fact_7161_ln__powr__bound,axiom,
% 0.22/0.53 ! [X: real,A: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.22/0.53 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.53 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ln_powr_bound
% 0.22/0.53 thf(fact_7162_ln__powr__bound2,axiom,
% 0.22/0.53 ! [X: real,A: real] :
% 0.22/0.53 ( ( ord_less_real @ one_one_real @ X )
% 0.22/0.53 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.53 => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ln_powr_bound2
% 0.22/0.53 thf(fact_7163_tan__45,axiom,
% 0.22/0.53 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.22/0.53 = one_one_real ) ).
% 0.22/0.53
% 0.22/0.53 % tan_45
% 0.22/0.53 thf(fact_7164_log__add__eq__powr,axiom,
% 0.22/0.53 ! [B: real,X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ zero_zero_real @ B )
% 0.22/0.53 => ( ( B != one_one_real )
% 0.22/0.53 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( plus_plus_real @ ( log @ B @ X ) @ Y4 )
% 0.22/0.53 = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y4 ) ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % log_add_eq_powr
% 0.22/0.53 thf(fact_7165_add__log__eq__powr,axiom,
% 0.22/0.53 ! [B: real,X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ zero_zero_real @ B )
% 0.22/0.53 => ( ( B != one_one_real )
% 0.22/0.53 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( plus_plus_real @ Y4 @ ( log @ B @ X ) )
% 0.22/0.53 = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y4 ) @ X ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % add_log_eq_powr
% 0.22/0.53 thf(fact_7166_minus__log__eq__powr,axiom,
% 0.22/0.53 ! [B: real,X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ zero_zero_real @ B )
% 0.22/0.53 => ( ( B != one_one_real )
% 0.22/0.53 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( minus_minus_real @ Y4 @ ( log @ B @ X ) )
% 0.22/0.53 = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y4 ) @ X ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % minus_log_eq_powr
% 0.22/0.53 thf(fact_7167_tan__60,axiom,
% 0.22/0.53 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.22/0.53 = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_60
% 0.22/0.53 thf(fact_7168_cos__arcsin__nonzero,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.22/0.53 => ( ( ord_less_real @ X @ one_one_real )
% 0.22/0.53 => ( ( cos_real @ ( arcsin @ X ) )
% 0.22/0.53 != zero_zero_real ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % cos_arcsin_nonzero
% 0.22/0.53 thf(fact_7169_powr__def,axiom,
% 0.22/0.53 ( powr_real
% 0.22/0.53 = ( ^ [X3: real,A3: real] : ( if_real @ ( X3 = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A3 @ ( ln_ln_real @ X3 ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_def
% 0.22/0.53 thf(fact_7170_lemma__tan__total,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.22/0.53 => ? [X4: real] :
% 0.22/0.53 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.22/0.53 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 & ( ord_less_real @ Y4 @ ( tan_real @ X4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % lemma_tan_total
% 0.22/0.53 thf(fact_7171_tan__gt__zero,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 => ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_gt_zero
% 0.22/0.53 thf(fact_7172_lemma__tan__total1,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ? [X4: real] :
% 0.22/0.53 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.22/0.53 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 & ( ( tan_real @ X4 )
% 0.22/0.53 = Y4 ) ) ).
% 0.22/0.53
% 0.22/0.53 % lemma_tan_total1
% 0.22/0.53 thf(fact_7173_tan__mono__lt__eq,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.22/0.53 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 0.22/0.53 => ( ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 => ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) )
% 0.22/0.53 = ( ord_less_real @ X @ Y4 ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_mono_lt_eq
% 0.22/0.53 thf(fact_7174_tan__monotone_H,axiom,
% 0.22/0.53 ! [Y4: real,X: real] :
% 0.22/0.53 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 0.22/0.53 => ( ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.22/0.53 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 => ( ( ord_less_real @ Y4 @ X )
% 0.22/0.53 = ( ord_less_real @ ( tan_real @ Y4 ) @ ( tan_real @ X ) ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_monotone'
% 0.22/0.53 thf(fact_7175_tan__monotone,axiom,
% 0.22/0.53 ! [Y4: real,X: real] :
% 0.22/0.53 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 0.22/0.53 => ( ( ord_less_real @ Y4 @ X )
% 0.22/0.53 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 => ( ord_less_real @ ( tan_real @ Y4 ) @ ( tan_real @ X ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_monotone
% 0.22/0.53 thf(fact_7176_tan__total,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ? [X4: real] :
% 0.22/0.53 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.22/0.53 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 & ( ( tan_real @ X4 )
% 0.22/0.53 = Y4 )
% 0.22/0.53 & ! [Y3: real] :
% 0.22/0.53 ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.22/0.53 & ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 & ( ( tan_real @ Y3 )
% 0.22/0.53 = Y4 ) )
% 0.22/0.53 => ( Y3 = X4 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_total
% 0.22/0.53 thf(fact_7177_tan__minus__45,axiom,
% 0.22/0.53 ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
% 0.22/0.53 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_minus_45
% 0.22/0.53 thf(fact_7178_tan__inverse,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y4 ) )
% 0.22/0.53 = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y4 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_inverse
% 0.22/0.53 thf(fact_7179_log__minus__eq__powr,axiom,
% 0.22/0.53 ! [B: real,X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ zero_zero_real @ B )
% 0.22/0.53 => ( ( B != one_one_real )
% 0.22/0.53 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( minus_minus_real @ ( log @ B @ X ) @ Y4 )
% 0.22/0.53 = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y4 ) ) ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % log_minus_eq_powr
% 0.22/0.53 thf(fact_7180_complex__norm,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( real_V1022390504157884413omplex @ ( complex2 @ X @ Y4 ) )
% 0.22/0.53 = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % complex_norm
% 0.22/0.53 thf(fact_7181_add__tan__eq,axiom,
% 0.22/0.53 ! [X: complex,Y4: complex] :
% 0.22/0.53 ( ( ( cos_complex @ X )
% 0.22/0.53 != zero_zero_complex )
% 0.22/0.53 => ( ( ( cos_complex @ Y4 )
% 0.22/0.53 != zero_zero_complex )
% 0.22/0.53 => ( ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y4 ) )
% 0.22/0.53 = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X @ Y4 ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y4 ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % add_tan_eq
% 0.22/0.53 thf(fact_7182_add__tan__eq,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ( cos_real @ X )
% 0.22/0.53 != zero_zero_real )
% 0.22/0.53 => ( ( ( cos_real @ Y4 )
% 0.22/0.53 != zero_zero_real )
% 0.22/0.53 => ( ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) )
% 0.22/0.53 = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X @ Y4 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % add_tan_eq
% 0.22/0.53 thf(fact_7183_powr__half__sqrt,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 = ( sqrt @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_half_sqrt
% 0.22/0.53 thf(fact_7184_powr__neg__numeral,axiom,
% 0.22/0.53 ! [X: real,N: num] :
% 0.22/0.53 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.22/0.53 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % powr_neg_numeral
% 0.22/0.53 thf(fact_7185_tan__total__pos,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.53 => ? [X4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.22/0.53 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 & ( ( tan_real @ X4 )
% 0.22/0.53 = Y4 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_total_pos
% 0.22/0.53 thf(fact_7186_tan__pos__pi2__le,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_pos_pi2_le
% 0.22/0.53 thf(fact_7187_tan__less__zero,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 0.22/0.53 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.22/0.53 => ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_less_zero
% 0.22/0.53 thf(fact_7188_tan__mono__le,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ X @ Y4 )
% 0.22/0.53 => ( ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 => ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_mono_le
% 0.22/0.53 thf(fact_7189_tan__mono__le__eq,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.22/0.53 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 0.22/0.53 => ( ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 => ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) )
% 0.22/0.53 = ( ord_less_eq_real @ X @ Y4 ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_mono_le_eq
% 0.22/0.53 thf(fact_7190_tan__bound__pi2,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.22/0.53 => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_bound_pi2
% 0.22/0.53 thf(fact_7191_tan__30,axiom,
% 0.22/0.53 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.22/0.53 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_30
% 0.22/0.53 thf(fact_7192_arctan__unique,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.22/0.53 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 => ( ( ( tan_real @ X )
% 0.22/0.53 = Y4 )
% 0.22/0.53 => ( ( arctan @ Y4 )
% 0.22/0.53 = X ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arctan_unique
% 0.22/0.53 thf(fact_7193_arctan__tan,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.22/0.53 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 => ( ( arctan @ ( tan_real @ X ) )
% 0.22/0.53 = X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arctan_tan
% 0.22/0.53 thf(fact_7194_arctan,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y4 ) )
% 0.22/0.53 & ( ord_less_real @ ( arctan @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 & ( ( tan_real @ ( arctan @ Y4 ) )
% 0.22/0.53 = Y4 ) ) ).
% 0.22/0.53
% 0.22/0.53 % arctan
% 0.22/0.53 thf(fact_7195_lemma__tan__add1,axiom,
% 0.22/0.53 ! [X: complex,Y4: complex] :
% 0.22/0.53 ( ( ( cos_complex @ X )
% 0.22/0.53 != zero_zero_complex )
% 0.22/0.53 => ( ( ( cos_complex @ Y4 )
% 0.22/0.53 != zero_zero_complex )
% 0.22/0.53 => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y4 ) ) )
% 0.22/0.53 = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X @ Y4 ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y4 ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % lemma_tan_add1
% 0.22/0.53 thf(fact_7196_lemma__tan__add1,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ( cos_real @ X )
% 0.22/0.53 != zero_zero_real )
% 0.22/0.53 => ( ( ( cos_real @ Y4 )
% 0.22/0.53 != zero_zero_real )
% 0.22/0.53 => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) ) )
% 0.22/0.53 = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X @ Y4 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % lemma_tan_add1
% 0.22/0.53 thf(fact_7197_tan__diff,axiom,
% 0.22/0.53 ! [X: complex,Y4: complex] :
% 0.22/0.53 ( ( ( cos_complex @ X )
% 0.22/0.53 != zero_zero_complex )
% 0.22/0.53 => ( ( ( cos_complex @ Y4 )
% 0.22/0.53 != zero_zero_complex )
% 0.22/0.53 => ( ( ( cos_complex @ ( minus_minus_complex @ X @ Y4 ) )
% 0.22/0.53 != zero_zero_complex )
% 0.22/0.53 => ( ( tan_complex @ ( minus_minus_complex @ X @ Y4 ) )
% 0.22/0.53 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y4 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y4 ) ) ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_diff
% 0.22/0.53 thf(fact_7198_tan__diff,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ( cos_real @ X )
% 0.22/0.53 != zero_zero_real )
% 0.22/0.53 => ( ( ( cos_real @ Y4 )
% 0.22/0.53 != zero_zero_real )
% 0.22/0.53 => ( ( ( cos_real @ ( minus_minus_real @ X @ Y4 ) )
% 0.22/0.53 != zero_zero_real )
% 0.22/0.53 => ( ( tan_real @ ( minus_minus_real @ X @ Y4 ) )
% 0.22/0.53 = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) ) ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_diff
% 0.22/0.53 thf(fact_7199_tan__add,axiom,
% 0.22/0.53 ! [X: complex,Y4: complex] :
% 0.22/0.53 ( ( ( cos_complex @ X )
% 0.22/0.53 != zero_zero_complex )
% 0.22/0.53 => ( ( ( cos_complex @ Y4 )
% 0.22/0.53 != zero_zero_complex )
% 0.22/0.53 => ( ( ( cos_complex @ ( plus_plus_complex @ X @ Y4 ) )
% 0.22/0.53 != zero_zero_complex )
% 0.22/0.53 => ( ( tan_complex @ ( plus_plus_complex @ X @ Y4 ) )
% 0.22/0.53 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y4 ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y4 ) ) ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_add
% 0.22/0.53 thf(fact_7200_tan__add,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ( cos_real @ X )
% 0.22/0.53 != zero_zero_real )
% 0.22/0.53 => ( ( ( cos_real @ Y4 )
% 0.22/0.53 != zero_zero_real )
% 0.22/0.53 => ( ( ( cos_real @ ( plus_plus_real @ X @ Y4 ) )
% 0.22/0.53 != zero_zero_real )
% 0.22/0.53 => ( ( tan_real @ ( plus_plus_real @ X @ Y4 ) )
% 0.22/0.53 = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) ) ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_add
% 0.22/0.53 thf(fact_7201_tan__total__pi4,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.22/0.53 => ? [Z2: real] :
% 0.22/0.53 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z2 )
% 0.22/0.53 & ( ord_less_real @ Z2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.22/0.53 & ( ( tan_real @ Z2 )
% 0.22/0.53 = X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_total_pi4
% 0.22/0.53 thf(fact_7202_arcsin__lt__bounded,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 0.22/0.53 => ( ( ord_less_real @ Y4 @ one_one_real )
% 0.22/0.53 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y4 ) )
% 0.22/0.53 & ( ord_less_real @ ( arcsin @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arcsin_lt_bounded
% 0.22/0.53 thf(fact_7203_arcsin__lbound,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 0.22/0.53 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.22/0.53 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arcsin_lbound
% 0.22/0.53 thf(fact_7204_arcsin__ubound,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 0.22/0.53 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.22/0.53 => ( ord_less_eq_real @ ( arcsin @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arcsin_ubound
% 0.22/0.53 thf(fact_7205_arcsin__bounded,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 0.22/0.53 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.22/0.53 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y4 ) )
% 0.22/0.53 & ( ord_less_eq_real @ ( arcsin @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arcsin_bounded
% 0.22/0.53 thf(fact_7206_arcsin__sin,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 => ( ( arcsin @ ( sin_real @ X ) )
% 0.22/0.53 = X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arcsin_sin
% 0.22/0.53 thf(fact_7207_tan__half,axiom,
% 0.22/0.53 ( tan_complex
% 0.22/0.53 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_complex ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_half
% 0.22/0.53 thf(fact_7208_tan__half,axiom,
% 0.22/0.53 ( tan_real
% 0.22/0.53 = ( ^ [X3: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_real ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_half
% 0.22/0.53 thf(fact_7209_arcsin,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 0.22/0.53 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.22/0.53 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y4 ) )
% 0.22/0.53 & ( ord_less_eq_real @ ( arcsin @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 & ( ( sin_real @ ( arcsin @ Y4 ) )
% 0.22/0.53 = Y4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arcsin
% 0.22/0.53 thf(fact_7210_arcsin__pi,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 0.22/0.53 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.22/0.53 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y4 ) )
% 0.22/0.53 & ( ord_less_eq_real @ ( arcsin @ Y4 ) @ pi )
% 0.22/0.53 & ( ( sin_real @ ( arcsin @ Y4 ) )
% 0.22/0.53 = Y4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arcsin_pi
% 0.22/0.53 thf(fact_7211_arcsin__le__iff,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.22/0.53 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y4 )
% 0.22/0.53 => ( ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 => ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y4 )
% 0.22/0.53 = ( ord_less_eq_real @ X @ ( sin_real @ Y4 ) ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arcsin_le_iff
% 0.22/0.53 thf(fact_7212_le__arcsin__iff,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.22/0.53 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y4 )
% 0.22/0.53 => ( ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 => ( ( ord_less_eq_real @ Y4 @ ( arcsin @ X ) )
% 0.22/0.53 = ( ord_less_eq_real @ ( sin_real @ Y4 ) @ X ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % le_arcsin_iff
% 0.22/0.53 thf(fact_7213_arcosh__def,axiom,
% 0.22/0.53 ( arcosh_real
% 0.22/0.53 = ( ^ [X3: real] : ( ln_ln_real @ ( plus_plus_real @ X3 @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arcosh_def
% 0.22/0.53 thf(fact_7214_sin__arccos__abs,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
% 0.22/0.53 => ( ( sin_real @ ( arccos @ Y4 ) )
% 0.22/0.53 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % sin_arccos_abs
% 0.22/0.53 thf(fact_7215_sin__arccos,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.22/0.53 => ( ( sin_real @ ( arccos @ X ) )
% 0.22/0.53 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % sin_arccos
% 0.22/0.53 thf(fact_7216_arsinh__def,axiom,
% 0.22/0.53 ( arsinh_real
% 0.22/0.53 = ( ^ [X3: real] : ( ln_ln_real @ ( plus_plus_real @ X3 @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arsinh_def
% 0.22/0.53 thf(fact_7217_cos__npi__int,axiom,
% 0.22/0.53 ! [N: int] :
% 0.22/0.53 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.22/0.53 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.22/0.53 = one_one_real ) )
% 0.22/0.53 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.22/0.53 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.22/0.53 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % cos_npi_int
% 0.22/0.53 thf(fact_7218_of__int__eq__iff,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ( ring_1_of_int_real @ W )
% 0.22/0.53 = ( ring_1_of_int_real @ Z ) )
% 0.22/0.53 = ( W = Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_iff
% 0.22/0.53 thf(fact_7219_of__int__eq__iff,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ( ring_17405671764205052669omplex @ W )
% 0.22/0.53 = ( ring_17405671764205052669omplex @ Z ) )
% 0.22/0.53 = ( W = Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_iff
% 0.22/0.53 thf(fact_7220_of__int__eq__iff,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ( ring_1_of_int_rat @ W )
% 0.22/0.53 = ( ring_1_of_int_rat @ Z ) )
% 0.22/0.53 = ( W = Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_iff
% 0.22/0.53 thf(fact_7221_of__int__floor__cancel,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) )
% 0.22/0.53 = X )
% 0.22/0.53 = ( ? [N4: int] :
% 0.22/0.53 ( X
% 0.22/0.53 = ( ring_1_of_int_real @ N4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_floor_cancel
% 0.22/0.53 thf(fact_7222_of__int__floor__cancel,axiom,
% 0.22/0.53 ! [X: rat] :
% 0.22/0.53 ( ( ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) )
% 0.22/0.53 = X )
% 0.22/0.53 = ( ? [N4: int] :
% 0.22/0.53 ( X
% 0.22/0.53 = ( ring_1_of_int_rat @ N4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_floor_cancel
% 0.22/0.53 thf(fact_7223_of__int__ceiling__cancel,axiom,
% 0.22/0.53 ! [X: rat] :
% 0.22/0.53 ( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) )
% 0.22/0.53 = X )
% 0.22/0.53 = ( ? [N4: int] :
% 0.22/0.53 ( X
% 0.22/0.53 = ( ring_1_of_int_rat @ N4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_ceiling_cancel
% 0.22/0.53 thf(fact_7224_of__int__ceiling__cancel,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) )
% 0.22/0.53 = X )
% 0.22/0.53 = ( ? [N4: int] :
% 0.22/0.53 ( X
% 0.22/0.53 = ( ring_1_of_int_real @ N4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_ceiling_cancel
% 0.22/0.53 thf(fact_7225_of__int__eq__0__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ( ring_1_of_int_int @ Z )
% 0.22/0.53 = zero_zero_int )
% 0.22/0.53 = ( Z = zero_zero_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_0_iff
% 0.22/0.53 thf(fact_7226_of__int__eq__0__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ( ring_1_of_int_real @ Z )
% 0.22/0.53 = zero_zero_real )
% 0.22/0.53 = ( Z = zero_zero_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_0_iff
% 0.22/0.53 thf(fact_7227_of__int__eq__0__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ( ring_17405671764205052669omplex @ Z )
% 0.22/0.53 = zero_zero_complex )
% 0.22/0.53 = ( Z = zero_zero_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_0_iff
% 0.22/0.53 thf(fact_7228_of__int__eq__0__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ( ring_1_of_int_rat @ Z )
% 0.22/0.53 = zero_zero_rat )
% 0.22/0.53 = ( Z = zero_zero_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_0_iff
% 0.22/0.53 thf(fact_7229_of__int__0__eq__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( zero_zero_int
% 0.22/0.53 = ( ring_1_of_int_int @ Z ) )
% 0.22/0.53 = ( Z = zero_zero_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_0_eq_iff
% 0.22/0.53 thf(fact_7230_of__int__0__eq__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( zero_zero_real
% 0.22/0.53 = ( ring_1_of_int_real @ Z ) )
% 0.22/0.53 = ( Z = zero_zero_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_0_eq_iff
% 0.22/0.53 thf(fact_7231_of__int__0__eq__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( zero_zero_complex
% 0.22/0.53 = ( ring_17405671764205052669omplex @ Z ) )
% 0.22/0.53 = ( Z = zero_zero_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_0_eq_iff
% 0.22/0.53 thf(fact_7232_of__int__0__eq__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( zero_zero_rat
% 0.22/0.53 = ( ring_1_of_int_rat @ Z ) )
% 0.22/0.53 = ( Z = zero_zero_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_0_eq_iff
% 0.22/0.53 thf(fact_7233_of__int__0,axiom,
% 0.22/0.53 ( ( ring_1_of_int_int @ zero_zero_int )
% 0.22/0.53 = zero_zero_int ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_0
% 0.22/0.53 thf(fact_7234_of__int__0,axiom,
% 0.22/0.53 ( ( ring_1_of_int_real @ zero_zero_int )
% 0.22/0.53 = zero_zero_real ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_0
% 0.22/0.53 thf(fact_7235_of__int__0,axiom,
% 0.22/0.53 ( ( ring_17405671764205052669omplex @ zero_zero_int )
% 0.22/0.53 = zero_zero_complex ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_0
% 0.22/0.53 thf(fact_7236_of__int__0,axiom,
% 0.22/0.53 ( ( ring_1_of_int_rat @ zero_zero_int )
% 0.22/0.53 = zero_zero_rat ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_0
% 0.22/0.53 thf(fact_7237_of__real__0,axiom,
% 0.22/0.53 ( ( real_V1803761363581548252l_real @ zero_zero_real )
% 0.22/0.53 = zero_zero_real ) ).
% 0.22/0.53
% 0.22/0.53 % of_real_0
% 0.22/0.53 thf(fact_7238_of__real__0,axiom,
% 0.22/0.53 ( ( real_V4546457046886955230omplex @ zero_zero_real )
% 0.22/0.53 = zero_zero_complex ) ).
% 0.22/0.53
% 0.22/0.53 % of_real_0
% 0.22/0.53 thf(fact_7239_of__real__eq__0__iff,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ( real_V1803761363581548252l_real @ X )
% 0.22/0.53 = zero_zero_real )
% 0.22/0.53 = ( X = zero_zero_real ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_real_eq_0_iff
% 0.22/0.53 thf(fact_7240_of__real__eq__0__iff,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ( real_V4546457046886955230omplex @ X )
% 0.22/0.53 = zero_zero_complex )
% 0.22/0.53 = ( X = zero_zero_real ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_real_eq_0_iff
% 0.22/0.53 thf(fact_7241_of__int__le__iff,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 0.22/0.53 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_iff
% 0.22/0.53 thf(fact_7242_of__int__le__iff,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.22/0.53 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_iff
% 0.22/0.53 thf(fact_7243_of__int__le__iff,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.22/0.53 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_iff
% 0.22/0.53 thf(fact_7244_of__int__eq__numeral__iff,axiom,
% 0.22/0.53 ! [Z: int,N: num] :
% 0.22/0.53 ( ( ( ring_17405671764205052669omplex @ Z )
% 0.22/0.53 = ( numera6690914467698888265omplex @ N ) )
% 0.22/0.53 = ( Z
% 0.22/0.53 = ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_numeral_iff
% 0.22/0.53 thf(fact_7245_of__int__eq__numeral__iff,axiom,
% 0.22/0.53 ! [Z: int,N: num] :
% 0.22/0.53 ( ( ( ring_1_of_int_real @ Z )
% 0.22/0.53 = ( numeral_numeral_real @ N ) )
% 0.22/0.53 = ( Z
% 0.22/0.53 = ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_numeral_iff
% 0.22/0.53 thf(fact_7246_of__int__eq__numeral__iff,axiom,
% 0.22/0.53 ! [Z: int,N: num] :
% 0.22/0.53 ( ( ( ring_1_of_int_rat @ Z )
% 0.22/0.53 = ( numeral_numeral_rat @ N ) )
% 0.22/0.53 = ( Z
% 0.22/0.53 = ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_numeral_iff
% 0.22/0.53 thf(fact_7247_of__int__eq__numeral__iff,axiom,
% 0.22/0.53 ! [Z: int,N: num] :
% 0.22/0.53 ( ( ( ring_1_of_int_int @ Z )
% 0.22/0.53 = ( numeral_numeral_int @ N ) )
% 0.22/0.53 = ( Z
% 0.22/0.53 = ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_numeral_iff
% 0.22/0.53 thf(fact_7248_of__int__numeral,axiom,
% 0.22/0.53 ! [K: num] :
% 0.22/0.53 ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
% 0.22/0.53 = ( numera6690914467698888265omplex @ K ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_numeral
% 0.22/0.53 thf(fact_7249_of__int__numeral,axiom,
% 0.22/0.53 ! [K: num] :
% 0.22/0.53 ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 0.22/0.53 = ( numeral_numeral_real @ K ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_numeral
% 0.22/0.53 thf(fact_7250_of__int__numeral,axiom,
% 0.22/0.53 ! [K: num] :
% 0.22/0.53 ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
% 0.22/0.53 = ( numeral_numeral_rat @ K ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_numeral
% 0.22/0.53 thf(fact_7251_of__int__numeral,axiom,
% 0.22/0.53 ! [K: num] :
% 0.22/0.53 ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 0.22/0.53 = ( numeral_numeral_int @ K ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_numeral
% 0.22/0.53 thf(fact_7252_of__int__less__iff,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.22/0.53 = ( ord_less_int @ W @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_iff
% 0.22/0.53 thf(fact_7253_of__int__less__iff,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 0.22/0.53 = ( ord_less_int @ W @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_iff
% 0.22/0.53 thf(fact_7254_of__int__less__iff,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.22/0.53 = ( ord_less_int @ W @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_iff
% 0.22/0.53 thf(fact_7255_of__int__eq__1__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ( ring_1_of_int_int @ Z )
% 0.22/0.53 = one_one_int )
% 0.22/0.53 = ( Z = one_one_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_1_iff
% 0.22/0.53 thf(fact_7256_of__int__eq__1__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ( ring_1_of_int_real @ Z )
% 0.22/0.53 = one_one_real )
% 0.22/0.53 = ( Z = one_one_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_1_iff
% 0.22/0.53 thf(fact_7257_of__int__eq__1__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ( ring_17405671764205052669omplex @ Z )
% 0.22/0.53 = one_one_complex )
% 0.22/0.53 = ( Z = one_one_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_1_iff
% 0.22/0.53 thf(fact_7258_of__int__eq__1__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ( ring_1_of_int_rat @ Z )
% 0.22/0.53 = one_one_rat )
% 0.22/0.53 = ( Z = one_one_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_1_iff
% 0.22/0.53 thf(fact_7259_of__int__1,axiom,
% 0.22/0.53 ( ( ring_1_of_int_int @ one_one_int )
% 0.22/0.53 = one_one_int ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_1
% 0.22/0.53 thf(fact_7260_of__int__1,axiom,
% 0.22/0.53 ( ( ring_1_of_int_real @ one_one_int )
% 0.22/0.53 = one_one_real ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_1
% 0.22/0.53 thf(fact_7261_of__int__1,axiom,
% 0.22/0.53 ( ( ring_17405671764205052669omplex @ one_one_int )
% 0.22/0.53 = one_one_complex ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_1
% 0.22/0.53 thf(fact_7262_of__int__1,axiom,
% 0.22/0.53 ( ( ring_1_of_int_rat @ one_one_int )
% 0.22/0.53 = one_one_rat ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_1
% 0.22/0.53 thf(fact_7263_of__int__mult,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ring_17405671764205052669omplex @ ( times_times_int @ W @ Z ) )
% 0.22/0.53 = ( times_times_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_mult
% 0.22/0.53 thf(fact_7264_of__int__mult,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
% 0.22/0.53 = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_mult
% 0.22/0.53 thf(fact_7265_of__int__mult,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
% 0.22/0.53 = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_mult
% 0.22/0.53 thf(fact_7266_of__int__mult,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
% 0.22/0.53 = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_mult
% 0.22/0.53 thf(fact_7267_of__int__add,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
% 0.22/0.53 = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_add
% 0.22/0.53 thf(fact_7268_of__int__add,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
% 0.22/0.53 = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_add
% 0.22/0.53 thf(fact_7269_of__int__add,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ring_17405671764205052669omplex @ ( plus_plus_int @ W @ Z ) )
% 0.22/0.53 = ( plus_plus_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_add
% 0.22/0.53 thf(fact_7270_of__int__add,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
% 0.22/0.53 = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_add
% 0.22/0.53 thf(fact_7271_of__int__minus,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z ) )
% 0.22/0.53 = ( uminus_uminus_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_minus
% 0.22/0.53 thf(fact_7272_of__int__minus,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z ) )
% 0.22/0.53 = ( uminus_uminus_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_minus
% 0.22/0.53 thf(fact_7273_of__int__minus,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ Z ) )
% 0.22/0.53 = ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_minus
% 0.22/0.53 thf(fact_7274_of__int__minus,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ Z ) )
% 0.22/0.53 = ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_minus
% 0.22/0.53 thf(fact_7275_of__int__minus,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ Z ) )
% 0.22/0.53 = ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_minus
% 0.22/0.53 thf(fact_7276_of__real__mult,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( real_V1803761363581548252l_real @ ( times_times_real @ X @ Y4 ) )
% 0.22/0.53 = ( times_times_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y4 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_real_mult
% 0.22/0.53 thf(fact_7277_of__real__mult,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( real_V4546457046886955230omplex @ ( times_times_real @ X @ Y4 ) )
% 0.22/0.53 = ( times_times_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y4 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_real_mult
% 0.22/0.53 thf(fact_7278_of__real__numeral,axiom,
% 0.22/0.53 ! [W: num] :
% 0.22/0.53 ( ( real_V1803761363581548252l_real @ ( numeral_numeral_real @ W ) )
% 0.22/0.53 = ( numeral_numeral_real @ W ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_real_numeral
% 0.22/0.53 thf(fact_7279_of__real__numeral,axiom,
% 0.22/0.53 ! [W: num] :
% 0.22/0.53 ( ( real_V4546457046886955230omplex @ ( numeral_numeral_real @ W ) )
% 0.22/0.53 = ( numera6690914467698888265omplex @ W ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_real_numeral
% 0.22/0.53 thf(fact_7280_of__real__divide,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X @ Y4 ) )
% 0.22/0.53 = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y4 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_real_divide
% 0.22/0.53 thf(fact_7281_of__real__divide,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X @ Y4 ) )
% 0.22/0.53 = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y4 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_real_divide
% 0.22/0.53 thf(fact_7282_of__int__diff,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ring_17405671764205052669omplex @ ( minus_minus_int @ W @ Z ) )
% 0.22/0.53 = ( minus_minus_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_diff
% 0.22/0.53 thf(fact_7283_of__int__diff,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
% 0.22/0.53 = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_diff
% 0.22/0.53 thf(fact_7284_of__int__diff,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
% 0.22/0.53 = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_diff
% 0.22/0.53 thf(fact_7285_of__int__diff,axiom,
% 0.22/0.53 ! [W: int,Z: int] :
% 0.22/0.53 ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
% 0.22/0.53 = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_diff
% 0.22/0.53 thf(fact_7286_of__int__of__nat__eq,axiom,
% 0.22/0.53 ! [N: nat] :
% 0.22/0.53 ( ( ring_17405671764205052669omplex @ ( semiri1314217659103216013at_int @ N ) )
% 0.22/0.53 = ( semiri8010041392384452111omplex @ N ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_of_nat_eq
% 0.22/0.53 thf(fact_7287_of__int__of__nat__eq,axiom,
% 0.22/0.53 ! [N: nat] :
% 0.22/0.53 ( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N ) )
% 0.22/0.53 = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_of_nat_eq
% 0.22/0.53 thf(fact_7288_of__int__of__nat__eq,axiom,
% 0.22/0.53 ! [N: nat] :
% 0.22/0.53 ( ( ring_1_of_int_rat @ ( semiri1314217659103216013at_int @ N ) )
% 0.22/0.53 = ( semiri681578069525770553at_rat @ N ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_of_nat_eq
% 0.22/0.53 thf(fact_7289_of__int__of__nat__eq,axiom,
% 0.22/0.53 ! [N: nat] :
% 0.22/0.53 ( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.22/0.53 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_of_nat_eq
% 0.22/0.53 thf(fact_7290_of__real__diff,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( real_V1803761363581548252l_real @ ( minus_minus_real @ X @ Y4 ) )
% 0.22/0.53 = ( minus_minus_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y4 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_real_diff
% 0.22/0.53 thf(fact_7291_of__real__diff,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( real_V4546457046886955230omplex @ ( minus_minus_real @ X @ Y4 ) )
% 0.22/0.53 = ( minus_minus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y4 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_real_diff
% 0.22/0.53 thf(fact_7292_of__real__of__nat__eq,axiom,
% 0.22/0.53 ! [N: nat] :
% 0.22/0.53 ( ( real_V4546457046886955230omplex @ ( semiri5074537144036343181t_real @ N ) )
% 0.22/0.53 = ( semiri8010041392384452111omplex @ N ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_real_of_nat_eq
% 0.22/0.53 thf(fact_7293_of__real__of__nat__eq,axiom,
% 0.22/0.53 ! [N: nat] :
% 0.22/0.53 ( ( real_V1803761363581548252l_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.22/0.53 = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_real_of_nat_eq
% 0.22/0.53 thf(fact_7294_of__int__abs,axiom,
% 0.22/0.53 ! [X: int] :
% 0.22/0.53 ( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
% 0.22/0.53 = ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_abs
% 0.22/0.53 thf(fact_7295_of__int__abs,axiom,
% 0.22/0.53 ! [X: int] :
% 0.22/0.53 ( ( ring_18347121197199848620nteger @ ( abs_abs_int @ X ) )
% 0.22/0.53 = ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_abs
% 0.22/0.53 thf(fact_7296_of__int__abs,axiom,
% 0.22/0.53 ! [X: int] :
% 0.22/0.53 ( ( ring_1_of_int_real @ ( abs_abs_int @ X ) )
% 0.22/0.53 = ( abs_abs_real @ ( ring_1_of_int_real @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_abs
% 0.22/0.53 thf(fact_7297_of__int__abs,axiom,
% 0.22/0.53 ! [X: int] :
% 0.22/0.53 ( ( ring_1_of_int_rat @ ( abs_abs_int @ X ) )
% 0.22/0.53 = ( abs_abs_rat @ ( ring_1_of_int_rat @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_abs
% 0.22/0.53 thf(fact_7298_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: int,B: int,W: nat] :
% 0.22/0.53 ( ( ( ring_1_of_int_rat @ X )
% 0.22/0.53 = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 0.22/0.53 = ( X
% 0.22/0.53 = ( power_power_int @ B @ W ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_power_eq_of_int_cancel_iff
% 0.22/0.53 thf(fact_7299_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: int,B: int,W: nat] :
% 0.22/0.53 ( ( ( ring_1_of_int_int @ X )
% 0.22/0.53 = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.22/0.53 = ( X
% 0.22/0.53 = ( power_power_int @ B @ W ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_power_eq_of_int_cancel_iff
% 0.22/0.53 thf(fact_7300_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: int,B: int,W: nat] :
% 0.22/0.53 ( ( ( ring_1_of_int_real @ X )
% 0.22/0.53 = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.22/0.53 = ( X
% 0.22/0.53 = ( power_power_int @ B @ W ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_power_eq_of_int_cancel_iff
% 0.22/0.53 thf(fact_7301_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: int,B: int,W: nat] :
% 0.22/0.53 ( ( ( ring_17405671764205052669omplex @ X )
% 0.22/0.53 = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
% 0.22/0.53 = ( X
% 0.22/0.53 = ( power_power_int @ B @ W ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_power_eq_of_int_cancel_iff
% 0.22/0.53 thf(fact_7302_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.22/0.53 ! [B: int,W: nat,X: int] :
% 0.22/0.53 ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
% 0.22/0.53 = ( ring_1_of_int_rat @ X ) )
% 0.22/0.53 = ( ( power_power_int @ B @ W )
% 0.22/0.53 = X ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_of_int_power_cancel_iff
% 0.22/0.53 thf(fact_7303_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.22/0.53 ! [B: int,W: nat,X: int] :
% 0.22/0.53 ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
% 0.22/0.53 = ( ring_1_of_int_int @ X ) )
% 0.22/0.53 = ( ( power_power_int @ B @ W )
% 0.22/0.53 = X ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_of_int_power_cancel_iff
% 0.22/0.53 thf(fact_7304_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.22/0.53 ! [B: int,W: nat,X: int] :
% 0.22/0.53 ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
% 0.22/0.53 = ( ring_1_of_int_real @ X ) )
% 0.22/0.53 = ( ( power_power_int @ B @ W )
% 0.22/0.53 = X ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_of_int_power_cancel_iff
% 0.22/0.53 thf(fact_7305_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.22/0.53 ! [B: int,W: nat,X: int] :
% 0.22/0.53 ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
% 0.22/0.53 = ( ring_17405671764205052669omplex @ X ) )
% 0.22/0.53 = ( ( power_power_int @ B @ W )
% 0.22/0.53 = X ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_of_int_power_cancel_iff
% 0.22/0.53 thf(fact_7306_of__int__power,axiom,
% 0.22/0.53 ! [Z: int,N: nat] :
% 0.22/0.53 ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N ) )
% 0.22/0.53 = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_power
% 0.22/0.53 thf(fact_7307_of__int__power,axiom,
% 0.22/0.53 ! [Z: int,N: nat] :
% 0.22/0.53 ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
% 0.22/0.53 = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_power
% 0.22/0.53 thf(fact_7308_of__int__power,axiom,
% 0.22/0.53 ! [Z: int,N: nat] :
% 0.22/0.53 ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
% 0.22/0.53 = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_power
% 0.22/0.53 thf(fact_7309_of__int__power,axiom,
% 0.22/0.53 ! [Z: int,N: nat] :
% 0.22/0.53 ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N ) )
% 0.22/0.53 = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_power
% 0.22/0.53 thf(fact_7310_arccos__1,axiom,
% 0.22/0.53 ( ( arccos @ one_one_real )
% 0.22/0.53 = zero_zero_real ) ).
% 0.22/0.53
% 0.22/0.53 % arccos_1
% 0.22/0.53 thf(fact_7311_of__int__of__bool,axiom,
% 0.22/0.53 ! [P: $o] :
% 0.22/0.53 ( ( ring_1_of_int_real @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.22/0.53 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_of_bool
% 0.22/0.53 thf(fact_7312_of__int__of__bool,axiom,
% 0.22/0.53 ! [P: $o] :
% 0.22/0.53 ( ( ring_17405671764205052669omplex @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.22/0.53 = ( zero_n1201886186963655149omplex @ P ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_of_bool
% 0.22/0.53 thf(fact_7313_of__int__of__bool,axiom,
% 0.22/0.53 ! [P: $o] :
% 0.22/0.53 ( ( ring_1_of_int_rat @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.22/0.53 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_of_bool
% 0.22/0.53 thf(fact_7314_of__int__of__bool,axiom,
% 0.22/0.53 ! [P: $o] :
% 0.22/0.53 ( ( ring_1_of_int_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.22/0.53 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_of_bool
% 0.22/0.53 thf(fact_7315_of__int__of__bool,axiom,
% 0.22/0.53 ! [P: $o] :
% 0.22/0.53 ( ( ring_18347121197199848620nteger @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.22/0.53 = ( zero_n356916108424825756nteger @ P ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_of_bool
% 0.22/0.53 thf(fact_7316_sin__of__real__pi,axiom,
% 0.22/0.53 ( ( sin_real @ ( real_V1803761363581548252l_real @ pi ) )
% 0.22/0.53 = zero_zero_real ) ).
% 0.22/0.53
% 0.22/0.53 % sin_of_real_pi
% 0.22/0.53 thf(fact_7317_sin__of__real__pi,axiom,
% 0.22/0.53 ( ( sin_complex @ ( real_V4546457046886955230omplex @ pi ) )
% 0.22/0.53 = zero_zero_complex ) ).
% 0.22/0.53
% 0.22/0.53 % sin_of_real_pi
% 0.22/0.53 thf(fact_7318_floor__diff__of__int,axiom,
% 0.22/0.53 ! [X: real,Z: int] :
% 0.22/0.53 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
% 0.22/0.53 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_diff_of_int
% 0.22/0.53 thf(fact_7319_floor__diff__of__int,axiom,
% 0.22/0.53 ! [X: rat,Z: int] :
% 0.22/0.53 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
% 0.22/0.53 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_diff_of_int
% 0.22/0.53 thf(fact_7320_ceiling__diff__of__int,axiom,
% 0.22/0.53 ! [X: rat,Z: int] :
% 0.22/0.53 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
% 0.22/0.53 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_diff_of_int
% 0.22/0.53 thf(fact_7321_ceiling__diff__of__int,axiom,
% 0.22/0.53 ! [X: real,Z: int] :
% 0.22/0.53 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
% 0.22/0.53 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_diff_of_int
% 0.22/0.53 thf(fact_7322_of__nat__nat__take__bit__eq,axiom,
% 0.22/0.53 ! [N: nat,K: int] :
% 0.22/0.53 ( ( semiri8010041392384452111omplex @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.22/0.53 = ( ring_17405671764205052669omplex @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_nat_nat_take_bit_eq
% 0.22/0.53 thf(fact_7323_of__nat__nat__take__bit__eq,axiom,
% 0.22/0.53 ! [N: nat,K: int] :
% 0.22/0.53 ( ( semiri5074537144036343181t_real @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.22/0.53 = ( ring_1_of_int_real @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_nat_nat_take_bit_eq
% 0.22/0.53 thf(fact_7324_of__nat__nat__take__bit__eq,axiom,
% 0.22/0.53 ! [N: nat,K: int] :
% 0.22/0.53 ( ( semiri681578069525770553at_rat @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.22/0.53 = ( ring_1_of_int_rat @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_nat_nat_take_bit_eq
% 0.22/0.53 thf(fact_7325_of__nat__nat__take__bit__eq,axiom,
% 0.22/0.53 ! [N: nat,K: int] :
% 0.22/0.53 ( ( semiri1314217659103216013at_int @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.22/0.53 = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_nat_nat_take_bit_eq
% 0.22/0.53 thf(fact_7326_of__int__le__0__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 0.22/0.53 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_0_iff
% 0.22/0.53 thf(fact_7327_of__int__le__0__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.22/0.53 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_0_iff
% 0.22/0.53 thf(fact_7328_of__int__le__0__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.22/0.53 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_0_iff
% 0.22/0.53 thf(fact_7329_of__int__0__le__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.22/0.53 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_0_le_iff
% 0.22/0.53 thf(fact_7330_of__int__0__le__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.22/0.53 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_0_le_iff
% 0.22/0.53 thf(fact_7331_of__int__0__le__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.22/0.53 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_0_le_iff
% 0.22/0.53 thf(fact_7332_of__int__less__0__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.22/0.53 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_0_iff
% 0.22/0.53 thf(fact_7333_of__int__less__0__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 0.22/0.53 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_0_iff
% 0.22/0.53 thf(fact_7334_of__int__less__0__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.22/0.53 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_0_iff
% 0.22/0.53 thf(fact_7335_of__int__0__less__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.22/0.53 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_0_less_iff
% 0.22/0.53 thf(fact_7336_of__int__0__less__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.22/0.53 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_0_less_iff
% 0.22/0.53 thf(fact_7337_of__int__0__less__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.22/0.53 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_0_less_iff
% 0.22/0.53 thf(fact_7338_of__int__numeral__le__iff,axiom,
% 0.22/0.53 ! [N: num,Z: int] :
% 0.22/0.53 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 0.22/0.53 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_numeral_le_iff
% 0.22/0.53 thf(fact_7339_of__int__numeral__le__iff,axiom,
% 0.22/0.53 ! [N: num,Z: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 0.22/0.53 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_numeral_le_iff
% 0.22/0.53 thf(fact_7340_of__int__numeral__le__iff,axiom,
% 0.22/0.53 ! [N: num,Z: int] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 0.22/0.53 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_numeral_le_iff
% 0.22/0.53 thf(fact_7341_of__int__le__numeral__iff,axiom,
% 0.22/0.53 ! [Z: int,N: num] :
% 0.22/0.53 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 0.22/0.53 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_numeral_iff
% 0.22/0.53 thf(fact_7342_of__int__le__numeral__iff,axiom,
% 0.22/0.53 ! [Z: int,N: num] :
% 0.22/0.53 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 0.22/0.53 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_numeral_iff
% 0.22/0.53 thf(fact_7343_of__int__le__numeral__iff,axiom,
% 0.22/0.53 ! [Z: int,N: num] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 0.22/0.53 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_numeral_iff
% 0.22/0.53 thf(fact_7344_of__int__less__numeral__iff,axiom,
% 0.22/0.53 ! [Z: int,N: num] :
% 0.22/0.53 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 0.22/0.53 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_numeral_iff
% 0.22/0.53 thf(fact_7345_of__int__less__numeral__iff,axiom,
% 0.22/0.53 ! [Z: int,N: num] :
% 0.22/0.53 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 0.22/0.53 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_numeral_iff
% 0.22/0.53 thf(fact_7346_of__int__less__numeral__iff,axiom,
% 0.22/0.53 ! [Z: int,N: num] :
% 0.22/0.53 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 0.22/0.53 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_numeral_iff
% 0.22/0.53 thf(fact_7347_of__int__numeral__less__iff,axiom,
% 0.22/0.53 ! [N: num,Z: int] :
% 0.22/0.53 ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 0.22/0.53 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_numeral_less_iff
% 0.22/0.53 thf(fact_7348_of__int__numeral__less__iff,axiom,
% 0.22/0.53 ! [N: num,Z: int] :
% 0.22/0.53 ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 0.22/0.53 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_numeral_less_iff
% 0.22/0.53 thf(fact_7349_of__int__numeral__less__iff,axiom,
% 0.22/0.53 ! [N: num,Z: int] :
% 0.22/0.53 ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 0.22/0.53 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_numeral_less_iff
% 0.22/0.53 thf(fact_7350_of__int__le__1__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 0.22/0.53 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_1_iff
% 0.22/0.53 thf(fact_7351_of__int__le__1__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.22/0.53 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_1_iff
% 0.22/0.53 thf(fact_7352_of__int__le__1__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.22/0.53 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_1_iff
% 0.22/0.53 thf(fact_7353_of__int__1__le__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.22/0.53 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_1_le_iff
% 0.22/0.53 thf(fact_7354_of__int__1__le__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.22/0.53 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_1_le_iff
% 0.22/0.53 thf(fact_7355_of__int__1__le__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.22/0.53 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_1_le_iff
% 0.22/0.53 thf(fact_7356_of__int__less__1__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.22/0.53 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_1_iff
% 0.22/0.53 thf(fact_7357_of__int__less__1__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 0.22/0.53 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_1_iff
% 0.22/0.53 thf(fact_7358_of__int__less__1__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.22/0.53 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_1_iff
% 0.22/0.53 thf(fact_7359_of__int__1__less__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.22/0.53 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_1_less_iff
% 0.22/0.53 thf(fact_7360_of__int__1__less__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.22/0.53 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_1_less_iff
% 0.22/0.53 thf(fact_7361_of__int__1__less__iff,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.22/0.53 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_1_less_iff
% 0.22/0.53 thf(fact_7362_of__real__neg__numeral,axiom,
% 0.22/0.53 ! [W: num] :
% 0.22/0.53 ( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.22/0.53 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_real_neg_numeral
% 0.22/0.53 thf(fact_7363_of__real__neg__numeral,axiom,
% 0.22/0.53 ! [W: num] :
% 0.22/0.53 ( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.22/0.53 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_real_neg_numeral
% 0.22/0.53 thf(fact_7364_of__int__power__le__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: int,B: int,W: nat] :
% 0.22/0.53 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 0.22/0.53 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_power_le_of_int_cancel_iff
% 0.22/0.53 thf(fact_7365_of__int__power__le__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: int,B: int,W: nat] :
% 0.22/0.53 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.22/0.53 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_power_le_of_int_cancel_iff
% 0.22/0.53 thf(fact_7366_of__int__power__le__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: int,B: int,W: nat] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.22/0.53 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_power_le_of_int_cancel_iff
% 0.22/0.53 thf(fact_7367_of__int__le__of__int__power__cancel__iff,axiom,
% 0.22/0.53 ! [B: int,W: nat,X: int] :
% 0.22/0.53 ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 0.22/0.53 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_of_int_power_cancel_iff
% 0.22/0.53 thf(fact_7368_of__int__le__of__int__power__cancel__iff,axiom,
% 0.22/0.53 ! [B: int,W: nat,X: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 0.22/0.53 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_of_int_power_cancel_iff
% 0.22/0.53 thf(fact_7369_of__int__le__of__int__power__cancel__iff,axiom,
% 0.22/0.53 ! [B: int,W: nat,X: int] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 0.22/0.53 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_of_int_power_cancel_iff
% 0.22/0.53 thf(fact_7370_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,Y4: int] :
% 0.22/0.53 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
% 0.22/0.53 = ( ring_17405671764205052669omplex @ Y4 ) )
% 0.22/0.53 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.22/0.53 = Y4 ) ) ).
% 0.22/0.53
% 0.22/0.53 % numeral_power_eq_of_int_cancel_iff
% 0.22/0.53 thf(fact_7371_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,Y4: int] :
% 0.22/0.53 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 0.22/0.53 = ( ring_1_of_int_real @ Y4 ) )
% 0.22/0.53 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.22/0.53 = Y4 ) ) ).
% 0.22/0.53
% 0.22/0.53 % numeral_power_eq_of_int_cancel_iff
% 0.22/0.53 thf(fact_7372_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,Y4: int] :
% 0.22/0.53 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
% 0.22/0.53 = ( ring_1_of_int_rat @ Y4 ) )
% 0.22/0.53 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.22/0.53 = Y4 ) ) ).
% 0.22/0.53
% 0.22/0.53 % numeral_power_eq_of_int_cancel_iff
% 0.22/0.53 thf(fact_7373_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,Y4: int] :
% 0.22/0.53 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.22/0.53 = ( ring_1_of_int_int @ Y4 ) )
% 0.22/0.53 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.22/0.53 = Y4 ) ) ).
% 0.22/0.53
% 0.22/0.53 % numeral_power_eq_of_int_cancel_iff
% 0.22/0.53 thf(fact_7374_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [Y4: int,X: num,N: nat] :
% 0.22/0.53 ( ( ( ring_17405671764205052669omplex @ Y4 )
% 0.22/0.53 = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
% 0.22/0.53 = ( Y4
% 0.22/0.53 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7375_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [Y4: int,X: num,N: nat] :
% 0.22/0.53 ( ( ( ring_1_of_int_real @ Y4 )
% 0.22/0.53 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.22/0.53 = ( Y4
% 0.22/0.53 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7376_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [Y4: int,X: num,N: nat] :
% 0.22/0.53 ( ( ( ring_1_of_int_rat @ Y4 )
% 0.22/0.53 = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.22/0.53 = ( Y4
% 0.22/0.53 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7377_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [Y4: int,X: num,N: nat] :
% 0.22/0.53 ( ( ( ring_1_of_int_int @ Y4 )
% 0.22/0.53 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.22/0.53 = ( Y4
% 0.22/0.53 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7378_of__int__power__less__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: int,B: int,W: nat] :
% 0.22/0.53 ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.22/0.53 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_power_less_of_int_cancel_iff
% 0.22/0.53 thf(fact_7379_of__int__power__less__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: int,B: int,W: nat] :
% 0.22/0.53 ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 0.22/0.53 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_power_less_of_int_cancel_iff
% 0.22/0.53 thf(fact_7380_of__int__power__less__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: int,B: int,W: nat] :
% 0.22/0.53 ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.22/0.53 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_power_less_of_int_cancel_iff
% 0.22/0.53 thf(fact_7381_of__int__less__of__int__power__cancel__iff,axiom,
% 0.22/0.53 ! [B: int,W: nat,X: int] :
% 0.22/0.53 ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 0.22/0.53 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_of_int_power_cancel_iff
% 0.22/0.53 thf(fact_7382_of__int__less__of__int__power__cancel__iff,axiom,
% 0.22/0.53 ! [B: int,W: nat,X: int] :
% 0.22/0.53 ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 0.22/0.53 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_of_int_power_cancel_iff
% 0.22/0.53 thf(fact_7383_of__int__less__of__int__power__cancel__iff,axiom,
% 0.22/0.53 ! [B: int,W: nat,X: int] :
% 0.22/0.53 ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 0.22/0.53 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_of_int_power_cancel_iff
% 0.22/0.53 thf(fact_7384_of__nat__nat,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.22/0.53 => ( ( semiri8010041392384452111omplex @ ( nat2 @ Z ) )
% 0.22/0.53 = ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_nat_nat
% 0.22/0.53 thf(fact_7385_of__nat__nat,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.22/0.53 => ( ( semiri5074537144036343181t_real @ ( nat2 @ Z ) )
% 0.22/0.53 = ( ring_1_of_int_real @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_nat_nat
% 0.22/0.53 thf(fact_7386_of__nat__nat,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.22/0.53 => ( ( semiri681578069525770553at_rat @ ( nat2 @ Z ) )
% 0.22/0.53 = ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_nat_nat
% 0.22/0.53 thf(fact_7387_of__nat__nat,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.22/0.53 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.22/0.53 = ( ring_1_of_int_int @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_nat_nat
% 0.22/0.53 thf(fact_7388_sin__npi__int,axiom,
% 0.22/0.53 ! [N: int] :
% 0.22/0.53 ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.22/0.53 = zero_zero_real ) ).
% 0.22/0.53
% 0.22/0.53 % sin_npi_int
% 0.22/0.53 thf(fact_7389_cos__arccos,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 0.22/0.53 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.22/0.53 => ( ( cos_real @ ( arccos @ Y4 ) )
% 0.22/0.53 = Y4 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % cos_arccos
% 0.22/0.53 thf(fact_7390_tan__periodic__int,axiom,
% 0.22/0.53 ! [X: real,I: int] :
% 0.22/0.53 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
% 0.22/0.53 = ( tan_real @ X ) ) ).
% 0.22/0.53
% 0.22/0.53 % tan_periodic_int
% 0.22/0.53 thf(fact_7391_of__int__le__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [A: int,X: num,N: nat] :
% 0.22/0.53 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.22/0.53 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7392_of__int__le__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [A: int,X: num,N: nat] :
% 0.22/0.53 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.22/0.53 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7393_of__int__le__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [A: int,X: num,N: nat] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.22/0.53 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7394_numeral__power__le__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,A: int] :
% 0.22/0.53 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.22/0.53 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.22/0.53
% 0.22/0.53 % numeral_power_le_of_int_cancel_iff
% 0.22/0.53 thf(fact_7395_numeral__power__le__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,A: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.22/0.53 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.22/0.53
% 0.22/0.53 % numeral_power_le_of_int_cancel_iff
% 0.22/0.53 thf(fact_7396_numeral__power__le__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,A: int] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.22/0.53 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.22/0.53
% 0.22/0.53 % numeral_power_le_of_int_cancel_iff
% 0.22/0.53 thf(fact_7397_of__int__less__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [A: int,X: num,N: nat] :
% 0.22/0.53 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.22/0.53 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7398_of__int__less__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [A: int,X: num,N: nat] :
% 0.22/0.53 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.22/0.53 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7399_of__int__less__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [A: int,X: num,N: nat] :
% 0.22/0.53 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.22/0.53 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7400_numeral__power__less__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,A: int] :
% 0.22/0.53 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.22/0.53 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.22/0.53
% 0.22/0.53 % numeral_power_less_of_int_cancel_iff
% 0.22/0.53 thf(fact_7401_numeral__power__less__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,A: int] :
% 0.22/0.53 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.22/0.53 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.22/0.53
% 0.22/0.53 % numeral_power_less_of_int_cancel_iff
% 0.22/0.53 thf(fact_7402_numeral__power__less__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,A: int] :
% 0.22/0.53 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.22/0.53 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.22/0.53
% 0.22/0.53 % numeral_power_less_of_int_cancel_iff
% 0.22/0.53 thf(fact_7403_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [Y4: int,X: num,N: nat] :
% 0.22/0.53 ( ( ( ring_1_of_int_real @ Y4 )
% 0.22/0.53 = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 0.22/0.53 = ( Y4
% 0.22/0.53 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_neg_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7404_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [Y4: int,X: num,N: nat] :
% 0.22/0.53 ( ( ( ring_1_of_int_int @ Y4 )
% 0.22/0.53 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 0.22/0.53 = ( Y4
% 0.22/0.53 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_neg_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7405_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [Y4: int,X: num,N: nat] :
% 0.22/0.53 ( ( ( ring_17405671764205052669omplex @ Y4 )
% 0.22/0.53 = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N ) )
% 0.22/0.53 = ( Y4
% 0.22/0.53 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_neg_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7406_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [Y4: int,X: num,N: nat] :
% 0.22/0.53 ( ( ( ring_18347121197199848620nteger @ Y4 )
% 0.22/0.53 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 0.22/0.53 = ( Y4
% 0.22/0.53 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_neg_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7407_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [Y4: int,X: num,N: nat] :
% 0.22/0.53 ( ( ( ring_1_of_int_rat @ Y4 )
% 0.22/0.53 = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 0.22/0.53 = ( Y4
% 0.22/0.53 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_eq_neg_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7408_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,Y4: int] :
% 0.22/0.53 ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N )
% 0.22/0.53 = ( ring_1_of_int_real @ Y4 ) )
% 0.22/0.53 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.22/0.53 = Y4 ) ) ).
% 0.22/0.53
% 0.22/0.53 % neg_numeral_power_eq_of_int_cancel_iff
% 0.22/0.53 thf(fact_7409_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,Y4: int] :
% 0.22/0.53 ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.22/0.53 = ( ring_1_of_int_int @ Y4 ) )
% 0.22/0.53 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.22/0.53 = Y4 ) ) ).
% 0.22/0.53
% 0.22/0.53 % neg_numeral_power_eq_of_int_cancel_iff
% 0.22/0.53 thf(fact_7410_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,Y4: int] :
% 0.22/0.53 ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N )
% 0.22/0.53 = ( ring_17405671764205052669omplex @ Y4 ) )
% 0.22/0.53 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.22/0.53 = Y4 ) ) ).
% 0.22/0.53
% 0.22/0.53 % neg_numeral_power_eq_of_int_cancel_iff
% 0.22/0.53 thf(fact_7411_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,Y4: int] :
% 0.22/0.53 ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N )
% 0.22/0.53 = ( ring_18347121197199848620nteger @ Y4 ) )
% 0.22/0.53 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.22/0.53 = Y4 ) ) ).
% 0.22/0.53
% 0.22/0.53 % neg_numeral_power_eq_of_int_cancel_iff
% 0.22/0.53 thf(fact_7412_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,Y4: int] :
% 0.22/0.53 ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N )
% 0.22/0.53 = ( ring_1_of_int_rat @ Y4 ) )
% 0.22/0.53 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.22/0.53 = Y4 ) ) ).
% 0.22/0.53
% 0.22/0.53 % neg_numeral_power_eq_of_int_cancel_iff
% 0.22/0.53 thf(fact_7413_norm__of__real__addn,axiom,
% 0.22/0.53 ! [X: real,B: num] :
% 0.22/0.53 ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( numeral_numeral_real @ B ) ) )
% 0.22/0.53 = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % norm_of_real_addn
% 0.22/0.53 thf(fact_7414_norm__of__real__addn,axiom,
% 0.22/0.53 ! [X: real,B: num] :
% 0.22/0.53 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( numera6690914467698888265omplex @ B ) ) )
% 0.22/0.53 = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % norm_of_real_addn
% 0.22/0.53 thf(fact_7415_arccos__0,axiom,
% 0.22/0.53 ( ( arccos @ zero_zero_real )
% 0.22/0.53 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arccos_0
% 0.22/0.53 thf(fact_7416_cos__of__real__pi__half,axiom,
% 0.22/0.53 ( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 = zero_zero_real ) ).
% 0.22/0.53
% 0.22/0.53 % cos_of_real_pi_half
% 0.22/0.53 thf(fact_7417_cos__of__real__pi__half,axiom,
% 0.22/0.53 ( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 0.22/0.53 = zero_zero_complex ) ).
% 0.22/0.53
% 0.22/0.53 % cos_of_real_pi_half
% 0.22/0.53 thf(fact_7418_sin__of__real__pi__half,axiom,
% 0.22/0.53 ( ( sin_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.53 = one_one_real ) ).
% 0.22/0.53
% 0.22/0.53 % sin_of_real_pi_half
% 0.22/0.53 thf(fact_7419_sin__of__real__pi__half,axiom,
% 0.22/0.53 ( ( sin_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 0.22/0.53 = one_one_complex ) ).
% 0.22/0.53
% 0.22/0.53 % sin_of_real_pi_half
% 0.22/0.53 thf(fact_7420_sin__int__2pin,axiom,
% 0.22/0.53 ! [N: int] :
% 0.22/0.53 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 0.22/0.53 = zero_zero_real ) ).
% 0.22/0.53
% 0.22/0.53 % sin_int_2pin
% 0.22/0.53 thf(fact_7421_cos__int__2pin,axiom,
% 0.22/0.53 ! [N: int] :
% 0.22/0.53 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 0.22/0.53 = one_one_real ) ).
% 0.22/0.53
% 0.22/0.53 % cos_int_2pin
% 0.22/0.53 thf(fact_7422_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,A: int] :
% 0.22/0.53 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.22/0.53 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.22/0.53
% 0.22/0.53 % neg_numeral_power_le_of_int_cancel_iff
% 0.22/0.53 thf(fact_7423_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,A: int] :
% 0.22/0.53 ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.22/0.53 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.22/0.53
% 0.22/0.53 % neg_numeral_power_le_of_int_cancel_iff
% 0.22/0.53 thf(fact_7424_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,A: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.22/0.53 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.22/0.53
% 0.22/0.53 % neg_numeral_power_le_of_int_cancel_iff
% 0.22/0.53 thf(fact_7425_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,A: int] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.22/0.53 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.22/0.53
% 0.22/0.53 % neg_numeral_power_le_of_int_cancel_iff
% 0.22/0.53 thf(fact_7426_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [A: int,X: num,N: nat] :
% 0.22/0.53 ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 0.22/0.53 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_neg_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7427_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [A: int,X: num,N: nat] :
% 0.22/0.53 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 0.22/0.53 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_neg_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7428_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [A: int,X: num,N: nat] :
% 0.22/0.53 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 0.22/0.53 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_neg_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7429_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [A: int,X: num,N: nat] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 0.22/0.53 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_le_neg_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7430_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [A: int,X: num,N: nat] :
% 0.22/0.53 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 0.22/0.53 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_neg_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7431_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [A: int,X: num,N: nat] :
% 0.22/0.53 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 0.22/0.53 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_neg_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7432_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [A: int,X: num,N: nat] :
% 0.22/0.53 ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 0.22/0.53 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_neg_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7433_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.22/0.53 ! [A: int,X: num,N: nat] :
% 0.22/0.53 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 0.22/0.53 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_less_neg_numeral_power_cancel_iff
% 0.22/0.53 thf(fact_7434_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,A: int] :
% 0.22/0.53 ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.22/0.53 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.22/0.53
% 0.22/0.53 % neg_numeral_power_less_of_int_cancel_iff
% 0.22/0.53 thf(fact_7435_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,A: int] :
% 0.22/0.53 ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.22/0.53 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.22/0.53
% 0.22/0.53 % neg_numeral_power_less_of_int_cancel_iff
% 0.22/0.53 thf(fact_7436_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,A: int] :
% 0.22/0.53 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.22/0.53 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.22/0.53
% 0.22/0.53 % neg_numeral_power_less_of_int_cancel_iff
% 0.22/0.53 thf(fact_7437_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.22/0.53 ! [X: num,N: nat,A: int] :
% 0.22/0.53 ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.22/0.53 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.22/0.53
% 0.22/0.53 % neg_numeral_power_less_of_int_cancel_iff
% 0.22/0.53 thf(fact_7438_ex__le__of__int,axiom,
% 0.22/0.53 ! [X: rat] :
% 0.22/0.53 ? [Z2: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ).
% 0.22/0.53
% 0.22/0.53 % ex_le_of_int
% 0.22/0.53 thf(fact_7439_ex__le__of__int,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ? [Z2: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).
% 0.22/0.53
% 0.22/0.53 % ex_le_of_int
% 0.22/0.53 thf(fact_7440_mult__of__int__commute,axiom,
% 0.22/0.53 ! [X: int,Y4: complex] :
% 0.22/0.53 ( ( times_times_complex @ ( ring_17405671764205052669omplex @ X ) @ Y4 )
% 0.22/0.53 = ( times_times_complex @ Y4 @ ( ring_17405671764205052669omplex @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % mult_of_int_commute
% 0.22/0.53 thf(fact_7441_mult__of__int__commute,axiom,
% 0.22/0.53 ! [X: int,Y4: real] :
% 0.22/0.53 ( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y4 )
% 0.22/0.53 = ( times_times_real @ Y4 @ ( ring_1_of_int_real @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % mult_of_int_commute
% 0.22/0.53 thf(fact_7442_mult__of__int__commute,axiom,
% 0.22/0.53 ! [X: int,Y4: rat] :
% 0.22/0.53 ( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y4 )
% 0.22/0.53 = ( times_times_rat @ Y4 @ ( ring_1_of_int_rat @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % mult_of_int_commute
% 0.22/0.53 thf(fact_7443_mult__of__int__commute,axiom,
% 0.22/0.53 ! [X: int,Y4: int] :
% 0.22/0.53 ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y4 )
% 0.22/0.53 = ( times_times_int @ Y4 @ ( ring_1_of_int_int @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % mult_of_int_commute
% 0.22/0.53 thf(fact_7444_cos__int__times__real,axiom,
% 0.22/0.53 ! [M: int,X: real] :
% 0.22/0.53 ( ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X ) ) )
% 0.22/0.53 = ( real_V1803761363581548252l_real @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % cos_int_times_real
% 0.22/0.53 thf(fact_7445_cos__int__times__real,axiom,
% 0.22/0.53 ! [M: int,X: real] :
% 0.22/0.53 ( ( cos_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X ) ) )
% 0.22/0.53 = ( real_V4546457046886955230omplex @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % cos_int_times_real
% 0.22/0.53 thf(fact_7446_sin__int__times__real,axiom,
% 0.22/0.53 ! [M: int,X: real] :
% 0.22/0.53 ( ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X ) ) )
% 0.22/0.53 = ( real_V1803761363581548252l_real @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % sin_int_times_real
% 0.22/0.53 thf(fact_7447_sin__int__times__real,axiom,
% 0.22/0.53 ! [M: int,X: real] :
% 0.22/0.53 ( ( sin_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X ) ) )
% 0.22/0.53 = ( real_V4546457046886955230omplex @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % sin_int_times_real
% 0.22/0.53 thf(fact_7448_of__int__floor__le,axiom,
% 0.22/0.53 ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_floor_le
% 0.22/0.53 thf(fact_7449_of__int__floor__le,axiom,
% 0.22/0.53 ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_floor_le
% 0.22/0.53 thf(fact_7450_le__of__int__ceiling,axiom,
% 0.22/0.53 ! [X: rat] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % le_of_int_ceiling
% 0.22/0.53 thf(fact_7451_le__of__int__ceiling,axiom,
% 0.22/0.53 ! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % le_of_int_ceiling
% 0.22/0.53 thf(fact_7452_complex__of__real__def,axiom,
% 0.22/0.53 ( real_V4546457046886955230omplex
% 0.22/0.53 = ( ^ [R5: real] : ( complex2 @ R5 @ zero_zero_real ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % complex_of_real_def
% 0.22/0.53 thf(fact_7453_complex__of__real__code,axiom,
% 0.22/0.53 ( real_V4546457046886955230omplex
% 0.22/0.53 = ( ^ [X3: real] : ( complex2 @ X3 @ zero_zero_real ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % complex_of_real_code
% 0.22/0.53 thf(fact_7454_complex__eq__cancel__iff2,axiom,
% 0.22/0.53 ! [X: real,Y4: real,Xa: real] :
% 0.22/0.53 ( ( ( complex2 @ X @ Y4 )
% 0.22/0.53 = ( real_V4546457046886955230omplex @ Xa ) )
% 0.22/0.53 = ( ( X = Xa )
% 0.22/0.53 & ( Y4 = zero_zero_real ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % complex_eq_cancel_iff2
% 0.22/0.53 thf(fact_7455_complex__of__real__mult__Complex,axiom,
% 0.22/0.53 ! [R2: real,X: real,Y4: real] :
% 0.22/0.53 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X @ Y4 ) )
% 0.22/0.53 = ( complex2 @ ( times_times_real @ R2 @ X ) @ ( times_times_real @ R2 @ Y4 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % complex_of_real_mult_Complex
% 0.22/0.53 thf(fact_7456_Complex__mult__complex__of__real,axiom,
% 0.22/0.53 ! [X: real,Y4: real,R2: real] :
% 0.22/0.53 ( ( times_times_complex @ ( complex2 @ X @ Y4 ) @ ( real_V4546457046886955230omplex @ R2 ) )
% 0.22/0.53 = ( complex2 @ ( times_times_real @ X @ R2 ) @ ( times_times_real @ Y4 @ R2 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % Complex_mult_complex_of_real
% 0.22/0.53 thf(fact_7457_take__bit__of__int,axiom,
% 0.22/0.53 ! [N: nat,K: int] :
% 0.22/0.53 ( ( bit_se2923211474154528505it_int @ N @ ( ring_1_of_int_int @ K ) )
% 0.22/0.53 = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % take_bit_of_int
% 0.22/0.53 thf(fact_7458_of__int__and__eq,axiom,
% 0.22/0.53 ! [K: int,L: int] :
% 0.22/0.53 ( ( ring_1_of_int_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 0.22/0.53 = ( bit_se725231765392027082nd_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_and_eq
% 0.22/0.53 thf(fact_7459_of__int__mask__eq,axiom,
% 0.22/0.53 ! [N: nat] :
% 0.22/0.53 ( ( ring_1_of_int_int @ ( bit_se2000444600071755411sk_int @ N ) )
% 0.22/0.53 = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_mask_eq
% 0.22/0.53 thf(fact_7460_nonzero__of__real__divide,axiom,
% 0.22/0.53 ! [Y4: real,X: real] :
% 0.22/0.53 ( ( Y4 != zero_zero_real )
% 0.22/0.53 => ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X @ Y4 ) )
% 0.22/0.53 = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % nonzero_of_real_divide
% 0.22/0.53 thf(fact_7461_nonzero__of__real__divide,axiom,
% 0.22/0.53 ! [Y4: real,X: real] :
% 0.22/0.53 ( ( Y4 != zero_zero_real )
% 0.22/0.53 => ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X @ Y4 ) )
% 0.22/0.53 = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % nonzero_of_real_divide
% 0.22/0.53 thf(fact_7462_le__floor__iff,axiom,
% 0.22/0.53 ! [Z: int,X: real] :
% 0.22/0.53 ( ( ord_less_eq_int @ Z @ ( archim6058952711729229775r_real @ X ) )
% 0.22/0.53 = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).
% 0.22/0.53
% 0.22/0.53 % le_floor_iff
% 0.22/0.53 thf(fact_7463_le__floor__iff,axiom,
% 0.22/0.53 ! [Z: int,X: rat] :
% 0.22/0.53 ( ( ord_less_eq_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
% 0.22/0.53 = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).
% 0.22/0.53
% 0.22/0.53 % le_floor_iff
% 0.22/0.53 thf(fact_7464_ceiling__le__iff,axiom,
% 0.22/0.53 ! [X: rat,Z: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
% 0.22/0.53 = ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_le_iff
% 0.22/0.53 thf(fact_7465_ceiling__le__iff,axiom,
% 0.22/0.53 ! [X: real,Z: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z )
% 0.22/0.53 = ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_le_iff
% 0.22/0.53 thf(fact_7466_ceiling__le,axiom,
% 0.22/0.53 ! [X: rat,A: int] :
% 0.22/0.53 ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) )
% 0.22/0.53 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ A ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_le
% 0.22/0.53 thf(fact_7467_ceiling__le,axiom,
% 0.22/0.53 ! [X: real,A: int] :
% 0.22/0.53 ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
% 0.22/0.53 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_le
% 0.22/0.53 thf(fact_7468_real__of__int__div4,axiom,
% 0.22/0.53 ! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % real_of_int_div4
% 0.22/0.53 thf(fact_7469_floor__divide__of__int__eq,axiom,
% 0.22/0.53 ! [K: int,L: int] :
% 0.22/0.53 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( ring_1_of_int_real @ K ) @ ( ring_1_of_int_real @ L ) ) )
% 0.22/0.53 = ( divide_divide_int @ K @ L ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_divide_of_int_eq
% 0.22/0.53 thf(fact_7470_floor__divide__of__int__eq,axiom,
% 0.22/0.53 ! [K: int,L: int] :
% 0.22/0.53 ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ K ) @ ( ring_1_of_int_rat @ L ) ) )
% 0.22/0.53 = ( divide_divide_int @ K @ L ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_divide_of_int_eq
% 0.22/0.53 thf(fact_7471_floor__power,axiom,
% 0.22/0.53 ! [X: real,N: nat] :
% 0.22/0.53 ( ( X
% 0.22/0.53 = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) )
% 0.22/0.53 => ( ( archim6058952711729229775r_real @ ( power_power_real @ X @ N ) )
% 0.22/0.53 = ( power_power_int @ ( archim6058952711729229775r_real @ X ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_power
% 0.22/0.53 thf(fact_7472_floor__power,axiom,
% 0.22/0.53 ! [X: rat,N: nat] :
% 0.22/0.53 ( ( X
% 0.22/0.53 = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) )
% 0.22/0.53 => ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X @ N ) )
% 0.22/0.53 = ( power_power_int @ ( archim3151403230148437115or_rat @ X ) @ N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_power
% 0.22/0.53 thf(fact_7473_real__of__int__div,axiom,
% 0.22/0.53 ! [D: int,N: int] :
% 0.22/0.53 ( ( dvd_dvd_int @ D @ N )
% 0.22/0.53 => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
% 0.22/0.53 = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % real_of_int_div
% 0.22/0.53 thf(fact_7474_arccos__le__arccos,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ X @ Y4 )
% 0.22/0.53 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.22/0.53 => ( ord_less_eq_real @ ( arccos @ Y4 ) @ ( arccos @ X ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arccos_le_arccos
% 0.22/0.53 thf(fact_7475_arccos__eq__iff,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.22/0.53 & ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real ) )
% 0.22/0.53 => ( ( ( arccos @ X )
% 0.22/0.53 = ( arccos @ Y4 ) )
% 0.22/0.53 = ( X = Y4 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arccos_eq_iff
% 0.22/0.53 thf(fact_7476_arccos__le__mono,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.22/0.53 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
% 0.22/0.53 => ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y4 ) )
% 0.22/0.53 = ( ord_less_eq_real @ Y4 @ X ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arccos_le_mono
% 0.22/0.53 thf(fact_7477_of__int__nonneg,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.22/0.53 => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_nonneg
% 0.22/0.53 thf(fact_7478_of__int__nonneg,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.22/0.53 => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_nonneg
% 0.22/0.53 thf(fact_7479_of__int__nonneg,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.22/0.53 => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_nonneg
% 0.22/0.53 thf(fact_7480_of__int__leD,axiom,
% 0.22/0.53 ! [N: int,X: code_integer] :
% 0.22/0.53 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 0.22/0.53 => ( ( N = zero_zero_int )
% 0.22/0.53 | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_leD
% 0.22/0.53 thf(fact_7481_of__int__leD,axiom,
% 0.22/0.53 ! [N: int,X: rat] :
% 0.22/0.53 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 0.22/0.53 => ( ( N = zero_zero_int )
% 0.22/0.53 | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_leD
% 0.22/0.53 thf(fact_7482_of__int__leD,axiom,
% 0.22/0.53 ! [N: int,X: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 0.22/0.53 => ( ( N = zero_zero_int )
% 0.22/0.53 | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_leD
% 0.22/0.53 thf(fact_7483_of__int__leD,axiom,
% 0.22/0.53 ! [N: int,X: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 0.22/0.53 => ( ( N = zero_zero_int )
% 0.22/0.53 | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_leD
% 0.22/0.53 thf(fact_7484_of__int__pos,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.22/0.53 => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_pos
% 0.22/0.53 thf(fact_7485_of__int__pos,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.22/0.53 => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_pos
% 0.22/0.53 thf(fact_7486_of__int__pos,axiom,
% 0.22/0.53 ! [Z: int] :
% 0.22/0.53 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.22/0.53 => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_pos
% 0.22/0.53 thf(fact_7487_of__int__lessD,axiom,
% 0.22/0.53 ! [N: int,X: code_integer] :
% 0.22/0.53 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 0.22/0.53 => ( ( N = zero_zero_int )
% 0.22/0.53 | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_lessD
% 0.22/0.53 thf(fact_7488_of__int__lessD,axiom,
% 0.22/0.53 ! [N: int,X: real] :
% 0.22/0.53 ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 0.22/0.53 => ( ( N = zero_zero_int )
% 0.22/0.53 | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_lessD
% 0.22/0.53 thf(fact_7489_of__int__lessD,axiom,
% 0.22/0.53 ! [N: int,X: rat] :
% 0.22/0.53 ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 0.22/0.53 => ( ( N = zero_zero_int )
% 0.22/0.53 | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_lessD
% 0.22/0.53 thf(fact_7490_of__int__lessD,axiom,
% 0.22/0.53 ! [N: int,X: int] :
% 0.22/0.53 ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 0.22/0.53 => ( ( N = zero_zero_int )
% 0.22/0.53 | ( ord_less_int @ one_one_int @ X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_lessD
% 0.22/0.53 thf(fact_7491_floor__exists1,axiom,
% 0.22/0.53 ! [X: rat] :
% 0.22/0.53 ? [X4: int] :
% 0.22/0.53 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ X )
% 0.22/0.53 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 0.22/0.53 & ! [Y3: int] :
% 0.22/0.53 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y3 ) @ X )
% 0.22/0.53 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y3 @ one_one_int ) ) ) )
% 0.22/0.53 => ( Y3 = X4 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_exists1
% 0.22/0.53 thf(fact_7492_floor__exists1,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ? [X4: int] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ X )
% 0.22/0.53 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 0.22/0.53 & ! [Y3: int] :
% 0.22/0.53 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y3 ) @ X )
% 0.22/0.53 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y3 @ one_one_int ) ) ) )
% 0.22/0.53 => ( Y3 = X4 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_exists1
% 0.22/0.53 thf(fact_7493_floor__exists,axiom,
% 0.22/0.53 ! [X: rat] :
% 0.22/0.53 ? [Z2: int] :
% 0.22/0.53 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
% 0.22/0.53 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_exists
% 0.22/0.53 thf(fact_7494_floor__exists,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ? [Z2: int] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
% 0.22/0.53 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_exists
% 0.22/0.53 thf(fact_7495_of__int__ceiling__le__add__one,axiom,
% 0.22/0.53 ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_ceiling_le_add_one
% 0.22/0.53 thf(fact_7496_of__int__ceiling__le__add__one,axiom,
% 0.22/0.53 ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_ceiling_le_add_one
% 0.22/0.53 thf(fact_7497_of__int__neg__numeral,axiom,
% 0.22/0.53 ! [K: num] :
% 0.22/0.53 ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.22/0.53 = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_neg_numeral
% 0.22/0.53 thf(fact_7498_of__int__neg__numeral,axiom,
% 0.22/0.53 ! [K: num] :
% 0.22/0.53 ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.22/0.53 = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_neg_numeral
% 0.22/0.53 thf(fact_7499_of__int__neg__numeral,axiom,
% 0.22/0.53 ! [K: num] :
% 0.22/0.53 ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.22/0.53 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_neg_numeral
% 0.22/0.53 thf(fact_7500_of__int__neg__numeral,axiom,
% 0.22/0.53 ! [K: num] :
% 0.22/0.53 ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.22/0.53 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_neg_numeral
% 0.22/0.53 thf(fact_7501_of__int__neg__numeral,axiom,
% 0.22/0.53 ! [K: num] :
% 0.22/0.53 ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.22/0.53 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_neg_numeral
% 0.22/0.53 thf(fact_7502_of__int__ceiling__diff__one__le,axiom,
% 0.22/0.53 ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_ceiling_diff_one_le
% 0.22/0.53 thf(fact_7503_of__int__ceiling__diff__one__le,axiom,
% 0.22/0.53 ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_ceiling_diff_one_le
% 0.22/0.53 thf(fact_7504_of__nat__less__of__int__iff,axiom,
% 0.22/0.53 ! [N: nat,X: int] :
% 0.22/0.53 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
% 0.22/0.53 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_nat_less_of_int_iff
% 0.22/0.53 thf(fact_7505_of__nat__less__of__int__iff,axiom,
% 0.22/0.53 ! [N: nat,X: int] :
% 0.22/0.53 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X ) )
% 0.22/0.53 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_nat_less_of_int_iff
% 0.22/0.53 thf(fact_7506_of__nat__less__of__int__iff,axiom,
% 0.22/0.53 ! [N: nat,X: int] :
% 0.22/0.53 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
% 0.22/0.53 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_nat_less_of_int_iff
% 0.22/0.53 thf(fact_7507_int__le__real__less,axiom,
% 0.22/0.53 ( ord_less_eq_int
% 0.22/0.53 = ( ^ [N4: int,M5: int] : ( ord_less_real @ ( ring_1_of_int_real @ N4 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M5 ) @ one_one_real ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % int_le_real_less
% 0.22/0.53 thf(fact_7508_int__less__real__le,axiom,
% 0.22/0.53 ( ord_less_int
% 0.22/0.53 = ( ^ [N4: int,M5: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N4 ) @ one_one_real ) @ ( ring_1_of_int_real @ M5 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % int_less_real_le
% 0.22/0.53 thf(fact_7509_ceiling__divide__eq__div,axiom,
% 0.22/0.53 ! [A: int,B: int] :
% 0.22/0.53 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) )
% 0.22/0.53 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_divide_eq_div
% 0.22/0.53 thf(fact_7510_ceiling__divide__eq__div,axiom,
% 0.22/0.53 ! [A: int,B: int] :
% 0.22/0.53 ( ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) )
% 0.22/0.53 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_divide_eq_div
% 0.22/0.53 thf(fact_7511_sin__zero__iff__int2,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ( sin_real @ X )
% 0.22/0.53 = zero_zero_real )
% 0.22/0.53 = ( ? [I2: int] :
% 0.22/0.53 ( X
% 0.22/0.53 = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ pi ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % sin_zero_iff_int2
% 0.22/0.53 thf(fact_7512_arccos__cos__eq__abs__2pi,axiom,
% 0.22/0.53 ! [Theta: real] :
% 0.22/0.53 ~ ! [K2: int] :
% 0.22/0.53 ( ( arccos @ ( cos_real @ Theta ) )
% 0.22/0.53 != ( 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 ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arccos_cos_eq_abs_2pi
% 0.22/0.53 thf(fact_7513_real__of__int__div__aux,axiom,
% 0.22/0.53 ! [X: int,D: int] :
% 0.22/0.53 ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
% 0.22/0.53 = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % real_of_int_div_aux
% 0.22/0.53 thf(fact_7514_floor__eq,axiom,
% 0.22/0.53 ! [N: int,X: real] :
% 0.22/0.53 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
% 0.22/0.53 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.22/0.53 => ( ( archim6058952711729229775r_real @ X )
% 0.22/0.53 = N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_eq
% 0.22/0.53 thf(fact_7515_real__of__int__floor__add__one__gt,axiom,
% 0.22/0.53 ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 0.22/0.53
% 0.22/0.53 % real_of_int_floor_add_one_gt
% 0.22/0.53 thf(fact_7516_real__of__int__floor__add__one__ge,axiom,
% 0.22/0.53 ! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 0.22/0.53
% 0.22/0.53 % real_of_int_floor_add_one_ge
% 0.22/0.53 thf(fact_7517_real__of__int__floor__gt__diff__one,axiom,
% 0.22/0.53 ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % real_of_int_floor_gt_diff_one
% 0.22/0.53 thf(fact_7518_real__of__int__floor__ge__diff__one,axiom,
% 0.22/0.53 ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % real_of_int_floor_ge_diff_one
% 0.22/0.53 thf(fact_7519_arccos__lbound,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 0.22/0.53 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.22/0.53 => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arccos_lbound
% 0.22/0.53 thf(fact_7520_arccos__less__arccos,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.22/0.53 => ( ( ord_less_real @ X @ Y4 )
% 0.22/0.53 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.22/0.53 => ( ord_less_real @ ( arccos @ Y4 ) @ ( arccos @ X ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arccos_less_arccos
% 0.22/0.53 thf(fact_7521_arccos__less__mono,axiom,
% 0.22/0.53 ! [X: real,Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.22/0.53 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
% 0.22/0.53 => ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y4 ) )
% 0.22/0.53 = ( ord_less_real @ Y4 @ X ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arccos_less_mono
% 0.22/0.53 thf(fact_7522_arccos__ubound,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 0.22/0.53 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.22/0.53 => ( ord_less_eq_real @ ( arccos @ Y4 ) @ pi ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arccos_ubound
% 0.22/0.53 thf(fact_7523_arccos__cos,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ X @ pi )
% 0.22/0.53 => ( ( arccos @ ( cos_real @ X ) )
% 0.22/0.53 = X ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arccos_cos
% 0.22/0.53 thf(fact_7524_floor__split,axiom,
% 0.22/0.53 ! [P: int > $o,T: real] :
% 0.22/0.53 ( ( P @ ( archim6058952711729229775r_real @ T ) )
% 0.22/0.53 = ( ! [I2: int] :
% 0.22/0.53 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I2 ) @ T )
% 0.22/0.53 & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) ) )
% 0.22/0.53 => ( P @ I2 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_split
% 0.22/0.53 thf(fact_7525_floor__split,axiom,
% 0.22/0.53 ! [P: int > $o,T: rat] :
% 0.22/0.53 ( ( P @ ( archim3151403230148437115or_rat @ T ) )
% 0.22/0.53 = ( ! [I2: int] :
% 0.22/0.53 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I2 ) @ T )
% 0.22/0.53 & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) ) )
% 0.22/0.53 => ( P @ I2 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_split
% 0.22/0.53 thf(fact_7526_floor__eq__iff,axiom,
% 0.22/0.53 ! [X: real,A: int] :
% 0.22/0.53 ( ( ( archim6058952711729229775r_real @ X )
% 0.22/0.53 = A )
% 0.22/0.53 = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X )
% 0.22/0.53 & ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_eq_iff
% 0.22/0.53 thf(fact_7527_floor__eq__iff,axiom,
% 0.22/0.53 ! [X: rat,A: int] :
% 0.22/0.53 ( ( ( archim3151403230148437115or_rat @ X )
% 0.22/0.53 = A )
% 0.22/0.53 = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X )
% 0.22/0.53 & ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_eq_iff
% 0.22/0.53 thf(fact_7528_floor__unique,axiom,
% 0.22/0.53 ! [Z: int,X: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X )
% 0.22/0.53 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) )
% 0.22/0.53 => ( ( archim6058952711729229775r_real @ X )
% 0.22/0.53 = Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_unique
% 0.22/0.53 thf(fact_7529_floor__unique,axiom,
% 0.22/0.53 ! [Z: int,X: rat] :
% 0.22/0.53 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X )
% 0.22/0.53 => ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
% 0.22/0.53 => ( ( archim3151403230148437115or_rat @ X )
% 0.22/0.53 = Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_unique
% 0.22/0.53 thf(fact_7530_cos__arccos__abs,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
% 0.22/0.53 => ( ( cos_real @ ( arccos @ Y4 ) )
% 0.22/0.53 = Y4 ) ) ).
% 0.22/0.53
% 0.22/0.53 % cos_arccos_abs
% 0.22/0.53 thf(fact_7531_norm__of__real__diff,axiom,
% 0.22/0.53 ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( real_V1803761363581548252l_real @ B ) @ ( real_V1803761363581548252l_real @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % norm_of_real_diff
% 0.22/0.53 thf(fact_7532_norm__of__real__diff,axiom,
% 0.22/0.53 ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( real_V4546457046886955230omplex @ B ) @ ( real_V4546457046886955230omplex @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % norm_of_real_diff
% 0.22/0.53 thf(fact_7533_ceiling__split,axiom,
% 0.22/0.53 ! [P: int > $o,T: rat] :
% 0.22/0.53 ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
% 0.22/0.53 = ( ! [I2: int] :
% 0.22/0.53 ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) @ T )
% 0.22/0.53 & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I2 ) ) )
% 0.22/0.53 => ( P @ I2 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_split
% 0.22/0.53 thf(fact_7534_ceiling__split,axiom,
% 0.22/0.53 ! [P: int > $o,T: real] :
% 0.22/0.53 ( ( P @ ( archim7802044766580827645g_real @ T ) )
% 0.22/0.53 = ( ! [I2: int] :
% 0.22/0.53 ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) @ T )
% 0.22/0.53 & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I2 ) ) )
% 0.22/0.53 => ( P @ I2 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_split
% 0.22/0.53 thf(fact_7535_ceiling__eq__iff,axiom,
% 0.22/0.53 ! [X: rat,A: int] :
% 0.22/0.53 ( ( ( archim2889992004027027881ng_rat @ X )
% 0.22/0.53 = A )
% 0.22/0.53 = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X )
% 0.22/0.53 & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_eq_iff
% 0.22/0.53 thf(fact_7536_ceiling__eq__iff,axiom,
% 0.22/0.53 ! [X: real,A: int] :
% 0.22/0.53 ( ( ( archim7802044766580827645g_real @ X )
% 0.22/0.53 = A )
% 0.22/0.53 = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X )
% 0.22/0.53 & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_eq_iff
% 0.22/0.53 thf(fact_7537_ceiling__unique,axiom,
% 0.22/0.53 ! [Z: int,X: rat] :
% 0.22/0.53 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X )
% 0.22/0.53 => ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) )
% 0.22/0.53 => ( ( archim2889992004027027881ng_rat @ X )
% 0.22/0.53 = Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_unique
% 0.22/0.53 thf(fact_7538_ceiling__unique,axiom,
% 0.22/0.53 ! [Z: int,X: real] :
% 0.22/0.53 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) )
% 0.22/0.53 => ( ( archim7802044766580827645g_real @ X )
% 0.22/0.53 = Z ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_unique
% 0.22/0.53 thf(fact_7539_ceiling__correct,axiom,
% 0.22/0.53 ! [X: rat] :
% 0.22/0.53 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) @ one_one_rat ) @ X )
% 0.22/0.53 & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_correct
% 0.22/0.53 thf(fact_7540_ceiling__correct,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) @ one_one_real ) @ X )
% 0.22/0.53 & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_correct
% 0.22/0.53 thf(fact_7541_arccos__cos__eq__abs,axiom,
% 0.22/0.53 ! [Theta: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
% 0.22/0.53 => ( ( arccos @ ( cos_real @ Theta ) )
% 0.22/0.53 = ( abs_abs_real @ Theta ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arccos_cos_eq_abs
% 0.22/0.53 thf(fact_7542_less__floor__iff,axiom,
% 0.22/0.53 ! [Z: int,X: real] :
% 0.22/0.53 ( ( ord_less_int @ Z @ ( archim6058952711729229775r_real @ X ) )
% 0.22/0.53 = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).
% 0.22/0.53
% 0.22/0.53 % less_floor_iff
% 0.22/0.53 thf(fact_7543_less__floor__iff,axiom,
% 0.22/0.53 ! [Z: int,X: rat] :
% 0.22/0.53 ( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
% 0.22/0.53 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).
% 0.22/0.53
% 0.22/0.53 % less_floor_iff
% 0.22/0.53 thf(fact_7544_floor__le__iff,axiom,
% 0.22/0.53 ! [X: real,Z: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ Z )
% 0.22/0.53 = ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_le_iff
% 0.22/0.53 thf(fact_7545_floor__le__iff,axiom,
% 0.22/0.53 ! [X: rat,Z: int] :
% 0.22/0.53 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
% 0.22/0.53 = ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_le_iff
% 0.22/0.53 thf(fact_7546_ceiling__less__iff,axiom,
% 0.22/0.53 ! [X: rat,Z: int] :
% 0.22/0.53 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
% 0.22/0.53 = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_less_iff
% 0.22/0.53 thf(fact_7547_ceiling__less__iff,axiom,
% 0.22/0.53 ! [X: real,Z: int] :
% 0.22/0.53 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ Z )
% 0.22/0.53 = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_less_iff
% 0.22/0.53 thf(fact_7548_le__ceiling__iff,axiom,
% 0.22/0.53 ! [Z: int,X: rat] :
% 0.22/0.53 ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
% 0.22/0.53 = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).
% 0.22/0.53
% 0.22/0.53 % le_ceiling_iff
% 0.22/0.53 thf(fact_7549_le__ceiling__iff,axiom,
% 0.22/0.53 ! [Z: int,X: real] :
% 0.22/0.53 ( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X ) )
% 0.22/0.53 = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).
% 0.22/0.53
% 0.22/0.53 % le_ceiling_iff
% 0.22/0.53 thf(fact_7550_floor__correct,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X )
% 0.22/0.53 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_correct
% 0.22/0.53 thf(fact_7551_floor__correct,axiom,
% 0.22/0.53 ! [X: rat] :
% 0.22/0.53 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X )
% 0.22/0.53 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_correct
% 0.22/0.53 thf(fact_7552_real__of__int__div2,axiom,
% 0.22/0.53 ! [N: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % real_of_int_div2
% 0.22/0.53 thf(fact_7553_real__of__int__div3,axiom,
% 0.22/0.53 ! [N: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) @ one_one_real ) ).
% 0.22/0.53
% 0.22/0.53 % real_of_int_div3
% 0.22/0.53 thf(fact_7554_floor__eq2,axiom,
% 0.22/0.53 ! [N: int,X: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
% 0.22/0.53 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.22/0.53 => ( ( archim6058952711729229775r_real @ X )
% 0.22/0.53 = N ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_eq2
% 0.22/0.53 thf(fact_7555_floor__divide__real__eq__div,axiom,
% 0.22/0.53 ! [B: int,A: real] :
% 0.22/0.53 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.22/0.53 => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
% 0.22/0.53 = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_divide_real_eq_div
% 0.22/0.53 thf(fact_7556_floor__divide__lower,axiom,
% 0.22/0.53 ! [Q2: real,P5: real] :
% 0.22/0.53 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 0.22/0.53 => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P5 @ Q2 ) ) ) @ Q2 ) @ P5 ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_divide_lower
% 0.22/0.53 thf(fact_7557_floor__divide__lower,axiom,
% 0.22/0.53 ! [Q2: rat,P5: rat] :
% 0.22/0.53 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 0.22/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P5 @ Q2 ) ) ) @ Q2 ) @ P5 ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_divide_lower
% 0.22/0.53 thf(fact_7558_ceiling__divide__upper,axiom,
% 0.22/0.53 ! [Q2: rat,P5: rat] :
% 0.22/0.53 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 0.22/0.53 => ( ord_less_eq_rat @ P5 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P5 @ Q2 ) ) ) @ Q2 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_divide_upper
% 0.22/0.53 thf(fact_7559_ceiling__divide__upper,axiom,
% 0.22/0.53 ! [Q2: real,P5: real] :
% 0.22/0.53 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 0.22/0.53 => ( ord_less_eq_real @ P5 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P5 @ Q2 ) ) ) @ Q2 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_divide_upper
% 0.22/0.53 thf(fact_7560_arccos__lt__bounded,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 0.22/0.53 => ( ( ord_less_real @ Y4 @ one_one_real )
% 0.22/0.53 => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y4 ) )
% 0.22/0.53 & ( ord_less_real @ ( arccos @ Y4 ) @ pi ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arccos_lt_bounded
% 0.22/0.53 thf(fact_7561_even__of__int__iff,axiom,
% 0.22/0.53 ! [K: int] :
% 0.22/0.53 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
% 0.22/0.53 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 0.22/0.53
% 0.22/0.53 % even_of_int_iff
% 0.22/0.53 thf(fact_7562_even__of__int__iff,axiom,
% 0.22/0.53 ! [K: int] :
% 0.22/0.53 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
% 0.22/0.53 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 0.22/0.53
% 0.22/0.53 % even_of_int_iff
% 0.22/0.53 thf(fact_7563_arccos__bounded,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 0.22/0.53 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.22/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y4 ) )
% 0.22/0.53 & ( ord_less_eq_real @ ( arccos @ Y4 ) @ pi ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arccos_bounded
% 0.22/0.53 thf(fact_7564_sin__arccos__nonzero,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.22/0.53 => ( ( ord_less_real @ X @ one_one_real )
% 0.22/0.53 => ( ( sin_real @ ( arccos @ X ) )
% 0.22/0.53 != zero_zero_real ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % sin_arccos_nonzero
% 0.22/0.53 thf(fact_7565_arccos__cos2,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.22/0.53 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X )
% 0.22/0.53 => ( ( arccos @ ( cos_real @ X ) )
% 0.22/0.53 = ( uminus_uminus_real @ X ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arccos_cos2
% 0.22/0.53 thf(fact_7566_arccos__minus,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.22/0.53 => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 0.22/0.53 = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arccos_minus
% 0.22/0.53 thf(fact_7567_of__int__of__nat,axiom,
% 0.22/0.53 ( ring_18347121197199848620nteger
% 0.22/0.53 = ( ^ [K3: int] : ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri4939895301339042750nteger @ ( nat2 @ K3 ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_of_nat
% 0.22/0.53 thf(fact_7568_of__int__of__nat,axiom,
% 0.22/0.53 ( ring_17405671764205052669omplex
% 0.22/0.53 = ( ^ [K3: int] : ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri8010041392384452111omplex @ ( nat2 @ K3 ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_of_nat
% 0.22/0.53 thf(fact_7569_of__int__of__nat,axiom,
% 0.22/0.53 ( ring_1_of_int_real
% 0.22/0.53 = ( ^ [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 ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_of_nat
% 0.22/0.53 thf(fact_7570_of__int__of__nat,axiom,
% 0.22/0.53 ( ring_1_of_int_rat
% 0.22/0.53 = ( ^ [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 ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_of_nat
% 0.22/0.53 thf(fact_7571_of__int__of__nat,axiom,
% 0.22/0.53 ( ring_1_of_int_int
% 0.22/0.53 = ( ^ [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 ) ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % of_int_of_nat
% 0.22/0.53 thf(fact_7572_floor__divide__upper,axiom,
% 0.22/0.53 ! [Q2: real,P5: real] :
% 0.22/0.53 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 0.22/0.53 => ( ord_less_real @ P5 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P5 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_divide_upper
% 0.22/0.53 thf(fact_7573_floor__divide__upper,axiom,
% 0.22/0.53 ! [Q2: rat,P5: rat] :
% 0.22/0.53 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 0.22/0.53 => ( ord_less_rat @ P5 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P5 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % floor_divide_upper
% 0.22/0.53 thf(fact_7574_ceiling__divide__lower,axiom,
% 0.22/0.53 ! [Q2: real,P5: real] :
% 0.22/0.53 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 0.22/0.53 => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P5 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) @ P5 ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_divide_lower
% 0.22/0.53 thf(fact_7575_ceiling__divide__lower,axiom,
% 0.22/0.53 ! [Q2: rat,P5: rat] :
% 0.22/0.53 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 0.22/0.53 => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P5 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) @ P5 ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_divide_lower
% 0.22/0.53 thf(fact_7576_ceiling__eq,axiom,
% 0.22/0.53 ! [N: int,X: rat] :
% 0.22/0.53 ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X )
% 0.22/0.53 => ( ( ord_less_eq_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
% 0.22/0.53 => ( ( archim2889992004027027881ng_rat @ X )
% 0.22/0.53 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_eq
% 0.22/0.53 thf(fact_7577_ceiling__eq,axiom,
% 0.22/0.53 ! [N: int,X: real] :
% 0.22/0.53 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
% 0.22/0.53 => ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.22/0.53 => ( ( archim7802044766580827645g_real @ X )
% 0.22/0.53 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % ceiling_eq
% 0.22/0.53 thf(fact_7578_cos__one__2pi__int,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ( cos_real @ X )
% 0.22/0.53 = one_one_real )
% 0.22/0.53 = ( ? [X3: int] :
% 0.22/0.53 ( X
% 0.22/0.53 = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % cos_one_2pi_int
% 0.22/0.53 thf(fact_7579_arccos,axiom,
% 0.22/0.53 ! [Y4: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 0.22/0.53 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.22/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y4 ) )
% 0.22/0.53 & ( ord_less_eq_real @ ( arccos @ Y4 ) @ pi )
% 0.22/0.53 & ( ( cos_real @ ( arccos @ Y4 ) )
% 0.22/0.53 = Y4 ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arccos
% 0.22/0.53 thf(fact_7580_arccos__minus__abs,axiom,
% 0.22/0.53 ! [X: real] :
% 0.22/0.53 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.22/0.53 => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 0.22/0.53 = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ).
% 0.22/0.53
% 0.22/0.53 % arccos_minus_abs
% 0.22/0.53 thf(fact_7581_sin__cos__eq,axiom,
% 0.22/0.53 ( sin_real
% 0.22/0.53 = ( ^ [X3: real] : ( cos_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % sin_cos_eq
% 0.22/0.54 thf(fact_7582_sin__cos__eq,axiom,
% 0.22/0.54 ( sin_complex
% 0.22/0.54 = ( ^ [X3: complex] : ( cos_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % sin_cos_eq
% 0.22/0.54 thf(fact_7583_cos__sin__eq,axiom,
% 0.22/0.54 ( cos_real
% 0.22/0.54 = ( ^ [X3: real] : ( sin_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % cos_sin_eq
% 0.22/0.54 thf(fact_7584_cos__sin__eq,axiom,
% 0.22/0.54 ( cos_complex
% 0.22/0.54 = ( ^ [X3: complex] : ( sin_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % cos_sin_eq
% 0.22/0.54 thf(fact_7585_minus__sin__cos__eq,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( uminus_uminus_real @ ( sin_real @ X ) )
% 0.22/0.54 = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % minus_sin_cos_eq
% 0.22/0.54 thf(fact_7586_minus__sin__cos__eq,axiom,
% 0.22/0.54 ! [X: complex] :
% 0.22/0.54 ( ( uminus1482373934393186551omplex @ ( sin_complex @ X ) )
% 0.22/0.54 = ( cos_complex @ ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % minus_sin_cos_eq
% 0.22/0.54 thf(fact_7587_arccos__le__pi2,axiom,
% 0.22/0.54 ! [Y4: real] :
% 0.22/0.54 ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.54 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 0.22/0.54 => ( ord_less_eq_real @ ( arccos @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % arccos_le_pi2
% 0.22/0.54 thf(fact_7588_floor__log__eq__powr__iff,axiom,
% 0.22/0.54 ! [X: real,B: real,K: int] :
% 0.22/0.54 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.54 => ( ( ord_less_real @ one_one_real @ B )
% 0.22/0.54 => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
% 0.22/0.54 = K )
% 0.22/0.54 = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
% 0.22/0.54 & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % floor_log_eq_powr_iff
% 0.22/0.54 thf(fact_7589_cos__zero__iff__int,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( ( cos_real @ X )
% 0.22/0.54 = zero_zero_real )
% 0.22/0.54 = ( ? [I2: int] :
% 0.22/0.54 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I2 )
% 0.22/0.54 & ( X
% 0.22/0.54 = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % cos_zero_iff_int
% 0.22/0.54 thf(fact_7590_sin__zero__iff__int,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( ( sin_real @ X )
% 0.22/0.54 = zero_zero_real )
% 0.22/0.54 = ( ? [I2: int] :
% 0.22/0.54 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I2 )
% 0.22/0.54 & ( X
% 0.22/0.54 = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % sin_zero_iff_int
% 0.22/0.54 thf(fact_7591_powr__int,axiom,
% 0.22/0.54 ! [X: real,I: int] :
% 0.22/0.54 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.54 => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.22/0.54 => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 0.22/0.54 = ( power_power_real @ X @ ( nat2 @ I ) ) ) )
% 0.22/0.54 & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
% 0.22/0.54 => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 0.22/0.54 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % powr_int
% 0.22/0.54 thf(fact_7592_round__unique,axiom,
% 0.22/0.54 ! [X: rat,Y4: int] :
% 0.22/0.54 ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y4 ) )
% 0.22/0.54 => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y4 ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.54 => ( ( archim7778729529865785530nd_rat @ X )
% 0.22/0.54 = Y4 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % round_unique
% 0.22/0.54 thf(fact_7593_round__unique,axiom,
% 0.22/0.54 ! [X: real,Y4: int] :
% 0.22/0.54 ( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y4 ) )
% 0.22/0.54 => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y4 ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.22/0.54 => ( ( archim8280529875227126926d_real @ X )
% 0.22/0.54 = Y4 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % round_unique
% 0.22/0.54 thf(fact_7594_round__unique_H,axiom,
% 0.22/0.54 ! [X: real,N: int] :
% 0.22/0.54 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.54 => ( ( archim8280529875227126926d_real @ X )
% 0.22/0.54 = N ) ) ).
% 0.22/0.54
% 0.22/0.54 % round_unique'
% 0.22/0.54 thf(fact_7595_round__unique_H,axiom,
% 0.22/0.54 ! [X: rat,N: int] :
% 0.22/0.54 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.22/0.54 => ( ( archim7778729529865785530nd_rat @ X )
% 0.22/0.54 = N ) ) ).
% 0.22/0.54
% 0.22/0.54 % round_unique'
% 0.22/0.54 thf(fact_7596_of__int__round__abs__le,axiom,
% 0.22/0.54 ! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % of_int_round_abs_le
% 0.22/0.54 thf(fact_7597_of__int__round__abs__le,axiom,
% 0.22/0.54 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ X ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % of_int_round_abs_le
% 0.22/0.54 thf(fact_7598_of__int__round__gt,axiom,
% 0.22/0.54 ! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % of_int_round_gt
% 0.22/0.54 thf(fact_7599_of__int__round__gt,axiom,
% 0.22/0.54 ! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % of_int_round_gt
% 0.22/0.54 thf(fact_7600_of__int__round__ge,axiom,
% 0.22/0.54 ! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % of_int_round_ge
% 0.22/0.54 thf(fact_7601_of__int__round__ge,axiom,
% 0.22/0.54 ! [X: real] : ( ord_less_eq_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % of_int_round_ge
% 0.22/0.54 thf(fact_7602_round__0,axiom,
% 0.22/0.54 ( ( archim8280529875227126926d_real @ zero_zero_real )
% 0.22/0.54 = zero_zero_int ) ).
% 0.22/0.54
% 0.22/0.54 % round_0
% 0.22/0.54 thf(fact_7603_round__0,axiom,
% 0.22/0.54 ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
% 0.22/0.54 = zero_zero_int ) ).
% 0.22/0.54
% 0.22/0.54 % round_0
% 0.22/0.54 thf(fact_7604_round__numeral,axiom,
% 0.22/0.54 ! [N: num] :
% 0.22/0.54 ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
% 0.22/0.54 = ( numeral_numeral_int @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % round_numeral
% 0.22/0.54 thf(fact_7605_round__numeral,axiom,
% 0.22/0.54 ! [N: num] :
% 0.22/0.54 ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
% 0.22/0.54 = ( numeral_numeral_int @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % round_numeral
% 0.22/0.54 thf(fact_7606_round__of__nat,axiom,
% 0.22/0.54 ! [N: nat] :
% 0.22/0.54 ( ( archim8280529875227126926d_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.22/0.54 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % round_of_nat
% 0.22/0.54 thf(fact_7607_round__of__nat,axiom,
% 0.22/0.54 ! [N: nat] :
% 0.22/0.54 ( ( archim7778729529865785530nd_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.22/0.54 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % round_of_nat
% 0.22/0.54 thf(fact_7608_round__neg__numeral,axiom,
% 0.22/0.54 ! [N: num] :
% 0.22/0.54 ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.22/0.54 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % round_neg_numeral
% 0.22/0.54 thf(fact_7609_round__neg__numeral,axiom,
% 0.22/0.54 ! [N: num] :
% 0.22/0.54 ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.22/0.54 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % round_neg_numeral
% 0.22/0.54 thf(fact_7610_round__mono,axiom,
% 0.22/0.54 ! [X: rat,Y4: rat] :
% 0.22/0.54 ( ( ord_less_eq_rat @ X @ Y4 )
% 0.22/0.54 => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y4 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % round_mono
% 0.22/0.54 thf(fact_7611_round__mono,axiom,
% 0.22/0.54 ! [X: real,Y4: real] :
% 0.22/0.54 ( ( ord_less_eq_real @ X @ Y4 )
% 0.22/0.54 => ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim8280529875227126926d_real @ Y4 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % round_mono
% 0.22/0.54 thf(fact_7612_floor__le__round,axiom,
% 0.22/0.54 ! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim8280529875227126926d_real @ X ) ) ).
% 0.22/0.54
% 0.22/0.54 % floor_le_round
% 0.22/0.54 thf(fact_7613_floor__le__round,axiom,
% 0.22/0.54 ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim7778729529865785530nd_rat @ X ) ) ).
% 0.22/0.54
% 0.22/0.54 % floor_le_round
% 0.22/0.54 thf(fact_7614_ceiling__ge__round,axiom,
% 0.22/0.54 ! [X: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% 0.22/0.54
% 0.22/0.54 % ceiling_ge_round
% 0.22/0.54 thf(fact_7615_round__diff__minimal,axiom,
% 0.22/0.54 ! [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 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % round_diff_minimal
% 0.22/0.54 thf(fact_7616_round__diff__minimal,axiom,
% 0.22/0.54 ! [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 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % round_diff_minimal
% 0.22/0.54 thf(fact_7617_round__def,axiom,
% 0.22/0.54 ( archim8280529875227126926d_real
% 0.22/0.54 = ( ^ [X3: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % round_def
% 0.22/0.54 thf(fact_7618_round__def,axiom,
% 0.22/0.54 ( archim7778729529865785530nd_rat
% 0.22/0.54 = ( ^ [X3: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % round_def
% 0.22/0.54 thf(fact_7619_of__int__round__le,axiom,
% 0.22/0.54 ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % of_int_round_le
% 0.22/0.54 thf(fact_7620_of__int__round__le,axiom,
% 0.22/0.54 ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % of_int_round_le
% 0.22/0.54 thf(fact_7621_round__altdef,axiom,
% 0.22/0.54 ( archim8280529875227126926d_real
% 0.22/0.54 = ( ^ [X3: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X3 ) ) @ ( archim7802044766580827645g_real @ X3 ) @ ( archim6058952711729229775r_real @ X3 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % round_altdef
% 0.22/0.54 thf(fact_7622_round__altdef,axiom,
% 0.22/0.54 ( archim7778729529865785530nd_rat
% 0.22/0.54 = ( ^ [X3: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X3 ) ) @ ( archim2889992004027027881ng_rat @ X3 ) @ ( archim3151403230148437115or_rat @ X3 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % round_altdef
% 0.22/0.54 thf(fact_7623_exp__two__pi__i,axiom,
% 0.22/0.54 ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
% 0.22/0.54 = one_one_complex ) ).
% 0.22/0.54
% 0.22/0.54 % exp_two_pi_i
% 0.22/0.54 thf(fact_7624_exp__two__pi__i_H,axiom,
% 0.22/0.54 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
% 0.22/0.54 = one_one_complex ) ).
% 0.22/0.54
% 0.22/0.54 % exp_two_pi_i'
% 0.22/0.54 thf(fact_7625_cot__less__zero,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 0.22/0.54 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.22/0.54 => ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % cot_less_zero
% 0.22/0.54 thf(fact_7626_cot__periodic,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( cot_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.22/0.54 = ( cot_real @ X ) ) ).
% 0.22/0.54
% 0.22/0.54 % cot_periodic
% 0.22/0.54 thf(fact_7627_cot__zero,axiom,
% 0.22/0.54 ( ( cot_complex @ zero_zero_complex )
% 0.22/0.54 = zero_zero_complex ) ).
% 0.22/0.54
% 0.22/0.54 % cot_zero
% 0.22/0.54 thf(fact_7628_cot__zero,axiom,
% 0.22/0.54 ( ( cot_real @ zero_zero_real )
% 0.22/0.54 = zero_zero_real ) ).
% 0.22/0.54
% 0.22/0.54 % cot_zero
% 0.22/0.54 thf(fact_7629_frac__of__int,axiom,
% 0.22/0.54 ! [Z: int] :
% 0.22/0.54 ( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z ) )
% 0.22/0.54 = zero_zero_real ) ).
% 0.22/0.54
% 0.22/0.54 % frac_of_int
% 0.22/0.54 thf(fact_7630_frac__of__int,axiom,
% 0.22/0.54 ! [Z: int] :
% 0.22/0.54 ( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.22/0.54 = zero_zero_rat ) ).
% 0.22/0.54
% 0.22/0.54 % frac_of_int
% 0.22/0.54 thf(fact_7631_cot__pi,axiom,
% 0.22/0.54 ( ( cot_real @ pi )
% 0.22/0.54 = zero_zero_real ) ).
% 0.22/0.54
% 0.22/0.54 % cot_pi
% 0.22/0.54 thf(fact_7632_cot__npi,axiom,
% 0.22/0.54 ! [N: nat] :
% 0.22/0.54 ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.22/0.54 = zero_zero_real ) ).
% 0.22/0.54
% 0.22/0.54 % cot_npi
% 0.22/0.54 thf(fact_7633_power2__i,axiom,
% 0.22/0.54 ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.54 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.22/0.54
% 0.22/0.54 % power2_i
% 0.22/0.54 thf(fact_7634_i__even__power,axiom,
% 0.22/0.54 ! [N: nat] :
% 0.22/0.54 ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.54 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % i_even_power
% 0.22/0.54 thf(fact_7635_complex__i__not__zero,axiom,
% 0.22/0.54 imaginary_unit != zero_zero_complex ).
% 0.22/0.54
% 0.22/0.54 % complex_i_not_zero
% 0.22/0.54 thf(fact_7636_frac__ge__0,axiom,
% 0.22/0.54 ! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).
% 0.22/0.54
% 0.22/0.54 % frac_ge_0
% 0.22/0.54 thf(fact_7637_frac__ge__0,axiom,
% 0.22/0.54 ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).
% 0.22/0.54
% 0.22/0.54 % frac_ge_0
% 0.22/0.54 thf(fact_7638_imaginary__unit_Ocode,axiom,
% 0.22/0.54 ( imaginary_unit
% 0.22/0.54 = ( complex2 @ zero_zero_real @ one_one_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % imaginary_unit.code
% 0.22/0.54 thf(fact_7639_Complex__eq__i,axiom,
% 0.22/0.54 ! [X: real,Y4: real] :
% 0.22/0.54 ( ( ( complex2 @ X @ Y4 )
% 0.22/0.54 = imaginary_unit )
% 0.22/0.54 = ( ( X = zero_zero_real )
% 0.22/0.54 & ( Y4 = one_one_real ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Complex_eq_i
% 0.22/0.54 thf(fact_7640_frac__def,axiom,
% 0.22/0.54 ( archim2898591450579166408c_real
% 0.22/0.54 = ( ^ [X3: real] : ( minus_minus_real @ X3 @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X3 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % frac_def
% 0.22/0.54 thf(fact_7641_frac__def,axiom,
% 0.22/0.54 ( archimedean_frac_rat
% 0.22/0.54 = ( ^ [X3: rat] : ( minus_minus_rat @ X3 @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X3 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % frac_def
% 0.22/0.54 thf(fact_7642_cot__def,axiom,
% 0.22/0.54 ( cot_complex
% 0.22/0.54 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( cos_complex @ X3 ) @ ( sin_complex @ X3 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % cot_def
% 0.22/0.54 thf(fact_7643_cot__def,axiom,
% 0.22/0.54 ( cot_real
% 0.22/0.54 = ( ^ [X3: real] : ( divide_divide_real @ ( cos_real @ X3 ) @ ( sin_real @ X3 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % cot_def
% 0.22/0.54 thf(fact_7644_complex__of__real__i,axiom,
% 0.22/0.54 ! [R2: real] :
% 0.22/0.54 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ imaginary_unit )
% 0.22/0.54 = ( complex2 @ zero_zero_real @ R2 ) ) ).
% 0.22/0.54
% 0.22/0.54 % complex_of_real_i
% 0.22/0.54 thf(fact_7645_i__complex__of__real,axiom,
% 0.22/0.54 ! [R2: real] :
% 0.22/0.54 ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R2 ) )
% 0.22/0.54 = ( complex2 @ zero_zero_real @ R2 ) ) ).
% 0.22/0.54
% 0.22/0.54 % i_complex_of_real
% 0.22/0.54 thf(fact_7646_frac__eq,axiom,
% 0.22/0.54 ! [X: rat] :
% 0.22/0.54 ( ( ( archimedean_frac_rat @ X )
% 0.22/0.54 = X )
% 0.22/0.54 = ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.22/0.54 & ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % frac_eq
% 0.22/0.54 thf(fact_7647_frac__eq,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( ( archim2898591450579166408c_real @ X )
% 0.22/0.54 = X )
% 0.22/0.54 = ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.54 & ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % frac_eq
% 0.22/0.54 thf(fact_7648_frac__add,axiom,
% 0.22/0.54 ! [X: real,Y4: real] :
% 0.22/0.54 ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y4 ) ) @ one_one_real )
% 0.22/0.54 => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y4 ) )
% 0.22/0.54 = ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y4 ) ) ) )
% 0.22/0.54 & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y4 ) ) @ one_one_real )
% 0.22/0.54 => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y4 ) )
% 0.22/0.54 = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y4 ) ) @ one_one_real ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % frac_add
% 0.22/0.54 thf(fact_7649_frac__add,axiom,
% 0.22/0.54 ! [X: rat,Y4: rat] :
% 0.22/0.54 ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y4 ) ) @ one_one_rat )
% 0.22/0.54 => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y4 ) )
% 0.22/0.54 = ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y4 ) ) ) )
% 0.22/0.54 & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y4 ) ) @ one_one_rat )
% 0.22/0.54 => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y4 ) )
% 0.22/0.54 = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y4 ) ) @ one_one_rat ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % frac_add
% 0.22/0.54 thf(fact_7650_cot__gt__zero,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.54 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.54 => ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % cot_gt_zero
% 0.22/0.54 thf(fact_7651_tan__cot_H,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 0.22/0.54 = ( cot_real @ X ) ) ).
% 0.22/0.54
% 0.22/0.54 % tan_cot'
% 0.22/0.54 thf(fact_7652_Arg__minus__ii,axiom,
% 0.22/0.54 ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
% 0.22/0.54 = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Arg_minus_ii
% 0.22/0.54 thf(fact_7653_csqrt__ii,axiom,
% 0.22/0.54 ( ( csqrt @ imaginary_unit )
% 0.22/0.54 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % csqrt_ii
% 0.22/0.54 thf(fact_7654_Arg__ii,axiom,
% 0.22/0.54 ( ( arg @ imaginary_unit )
% 0.22/0.54 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Arg_ii
% 0.22/0.54 thf(fact_7655_cis__minus__pi__half,axiom,
% 0.22/0.54 ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.22/0.54 = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% 0.22/0.54
% 0.22/0.54 % cis_minus_pi_half
% 0.22/0.54 thf(fact_7656_signed__take__bit__eq__take__bit__minus,axiom,
% 0.22/0.54 ( bit_ri631733984087533419it_int
% 0.22/0.54 = ( ^ [N4: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N4 ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % signed_take_bit_eq_take_bit_minus
% 0.22/0.54 thf(fact_7657_csqrt__eq__0,axiom,
% 0.22/0.54 ! [Z: complex] :
% 0.22/0.54 ( ( ( csqrt @ Z )
% 0.22/0.54 = zero_zero_complex )
% 0.22/0.54 = ( Z = zero_zero_complex ) ) ).
% 0.22/0.54
% 0.22/0.54 % csqrt_eq_0
% 0.22/0.54 thf(fact_7658_csqrt__0,axiom,
% 0.22/0.54 ( ( csqrt @ zero_zero_complex )
% 0.22/0.54 = zero_zero_complex ) ).
% 0.22/0.54
% 0.22/0.54 % csqrt_0
% 0.22/0.54 thf(fact_7659_cis__zero,axiom,
% 0.22/0.54 ( ( cis @ zero_zero_real )
% 0.22/0.54 = one_one_complex ) ).
% 0.22/0.54
% 0.22/0.54 % cis_zero
% 0.22/0.54 thf(fact_7660_bit__numeral__Bit0__Suc__iff,axiom,
% 0.22/0.54 ! [M: num,N: nat] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 0.22/0.54 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_numeral_Bit0_Suc_iff
% 0.22/0.54 thf(fact_7661_bit__numeral__Bit0__Suc__iff,axiom,
% 0.22/0.54 ! [M: num,N: nat] :
% 0.22/0.54 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 0.22/0.54 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_numeral_Bit0_Suc_iff
% 0.22/0.54 thf(fact_7662_bit__numeral__Bit1__Suc__iff,axiom,
% 0.22/0.54 ! [M: num,N: nat] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 0.22/0.54 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_numeral_Bit1_Suc_iff
% 0.22/0.54 thf(fact_7663_bit__numeral__Bit1__Suc__iff,axiom,
% 0.22/0.54 ! [M: num,N: nat] :
% 0.22/0.54 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 0.22/0.54 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_numeral_Bit1_Suc_iff
% 0.22/0.54 thf(fact_7664_signed__take__bit__nonnegative__iff,axiom,
% 0.22/0.54 ! [N: nat,K: int] :
% 0.22/0.54 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.22/0.54 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % signed_take_bit_nonnegative_iff
% 0.22/0.54 thf(fact_7665_signed__take__bit__negative__iff,axiom,
% 0.22/0.54 ! [N: nat,K: int] :
% 0.22/0.54 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
% 0.22/0.54 = ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % signed_take_bit_negative_iff
% 0.22/0.54 thf(fact_7666_bit__numeral__simps_I2_J,axiom,
% 0.22/0.54 ! [W: num,N: num] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.22/0.54 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_numeral_simps(2)
% 0.22/0.54 thf(fact_7667_bit__numeral__simps_I2_J,axiom,
% 0.22/0.54 ! [W: num,N: num] :
% 0.22/0.54 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.22/0.54 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_numeral_simps(2)
% 0.22/0.54 thf(fact_7668_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 0.22/0.54 ! [W: num,N: nat] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
% 0.22/0.54 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_minus_numeral_Bit0_Suc_iff
% 0.22/0.54 thf(fact_7669_bit__numeral__simps_I3_J,axiom,
% 0.22/0.54 ! [W: num,N: num] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.22/0.54 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_numeral_simps(3)
% 0.22/0.54 thf(fact_7670_bit__numeral__simps_I3_J,axiom,
% 0.22/0.54 ! [W: num,N: num] :
% 0.22/0.54 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.22/0.54 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_numeral_simps(3)
% 0.22/0.54 thf(fact_7671_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 0.22/0.54 ! [W: num,N: nat] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
% 0.22/0.54 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_minus_numeral_Bit1_Suc_iff
% 0.22/0.54 thf(fact_7672_power2__csqrt,axiom,
% 0.22/0.54 ! [Z: complex] :
% 0.22/0.54 ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.54 = Z ) ).
% 0.22/0.54
% 0.22/0.54 % power2_csqrt
% 0.22/0.54 thf(fact_7673_bit__0,axiom,
% 0.22/0.54 ! [A: code_integer] :
% 0.22/0.54 ( ( bit_se9216721137139052372nteger @ A @ zero_zero_nat )
% 0.22/0.54 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_0
% 0.22/0.54 thf(fact_7674_bit__0,axiom,
% 0.22/0.54 ! [A: int] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ A @ zero_zero_nat )
% 0.22/0.54 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_0
% 0.22/0.54 thf(fact_7675_bit__0,axiom,
% 0.22/0.54 ! [A: nat] :
% 0.22/0.54 ( ( bit_se1148574629649215175it_nat @ A @ zero_zero_nat )
% 0.22/0.54 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_0
% 0.22/0.54 thf(fact_7676_bit__minus__numeral__int_I1_J,axiom,
% 0.22/0.54 ! [W: num,N: num] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 0.22/0.54 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_minus_numeral_int(1)
% 0.22/0.54 thf(fact_7677_bit__minus__numeral__int_I2_J,axiom,
% 0.22/0.54 ! [W: num,N: num] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 0.22/0.54 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_minus_numeral_int(2)
% 0.22/0.54 thf(fact_7678_bit__mod__2__iff,axiom,
% 0.22/0.54 ! [A: code_integer,N: nat] :
% 0.22/0.54 ( ( bit_se9216721137139052372nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N )
% 0.22/0.54 = ( ( N = zero_zero_nat )
% 0.22/0.54 & ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_mod_2_iff
% 0.22/0.54 thf(fact_7679_bit__mod__2__iff,axiom,
% 0.22/0.54 ! [A: int,N: nat] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N )
% 0.22/0.54 = ( ( N = zero_zero_nat )
% 0.22/0.54 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_mod_2_iff
% 0.22/0.54 thf(fact_7680_bit__mod__2__iff,axiom,
% 0.22/0.54 ! [A: nat,N: nat] :
% 0.22/0.54 ( ( bit_se1148574629649215175it_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 0.22/0.54 = ( ( N = zero_zero_nat )
% 0.22/0.54 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_mod_2_iff
% 0.22/0.54 thf(fact_7681_cis__pi__half,axiom,
% 0.22/0.54 ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.54 = imaginary_unit ) ).
% 0.22/0.54
% 0.22/0.54 % cis_pi_half
% 0.22/0.54 thf(fact_7682_cis__2pi,axiom,
% 0.22/0.54 ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.22/0.54 = one_one_complex ) ).
% 0.22/0.54
% 0.22/0.54 % cis_2pi
% 0.22/0.54 thf(fact_7683_cis__Arg,axiom,
% 0.22/0.54 ! [Z: complex] :
% 0.22/0.54 ( ( Z != zero_zero_complex )
% 0.22/0.54 => ( ( cis @ ( arg @ Z ) )
% 0.22/0.54 = ( sgn_sgn_complex @ Z ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % cis_Arg
% 0.22/0.54 thf(fact_7684_bit__numeral__iff,axiom,
% 0.22/0.54 ! [M: num,N: nat] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N )
% 0.22/0.54 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_numeral_iff
% 0.22/0.54 thf(fact_7685_bit__numeral__iff,axiom,
% 0.22/0.54 ! [M: num,N: nat] :
% 0.22/0.54 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N )
% 0.22/0.54 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_numeral_iff
% 0.22/0.54 thf(fact_7686_bit__of__nat__iff__bit,axiom,
% 0.22/0.54 ! [M: nat,N: nat] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ ( semiri1314217659103216013at_int @ M ) @ N )
% 0.22/0.54 = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_of_nat_iff_bit
% 0.22/0.54 thf(fact_7687_bit__of__nat__iff__bit,axiom,
% 0.22/0.54 ! [M: nat,N: nat] :
% 0.22/0.54 ( ( bit_se1148574629649215175it_nat @ ( semiri1316708129612266289at_nat @ M ) @ N )
% 0.22/0.54 = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_of_nat_iff_bit
% 0.22/0.54 thf(fact_7688_bit__disjunctive__add__iff,axiom,
% 0.22/0.54 ! [A: int,B: int,N: nat] :
% 0.22/0.54 ( ! [N2: nat] :
% 0.22/0.54 ( ~ ( bit_se1146084159140164899it_int @ A @ N2 )
% 0.22/0.54 | ~ ( bit_se1146084159140164899it_int @ B @ N2 ) )
% 0.22/0.54 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B ) @ N )
% 0.22/0.54 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.22/0.54 | ( bit_se1146084159140164899it_int @ B @ N ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_disjunctive_add_iff
% 0.22/0.54 thf(fact_7689_bit__disjunctive__add__iff,axiom,
% 0.22/0.54 ! [A: nat,B: nat,N: nat] :
% 0.22/0.54 ( ! [N2: nat] :
% 0.22/0.54 ( ~ ( bit_se1148574629649215175it_nat @ A @ N2 )
% 0.22/0.54 | ~ ( bit_se1148574629649215175it_nat @ B @ N2 ) )
% 0.22/0.54 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B ) @ N )
% 0.22/0.54 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.22/0.54 | ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_disjunctive_add_iff
% 0.22/0.54 thf(fact_7690_bit__and__iff,axiom,
% 0.22/0.54 ! [A: int,B: int,N: nat] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ N )
% 0.22/0.54 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.22/0.54 & ( bit_se1146084159140164899it_int @ B @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_and_iff
% 0.22/0.54 thf(fact_7691_bit__and__iff,axiom,
% 0.22/0.54 ! [A: nat,B: nat,N: nat] :
% 0.22/0.54 ( ( bit_se1148574629649215175it_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ N )
% 0.22/0.54 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.22/0.54 & ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_and_iff
% 0.22/0.54 thf(fact_7692_bit__and__int__iff,axiom,
% 0.22/0.54 ! [K: int,L: int,N: nat] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ N )
% 0.22/0.54 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.22/0.54 & ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_and_int_iff
% 0.22/0.54 thf(fact_7693_bit__unset__bit__iff,axiom,
% 0.22/0.54 ! [M: nat,A: int,N: nat] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ ( bit_se4203085406695923979it_int @ M @ A ) @ N )
% 0.22/0.54 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.22/0.54 & ( M != N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_unset_bit_iff
% 0.22/0.54 thf(fact_7694_bit__unset__bit__iff,axiom,
% 0.22/0.54 ! [M: nat,A: nat,N: nat] :
% 0.22/0.54 ( ( bit_se1148574629649215175it_nat @ ( bit_se4205575877204974255it_nat @ M @ A ) @ N )
% 0.22/0.54 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.22/0.54 & ( M != N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_unset_bit_iff
% 0.22/0.54 thf(fact_7695_bit__unset__bit__iff,axiom,
% 0.22/0.54 ! [M: nat,A: code_integer,N: nat] :
% 0.22/0.54 ( ( bit_se9216721137139052372nteger @ ( bit_se8260200283734997820nteger @ M @ A ) @ N )
% 0.22/0.54 = ( ( bit_se9216721137139052372nteger @ A @ N )
% 0.22/0.54 & ( M != N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_unset_bit_iff
% 0.22/0.54 thf(fact_7696_cis__divide,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B ) )
% 0.22/0.54 = ( cis @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % cis_divide
% 0.22/0.54 thf(fact_7697_cis__neq__zero,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( cis @ A )
% 0.22/0.54 != zero_zero_complex ) ).
% 0.22/0.54
% 0.22/0.54 % cis_neq_zero
% 0.22/0.54 thf(fact_7698_not__bit__1__Suc,axiom,
% 0.22/0.54 ! [N: nat] :
% 0.22/0.54 ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % not_bit_1_Suc
% 0.22/0.54 thf(fact_7699_not__bit__1__Suc,axiom,
% 0.22/0.54 ! [N: nat] :
% 0.22/0.54 ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % not_bit_1_Suc
% 0.22/0.54 thf(fact_7700_bit__1__iff,axiom,
% 0.22/0.54 ! [N: nat] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ one_one_int @ N )
% 0.22/0.54 = ( N = zero_zero_nat ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_1_iff
% 0.22/0.54 thf(fact_7701_bit__1__iff,axiom,
% 0.22/0.54 ! [N: nat] :
% 0.22/0.54 ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N )
% 0.22/0.54 = ( N = zero_zero_nat ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_1_iff
% 0.22/0.54 thf(fact_7702_bit__numeral__simps_I1_J,axiom,
% 0.22/0.54 ! [N: num] :
% 0.22/0.54 ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( numeral_numeral_nat @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_numeral_simps(1)
% 0.22/0.54 thf(fact_7703_bit__numeral__simps_I1_J,axiom,
% 0.22/0.54 ! [N: num] :
% 0.22/0.54 ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_numeral_simps(1)
% 0.22/0.54 thf(fact_7704_bit__take__bit__iff,axiom,
% 0.22/0.54 ! [M: nat,A: int,N: nat] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ M @ A ) @ N )
% 0.22/0.54 = ( ( ord_less_nat @ N @ M )
% 0.22/0.54 & ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_take_bit_iff
% 0.22/0.54 thf(fact_7705_bit__take__bit__iff,axiom,
% 0.22/0.54 ! [M: nat,A: nat,N: nat] :
% 0.22/0.54 ( ( bit_se1148574629649215175it_nat @ ( bit_se2925701944663578781it_nat @ M @ A ) @ N )
% 0.22/0.54 = ( ( ord_less_nat @ N @ M )
% 0.22/0.54 & ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_take_bit_iff
% 0.22/0.54 thf(fact_7706_bit__of__bool__iff,axiom,
% 0.22/0.54 ! [B: $o,N: nat] :
% 0.22/0.54 ( ( bit_se9216721137139052372nteger @ ( zero_n356916108424825756nteger @ B ) @ N )
% 0.22/0.54 = ( B
% 0.22/0.54 & ( N = zero_zero_nat ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_of_bool_iff
% 0.22/0.54 thf(fact_7707_bit__of__bool__iff,axiom,
% 0.22/0.54 ! [B: $o,N: nat] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ ( zero_n2684676970156552555ol_int @ B ) @ N )
% 0.22/0.54 = ( B
% 0.22/0.54 & ( N = zero_zero_nat ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_of_bool_iff
% 0.22/0.54 thf(fact_7708_bit__of__bool__iff,axiom,
% 0.22/0.54 ! [B: $o,N: nat] :
% 0.22/0.54 ( ( bit_se1148574629649215175it_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ N )
% 0.22/0.54 = ( B
% 0.22/0.54 & ( N = zero_zero_nat ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_of_bool_iff
% 0.22/0.54 thf(fact_7709_signed__take__bit__eq__if__positive,axiom,
% 0.22/0.54 ! [A: int,N: nat] :
% 0.22/0.54 ( ~ ( bit_se1146084159140164899it_int @ A @ N )
% 0.22/0.54 => ( ( bit_ri631733984087533419it_int @ N @ A )
% 0.22/0.54 = ( bit_se2923211474154528505it_int @ N @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % signed_take_bit_eq_if_positive
% 0.22/0.54 thf(fact_7710_DeMoivre,axiom,
% 0.22/0.54 ! [A: real,N: nat] :
% 0.22/0.54 ( ( power_power_complex @ ( cis @ A ) @ N )
% 0.22/0.54 = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % DeMoivre
% 0.22/0.54 thf(fact_7711_cis__Arg__unique,axiom,
% 0.22/0.54 ! [Z: complex,X: real] :
% 0.22/0.54 ( ( ( sgn_sgn_complex @ Z )
% 0.22/0.54 = ( cis @ X ) )
% 0.22/0.54 => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 0.22/0.54 => ( ( ord_less_eq_real @ X @ pi )
% 0.22/0.54 => ( ( arg @ Z )
% 0.22/0.54 = X ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % cis_Arg_unique
% 0.22/0.54 thf(fact_7712_bit__not__int__iff_H,axiom,
% 0.22/0.54 ! [K: int,N: nat] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
% 0.22/0.54 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_not_int_iff'
% 0.22/0.54 thf(fact_7713_flip__bit__eq__if,axiom,
% 0.22/0.54 ( bit_se2159334234014336723it_int
% 0.22/0.54 = ( ^ [N4: nat,A3: int] : ( if_nat_int_int @ ( bit_se1146084159140164899it_int @ A3 @ N4 ) @ bit_se4203085406695923979it_int @ bit_se7879613467334960850it_int @ N4 @ A3 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % flip_bit_eq_if
% 0.22/0.54 thf(fact_7714_flip__bit__eq__if,axiom,
% 0.22/0.54 ( bit_se2161824704523386999it_nat
% 0.22/0.54 = ( ^ [N4: nat,A3: nat] : ( if_nat_nat_nat @ ( bit_se1148574629649215175it_nat @ A3 @ N4 ) @ bit_se4205575877204974255it_nat @ bit_se7882103937844011126it_nat @ N4 @ A3 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % flip_bit_eq_if
% 0.22/0.54 thf(fact_7715_flip__bit__eq__if,axiom,
% 0.22/0.54 ( bit_se1345352211410354436nteger
% 0.22/0.54 = ( ^ [N4: nat,A3: code_integer] : ( if_nat5617392847756311170nteger @ ( bit_se9216721137139052372nteger @ A3 @ N4 ) @ bit_se8260200283734997820nteger @ bit_se2793503036327961859nteger @ N4 @ A3 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % flip_bit_eq_if
% 0.22/0.54 thf(fact_7716_Arg__correct,axiom,
% 0.22/0.54 ! [Z: complex] :
% 0.22/0.54 ( ( Z != zero_zero_complex )
% 0.22/0.54 => ( ( ( sgn_sgn_complex @ Z )
% 0.22/0.54 = ( cis @ ( arg @ Z ) ) )
% 0.22/0.54 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 0.22/0.54 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Arg_correct
% 0.22/0.54 thf(fact_7717_Arg__zero,axiom,
% 0.22/0.54 ( ( arg @ zero_zero_complex )
% 0.22/0.54 = zero_zero_real ) ).
% 0.22/0.54
% 0.22/0.54 % Arg_zero
% 0.22/0.54 thf(fact_7718_bit__imp__take__bit__positive,axiom,
% 0.22/0.54 ! [N: nat,M: nat,K: int] :
% 0.22/0.54 ( ( ord_less_nat @ N @ M )
% 0.22/0.54 => ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.22/0.54 => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_imp_take_bit_positive
% 0.22/0.54 thf(fact_7719_bit__concat__bit__iff,axiom,
% 0.22/0.54 ! [M: nat,K: int,L: int,N: nat] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
% 0.22/0.54 = ( ( ( ord_less_nat @ N @ M )
% 0.22/0.54 & ( bit_se1146084159140164899it_int @ K @ N ) )
% 0.22/0.54 | ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.54 & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_concat_bit_iff
% 0.22/0.54 thf(fact_7720_signed__take__bit__eq__concat__bit,axiom,
% 0.22/0.54 ( bit_ri631733984087533419it_int
% 0.22/0.54 = ( ^ [N4: nat,K3: int] : ( bit_concat_bit @ N4 @ K3 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N4 ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % signed_take_bit_eq_concat_bit
% 0.22/0.54 thf(fact_7721_exp__eq__0__imp__not__bit,axiom,
% 0.22/0.54 ! [N: nat,A: int] :
% 0.22/0.54 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.22/0.54 = zero_zero_int )
% 0.22/0.54 => ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % exp_eq_0_imp_not_bit
% 0.22/0.54 thf(fact_7722_exp__eq__0__imp__not__bit,axiom,
% 0.22/0.54 ! [N: nat,A: nat] :
% 0.22/0.54 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.54 = zero_zero_nat )
% 0.22/0.54 => ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % exp_eq_0_imp_not_bit
% 0.22/0.54 thf(fact_7723_bit__Suc,axiom,
% 0.22/0.54 ! [A: int,N: nat] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ A @ ( suc @ N ) )
% 0.22/0.54 = ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_Suc
% 0.22/0.54 thf(fact_7724_bit__Suc,axiom,
% 0.22/0.54 ! [A: nat,N: nat] :
% 0.22/0.54 ( ( bit_se1148574629649215175it_nat @ A @ ( suc @ N ) )
% 0.22/0.54 = ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_Suc
% 0.22/0.54 thf(fact_7725_bit__iff__idd__imp__stable,axiom,
% 0.22/0.54 ! [A: code_integer] :
% 0.22/0.54 ( ! [N2: nat] :
% 0.22/0.54 ( ( bit_se9216721137139052372nteger @ A @ N2 )
% 0.22/0.54 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 0.22/0.54 => ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.54 = A ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_iff_idd_imp_stable
% 0.22/0.54 thf(fact_7726_bit__iff__idd__imp__stable,axiom,
% 0.22/0.54 ! [A: int] :
% 0.22/0.54 ( ! [N2: nat] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ A @ N2 )
% 0.22/0.54 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 0.22/0.54 => ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.54 = A ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_iff_idd_imp_stable
% 0.22/0.54 thf(fact_7727_bit__iff__idd__imp__stable,axiom,
% 0.22/0.54 ! [A: nat] :
% 0.22/0.54 ( ! [N2: nat] :
% 0.22/0.54 ( ( bit_se1148574629649215175it_nat @ A @ N2 )
% 0.22/0.54 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 0.22/0.54 => ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.54 = A ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_iff_idd_imp_stable
% 0.22/0.54 thf(fact_7728_stable__imp__bit__iff__odd,axiom,
% 0.22/0.54 ! [A: code_integer,N: nat] :
% 0.22/0.54 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.54 = A )
% 0.22/0.54 => ( ( bit_se9216721137139052372nteger @ A @ N )
% 0.22/0.54 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % stable_imp_bit_iff_odd
% 0.22/0.54 thf(fact_7729_stable__imp__bit__iff__odd,axiom,
% 0.22/0.54 ! [A: int,N: nat] :
% 0.22/0.54 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.54 = A )
% 0.22/0.54 => ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.22/0.54 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % stable_imp_bit_iff_odd
% 0.22/0.54 thf(fact_7730_stable__imp__bit__iff__odd,axiom,
% 0.22/0.54 ! [A: nat,N: nat] :
% 0.22/0.54 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.54 = A )
% 0.22/0.54 => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.22/0.54 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % stable_imp_bit_iff_odd
% 0.22/0.54 thf(fact_7731_int__bit__bound,axiom,
% 0.22/0.54 ! [K: int] :
% 0.22/0.54 ~ ! [N2: nat] :
% 0.22/0.54 ( ! [M2: nat] :
% 0.22/0.54 ( ( ord_less_eq_nat @ N2 @ M2 )
% 0.22/0.54 => ( ( bit_se1146084159140164899it_int @ K @ M2 )
% 0.22/0.54 = ( bit_se1146084159140164899it_int @ K @ N2 ) ) )
% 0.22/0.54 => ~ ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.22/0.54 => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 0.22/0.54 = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % int_bit_bound
% 0.22/0.54 thf(fact_7732_bit__iff__odd,axiom,
% 0.22/0.54 ( bit_se9216721137139052372nteger
% 0.22/0.54 = ( ^ [A3: code_integer,N4: nat] :
% 0.22/0.54 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_iff_odd
% 0.22/0.54 thf(fact_7733_bit__iff__odd,axiom,
% 0.22/0.54 ( bit_se1146084159140164899it_int
% 0.22/0.54 = ( ^ [A3: int,N4: nat] :
% 0.22/0.54 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_iff_odd
% 0.22/0.54 thf(fact_7734_bit__iff__odd,axiom,
% 0.22/0.54 ( bit_se1148574629649215175it_nat
% 0.22/0.54 = ( ^ [A3: nat,N4: nat] :
% 0.22/0.54 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_iff_odd
% 0.22/0.54 thf(fact_7735_and__exp__eq__0__iff__not__bit,axiom,
% 0.22/0.54 ! [A: int,N: nat] :
% 0.22/0.54 ( ( ( bit_se725231765392027082nd_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.54 = zero_zero_int )
% 0.22/0.54 = ( ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % and_exp_eq_0_iff_not_bit
% 0.22/0.54 thf(fact_7736_and__exp__eq__0__iff__not__bit,axiom,
% 0.22/0.54 ! [A: nat,N: nat] :
% 0.22/0.54 ( ( ( bit_se727722235901077358nd_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.54 = zero_zero_nat )
% 0.22/0.54 = ( ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % and_exp_eq_0_iff_not_bit
% 0.22/0.54 thf(fact_7737_bit__int__def,axiom,
% 0.22/0.54 ( bit_se1146084159140164899it_int
% 0.22/0.54 = ( ^ [K3: int,N4: nat] :
% 0.22/0.54 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_int_def
% 0.22/0.54 thf(fact_7738_of__real__sqrt,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.54 => ( ( real_V4546457046886955230omplex @ ( sqrt @ X ) )
% 0.22/0.54 = ( csqrt @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % of_real_sqrt
% 0.22/0.54 thf(fact_7739_even__bit__succ__iff,axiom,
% 0.22/0.54 ! [A: code_integer,N: nat] :
% 0.22/0.54 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.54 => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ N )
% 0.22/0.54 = ( ( bit_se9216721137139052372nteger @ A @ N )
% 0.22/0.54 | ( N = zero_zero_nat ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % even_bit_succ_iff
% 0.22/0.54 thf(fact_7740_even__bit__succ__iff,axiom,
% 0.22/0.54 ! [A: int,N: nat] :
% 0.22/0.54 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.54 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N )
% 0.22/0.54 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.22/0.54 | ( N = zero_zero_nat ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % even_bit_succ_iff
% 0.22/0.54 thf(fact_7741_even__bit__succ__iff,axiom,
% 0.22/0.54 ! [A: nat,N: nat] :
% 0.22/0.54 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.54 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N )
% 0.22/0.54 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.22/0.54 | ( N = zero_zero_nat ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % even_bit_succ_iff
% 0.22/0.54 thf(fact_7742_odd__bit__iff__bit__pred,axiom,
% 0.22/0.54 ! [A: code_integer,N: nat] :
% 0.22/0.54 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.54 => ( ( bit_se9216721137139052372nteger @ A @ N )
% 0.22/0.54 = ( ( bit_se9216721137139052372nteger @ ( minus_8373710615458151222nteger @ A @ one_one_Code_integer ) @ N )
% 0.22/0.54 | ( N = zero_zero_nat ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % odd_bit_iff_bit_pred
% 0.22/0.54 thf(fact_7743_odd__bit__iff__bit__pred,axiom,
% 0.22/0.54 ! [A: int,N: nat] :
% 0.22/0.54 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.54 => ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.22/0.54 = ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A @ one_one_int ) @ N )
% 0.22/0.54 | ( N = zero_zero_nat ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % odd_bit_iff_bit_pred
% 0.22/0.54 thf(fact_7744_odd__bit__iff__bit__pred,axiom,
% 0.22/0.54 ! [A: nat,N: nat] :
% 0.22/0.54 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.54 => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.22/0.54 = ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A @ one_one_nat ) @ N )
% 0.22/0.54 | ( N = zero_zero_nat ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % odd_bit_iff_bit_pred
% 0.22/0.54 thf(fact_7745_Arg__bounded,axiom,
% 0.22/0.54 ! [Z: complex] :
% 0.22/0.54 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 0.22/0.54 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).
% 0.22/0.54
% 0.22/0.54 % Arg_bounded
% 0.22/0.54 thf(fact_7746_bit__sum__mult__2__cases,axiom,
% 0.22/0.54 ! [A: code_integer,B: code_integer,N: nat] :
% 0.22/0.54 ( ! [J2: nat] :
% 0.22/0.54 ~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J2 ) )
% 0.22/0.54 => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ N )
% 0.22/0.54 = ( ( ( N = zero_zero_nat )
% 0.22/0.54 => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.22/0.54 & ( ( N != zero_zero_nat )
% 0.22/0.54 => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_sum_mult_2_cases
% 0.22/0.54 thf(fact_7747_bit__sum__mult__2__cases,axiom,
% 0.22/0.54 ! [A: int,B: int,N: nat] :
% 0.22/0.54 ( ! [J2: nat] :
% 0.22/0.54 ~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J2 ) )
% 0.22/0.54 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ N )
% 0.22/0.54 = ( ( ( N = zero_zero_nat )
% 0.22/0.54 => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.22/0.54 & ( ( N != zero_zero_nat )
% 0.22/0.54 => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_sum_mult_2_cases
% 0.22/0.54 thf(fact_7748_bit__sum__mult__2__cases,axiom,
% 0.22/0.54 ! [A: nat,B: nat,N: nat] :
% 0.22/0.54 ( ! [J2: nat] :
% 0.22/0.54 ~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J2 ) )
% 0.22/0.54 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ N )
% 0.22/0.54 = ( ( ( N = zero_zero_nat )
% 0.22/0.54 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.22/0.54 & ( ( N != zero_zero_nat )
% 0.22/0.54 => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_sum_mult_2_cases
% 0.22/0.54 thf(fact_7749_bit__rec,axiom,
% 0.22/0.54 ( bit_se9216721137139052372nteger
% 0.22/0.54 = ( ^ [A3: code_integer,N4: nat] :
% 0.22/0.54 ( ( ( N4 = zero_zero_nat )
% 0.22/0.54 => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) )
% 0.22/0.54 & ( ( N4 != zero_zero_nat )
% 0.22/0.54 => ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_rec
% 0.22/0.54 thf(fact_7750_bit__rec,axiom,
% 0.22/0.54 ( bit_se1146084159140164899it_int
% 0.22/0.54 = ( ^ [A3: int,N4: nat] :
% 0.22/0.54 ( ( ( N4 = zero_zero_nat )
% 0.22/0.54 => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
% 0.22/0.54 & ( ( N4 != zero_zero_nat )
% 0.22/0.54 => ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_rec
% 0.22/0.54 thf(fact_7751_bit__rec,axiom,
% 0.22/0.54 ( bit_se1148574629649215175it_nat
% 0.22/0.54 = ( ^ [A3: nat,N4: nat] :
% 0.22/0.54 ( ( ( N4 = zero_zero_nat )
% 0.22/0.54 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) )
% 0.22/0.54 & ( ( N4 != zero_zero_nat )
% 0.22/0.54 => ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_rec
% 0.22/0.54 thf(fact_7752_set__bit__eq,axiom,
% 0.22/0.54 ( bit_se7879613467334960850it_int
% 0.22/0.54 = ( ^ [N4: nat,K3: int] :
% 0.22/0.54 ( plus_plus_int @ K3
% 0.22/0.54 @ ( times_times_int
% 0.22/0.54 @ ( zero_n2684676970156552555ol_int
% 0.22/0.54 @ ~ ( bit_se1146084159140164899it_int @ K3 @ N4 ) )
% 0.22/0.54 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % set_bit_eq
% 0.22/0.54 thf(fact_7753_unset__bit__eq,axiom,
% 0.22/0.54 ( bit_se4203085406695923979it_int
% 0.22/0.54 = ( ^ [N4: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N4 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % unset_bit_eq
% 0.22/0.54 thf(fact_7754_take__bit__Suc__from__most,axiom,
% 0.22/0.54 ! [N: nat,K: int] :
% 0.22/0.54 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
% 0.22/0.54 = ( 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 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % take_bit_Suc_from_most
% 0.22/0.54 thf(fact_7755_cis__multiple__2pi,axiom,
% 0.22/0.54 ! [N: real] :
% 0.22/0.54 ( ( member_real @ N @ ring_1_Ints_real )
% 0.22/0.54 => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.22/0.54 = one_one_complex ) ) ).
% 0.22/0.54
% 0.22/0.54 % cis_multiple_2pi
% 0.22/0.54 thf(fact_7756_powr__real__of__int,axiom,
% 0.22/0.54 ! [X: real,N: int] :
% 0.22/0.54 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.54 => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
% 0.22/0.54 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 0.22/0.54 = ( power_power_real @ X @ ( nat2 @ N ) ) ) )
% 0.22/0.54 & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
% 0.22/0.54 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 0.22/0.54 = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % powr_real_of_int
% 0.22/0.54 thf(fact_7757_Suc__0__xor__eq,axiom,
% 0.22/0.54 ! [N: nat] :
% 0.22/0.54 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.22/0.54 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.22/0.54 @ ( zero_n2687167440665602831ol_nat
% 0.22/0.54 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Suc_0_xor_eq
% 0.22/0.54 thf(fact_7758_xor__Suc__0__eq,axiom,
% 0.22/0.54 ! [N: nat] :
% 0.22/0.54 ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.22/0.54 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.22/0.54 @ ( zero_n2687167440665602831ol_nat
% 0.22/0.54 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_Suc_0_eq
% 0.22/0.54 thf(fact_7759_gbinomial__absorption_H,axiom,
% 0.22/0.54 ! [K: nat,A: complex] :
% 0.22/0.54 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.22/0.54 => ( ( gbinomial_complex @ A @ K )
% 0.22/0.54 = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_absorption'
% 0.22/0.54 thf(fact_7760_gbinomial__absorption_H,axiom,
% 0.22/0.54 ! [K: nat,A: real] :
% 0.22/0.54 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.22/0.54 => ( ( gbinomial_real @ A @ K )
% 0.22/0.54 = ( 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 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_absorption'
% 0.22/0.54 thf(fact_7761_gbinomial__absorption_H,axiom,
% 0.22/0.54 ! [K: nat,A: rat] :
% 0.22/0.54 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.22/0.54 => ( ( gbinomial_rat @ A @ K )
% 0.22/0.54 = ( 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 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_absorption'
% 0.22/0.54 thf(fact_7762_inverse__inverse__eq,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
% 0.22/0.54 = A ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_inverse_eq
% 0.22/0.54 thf(fact_7763_inverse__inverse__eq,axiom,
% 0.22/0.54 ! [A: complex] :
% 0.22/0.54 ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
% 0.22/0.54 = A ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_inverse_eq
% 0.22/0.54 thf(fact_7764_inverse__inverse__eq,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
% 0.22/0.54 = A ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_inverse_eq
% 0.22/0.54 thf(fact_7765_inverse__eq__iff__eq,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( ( inverse_inverse_real @ A )
% 0.22/0.54 = ( inverse_inverse_real @ B ) )
% 0.22/0.54 = ( A = B ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_eq_iff_eq
% 0.22/0.54 thf(fact_7766_inverse__eq__iff__eq,axiom,
% 0.22/0.54 ! [A: complex,B: complex] :
% 0.22/0.54 ( ( ( invers8013647133539491842omplex @ A )
% 0.22/0.54 = ( invers8013647133539491842omplex @ B ) )
% 0.22/0.54 = ( A = B ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_eq_iff_eq
% 0.22/0.54 thf(fact_7767_inverse__eq__iff__eq,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( ( inverse_inverse_rat @ A )
% 0.22/0.54 = ( inverse_inverse_rat @ B ) )
% 0.22/0.54 = ( A = B ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_eq_iff_eq
% 0.22/0.54 thf(fact_7768_bit_Oxor__left__self,axiom,
% 0.22/0.54 ! [X: int,Y4: int] :
% 0.22/0.54 ( ( bit_se6526347334894502574or_int @ X @ ( bit_se6526347334894502574or_int @ X @ Y4 ) )
% 0.22/0.54 = Y4 ) ).
% 0.22/0.54
% 0.22/0.54 % bit.xor_left_self
% 0.22/0.54 thf(fact_7769_inverse__nonzero__iff__nonzero,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( ( inverse_inverse_real @ A )
% 0.22/0.54 = zero_zero_real )
% 0.22/0.54 = ( A = zero_zero_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_nonzero_iff_nonzero
% 0.22/0.54 thf(fact_7770_inverse__nonzero__iff__nonzero,axiom,
% 0.22/0.54 ! [A: complex] :
% 0.22/0.54 ( ( ( invers8013647133539491842omplex @ A )
% 0.22/0.54 = zero_zero_complex )
% 0.22/0.54 = ( A = zero_zero_complex ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_nonzero_iff_nonzero
% 0.22/0.54 thf(fact_7771_inverse__nonzero__iff__nonzero,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( ( inverse_inverse_rat @ A )
% 0.22/0.54 = zero_zero_rat )
% 0.22/0.54 = ( A = zero_zero_rat ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_nonzero_iff_nonzero
% 0.22/0.54 thf(fact_7772_inverse__zero,axiom,
% 0.22/0.54 ( ( inverse_inverse_real @ zero_zero_real )
% 0.22/0.54 = zero_zero_real ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_zero
% 0.22/0.54 thf(fact_7773_inverse__zero,axiom,
% 0.22/0.54 ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 0.22/0.54 = zero_zero_complex ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_zero
% 0.22/0.54 thf(fact_7774_inverse__zero,axiom,
% 0.22/0.54 ( ( inverse_inverse_rat @ zero_zero_rat )
% 0.22/0.54 = zero_zero_rat ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_zero
% 0.22/0.54 thf(fact_7775_inverse__mult__distrib,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
% 0.22/0.54 = ( times_times_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_mult_distrib
% 0.22/0.54 thf(fact_7776_inverse__mult__distrib,axiom,
% 0.22/0.54 ! [A: complex,B: complex] :
% 0.22/0.54 ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
% 0.22/0.54 = ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_mult_distrib
% 0.22/0.54 thf(fact_7777_inverse__mult__distrib,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
% 0.22/0.54 = ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_mult_distrib
% 0.22/0.54 thf(fact_7778_inverse__1,axiom,
% 0.22/0.54 ( ( inverse_inverse_real @ one_one_real )
% 0.22/0.54 = one_one_real ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_1
% 0.22/0.54 thf(fact_7779_inverse__1,axiom,
% 0.22/0.54 ( ( invers8013647133539491842omplex @ one_one_complex )
% 0.22/0.54 = one_one_complex ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_1
% 0.22/0.54 thf(fact_7780_inverse__1,axiom,
% 0.22/0.54 ( ( inverse_inverse_rat @ one_one_rat )
% 0.22/0.54 = one_one_rat ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_1
% 0.22/0.54 thf(fact_7781_inverse__eq__1__iff,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( ( inverse_inverse_real @ X )
% 0.22/0.54 = one_one_real )
% 0.22/0.54 = ( X = one_one_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_eq_1_iff
% 0.22/0.54 thf(fact_7782_inverse__eq__1__iff,axiom,
% 0.22/0.54 ! [X: complex] :
% 0.22/0.54 ( ( ( invers8013647133539491842omplex @ X )
% 0.22/0.54 = one_one_complex )
% 0.22/0.54 = ( X = one_one_complex ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_eq_1_iff
% 0.22/0.54 thf(fact_7783_inverse__eq__1__iff,axiom,
% 0.22/0.54 ! [X: rat] :
% 0.22/0.54 ( ( ( inverse_inverse_rat @ X )
% 0.22/0.54 = one_one_rat )
% 0.22/0.54 = ( X = one_one_rat ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_eq_1_iff
% 0.22/0.54 thf(fact_7784_inverse__divide,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( inverse_inverse_real @ ( divide_divide_real @ A @ B ) )
% 0.22/0.54 = ( divide_divide_real @ B @ A ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_divide
% 0.22/0.54 thf(fact_7785_inverse__divide,axiom,
% 0.22/0.54 ! [A: complex,B: complex] :
% 0.22/0.54 ( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.22/0.54 = ( divide1717551699836669952omplex @ B @ A ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_divide
% 0.22/0.54 thf(fact_7786_inverse__divide,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( inverse_inverse_rat @ ( divide_divide_rat @ A @ B ) )
% 0.22/0.54 = ( divide_divide_rat @ B @ A ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_divide
% 0.22/0.54 thf(fact_7787_inverse__minus__eq,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
% 0.22/0.54 = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_minus_eq
% 0.22/0.54 thf(fact_7788_inverse__minus__eq,axiom,
% 0.22/0.54 ! [A: complex] :
% 0.22/0.54 ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
% 0.22/0.54 = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_minus_eq
% 0.22/0.54 thf(fact_7789_inverse__minus__eq,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
% 0.22/0.54 = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_minus_eq
% 0.22/0.54 thf(fact_7790_xor_Oright__neutral,axiom,
% 0.22/0.54 ! [A: nat] :
% 0.22/0.54 ( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
% 0.22/0.54 = A ) ).
% 0.22/0.54
% 0.22/0.54 % xor.right_neutral
% 0.22/0.54 thf(fact_7791_xor_Oright__neutral,axiom,
% 0.22/0.54 ! [A: int] :
% 0.22/0.54 ( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
% 0.22/0.54 = A ) ).
% 0.22/0.54
% 0.22/0.54 % xor.right_neutral
% 0.22/0.54 thf(fact_7792_xor_Oleft__neutral,axiom,
% 0.22/0.54 ! [A: nat] :
% 0.22/0.54 ( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
% 0.22/0.54 = A ) ).
% 0.22/0.54
% 0.22/0.54 % xor.left_neutral
% 0.22/0.54 thf(fact_7793_xor_Oleft__neutral,axiom,
% 0.22/0.54 ! [A: int] :
% 0.22/0.54 ( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
% 0.22/0.54 = A ) ).
% 0.22/0.54
% 0.22/0.54 % xor.left_neutral
% 0.22/0.54 thf(fact_7794_xor__self__eq,axiom,
% 0.22/0.54 ! [A: nat] :
% 0.22/0.54 ( ( bit_se6528837805403552850or_nat @ A @ A )
% 0.22/0.54 = zero_zero_nat ) ).
% 0.22/0.54
% 0.22/0.54 % xor_self_eq
% 0.22/0.54 thf(fact_7795_xor__self__eq,axiom,
% 0.22/0.54 ! [A: int] :
% 0.22/0.54 ( ( bit_se6526347334894502574or_int @ A @ A )
% 0.22/0.54 = zero_zero_int ) ).
% 0.22/0.54
% 0.22/0.54 % xor_self_eq
% 0.22/0.54 thf(fact_7796_bit_Oxor__self,axiom,
% 0.22/0.54 ! [X: int] :
% 0.22/0.54 ( ( bit_se6526347334894502574or_int @ X @ X )
% 0.22/0.54 = zero_zero_int ) ).
% 0.22/0.54
% 0.22/0.54 % bit.xor_self
% 0.22/0.54 thf(fact_7797_abs__inverse,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
% 0.22/0.54 = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % abs_inverse
% 0.22/0.54 thf(fact_7798_abs__inverse,axiom,
% 0.22/0.54 ! [A: complex] :
% 0.22/0.54 ( ( abs_abs_complex @ ( invers8013647133539491842omplex @ A ) )
% 0.22/0.54 = ( invers8013647133539491842omplex @ ( abs_abs_complex @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % abs_inverse
% 0.22/0.54 thf(fact_7799_abs__inverse,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
% 0.22/0.54 = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % abs_inverse
% 0.22/0.54 thf(fact_7800_inverse__sgn,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( inverse_inverse_real @ ( sgn_sgn_real @ A ) )
% 0.22/0.54 = ( sgn_sgn_real @ A ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_sgn
% 0.22/0.54 thf(fact_7801_inverse__sgn,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) )
% 0.22/0.54 = ( sgn_sgn_rat @ A ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_sgn
% 0.22/0.54 thf(fact_7802_sgn__inverse,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( sgn_sgn_real @ ( inverse_inverse_real @ A ) )
% 0.22/0.54 = ( inverse_inverse_real @ ( sgn_sgn_real @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % sgn_inverse
% 0.22/0.54 thf(fact_7803_sgn__inverse,axiom,
% 0.22/0.54 ! [A: complex] :
% 0.22/0.54 ( ( sgn_sgn_complex @ ( invers8013647133539491842omplex @ A ) )
% 0.22/0.54 = ( invers8013647133539491842omplex @ ( sgn_sgn_complex @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % sgn_inverse
% 0.22/0.54 thf(fact_7804_sgn__inverse,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( sgn_sgn_rat @ ( inverse_inverse_rat @ A ) )
% 0.22/0.54 = ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % sgn_inverse
% 0.22/0.54 thf(fact_7805_take__bit__xor,axiom,
% 0.22/0.54 ! [N: nat,A: int,B: int] :
% 0.22/0.54 ( ( bit_se2923211474154528505it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B ) )
% 0.22/0.54 = ( bit_se6526347334894502574or_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % take_bit_xor
% 0.22/0.54 thf(fact_7806_take__bit__xor,axiom,
% 0.22/0.54 ! [N: nat,A: nat,B: nat] :
% 0.22/0.54 ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B ) )
% 0.22/0.54 = ( bit_se6528837805403552850or_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % take_bit_xor
% 0.22/0.54 thf(fact_7807_inverse__nonnegative__iff__nonnegative,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.22/0.54 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_nonnegative_iff_nonnegative
% 0.22/0.54 thf(fact_7808_inverse__nonnegative__iff__nonnegative,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 0.22/0.54 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_nonnegative_iff_nonnegative
% 0.22/0.54 thf(fact_7809_inverse__nonpositive__iff__nonpositive,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.22/0.54 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_nonpositive_iff_nonpositive
% 0.22/0.54 thf(fact_7810_inverse__nonpositive__iff__nonpositive,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 0.22/0.54 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_nonpositive_iff_nonpositive
% 0.22/0.54 thf(fact_7811_inverse__positive__iff__positive,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.22/0.54 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_positive_iff_positive
% 0.22/0.54 thf(fact_7812_inverse__positive__iff__positive,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 0.22/0.54 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_positive_iff_positive
% 0.22/0.54 thf(fact_7813_inverse__negative__iff__negative,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.22/0.54 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_negative_iff_negative
% 0.22/0.54 thf(fact_7814_inverse__negative__iff__negative,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 0.22/0.54 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_negative_iff_negative
% 0.22/0.54 thf(fact_7815_inverse__less__iff__less__neg,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( ord_less_real @ A @ zero_zero_real )
% 0.22/0.54 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.22/0.54 => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.22/0.54 = ( ord_less_real @ B @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_less_iff_less_neg
% 0.22/0.54 thf(fact_7816_inverse__less__iff__less__neg,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.22/0.54 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.22/0.54 => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.22/0.54 = ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_less_iff_less_neg
% 0.22/0.54 thf(fact_7817_inverse__less__iff__less,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.54 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.22/0.54 => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.22/0.54 = ( ord_less_real @ B @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_less_iff_less
% 0.22/0.54 thf(fact_7818_inverse__less__iff__less,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.22/0.54 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.22/0.54 => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.22/0.54 = ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_less_iff_less
% 0.22/0.54 thf(fact_7819_gbinomial__0_I2_J,axiom,
% 0.22/0.54 ! [K: nat] :
% 0.22/0.54 ( ( gbinomial_complex @ zero_zero_complex @ ( suc @ K ) )
% 0.22/0.54 = zero_zero_complex ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_0(2)
% 0.22/0.54 thf(fact_7820_gbinomial__0_I2_J,axiom,
% 0.22/0.54 ! [K: nat] :
% 0.22/0.54 ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
% 0.22/0.54 = zero_zero_real ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_0(2)
% 0.22/0.54 thf(fact_7821_gbinomial__0_I2_J,axiom,
% 0.22/0.54 ! [K: nat] :
% 0.22/0.54 ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
% 0.22/0.54 = zero_zero_rat ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_0(2)
% 0.22/0.54 thf(fact_7822_gbinomial__0_I2_J,axiom,
% 0.22/0.54 ! [K: nat] :
% 0.22/0.54 ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
% 0.22/0.54 = zero_zero_nat ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_0(2)
% 0.22/0.54 thf(fact_7823_gbinomial__0_I2_J,axiom,
% 0.22/0.54 ! [K: nat] :
% 0.22/0.54 ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
% 0.22/0.54 = zero_zero_int ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_0(2)
% 0.22/0.54 thf(fact_7824_gbinomial__0_I1_J,axiom,
% 0.22/0.54 ! [A: complex] :
% 0.22/0.54 ( ( gbinomial_complex @ A @ zero_zero_nat )
% 0.22/0.54 = one_one_complex ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_0(1)
% 0.22/0.54 thf(fact_7825_gbinomial__0_I1_J,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( gbinomial_real @ A @ zero_zero_nat )
% 0.22/0.54 = one_one_real ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_0(1)
% 0.22/0.54 thf(fact_7826_gbinomial__0_I1_J,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( gbinomial_rat @ A @ zero_zero_nat )
% 0.22/0.54 = one_one_rat ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_0(1)
% 0.22/0.54 thf(fact_7827_gbinomial__0_I1_J,axiom,
% 0.22/0.54 ! [A: nat] :
% 0.22/0.54 ( ( gbinomial_nat @ A @ zero_zero_nat )
% 0.22/0.54 = one_one_nat ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_0(1)
% 0.22/0.54 thf(fact_7828_gbinomial__0_I1_J,axiom,
% 0.22/0.54 ! [A: int] :
% 0.22/0.54 ( ( gbinomial_int @ A @ zero_zero_nat )
% 0.22/0.54 = one_one_int ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_0(1)
% 0.22/0.54 thf(fact_7829_frac__eq__0__iff,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( ( archim2898591450579166408c_real @ X )
% 0.22/0.54 = zero_zero_real )
% 0.22/0.54 = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % frac_eq_0_iff
% 0.22/0.54 thf(fact_7830_frac__eq__0__iff,axiom,
% 0.22/0.54 ! [X: rat] :
% 0.22/0.54 ( ( ( archimedean_frac_rat @ X )
% 0.22/0.54 = zero_zero_rat )
% 0.22/0.54 = ( member_rat @ X @ ring_1_Ints_rat ) ) ).
% 0.22/0.54
% 0.22/0.54 % frac_eq_0_iff
% 0.22/0.54 thf(fact_7831_inverse__le__iff__le__neg,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( ord_less_real @ A @ zero_zero_real )
% 0.22/0.54 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.22/0.54 => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.22/0.54 = ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_le_iff_le_neg
% 0.22/0.54 thf(fact_7832_inverse__le__iff__le__neg,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.22/0.54 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.22/0.54 => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.22/0.54 = ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_le_iff_le_neg
% 0.22/0.54 thf(fact_7833_inverse__le__iff__le,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.54 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.22/0.54 => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.22/0.54 = ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_le_iff_le
% 0.22/0.54 thf(fact_7834_inverse__le__iff__le,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.22/0.54 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.22/0.54 => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.22/0.54 = ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_le_iff_le
% 0.22/0.54 thf(fact_7835_right__inverse,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( A != zero_zero_real )
% 0.22/0.54 => ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
% 0.22/0.54 = one_one_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % right_inverse
% 0.22/0.54 thf(fact_7836_right__inverse,axiom,
% 0.22/0.54 ! [A: complex] :
% 0.22/0.54 ( ( A != zero_zero_complex )
% 0.22/0.54 => ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
% 0.22/0.54 = one_one_complex ) ) ).
% 0.22/0.54
% 0.22/0.54 % right_inverse
% 0.22/0.54 thf(fact_7837_right__inverse,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( A != zero_zero_rat )
% 0.22/0.54 => ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
% 0.22/0.54 = one_one_rat ) ) ).
% 0.22/0.54
% 0.22/0.54 % right_inverse
% 0.22/0.54 thf(fact_7838_left__inverse,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( A != zero_zero_real )
% 0.22/0.54 => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 0.22/0.54 = one_one_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % left_inverse
% 0.22/0.54 thf(fact_7839_left__inverse,axiom,
% 0.22/0.54 ! [A: complex] :
% 0.22/0.54 ( ( A != zero_zero_complex )
% 0.22/0.54 => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 0.22/0.54 = one_one_complex ) ) ).
% 0.22/0.54
% 0.22/0.54 % left_inverse
% 0.22/0.54 thf(fact_7840_left__inverse,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( A != zero_zero_rat )
% 0.22/0.54 => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 0.22/0.54 = one_one_rat ) ) ).
% 0.22/0.54
% 0.22/0.54 % left_inverse
% 0.22/0.54 thf(fact_7841_inverse__eq__divide__numeral,axiom,
% 0.22/0.54 ! [W: num] :
% 0.22/0.54 ( ( inverse_inverse_real @ ( numeral_numeral_real @ W ) )
% 0.22/0.54 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ W ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_eq_divide_numeral
% 0.22/0.54 thf(fact_7842_inverse__eq__divide__numeral,axiom,
% 0.22/0.54 ! [W: num] :
% 0.22/0.54 ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.22/0.54 = ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_eq_divide_numeral
% 0.22/0.54 thf(fact_7843_inverse__eq__divide__numeral,axiom,
% 0.22/0.54 ! [W: num] :
% 0.22/0.54 ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ W ) )
% 0.22/0.54 = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_eq_divide_numeral
% 0.22/0.54 thf(fact_7844_frac__gt__0__iff,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( ord_less_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) )
% 0.22/0.54 = ( ~ ( member_real @ X @ ring_1_Ints_real ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % frac_gt_0_iff
% 0.22/0.54 thf(fact_7845_frac__gt__0__iff,axiom,
% 0.22/0.54 ! [X: rat] :
% 0.22/0.54 ( ( ord_less_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) )
% 0.22/0.54 = ( ~ ( member_rat @ X @ ring_1_Ints_rat ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % frac_gt_0_iff
% 0.22/0.54 thf(fact_7846_inverse__eq__divide__neg__numeral,axiom,
% 0.22/0.54 ! [W: num] :
% 0.22/0.54 ( ( inverse_inverse_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.22/0.54 = ( divide_divide_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_eq_divide_neg_numeral
% 0.22/0.54 thf(fact_7847_inverse__eq__divide__neg__numeral,axiom,
% 0.22/0.54 ! [W: num] :
% 0.22/0.54 ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.22/0.54 = ( divide1717551699836669952omplex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_eq_divide_neg_numeral
% 0.22/0.54 thf(fact_7848_inverse__eq__divide__neg__numeral,axiom,
% 0.22/0.54 ! [W: num] :
% 0.22/0.54 ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.22/0.54 = ( divide_divide_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_eq_divide_neg_numeral
% 0.22/0.54 thf(fact_7849_xor__numerals_I3_J,axiom,
% 0.22/0.54 ! [X: num,Y4: num] :
% 0.22/0.54 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 0.22/0.54 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_numerals(3)
% 0.22/0.54 thf(fact_7850_xor__numerals_I3_J,axiom,
% 0.22/0.54 ! [X: num,Y4: num] :
% 0.22/0.54 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y4 ) ) )
% 0.22/0.54 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_numerals(3)
% 0.22/0.54 thf(fact_7851_xor__numerals_I1_J,axiom,
% 0.22/0.54 ! [Y4: num] :
% 0.22/0.54 ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 0.22/0.54 = ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_numerals(1)
% 0.22/0.54 thf(fact_7852_xor__numerals_I1_J,axiom,
% 0.22/0.54 ! [Y4: num] :
% 0.22/0.54 ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y4 ) ) )
% 0.22/0.54 = ( numeral_numeral_int @ ( bit1 @ Y4 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_numerals(1)
% 0.22/0.54 thf(fact_7853_xor__numerals_I2_J,axiom,
% 0.22/0.54 ! [Y4: num] :
% 0.22/0.54 ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 0.22/0.54 = ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_numerals(2)
% 0.22/0.54 thf(fact_7854_xor__numerals_I2_J,axiom,
% 0.22/0.54 ! [Y4: num] :
% 0.22/0.54 ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y4 ) ) )
% 0.22/0.54 = ( numeral_numeral_int @ ( bit0 @ Y4 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_numerals(2)
% 0.22/0.54 thf(fact_7855_xor__numerals_I5_J,axiom,
% 0.22/0.54 ! [X: num] :
% 0.22/0.54 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
% 0.22/0.54 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_numerals(5)
% 0.22/0.54 thf(fact_7856_xor__numerals_I5_J,axiom,
% 0.22/0.54 ! [X: num] :
% 0.22/0.54 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
% 0.22/0.54 = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_numerals(5)
% 0.22/0.54 thf(fact_7857_xor__numerals_I8_J,axiom,
% 0.22/0.54 ! [X: num] :
% 0.22/0.54 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
% 0.22/0.54 = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_numerals(8)
% 0.22/0.54 thf(fact_7858_xor__numerals_I8_J,axiom,
% 0.22/0.54 ! [X: num] :
% 0.22/0.54 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
% 0.22/0.54 = ( numeral_numeral_int @ ( bit0 @ X ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_numerals(8)
% 0.22/0.54 thf(fact_7859_xor__numerals_I7_J,axiom,
% 0.22/0.54 ! [X: num,Y4: num] :
% 0.22/0.54 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 0.22/0.54 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_numerals(7)
% 0.22/0.54 thf(fact_7860_xor__numerals_I7_J,axiom,
% 0.22/0.54 ! [X: num,Y4: num] :
% 0.22/0.54 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y4 ) ) )
% 0.22/0.54 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_numerals(7)
% 0.22/0.54 thf(fact_7861_xor__nat__numerals_I1_J,axiom,
% 0.22/0.54 ! [Y4: num] :
% 0.22/0.54 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 0.22/0.54 = ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_nat_numerals(1)
% 0.22/0.54 thf(fact_7862_xor__nat__numerals_I2_J,axiom,
% 0.22/0.54 ! [Y4: num] :
% 0.22/0.54 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 0.22/0.54 = ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_nat_numerals(2)
% 0.22/0.54 thf(fact_7863_xor__nat__numerals_I3_J,axiom,
% 0.22/0.54 ! [X: num] :
% 0.22/0.54 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.22/0.54 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_nat_numerals(3)
% 0.22/0.54 thf(fact_7864_xor__nat__numerals_I4_J,axiom,
% 0.22/0.54 ! [X: num] :
% 0.22/0.54 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.22/0.54 = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_nat_numerals(4)
% 0.22/0.54 thf(fact_7865_xor__numerals_I4_J,axiom,
% 0.22/0.54 ! [X: num,Y4: num] :
% 0.22/0.54 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 0.22/0.54 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_numerals(4)
% 0.22/0.54 thf(fact_7866_xor__numerals_I4_J,axiom,
% 0.22/0.54 ! [X: num,Y4: num] :
% 0.22/0.54 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y4 ) ) )
% 0.22/0.54 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_numerals(4)
% 0.22/0.54 thf(fact_7867_xor__numerals_I6_J,axiom,
% 0.22/0.54 ! [X: num,Y4: num] :
% 0.22/0.54 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 0.22/0.54 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_numerals(6)
% 0.22/0.54 thf(fact_7868_xor__numerals_I6_J,axiom,
% 0.22/0.54 ! [X: num,Y4: num] :
% 0.22/0.54 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y4 ) ) )
% 0.22/0.54 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor_numerals(6)
% 0.22/0.54 thf(fact_7869_Ints__power,axiom,
% 0.22/0.54 ! [A: int,N: nat] :
% 0.22/0.54 ( ( member_int @ A @ ring_1_Ints_int )
% 0.22/0.54 => ( member_int @ ( power_power_int @ A @ N ) @ ring_1_Ints_int ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_power
% 0.22/0.54 thf(fact_7870_Ints__power,axiom,
% 0.22/0.54 ! [A: real,N: nat] :
% 0.22/0.54 ( ( member_real @ A @ ring_1_Ints_real )
% 0.22/0.54 => ( member_real @ ( power_power_real @ A @ N ) @ ring_1_Ints_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_power
% 0.22/0.54 thf(fact_7871_Ints__power,axiom,
% 0.22/0.54 ! [A: complex,N: nat] :
% 0.22/0.54 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.22/0.54 => ( member_complex @ ( power_power_complex @ A @ N ) @ ring_1_Ints_complex ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_power
% 0.22/0.54 thf(fact_7872_nonzero__of__real__inverse,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( X != zero_zero_real )
% 0.22/0.54 => ( ( real_V1803761363581548252l_real @ ( inverse_inverse_real @ X ) )
% 0.22/0.54 = ( inverse_inverse_real @ ( real_V1803761363581548252l_real @ X ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_of_real_inverse
% 0.22/0.54 thf(fact_7873_nonzero__of__real__inverse,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( X != zero_zero_real )
% 0.22/0.54 => ( ( real_V4546457046886955230omplex @ ( inverse_inverse_real @ X ) )
% 0.22/0.54 = ( invers8013647133539491842omplex @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_of_real_inverse
% 0.22/0.54 thf(fact_7874_of__int__xor__eq,axiom,
% 0.22/0.54 ! [K: int,L: int] :
% 0.22/0.54 ( ( ring_1_of_int_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 0.22/0.54 = ( bit_se6526347334894502574or_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % of_int_xor_eq
% 0.22/0.54 thf(fact_7875_Ints__0,axiom,
% 0.22/0.54 member_complex @ zero_zero_complex @ ring_1_Ints_complex ).
% 0.22/0.54
% 0.22/0.54 % Ints_0
% 0.22/0.54 thf(fact_7876_Ints__0,axiom,
% 0.22/0.54 member_real @ zero_zero_real @ ring_1_Ints_real ).
% 0.22/0.54
% 0.22/0.54 % Ints_0
% 0.22/0.54 thf(fact_7877_Ints__0,axiom,
% 0.22/0.54 member_rat @ zero_zero_rat @ ring_1_Ints_rat ).
% 0.22/0.54
% 0.22/0.54 % Ints_0
% 0.22/0.54 thf(fact_7878_Ints__0,axiom,
% 0.22/0.54 member_int @ zero_zero_int @ ring_1_Ints_int ).
% 0.22/0.54
% 0.22/0.54 % Ints_0
% 0.22/0.54 thf(fact_7879_Ints__numeral,axiom,
% 0.22/0.54 ! [N: num] : ( member_complex @ ( numera6690914467698888265omplex @ N ) @ ring_1_Ints_complex ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_numeral
% 0.22/0.54 thf(fact_7880_Ints__numeral,axiom,
% 0.22/0.54 ! [N: num] : ( member_real @ ( numeral_numeral_real @ N ) @ ring_1_Ints_real ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_numeral
% 0.22/0.54 thf(fact_7881_Ints__numeral,axiom,
% 0.22/0.54 ! [N: num] : ( member_rat @ ( numeral_numeral_rat @ N ) @ ring_1_Ints_rat ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_numeral
% 0.22/0.54 thf(fact_7882_Ints__numeral,axiom,
% 0.22/0.54 ! [N: num] : ( member_int @ ( numeral_numeral_int @ N ) @ ring_1_Ints_int ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_numeral
% 0.22/0.54 thf(fact_7883_Ints__mult,axiom,
% 0.22/0.54 ! [A: complex,B: complex] :
% 0.22/0.54 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.22/0.54 => ( ( member_complex @ B @ ring_1_Ints_complex )
% 0.22/0.54 => ( member_complex @ ( times_times_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_mult
% 0.22/0.54 thf(fact_7884_Ints__mult,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( member_real @ A @ ring_1_Ints_real )
% 0.22/0.54 => ( ( member_real @ B @ ring_1_Ints_real )
% 0.22/0.54 => ( member_real @ ( times_times_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_mult
% 0.22/0.54 thf(fact_7885_Ints__mult,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.22/0.54 => ( ( member_rat @ B @ ring_1_Ints_rat )
% 0.22/0.54 => ( member_rat @ ( times_times_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_mult
% 0.22/0.54 thf(fact_7886_Ints__mult,axiom,
% 0.22/0.54 ! [A: int,B: int] :
% 0.22/0.54 ( ( member_int @ A @ ring_1_Ints_int )
% 0.22/0.54 => ( ( member_int @ B @ ring_1_Ints_int )
% 0.22/0.54 => ( member_int @ ( times_times_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_mult
% 0.22/0.54 thf(fact_7887_Ints__add,axiom,
% 0.22/0.54 ! [A: complex,B: complex] :
% 0.22/0.54 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.22/0.54 => ( ( member_complex @ B @ ring_1_Ints_complex )
% 0.22/0.54 => ( member_complex @ ( plus_plus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_add
% 0.22/0.54 thf(fact_7888_Ints__add,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( member_real @ A @ ring_1_Ints_real )
% 0.22/0.54 => ( ( member_real @ B @ ring_1_Ints_real )
% 0.22/0.54 => ( member_real @ ( plus_plus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_add
% 0.22/0.54 thf(fact_7889_Ints__add,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.22/0.54 => ( ( member_rat @ B @ ring_1_Ints_rat )
% 0.22/0.54 => ( member_rat @ ( plus_plus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_add
% 0.22/0.54 thf(fact_7890_Ints__add,axiom,
% 0.22/0.54 ! [A: int,B: int] :
% 0.22/0.54 ( ( member_int @ A @ ring_1_Ints_int )
% 0.22/0.54 => ( ( member_int @ B @ ring_1_Ints_int )
% 0.22/0.54 => ( member_int @ ( plus_plus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_add
% 0.22/0.54 thf(fact_7891_Ints__1,axiom,
% 0.22/0.54 member_rat @ one_one_rat @ ring_1_Ints_rat ).
% 0.22/0.54
% 0.22/0.54 % Ints_1
% 0.22/0.54 thf(fact_7892_Ints__1,axiom,
% 0.22/0.54 member_int @ one_one_int @ ring_1_Ints_int ).
% 0.22/0.54
% 0.22/0.54 % Ints_1
% 0.22/0.54 thf(fact_7893_Ints__1,axiom,
% 0.22/0.54 member_real @ one_one_real @ ring_1_Ints_real ).
% 0.22/0.54
% 0.22/0.54 % Ints_1
% 0.22/0.54 thf(fact_7894_Ints__1,axiom,
% 0.22/0.54 member_complex @ one_one_complex @ ring_1_Ints_complex ).
% 0.22/0.54
% 0.22/0.54 % Ints_1
% 0.22/0.54 thf(fact_7895_real__sqrt__inverse,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( sqrt @ ( inverse_inverse_real @ X ) )
% 0.22/0.54 = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % real_sqrt_inverse
% 0.22/0.54 thf(fact_7896_Ints__diff,axiom,
% 0.22/0.54 ! [A: complex,B: complex] :
% 0.22/0.54 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.22/0.54 => ( ( member_complex @ B @ ring_1_Ints_complex )
% 0.22/0.54 => ( member_complex @ ( minus_minus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_diff
% 0.22/0.54 thf(fact_7897_Ints__diff,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( member_real @ A @ ring_1_Ints_real )
% 0.22/0.54 => ( ( member_real @ B @ ring_1_Ints_real )
% 0.22/0.54 => ( member_real @ ( minus_minus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_diff
% 0.22/0.54 thf(fact_7898_Ints__diff,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.22/0.54 => ( ( member_rat @ B @ ring_1_Ints_rat )
% 0.22/0.54 => ( member_rat @ ( minus_minus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_diff
% 0.22/0.54 thf(fact_7899_Ints__diff,axiom,
% 0.22/0.54 ! [A: int,B: int] :
% 0.22/0.54 ( ( member_int @ A @ ring_1_Ints_int )
% 0.22/0.54 => ( ( member_int @ B @ ring_1_Ints_int )
% 0.22/0.54 => ( member_int @ ( minus_minus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_diff
% 0.22/0.54 thf(fact_7900_Ints__minus,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( member_real @ A @ ring_1_Ints_real )
% 0.22/0.54 => ( member_real @ ( uminus_uminus_real @ A ) @ ring_1_Ints_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_minus
% 0.22/0.54 thf(fact_7901_Ints__minus,axiom,
% 0.22/0.54 ! [A: int] :
% 0.22/0.54 ( ( member_int @ A @ ring_1_Ints_int )
% 0.22/0.54 => ( member_int @ ( uminus_uminus_int @ A ) @ ring_1_Ints_int ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_minus
% 0.22/0.54 thf(fact_7902_Ints__minus,axiom,
% 0.22/0.54 ! [A: complex] :
% 0.22/0.54 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.22/0.54 => ( member_complex @ ( uminus1482373934393186551omplex @ A ) @ ring_1_Ints_complex ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_minus
% 0.22/0.54 thf(fact_7903_Ints__minus,axiom,
% 0.22/0.54 ! [A: code_integer] :
% 0.22/0.54 ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
% 0.22/0.54 => ( member_Code_integer @ ( uminus1351360451143612070nteger @ A ) @ ring_11222124179247155820nteger ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_minus
% 0.22/0.54 thf(fact_7904_Ints__minus,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.22/0.54 => ( member_rat @ ( uminus_uminus_rat @ A ) @ ring_1_Ints_rat ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_minus
% 0.22/0.54 thf(fact_7905_minus__in__Ints__iff,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( member_real @ ( uminus_uminus_real @ X ) @ ring_1_Ints_real )
% 0.22/0.54 = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % minus_in_Ints_iff
% 0.22/0.54 thf(fact_7906_minus__in__Ints__iff,axiom,
% 0.22/0.54 ! [X: int] :
% 0.22/0.54 ( ( member_int @ ( uminus_uminus_int @ X ) @ ring_1_Ints_int )
% 0.22/0.54 = ( member_int @ X @ ring_1_Ints_int ) ) ).
% 0.22/0.54
% 0.22/0.54 % minus_in_Ints_iff
% 0.22/0.54 thf(fact_7907_minus__in__Ints__iff,axiom,
% 0.22/0.54 ! [X: complex] :
% 0.22/0.54 ( ( member_complex @ ( uminus1482373934393186551omplex @ X ) @ ring_1_Ints_complex )
% 0.22/0.54 = ( member_complex @ X @ ring_1_Ints_complex ) ) ).
% 0.22/0.54
% 0.22/0.54 % minus_in_Ints_iff
% 0.22/0.54 thf(fact_7908_minus__in__Ints__iff,axiom,
% 0.22/0.54 ! [X: code_integer] :
% 0.22/0.54 ( ( member_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ ring_11222124179247155820nteger )
% 0.22/0.54 = ( member_Code_integer @ X @ ring_11222124179247155820nteger ) ) ).
% 0.22/0.54
% 0.22/0.54 % minus_in_Ints_iff
% 0.22/0.54 thf(fact_7909_minus__in__Ints__iff,axiom,
% 0.22/0.54 ! [X: rat] :
% 0.22/0.54 ( ( member_rat @ ( uminus_uminus_rat @ X ) @ ring_1_Ints_rat )
% 0.22/0.54 = ( member_rat @ X @ ring_1_Ints_rat ) ) ).
% 0.22/0.54
% 0.22/0.54 % minus_in_Ints_iff
% 0.22/0.54 thf(fact_7910_of__nat__gbinomial,axiom,
% 0.22/0.54 ! [N: nat,K: nat] :
% 0.22/0.54 ( ( semiri8010041392384452111omplex @ ( gbinomial_nat @ N @ K ) )
% 0.22/0.54 = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K ) ) ).
% 0.22/0.54
% 0.22/0.54 % of_nat_gbinomial
% 0.22/0.54 thf(fact_7911_of__nat__gbinomial,axiom,
% 0.22/0.54 ! [N: nat,K: nat] :
% 0.22/0.54 ( ( semiri5074537144036343181t_real @ ( gbinomial_nat @ N @ K ) )
% 0.22/0.54 = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K ) ) ).
% 0.22/0.54
% 0.22/0.54 % of_nat_gbinomial
% 0.22/0.54 thf(fact_7912_of__nat__gbinomial,axiom,
% 0.22/0.54 ! [N: nat,K: nat] :
% 0.22/0.54 ( ( semiri681578069525770553at_rat @ ( gbinomial_nat @ N @ K ) )
% 0.22/0.54 = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K ) ) ).
% 0.22/0.54
% 0.22/0.54 % of_nat_gbinomial
% 0.22/0.54 thf(fact_7913_xor_Oleft__commute,axiom,
% 0.22/0.54 ! [B: nat,A: nat,C: nat] :
% 0.22/0.54 ( ( bit_se6528837805403552850or_nat @ B @ ( bit_se6528837805403552850or_nat @ A @ C ) )
% 0.22/0.54 = ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B @ C ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor.left_commute
% 0.22/0.54 thf(fact_7914_xor_Oleft__commute,axiom,
% 0.22/0.54 ! [B: int,A: int,C: int] :
% 0.22/0.54 ( ( bit_se6526347334894502574or_int @ B @ ( bit_se6526347334894502574or_int @ A @ C ) )
% 0.22/0.54 = ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B @ C ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor.left_commute
% 0.22/0.54 thf(fact_7915_xor_Ocommute,axiom,
% 0.22/0.54 ( bit_se6528837805403552850or_nat
% 0.22/0.54 = ( ^ [A3: nat,B2: nat] : ( bit_se6528837805403552850or_nat @ B2 @ A3 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor.commute
% 0.22/0.54 thf(fact_7916_xor_Ocommute,axiom,
% 0.22/0.54 ( bit_se6526347334894502574or_int
% 0.22/0.54 = ( ^ [A3: int,B2: int] : ( bit_se6526347334894502574or_int @ B2 @ A3 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor.commute
% 0.22/0.54 thf(fact_7917_xor_Oassoc,axiom,
% 0.22/0.54 ! [A: nat,B: nat,C: nat] :
% 0.22/0.54 ( ( bit_se6528837805403552850or_nat @ ( bit_se6528837805403552850or_nat @ A @ B ) @ C )
% 0.22/0.54 = ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B @ C ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor.assoc
% 0.22/0.54 thf(fact_7918_xor_Oassoc,axiom,
% 0.22/0.54 ! [A: int,B: int,C: int] :
% 0.22/0.54 ( ( bit_se6526347334894502574or_int @ ( bit_se6526347334894502574or_int @ A @ B ) @ C )
% 0.22/0.54 = ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B @ C ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % xor.assoc
% 0.22/0.54 thf(fact_7919_inverse__eq__imp__eq,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( ( inverse_inverse_real @ A )
% 0.22/0.54 = ( inverse_inverse_real @ B ) )
% 0.22/0.54 => ( A = B ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_eq_imp_eq
% 0.22/0.54 thf(fact_7920_inverse__eq__imp__eq,axiom,
% 0.22/0.54 ! [A: complex,B: complex] :
% 0.22/0.54 ( ( ( invers8013647133539491842omplex @ A )
% 0.22/0.54 = ( invers8013647133539491842omplex @ B ) )
% 0.22/0.54 => ( A = B ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_eq_imp_eq
% 0.22/0.54 thf(fact_7921_inverse__eq__imp__eq,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( ( inverse_inverse_rat @ A )
% 0.22/0.54 = ( inverse_inverse_rat @ B ) )
% 0.22/0.54 => ( A = B ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_eq_imp_eq
% 0.22/0.54 thf(fact_7922_of__nat__xor__eq,axiom,
% 0.22/0.54 ! [M: nat,N: nat] :
% 0.22/0.54 ( ( semiri1316708129612266289at_nat @ ( bit_se6528837805403552850or_nat @ M @ N ) )
% 0.22/0.54 = ( bit_se6528837805403552850or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % of_nat_xor_eq
% 0.22/0.54 thf(fact_7923_of__nat__xor__eq,axiom,
% 0.22/0.54 ! [M: nat,N: nat] :
% 0.22/0.54 ( ( semiri1314217659103216013at_int @ ( bit_se6528837805403552850or_nat @ M @ N ) )
% 0.22/0.54 = ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % of_nat_xor_eq
% 0.22/0.54 thf(fact_7924_nonzero__imp__inverse__nonzero,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( A != zero_zero_real )
% 0.22/0.54 => ( ( inverse_inverse_real @ A )
% 0.22/0.54 != zero_zero_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_imp_inverse_nonzero
% 0.22/0.54 thf(fact_7925_nonzero__imp__inverse__nonzero,axiom,
% 0.22/0.54 ! [A: complex] :
% 0.22/0.54 ( ( A != zero_zero_complex )
% 0.22/0.54 => ( ( invers8013647133539491842omplex @ A )
% 0.22/0.54 != zero_zero_complex ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_imp_inverse_nonzero
% 0.22/0.54 thf(fact_7926_nonzero__imp__inverse__nonzero,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( A != zero_zero_rat )
% 0.22/0.54 => ( ( inverse_inverse_rat @ A )
% 0.22/0.54 != zero_zero_rat ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_imp_inverse_nonzero
% 0.22/0.54 thf(fact_7927_nonzero__inverse__inverse__eq,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( A != zero_zero_real )
% 0.22/0.54 => ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
% 0.22/0.54 = A ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_inverse_inverse_eq
% 0.22/0.54 thf(fact_7928_nonzero__inverse__inverse__eq,axiom,
% 0.22/0.54 ! [A: complex] :
% 0.22/0.54 ( ( A != zero_zero_complex )
% 0.22/0.54 => ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
% 0.22/0.54 = A ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_inverse_inverse_eq
% 0.22/0.54 thf(fact_7929_nonzero__inverse__inverse__eq,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( A != zero_zero_rat )
% 0.22/0.54 => ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
% 0.22/0.54 = A ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_inverse_inverse_eq
% 0.22/0.54 thf(fact_7930_nonzero__inverse__eq__imp__eq,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( ( inverse_inverse_real @ A )
% 0.22/0.54 = ( inverse_inverse_real @ B ) )
% 0.22/0.54 => ( ( A != zero_zero_real )
% 0.22/0.54 => ( ( B != zero_zero_real )
% 0.22/0.54 => ( A = B ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_inverse_eq_imp_eq
% 0.22/0.54 thf(fact_7931_nonzero__inverse__eq__imp__eq,axiom,
% 0.22/0.54 ! [A: complex,B: complex] :
% 0.22/0.54 ( ( ( invers8013647133539491842omplex @ A )
% 0.22/0.54 = ( invers8013647133539491842omplex @ B ) )
% 0.22/0.54 => ( ( A != zero_zero_complex )
% 0.22/0.54 => ( ( B != zero_zero_complex )
% 0.22/0.54 => ( A = B ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_inverse_eq_imp_eq
% 0.22/0.54 thf(fact_7932_nonzero__inverse__eq__imp__eq,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( ( inverse_inverse_rat @ A )
% 0.22/0.54 = ( inverse_inverse_rat @ B ) )
% 0.22/0.54 => ( ( A != zero_zero_rat )
% 0.22/0.54 => ( ( B != zero_zero_rat )
% 0.22/0.54 => ( A = B ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_inverse_eq_imp_eq
% 0.22/0.54 thf(fact_7933_inverse__zero__imp__zero,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( ( inverse_inverse_real @ A )
% 0.22/0.54 = zero_zero_real )
% 0.22/0.54 => ( A = zero_zero_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_zero_imp_zero
% 0.22/0.54 thf(fact_7934_inverse__zero__imp__zero,axiom,
% 0.22/0.54 ! [A: complex] :
% 0.22/0.54 ( ( ( invers8013647133539491842omplex @ A )
% 0.22/0.54 = zero_zero_complex )
% 0.22/0.54 => ( A = zero_zero_complex ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_zero_imp_zero
% 0.22/0.54 thf(fact_7935_inverse__zero__imp__zero,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( ( inverse_inverse_rat @ A )
% 0.22/0.54 = zero_zero_rat )
% 0.22/0.54 => ( A = zero_zero_rat ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_zero_imp_zero
% 0.22/0.54 thf(fact_7936_field__class_Ofield__inverse__zero,axiom,
% 0.22/0.54 ( ( inverse_inverse_real @ zero_zero_real )
% 0.22/0.54 = zero_zero_real ) ).
% 0.22/0.54
% 0.22/0.54 % field_class.field_inverse_zero
% 0.22/0.54 thf(fact_7937_field__class_Ofield__inverse__zero,axiom,
% 0.22/0.54 ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 0.22/0.54 = zero_zero_complex ) ).
% 0.22/0.54
% 0.22/0.54 % field_class.field_inverse_zero
% 0.22/0.54 thf(fact_7938_field__class_Ofield__inverse__zero,axiom,
% 0.22/0.54 ( ( inverse_inverse_rat @ zero_zero_rat )
% 0.22/0.54 = zero_zero_rat ) ).
% 0.22/0.54
% 0.22/0.54 % field_class.field_inverse_zero
% 0.22/0.54 thf(fact_7939_mult__commute__imp__mult__inverse__commute,axiom,
% 0.22/0.54 ! [Y4: real,X: real] :
% 0.22/0.54 ( ( ( times_times_real @ Y4 @ X )
% 0.22/0.54 = ( times_times_real @ X @ Y4 ) )
% 0.22/0.54 => ( ( times_times_real @ ( inverse_inverse_real @ Y4 ) @ X )
% 0.22/0.54 = ( times_times_real @ X @ ( inverse_inverse_real @ Y4 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % mult_commute_imp_mult_inverse_commute
% 0.22/0.54 thf(fact_7940_mult__commute__imp__mult__inverse__commute,axiom,
% 0.22/0.54 ! [Y4: complex,X: complex] :
% 0.22/0.54 ( ( ( times_times_complex @ Y4 @ X )
% 0.22/0.54 = ( times_times_complex @ X @ Y4 ) )
% 0.22/0.54 => ( ( times_times_complex @ ( invers8013647133539491842omplex @ Y4 ) @ X )
% 0.22/0.54 = ( times_times_complex @ X @ ( invers8013647133539491842omplex @ Y4 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % mult_commute_imp_mult_inverse_commute
% 0.22/0.54 thf(fact_7941_mult__commute__imp__mult__inverse__commute,axiom,
% 0.22/0.54 ! [Y4: rat,X: rat] :
% 0.22/0.54 ( ( ( times_times_rat @ Y4 @ X )
% 0.22/0.54 = ( times_times_rat @ X @ Y4 ) )
% 0.22/0.54 => ( ( times_times_rat @ ( inverse_inverse_rat @ Y4 ) @ X )
% 0.22/0.54 = ( times_times_rat @ X @ ( inverse_inverse_rat @ Y4 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % mult_commute_imp_mult_inverse_commute
% 0.22/0.54 thf(fact_7942_nonzero__norm__inverse,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( A != zero_zero_real )
% 0.22/0.54 => ( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A ) )
% 0.22/0.54 = ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_norm_inverse
% 0.22/0.54 thf(fact_7943_nonzero__norm__inverse,axiom,
% 0.22/0.54 ! [A: complex] :
% 0.22/0.54 ( ( A != zero_zero_complex )
% 0.22/0.54 => ( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A ) )
% 0.22/0.54 = ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_norm_inverse
% 0.22/0.54 thf(fact_7944_Ints__of__nat,axiom,
% 0.22/0.54 ! [N: nat] : ( member_complex @ ( semiri8010041392384452111omplex @ N ) @ ring_1_Ints_complex ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_of_nat
% 0.22/0.54 thf(fact_7945_Ints__of__nat,axiom,
% 0.22/0.54 ! [N: nat] : ( member_real @ ( semiri5074537144036343181t_real @ N ) @ ring_1_Ints_real ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_of_nat
% 0.22/0.54 thf(fact_7946_Ints__of__nat,axiom,
% 0.22/0.54 ! [N: nat] : ( member_rat @ ( semiri681578069525770553at_rat @ N ) @ ring_1_Ints_rat ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_of_nat
% 0.22/0.54 thf(fact_7947_Ints__of__nat,axiom,
% 0.22/0.54 ! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ring_1_Ints_int ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_of_nat
% 0.22/0.54 thf(fact_7948_bit_Oconj__xor__distrib2,axiom,
% 0.22/0.54 ! [Y4: int,Z: int,X: int] :
% 0.22/0.54 ( ( bit_se725231765392027082nd_int @ ( bit_se6526347334894502574or_int @ Y4 @ Z ) @ X )
% 0.22/0.54 = ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ Y4 @ X ) @ ( bit_se725231765392027082nd_int @ Z @ X ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit.conj_xor_distrib2
% 0.22/0.54 thf(fact_7949_bit_Oconj__xor__distrib,axiom,
% 0.22/0.54 ! [X: int,Y4: int,Z: int] :
% 0.22/0.54 ( ( bit_se725231765392027082nd_int @ X @ ( bit_se6526347334894502574or_int @ Y4 @ Z ) )
% 0.22/0.54 = ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ X @ Y4 ) @ ( bit_se725231765392027082nd_int @ X @ Z ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit.conj_xor_distrib
% 0.22/0.54 thf(fact_7950_bit__xor__iff,axiom,
% 0.22/0.54 ! [A: nat,B: nat,N: nat] :
% 0.22/0.54 ( ( bit_se1148574629649215175it_nat @ ( bit_se6528837805403552850or_nat @ A @ B ) @ N )
% 0.22/0.54 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.22/0.54 != ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_xor_iff
% 0.22/0.54 thf(fact_7951_bit__xor__iff,axiom,
% 0.22/0.54 ! [A: int,B: int,N: nat] :
% 0.22/0.54 ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ A @ B ) @ N )
% 0.22/0.54 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.22/0.54 != ( bit_se1146084159140164899it_int @ B @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_xor_iff
% 0.22/0.54 thf(fact_7952_Ints__abs,axiom,
% 0.22/0.54 ! [A: int] :
% 0.22/0.54 ( ( member_int @ A @ ring_1_Ints_int )
% 0.22/0.54 => ( member_int @ ( abs_abs_int @ A ) @ ring_1_Ints_int ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_abs
% 0.22/0.54 thf(fact_7953_Ints__abs,axiom,
% 0.22/0.54 ! [A: code_integer] :
% 0.22/0.54 ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
% 0.22/0.54 => ( member_Code_integer @ ( abs_abs_Code_integer @ A ) @ ring_11222124179247155820nteger ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_abs
% 0.22/0.54 thf(fact_7954_Ints__abs,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.22/0.54 => ( member_rat @ ( abs_abs_rat @ A ) @ ring_1_Ints_rat ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_abs
% 0.22/0.54 thf(fact_7955_Ints__abs,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( member_real @ A @ ring_1_Ints_real )
% 0.22/0.54 => ( member_real @ ( abs_abs_real @ A ) @ ring_1_Ints_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_abs
% 0.22/0.54 thf(fact_7956_Ints__of__int,axiom,
% 0.22/0.54 ! [Z: int] : ( member_int @ ( ring_1_of_int_int @ Z ) @ ring_1_Ints_int ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_of_int
% 0.22/0.54 thf(fact_7957_Ints__of__int,axiom,
% 0.22/0.54 ! [Z: int] : ( member_real @ ( ring_1_of_int_real @ Z ) @ ring_1_Ints_real ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_of_int
% 0.22/0.54 thf(fact_7958_Ints__of__int,axiom,
% 0.22/0.54 ! [Z: int] : ( member_complex @ ( ring_17405671764205052669omplex @ Z ) @ ring_1_Ints_complex ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_of_int
% 0.22/0.54 thf(fact_7959_Ints__of__int,axiom,
% 0.22/0.54 ! [Z: int] : ( member_rat @ ( ring_1_of_int_rat @ Z ) @ ring_1_Ints_rat ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_of_int
% 0.22/0.54 thf(fact_7960_Ints__induct,axiom,
% 0.22/0.54 ! [Q2: int,P: int > $o] :
% 0.22/0.54 ( ( member_int @ Q2 @ ring_1_Ints_int )
% 0.22/0.54 => ( ! [Z2: int] : ( P @ ( ring_1_of_int_int @ Z2 ) )
% 0.22/0.54 => ( P @ Q2 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_induct
% 0.22/0.54 thf(fact_7961_Ints__induct,axiom,
% 0.22/0.54 ! [Q2: real,P: real > $o] :
% 0.22/0.54 ( ( member_real @ Q2 @ ring_1_Ints_real )
% 0.22/0.54 => ( ! [Z2: int] : ( P @ ( ring_1_of_int_real @ Z2 ) )
% 0.22/0.54 => ( P @ Q2 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_induct
% 0.22/0.54 thf(fact_7962_Ints__induct,axiom,
% 0.22/0.54 ! [Q2: complex,P: complex > $o] :
% 0.22/0.54 ( ( member_complex @ Q2 @ ring_1_Ints_complex )
% 0.22/0.54 => ( ! [Z2: int] : ( P @ ( ring_17405671764205052669omplex @ Z2 ) )
% 0.22/0.54 => ( P @ Q2 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_induct
% 0.22/0.54 thf(fact_7963_Ints__induct,axiom,
% 0.22/0.54 ! [Q2: rat,P: rat > $o] :
% 0.22/0.54 ( ( member_rat @ Q2 @ ring_1_Ints_rat )
% 0.22/0.54 => ( ! [Z2: int] : ( P @ ( ring_1_of_int_rat @ Z2 ) )
% 0.22/0.54 => ( P @ Q2 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_induct
% 0.22/0.54 thf(fact_7964_Ints__cases,axiom,
% 0.22/0.54 ! [Q2: int] :
% 0.22/0.54 ( ( member_int @ Q2 @ ring_1_Ints_int )
% 0.22/0.54 => ~ ! [Z2: int] :
% 0.22/0.54 ( Q2
% 0.22/0.54 != ( ring_1_of_int_int @ Z2 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_cases
% 0.22/0.54 thf(fact_7965_Ints__cases,axiom,
% 0.22/0.54 ! [Q2: real] :
% 0.22/0.54 ( ( member_real @ Q2 @ ring_1_Ints_real )
% 0.22/0.54 => ~ ! [Z2: int] :
% 0.22/0.54 ( Q2
% 0.22/0.54 != ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_cases
% 0.22/0.54 thf(fact_7966_Ints__cases,axiom,
% 0.22/0.54 ! [Q2: complex] :
% 0.22/0.54 ( ( member_complex @ Q2 @ ring_1_Ints_complex )
% 0.22/0.54 => ~ ! [Z2: int] :
% 0.22/0.54 ( Q2
% 0.22/0.54 != ( ring_17405671764205052669omplex @ Z2 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_cases
% 0.22/0.54 thf(fact_7967_Ints__cases,axiom,
% 0.22/0.54 ! [Q2: rat] :
% 0.22/0.54 ( ( member_rat @ Q2 @ ring_1_Ints_rat )
% 0.22/0.54 => ~ ! [Z2: int] :
% 0.22/0.54 ( Q2
% 0.22/0.54 != ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_cases
% 0.22/0.54 thf(fact_7968_norm__inverse__le__norm,axiom,
% 0.22/0.54 ! [R2: real,X: real] :
% 0.22/0.54 ( ( ord_less_eq_real @ R2 @ ( real_V7735802525324610683m_real @ X ) )
% 0.22/0.54 => ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.22/0.54 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % norm_inverse_le_norm
% 0.22/0.54 thf(fact_7969_norm__inverse__le__norm,axiom,
% 0.22/0.54 ! [R2: real,X: complex] :
% 0.22/0.54 ( ( ord_less_eq_real @ R2 @ ( real_V1022390504157884413omplex @ X ) )
% 0.22/0.54 => ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.22/0.54 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % norm_inverse_le_norm
% 0.22/0.54 thf(fact_7970_positive__imp__inverse__positive,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.54 => ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % positive_imp_inverse_positive
% 0.22/0.54 thf(fact_7971_positive__imp__inverse__positive,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.22/0.54 => ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % positive_imp_inverse_positive
% 0.22/0.54 thf(fact_7972_negative__imp__inverse__negative,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( ord_less_real @ A @ zero_zero_real )
% 0.22/0.54 => ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % negative_imp_inverse_negative
% 0.22/0.54 thf(fact_7973_negative__imp__inverse__negative,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.22/0.54 => ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat ) ) ).
% 0.22/0.54
% 0.22/0.54 % negative_imp_inverse_negative
% 0.22/0.54 thf(fact_7974_inverse__positive__imp__positive,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.22/0.54 => ( ( A != zero_zero_real )
% 0.22/0.54 => ( ord_less_real @ zero_zero_real @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_positive_imp_positive
% 0.22/0.54 thf(fact_7975_inverse__positive__imp__positive,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 0.22/0.54 => ( ( A != zero_zero_rat )
% 0.22/0.54 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_positive_imp_positive
% 0.22/0.54 thf(fact_7976_inverse__negative__imp__negative,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.22/0.54 => ( ( A != zero_zero_real )
% 0.22/0.54 => ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_negative_imp_negative
% 0.22/0.54 thf(fact_7977_inverse__negative__imp__negative,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 0.22/0.54 => ( ( A != zero_zero_rat )
% 0.22/0.54 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_negative_imp_negative
% 0.22/0.54 thf(fact_7978_less__imp__inverse__less__neg,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( ord_less_real @ A @ B )
% 0.22/0.54 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.22/0.54 => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % less_imp_inverse_less_neg
% 0.22/0.54 thf(fact_7979_less__imp__inverse__less__neg,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( ord_less_rat @ A @ B )
% 0.22/0.54 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.22/0.54 => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % less_imp_inverse_less_neg
% 0.22/0.54 thf(fact_7980_inverse__less__imp__less__neg,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.22/0.54 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.22/0.54 => ( ord_less_real @ B @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_less_imp_less_neg
% 0.22/0.54 thf(fact_7981_inverse__less__imp__less__neg,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.22/0.54 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.22/0.54 => ( ord_less_rat @ B @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_less_imp_less_neg
% 0.22/0.54 thf(fact_7982_less__imp__inverse__less,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( ord_less_real @ A @ B )
% 0.22/0.54 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.54 => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % less_imp_inverse_less
% 0.22/0.54 thf(fact_7983_less__imp__inverse__less,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( ord_less_rat @ A @ B )
% 0.22/0.54 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.22/0.54 => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % less_imp_inverse_less
% 0.22/0.54 thf(fact_7984_inverse__less__imp__less,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.22/0.54 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.54 => ( ord_less_real @ B @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_less_imp_less
% 0.22/0.54 thf(fact_7985_inverse__less__imp__less,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.22/0.54 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.22/0.54 => ( ord_less_rat @ B @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_less_imp_less
% 0.22/0.54 thf(fact_7986_nonzero__inverse__mult__distrib,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( A != zero_zero_real )
% 0.22/0.54 => ( ( B != zero_zero_real )
% 0.22/0.54 => ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
% 0.22/0.54 = ( times_times_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_inverse_mult_distrib
% 0.22/0.54 thf(fact_7987_nonzero__inverse__mult__distrib,axiom,
% 0.22/0.54 ! [A: complex,B: complex] :
% 0.22/0.54 ( ( A != zero_zero_complex )
% 0.22/0.54 => ( ( B != zero_zero_complex )
% 0.22/0.54 => ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
% 0.22/0.54 = ( times_times_complex @ ( invers8013647133539491842omplex @ B ) @ ( invers8013647133539491842omplex @ A ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_inverse_mult_distrib
% 0.22/0.54 thf(fact_7988_nonzero__inverse__mult__distrib,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( A != zero_zero_rat )
% 0.22/0.54 => ( ( B != zero_zero_rat )
% 0.22/0.54 => ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
% 0.22/0.54 = ( times_times_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_inverse_mult_distrib
% 0.22/0.54 thf(fact_7989_bit__Suc__0__iff,axiom,
% 0.22/0.54 ! [N: nat] :
% 0.22/0.54 ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.22/0.54 = ( N = zero_zero_nat ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_Suc_0_iff
% 0.22/0.54 thf(fact_7990_not__bit__Suc__0__Suc,axiom,
% 0.22/0.54 ! [N: nat] :
% 0.22/0.54 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % not_bit_Suc_0_Suc
% 0.22/0.54 thf(fact_7991_nonzero__inverse__minus__eq,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( A != zero_zero_real )
% 0.22/0.54 => ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
% 0.22/0.54 = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_inverse_minus_eq
% 0.22/0.54 thf(fact_7992_nonzero__inverse__minus__eq,axiom,
% 0.22/0.54 ! [A: complex] :
% 0.22/0.54 ( ( A != zero_zero_complex )
% 0.22/0.54 => ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
% 0.22/0.54 = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_inverse_minus_eq
% 0.22/0.54 thf(fact_7993_nonzero__inverse__minus__eq,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( A != zero_zero_rat )
% 0.22/0.54 => ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
% 0.22/0.54 = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_inverse_minus_eq
% 0.22/0.54 thf(fact_7994_inverse__numeral__1,axiom,
% 0.22/0.54 ( ( inverse_inverse_real @ ( numeral_numeral_real @ one ) )
% 0.22/0.54 = ( numeral_numeral_real @ one ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_numeral_1
% 0.22/0.54 thf(fact_7995_inverse__numeral__1,axiom,
% 0.22/0.54 ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ one ) )
% 0.22/0.54 = ( numera6690914467698888265omplex @ one ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_numeral_1
% 0.22/0.54 thf(fact_7996_inverse__numeral__1,axiom,
% 0.22/0.54 ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ one ) )
% 0.22/0.54 = ( numeral_numeral_rat @ one ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_numeral_1
% 0.22/0.54 thf(fact_7997_inverse__unique,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( ( times_times_real @ A @ B )
% 0.22/0.54 = one_one_real )
% 0.22/0.54 => ( ( inverse_inverse_real @ A )
% 0.22/0.54 = B ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_unique
% 0.22/0.54 thf(fact_7998_inverse__unique,axiom,
% 0.22/0.54 ! [A: complex,B: complex] :
% 0.22/0.54 ( ( ( times_times_complex @ A @ B )
% 0.22/0.54 = one_one_complex )
% 0.22/0.54 => ( ( invers8013647133539491842omplex @ A )
% 0.22/0.54 = B ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_unique
% 0.22/0.54 thf(fact_7999_inverse__unique,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( ( times_times_rat @ A @ B )
% 0.22/0.54 = one_one_rat )
% 0.22/0.54 => ( ( inverse_inverse_rat @ A )
% 0.22/0.54 = B ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_unique
% 0.22/0.54 thf(fact_8000_divide__inverse__commute,axiom,
% 0.22/0.54 ( divide_divide_real
% 0.22/0.54 = ( ^ [A3: real,B2: real] : ( times_times_real @ ( inverse_inverse_real @ B2 ) @ A3 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % divide_inverse_commute
% 0.22/0.54 thf(fact_8001_divide__inverse__commute,axiom,
% 0.22/0.54 ( divide1717551699836669952omplex
% 0.22/0.54 = ( ^ [A3: complex,B2: complex] : ( times_times_complex @ ( invers8013647133539491842omplex @ B2 ) @ A3 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % divide_inverse_commute
% 0.22/0.54 thf(fact_8002_divide__inverse__commute,axiom,
% 0.22/0.54 ( divide_divide_rat
% 0.22/0.54 = ( ^ [A3: rat,B2: rat] : ( times_times_rat @ ( inverse_inverse_rat @ B2 ) @ A3 ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % divide_inverse_commute
% 0.22/0.54 thf(fact_8003_divide__inverse,axiom,
% 0.22/0.54 ( divide_divide_real
% 0.22/0.54 = ( ^ [A3: real,B2: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B2 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % divide_inverse
% 0.22/0.54 thf(fact_8004_divide__inverse,axiom,
% 0.22/0.54 ( divide1717551699836669952omplex
% 0.22/0.54 = ( ^ [A3: complex,B2: complex] : ( times_times_complex @ A3 @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % divide_inverse
% 0.22/0.54 thf(fact_8005_divide__inverse,axiom,
% 0.22/0.54 ( divide_divide_rat
% 0.22/0.54 = ( ^ [A3: rat,B2: rat] : ( times_times_rat @ A3 @ ( inverse_inverse_rat @ B2 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % divide_inverse
% 0.22/0.54 thf(fact_8006_field__class_Ofield__divide__inverse,axiom,
% 0.22/0.54 ( divide_divide_real
% 0.22/0.54 = ( ^ [A3: real,B2: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B2 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % field_class.field_divide_inverse
% 0.22/0.54 thf(fact_8007_field__class_Ofield__divide__inverse,axiom,
% 0.22/0.54 ( divide1717551699836669952omplex
% 0.22/0.54 = ( ^ [A3: complex,B2: complex] : ( times_times_complex @ A3 @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % field_class.field_divide_inverse
% 0.22/0.54 thf(fact_8008_field__class_Ofield__divide__inverse,axiom,
% 0.22/0.54 ( divide_divide_rat
% 0.22/0.54 = ( ^ [A3: rat,B2: rat] : ( times_times_rat @ A3 @ ( inverse_inverse_rat @ B2 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % field_class.field_divide_inverse
% 0.22/0.54 thf(fact_8009_inverse__eq__divide,axiom,
% 0.22/0.54 ( inverse_inverse_real
% 0.22/0.54 = ( divide_divide_real @ one_one_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_eq_divide
% 0.22/0.54 thf(fact_8010_inverse__eq__divide,axiom,
% 0.22/0.54 ( invers8013647133539491842omplex
% 0.22/0.54 = ( divide1717551699836669952omplex @ one_one_complex ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_eq_divide
% 0.22/0.54 thf(fact_8011_inverse__eq__divide,axiom,
% 0.22/0.54 ( inverse_inverse_rat
% 0.22/0.54 = ( divide_divide_rat @ one_one_rat ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_eq_divide
% 0.22/0.54 thf(fact_8012_power__mult__inverse__distrib,axiom,
% 0.22/0.54 ! [X: real,M: nat] :
% 0.22/0.54 ( ( times_times_real @ ( power_power_real @ X @ M ) @ ( inverse_inverse_real @ X ) )
% 0.22/0.54 = ( times_times_real @ ( inverse_inverse_real @ X ) @ ( power_power_real @ X @ M ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % power_mult_inverse_distrib
% 0.22/0.54 thf(fact_8013_power__mult__inverse__distrib,axiom,
% 0.22/0.54 ! [X: complex,M: nat] :
% 0.22/0.54 ( ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( invers8013647133539491842omplex @ X ) )
% 0.22/0.54 = ( times_times_complex @ ( invers8013647133539491842omplex @ X ) @ ( power_power_complex @ X @ M ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % power_mult_inverse_distrib
% 0.22/0.54 thf(fact_8014_power__mult__inverse__distrib,axiom,
% 0.22/0.54 ! [X: rat,M: nat] :
% 0.22/0.54 ( ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( inverse_inverse_rat @ X ) )
% 0.22/0.54 = ( times_times_rat @ ( inverse_inverse_rat @ X ) @ ( power_power_rat @ X @ M ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % power_mult_inverse_distrib
% 0.22/0.54 thf(fact_8015_power__mult__power__inverse__commute,axiom,
% 0.22/0.54 ! [X: real,M: nat,N: nat] :
% 0.22/0.54 ( ( times_times_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N ) )
% 0.22/0.54 = ( times_times_real @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N ) @ ( power_power_real @ X @ M ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % power_mult_power_inverse_commute
% 0.22/0.54 thf(fact_8016_power__mult__power__inverse__commute,axiom,
% 0.22/0.54 ! [X: complex,M: nat,N: nat] :
% 0.22/0.54 ( ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N ) )
% 0.22/0.54 = ( times_times_complex @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N ) @ ( power_power_complex @ X @ M ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % power_mult_power_inverse_commute
% 0.22/0.54 thf(fact_8017_power__mult__power__inverse__commute,axiom,
% 0.22/0.54 ! [X: rat,M: nat,N: nat] :
% 0.22/0.54 ( ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) )
% 0.22/0.54 = ( times_times_rat @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) @ ( power_power_rat @ X @ M ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % power_mult_power_inverse_commute
% 0.22/0.54 thf(fact_8018_mult__inverse__of__nat__commute,axiom,
% 0.22/0.54 ! [Xa: nat,X: real] :
% 0.22/0.54 ( ( times_times_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) @ X )
% 0.22/0.54 = ( times_times_real @ X @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % mult_inverse_of_nat_commute
% 0.22/0.54 thf(fact_8019_mult__inverse__of__nat__commute,axiom,
% 0.22/0.54 ! [Xa: nat,X: complex] :
% 0.22/0.54 ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) @ X )
% 0.22/0.54 = ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % mult_inverse_of_nat_commute
% 0.22/0.54 thf(fact_8020_mult__inverse__of__nat__commute,axiom,
% 0.22/0.54 ! [Xa: nat,X: rat] :
% 0.22/0.54 ( ( times_times_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) @ X )
% 0.22/0.54 = ( times_times_rat @ X @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % mult_inverse_of_nat_commute
% 0.22/0.54 thf(fact_8021_nonzero__abs__inverse,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( A != zero_zero_real )
% 0.22/0.54 => ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
% 0.22/0.54 = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_abs_inverse
% 0.22/0.54 thf(fact_8022_nonzero__abs__inverse,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( A != zero_zero_rat )
% 0.22/0.54 => ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
% 0.22/0.54 = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_abs_inverse
% 0.22/0.54 thf(fact_8023_mult__inverse__of__int__commute,axiom,
% 0.22/0.54 ! [Xa: int,X: real] :
% 0.22/0.54 ( ( times_times_real @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) @ X )
% 0.22/0.54 = ( times_times_real @ X @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % mult_inverse_of_int_commute
% 0.22/0.54 thf(fact_8024_mult__inverse__of__int__commute,axiom,
% 0.22/0.54 ! [Xa: int,X: complex] :
% 0.22/0.54 ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) @ X )
% 0.22/0.54 = ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % mult_inverse_of_int_commute
% 0.22/0.54 thf(fact_8025_mult__inverse__of__int__commute,axiom,
% 0.22/0.54 ! [Xa: int,X: rat] :
% 0.22/0.54 ( ( times_times_rat @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) @ X )
% 0.22/0.54 = ( times_times_rat @ X @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % mult_inverse_of_int_commute
% 0.22/0.54 thf(fact_8026_divide__real__def,axiom,
% 0.22/0.54 ( divide_divide_real
% 0.22/0.54 = ( ^ [X3: real,Y6: real] : ( times_times_real @ X3 @ ( inverse_inverse_real @ Y6 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % divide_real_def
% 0.22/0.54 thf(fact_8027_Ints__double__eq__0__iff,axiom,
% 0.22/0.54 ! [A: complex] :
% 0.22/0.54 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.22/0.54 => ( ( ( plus_plus_complex @ A @ A )
% 0.22/0.54 = zero_zero_complex )
% 0.22/0.54 = ( A = zero_zero_complex ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_double_eq_0_iff
% 0.22/0.54 thf(fact_8028_Ints__double__eq__0__iff,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( member_real @ A @ ring_1_Ints_real )
% 0.22/0.54 => ( ( ( plus_plus_real @ A @ A )
% 0.22/0.54 = zero_zero_real )
% 0.22/0.54 = ( A = zero_zero_real ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_double_eq_0_iff
% 0.22/0.54 thf(fact_8029_Ints__double__eq__0__iff,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.22/0.54 => ( ( ( plus_plus_rat @ A @ A )
% 0.22/0.54 = zero_zero_rat )
% 0.22/0.54 = ( A = zero_zero_rat ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_double_eq_0_iff
% 0.22/0.54 thf(fact_8030_Ints__double__eq__0__iff,axiom,
% 0.22/0.54 ! [A: int] :
% 0.22/0.54 ( ( member_int @ A @ ring_1_Ints_int )
% 0.22/0.54 => ( ( ( plus_plus_int @ A @ A )
% 0.22/0.54 = zero_zero_int )
% 0.22/0.54 = ( A = zero_zero_int ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_double_eq_0_iff
% 0.22/0.54 thf(fact_8031_binomial__gbinomial,axiom,
% 0.22/0.54 ! [N: nat,K: nat] :
% 0.22/0.54 ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 0.22/0.54 = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K ) ) ).
% 0.22/0.54
% 0.22/0.54 % binomial_gbinomial
% 0.22/0.54 thf(fact_8032_binomial__gbinomial,axiom,
% 0.22/0.54 ! [N: nat,K: nat] :
% 0.22/0.54 ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 0.22/0.54 = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K ) ) ).
% 0.22/0.54
% 0.22/0.54 % binomial_gbinomial
% 0.22/0.54 thf(fact_8033_binomial__gbinomial,axiom,
% 0.22/0.54 ! [N: nat,K: nat] :
% 0.22/0.54 ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 0.22/0.54 = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K ) ) ).
% 0.22/0.54
% 0.22/0.54 % binomial_gbinomial
% 0.22/0.54 thf(fact_8034_le__imp__inverse__le__neg,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( ord_less_eq_real @ A @ B )
% 0.22/0.54 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.22/0.54 => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % le_imp_inverse_le_neg
% 0.22/0.54 thf(fact_8035_le__imp__inverse__le__neg,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( ord_less_eq_rat @ A @ B )
% 0.22/0.54 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.22/0.54 => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % le_imp_inverse_le_neg
% 0.22/0.54 thf(fact_8036_inverse__le__imp__le__neg,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.22/0.54 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.22/0.54 => ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_le_imp_le_neg
% 0.22/0.54 thf(fact_8037_inverse__le__imp__le__neg,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.22/0.54 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.22/0.54 => ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_le_imp_le_neg
% 0.22/0.54 thf(fact_8038_le__imp__inverse__le,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( ord_less_eq_real @ A @ B )
% 0.22/0.54 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.54 => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % le_imp_inverse_le
% 0.22/0.54 thf(fact_8039_le__imp__inverse__le,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( ord_less_eq_rat @ A @ B )
% 0.22/0.54 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.22/0.54 => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % le_imp_inverse_le
% 0.22/0.54 thf(fact_8040_inverse__le__imp__le,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.22/0.54 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.54 => ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_le_imp_le
% 0.22/0.54 thf(fact_8041_inverse__le__imp__le,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.22/0.54 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.22/0.54 => ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_le_imp_le
% 0.22/0.54 thf(fact_8042_inverse__le__1__iff,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( ord_less_eq_real @ ( inverse_inverse_real @ X ) @ one_one_real )
% 0.22/0.54 = ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.22/0.54 | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_le_1_iff
% 0.22/0.54 thf(fact_8043_inverse__le__1__iff,axiom,
% 0.22/0.54 ! [X: rat] :
% 0.22/0.54 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
% 0.22/0.54 = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.22/0.54 | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_le_1_iff
% 0.22/0.54 thf(fact_8044_one__less__inverse__iff,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X ) )
% 0.22/0.54 = ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.54 & ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % one_less_inverse_iff
% 0.22/0.54 thf(fact_8045_one__less__inverse__iff,axiom,
% 0.22/0.54 ! [X: rat] :
% 0.22/0.54 ( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
% 0.22/0.54 = ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.22/0.54 & ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % one_less_inverse_iff
% 0.22/0.54 thf(fact_8046_one__less__inverse,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.54 => ( ( ord_less_real @ A @ one_one_real )
% 0.22/0.54 => ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % one_less_inverse
% 0.22/0.54 thf(fact_8047_one__less__inverse,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.22/0.54 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.22/0.54 => ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % one_less_inverse
% 0.22/0.54 thf(fact_8048_division__ring__inverse__add,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( A != zero_zero_real )
% 0.22/0.54 => ( ( B != zero_zero_real )
% 0.22/0.54 => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.22/0.54 = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( plus_plus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % division_ring_inverse_add
% 0.22/0.54 thf(fact_8049_division__ring__inverse__add,axiom,
% 0.22/0.54 ! [A: complex,B: complex] :
% 0.22/0.54 ( ( A != zero_zero_complex )
% 0.22/0.54 => ( ( B != zero_zero_complex )
% 0.22/0.54 => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 0.22/0.54 = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( plus_plus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % division_ring_inverse_add
% 0.22/0.54 thf(fact_8050_division__ring__inverse__add,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( A != zero_zero_rat )
% 0.22/0.54 => ( ( B != zero_zero_rat )
% 0.22/0.54 => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.22/0.54 = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( plus_plus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % division_ring_inverse_add
% 0.22/0.54 thf(fact_8051_inverse__add,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( A != zero_zero_real )
% 0.22/0.54 => ( ( B != zero_zero_real )
% 0.22/0.54 => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.22/0.54 = ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( inverse_inverse_real @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_add
% 0.22/0.54 thf(fact_8052_inverse__add,axiom,
% 0.22/0.54 ! [A: complex,B: complex] :
% 0.22/0.54 ( ( A != zero_zero_complex )
% 0.22/0.54 => ( ( B != zero_zero_complex )
% 0.22/0.54 => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 0.22/0.54 = ( times_times_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( invers8013647133539491842omplex @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_add
% 0.22/0.54 thf(fact_8053_inverse__add,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( A != zero_zero_rat )
% 0.22/0.54 => ( ( B != zero_zero_rat )
% 0.22/0.54 => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.22/0.54 = ( times_times_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( inverse_inverse_rat @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_add
% 0.22/0.54 thf(fact_8054_field__class_Ofield__inverse,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( A != zero_zero_real )
% 0.22/0.54 => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 0.22/0.54 = one_one_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % field_class.field_inverse
% 0.22/0.54 thf(fact_8055_field__class_Ofield__inverse,axiom,
% 0.22/0.54 ! [A: complex] :
% 0.22/0.54 ( ( A != zero_zero_complex )
% 0.22/0.54 => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 0.22/0.54 = one_one_complex ) ) ).
% 0.22/0.54
% 0.22/0.54 % field_class.field_inverse
% 0.22/0.54 thf(fact_8056_field__class_Ofield__inverse,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( A != zero_zero_rat )
% 0.22/0.54 => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 0.22/0.54 = one_one_rat ) ) ).
% 0.22/0.54
% 0.22/0.54 % field_class.field_inverse
% 0.22/0.54 thf(fact_8057_division__ring__inverse__diff,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( A != zero_zero_real )
% 0.22/0.54 => ( ( B != zero_zero_real )
% 0.22/0.54 => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.22/0.54 = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ B @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % division_ring_inverse_diff
% 0.22/0.54 thf(fact_8058_division__ring__inverse__diff,axiom,
% 0.22/0.54 ! [A: complex,B: complex] :
% 0.22/0.54 ( ( A != zero_zero_complex )
% 0.22/0.54 => ( ( B != zero_zero_complex )
% 0.22/0.54 => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 0.22/0.54 = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ B @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % division_ring_inverse_diff
% 0.22/0.54 thf(fact_8059_division__ring__inverse__diff,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( A != zero_zero_rat )
% 0.22/0.54 => ( ( B != zero_zero_rat )
% 0.22/0.54 => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.22/0.54 = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ B @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % division_ring_inverse_diff
% 0.22/0.54 thf(fact_8060_not__bit__Suc__0__numeral,axiom,
% 0.22/0.54 ! [N: num] :
% 0.22/0.54 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% 0.22/0.54
% 0.22/0.54 % not_bit_Suc_0_numeral
% 0.22/0.54 thf(fact_8061_nonzero__inverse__eq__divide,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( A != zero_zero_real )
% 0.22/0.54 => ( ( inverse_inverse_real @ A )
% 0.22/0.54 = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_inverse_eq_divide
% 0.22/0.54 thf(fact_8062_nonzero__inverse__eq__divide,axiom,
% 0.22/0.54 ! [A: complex] :
% 0.22/0.54 ( ( A != zero_zero_complex )
% 0.22/0.54 => ( ( invers8013647133539491842omplex @ A )
% 0.22/0.54 = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_inverse_eq_divide
% 0.22/0.54 thf(fact_8063_nonzero__inverse__eq__divide,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( A != zero_zero_rat )
% 0.22/0.54 => ( ( inverse_inverse_rat @ A )
% 0.22/0.54 = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % nonzero_inverse_eq_divide
% 0.22/0.54 thf(fact_8064_gbinomial__Suc__Suc,axiom,
% 0.22/0.54 ! [A: complex,K: nat] :
% 0.22/0.54 ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 0.22/0.54 = ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_Suc_Suc
% 0.22/0.54 thf(fact_8065_gbinomial__Suc__Suc,axiom,
% 0.22/0.54 ! [A: real,K: nat] :
% 0.22/0.54 ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 0.22/0.54 = ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_Suc_Suc
% 0.22/0.54 thf(fact_8066_gbinomial__Suc__Suc,axiom,
% 0.22/0.54 ! [A: rat,K: nat] :
% 0.22/0.54 ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 0.22/0.54 = ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_Suc_Suc
% 0.22/0.54 thf(fact_8067_gbinomial__of__nat__symmetric,axiom,
% 0.22/0.54 ! [K: nat,N: nat] :
% 0.22/0.54 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.54 => ( ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K )
% 0.22/0.54 = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_of_nat_symmetric
% 0.22/0.54 thf(fact_8068_gbinomial__of__nat__symmetric,axiom,
% 0.22/0.54 ! [K: nat,N: nat] :
% 0.22/0.54 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.54 => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
% 0.22/0.54 = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_of_nat_symmetric
% 0.22/0.54 thf(fact_8069_gbinomial__of__nat__symmetric,axiom,
% 0.22/0.54 ! [K: nat,N: nat] :
% 0.22/0.54 ( ( ord_less_eq_nat @ K @ N )
% 0.22/0.54 => ( ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K )
% 0.22/0.54 = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_of_nat_symmetric
% 0.22/0.54 thf(fact_8070_inverse__powr,axiom,
% 0.22/0.54 ! [Y4: real,A: real] :
% 0.22/0.54 ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.54 => ( ( powr_real @ ( inverse_inverse_real @ Y4 ) @ A )
% 0.22/0.54 = ( inverse_inverse_real @ ( powr_real @ Y4 @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_powr
% 0.22/0.54 thf(fact_8071_Ints__odd__nonzero,axiom,
% 0.22/0.54 ! [A: complex] :
% 0.22/0.54 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.22/0.54 => ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A ) @ A )
% 0.22/0.54 != zero_zero_complex ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_odd_nonzero
% 0.22/0.54 thf(fact_8072_Ints__odd__nonzero,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( member_real @ A @ ring_1_Ints_real )
% 0.22/0.54 => ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A )
% 0.22/0.54 != zero_zero_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_odd_nonzero
% 0.22/0.54 thf(fact_8073_Ints__odd__nonzero,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.22/0.54 => ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A )
% 0.22/0.54 != zero_zero_rat ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_odd_nonzero
% 0.22/0.54 thf(fact_8074_Ints__odd__nonzero,axiom,
% 0.22/0.54 ! [A: int] :
% 0.22/0.54 ( ( member_int @ A @ ring_1_Ints_int )
% 0.22/0.54 => ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A )
% 0.22/0.54 != zero_zero_int ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_odd_nonzero
% 0.22/0.54 thf(fact_8075_of__int__divide__in__Ints,axiom,
% 0.22/0.54 ! [B: int,A: int] :
% 0.22/0.54 ( ( dvd_dvd_int @ B @ A )
% 0.22/0.54 => ( member_complex @ ( divide1717551699836669952omplex @ ( ring_17405671764205052669omplex @ A ) @ ( ring_17405671764205052669omplex @ B ) ) @ ring_1_Ints_complex ) ) ).
% 0.22/0.54
% 0.22/0.54 % of_int_divide_in_Ints
% 0.22/0.54 thf(fact_8076_of__int__divide__in__Ints,axiom,
% 0.22/0.54 ! [B: int,A: int] :
% 0.22/0.54 ( ( dvd_dvd_int @ B @ A )
% 0.22/0.54 => ( member_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) @ ring_1_Ints_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % of_int_divide_in_Ints
% 0.22/0.54 thf(fact_8077_of__int__divide__in__Ints,axiom,
% 0.22/0.54 ! [B: int,A: int] :
% 0.22/0.54 ( ( dvd_dvd_int @ B @ A )
% 0.22/0.54 => ( member_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) @ ring_1_Ints_rat ) ) ).
% 0.22/0.54
% 0.22/0.54 % of_int_divide_in_Ints
% 0.22/0.54 thf(fact_8078_of__int__divide__in__Ints,axiom,
% 0.22/0.54 ! [B: int,A: int] :
% 0.22/0.54 ( ( dvd_dvd_int @ B @ A )
% 0.22/0.54 => ( member_int @ ( divide_divide_int @ ( ring_1_of_int_int @ A ) @ ( ring_1_of_int_int @ B ) ) @ ring_1_Ints_int ) ) ).
% 0.22/0.54
% 0.22/0.54 % of_int_divide_in_Ints
% 0.22/0.54 thf(fact_8079_inverse__le__iff,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.22/0.54 = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.22/0.54 => ( ord_less_eq_real @ B @ A ) )
% 0.22/0.54 & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.22/0.54 => ( ord_less_eq_real @ A @ B ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_le_iff
% 0.22/0.54 thf(fact_8080_inverse__le__iff,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.22/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.22/0.54 => ( ord_less_eq_rat @ B @ A ) )
% 0.22/0.54 & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.22/0.54 => ( ord_less_eq_rat @ A @ B ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_le_iff
% 0.22/0.54 thf(fact_8081_inverse__less__iff,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.22/0.54 = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.22/0.54 => ( ord_less_real @ B @ A ) )
% 0.22/0.54 & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.22/0.54 => ( ord_less_real @ A @ B ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_less_iff
% 0.22/0.54 thf(fact_8082_inverse__less__iff,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.22/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.22/0.54 => ( ord_less_rat @ B @ A ) )
% 0.22/0.54 & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.22/0.54 => ( ord_less_rat @ A @ B ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_less_iff
% 0.22/0.54 thf(fact_8083_one__le__inverse__iff,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X ) )
% 0.22/0.54 = ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.54 & ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % one_le_inverse_iff
% 0.22/0.54 thf(fact_8084_one__le__inverse__iff,axiom,
% 0.22/0.54 ! [X: rat] :
% 0.22/0.54 ( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
% 0.22/0.54 = ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.22/0.54 & ( ord_less_eq_rat @ X @ one_one_rat ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % one_le_inverse_iff
% 0.22/0.54 thf(fact_8085_inverse__less__1__iff,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( ord_less_real @ ( inverse_inverse_real @ X ) @ one_one_real )
% 0.22/0.54 = ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.22/0.54 | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_less_1_iff
% 0.22/0.54 thf(fact_8086_inverse__less__1__iff,axiom,
% 0.22/0.54 ! [X: rat] :
% 0.22/0.54 ( ( ord_less_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
% 0.22/0.54 = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.22/0.54 | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_less_1_iff
% 0.22/0.54 thf(fact_8087_one__le__inverse,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.54 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.22/0.54 => ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % one_le_inverse
% 0.22/0.54 thf(fact_8088_one__le__inverse,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.22/0.54 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.22/0.54 => ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % one_le_inverse
% 0.22/0.54 thf(fact_8089_inverse__diff__inverse,axiom,
% 0.22/0.54 ! [A: real,B: real] :
% 0.22/0.54 ( ( A != zero_zero_real )
% 0.22/0.54 => ( ( B != zero_zero_real )
% 0.22/0.54 => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.22/0.54 = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_diff_inverse
% 0.22/0.54 thf(fact_8090_inverse__diff__inverse,axiom,
% 0.22/0.54 ! [A: complex,B: complex] :
% 0.22/0.54 ( ( A != zero_zero_complex )
% 0.22/0.54 => ( ( B != zero_zero_complex )
% 0.22/0.54 => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 0.22/0.54 = ( uminus1482373934393186551omplex @ ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_diff_inverse
% 0.22/0.54 thf(fact_8091_inverse__diff__inverse,axiom,
% 0.22/0.54 ! [A: rat,B: rat] :
% 0.22/0.54 ( ( A != zero_zero_rat )
% 0.22/0.54 => ( ( B != zero_zero_rat )
% 0.22/0.54 => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.22/0.54 = ( uminus_uminus_rat @ ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % inverse_diff_inverse
% 0.22/0.54 thf(fact_8092_reals__Archimedean,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.54 => ? [N2: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ X ) ) ).
% 0.22/0.54
% 0.22/0.54 % reals_Archimedean
% 0.22/0.54 thf(fact_8093_reals__Archimedean,axiom,
% 0.22/0.54 ! [X: rat] :
% 0.22/0.54 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.22/0.54 => ? [N2: nat] : ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ X ) ) ).
% 0.22/0.54
% 0.22/0.54 % reals_Archimedean
% 0.22/0.54 thf(fact_8094_gbinomial__addition__formula,axiom,
% 0.22/0.54 ! [A: complex,K: nat] :
% 0.22/0.54 ( ( gbinomial_complex @ A @ ( suc @ K ) )
% 0.22/0.54 = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_addition_formula
% 0.22/0.54 thf(fact_8095_gbinomial__addition__formula,axiom,
% 0.22/0.54 ! [A: real,K: nat] :
% 0.22/0.54 ( ( gbinomial_real @ A @ ( suc @ K ) )
% 0.22/0.54 = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_addition_formula
% 0.22/0.54 thf(fact_8096_gbinomial__addition__formula,axiom,
% 0.22/0.54 ! [A: rat,K: nat] :
% 0.22/0.54 ( ( gbinomial_rat @ A @ ( suc @ K ) )
% 0.22/0.54 = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_addition_formula
% 0.22/0.54 thf(fact_8097_gbinomial__absorb__comp,axiom,
% 0.22/0.54 ! [A: complex,K: nat] :
% 0.22/0.54 ( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A @ K ) )
% 0.22/0.54 = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_absorb_comp
% 0.22/0.54 thf(fact_8098_gbinomial__absorb__comp,axiom,
% 0.22/0.54 ! [A: real,K: nat] :
% 0.22/0.54 ( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A @ K ) )
% 0.22/0.54 = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_absorb_comp
% 0.22/0.54 thf(fact_8099_gbinomial__absorb__comp,axiom,
% 0.22/0.54 ! [A: rat,K: nat] :
% 0.22/0.54 ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
% 0.22/0.54 = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_absorb_comp
% 0.22/0.54 thf(fact_8100_gbinomial__ge__n__over__k__pow__k,axiom,
% 0.22/0.54 ! [K: nat,A: rat] :
% 0.22/0.54 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
% 0.22/0.54 => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_ge_n_over_k_pow_k
% 0.22/0.54 thf(fact_8101_gbinomial__ge__n__over__k__pow__k,axiom,
% 0.22/0.54 ! [K: nat,A: real] :
% 0.22/0.54 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
% 0.22/0.54 => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_ge_n_over_k_pow_k
% 0.22/0.54 thf(fact_8102_gbinomial__mult__1_H,axiom,
% 0.22/0.54 ! [A: complex,K: nat] :
% 0.22/0.54 ( ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ A )
% 0.22/0.54 = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_mult_1'
% 0.22/0.54 thf(fact_8103_gbinomial__mult__1_H,axiom,
% 0.22/0.54 ! [A: real,K: nat] :
% 0.22/0.54 ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
% 0.22/0.54 = ( 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 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_mult_1'
% 0.22/0.54 thf(fact_8104_gbinomial__mult__1_H,axiom,
% 0.22/0.54 ! [A: rat,K: nat] :
% 0.22/0.54 ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
% 0.22/0.54 = ( 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 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_mult_1'
% 0.22/0.54 thf(fact_8105_gbinomial__mult__1,axiom,
% 0.22/0.54 ! [A: complex,K: nat] :
% 0.22/0.54 ( ( times_times_complex @ A @ ( gbinomial_complex @ A @ K ) )
% 0.22/0.54 = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_mult_1
% 0.22/0.54 thf(fact_8106_gbinomial__mult__1,axiom,
% 0.22/0.54 ! [A: real,K: nat] :
% 0.22/0.54 ( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
% 0.22/0.54 = ( 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 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_mult_1
% 0.22/0.54 thf(fact_8107_gbinomial__mult__1,axiom,
% 0.22/0.54 ! [A: rat,K: nat] :
% 0.22/0.54 ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
% 0.22/0.54 = ( 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 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_mult_1
% 0.22/0.54 thf(fact_8108_forall__pos__mono__1,axiom,
% 0.22/0.54 ! [P: real > $o,E2: real] :
% 0.22/0.54 ( ! [D2: real,E: real] :
% 0.22/0.54 ( ( ord_less_real @ D2 @ E )
% 0.22/0.54 => ( ( P @ D2 )
% 0.22/0.54 => ( P @ E ) ) )
% 0.22/0.54 => ( ! [N2: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
% 0.22/0.54 => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.22/0.54 => ( P @ E2 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % forall_pos_mono_1
% 0.22/0.54 thf(fact_8109_real__arch__inverse,axiom,
% 0.22/0.54 ! [E2: real] :
% 0.22/0.54 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.22/0.54 = ( ? [N4: nat] :
% 0.22/0.54 ( ( N4 != zero_zero_nat )
% 0.22/0.54 & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) )
% 0.22/0.54 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) @ E2 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % real_arch_inverse
% 0.22/0.54 thf(fact_8110_forall__pos__mono,axiom,
% 0.22/0.54 ! [P: real > $o,E2: real] :
% 0.22/0.54 ( ! [D2: real,E: real] :
% 0.22/0.54 ( ( ord_less_real @ D2 @ E )
% 0.22/0.54 => ( ( P @ D2 )
% 0.22/0.54 => ( P @ E ) ) )
% 0.22/0.54 => ( ! [N2: nat] :
% 0.22/0.54 ( ( N2 != zero_zero_nat )
% 0.22/0.54 => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
% 0.22/0.54 => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.22/0.54 => ( P @ E2 ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % forall_pos_mono
% 0.22/0.54 thf(fact_8111_bit__nat__iff,axiom,
% 0.22/0.54 ! [K: int,N: nat] :
% 0.22/0.54 ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
% 0.22/0.54 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.22/0.54 & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_nat_iff
% 0.22/0.54 thf(fact_8112_even__xor__iff,axiom,
% 0.22/0.54 ! [A: code_integer,B: code_integer] :
% 0.22/0.54 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ A @ B ) )
% 0.22/0.54 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.54 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % even_xor_iff
% 0.22/0.54 thf(fact_8113_even__xor__iff,axiom,
% 0.22/0.54 ! [A: nat,B: nat] :
% 0.22/0.54 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ A @ B ) )
% 0.22/0.54 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.54 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % even_xor_iff
% 0.22/0.54 thf(fact_8114_even__xor__iff,axiom,
% 0.22/0.54 ! [A: int,B: int] :
% 0.22/0.54 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ A @ B ) )
% 0.22/0.54 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.54 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % even_xor_iff
% 0.22/0.54 thf(fact_8115_sqrt__divide__self__eq,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.54 => ( ( divide_divide_real @ ( sqrt @ X ) @ X )
% 0.22/0.54 = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % sqrt_divide_self_eq
% 0.22/0.54 thf(fact_8116_ln__inverse,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.54 => ( ( ln_ln_real @ ( inverse_inverse_real @ X ) )
% 0.22/0.54 = ( uminus_uminus_real @ ( ln_ln_real @ X ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % ln_inverse
% 0.22/0.54 thf(fact_8117_Ints__odd__less__0,axiom,
% 0.22/0.54 ! [A: real] :
% 0.22/0.54 ( ( member_real @ A @ ring_1_Ints_real )
% 0.22/0.54 => ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A ) @ zero_zero_real )
% 0.22/0.54 = ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_odd_less_0
% 0.22/0.54 thf(fact_8118_Ints__odd__less__0,axiom,
% 0.22/0.54 ! [A: rat] :
% 0.22/0.54 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.22/0.54 => ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A ) @ zero_zero_rat )
% 0.22/0.54 = ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_odd_less_0
% 0.22/0.54 thf(fact_8119_Ints__odd__less__0,axiom,
% 0.22/0.54 ! [A: int] :
% 0.22/0.54 ( ( member_int @ A @ ring_1_Ints_int )
% 0.22/0.54 => ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A ) @ zero_zero_int )
% 0.22/0.54 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_odd_less_0
% 0.22/0.54 thf(fact_8120_Ints__nonzero__abs__ge1,axiom,
% 0.22/0.54 ! [X: code_integer] :
% 0.22/0.54 ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
% 0.22/0.54 => ( ( X != zero_z3403309356797280102nteger )
% 0.22/0.54 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_nonzero_abs_ge1
% 0.22/0.54 thf(fact_8121_Ints__nonzero__abs__ge1,axiom,
% 0.22/0.54 ! [X: rat] :
% 0.22/0.54 ( ( member_rat @ X @ ring_1_Ints_rat )
% 0.22/0.54 => ( ( X != zero_zero_rat )
% 0.22/0.54 => ( ord_less_eq_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_nonzero_abs_ge1
% 0.22/0.54 thf(fact_8122_Ints__nonzero__abs__ge1,axiom,
% 0.22/0.54 ! [X: int] :
% 0.22/0.54 ( ( member_int @ X @ ring_1_Ints_int )
% 0.22/0.54 => ( ( X != zero_zero_int )
% 0.22/0.54 => ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_nonzero_abs_ge1
% 0.22/0.54 thf(fact_8123_Ints__nonzero__abs__ge1,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( member_real @ X @ ring_1_Ints_real )
% 0.22/0.54 => ( ( X != zero_zero_real )
% 0.22/0.54 => ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_nonzero_abs_ge1
% 0.22/0.54 thf(fact_8124_Ints__nonzero__abs__less1,axiom,
% 0.22/0.54 ! [X: code_integer] :
% 0.22/0.54 ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
% 0.22/0.54 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer )
% 0.22/0.54 => ( X = zero_z3403309356797280102nteger ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_nonzero_abs_less1
% 0.22/0.54 thf(fact_8125_Ints__nonzero__abs__less1,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( member_real @ X @ ring_1_Ints_real )
% 0.22/0.54 => ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.22/0.54 => ( X = zero_zero_real ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_nonzero_abs_less1
% 0.22/0.54 thf(fact_8126_Ints__nonzero__abs__less1,axiom,
% 0.22/0.54 ! [X: rat] :
% 0.22/0.54 ( ( member_rat @ X @ ring_1_Ints_rat )
% 0.22/0.54 => ( ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat )
% 0.22/0.54 => ( X = zero_zero_rat ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_nonzero_abs_less1
% 0.22/0.54 thf(fact_8127_Ints__nonzero__abs__less1,axiom,
% 0.22/0.54 ! [X: int] :
% 0.22/0.54 ( ( member_int @ X @ ring_1_Ints_int )
% 0.22/0.54 => ( ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int )
% 0.22/0.54 => ( X = zero_zero_int ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_nonzero_abs_less1
% 0.22/0.54 thf(fact_8128_Ints__eq__abs__less1,axiom,
% 0.22/0.54 ! [X: code_integer,Y4: code_integer] :
% 0.22/0.54 ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
% 0.22/0.54 => ( ( member_Code_integer @ Y4 @ ring_11222124179247155820nteger )
% 0.22/0.54 => ( ( X = Y4 )
% 0.22/0.54 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ Y4 ) ) @ one_one_Code_integer ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_eq_abs_less1
% 0.22/0.54 thf(fact_8129_Ints__eq__abs__less1,axiom,
% 0.22/0.54 ! [X: real,Y4: real] :
% 0.22/0.54 ( ( member_real @ X @ ring_1_Ints_real )
% 0.22/0.54 => ( ( member_real @ Y4 @ ring_1_Ints_real )
% 0.22/0.54 => ( ( X = Y4 )
% 0.22/0.54 = ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ one_one_real ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_eq_abs_less1
% 0.22/0.54 thf(fact_8130_Ints__eq__abs__less1,axiom,
% 0.22/0.54 ! [X: rat,Y4: rat] :
% 0.22/0.54 ( ( member_rat @ X @ ring_1_Ints_rat )
% 0.22/0.54 => ( ( member_rat @ Y4 @ ring_1_Ints_rat )
% 0.22/0.54 => ( ( X = Y4 )
% 0.22/0.54 = ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ Y4 ) ) @ one_one_rat ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_eq_abs_less1
% 0.22/0.54 thf(fact_8131_Ints__eq__abs__less1,axiom,
% 0.22/0.54 ! [X: int,Y4: int] :
% 0.22/0.54 ( ( member_int @ X @ ring_1_Ints_int )
% 0.22/0.54 => ( ( member_int @ Y4 @ ring_1_Ints_int )
% 0.22/0.54 => ( ( X = Y4 )
% 0.22/0.54 = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Y4 ) ) @ one_one_int ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Ints_eq_abs_less1
% 0.22/0.54 thf(fact_8132_sin__times__pi__eq__0,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( ( sin_real @ ( times_times_real @ X @ pi ) )
% 0.22/0.54 = zero_zero_real )
% 0.22/0.54 = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 0.22/0.54
% 0.22/0.54 % sin_times_pi_eq_0
% 0.22/0.54 thf(fact_8133_ex__inverse__of__nat__less,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.54 => ? [N2: nat] :
% 0.22/0.54 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.22/0.54 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % ex_inverse_of_nat_less
% 0.22/0.54 thf(fact_8134_ex__inverse__of__nat__less,axiom,
% 0.22/0.54 ! [X: rat] :
% 0.22/0.54 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.22/0.54 => ? [N2: nat] :
% 0.22/0.54 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.22/0.54 & ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ X ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % ex_inverse_of_nat_less
% 0.22/0.54 thf(fact_8135_power__diff__conv__inverse,axiom,
% 0.22/0.54 ! [X: real,M: nat,N: nat] :
% 0.22/0.54 ( ( X != zero_zero_real )
% 0.22/0.54 => ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.54 => ( ( power_power_real @ X @ ( minus_minus_nat @ N @ M ) )
% 0.22/0.54 = ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ M ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % power_diff_conv_inverse
% 0.22/0.54 thf(fact_8136_power__diff__conv__inverse,axiom,
% 0.22/0.54 ! [X: complex,M: nat,N: nat] :
% 0.22/0.54 ( ( X != zero_zero_complex )
% 0.22/0.54 => ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.54 => ( ( power_power_complex @ X @ ( minus_minus_nat @ N @ M ) )
% 0.22/0.54 = ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ M ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % power_diff_conv_inverse
% 0.22/0.54 thf(fact_8137_power__diff__conv__inverse,axiom,
% 0.22/0.54 ! [X: rat,M: nat,N: nat] :
% 0.22/0.54 ( ( X != zero_zero_rat )
% 0.22/0.54 => ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.54 => ( ( power_power_rat @ X @ ( minus_minus_nat @ N @ M ) )
% 0.22/0.54 = ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ M ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % power_diff_conv_inverse
% 0.22/0.54 thf(fact_8138_Suc__times__gbinomial,axiom,
% 0.22/0.54 ! [K: nat,A: complex] :
% 0.22/0.54 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
% 0.22/0.54 = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Suc_times_gbinomial
% 0.22/0.54 thf(fact_8139_Suc__times__gbinomial,axiom,
% 0.22/0.54 ! [K: nat,A: real] :
% 0.22/0.54 ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
% 0.22/0.54 = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Suc_times_gbinomial
% 0.22/0.54 thf(fact_8140_Suc__times__gbinomial,axiom,
% 0.22/0.54 ! [K: nat,A: rat] :
% 0.22/0.54 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
% 0.22/0.54 = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % Suc_times_gbinomial
% 0.22/0.54 thf(fact_8141_gbinomial__absorption,axiom,
% 0.22/0.54 ! [K: nat,A: complex] :
% 0.22/0.54 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
% 0.22/0.54 = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_absorption
% 0.22/0.54 thf(fact_8142_gbinomial__absorption,axiom,
% 0.22/0.54 ! [K: nat,A: real] :
% 0.22/0.54 ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
% 0.22/0.54 = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_absorption
% 0.22/0.54 thf(fact_8143_gbinomial__absorption,axiom,
% 0.22/0.54 ! [K: nat,A: rat] :
% 0.22/0.54 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
% 0.22/0.54 = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_absorption
% 0.22/0.54 thf(fact_8144_gbinomial__trinomial__revision,axiom,
% 0.22/0.54 ! [K: nat,M: nat,A: complex] :
% 0.22/0.54 ( ( ord_less_eq_nat @ K @ M )
% 0.22/0.54 => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
% 0.22/0.54 = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_trinomial_revision
% 0.22/0.54 thf(fact_8145_gbinomial__trinomial__revision,axiom,
% 0.22/0.54 ! [K: nat,M: nat,A: real] :
% 0.22/0.54 ( ( ord_less_eq_nat @ K @ M )
% 0.22/0.54 => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
% 0.22/0.54 = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_trinomial_revision
% 0.22/0.54 thf(fact_8146_gbinomial__trinomial__revision,axiom,
% 0.22/0.54 ! [K: nat,M: nat,A: rat] :
% 0.22/0.54 ( ( ord_less_eq_nat @ K @ M )
% 0.22/0.54 => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
% 0.22/0.54 = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_trinomial_revision
% 0.22/0.54 thf(fact_8147_log__inverse,axiom,
% 0.22/0.54 ! [A: real,X: real] :
% 0.22/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.54 => ( ( A != one_one_real )
% 0.22/0.54 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.54 => ( ( log @ A @ ( inverse_inverse_real @ X ) )
% 0.22/0.54 = ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % log_inverse
% 0.22/0.54 thf(fact_8148_frac__neg,axiom,
% 0.22/0.54 ! [X: real] :
% 0.22/0.54 ( ( ( member_real @ X @ ring_1_Ints_real )
% 0.22/0.54 => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
% 0.22/0.54 = zero_zero_real ) )
% 0.22/0.54 & ( ~ ( member_real @ X @ ring_1_Ints_real )
% 0.22/0.54 => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
% 0.22/0.54 = ( minus_minus_real @ one_one_real @ ( archim2898591450579166408c_real @ X ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % frac_neg
% 0.22/0.54 thf(fact_8149_frac__neg,axiom,
% 0.22/0.54 ! [X: rat] :
% 0.22/0.54 ( ( ( member_rat @ X @ ring_1_Ints_rat )
% 0.22/0.54 => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
% 0.22/0.54 = zero_zero_rat ) )
% 0.22/0.54 & ( ~ ( member_rat @ X @ ring_1_Ints_rat )
% 0.22/0.54 => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
% 0.22/0.54 = ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % frac_neg
% 0.22/0.54 thf(fact_8150_bit__nat__def,axiom,
% 0.22/0.54 ( bit_se1148574629649215175it_nat
% 0.22/0.54 = ( ^ [M5: nat,N4: nat] :
% 0.22/0.54 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % bit_nat_def
% 0.22/0.54 thf(fact_8151_gbinomial__factors,axiom,
% 0.22/0.54 ! [A: complex,K: nat] :
% 0.22/0.54 ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 0.22/0.54 = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_factors
% 0.22/0.54 thf(fact_8152_gbinomial__factors,axiom,
% 0.22/0.54 ! [A: real,K: nat] :
% 0.22/0.54 ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 0.22/0.54 = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_factors
% 0.22/0.54 thf(fact_8153_gbinomial__factors,axiom,
% 0.22/0.54 ! [A: rat,K: nat] :
% 0.22/0.54 ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 0.22/0.54 = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_factors
% 0.22/0.54 thf(fact_8154_gbinomial__rec,axiom,
% 0.22/0.54 ! [A: complex,K: nat] :
% 0.22/0.54 ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 0.22/0.54 = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).
% 0.22/0.54
% 0.22/0.54 % gbinomial_rec
% 0.22/0.55 thf(fact_8155_gbinomial__rec,axiom,
% 0.22/0.55 ! [A: real,K: nat] :
% 0.22/0.55 ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 0.22/0.55 = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_rec
% 0.22/0.55 thf(fact_8156_gbinomial__rec,axiom,
% 0.22/0.55 ! [A: rat,K: nat] :
% 0.22/0.55 ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 0.22/0.55 = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_rec
% 0.22/0.55 thf(fact_8157_gbinomial__index__swap,axiom,
% 0.22/0.55 ! [K: nat,N: nat] :
% 0.22/0.55 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ K ) )
% 0.22/0.55 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_index_swap
% 0.22/0.55 thf(fact_8158_gbinomial__index__swap,axiom,
% 0.22/0.55 ! [K: nat,N: nat] :
% 0.22/0.55 ( ( 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 ) )
% 0.22/0.55 = ( 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 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_index_swap
% 0.22/0.55 thf(fact_8159_gbinomial__index__swap,axiom,
% 0.22/0.55 ! [K: nat,N: nat] :
% 0.22/0.55 ( ( 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 ) )
% 0.22/0.55 = ( 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 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_index_swap
% 0.22/0.55 thf(fact_8160_gbinomial__negated__upper,axiom,
% 0.22/0.55 ( gbinomial_complex
% 0.22/0.55 = ( ^ [A3: complex,K3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A3 ) @ one_one_complex ) @ K3 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_negated_upper
% 0.22/0.55 thf(fact_8161_gbinomial__negated__upper,axiom,
% 0.22/0.55 ( gbinomial_real
% 0.22/0.55 = ( ^ [A3: 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 ) @ A3 ) @ one_one_real ) @ K3 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_negated_upper
% 0.22/0.55 thf(fact_8162_gbinomial__negated__upper,axiom,
% 0.22/0.55 ( gbinomial_rat
% 0.22/0.55 = ( ^ [A3: 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 ) @ A3 ) @ one_one_rat ) @ K3 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_negated_upper
% 0.22/0.55 thf(fact_8163_exp__plus__inverse__exp,axiom,
% 0.22/0.55 ! [X: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % exp_plus_inverse_exp
% 0.22/0.55 thf(fact_8164_le__mult__floor__Ints,axiom,
% 0.22/0.55 ! [A: real,B: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.55 => ( ( member_real @ A @ ring_1_Ints_real )
% 0.22/0.55 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_rat @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % le_mult_floor_Ints
% 0.22/0.55 thf(fact_8165_le__mult__floor__Ints,axiom,
% 0.22/0.55 ! [A: real,B: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.55 => ( ( member_real @ A @ ring_1_Ints_real )
% 0.22/0.55 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_int @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % le_mult_floor_Ints
% 0.22/0.55 thf(fact_8166_le__mult__floor__Ints,axiom,
% 0.22/0.55 ! [A: real,B: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.55 => ( ( member_real @ A @ ring_1_Ints_real )
% 0.22/0.55 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % le_mult_floor_Ints
% 0.22/0.55 thf(fact_8167_le__mult__floor__Ints,axiom,
% 0.22/0.55 ! [A: rat,B: rat] :
% 0.22/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.22/0.55 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.22/0.55 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % le_mult_floor_Ints
% 0.22/0.55 thf(fact_8168_le__mult__floor__Ints,axiom,
% 0.22/0.55 ! [A: rat,B: rat] :
% 0.22/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.22/0.55 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.22/0.55 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_int @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % le_mult_floor_Ints
% 0.22/0.55 thf(fact_8169_le__mult__floor__Ints,axiom,
% 0.22/0.55 ! [A: rat,B: rat] :
% 0.22/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.22/0.55 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.22/0.55 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_real @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % le_mult_floor_Ints
% 0.22/0.55 thf(fact_8170_frac__unique__iff,axiom,
% 0.22/0.55 ! [X: rat,A: rat] :
% 0.22/0.55 ( ( ( archimedean_frac_rat @ X )
% 0.22/0.55 = A )
% 0.22/0.55 = ( ( member_rat @ ( minus_minus_rat @ X @ A ) @ ring_1_Ints_rat )
% 0.22/0.55 & ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.22/0.55 & ( ord_less_rat @ A @ one_one_rat ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % frac_unique_iff
% 0.22/0.55 thf(fact_8171_frac__unique__iff,axiom,
% 0.22/0.55 ! [X: real,A: real] :
% 0.22/0.55 ( ( ( archim2898591450579166408c_real @ X )
% 0.22/0.55 = A )
% 0.22/0.55 = ( ( member_real @ ( minus_minus_real @ X @ A ) @ ring_1_Ints_real )
% 0.22/0.55 & ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.55 & ( ord_less_real @ A @ one_one_real ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % frac_unique_iff
% 0.22/0.55 thf(fact_8172_mult__ceiling__le__Ints,axiom,
% 0.22/0.55 ! [A: rat,B: rat] :
% 0.22/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.22/0.55 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.22/0.55 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % mult_ceiling_le_Ints
% 0.22/0.55 thf(fact_8173_mult__ceiling__le__Ints,axiom,
% 0.22/0.55 ! [A: rat,B: rat] :
% 0.22/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.22/0.55 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.22/0.55 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % mult_ceiling_le_Ints
% 0.22/0.55 thf(fact_8174_mult__ceiling__le__Ints,axiom,
% 0.22/0.55 ! [A: rat,B: rat] :
% 0.22/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.22/0.55 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.22/0.55 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % mult_ceiling_le_Ints
% 0.22/0.55 thf(fact_8175_mult__ceiling__le__Ints,axiom,
% 0.22/0.55 ! [A: real,B: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.55 => ( ( member_real @ A @ ring_1_Ints_real )
% 0.22/0.55 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % mult_ceiling_le_Ints
% 0.22/0.55 thf(fact_8176_mult__ceiling__le__Ints,axiom,
% 0.22/0.55 ! [A: real,B: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.55 => ( ( member_real @ A @ ring_1_Ints_real )
% 0.22/0.55 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % mult_ceiling_le_Ints
% 0.22/0.55 thf(fact_8177_mult__ceiling__le__Ints,axiom,
% 0.22/0.55 ! [A: real,B: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.55 => ( ( member_real @ A @ ring_1_Ints_real )
% 0.22/0.55 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % mult_ceiling_le_Ints
% 0.22/0.55 thf(fact_8178_gbinomial__minus,axiom,
% 0.22/0.55 ! [A: complex,K: nat] :
% 0.22/0.55 ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
% 0.22/0.55 = ( 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 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_minus
% 0.22/0.55 thf(fact_8179_gbinomial__minus,axiom,
% 0.22/0.55 ! [A: real,K: nat] :
% 0.22/0.55 ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
% 0.22/0.55 = ( 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 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_minus
% 0.22/0.55 thf(fact_8180_gbinomial__minus,axiom,
% 0.22/0.55 ! [A: rat,K: nat] :
% 0.22/0.55 ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
% 0.22/0.55 = ( 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 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_minus
% 0.22/0.55 thf(fact_8181_plus__inverse__ge__2,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.55 => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % plus_inverse_ge_2
% 0.22/0.55 thf(fact_8182_real__inv__sqrt__pow2,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.55 => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.55 = ( inverse_inverse_real @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % real_inv_sqrt_pow2
% 0.22/0.55 thf(fact_8183_gbinomial__reduce__nat,axiom,
% 0.22/0.55 ! [K: nat,A: complex] :
% 0.22/0.55 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.22/0.55 => ( ( gbinomial_complex @ A @ K )
% 0.22/0.55 = ( 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 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_reduce_nat
% 0.22/0.55 thf(fact_8184_gbinomial__reduce__nat,axiom,
% 0.22/0.55 ! [K: nat,A: real] :
% 0.22/0.55 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.22/0.55 => ( ( gbinomial_real @ A @ K )
% 0.22/0.55 = ( 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 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_reduce_nat
% 0.22/0.55 thf(fact_8185_gbinomial__reduce__nat,axiom,
% 0.22/0.55 ! [K: nat,A: rat] :
% 0.22/0.55 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.22/0.55 => ( ( gbinomial_rat @ A @ K )
% 0.22/0.55 = ( 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 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_reduce_nat
% 0.22/0.55 thf(fact_8186_gbinomial__pochhammer,axiom,
% 0.22/0.55 ( gbinomial_rat
% 0.22/0.55 = ( ^ [A3: 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 @ A3 ) @ K3 ) ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_pochhammer
% 0.22/0.55 thf(fact_8187_gbinomial__pochhammer,axiom,
% 0.22/0.55 ( gbinomial_real
% 0.22/0.55 = ( ^ [A3: 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 @ A3 ) @ K3 ) ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_pochhammer
% 0.22/0.55 thf(fact_8188_gbinomial__pochhammer,axiom,
% 0.22/0.55 ( gbinomial_complex
% 0.22/0.55 = ( ^ [A3: complex,K3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A3 ) @ K3 ) ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_pochhammer
% 0.22/0.55 thf(fact_8189_gbinomial__pochhammer_H,axiom,
% 0.22/0.55 ( gbinomial_rat
% 0.22/0.55 = ( ^ [A3: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_pochhammer'
% 0.22/0.55 thf(fact_8190_gbinomial__pochhammer_H,axiom,
% 0.22/0.55 ( gbinomial_real
% 0.22/0.55 = ( ^ [A3: real,K3: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ K3 ) ) @ one_one_real ) @ K3 ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_pochhammer'
% 0.22/0.55 thf(fact_8191_gbinomial__pochhammer_H,axiom,
% 0.22/0.55 ( gbinomial_complex
% 0.22/0.55 = ( ^ [A3: complex,K3: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ K3 ) ) @ one_one_complex ) @ K3 ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % gbinomial_pochhammer'
% 0.22/0.55 thf(fact_8192_tan__cot,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 0.22/0.55 = ( inverse_inverse_real @ ( tan_real @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % tan_cot
% 0.22/0.55 thf(fact_8193_sin__integer__2pi,axiom,
% 0.22/0.55 ! [N: real] :
% 0.22/0.55 ( ( member_real @ N @ ring_1_Ints_real )
% 0.22/0.55 => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.22/0.55 = zero_zero_real ) ) ).
% 0.22/0.55
% 0.22/0.55 % sin_integer_2pi
% 0.22/0.55 thf(fact_8194_cos__integer__2pi,axiom,
% 0.22/0.55 ! [N: real] :
% 0.22/0.55 ( ( member_real @ N @ ring_1_Ints_real )
% 0.22/0.55 => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.22/0.55 = one_one_real ) ) ).
% 0.22/0.55
% 0.22/0.55 % cos_integer_2pi
% 0.22/0.55 thf(fact_8195_xor__nat__unfold,axiom,
% 0.22/0.55 ( bit_se6528837805403552850or_nat
% 0.22/0.55 = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M5 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % xor_nat_unfold
% 0.22/0.55 thf(fact_8196_real__le__x__sinh,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.55 => ( ord_less_eq_real @ X @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % real_le_x_sinh
% 0.22/0.55 thf(fact_8197_xor__nat__rec,axiom,
% 0.22/0.55 ( bit_se6528837805403552850or_nat
% 0.22/0.55 = ( ^ [M5: nat,N4: nat] :
% 0.22/0.55 ( plus_plus_nat
% 0.22/0.55 @ ( zero_n2687167440665602831ol_nat
% 0.22/0.55 @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 0.22/0.55 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
% 0.22/0.55 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % xor_nat_rec
% 0.22/0.55 thf(fact_8198_one__xor__eq,axiom,
% 0.22/0.55 ! [A: code_integer] :
% 0.22/0.55 ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ A )
% 0.22/0.55 = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 0.22/0.55 @ ( zero_n356916108424825756nteger
% 0.22/0.55 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % one_xor_eq
% 0.22/0.55 thf(fact_8199_one__xor__eq,axiom,
% 0.22/0.55 ! [A: nat] :
% 0.22/0.55 ( ( bit_se6528837805403552850or_nat @ one_one_nat @ A )
% 0.22/0.55 = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 0.22/0.55 @ ( zero_n2687167440665602831ol_nat
% 0.22/0.55 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % one_xor_eq
% 0.22/0.55 thf(fact_8200_one__xor__eq,axiom,
% 0.22/0.55 ! [A: int] :
% 0.22/0.55 ( ( bit_se6526347334894502574or_int @ one_one_int @ A )
% 0.22/0.55 = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 0.22/0.55 @ ( zero_n2684676970156552555ol_int
% 0.22/0.55 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % one_xor_eq
% 0.22/0.55 thf(fact_8201_xor__one__eq,axiom,
% 0.22/0.55 ! [A: code_integer] :
% 0.22/0.55 ( ( bit_se3222712562003087583nteger @ A @ one_one_Code_integer )
% 0.22/0.55 = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 0.22/0.55 @ ( zero_n356916108424825756nteger
% 0.22/0.55 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % xor_one_eq
% 0.22/0.55 thf(fact_8202_xor__one__eq,axiom,
% 0.22/0.55 ! [A: nat] :
% 0.22/0.55 ( ( bit_se6528837805403552850or_nat @ A @ one_one_nat )
% 0.22/0.55 = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 0.22/0.55 @ ( zero_n2687167440665602831ol_nat
% 0.22/0.55 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % xor_one_eq
% 0.22/0.55 thf(fact_8203_xor__one__eq,axiom,
% 0.22/0.55 ! [A: int] :
% 0.22/0.55 ( ( bit_se6526347334894502574or_int @ A @ one_one_int )
% 0.22/0.55 = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 0.22/0.55 @ ( zero_n2684676970156552555ol_int
% 0.22/0.55 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % xor_one_eq
% 0.22/0.55 thf(fact_8204_real__le__abs__sinh,axiom,
% 0.22/0.55 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % real_le_abs_sinh
% 0.22/0.55 thf(fact_8205_tan__sec,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( ( cos_real @ X )
% 0.22/0.55 != zero_zero_real )
% 0.22/0.55 => ( ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.55 = ( power_power_real @ ( inverse_inverse_real @ ( cos_real @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % tan_sec
% 0.22/0.55 thf(fact_8206_tan__sec,axiom,
% 0.22/0.55 ! [X: complex] :
% 0.22/0.55 ( ( ( cos_complex @ X )
% 0.22/0.55 != zero_zero_complex )
% 0.22/0.55 => ( ( plus_plus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.55 = ( power_power_complex @ ( invers8013647133539491842omplex @ ( cos_complex @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % tan_sec
% 0.22/0.55 thf(fact_8207_sinh__ln__real,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.55 => ( ( sinh_real @ ( ln_ln_real @ X ) )
% 0.22/0.55 = ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_ln_real
% 0.22/0.55 thf(fact_8208_cosh__ln__real,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.55 => ( ( cosh_real @ ( ln_ln_real @ X ) )
% 0.22/0.55 = ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_ln_real
% 0.22/0.55 thf(fact_8209_horner__sum__of__bool__2__less,axiom,
% 0.22/0.55 ! [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 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % horner_sum_of_bool_2_less
% 0.22/0.55 thf(fact_8210_cosh__zero__iff,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( ( cosh_real @ X )
% 0.22/0.55 = zero_zero_real )
% 0.22/0.55 = ( ( power_power_real @ ( exp_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.55 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_zero_iff
% 0.22/0.55 thf(fact_8211_cosh__zero__iff,axiom,
% 0.22/0.55 ! [X: complex] :
% 0.22/0.55 ( ( ( cosh_complex @ X )
% 0.22/0.55 = zero_zero_complex )
% 0.22/0.55 = ( ( power_power_complex @ ( exp_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.55 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_zero_iff
% 0.22/0.55 thf(fact_8212_push__bit__numeral__minus__1,axiom,
% 0.22/0.55 ! [N: num] :
% 0.22/0.55 ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.55 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_numeral_minus_1
% 0.22/0.55 thf(fact_8213_push__bit__numeral__minus__1,axiom,
% 0.22/0.55 ! [N: num] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ N ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.55 = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_numeral_minus_1
% 0.22/0.55 thf(fact_8214_push__bit__nonnegative__int__iff,axiom,
% 0.22/0.55 ! [N: nat,K: int] :
% 0.22/0.55 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
% 0.22/0.55 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_nonnegative_int_iff
% 0.22/0.55 thf(fact_8215_push__bit__negative__int__iff,axiom,
% 0.22/0.55 ! [N: nat,K: int] :
% 0.22/0.55 ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
% 0.22/0.55 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_negative_int_iff
% 0.22/0.55 thf(fact_8216_push__bit__of__0,axiom,
% 0.22/0.55 ! [N: nat] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ N @ zero_zero_int )
% 0.22/0.55 = zero_zero_int ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_of_0
% 0.22/0.55 thf(fact_8217_push__bit__of__0,axiom,
% 0.22/0.55 ! [N: nat] :
% 0.22/0.55 ( ( bit_se547839408752420682it_nat @ N @ zero_zero_nat )
% 0.22/0.55 = zero_zero_nat ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_of_0
% 0.22/0.55 thf(fact_8218_push__bit__eq__0__iff,axiom,
% 0.22/0.55 ! [N: nat,A: int] :
% 0.22/0.55 ( ( ( bit_se545348938243370406it_int @ N @ A )
% 0.22/0.55 = zero_zero_int )
% 0.22/0.55 = ( A = zero_zero_int ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_eq_0_iff
% 0.22/0.55 thf(fact_8219_push__bit__eq__0__iff,axiom,
% 0.22/0.55 ! [N: nat,A: nat] :
% 0.22/0.55 ( ( ( bit_se547839408752420682it_nat @ N @ A )
% 0.22/0.55 = zero_zero_nat )
% 0.22/0.55 = ( A = zero_zero_nat ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_eq_0_iff
% 0.22/0.55 thf(fact_8220_sinh__0,axiom,
% 0.22/0.55 ( ( sinh_complex @ zero_zero_complex )
% 0.22/0.55 = zero_zero_complex ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_0
% 0.22/0.55 thf(fact_8221_sinh__0,axiom,
% 0.22/0.55 ( ( sinh_real @ zero_zero_real )
% 0.22/0.55 = zero_zero_real ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_0
% 0.22/0.55 thf(fact_8222_push__bit__push__bit,axiom,
% 0.22/0.55 ! [M: nat,N: nat,A: int] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
% 0.22/0.55 = ( bit_se545348938243370406it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_push_bit
% 0.22/0.55 thf(fact_8223_push__bit__push__bit,axiom,
% 0.22/0.55 ! [M: nat,N: nat,A: nat] :
% 0.22/0.55 ( ( bit_se547839408752420682it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 0.22/0.55 = ( bit_se547839408752420682it_nat @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_push_bit
% 0.22/0.55 thf(fact_8224_push__bit__and,axiom,
% 0.22/0.55 ! [N: nat,A: int,B: int] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 0.22/0.55 = ( bit_se725231765392027082nd_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_and
% 0.22/0.55 thf(fact_8225_push__bit__and,axiom,
% 0.22/0.55 ! [N: nat,A: nat,B: nat] :
% 0.22/0.55 ( ( bit_se547839408752420682it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 0.22/0.55 = ( bit_se727722235901077358nd_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_and
% 0.22/0.55 thf(fact_8226_push__bit__xor,axiom,
% 0.22/0.55 ! [N: nat,A: int,B: int] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B ) )
% 0.22/0.55 = ( bit_se6526347334894502574or_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_xor
% 0.22/0.55 thf(fact_8227_push__bit__xor,axiom,
% 0.22/0.55 ! [N: nat,A: nat,B: nat] :
% 0.22/0.55 ( ( bit_se547839408752420682it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B ) )
% 0.22/0.55 = ( bit_se6528837805403552850or_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_xor
% 0.22/0.55 thf(fact_8228_concat__bit__of__zero__1,axiom,
% 0.22/0.55 ! [N: nat,L: int] :
% 0.22/0.55 ( ( bit_concat_bit @ N @ zero_zero_int @ L )
% 0.22/0.55 = ( bit_se545348938243370406it_int @ N @ L ) ) ).
% 0.22/0.55
% 0.22/0.55 % concat_bit_of_zero_1
% 0.22/0.55 thf(fact_8229_sinh__real__zero__iff,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( ( sinh_real @ X )
% 0.22/0.55 = zero_zero_real )
% 0.22/0.55 = ( X = zero_zero_real ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_real_zero_iff
% 0.22/0.55 thf(fact_8230_sinh__real__le__iff,axiom,
% 0.22/0.55 ! [X: real,Y4: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ ( sinh_real @ X ) @ ( sinh_real @ Y4 ) )
% 0.22/0.55 = ( ord_less_eq_real @ X @ Y4 ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_real_le_iff
% 0.22/0.55 thf(fact_8231_cosh__0,axiom,
% 0.22/0.55 ( ( cosh_complex @ zero_zero_complex )
% 0.22/0.55 = one_one_complex ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_0
% 0.22/0.55 thf(fact_8232_cosh__0,axiom,
% 0.22/0.55 ( ( cosh_real @ zero_zero_real )
% 0.22/0.55 = one_one_real ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_0
% 0.22/0.55 thf(fact_8233_sinh__real__neg__iff,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( ord_less_real @ ( sinh_real @ X ) @ zero_zero_real )
% 0.22/0.55 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_real_neg_iff
% 0.22/0.55 thf(fact_8234_sinh__real__pos__iff,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X ) )
% 0.22/0.55 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_real_pos_iff
% 0.22/0.55 thf(fact_8235_sinh__real__nonpos__iff,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ ( sinh_real @ X ) @ zero_zero_real )
% 0.22/0.55 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_real_nonpos_iff
% 0.22/0.55 thf(fact_8236_sinh__real__nonneg__iff,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X ) )
% 0.22/0.55 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_real_nonneg_iff
% 0.22/0.55 thf(fact_8237_xor__nonnegative__int__iff,axiom,
% 0.22/0.55 ! [K: int,L: int] :
% 0.22/0.55 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 0.22/0.55 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.22/0.55 = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % xor_nonnegative_int_iff
% 0.22/0.55 thf(fact_8238_xor__negative__int__iff,axiom,
% 0.22/0.55 ! [K: int,L: int] :
% 0.22/0.55 ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
% 0.22/0.55 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.22/0.55 != ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % xor_negative_int_iff
% 0.22/0.55 thf(fact_8239_push__bit__Suc__numeral,axiom,
% 0.22/0.55 ! [N: nat,K: num] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( numeral_numeral_int @ K ) )
% 0.22/0.55 = ( bit_se545348938243370406it_int @ N @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_Suc_numeral
% 0.22/0.55 thf(fact_8240_push__bit__Suc__numeral,axiom,
% 0.22/0.55 ! [N: nat,K: num] :
% 0.22/0.55 ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.22/0.55 = ( bit_se547839408752420682it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_Suc_numeral
% 0.22/0.55 thf(fact_8241_push__bit__Suc__minus__numeral,axiom,
% 0.22/0.55 ! [N: nat,K: num] :
% 0.22/0.55 ( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.22/0.55 = ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_Suc_minus_numeral
% 0.22/0.55 thf(fact_8242_push__bit__Suc__minus__numeral,axiom,
% 0.22/0.55 ! [N: nat,K: num] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.22/0.55 = ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_Suc_minus_numeral
% 0.22/0.55 thf(fact_8243_push__bit__numeral,axiom,
% 0.22/0.55 ! [L: num,K: num] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ K ) )
% 0.22/0.55 = ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_numeral
% 0.22/0.55 thf(fact_8244_push__bit__numeral,axiom,
% 0.22/0.55 ! [L: num,K: num] :
% 0.22/0.55 ( ( bit_se547839408752420682it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ K ) )
% 0.22/0.55 = ( bit_se547839408752420682it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_numeral
% 0.22/0.55 thf(fact_8245_push__bit__Suc,axiom,
% 0.22/0.55 ! [N: nat,A: int] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ A )
% 0.22/0.55 = ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_Suc
% 0.22/0.55 thf(fact_8246_push__bit__Suc,axiom,
% 0.22/0.55 ! [N: nat,A: nat] :
% 0.22/0.55 ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ A )
% 0.22/0.55 = ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_Suc
% 0.22/0.55 thf(fact_8247_push__bit__of__Suc__0,axiom,
% 0.22/0.55 ! [N: nat] :
% 0.22/0.55 ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.22/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_of_Suc_0
% 0.22/0.55 thf(fact_8248_push__bit__of__1,axiom,
% 0.22/0.55 ! [N: nat] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ N @ one_one_int )
% 0.22/0.55 = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_of_1
% 0.22/0.55 thf(fact_8249_push__bit__of__1,axiom,
% 0.22/0.55 ! [N: nat] :
% 0.22/0.55 ( ( bit_se547839408752420682it_nat @ N @ one_one_nat )
% 0.22/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_of_1
% 0.22/0.55 thf(fact_8250_even__push__bit__iff,axiom,
% 0.22/0.55 ! [N: nat,A: code_integer] :
% 0.22/0.55 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se7788150548672797655nteger @ N @ A ) )
% 0.22/0.55 = ( ( N != zero_zero_nat )
% 0.22/0.55 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % even_push_bit_iff
% 0.22/0.55 thf(fact_8251_even__push__bit__iff,axiom,
% 0.22/0.55 ! [N: nat,A: int] :
% 0.22/0.55 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se545348938243370406it_int @ N @ A ) )
% 0.22/0.55 = ( ( N != zero_zero_nat )
% 0.22/0.55 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % even_push_bit_iff
% 0.22/0.55 thf(fact_8252_even__push__bit__iff,axiom,
% 0.22/0.55 ! [N: nat,A: nat] :
% 0.22/0.55 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 0.22/0.55 = ( ( N != zero_zero_nat )
% 0.22/0.55 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % even_push_bit_iff
% 0.22/0.55 thf(fact_8253_push__bit__minus__numeral,axiom,
% 0.22/0.55 ! [L: num,K: num] :
% 0.22/0.55 ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.22/0.55 = ( bit_se7788150548672797655nteger @ ( pred_numeral @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_minus_numeral
% 0.22/0.55 thf(fact_8254_push__bit__minus__numeral,axiom,
% 0.22/0.55 ! [L: num,K: num] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.22/0.55 = ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_minus_numeral
% 0.22/0.55 thf(fact_8255_flip__bit__int__def,axiom,
% 0.22/0.55 ( bit_se2159334234014336723it_int
% 0.22/0.55 = ( ^ [N4: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % flip_bit_int_def
% 0.22/0.55 thf(fact_8256_bit__xor__int__iff,axiom,
% 0.22/0.55 ! [K: int,L: int,N: nat] :
% 0.22/0.55 ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ N )
% 0.22/0.55 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.22/0.55 != ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit_xor_int_iff
% 0.22/0.55 thf(fact_8257_sinh__add,axiom,
% 0.22/0.55 ! [X: real,Y4: real] :
% 0.22/0.55 ( ( sinh_real @ ( plus_plus_real @ X @ Y4 ) )
% 0.22/0.55 = ( plus_plus_real @ ( times_times_real @ ( sinh_real @ X ) @ ( cosh_real @ Y4 ) ) @ ( times_times_real @ ( cosh_real @ X ) @ ( sinh_real @ Y4 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_add
% 0.22/0.55 thf(fact_8258_cosh__add,axiom,
% 0.22/0.55 ! [X: real,Y4: real] :
% 0.22/0.55 ( ( cosh_real @ ( plus_plus_real @ X @ Y4 ) )
% 0.22/0.55 = ( plus_plus_real @ ( times_times_real @ ( cosh_real @ X ) @ ( cosh_real @ Y4 ) ) @ ( times_times_real @ ( sinh_real @ X ) @ ( sinh_real @ Y4 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_add
% 0.22/0.55 thf(fact_8259_cosh__diff,axiom,
% 0.22/0.55 ! [X: real,Y4: real] :
% 0.22/0.55 ( ( cosh_real @ ( minus_minus_real @ X @ Y4 ) )
% 0.22/0.55 = ( minus_minus_real @ ( times_times_real @ ( cosh_real @ X ) @ ( cosh_real @ Y4 ) ) @ ( times_times_real @ ( sinh_real @ X ) @ ( sinh_real @ Y4 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_diff
% 0.22/0.55 thf(fact_8260_sinh__diff,axiom,
% 0.22/0.55 ! [X: real,Y4: real] :
% 0.22/0.55 ( ( sinh_real @ ( minus_minus_real @ X @ Y4 ) )
% 0.22/0.55 = ( minus_minus_real @ ( times_times_real @ ( sinh_real @ X ) @ ( cosh_real @ Y4 ) ) @ ( times_times_real @ ( cosh_real @ X ) @ ( sinh_real @ Y4 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_diff
% 0.22/0.55 thf(fact_8261_cosh__real__nonzero,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( cosh_real @ X )
% 0.22/0.55 != zero_zero_real ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_real_nonzero
% 0.22/0.55 thf(fact_8262_push__bit__nat__eq,axiom,
% 0.22/0.55 ! [N: nat,K: int] :
% 0.22/0.55 ( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K ) )
% 0.22/0.55 = ( nat2 @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_nat_eq
% 0.22/0.55 thf(fact_8263_sinh__le__cosh__real,axiom,
% 0.22/0.55 ! [X: real] : ( ord_less_eq_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_le_cosh_real
% 0.22/0.55 thf(fact_8264_push__bit__of__nat,axiom,
% 0.22/0.55 ! [N: nat,M: nat] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 0.22/0.55 = ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_of_nat
% 0.22/0.55 thf(fact_8265_push__bit__of__nat,axiom,
% 0.22/0.55 ! [N: nat,M: nat] :
% 0.22/0.55 ( ( bit_se547839408752420682it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 0.22/0.55 = ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_of_nat
% 0.22/0.55 thf(fact_8266_of__nat__push__bit,axiom,
% 0.22/0.55 ! [M: nat,N: nat] :
% 0.22/0.55 ( ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 0.22/0.55 = ( bit_se545348938243370406it_int @ M @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % of_nat_push_bit
% 0.22/0.55 thf(fact_8267_of__nat__push__bit,axiom,
% 0.22/0.55 ! [M: nat,N: nat] :
% 0.22/0.55 ( ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 0.22/0.55 = ( bit_se547839408752420682it_nat @ M @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % of_nat_push_bit
% 0.22/0.55 thf(fact_8268_push__bit__minus,axiom,
% 0.22/0.55 ! [N: nat,A: code_integer] :
% 0.22/0.55 ( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ A ) )
% 0.22/0.55 = ( uminus1351360451143612070nteger @ ( bit_se7788150548672797655nteger @ N @ A ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_minus
% 0.22/0.55 thf(fact_8269_push__bit__minus,axiom,
% 0.22/0.55 ! [N: nat,A: int] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ A ) )
% 0.22/0.55 = ( uminus_uminus_int @ ( bit_se545348938243370406it_int @ N @ A ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_minus
% 0.22/0.55 thf(fact_8270_push__bit__add,axiom,
% 0.22/0.55 ! [N: nat,A: int,B: int] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ N @ ( plus_plus_int @ A @ B ) )
% 0.22/0.55 = ( plus_plus_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_add
% 0.22/0.55 thf(fact_8271_push__bit__add,axiom,
% 0.22/0.55 ! [N: nat,A: nat,B: nat] :
% 0.22/0.55 ( ( bit_se547839408752420682it_nat @ N @ ( plus_plus_nat @ A @ B ) )
% 0.22/0.55 = ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_add
% 0.22/0.55 thf(fact_8272_push__bit__of__int,axiom,
% 0.22/0.55 ! [N: nat,K: int] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ N @ ( ring_1_of_int_int @ K ) )
% 0.22/0.55 = ( ring_1_of_int_int @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_of_int
% 0.22/0.55 thf(fact_8273_tanh__def,axiom,
% 0.22/0.55 ( tanh_complex
% 0.22/0.55 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( sinh_complex @ X3 ) @ ( cosh_complex @ X3 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % tanh_def
% 0.22/0.55 thf(fact_8274_tanh__def,axiom,
% 0.22/0.55 ( tanh_real
% 0.22/0.55 = ( ^ [X3: real] : ( divide_divide_real @ ( sinh_real @ X3 ) @ ( cosh_real @ X3 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % tanh_def
% 0.22/0.55 thf(fact_8275_sinh__minus__cosh,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( minus_minus_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) )
% 0.22/0.55 = ( uminus_uminus_real @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_minus_cosh
% 0.22/0.55 thf(fact_8276_sinh__minus__cosh,axiom,
% 0.22/0.55 ! [X: complex] :
% 0.22/0.55 ( ( minus_minus_complex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) )
% 0.22/0.55 = ( uminus1482373934393186551omplex @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_minus_cosh
% 0.22/0.55 thf(fact_8277_cosh__minus__sinh,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( minus_minus_real @ ( cosh_real @ X ) @ ( sinh_real @ X ) )
% 0.22/0.55 = ( exp_real @ ( uminus_uminus_real @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_minus_sinh
% 0.22/0.55 thf(fact_8278_cosh__minus__sinh,axiom,
% 0.22/0.55 ! [X: complex] :
% 0.22/0.55 ( ( minus_minus_complex @ ( cosh_complex @ X ) @ ( sinh_complex @ X ) )
% 0.22/0.55 = ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_minus_sinh
% 0.22/0.55 thf(fact_8279_XOR__lower,axiom,
% 0.22/0.55 ! [X: int,Y4: int] :
% 0.22/0.55 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.22/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.22/0.55 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y4 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % XOR_lower
% 0.22/0.55 thf(fact_8280_cosh__real__pos,axiom,
% 0.22/0.55 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_real_pos
% 0.22/0.55 thf(fact_8281_cosh__real__nonpos__le__iff,axiom,
% 0.22/0.55 ! [X: real,Y4: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.22/0.55 => ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
% 0.22/0.55 => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y4 ) )
% 0.22/0.55 = ( ord_less_eq_real @ Y4 @ X ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_real_nonpos_le_iff
% 0.22/0.55 thf(fact_8282_cosh__real__nonneg__le__iff,axiom,
% 0.22/0.55 ! [X: real,Y4: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.55 => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y4 ) )
% 0.22/0.55 = ( ord_less_eq_real @ X @ Y4 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_real_nonneg_le_iff
% 0.22/0.55 thf(fact_8283_cosh__real__nonneg,axiom,
% 0.22/0.55 ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_real_nonneg
% 0.22/0.55 thf(fact_8284_cosh__real__ge__1,axiom,
% 0.22/0.55 ! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_real_ge_1
% 0.22/0.55 thf(fact_8285_sinh__double,axiom,
% 0.22/0.55 ! [X: complex] :
% 0.22/0.55 ( ( sinh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.22/0.55 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sinh_complex @ X ) ) @ ( cosh_complex @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_double
% 0.22/0.55 thf(fact_8286_sinh__double,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( sinh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.22/0.55 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sinh_real @ X ) ) @ ( cosh_real @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_double
% 0.22/0.55 thf(fact_8287_push__bit__take__bit,axiom,
% 0.22/0.55 ! [M: nat,N: nat,A: int] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.22/0.55 = ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ M @ A ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_take_bit
% 0.22/0.55 thf(fact_8288_push__bit__take__bit,axiom,
% 0.22/0.55 ! [M: nat,N: nat,A: nat] :
% 0.22/0.55 ( ( bit_se547839408752420682it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 0.22/0.55 = ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N ) @ ( bit_se547839408752420682it_nat @ M @ A ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_take_bit
% 0.22/0.55 thf(fact_8289_take__bit__push__bit,axiom,
% 0.22/0.55 ! [M: nat,N: nat,A: int] :
% 0.22/0.55 ( ( bit_se2923211474154528505it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
% 0.22/0.55 = ( bit_se545348938243370406it_int @ N @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % take_bit_push_bit
% 0.22/0.55 thf(fact_8290_take__bit__push__bit,axiom,
% 0.22/0.55 ! [M: nat,N: nat,A: nat] :
% 0.22/0.55 ( ( bit_se2925701944663578781it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 0.22/0.55 = ( bit_se547839408752420682it_nat @ N @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % take_bit_push_bit
% 0.22/0.55 thf(fact_8291_flip__bit__nat__def,axiom,
% 0.22/0.55 ( bit_se2161824704523386999it_nat
% 0.22/0.55 = ( ^ [M5: nat,N4: nat] : ( bit_se6528837805403552850or_nat @ N4 @ ( bit_se547839408752420682it_nat @ M5 @ one_one_nat ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % flip_bit_nat_def
% 0.22/0.55 thf(fact_8292_cosh__real__nonpos__less__iff,axiom,
% 0.22/0.55 ! [X: real,Y4: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.22/0.55 => ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
% 0.22/0.55 => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y4 ) )
% 0.22/0.55 = ( ord_less_real @ Y4 @ X ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_real_nonpos_less_iff
% 0.22/0.55 thf(fact_8293_cosh__real__nonneg__less__iff,axiom,
% 0.22/0.55 ! [X: real,Y4: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.55 => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y4 ) )
% 0.22/0.55 = ( ord_less_real @ X @ Y4 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_real_nonneg_less_iff
% 0.22/0.55 thf(fact_8294_cosh__real__strict__mono,axiom,
% 0.22/0.55 ! [X: real,Y4: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.55 => ( ( ord_less_real @ X @ Y4 )
% 0.22/0.55 => ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y4 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_real_strict_mono
% 0.22/0.55 thf(fact_8295_bit__push__bit__iff__int,axiom,
% 0.22/0.55 ! [M: nat,K: int,N: nat] :
% 0.22/0.55 ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
% 0.22/0.55 = ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.55 & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit_push_bit_iff_int
% 0.22/0.55 thf(fact_8296_cosh__square__eq,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.55 = ( plus_plus_real @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_square_eq
% 0.22/0.55 thf(fact_8297_cosh__square__eq,axiom,
% 0.22/0.55 ! [X: complex] :
% 0.22/0.55 ( ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.55 = ( plus_plus_complex @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_square_eq
% 0.22/0.55 thf(fact_8298_sinh__square__eq,axiom,
% 0.22/0.55 ! [X: complex] :
% 0.22/0.55 ( ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.55 = ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_square_eq
% 0.22/0.55 thf(fact_8299_sinh__square__eq,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.55 = ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_square_eq
% 0.22/0.55 thf(fact_8300_hyperbolic__pythagoras,axiom,
% 0.22/0.55 ! [X: complex] :
% 0.22/0.55 ( ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.55 = one_one_complex ) ).
% 0.22/0.55
% 0.22/0.55 % hyperbolic_pythagoras
% 0.22/0.55 thf(fact_8301_hyperbolic__pythagoras,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.55 = one_one_real ) ).
% 0.22/0.55
% 0.22/0.55 % hyperbolic_pythagoras
% 0.22/0.55 thf(fact_8302_xor__nat__def,axiom,
% 0.22/0.55 ( bit_se6528837805403552850or_nat
% 0.22/0.55 = ( ^ [M5: nat,N4: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % xor_nat_def
% 0.22/0.55 thf(fact_8303_bit__push__bit__iff__nat,axiom,
% 0.22/0.55 ! [M: nat,Q2: nat,N: nat] :
% 0.22/0.55 ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N )
% 0.22/0.55 = ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.55 & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit_push_bit_iff_nat
% 0.22/0.55 thf(fact_8304_arcosh__cosh__real,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.55 => ( ( arcosh_real @ ( cosh_real @ X ) )
% 0.22/0.55 = X ) ) ).
% 0.22/0.55
% 0.22/0.55 % arcosh_cosh_real
% 0.22/0.55 thf(fact_8305_concat__bit__eq,axiom,
% 0.22/0.55 ( bit_concat_bit
% 0.22/0.55 = ( ^ [N4: nat,K3: int,L3: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N4 @ K3 ) @ ( bit_se545348938243370406it_int @ N4 @ L3 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % concat_bit_eq
% 0.22/0.55 thf(fact_8306_flip__bit__eq__xor,axiom,
% 0.22/0.55 ( bit_se1345352211410354436nteger
% 0.22/0.55 = ( ^ [N4: nat,A3: code_integer] : ( bit_se3222712562003087583nteger @ A3 @ ( bit_se7788150548672797655nteger @ N4 @ one_one_Code_integer ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % flip_bit_eq_xor
% 0.22/0.55 thf(fact_8307_flip__bit__eq__xor,axiom,
% 0.22/0.55 ( bit_se2159334234014336723it_int
% 0.22/0.55 = ( ^ [N4: nat,A3: int] : ( bit_se6526347334894502574or_int @ A3 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % flip_bit_eq_xor
% 0.22/0.55 thf(fact_8308_flip__bit__eq__xor,axiom,
% 0.22/0.55 ( bit_se2161824704523386999it_nat
% 0.22/0.55 = ( ^ [N4: nat,A3: nat] : ( bit_se6528837805403552850or_nat @ A3 @ ( bit_se547839408752420682it_nat @ N4 @ one_one_nat ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % flip_bit_eq_xor
% 0.22/0.55 thf(fact_8309_cosh__double,axiom,
% 0.22/0.55 ! [X: complex] :
% 0.22/0.55 ( ( cosh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.22/0.55 = ( plus_plus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_double
% 0.22/0.55 thf(fact_8310_cosh__double,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( cosh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.22/0.55 = ( plus_plus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_double
% 0.22/0.55 thf(fact_8311_push__bit__double,axiom,
% 0.22/0.55 ! [N: nat,A: int] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.22/0.55 = ( times_times_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_double
% 0.22/0.55 thf(fact_8312_push__bit__double,axiom,
% 0.22/0.55 ! [N: nat,A: nat] :
% 0.22/0.55 ( ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.55 = ( times_times_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_double
% 0.22/0.55 thf(fact_8313_bit__iff__and__push__bit__not__eq__0,axiom,
% 0.22/0.55 ( bit_se1146084159140164899it_int
% 0.22/0.55 = ( ^ [A3: int,N4: nat] :
% 0.22/0.55 ( ( bit_se725231765392027082nd_int @ A3 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) )
% 0.22/0.55 != zero_zero_int ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit_iff_and_push_bit_not_eq_0
% 0.22/0.55 thf(fact_8314_bit__iff__and__push__bit__not__eq__0,axiom,
% 0.22/0.55 ( bit_se1148574629649215175it_nat
% 0.22/0.55 = ( ^ [A3: nat,N4: nat] :
% 0.22/0.55 ( ( bit_se727722235901077358nd_nat @ A3 @ ( bit_se547839408752420682it_nat @ N4 @ one_one_nat ) )
% 0.22/0.55 != zero_zero_nat ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit_iff_and_push_bit_not_eq_0
% 0.22/0.55 thf(fact_8315_push__bit__int__def,axiom,
% 0.22/0.55 ( bit_se545348938243370406it_int
% 0.22/0.55 = ( ^ [N4: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_int_def
% 0.22/0.55 thf(fact_8316_push__bit__nat__def,axiom,
% 0.22/0.55 ( bit_se547839408752420682it_nat
% 0.22/0.55 = ( ^ [N4: nat,M5: nat] : ( times_times_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_nat_def
% 0.22/0.55 thf(fact_8317_push__bit__eq__mult,axiom,
% 0.22/0.55 ( bit_se545348938243370406it_int
% 0.22/0.55 = ( ^ [N4: nat,A3: int] : ( times_times_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_eq_mult
% 0.22/0.55 thf(fact_8318_push__bit__eq__mult,axiom,
% 0.22/0.55 ( bit_se547839408752420682it_nat
% 0.22/0.55 = ( ^ [N4: nat,A3: nat] : ( times_times_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_eq_mult
% 0.22/0.55 thf(fact_8319_exp__dvdE,axiom,
% 0.22/0.55 ! [N: nat,A: code_integer] :
% 0.22/0.55 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A )
% 0.22/0.55 => ~ ! [B3: code_integer] :
% 0.22/0.55 ( A
% 0.22/0.55 != ( bit_se7788150548672797655nteger @ N @ B3 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % exp_dvdE
% 0.22/0.55 thf(fact_8320_exp__dvdE,axiom,
% 0.22/0.55 ! [N: nat,A: int] :
% 0.22/0.55 ( ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A )
% 0.22/0.55 => ~ ! [B3: int] :
% 0.22/0.55 ( A
% 0.22/0.55 != ( bit_se545348938243370406it_int @ N @ B3 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % exp_dvdE
% 0.22/0.55 thf(fact_8321_exp__dvdE,axiom,
% 0.22/0.55 ! [N: nat,A: nat] :
% 0.22/0.55 ( ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A )
% 0.22/0.55 => ~ ! [B3: nat] :
% 0.22/0.55 ( A
% 0.22/0.55 != ( bit_se547839408752420682it_nat @ N @ B3 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % exp_dvdE
% 0.22/0.55 thf(fact_8322_push__bit__minus__one,axiom,
% 0.22/0.55 ! [N: nat] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.55 = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_minus_one
% 0.22/0.55 thf(fact_8323_tanh__add,axiom,
% 0.22/0.55 ! [X: complex,Y4: complex] :
% 0.22/0.55 ( ( ( cosh_complex @ X )
% 0.22/0.55 != zero_zero_complex )
% 0.22/0.55 => ( ( ( cosh_complex @ Y4 )
% 0.22/0.55 != zero_zero_complex )
% 0.22/0.55 => ( ( tanh_complex @ ( plus_plus_complex @ X @ Y4 ) )
% 0.22/0.55 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y4 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y4 ) ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % tanh_add
% 0.22/0.55 thf(fact_8324_tanh__add,axiom,
% 0.22/0.55 ! [X: real,Y4: real] :
% 0.22/0.55 ( ( ( cosh_real @ X )
% 0.22/0.55 != zero_zero_real )
% 0.22/0.55 => ( ( ( cosh_real @ Y4 )
% 0.22/0.55 != zero_zero_real )
% 0.22/0.55 => ( ( tanh_real @ ( plus_plus_real @ X @ Y4 ) )
% 0.22/0.55 = ( divide_divide_real @ ( plus_plus_real @ ( tanh_real @ X ) @ ( tanh_real @ Y4 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tanh_real @ X ) @ ( tanh_real @ Y4 ) ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % tanh_add
% 0.22/0.55 thf(fact_8325_XOR__upper,axiom,
% 0.22/0.55 ! [X: int,N: nat,Y4: int] :
% 0.22/0.55 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.22/0.55 => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.55 => ( ( ord_less_int @ Y4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.55 => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y4 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % XOR_upper
% 0.22/0.55 thf(fact_8326_cosh__field__def,axiom,
% 0.22/0.55 ( cosh_real
% 0.22/0.55 = ( ^ [Z5: real] : ( divide_divide_real @ ( plus_plus_real @ ( exp_real @ Z5 ) @ ( exp_real @ ( uminus_uminus_real @ Z5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_field_def
% 0.22/0.55 thf(fact_8327_cosh__field__def,axiom,
% 0.22/0.55 ( cosh_complex
% 0.22/0.55 = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( exp_complex @ Z5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z5 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_field_def
% 0.22/0.55 thf(fact_8328_complex__inverse,axiom,
% 0.22/0.55 ! [A: real,B: real] :
% 0.22/0.55 ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
% 0.22/0.55 = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_inverse
% 0.22/0.55 thf(fact_8329_xor__int__rec,axiom,
% 0.22/0.55 ( bit_se6526347334894502574or_int
% 0.22/0.55 = ( ^ [K3: int,L3: int] :
% 0.22/0.55 ( plus_plus_int
% 0.22/0.55 @ ( zero_n2684676970156552555ol_int
% 0.22/0.55 @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) )
% 0.22/0.55 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) ) )
% 0.22/0.55 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % xor_int_rec
% 0.22/0.55 thf(fact_8330_sinh__field__def,axiom,
% 0.22/0.55 ( sinh_real
% 0.22/0.55 = ( ^ [Z5: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ Z5 ) @ ( exp_real @ ( uminus_uminus_real @ Z5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_field_def
% 0.22/0.55 thf(fact_8331_sinh__field__def,axiom,
% 0.22/0.55 ( sinh_complex
% 0.22/0.55 = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ Z5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z5 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_field_def
% 0.22/0.55 thf(fact_8332_xor__int__unfold,axiom,
% 0.22/0.55 ( bit_se6526347334894502574or_int
% 0.22/0.55 = ( ^ [K3: int,L3: int] :
% 0.22/0.55 ( if_int
% 0.22/0.55 @ ( K3
% 0.22/0.55 = ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.55 @ ( bit_ri7919022796975470100ot_int @ L3 )
% 0.22/0.55 @ ( if_int
% 0.22/0.55 @ ( L3
% 0.22/0.55 = ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.55 @ ( bit_ri7919022796975470100ot_int @ K3 )
% 0.22/0.55 @ ( if_int @ ( K3 = zero_zero_int ) @ L3 @ ( if_int @ ( L3 = 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 @ L3 @ ( 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 @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % xor_int_unfold
% 0.22/0.55 thf(fact_8333_bit__horner__sum__bit__iff,axiom,
% 0.22/0.55 ! [Bs: list_o,N: nat] :
% 0.22/0.55 ( ( bit_se9216721137139052372nteger @ ( groups3417619833198082522nteger @ zero_n356916108424825756nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Bs ) @ N )
% 0.22/0.55 = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
% 0.22/0.55 & ( nth_o @ Bs @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit_horner_sum_bit_iff
% 0.22/0.55 thf(fact_8334_bit__horner__sum__bit__iff,axiom,
% 0.22/0.55 ! [Bs: list_o,N: nat] :
% 0.22/0.55 ( ( bit_se1148574629649215175it_nat @ ( groups9119017779487936845_o_nat @ zero_n2687167440665602831ol_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Bs ) @ N )
% 0.22/0.55 = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
% 0.22/0.55 & ( nth_o @ Bs @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit_horner_sum_bit_iff
% 0.22/0.55 thf(fact_8335_bit__horner__sum__bit__iff,axiom,
% 0.22/0.55 ! [Bs: list_o,N: nat] :
% 0.22/0.55 ( ( bit_se1146084159140164899it_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ N )
% 0.22/0.55 = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
% 0.22/0.55 & ( nth_o @ Bs @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit_horner_sum_bit_iff
% 0.22/0.55 thf(fact_8336_valid__eq2,axiom,
% 0.22/0.55 ! [T: vEBT_VEBT,D: nat] :
% 0.22/0.55 ( ( vEBT_VEBT_valid @ T @ D )
% 0.22/0.55 => ( vEBT_invar_vebt @ T @ D ) ) ).
% 0.22/0.55
% 0.22/0.55 % valid_eq2
% 0.22/0.55 thf(fact_8337_valid__eq1,axiom,
% 0.22/0.55 ! [T: vEBT_VEBT,D: nat] :
% 0.22/0.55 ( ( vEBT_invar_vebt @ T @ D )
% 0.22/0.55 => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 0.22/0.55
% 0.22/0.55 % valid_eq1
% 0.22/0.55 thf(fact_8338_valid__eq,axiom,
% 0.22/0.55 vEBT_VEBT_valid = vEBT_invar_vebt ).
% 0.22/0.55
% 0.22/0.55 % valid_eq
% 0.22/0.55 thf(fact_8339_intind,axiom,
% 0.22/0.55 ! [I: nat,N: nat,P: nat > $o,X: nat] :
% 0.22/0.55 ( ( ord_less_nat @ I @ N )
% 0.22/0.55 => ( ( P @ X )
% 0.22/0.55 => ( P @ ( nth_nat @ ( replicate_nat @ N @ X ) @ I ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % intind
% 0.22/0.55 thf(fact_8340_intind,axiom,
% 0.22/0.55 ! [I: nat,N: nat,P: int > $o,X: int] :
% 0.22/0.55 ( ( ord_less_nat @ I @ N )
% 0.22/0.55 => ( ( P @ X )
% 0.22/0.55 => ( P @ ( nth_int @ ( replicate_int @ N @ X ) @ I ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % intind
% 0.22/0.55 thf(fact_8341_intind,axiom,
% 0.22/0.55 ! [I: nat,N: nat,P: real > $o,X: real] :
% 0.22/0.55 ( ( ord_less_nat @ I @ N )
% 0.22/0.55 => ( ( P @ X )
% 0.22/0.55 => ( P @ ( nth_real @ ( replicate_real @ N @ X ) @ I ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % intind
% 0.22/0.55 thf(fact_8342_intind,axiom,
% 0.22/0.55 ! [I: nat,N: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
% 0.22/0.55 ( ( ord_less_nat @ I @ N )
% 0.22/0.55 => ( ( P @ X )
% 0.22/0.55 => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % intind
% 0.22/0.55 thf(fact_8343_inthall,axiom,
% 0.22/0.55 ! [Xs: list_complex,P: complex > $o,N: nat] :
% 0.22/0.55 ( ! [X4: complex] :
% 0.22/0.55 ( ( member_complex @ X4 @ ( set_complex2 @ Xs ) )
% 0.22/0.55 => ( P @ X4 ) )
% 0.22/0.55 => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
% 0.22/0.55 => ( P @ ( nth_complex @ Xs @ N ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % inthall
% 0.22/0.55 thf(fact_8344_inthall,axiom,
% 0.22/0.55 ! [Xs: list_set_nat,P: set_nat > $o,N: nat] :
% 0.22/0.55 ( ! [X4: set_nat] :
% 0.22/0.55 ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
% 0.22/0.55 => ( P @ X4 ) )
% 0.22/0.55 => ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.22/0.55 => ( P @ ( nth_set_nat @ Xs @ N ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % inthall
% 0.22/0.55 thf(fact_8345_inthall,axiom,
% 0.22/0.55 ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
% 0.22/0.55 ( ! [X4: vEBT_VEBT] :
% 0.22/0.55 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.22/0.55 => ( P @ X4 ) )
% 0.22/0.55 => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.22/0.55 => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % inthall
% 0.22/0.55 thf(fact_8346_inthall,axiom,
% 0.22/0.55 ! [Xs: list_real,P: real > $o,N: nat] :
% 0.22/0.55 ( ! [X4: real] :
% 0.22/0.55 ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
% 0.22/0.55 => ( P @ X4 ) )
% 0.22/0.55 => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 0.22/0.55 => ( P @ ( nth_real @ Xs @ N ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % inthall
% 0.22/0.55 thf(fact_8347_inthall,axiom,
% 0.22/0.55 ! [Xs: list_o,P: $o > $o,N: nat] :
% 0.22/0.55 ( ! [X4: $o] :
% 0.22/0.55 ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 0.22/0.55 => ( P @ X4 ) )
% 0.22/0.55 => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 0.22/0.55 => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % inthall
% 0.22/0.55 thf(fact_8348_inthall,axiom,
% 0.22/0.55 ! [Xs: list_nat,P: nat > $o,N: nat] :
% 0.22/0.55 ( ! [X4: nat] :
% 0.22/0.55 ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 0.22/0.55 => ( P @ X4 ) )
% 0.22/0.55 => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.22/0.55 => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % inthall
% 0.22/0.55 thf(fact_8349_inthall,axiom,
% 0.22/0.55 ! [Xs: list_int,P: int > $o,N: nat] :
% 0.22/0.55 ( ! [X4: int] :
% 0.22/0.55 ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 0.22/0.55 => ( P @ X4 ) )
% 0.22/0.55 => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 0.22/0.55 => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % inthall
% 0.22/0.55 thf(fact_8350_bit_Ocompl__eq__compl__iff,axiom,
% 0.22/0.55 ! [X: int,Y4: int] :
% 0.22/0.55 ( ( ( bit_ri7919022796975470100ot_int @ X )
% 0.22/0.55 = ( bit_ri7919022796975470100ot_int @ Y4 ) )
% 0.22/0.55 = ( X = Y4 ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.compl_eq_compl_iff
% 0.22/0.55 thf(fact_8351_bit_Odouble__compl,axiom,
% 0.22/0.55 ! [X: int] :
% 0.22/0.55 ( ( bit_ri7919022796975470100ot_int @ ( bit_ri7919022796975470100ot_int @ X ) )
% 0.22/0.55 = X ) ).
% 0.22/0.55
% 0.22/0.55 % bit.double_compl
% 0.22/0.55 thf(fact_8352_bit_Oxor__compl__right,axiom,
% 0.22/0.55 ! [X: int,Y4: int] :
% 0.22/0.55 ( ( bit_se6526347334894502574or_int @ X @ ( bit_ri7919022796975470100ot_int @ Y4 ) )
% 0.22/0.55 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ X @ Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.xor_compl_right
% 0.22/0.55 thf(fact_8353_bit_Oxor__compl__left,axiom,
% 0.22/0.55 ! [X: int,Y4: int] :
% 0.22/0.55 ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ Y4 )
% 0.22/0.55 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ X @ Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.xor_compl_left
% 0.22/0.55 thf(fact_8354_bit_Oconj__cancel__left,axiom,
% 0.22/0.55 ! [X: int] :
% 0.22/0.55 ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
% 0.22/0.55 = zero_zero_int ) ).
% 0.22/0.55
% 0.22/0.55 % bit.conj_cancel_left
% 0.22/0.55 thf(fact_8355_bit_Oconj__cancel__right,axiom,
% 0.22/0.55 ! [X: int] :
% 0.22/0.55 ( ( bit_se725231765392027082nd_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
% 0.22/0.55 = zero_zero_int ) ).
% 0.22/0.55
% 0.22/0.55 % bit.conj_cancel_right
% 0.22/0.55 thf(fact_8356_bit_Ocompl__zero,axiom,
% 0.22/0.55 ( ( bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger )
% 0.22/0.55 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.compl_zero
% 0.22/0.55 thf(fact_8357_bit_Ocompl__zero,axiom,
% 0.22/0.55 ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
% 0.22/0.55 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.compl_zero
% 0.22/0.55 thf(fact_8358_bit_Ocompl__one,axiom,
% 0.22/0.55 ( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.55 = zero_z3403309356797280102nteger ) ).
% 0.22/0.55
% 0.22/0.55 % bit.compl_one
% 0.22/0.55 thf(fact_8359_bit_Ocompl__one,axiom,
% 0.22/0.55 ( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.55 = zero_zero_int ) ).
% 0.22/0.55
% 0.22/0.55 % bit.compl_one
% 0.22/0.55 thf(fact_8360_bit_Oxor__one__left,axiom,
% 0.22/0.55 ! [X: code_integer] :
% 0.22/0.55 ( ( bit_se3222712562003087583nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 0.22/0.55 = ( bit_ri7632146776885996613nteger @ X ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.xor_one_left
% 0.22/0.55 thf(fact_8361_bit_Oxor__one__left,axiom,
% 0.22/0.55 ! [X: int] :
% 0.22/0.55 ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 0.22/0.55 = ( bit_ri7919022796975470100ot_int @ X ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.xor_one_left
% 0.22/0.55 thf(fact_8362_bit_Oxor__one__right,axiom,
% 0.22/0.55 ! [X: code_integer] :
% 0.22/0.55 ( ( bit_se3222712562003087583nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.55 = ( bit_ri7632146776885996613nteger @ X ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.xor_one_right
% 0.22/0.55 thf(fact_8363_bit_Oxor__one__right,axiom,
% 0.22/0.55 ! [X: int] :
% 0.22/0.55 ( ( bit_se6526347334894502574or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.55 = ( bit_ri7919022796975470100ot_int @ X ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.xor_one_right
% 0.22/0.55 thf(fact_8364_bit_Oxor__cancel__left,axiom,
% 0.22/0.55 ! [X: code_integer] :
% 0.22/0.55 ( ( bit_se3222712562003087583nteger @ ( bit_ri7632146776885996613nteger @ X ) @ X )
% 0.22/0.55 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.xor_cancel_left
% 0.22/0.55 thf(fact_8365_bit_Oxor__cancel__left,axiom,
% 0.22/0.55 ! [X: int] :
% 0.22/0.55 ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
% 0.22/0.55 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.xor_cancel_left
% 0.22/0.55 thf(fact_8366_bit_Oxor__cancel__right,axiom,
% 0.22/0.55 ! [X: code_integer] :
% 0.22/0.55 ( ( bit_se3222712562003087583nteger @ X @ ( bit_ri7632146776885996613nteger @ X ) )
% 0.22/0.55 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.xor_cancel_right
% 0.22/0.55 thf(fact_8367_bit_Oxor__cancel__right,axiom,
% 0.22/0.55 ! [X: int] :
% 0.22/0.55 ( ( bit_se6526347334894502574or_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
% 0.22/0.55 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.xor_cancel_right
% 0.22/0.55 thf(fact_8368_not__negative__int__iff,axiom,
% 0.22/0.55 ! [K: int] :
% 0.22/0.55 ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 0.22/0.55 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.22/0.55
% 0.22/0.55 % not_negative_int_iff
% 0.22/0.55 thf(fact_8369_not__nonnegative__int__iff,axiom,
% 0.22/0.55 ! [K: int] :
% 0.22/0.55 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.22/0.55 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.22/0.55
% 0.22/0.55 % not_nonnegative_int_iff
% 0.22/0.55 thf(fact_8370_minus__not__numeral__eq,axiom,
% 0.22/0.55 ! [N: num] :
% 0.22/0.55 ( ( uminus1351360451143612070nteger @ ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.22/0.55 = ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % minus_not_numeral_eq
% 0.22/0.55 thf(fact_8371_minus__not__numeral__eq,axiom,
% 0.22/0.55 ! [N: num] :
% 0.22/0.55 ( ( uminus_uminus_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.55 = ( numeral_numeral_int @ ( inc @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % minus_not_numeral_eq
% 0.22/0.55 thf(fact_8372_even__not__iff,axiom,
% 0.22/0.55 ! [A: code_integer] :
% 0.22/0.55 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri7632146776885996613nteger @ A ) )
% 0.22/0.55 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % even_not_iff
% 0.22/0.55 thf(fact_8373_even__not__iff,axiom,
% 0.22/0.55 ! [A: int] :
% 0.22/0.55 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ A ) )
% 0.22/0.55 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % even_not_iff
% 0.22/0.55 thf(fact_8374_push__bit__minus__one__eq__not__mask,axiom,
% 0.22/0.55 ! [N: nat] :
% 0.22/0.55 ( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.55 = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_minus_one_eq_not_mask
% 0.22/0.55 thf(fact_8375_push__bit__minus__one__eq__not__mask,axiom,
% 0.22/0.55 ! [N: nat] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.55 = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_minus_one_eq_not_mask
% 0.22/0.55 thf(fact_8376_not__one__eq,axiom,
% 0.22/0.55 ( ( bit_ri7632146776885996613nteger @ one_one_Code_integer )
% 0.22/0.55 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % not_one_eq
% 0.22/0.55 thf(fact_8377_not__one__eq,axiom,
% 0.22/0.55 ( ( bit_ri7919022796975470100ot_int @ one_one_int )
% 0.22/0.55 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % not_one_eq
% 0.22/0.55 thf(fact_8378_of__int__not__eq,axiom,
% 0.22/0.55 ! [K: int] :
% 0.22/0.55 ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.22/0.55 = ( bit_ri7919022796975470100ot_int @ ( ring_1_of_int_int @ K ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % of_int_not_eq
% 0.22/0.55 thf(fact_8379_take__bit__not__iff,axiom,
% 0.22/0.55 ! [N: nat,A: int,B: int] :
% 0.22/0.55 ( ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) )
% 0.22/0.55 = ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ B ) ) )
% 0.22/0.55 = ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.22/0.55 = ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % take_bit_not_iff
% 0.22/0.55 thf(fact_8380_take__bit__not__take__bit,axiom,
% 0.22/0.55 ! [N: nat,A: int] :
% 0.22/0.55 ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.22/0.55 = ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % take_bit_not_take_bit
% 0.22/0.55 thf(fact_8381_bit__not__int__iff,axiom,
% 0.22/0.55 ! [K: int,N: nat] :
% 0.22/0.55 ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N )
% 0.22/0.55 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit_not_int_iff
% 0.22/0.55 thf(fact_8382_of__int__not__numeral,axiom,
% 0.22/0.55 ! [K: num] :
% 0.22/0.55 ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) )
% 0.22/0.55 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % of_int_not_numeral
% 0.22/0.55 thf(fact_8383_not__add__distrib,axiom,
% 0.22/0.55 ! [A: int,B: int] :
% 0.22/0.55 ( ( bit_ri7919022796975470100ot_int @ ( plus_plus_int @ A @ B ) )
% 0.22/0.55 = ( minus_minus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).
% 0.22/0.55
% 0.22/0.55 % not_add_distrib
% 0.22/0.55 thf(fact_8384_not__diff__distrib,axiom,
% 0.22/0.55 ! [A: int,B: int] :
% 0.22/0.55 ( ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A @ B ) )
% 0.22/0.55 = ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).
% 0.22/0.55
% 0.22/0.55 % not_diff_distrib
% 0.22/0.55 thf(fact_8385_minus__eq__not__plus__1,axiom,
% 0.22/0.55 ( uminus1351360451143612070nteger
% 0.22/0.55 = ( ^ [A3: code_integer] : ( plus_p5714425477246183910nteger @ ( bit_ri7632146776885996613nteger @ A3 ) @ one_one_Code_integer ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % minus_eq_not_plus_1
% 0.22/0.55 thf(fact_8386_minus__eq__not__plus__1,axiom,
% 0.22/0.55 ( uminus_uminus_int
% 0.22/0.55 = ( ^ [A3: int] : ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A3 ) @ one_one_int ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % minus_eq_not_plus_1
% 0.22/0.55 thf(fact_8387_not__eq__complement,axiom,
% 0.22/0.55 ( bit_ri7632146776885996613nteger
% 0.22/0.55 = ( ^ [A3: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A3 ) @ one_one_Code_integer ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % not_eq_complement
% 0.22/0.55 thf(fact_8388_not__eq__complement,axiom,
% 0.22/0.55 ( bit_ri7919022796975470100ot_int
% 0.22/0.55 = ( ^ [A3: int] : ( minus_minus_int @ ( uminus_uminus_int @ A3 ) @ one_one_int ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % not_eq_complement
% 0.22/0.55 thf(fact_8389_minus__eq__not__minus__1,axiom,
% 0.22/0.55 ( uminus1351360451143612070nteger
% 0.22/0.55 = ( ^ [A3: code_integer] : ( bit_ri7632146776885996613nteger @ ( minus_8373710615458151222nteger @ A3 @ one_one_Code_integer ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % minus_eq_not_minus_1
% 0.22/0.55 thf(fact_8390_minus__eq__not__minus__1,axiom,
% 0.22/0.55 ( uminus_uminus_int
% 0.22/0.55 = ( ^ [A3: int] : ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A3 @ one_one_int ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % minus_eq_not_minus_1
% 0.22/0.55 thf(fact_8391_not__int__def,axiom,
% 0.22/0.55 ( bit_ri7919022796975470100ot_int
% 0.22/0.55 = ( ^ [K3: int] : ( minus_minus_int @ ( uminus_uminus_int @ K3 ) @ one_one_int ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % not_int_def
% 0.22/0.55 thf(fact_8392_and__not__numerals_I1_J,axiom,
% 0.22/0.55 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.22/0.55 = zero_zero_int ) ).
% 0.22/0.55
% 0.22/0.55 % and_not_numerals(1)
% 0.22/0.55 thf(fact_8393_disjunctive__diff,axiom,
% 0.22/0.55 ! [B: int,A: int] :
% 0.22/0.55 ( ! [N2: nat] :
% 0.22/0.55 ( ( bit_se1146084159140164899it_int @ B @ N2 )
% 0.22/0.55 => ( bit_se1146084159140164899it_int @ A @ N2 ) )
% 0.22/0.55 => ( ( minus_minus_int @ A @ B )
% 0.22/0.55 = ( bit_se725231765392027082nd_int @ A @ ( bit_ri7919022796975470100ot_int @ B ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % disjunctive_diff
% 0.22/0.55 thf(fact_8394_take__bit__not__eq__mask__diff,axiom,
% 0.22/0.55 ! [N: nat,A: int] :
% 0.22/0.55 ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) )
% 0.22/0.55 = ( minus_minus_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % take_bit_not_eq_mask_diff
% 0.22/0.55 thf(fact_8395_minus__numeral__inc__eq,axiom,
% 0.22/0.55 ! [N: num] :
% 0.22/0.55 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) )
% 0.22/0.55 = ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % minus_numeral_inc_eq
% 0.22/0.55 thf(fact_8396_minus__numeral__inc__eq,axiom,
% 0.22/0.55 ! [N: num] :
% 0.22/0.55 ( ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) )
% 0.22/0.55 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % minus_numeral_inc_eq
% 0.22/0.55 thf(fact_8397_unset__bit__int__def,axiom,
% 0.22/0.55 ( bit_se4203085406695923979it_int
% 0.22/0.55 = ( ^ [N4: nat,K3: int] : ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % unset_bit_int_def
% 0.22/0.55 thf(fact_8398_not__int__div__2,axiom,
% 0.22/0.55 ! [K: int] :
% 0.22/0.55 ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.55 = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % not_int_div_2
% 0.22/0.55 thf(fact_8399_even__not__iff__int,axiom,
% 0.22/0.55 ! [K: int] :
% 0.22/0.55 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.22/0.55 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % even_not_iff_int
% 0.22/0.55 thf(fact_8400_not__numeral__Bit0__eq,axiom,
% 0.22/0.55 ! [N: num] :
% 0.22/0.55 ( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) )
% 0.22/0.55 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % not_numeral_Bit0_eq
% 0.22/0.55 thf(fact_8401_not__numeral__Bit0__eq,axiom,
% 0.22/0.55 ! [N: num] :
% 0.22/0.55 ( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) )
% 0.22/0.55 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % not_numeral_Bit0_eq
% 0.22/0.55 thf(fact_8402_and__not__numerals_I4_J,axiom,
% 0.22/0.55 ! [M: num] :
% 0.22/0.55 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.22/0.55 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % and_not_numerals(4)
% 0.22/0.55 thf(fact_8403_and__not__numerals_I2_J,axiom,
% 0.22/0.55 ! [N: num] :
% 0.22/0.55 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.22/0.55 = one_one_int ) ).
% 0.22/0.55
% 0.22/0.55 % and_not_numerals(2)
% 0.22/0.55 thf(fact_8404_bit__minus__int__iff,axiom,
% 0.22/0.55 ! [K: int,N: nat] :
% 0.22/0.55 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
% 0.22/0.55 = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit_minus_int_iff
% 0.22/0.55 thf(fact_8405_take__bit__not__mask__eq__0,axiom,
% 0.22/0.55 ! [M: nat,N: nat] :
% 0.22/0.55 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.55 => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) )
% 0.22/0.55 = zero_zero_int ) ) ).
% 0.22/0.55
% 0.22/0.55 % take_bit_not_mask_eq_0
% 0.22/0.55 thf(fact_8406_push__bit__mask__eq,axiom,
% 0.22/0.55 ! [M: nat,N: nat] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ M @ ( bit_se2000444600071755411sk_int @ N ) )
% 0.22/0.55 = ( bit_se725231765392027082nd_int @ ( bit_se2000444600071755411sk_int @ ( plus_plus_nat @ N @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ M ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_mask_eq
% 0.22/0.55 thf(fact_8407_unset__bit__eq__and__not,axiom,
% 0.22/0.55 ( bit_se4203085406695923979it_int
% 0.22/0.55 = ( ^ [N4: nat,A3: int] : ( bit_se725231765392027082nd_int @ A3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % unset_bit_eq_and_not
% 0.22/0.55 thf(fact_8408_unset__bit__eq__and__not,axiom,
% 0.22/0.55 ( bit_se8260200283734997820nteger
% 0.22/0.55 = ( ^ [N4: nat,A3: code_integer] : ( bit_se3949692690581998587nteger @ A3 @ ( bit_ri7632146776885996613nteger @ ( bit_se7788150548672797655nteger @ N4 @ one_one_Code_integer ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % unset_bit_eq_and_not
% 0.22/0.55 thf(fact_8409_and__not__numerals_I5_J,axiom,
% 0.22/0.55 ! [M: num,N: num] :
% 0.22/0.55 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.22/0.55 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % and_not_numerals(5)
% 0.22/0.55 thf(fact_8410_and__not__numerals_I7_J,axiom,
% 0.22/0.55 ! [M: num] :
% 0.22/0.55 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.22/0.55 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % and_not_numerals(7)
% 0.22/0.55 thf(fact_8411_and__not__numerals_I3_J,axiom,
% 0.22/0.55 ! [N: num] :
% 0.22/0.55 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.22/0.55 = zero_zero_int ) ).
% 0.22/0.55
% 0.22/0.55 % and_not_numerals(3)
% 0.22/0.55 thf(fact_8412_and__not__numerals_I6_J,axiom,
% 0.22/0.55 ! [M: num,N: num] :
% 0.22/0.55 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.22/0.55 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % and_not_numerals(6)
% 0.22/0.55 thf(fact_8413_and__not__numerals_I9_J,axiom,
% 0.22/0.55 ! [M: num,N: num] :
% 0.22/0.55 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.22/0.55 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % and_not_numerals(9)
% 0.22/0.55 thf(fact_8414_bit__not__iff__eq,axiom,
% 0.22/0.55 ! [A: int,N: nat] :
% 0.22/0.55 ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ A ) @ N )
% 0.22/0.55 = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.22/0.55 != zero_zero_int )
% 0.22/0.55 & ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit_not_iff_eq
% 0.22/0.55 thf(fact_8415_minus__exp__eq__not__mask,axiom,
% 0.22/0.55 ! [N: nat] :
% 0.22/0.55 ( ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.55 = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % minus_exp_eq_not_mask
% 0.22/0.55 thf(fact_8416_minus__exp__eq__not__mask,axiom,
% 0.22/0.55 ! [N: nat] :
% 0.22/0.55 ( ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.55 = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % minus_exp_eq_not_mask
% 0.22/0.55 thf(fact_8417_and__not__numerals_I8_J,axiom,
% 0.22/0.55 ! [M: num,N: num] :
% 0.22/0.55 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.22/0.55 = ( 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 ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % and_not_numerals(8)
% 0.22/0.55 thf(fact_8418_not__int__rec,axiom,
% 0.22/0.55 ( bit_ri7919022796975470100ot_int
% 0.22/0.55 = ( ^ [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 ) ) ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % not_int_rec
% 0.22/0.55 thf(fact_8419_Cauchy__iff2,axiom,
% 0.22/0.55 ( topolo4055970368930404560y_real
% 0.22/0.55 = ( ^ [X8: nat > real] :
% 0.22/0.55 ! [J3: nat] :
% 0.22/0.55 ? [M7: nat] :
% 0.22/0.55 ! [M5: nat] :
% 0.22/0.55 ( ( ord_less_eq_nat @ M7 @ M5 )
% 0.22/0.55 => ! [N4: nat] :
% 0.22/0.55 ( ( ord_less_eq_nat @ M7 @ N4 )
% 0.22/0.55 => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X8 @ M5 ) @ ( X8 @ N4 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Cauchy_iff2
% 0.22/0.55 thf(fact_8420_length__subseqs,axiom,
% 0.22/0.55 ! [Xs: list_real] :
% 0.22/0.55 ( ( size_s6660260683639930848t_real @ ( subseqs_real @ Xs ) )
% 0.22/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_real @ Xs ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % length_subseqs
% 0.22/0.55 thf(fact_8421_length__subseqs,axiom,
% 0.22/0.55 ! [Xs: list_o] :
% 0.22/0.55 ( ( size_s2710708370519433104list_o @ ( subseqs_o @ Xs ) )
% 0.22/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_o @ Xs ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % length_subseqs
% 0.22/0.55 thf(fact_8422_length__subseqs,axiom,
% 0.22/0.55 ! [Xs: list_nat] :
% 0.22/0.55 ( ( size_s3023201423986296836st_nat @ ( subseqs_nat @ Xs ) )
% 0.22/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_nat @ Xs ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % length_subseqs
% 0.22/0.55 thf(fact_8423_length__subseqs,axiom,
% 0.22/0.55 ! [Xs: list_int] :
% 0.22/0.55 ( ( size_s533118279054570080st_int @ ( subseqs_int @ Xs ) )
% 0.22/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_int @ Xs ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % length_subseqs
% 0.22/0.55 thf(fact_8424_csqrt_Osimps_I1_J,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( re @ ( csqrt @ Z ) )
% 0.22/0.55 = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % csqrt.simps(1)
% 0.22/0.55 thf(fact_8425_sinh__def,axiom,
% 0.22/0.55 ( sinh_complex
% 0.22/0.55 = ( ^ [X3: complex] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( minus_minus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_def
% 0.22/0.55 thf(fact_8426_sinh__def,axiom,
% 0.22/0.55 ( sinh_real
% 0.22/0.55 = ( ^ [X3: real] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % sinh_def
% 0.22/0.55 thf(fact_8427_scaleR__cancel__right,axiom,
% 0.22/0.55 ! [A: real,X: complex,B: real] :
% 0.22/0.55 ( ( ( real_V2046097035970521341omplex @ A @ X )
% 0.22/0.55 = ( real_V2046097035970521341omplex @ B @ X ) )
% 0.22/0.55 = ( ( A = B )
% 0.22/0.55 | ( X = zero_zero_complex ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_cancel_right
% 0.22/0.55 thf(fact_8428_scaleR__cancel__right,axiom,
% 0.22/0.55 ! [A: real,X: real,B: real] :
% 0.22/0.55 ( ( ( real_V1485227260804924795R_real @ A @ X )
% 0.22/0.55 = ( real_V1485227260804924795R_real @ B @ X ) )
% 0.22/0.55 = ( ( A = B )
% 0.22/0.55 | ( X = zero_zero_real ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_cancel_right
% 0.22/0.55 thf(fact_8429_scaleR__zero__right,axiom,
% 0.22/0.55 ! [A: real] :
% 0.22/0.55 ( ( real_V2046097035970521341omplex @ A @ zero_zero_complex )
% 0.22/0.55 = zero_zero_complex ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_zero_right
% 0.22/0.55 thf(fact_8430_scaleR__zero__right,axiom,
% 0.22/0.55 ! [A: real] :
% 0.22/0.55 ( ( real_V1485227260804924795R_real @ A @ zero_zero_real )
% 0.22/0.55 = zero_zero_real ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_zero_right
% 0.22/0.55 thf(fact_8431_mult__scaleR__right,axiom,
% 0.22/0.55 ! [X: complex,A: real,Y4: complex] :
% 0.22/0.55 ( ( times_times_complex @ X @ ( real_V2046097035970521341omplex @ A @ Y4 ) )
% 0.22/0.55 = ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X @ Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % mult_scaleR_right
% 0.22/0.55 thf(fact_8432_mult__scaleR__right,axiom,
% 0.22/0.55 ! [X: real,A: real,Y4: real] :
% 0.22/0.55 ( ( times_times_real @ X @ ( real_V1485227260804924795R_real @ A @ Y4 ) )
% 0.22/0.55 = ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X @ Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % mult_scaleR_right
% 0.22/0.55 thf(fact_8433_mult__scaleR__left,axiom,
% 0.22/0.55 ! [A: real,X: complex,Y4: complex] :
% 0.22/0.55 ( ( times_times_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ Y4 )
% 0.22/0.55 = ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X @ Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % mult_scaleR_left
% 0.22/0.55 thf(fact_8434_mult__scaleR__left,axiom,
% 0.22/0.55 ! [A: real,X: real,Y4: real] :
% 0.22/0.55 ( ( times_times_real @ ( real_V1485227260804924795R_real @ A @ X ) @ Y4 )
% 0.22/0.55 = ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X @ Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % mult_scaleR_left
% 0.22/0.55 thf(fact_8435_scaleR__cancel__left,axiom,
% 0.22/0.55 ! [A: real,X: complex,Y4: complex] :
% 0.22/0.55 ( ( ( real_V2046097035970521341omplex @ A @ X )
% 0.22/0.55 = ( real_V2046097035970521341omplex @ A @ Y4 ) )
% 0.22/0.55 = ( ( X = Y4 )
% 0.22/0.55 | ( A = zero_zero_real ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_cancel_left
% 0.22/0.55 thf(fact_8436_scaleR__cancel__left,axiom,
% 0.22/0.55 ! [A: real,X: real,Y4: real] :
% 0.22/0.55 ( ( ( real_V1485227260804924795R_real @ A @ X )
% 0.22/0.55 = ( real_V1485227260804924795R_real @ A @ Y4 ) )
% 0.22/0.55 = ( ( X = Y4 )
% 0.22/0.55 | ( A = zero_zero_real ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_cancel_left
% 0.22/0.55 thf(fact_8437_scaleR__scaleR,axiom,
% 0.22/0.55 ! [A: real,B: real,X: complex] :
% 0.22/0.55 ( ( real_V2046097035970521341omplex @ A @ ( real_V2046097035970521341omplex @ B @ X ) )
% 0.22/0.55 = ( real_V2046097035970521341omplex @ ( times_times_real @ A @ B ) @ X ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_scaleR
% 0.22/0.55 thf(fact_8438_scaleR__scaleR,axiom,
% 0.22/0.55 ! [A: real,B: real,X: real] :
% 0.22/0.55 ( ( real_V1485227260804924795R_real @ A @ ( real_V1485227260804924795R_real @ B @ X ) )
% 0.22/0.55 = ( real_V1485227260804924795R_real @ ( times_times_real @ A @ B ) @ X ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_scaleR
% 0.22/0.55 thf(fact_8439_complex__Re__of__nat,axiom,
% 0.22/0.55 ! [N: nat] :
% 0.22/0.55 ( ( re @ ( semiri8010041392384452111omplex @ N ) )
% 0.22/0.55 = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_Re_of_nat
% 0.22/0.55 thf(fact_8440_scaleR__zero__left,axiom,
% 0.22/0.55 ! [X: complex] :
% 0.22/0.55 ( ( real_V2046097035970521341omplex @ zero_zero_real @ X )
% 0.22/0.55 = zero_zero_complex ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_zero_left
% 0.22/0.55 thf(fact_8441_scaleR__zero__left,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( real_V1485227260804924795R_real @ zero_zero_real @ X )
% 0.22/0.55 = zero_zero_real ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_zero_left
% 0.22/0.55 thf(fact_8442_scaleR__eq__0__iff,axiom,
% 0.22/0.55 ! [A: real,X: complex] :
% 0.22/0.55 ( ( ( real_V2046097035970521341omplex @ A @ X )
% 0.22/0.55 = zero_zero_complex )
% 0.22/0.55 = ( ( A = zero_zero_real )
% 0.22/0.55 | ( X = zero_zero_complex ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_eq_0_iff
% 0.22/0.55 thf(fact_8443_scaleR__eq__0__iff,axiom,
% 0.22/0.55 ! [A: real,X: real] :
% 0.22/0.55 ( ( ( real_V1485227260804924795R_real @ A @ X )
% 0.22/0.55 = zero_zero_real )
% 0.22/0.55 = ( ( A = zero_zero_real )
% 0.22/0.55 | ( X = zero_zero_real ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_eq_0_iff
% 0.22/0.55 thf(fact_8444_complex__Re__numeral,axiom,
% 0.22/0.55 ! [V: num] :
% 0.22/0.55 ( ( re @ ( numera6690914467698888265omplex @ V ) )
% 0.22/0.55 = ( numeral_numeral_real @ V ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_Re_numeral
% 0.22/0.55 thf(fact_8445_scaleR__collapse,axiom,
% 0.22/0.55 ! [U: real,A: complex] :
% 0.22/0.55 ( ( plus_plus_complex @ ( real_V2046097035970521341omplex @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V2046097035970521341omplex @ U @ A ) )
% 0.22/0.55 = A ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_collapse
% 0.22/0.55 thf(fact_8446_scaleR__collapse,axiom,
% 0.22/0.55 ! [U: real,A: real] :
% 0.22/0.55 ( ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V1485227260804924795R_real @ U @ A ) )
% 0.22/0.55 = A ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_collapse
% 0.22/0.55 thf(fact_8447_norm__scaleR,axiom,
% 0.22/0.55 ! [A: real,X: complex] :
% 0.22/0.55 ( ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ A @ X ) )
% 0.22/0.55 = ( times_times_real @ ( abs_abs_real @ A ) @ ( real_V1022390504157884413omplex @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % norm_scaleR
% 0.22/0.55 thf(fact_8448_norm__scaleR,axiom,
% 0.22/0.55 ! [A: real,X: real] :
% 0.22/0.55 ( ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ A @ X ) )
% 0.22/0.55 = ( times_times_real @ ( abs_abs_real @ A ) @ ( real_V7735802525324610683m_real @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % norm_scaleR
% 0.22/0.55 thf(fact_8449_Re__divide__of__nat,axiom,
% 0.22/0.55 ! [Z: complex,N: nat] :
% 0.22/0.55 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
% 0.22/0.55 = ( divide_divide_real @ ( re @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Re_divide_of_nat
% 0.22/0.55 thf(fact_8450_Re__divide__of__real,axiom,
% 0.22/0.55 ! [Z: complex,R2: real] :
% 0.22/0.55 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R2 ) ) )
% 0.22/0.55 = ( divide_divide_real @ ( re @ Z ) @ R2 ) ) ).
% 0.22/0.55
% 0.22/0.55 % Re_divide_of_real
% 0.22/0.55 thf(fact_8451_Re__sgn,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( re @ ( sgn_sgn_complex @ Z ) )
% 0.22/0.55 = ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Re_sgn
% 0.22/0.55 thf(fact_8452_scaleR__times,axiom,
% 0.22/0.55 ! [U: num,W: num,A: complex] :
% 0.22/0.55 ( ( real_V2046097035970521341omplex @ ( numeral_numeral_real @ U ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 0.22/0.55 = ( real_V2046097035970521341omplex @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ A ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_times
% 0.22/0.55 thf(fact_8453_scaleR__times,axiom,
% 0.22/0.55 ! [U: num,W: num,A: real] :
% 0.22/0.55 ( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ U ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 0.22/0.55 = ( real_V1485227260804924795R_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ A ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_times
% 0.22/0.55 thf(fact_8454_Re__divide__numeral,axiom,
% 0.22/0.55 ! [Z: complex,W: num] :
% 0.22/0.55 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 0.22/0.55 = ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Re_divide_numeral
% 0.22/0.55 thf(fact_8455_cos__Arg__i__mult__zero,axiom,
% 0.22/0.55 ! [Y4: complex] :
% 0.22/0.55 ( ( Y4 != zero_zero_complex )
% 0.22/0.55 => ( ( ( re @ Y4 )
% 0.22/0.55 = zero_zero_real )
% 0.22/0.55 => ( ( cos_real @ ( arg @ Y4 ) )
% 0.22/0.55 = zero_zero_real ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cos_Arg_i_mult_zero
% 0.22/0.55 thf(fact_8456_inverse__scaleR__times,axiom,
% 0.22/0.55 ! [V: num,W: num,A: complex] :
% 0.22/0.55 ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 0.22/0.55 = ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ W ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% 0.22/0.55
% 0.22/0.55 % inverse_scaleR_times
% 0.22/0.55 thf(fact_8457_inverse__scaleR__times,axiom,
% 0.22/0.55 ! [V: num,W: num,A: real] :
% 0.22/0.55 ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 0.22/0.55 = ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ W ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% 0.22/0.55
% 0.22/0.55 % inverse_scaleR_times
% 0.22/0.55 thf(fact_8458_fraction__scaleR__times,axiom,
% 0.22/0.55 ! [U: num,V: num,W: num,A: complex] :
% 0.22/0.55 ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 0.22/0.55 = ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% 0.22/0.55
% 0.22/0.55 % fraction_scaleR_times
% 0.22/0.55 thf(fact_8459_fraction__scaleR__times,axiom,
% 0.22/0.55 ! [U: num,V: num,W: num,A: real] :
% 0.22/0.55 ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 0.22/0.55 = ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% 0.22/0.55
% 0.22/0.55 % fraction_scaleR_times
% 0.22/0.55 thf(fact_8460_scaleR__half__double,axiom,
% 0.22/0.55 ! [A: complex] :
% 0.22/0.55 ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_complex @ A @ A ) )
% 0.22/0.55 = A ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_half_double
% 0.22/0.55 thf(fact_8461_scaleR__half__double,axiom,
% 0.22/0.55 ! [A: real] :
% 0.22/0.55 ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ A @ A ) )
% 0.22/0.55 = A ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_half_double
% 0.22/0.55 thf(fact_8462_scaleR__right__imp__eq,axiom,
% 0.22/0.55 ! [X: complex,A: real,B: real] :
% 0.22/0.55 ( ( X != zero_zero_complex )
% 0.22/0.55 => ( ( ( real_V2046097035970521341omplex @ A @ X )
% 0.22/0.55 = ( real_V2046097035970521341omplex @ B @ X ) )
% 0.22/0.55 => ( A = B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_right_imp_eq
% 0.22/0.55 thf(fact_8463_scaleR__right__imp__eq,axiom,
% 0.22/0.55 ! [X: real,A: real,B: real] :
% 0.22/0.55 ( ( X != zero_zero_real )
% 0.22/0.55 => ( ( ( real_V1485227260804924795R_real @ A @ X )
% 0.22/0.55 = ( real_V1485227260804924795R_real @ B @ X ) )
% 0.22/0.55 => ( A = B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_right_imp_eq
% 0.22/0.55 thf(fact_8464_scaleR__right__diff__distrib,axiom,
% 0.22/0.55 ! [A: real,X: complex,Y4: complex] :
% 0.22/0.55 ( ( real_V2046097035970521341omplex @ A @ ( minus_minus_complex @ X @ Y4 ) )
% 0.22/0.55 = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ A @ Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_right_diff_distrib
% 0.22/0.55 thf(fact_8465_scaleR__right__diff__distrib,axiom,
% 0.22/0.55 ! [A: real,X: real,Y4: real] :
% 0.22/0.55 ( ( real_V1485227260804924795R_real @ A @ ( minus_minus_real @ X @ Y4 ) )
% 0.22/0.55 = ( minus_minus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_right_diff_distrib
% 0.22/0.55 thf(fact_8466_scaleR__complex_Osimps_I1_J,axiom,
% 0.22/0.55 ! [R2: real,X: complex] :
% 0.22/0.55 ( ( re @ ( real_V2046097035970521341omplex @ R2 @ X ) )
% 0.22/0.55 = ( times_times_real @ R2 @ ( re @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_complex.simps(1)
% 0.22/0.55 thf(fact_8467_scaleR__left__imp__eq,axiom,
% 0.22/0.55 ! [A: real,X: complex,Y4: complex] :
% 0.22/0.55 ( ( A != zero_zero_real )
% 0.22/0.55 => ( ( ( real_V2046097035970521341omplex @ A @ X )
% 0.22/0.55 = ( real_V2046097035970521341omplex @ A @ Y4 ) )
% 0.22/0.55 => ( X = Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_left_imp_eq
% 0.22/0.55 thf(fact_8468_scaleR__left__imp__eq,axiom,
% 0.22/0.55 ! [A: real,X: real,Y4: real] :
% 0.22/0.55 ( ( A != zero_zero_real )
% 0.22/0.55 => ( ( ( real_V1485227260804924795R_real @ A @ X )
% 0.22/0.55 = ( real_V1485227260804924795R_real @ A @ Y4 ) )
% 0.22/0.55 => ( X = Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_left_imp_eq
% 0.22/0.55 thf(fact_8469_real__scaleR__def,axiom,
% 0.22/0.55 real_V1485227260804924795R_real = times_times_real ).
% 0.22/0.55
% 0.22/0.55 % real_scaleR_def
% 0.22/0.55 thf(fact_8470_scaleR__conv__of__real,axiom,
% 0.22/0.55 ( real_V2046097035970521341omplex
% 0.22/0.55 = ( ^ [R5: real] : ( times_times_complex @ ( real_V4546457046886955230omplex @ R5 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_conv_of_real
% 0.22/0.55 thf(fact_8471_scaleR__conv__of__real,axiom,
% 0.22/0.55 ( real_V1485227260804924795R_real
% 0.22/0.55 = ( ^ [R5: real] : ( times_times_real @ ( real_V1803761363581548252l_real @ R5 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_conv_of_real
% 0.22/0.55 thf(fact_8472_scaleR__left__diff__distrib,axiom,
% 0.22/0.55 ! [A: real,B: real,X: complex] :
% 0.22/0.55 ( ( real_V2046097035970521341omplex @ ( minus_minus_real @ A @ B ) @ X )
% 0.22/0.55 = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ B @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_left_diff_distrib
% 0.22/0.55 thf(fact_8473_scaleR__left__diff__distrib,axiom,
% 0.22/0.55 ! [A: real,B: real,X: real] :
% 0.22/0.55 ( ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B ) @ X )
% 0.22/0.55 = ( minus_minus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_left_diff_distrib
% 0.22/0.55 thf(fact_8474_scaleR__left_Odiff,axiom,
% 0.22/0.55 ! [X: real,Y4: real,Xa: complex] :
% 0.22/0.55 ( ( real_V2046097035970521341omplex @ ( minus_minus_real @ X @ Y4 ) @ Xa )
% 0.22/0.55 = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ X @ Xa ) @ ( real_V2046097035970521341omplex @ Y4 @ Xa ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_left.diff
% 0.22/0.55 thf(fact_8475_scaleR__left_Odiff,axiom,
% 0.22/0.55 ! [X: real,Y4: real,Xa: real] :
% 0.22/0.55 ( ( real_V1485227260804924795R_real @ ( minus_minus_real @ X @ Y4 ) @ Xa )
% 0.22/0.55 = ( minus_minus_real @ ( real_V1485227260804924795R_real @ X @ Xa ) @ ( real_V1485227260804924795R_real @ Y4 @ Xa ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_left.diff
% 0.22/0.55 thf(fact_8476_imaginary__unit_Osimps_I1_J,axiom,
% 0.22/0.55 ( ( re @ imaginary_unit )
% 0.22/0.55 = zero_zero_real ) ).
% 0.22/0.55
% 0.22/0.55 % imaginary_unit.simps(1)
% 0.22/0.55 thf(fact_8477_complex__Re__le__cmod,axiom,
% 0.22/0.55 ! [X: complex] : ( ord_less_eq_real @ ( re @ X ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_Re_le_cmod
% 0.22/0.55 thf(fact_8478_zero__complex_Osimps_I1_J,axiom,
% 0.22/0.55 ( ( re @ zero_zero_complex )
% 0.22/0.55 = zero_zero_real ) ).
% 0.22/0.55
% 0.22/0.55 % zero_complex.simps(1)
% 0.22/0.55 thf(fact_8479_complex__scaleR,axiom,
% 0.22/0.55 ! [R2: real,A: real,B: real] :
% 0.22/0.55 ( ( real_V2046097035970521341omplex @ R2 @ ( complex2 @ A @ B ) )
% 0.22/0.55 = ( complex2 @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_scaleR
% 0.22/0.55 thf(fact_8480_minus__complex_Osimps_I1_J,axiom,
% 0.22/0.55 ! [X: complex,Y4: complex] :
% 0.22/0.55 ( ( re @ ( minus_minus_complex @ X @ Y4 ) )
% 0.22/0.55 = ( minus_minus_real @ ( re @ X ) @ ( re @ Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % minus_complex.simps(1)
% 0.22/0.55 thf(fact_8481_scaleR__right__mono,axiom,
% 0.22/0.55 ! [A: real,B: real,X: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ A @ B )
% 0.22/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.55 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_right_mono
% 0.22/0.55 thf(fact_8482_scaleR__right__mono__neg,axiom,
% 0.22/0.55 ! [B: real,A: real,C: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ B @ A )
% 0.22/0.55 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.22/0.55 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ C ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_right_mono_neg
% 0.22/0.55 thf(fact_8483_scaleR__le__cancel__left__pos,axiom,
% 0.22/0.55 ! [C: real,A: real,B: real] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.55 => ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) )
% 0.22/0.55 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_le_cancel_left_pos
% 0.22/0.55 thf(fact_8484_scaleR__le__cancel__left__neg,axiom,
% 0.22/0.55 ! [C: real,A: real,B: real] :
% 0.22/0.55 ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.55 => ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) )
% 0.22/0.55 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_le_cancel_left_neg
% 0.22/0.55 thf(fact_8485_scaleR__le__cancel__left,axiom,
% 0.22/0.55 ! [C: real,A: real,B: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) )
% 0.22/0.55 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.55 => ( ord_less_eq_real @ A @ B ) )
% 0.22/0.55 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.55 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_le_cancel_left
% 0.22/0.55 thf(fact_8486_scaleR__left__mono,axiom,
% 0.22/0.55 ! [X: real,Y4: real,A: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ X @ Y4 )
% 0.22/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.55 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y4 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_left_mono
% 0.22/0.55 thf(fact_8487_scaleR__left__mono__neg,axiom,
% 0.22/0.55 ! [B: real,A: real,C: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ B @ A )
% 0.22/0.55 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.22/0.55 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_left_mono_neg
% 0.22/0.55 thf(fact_8488_vector__fraction__eq__iff,axiom,
% 0.22/0.55 ! [U: real,V: real,A: complex,X: complex] :
% 0.22/0.55 ( ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A )
% 0.22/0.55 = X )
% 0.22/0.55 = ( ( ( V = zero_zero_real )
% 0.22/0.55 => ( X = zero_zero_complex ) )
% 0.22/0.55 & ( ( V != zero_zero_real )
% 0.22/0.55 => ( ( real_V2046097035970521341omplex @ U @ A )
% 0.22/0.55 = ( real_V2046097035970521341omplex @ V @ X ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % vector_fraction_eq_iff
% 0.22/0.55 thf(fact_8489_vector__fraction__eq__iff,axiom,
% 0.22/0.55 ! [U: real,V: real,A: real,X: real] :
% 0.22/0.55 ( ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A )
% 0.22/0.55 = X )
% 0.22/0.55 = ( ( ( V = zero_zero_real )
% 0.22/0.55 => ( X = zero_zero_real ) )
% 0.22/0.55 & ( ( V != zero_zero_real )
% 0.22/0.55 => ( ( real_V1485227260804924795R_real @ U @ A )
% 0.22/0.55 = ( real_V1485227260804924795R_real @ V @ X ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % vector_fraction_eq_iff
% 0.22/0.55 thf(fact_8490_eq__vector__fraction__iff,axiom,
% 0.22/0.55 ! [X: complex,U: real,V: real,A: complex] :
% 0.22/0.55 ( ( X
% 0.22/0.55 = ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A ) )
% 0.22/0.55 = ( ( ( V = zero_zero_real )
% 0.22/0.55 => ( X = zero_zero_complex ) )
% 0.22/0.55 & ( ( V != zero_zero_real )
% 0.22/0.55 => ( ( real_V2046097035970521341omplex @ V @ X )
% 0.22/0.55 = ( real_V2046097035970521341omplex @ U @ A ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % eq_vector_fraction_iff
% 0.22/0.55 thf(fact_8491_eq__vector__fraction__iff,axiom,
% 0.22/0.55 ! [X: real,U: real,V: real,A: real] :
% 0.22/0.55 ( ( X
% 0.22/0.55 = ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A ) )
% 0.22/0.55 = ( ( ( V = zero_zero_real )
% 0.22/0.55 => ( X = zero_zero_real ) )
% 0.22/0.55 & ( ( V != zero_zero_real )
% 0.22/0.55 => ( ( real_V1485227260804924795R_real @ V @ X )
% 0.22/0.55 = ( real_V1485227260804924795R_real @ U @ A ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % eq_vector_fraction_iff
% 0.22/0.55 thf(fact_8492_Real__Vector__Spaces_Ole__add__iff1,axiom,
% 0.22/0.55 ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E2 ) @ D ) )
% 0.22/0.55 = ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.22/0.55
% 0.22/0.55 % Real_Vector_Spaces.le_add_iff1
% 0.22/0.55 thf(fact_8493_Real__Vector__Spaces_Ole__add__iff2,axiom,
% 0.22/0.55 ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E2 ) @ D ) )
% 0.22/0.55 = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Real_Vector_Spaces.le_add_iff2
% 0.22/0.55 thf(fact_8494_abs__Re__le__cmod,axiom,
% 0.22/0.55 ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.22/0.55
% 0.22/0.55 % abs_Re_le_cmod
% 0.22/0.55 thf(fact_8495_Re__csqrt,axiom,
% 0.22/0.55 ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Re_csqrt
% 0.22/0.55 thf(fact_8496_zero__le__scaleR__iff,axiom,
% 0.22/0.55 ! [A: real,B: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) )
% 0.22/0.55 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.55 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.22/0.55 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.22/0.55 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.22/0.55 | ( A = zero_zero_real ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % zero_le_scaleR_iff
% 0.22/0.55 thf(fact_8497_scaleR__le__0__iff,axiom,
% 0.22/0.55 ! [A: real,B: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ B ) @ zero_zero_real )
% 0.22/0.55 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.55 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.22/0.55 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.22/0.55 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.22/0.55 | ( A = zero_zero_real ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_le_0_iff
% 0.22/0.55 thf(fact_8498_scaleR__mono,axiom,
% 0.22/0.55 ! [A: real,B: real,X: real,Y4: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ A @ B )
% 0.22/0.55 => ( ( ord_less_eq_real @ X @ Y4 )
% 0.22/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.22/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.55 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ Y4 ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_mono
% 0.22/0.55 thf(fact_8499_scaleR__mono_H,axiom,
% 0.22/0.55 ! [A: real,B: real,C: real,D: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ A @ B )
% 0.22/0.55 => ( ( ord_less_eq_real @ C @ D )
% 0.22/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.22/0.55 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ D ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_mono'
% 0.22/0.55 thf(fact_8500_split__scaleR__neg__le,axiom,
% 0.22/0.55 ! [A: real,X: real] :
% 0.22/0.55 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.55 & ( ord_less_eq_real @ X @ zero_zero_real ) )
% 0.22/0.55 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.22/0.55 & ( ord_less_eq_real @ zero_zero_real @ X ) ) )
% 0.22/0.55 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ).
% 0.22/0.55
% 0.22/0.55 % split_scaleR_neg_le
% 0.22/0.55 thf(fact_8501_split__scaleR__pos__le,axiom,
% 0.22/0.55 ! [A: real,B: real] :
% 0.22/0.55 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.55 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.22/0.55 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.22/0.55 & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.22/0.55 => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % split_scaleR_pos_le
% 0.22/0.55 thf(fact_8502_scaleR__nonneg__nonneg,axiom,
% 0.22/0.55 ! [A: real,X: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.55 => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_nonneg_nonneg
% 0.22/0.55 thf(fact_8503_scaleR__nonneg__nonpos,axiom,
% 0.22/0.55 ! [A: real,X: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.22/0.55 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.22/0.55 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_nonneg_nonpos
% 0.22/0.55 thf(fact_8504_scaleR__nonpos__nonneg,axiom,
% 0.22/0.55 ! [A: real,X: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.22/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.55 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_nonpos_nonneg
% 0.22/0.55 thf(fact_8505_scaleR__nonpos__nonpos,axiom,
% 0.22/0.55 ! [A: real,B: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.22/0.55 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.22/0.55 => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_nonpos_nonpos
% 0.22/0.55 thf(fact_8506_scaleR__left__le__one__le,axiom,
% 0.22/0.55 ! [X: real,A: real] :
% 0.22/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.55 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.22/0.55 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_left_le_one_le
% 0.22/0.55 thf(fact_8507_scaleR__2,axiom,
% 0.22/0.55 ! [X: complex] :
% 0.22/0.55 ( ( real_V2046097035970521341omplex @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X )
% 0.22/0.55 = ( plus_plus_complex @ X @ X ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_2
% 0.22/0.55 thf(fact_8508_scaleR__2,axiom,
% 0.22/0.55 ! [X: real] :
% 0.22/0.55 ( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X )
% 0.22/0.55 = ( plus_plus_real @ X @ X ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_2
% 0.22/0.55 thf(fact_8509_real__vector__eq__affinity,axiom,
% 0.22/0.55 ! [M: real,Y4: complex,X: complex,C: complex] :
% 0.22/0.55 ( ( M != zero_zero_real )
% 0.22/0.55 => ( ( Y4
% 0.22/0.55 = ( plus_plus_complex @ ( real_V2046097035970521341omplex @ M @ X ) @ C ) )
% 0.22/0.55 = ( ( minus_minus_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ Y4 ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ C ) )
% 0.22/0.55 = X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % real_vector_eq_affinity
% 0.22/0.55 thf(fact_8510_real__vector__eq__affinity,axiom,
% 0.22/0.55 ! [M: real,Y4: real,X: real,C: real] :
% 0.22/0.55 ( ( M != zero_zero_real )
% 0.22/0.55 => ( ( Y4
% 0.22/0.55 = ( plus_plus_real @ ( real_V1485227260804924795R_real @ M @ X ) @ C ) )
% 0.22/0.55 = ( ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ Y4 ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ C ) )
% 0.22/0.55 = X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % real_vector_eq_affinity
% 0.22/0.55 thf(fact_8511_real__vector__affinity__eq,axiom,
% 0.22/0.55 ! [M: real,X: complex,C: complex,Y4: complex] :
% 0.22/0.55 ( ( M != zero_zero_real )
% 0.22/0.55 => ( ( ( plus_plus_complex @ ( real_V2046097035970521341omplex @ M @ X ) @ C )
% 0.22/0.55 = Y4 )
% 0.22/0.55 = ( X
% 0.22/0.55 = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ Y4 ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ C ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % real_vector_affinity_eq
% 0.22/0.55 thf(fact_8512_real__vector__affinity__eq,axiom,
% 0.22/0.55 ! [M: real,X: real,C: real,Y4: real] :
% 0.22/0.55 ( ( M != zero_zero_real )
% 0.22/0.55 => ( ( ( plus_plus_real @ ( real_V1485227260804924795R_real @ M @ X ) @ C )
% 0.22/0.55 = Y4 )
% 0.22/0.55 = ( X
% 0.22/0.55 = ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ Y4 ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ C ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % real_vector_affinity_eq
% 0.22/0.55 thf(fact_8513_pos__divideR__le__eq,axiom,
% 0.22/0.55 ! [C: real,B: real,A: real] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.55 => ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
% 0.22/0.55 = ( ord_less_eq_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % pos_divideR_le_eq
% 0.22/0.55 thf(fact_8514_pos__le__divideR__eq,axiom,
% 0.22/0.55 ! [C: real,A: real,B: real] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.55 => ( ( ord_less_eq_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
% 0.22/0.55 = ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % pos_le_divideR_eq
% 0.22/0.55 thf(fact_8515_neg__divideR__le__eq,axiom,
% 0.22/0.55 ! [C: real,B: real,A: real] :
% 0.22/0.55 ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.55 => ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
% 0.22/0.55 = ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % neg_divideR_le_eq
% 0.22/0.55 thf(fact_8516_neg__le__divideR__eq,axiom,
% 0.22/0.55 ! [C: real,A: real,B: real] :
% 0.22/0.55 ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.55 => ( ( ord_less_eq_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
% 0.22/0.55 = ( ord_less_eq_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % neg_le_divideR_eq
% 0.22/0.55 thf(fact_8517_neg__less__divideR__eq,axiom,
% 0.22/0.55 ! [C: real,A: real,B: real] :
% 0.22/0.55 ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.55 => ( ( ord_less_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
% 0.22/0.55 = ( ord_less_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % neg_less_divideR_eq
% 0.22/0.55 thf(fact_8518_neg__divideR__less__eq,axiom,
% 0.22/0.55 ! [C: real,B: real,A: real] :
% 0.22/0.55 ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.55 => ( ( ord_less_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
% 0.22/0.55 = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % neg_divideR_less_eq
% 0.22/0.55 thf(fact_8519_pos__less__divideR__eq,axiom,
% 0.22/0.55 ! [C: real,A: real,B: real] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.55 => ( ( ord_less_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
% 0.22/0.55 = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % pos_less_divideR_eq
% 0.22/0.55 thf(fact_8520_pos__divideR__less__eq,axiom,
% 0.22/0.55 ! [C: real,B: real,A: real] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.55 => ( ( ord_less_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
% 0.22/0.55 = ( ord_less_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % pos_divideR_less_eq
% 0.22/0.55 thf(fact_8521_nonzero__inverse__scaleR__distrib,axiom,
% 0.22/0.55 ! [A: real,X: complex] :
% 0.22/0.55 ( ( A != zero_zero_real )
% 0.22/0.55 => ( ( X != zero_zero_complex )
% 0.22/0.55 => ( ( invers8013647133539491842omplex @ ( real_V2046097035970521341omplex @ A @ X ) )
% 0.22/0.55 = ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ A ) @ ( invers8013647133539491842omplex @ X ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % nonzero_inverse_scaleR_distrib
% 0.22/0.55 thf(fact_8522_nonzero__inverse__scaleR__distrib,axiom,
% 0.22/0.55 ! [A: real,X: real] :
% 0.22/0.55 ( ( A != zero_zero_real )
% 0.22/0.55 => ( ( X != zero_zero_real )
% 0.22/0.55 => ( ( inverse_inverse_real @ ( real_V1485227260804924795R_real @ A @ X ) )
% 0.22/0.55 = ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ X ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % nonzero_inverse_scaleR_distrib
% 0.22/0.55 thf(fact_8523_pos__le__minus__divideR__eq,axiom,
% 0.22/0.55 ! [C: real,A: real,B: real] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.55 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
% 0.22/0.55 = ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % pos_le_minus_divideR_eq
% 0.22/0.55 thf(fact_8524_pos__minus__divideR__le__eq,axiom,
% 0.22/0.55 ! [C: real,B: real,A: real] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.55 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
% 0.22/0.55 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % pos_minus_divideR_le_eq
% 0.22/0.55 thf(fact_8525_neg__le__minus__divideR__eq,axiom,
% 0.22/0.55 ! [C: real,A: real,B: real] :
% 0.22/0.55 ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.55 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
% 0.22/0.55 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % neg_le_minus_divideR_eq
% 0.22/0.55 thf(fact_8526_neg__minus__divideR__le__eq,axiom,
% 0.22/0.55 ! [C: real,B: real,A: real] :
% 0.22/0.55 ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.55 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
% 0.22/0.55 = ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % neg_minus_divideR_le_eq
% 0.22/0.55 thf(fact_8527_pos__less__minus__divideR__eq,axiom,
% 0.22/0.55 ! [C: real,A: real,B: real] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.55 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
% 0.22/0.55 = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % pos_less_minus_divideR_eq
% 0.22/0.55 thf(fact_8528_pos__minus__divideR__less__eq,axiom,
% 0.22/0.55 ! [C: real,B: real,A: real] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.55 => ( ( ord_less_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
% 0.22/0.55 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % pos_minus_divideR_less_eq
% 0.22/0.55 thf(fact_8529_neg__less__minus__divideR__eq,axiom,
% 0.22/0.55 ! [C: real,A: real,B: real] :
% 0.22/0.55 ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.55 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
% 0.22/0.55 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % neg_less_minus_divideR_eq
% 0.22/0.55 thf(fact_8530_neg__minus__divideR__less__eq,axiom,
% 0.22/0.55 ! [C: real,B: real,A: real] :
% 0.22/0.55 ( ( ord_less_real @ C @ zero_zero_real )
% 0.22/0.55 => ( ( ord_less_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
% 0.22/0.55 = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % neg_minus_divideR_less_eq
% 0.22/0.55 thf(fact_8531_cmod__plus__Re__le__0__iff,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
% 0.22/0.55 = ( ( re @ Z )
% 0.22/0.55 = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cmod_plus_Re_le_0_iff
% 0.22/0.55 thf(fact_8532_cos__n__Re__cis__pow__n,axiom,
% 0.22/0.55 ! [N: nat,A: real] :
% 0.22/0.55 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 0.22/0.55 = ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cos_n_Re_cis_pow_n
% 0.22/0.55 thf(fact_8533_CauchyD,axiom,
% 0.22/0.55 ! [X9: nat > complex,E2: real] :
% 0.22/0.55 ( ( topolo6517432010174082258omplex @ X9 )
% 0.22/0.55 => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.22/0.55 => ? [M8: nat] :
% 0.22/0.55 ! [M2: nat] :
% 0.22/0.55 ( ( ord_less_eq_nat @ M8 @ M2 )
% 0.22/0.55 => ! [N6: nat] :
% 0.22/0.55 ( ( ord_less_eq_nat @ M8 @ N6 )
% 0.22/0.55 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X9 @ M2 ) @ ( X9 @ N6 ) ) ) @ E2 ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % CauchyD
% 0.22/0.55 thf(fact_8534_CauchyD,axiom,
% 0.22/0.55 ! [X9: nat > real,E2: real] :
% 0.22/0.55 ( ( topolo4055970368930404560y_real @ X9 )
% 0.22/0.55 => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.22/0.55 => ? [M8: nat] :
% 0.22/0.55 ! [M2: nat] :
% 0.22/0.55 ( ( ord_less_eq_nat @ M8 @ M2 )
% 0.22/0.55 => ! [N6: nat] :
% 0.22/0.55 ( ( ord_less_eq_nat @ M8 @ N6 )
% 0.22/0.55 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X9 @ M2 ) @ ( X9 @ N6 ) ) ) @ E2 ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % CauchyD
% 0.22/0.55 thf(fact_8535_CauchyI,axiom,
% 0.22/0.55 ! [X9: nat > complex] :
% 0.22/0.55 ( ! [E: real] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ E )
% 0.22/0.55 => ? [M9: nat] :
% 0.22/0.55 ! [M4: nat] :
% 0.22/0.55 ( ( ord_less_eq_nat @ M9 @ M4 )
% 0.22/0.55 => ! [N2: nat] :
% 0.22/0.55 ( ( ord_less_eq_nat @ M9 @ N2 )
% 0.22/0.55 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X9 @ M4 ) @ ( X9 @ N2 ) ) ) @ E ) ) ) )
% 0.22/0.55 => ( topolo6517432010174082258omplex @ X9 ) ) ).
% 0.22/0.55
% 0.22/0.55 % CauchyI
% 0.22/0.55 thf(fact_8536_CauchyI,axiom,
% 0.22/0.55 ! [X9: nat > real] :
% 0.22/0.55 ( ! [E: real] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ E )
% 0.22/0.55 => ? [M9: nat] :
% 0.22/0.55 ! [M4: nat] :
% 0.22/0.55 ( ( ord_less_eq_nat @ M9 @ M4 )
% 0.22/0.55 => ! [N2: nat] :
% 0.22/0.55 ( ( ord_less_eq_nat @ M9 @ N2 )
% 0.22/0.55 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X9 @ M4 ) @ ( X9 @ N2 ) ) ) @ E ) ) ) )
% 0.22/0.55 => ( topolo4055970368930404560y_real @ X9 ) ) ).
% 0.22/0.55
% 0.22/0.55 % CauchyI
% 0.22/0.55 thf(fact_8537_Cauchy__iff,axiom,
% 0.22/0.55 ( topolo6517432010174082258omplex
% 0.22/0.55 = ( ^ [X8: nat > complex] :
% 0.22/0.55 ! [E3: real] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ E3 )
% 0.22/0.55 => ? [M7: nat] :
% 0.22/0.55 ! [M5: nat] :
% 0.22/0.55 ( ( ord_less_eq_nat @ M7 @ M5 )
% 0.22/0.55 => ! [N4: nat] :
% 0.22/0.55 ( ( ord_less_eq_nat @ M7 @ N4 )
% 0.22/0.55 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X8 @ M5 ) @ ( X8 @ N4 ) ) ) @ E3 ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Cauchy_iff
% 0.22/0.55 thf(fact_8538_Cauchy__iff,axiom,
% 0.22/0.55 ( topolo4055970368930404560y_real
% 0.22/0.55 = ( ^ [X8: nat > real] :
% 0.22/0.55 ! [E3: real] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ E3 )
% 0.22/0.55 => ? [M7: nat] :
% 0.22/0.55 ! [M5: nat] :
% 0.22/0.55 ( ( ord_less_eq_nat @ M7 @ M5 )
% 0.22/0.55 => ! [N4: nat] :
% 0.22/0.55 ( ( ord_less_eq_nat @ M7 @ N4 )
% 0.22/0.55 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X8 @ M5 ) @ ( X8 @ N4 ) ) ) @ E3 ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Cauchy_iff
% 0.22/0.55 thf(fact_8539_cosh__def,axiom,
% 0.22/0.55 ( cosh_complex
% 0.22/0.55 = ( ^ [X3: complex] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_def
% 0.22/0.55 thf(fact_8540_cosh__def,axiom,
% 0.22/0.55 ( cosh_real
% 0.22/0.55 = ( ^ [X3: real] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cosh_def
% 0.22/0.55 thf(fact_8541_csqrt_Ocode,axiom,
% 0.22/0.55 ( csqrt
% 0.22/0.55 = ( ^ [Z5: complex] :
% 0.22/0.55 ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.55 @ ( times_times_real
% 0.22/0.55 @ ( if_real
% 0.22/0.55 @ ( ( im @ Z5 )
% 0.22/0.55 = zero_zero_real )
% 0.22/0.55 @ one_one_real
% 0.22/0.55 @ ( sgn_sgn_real @ ( im @ Z5 ) ) )
% 0.22/0.55 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % csqrt.code
% 0.22/0.55 thf(fact_8542_csqrt_Osimps_I2_J,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( im @ ( csqrt @ Z ) )
% 0.22/0.55 = ( times_times_real
% 0.22/0.55 @ ( if_real
% 0.22/0.55 @ ( ( im @ Z )
% 0.22/0.55 = zero_zero_real )
% 0.22/0.55 @ one_one_real
% 0.22/0.55 @ ( sgn_sgn_real @ ( im @ Z ) ) )
% 0.22/0.55 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % csqrt.simps(2)
% 0.22/0.55 thf(fact_8543_csqrt__of__real__nonpos,axiom,
% 0.22/0.55 ! [X: complex] :
% 0.22/0.55 ( ( ( im @ X )
% 0.22/0.55 = zero_zero_real )
% 0.22/0.55 => ( ( ord_less_eq_real @ ( re @ X ) @ zero_zero_real )
% 0.22/0.55 => ( ( csqrt @ X )
% 0.22/0.55 = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X ) ) ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % csqrt_of_real_nonpos
% 0.22/0.55 thf(fact_8544_Complex__divide,axiom,
% 0.22/0.55 ( divide1717551699836669952omplex
% 0.22/0.55 = ( ^ [X3: complex,Y6: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y6 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y6 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y6 ) ) @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y6 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Complex_divide
% 0.22/0.55 thf(fact_8545_complex__Im__of__int,axiom,
% 0.22/0.55 ! [Z: int] :
% 0.22/0.55 ( ( im @ ( ring_17405671764205052669omplex @ Z ) )
% 0.22/0.55 = zero_zero_real ) ).
% 0.22/0.55
% 0.22/0.55 % complex_Im_of_int
% 0.22/0.55 thf(fact_8546_complex__Im__fact,axiom,
% 0.22/0.55 ! [N: nat] :
% 0.22/0.55 ( ( im @ ( semiri5044797733671781792omplex @ N ) )
% 0.22/0.55 = zero_zero_real ) ).
% 0.22/0.55
% 0.22/0.55 % complex_Im_fact
% 0.22/0.55 thf(fact_8547_Im__complex__of__real,axiom,
% 0.22/0.55 ! [Z: real] :
% 0.22/0.55 ( ( im @ ( real_V4546457046886955230omplex @ Z ) )
% 0.22/0.55 = zero_zero_real ) ).
% 0.22/0.55
% 0.22/0.55 % Im_complex_of_real
% 0.22/0.55 thf(fact_8548_Im__power__real,axiom,
% 0.22/0.55 ! [X: complex,N: nat] :
% 0.22/0.55 ( ( ( im @ X )
% 0.22/0.55 = zero_zero_real )
% 0.22/0.55 => ( ( im @ ( power_power_complex @ X @ N ) )
% 0.22/0.55 = zero_zero_real ) ) ).
% 0.22/0.55
% 0.22/0.55 % Im_power_real
% 0.22/0.55 thf(fact_8549_complex__Im__numeral,axiom,
% 0.22/0.55 ! [V: num] :
% 0.22/0.55 ( ( im @ ( numera6690914467698888265omplex @ V ) )
% 0.22/0.55 = zero_zero_real ) ).
% 0.22/0.55
% 0.22/0.55 % complex_Im_numeral
% 0.22/0.55 thf(fact_8550_complex__Im__of__nat,axiom,
% 0.22/0.55 ! [N: nat] :
% 0.22/0.55 ( ( im @ ( semiri8010041392384452111omplex @ N ) )
% 0.22/0.55 = zero_zero_real ) ).
% 0.22/0.55
% 0.22/0.55 % complex_Im_of_nat
% 0.22/0.55 thf(fact_8551_Im__divide__of__real,axiom,
% 0.22/0.55 ! [Z: complex,R2: real] :
% 0.22/0.55 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R2 ) ) )
% 0.22/0.55 = ( divide_divide_real @ ( im @ Z ) @ R2 ) ) ).
% 0.22/0.55
% 0.22/0.55 % Im_divide_of_real
% 0.22/0.55 thf(fact_8552_Im__sgn,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( im @ ( sgn_sgn_complex @ Z ) )
% 0.22/0.55 = ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Im_sgn
% 0.22/0.55 thf(fact_8553_Re__power__real,axiom,
% 0.22/0.55 ! [X: complex,N: nat] :
% 0.22/0.55 ( ( ( im @ X )
% 0.22/0.55 = zero_zero_real )
% 0.22/0.55 => ( ( re @ ( power_power_complex @ X @ N ) )
% 0.22/0.55 = ( power_power_real @ ( re @ X ) @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Re_power_real
% 0.22/0.55 thf(fact_8554_Im__divide__numeral,axiom,
% 0.22/0.55 ! [Z: complex,W: num] :
% 0.22/0.55 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 0.22/0.55 = ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Im_divide_numeral
% 0.22/0.55 thf(fact_8555_Im__divide__of__nat,axiom,
% 0.22/0.55 ! [Z: complex,N: nat] :
% 0.22/0.55 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
% 0.22/0.55 = ( divide_divide_real @ ( im @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Im_divide_of_nat
% 0.22/0.55 thf(fact_8556_csqrt__of__real__nonneg,axiom,
% 0.22/0.55 ! [X: complex] :
% 0.22/0.55 ( ( ( im @ X )
% 0.22/0.55 = zero_zero_real )
% 0.22/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) )
% 0.22/0.55 => ( ( csqrt @ X )
% 0.22/0.55 = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % csqrt_of_real_nonneg
% 0.22/0.55 thf(fact_8557_csqrt__minus,axiom,
% 0.22/0.55 ! [X: complex] :
% 0.22/0.55 ( ( ( ord_less_real @ ( im @ X ) @ zero_zero_real )
% 0.22/0.55 | ( ( ( im @ X )
% 0.22/0.55 = zero_zero_real )
% 0.22/0.55 & ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) ) ) )
% 0.22/0.55 => ( ( csqrt @ ( uminus1482373934393186551omplex @ X ) )
% 0.22/0.55 = ( times_times_complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % csqrt_minus
% 0.22/0.55 thf(fact_8558_scaleR__complex_Osimps_I2_J,axiom,
% 0.22/0.55 ! [R2: real,X: complex] :
% 0.22/0.55 ( ( im @ ( real_V2046097035970521341omplex @ R2 @ X ) )
% 0.22/0.55 = ( times_times_real @ R2 @ ( im @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_complex.simps(2)
% 0.22/0.55 thf(fact_8559_zero__complex_Osimps_I2_J,axiom,
% 0.22/0.55 ( ( im @ zero_zero_complex )
% 0.22/0.55 = zero_zero_real ) ).
% 0.22/0.55
% 0.22/0.55 % zero_complex.simps(2)
% 0.22/0.55 thf(fact_8560_one__complex_Osimps_I2_J,axiom,
% 0.22/0.55 ( ( im @ one_one_complex )
% 0.22/0.55 = zero_zero_real ) ).
% 0.22/0.55
% 0.22/0.55 % one_complex.simps(2)
% 0.22/0.55 thf(fact_8561_minus__complex_Osimps_I2_J,axiom,
% 0.22/0.55 ! [X: complex,Y4: complex] :
% 0.22/0.55 ( ( im @ ( minus_minus_complex @ X @ Y4 ) )
% 0.22/0.55 = ( minus_minus_real @ ( im @ X ) @ ( im @ Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % minus_complex.simps(2)
% 0.22/0.55 thf(fact_8562_complex__is__Int__iff,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( member_complex @ Z @ ring_1_Ints_complex )
% 0.22/0.55 = ( ( ( im @ Z )
% 0.22/0.55 = zero_zero_real )
% 0.22/0.55 & ? [I2: int] :
% 0.22/0.55 ( ( re @ Z )
% 0.22/0.55 = ( ring_1_of_int_real @ I2 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_is_Int_iff
% 0.22/0.55 thf(fact_8563_scaleR__complex_Ocode,axiom,
% 0.22/0.55 ( real_V2046097035970521341omplex
% 0.22/0.55 = ( ^ [R5: real,X3: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X3 ) ) @ ( times_times_real @ R5 @ ( im @ X3 ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % scaleR_complex.code
% 0.22/0.55 thf(fact_8564_abs__Im__le__cmod,axiom,
% 0.22/0.55 ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.22/0.55
% 0.22/0.55 % abs_Im_le_cmod
% 0.22/0.55 thf(fact_8565_times__complex_Osimps_I2_J,axiom,
% 0.22/0.55 ! [X: complex,Y4: complex] :
% 0.22/0.55 ( ( im @ ( times_times_complex @ X @ Y4 ) )
% 0.22/0.55 = ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y4 ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y4 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % times_complex.simps(2)
% 0.22/0.55 thf(fact_8566_cmod__eq__Re,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( ( im @ Z )
% 0.22/0.55 = zero_zero_real )
% 0.22/0.55 => ( ( real_V1022390504157884413omplex @ Z )
% 0.22/0.55 = ( abs_abs_real @ ( re @ Z ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cmod_eq_Re
% 0.22/0.55 thf(fact_8567_cmod__eq__Im,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( ( re @ Z )
% 0.22/0.55 = zero_zero_real )
% 0.22/0.55 => ( ( real_V1022390504157884413omplex @ Z )
% 0.22/0.55 = ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cmod_eq_Im
% 0.22/0.55 thf(fact_8568_Im__eq__0,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( ( abs_abs_real @ ( re @ Z ) )
% 0.22/0.55 = ( real_V1022390504157884413omplex @ Z ) )
% 0.22/0.55 => ( ( im @ Z )
% 0.22/0.55 = zero_zero_real ) ) ).
% 0.22/0.55
% 0.22/0.55 % Im_eq_0
% 0.22/0.55 thf(fact_8569_cmod__Im__le__iff,axiom,
% 0.22/0.55 ! [X: complex,Y4: complex] :
% 0.22/0.55 ( ( ( re @ X )
% 0.22/0.55 = ( re @ Y4 ) )
% 0.22/0.55 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) )
% 0.22/0.55 = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( abs_abs_real @ ( im @ Y4 ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cmod_Im_le_iff
% 0.22/0.55 thf(fact_8570_cmod__Re__le__iff,axiom,
% 0.22/0.55 ! [X: complex,Y4: complex] :
% 0.22/0.55 ( ( ( im @ X )
% 0.22/0.55 = ( im @ Y4 ) )
% 0.22/0.55 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) )
% 0.22/0.55 = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( abs_abs_real @ ( re @ Y4 ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cmod_Re_le_iff
% 0.22/0.55 thf(fact_8571_times__complex_Osimps_I1_J,axiom,
% 0.22/0.55 ! [X: complex,Y4: complex] :
% 0.22/0.55 ( ( re @ ( times_times_complex @ X @ Y4 ) )
% 0.22/0.55 = ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y4 ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y4 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % times_complex.simps(1)
% 0.22/0.55 thf(fact_8572_minus__complex_Ocode,axiom,
% 0.22/0.55 ( minus_minus_complex
% 0.22/0.55 = ( ^ [X3: complex,Y6: complex] : ( complex2 @ ( minus_minus_real @ ( re @ X3 ) @ ( re @ Y6 ) ) @ ( minus_minus_real @ ( im @ X3 ) @ ( im @ Y6 ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % minus_complex.code
% 0.22/0.55 thf(fact_8573_csqrt__principal,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
% 0.22/0.55 | ( ( ( re @ ( csqrt @ Z ) )
% 0.22/0.55 = zero_zero_real )
% 0.22/0.55 & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % csqrt_principal
% 0.22/0.55 thf(fact_8574_cmod__le,axiom,
% 0.22/0.55 ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cmod_le
% 0.22/0.55 thf(fact_8575_sin__n__Im__cis__pow__n,axiom,
% 0.22/0.55 ! [N: nat,A: real] :
% 0.22/0.55 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 0.22/0.55 = ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % sin_n_Im_cis_pow_n
% 0.22/0.55 thf(fact_8576_Re__exp,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( re @ ( exp_complex @ Z ) )
% 0.22/0.55 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Re_exp
% 0.22/0.55 thf(fact_8577_Im__exp,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( im @ ( exp_complex @ Z ) )
% 0.22/0.55 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Im_exp
% 0.22/0.55 thf(fact_8578_times__complex_Ocode,axiom,
% 0.22/0.55 ( times_times_complex
% 0.22/0.55 = ( ^ [X3: complex,Y6: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y6 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y6 ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y6 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y6 ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % times_complex.code
% 0.22/0.55 thf(fact_8579_cmod__power2,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.55 = ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cmod_power2
% 0.22/0.55 thf(fact_8580_Im__power2,axiom,
% 0.22/0.55 ! [X: complex] :
% 0.22/0.55 ( ( im @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.55 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Im_power2
% 0.22/0.55 thf(fact_8581_Re__power2,axiom,
% 0.22/0.55 ! [X: complex] :
% 0.22/0.55 ( ( re @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.55 = ( minus_minus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Re_power2
% 0.22/0.55 thf(fact_8582_complex__eq__0,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( Z = zero_zero_complex )
% 0.22/0.55 = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.55 = zero_zero_real ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_eq_0
% 0.22/0.55 thf(fact_8583_norm__complex__def,axiom,
% 0.22/0.55 ( real_V1022390504157884413omplex
% 0.22/0.55 = ( ^ [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 ) ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % norm_complex_def
% 0.22/0.55 thf(fact_8584_inverse__complex_Osimps_I1_J,axiom,
% 0.22/0.55 ! [X: complex] :
% 0.22/0.55 ( ( re @ ( invers8013647133539491842omplex @ X ) )
% 0.22/0.55 = ( divide_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % inverse_complex.simps(1)
% 0.22/0.55 thf(fact_8585_complex__neq__0,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( Z != zero_zero_complex )
% 0.22/0.55 = ( 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 ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_neq_0
% 0.22/0.55 thf(fact_8586_Re__divide,axiom,
% 0.22/0.55 ! [X: complex,Y4: complex] :
% 0.22/0.55 ( ( re @ ( divide1717551699836669952omplex @ X @ Y4 ) )
% 0.22/0.55 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y4 ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y4 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Re_divide
% 0.22/0.55 thf(fact_8587_csqrt__unique,axiom,
% 0.22/0.55 ! [W: complex,Z: complex] :
% 0.22/0.55 ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.55 = Z )
% 0.22/0.55 => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
% 0.22/0.55 | ( ( ( re @ W )
% 0.22/0.55 = zero_zero_real )
% 0.22/0.55 & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
% 0.22/0.55 => ( ( csqrt @ Z )
% 0.22/0.55 = W ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % csqrt_unique
% 0.22/0.55 thf(fact_8588_csqrt__square,axiom,
% 0.22/0.55 ! [B: complex] :
% 0.22/0.55 ( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
% 0.22/0.55 | ( ( ( re @ B )
% 0.22/0.55 = zero_zero_real )
% 0.22/0.55 & ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
% 0.22/0.55 => ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.55 = B ) ) ).
% 0.22/0.55
% 0.22/0.55 % csqrt_square
% 0.22/0.55 thf(fact_8589_inverse__complex_Osimps_I2_J,axiom,
% 0.22/0.55 ! [X: complex] :
% 0.22/0.55 ( ( im @ ( invers8013647133539491842omplex @ X ) )
% 0.22/0.55 = ( 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 ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % inverse_complex.simps(2)
% 0.22/0.55 thf(fact_8590_Im__divide,axiom,
% 0.22/0.55 ! [X: complex,Y4: complex] :
% 0.22/0.55 ( ( im @ ( divide1717551699836669952omplex @ X @ Y4 ) )
% 0.22/0.55 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y4 ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y4 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Im_divide
% 0.22/0.55 thf(fact_8591_complex__abs__le__norm,axiom,
% 0.22/0.55 ! [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 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_abs_le_norm
% 0.22/0.55 thf(fact_8592_complex__unit__circle,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( Z != zero_zero_complex )
% 0.22/0.55 => ( ( 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 ) ) ) )
% 0.22/0.55 = one_one_real ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_unit_circle
% 0.22/0.55 thf(fact_8593_inverse__complex_Ocode,axiom,
% 0.22/0.55 ( invers8013647133539491842omplex
% 0.22/0.55 = ( ^ [X3: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X3 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X3 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % inverse_complex.code
% 0.22/0.55 thf(fact_8594_Im__Reals__divide,axiom,
% 0.22/0.55 ! [R2: complex,Z: complex] :
% 0.22/0.55 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 0.22/0.55 => ( ( im @ ( divide1717551699836669952omplex @ R2 @ Z ) )
% 0.22/0.55 = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R2 ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Im_Reals_divide
% 0.22/0.55 thf(fact_8595_length__mul__elem,axiom,
% 0.22/0.55 ! [Xs: list_list_real,N: nat] :
% 0.22/0.55 ( ! [X4: list_real] :
% 0.22/0.55 ( ( member_list_real @ X4 @ ( set_list_real2 @ Xs ) )
% 0.22/0.55 => ( ( size_size_list_real @ X4 )
% 0.22/0.55 = N ) )
% 0.22/0.55 => ( ( size_size_list_real @ ( concat_real @ Xs ) )
% 0.22/0.55 = ( times_times_nat @ ( size_s6660260683639930848t_real @ Xs ) @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % length_mul_elem
% 0.22/0.55 thf(fact_8596_length__mul__elem,axiom,
% 0.22/0.55 ! [Xs: list_list_o,N: nat] :
% 0.22/0.55 ( ! [X4: list_o] :
% 0.22/0.55 ( ( member_list_o @ X4 @ ( set_list_o2 @ Xs ) )
% 0.22/0.55 => ( ( size_size_list_o @ X4 )
% 0.22/0.55 = N ) )
% 0.22/0.55 => ( ( size_size_list_o @ ( concat_o @ Xs ) )
% 0.22/0.55 = ( times_times_nat @ ( size_s2710708370519433104list_o @ Xs ) @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % length_mul_elem
% 0.22/0.55 thf(fact_8597_length__mul__elem,axiom,
% 0.22/0.55 ! [Xs: list_list_nat,N: nat] :
% 0.22/0.55 ( ! [X4: list_nat] :
% 0.22/0.55 ( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs ) )
% 0.22/0.55 => ( ( size_size_list_nat @ X4 )
% 0.22/0.55 = N ) )
% 0.22/0.55 => ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
% 0.22/0.55 = ( times_times_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % length_mul_elem
% 0.22/0.55 thf(fact_8598_length__mul__elem,axiom,
% 0.22/0.55 ! [Xs: list_list_int,N: nat] :
% 0.22/0.55 ( ! [X4: list_int] :
% 0.22/0.55 ( ( member_list_int @ X4 @ ( set_list_int2 @ Xs ) )
% 0.22/0.55 => ( ( size_size_list_int @ X4 )
% 0.22/0.55 = N ) )
% 0.22/0.55 => ( ( size_size_list_int @ ( concat_int @ Xs ) )
% 0.22/0.55 = ( times_times_nat @ ( size_s533118279054570080st_int @ Xs ) @ N ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % length_mul_elem
% 0.22/0.55 thf(fact_8599_Re__Reals__divide,axiom,
% 0.22/0.55 ! [R2: complex,Z: complex] :
% 0.22/0.55 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 0.22/0.55 => ( ( re @ ( divide1717551699836669952omplex @ R2 @ Z ) )
% 0.22/0.55 = ( divide_divide_real @ ( times_times_real @ ( re @ R2 ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Re_Reals_divide
% 0.22/0.55 thf(fact_8600_complex__diff__cnj,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
% 0.22/0.55 = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_diff_cnj
% 0.22/0.55 thf(fact_8601_complex__cnj__zero__iff,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( ( cnj @ Z )
% 0.22/0.55 = zero_zero_complex )
% 0.22/0.55 = ( Z = zero_zero_complex ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_cnj_zero_iff
% 0.22/0.55 thf(fact_8602_complex__cnj__zero,axiom,
% 0.22/0.55 ( ( cnj @ zero_zero_complex )
% 0.22/0.55 = zero_zero_complex ) ).
% 0.22/0.55
% 0.22/0.55 % complex_cnj_zero
% 0.22/0.55 thf(fact_8603_complex__cnj__diff,axiom,
% 0.22/0.55 ! [X: complex,Y4: complex] :
% 0.22/0.55 ( ( cnj @ ( minus_minus_complex @ X @ Y4 ) )
% 0.22/0.55 = ( minus_minus_complex @ ( cnj @ X ) @ ( cnj @ Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_cnj_diff
% 0.22/0.55 thf(fact_8604_complex__In__mult__cnj__zero,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( im @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 0.22/0.55 = zero_zero_real ) ).
% 0.22/0.55
% 0.22/0.55 % complex_In_mult_cnj_zero
% 0.22/0.55 thf(fact_8605_Re__divide__Reals,axiom,
% 0.22/0.55 ! [R2: complex,Z: complex] :
% 0.22/0.55 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 0.22/0.55 => ( ( re @ ( divide1717551699836669952omplex @ Z @ R2 ) )
% 0.22/0.55 = ( divide_divide_real @ ( re @ Z ) @ ( re @ R2 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Re_divide_Reals
% 0.22/0.55 thf(fact_8606_imaginary__eq__real__iff,axiom,
% 0.22/0.55 ! [Y4: complex,X: complex] :
% 0.22/0.55 ( ( member_complex @ Y4 @ real_V2521375963428798218omplex )
% 0.22/0.55 => ( ( member_complex @ X @ real_V2521375963428798218omplex )
% 0.22/0.55 => ( ( ( times_times_complex @ imaginary_unit @ Y4 )
% 0.22/0.55 = X )
% 0.22/0.55 = ( ( X = zero_zero_complex )
% 0.22/0.55 & ( Y4 = zero_zero_complex ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % imaginary_eq_real_iff
% 0.22/0.55 thf(fact_8607_real__eq__imaginary__iff,axiom,
% 0.22/0.55 ! [Y4: complex,X: complex] :
% 0.22/0.55 ( ( member_complex @ Y4 @ real_V2521375963428798218omplex )
% 0.22/0.55 => ( ( member_complex @ X @ real_V2521375963428798218omplex )
% 0.22/0.55 => ( ( X
% 0.22/0.55 = ( times_times_complex @ imaginary_unit @ Y4 ) )
% 0.22/0.55 = ( ( X = zero_zero_complex )
% 0.22/0.55 & ( Y4 = zero_zero_complex ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % real_eq_imaginary_iff
% 0.22/0.55 thf(fact_8608_Im__divide__Reals,axiom,
% 0.22/0.55 ! [R2: complex,Z: complex] :
% 0.22/0.55 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 0.22/0.55 => ( ( im @ ( divide1717551699836669952omplex @ Z @ R2 ) )
% 0.22/0.55 = ( divide_divide_real @ ( im @ Z ) @ ( re @ R2 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Im_divide_Reals
% 0.22/0.55 thf(fact_8609_Reals__of__nat,axiom,
% 0.22/0.55 ! [N: nat] : ( member_complex @ ( semiri8010041392384452111omplex @ N ) @ real_V2521375963428798218omplex ) ).
% 0.22/0.55
% 0.22/0.55 % Reals_of_nat
% 0.22/0.55 thf(fact_8610_Reals__of__nat,axiom,
% 0.22/0.55 ! [N: nat] : ( member_real @ ( semiri5074537144036343181t_real @ N ) @ real_V470468836141973256s_real ) ).
% 0.22/0.55
% 0.22/0.55 % Reals_of_nat
% 0.22/0.55 thf(fact_8611_Reals__divide,axiom,
% 0.22/0.55 ! [A: complex,B: complex] :
% 0.22/0.55 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 0.22/0.55 => ( ( member_complex @ B @ real_V2521375963428798218omplex )
% 0.22/0.55 => ( member_complex @ ( divide1717551699836669952omplex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Reals_divide
% 0.22/0.55 thf(fact_8612_Reals__divide,axiom,
% 0.22/0.55 ! [A: real,B: real] :
% 0.22/0.55 ( ( member_real @ A @ real_V470468836141973256s_real )
% 0.22/0.55 => ( ( member_real @ B @ real_V470468836141973256s_real )
% 0.22/0.55 => ( member_real @ ( divide_divide_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Reals_divide
% 0.22/0.55 thf(fact_8613_Reals__diff,axiom,
% 0.22/0.55 ! [A: real,B: real] :
% 0.22/0.55 ( ( member_real @ A @ real_V470468836141973256s_real )
% 0.22/0.55 => ( ( member_real @ B @ real_V470468836141973256s_real )
% 0.22/0.55 => ( member_real @ ( minus_minus_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Reals_diff
% 0.22/0.55 thf(fact_8614_Reals__diff,axiom,
% 0.22/0.55 ! [A: complex,B: complex] :
% 0.22/0.55 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 0.22/0.55 => ( ( member_complex @ B @ real_V2521375963428798218omplex )
% 0.22/0.55 => ( member_complex @ ( minus_minus_complex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Reals_diff
% 0.22/0.55 thf(fact_8615_Reals__mult,axiom,
% 0.22/0.55 ! [A: real,B: real] :
% 0.22/0.55 ( ( member_real @ A @ real_V470468836141973256s_real )
% 0.22/0.55 => ( ( member_real @ B @ real_V470468836141973256s_real )
% 0.22/0.55 => ( member_real @ ( times_times_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Reals_mult
% 0.22/0.55 thf(fact_8616_Reals__mult,axiom,
% 0.22/0.55 ! [A: complex,B: complex] :
% 0.22/0.55 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 0.22/0.55 => ( ( member_complex @ B @ real_V2521375963428798218omplex )
% 0.22/0.55 => ( member_complex @ ( times_times_complex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Reals_mult
% 0.22/0.55 thf(fact_8617_Reals__numeral,axiom,
% 0.22/0.55 ! [W: num] : ( member_complex @ ( numera6690914467698888265omplex @ W ) @ real_V2521375963428798218omplex ) ).
% 0.22/0.55
% 0.22/0.55 % Reals_numeral
% 0.22/0.55 thf(fact_8618_Reals__numeral,axiom,
% 0.22/0.55 ! [W: num] : ( member_real @ ( numeral_numeral_real @ W ) @ real_V470468836141973256s_real ) ).
% 0.22/0.55
% 0.22/0.55 % Reals_numeral
% 0.22/0.55 thf(fact_8619_Reals__0,axiom,
% 0.22/0.55 member_real @ zero_zero_real @ real_V470468836141973256s_real ).
% 0.22/0.55
% 0.22/0.55 % Reals_0
% 0.22/0.55 thf(fact_8620_Reals__0,axiom,
% 0.22/0.55 member_complex @ zero_zero_complex @ real_V2521375963428798218omplex ).
% 0.22/0.55
% 0.22/0.55 % Reals_0
% 0.22/0.55 thf(fact_8621_complex__is__Real__iff,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( member_complex @ Z @ real_V2521375963428798218omplex )
% 0.22/0.55 = ( ( im @ Z )
% 0.22/0.55 = zero_zero_real ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_is_Real_iff
% 0.22/0.55 thf(fact_8622_nonzero__Reals__divide,axiom,
% 0.22/0.55 ! [A: complex,B: complex] :
% 0.22/0.55 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 0.22/0.55 => ( ( member_complex @ B @ real_V2521375963428798218omplex )
% 0.22/0.55 => ( ( B != zero_zero_complex )
% 0.22/0.55 => ( member_complex @ ( divide1717551699836669952omplex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % nonzero_Reals_divide
% 0.22/0.55 thf(fact_8623_nonzero__Reals__divide,axiom,
% 0.22/0.55 ! [A: real,B: real] :
% 0.22/0.55 ( ( member_real @ A @ real_V470468836141973256s_real )
% 0.22/0.55 => ( ( member_real @ B @ real_V470468836141973256s_real )
% 0.22/0.55 => ( ( B != zero_zero_real )
% 0.22/0.55 => ( member_real @ ( divide_divide_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % nonzero_Reals_divide
% 0.22/0.55 thf(fact_8624_Complex__in__Reals,axiom,
% 0.22/0.55 ! [X: real] : ( member_complex @ ( complex2 @ X @ zero_zero_real ) @ real_V2521375963428798218omplex ) ).
% 0.22/0.55
% 0.22/0.55 % Complex_in_Reals
% 0.22/0.55 thf(fact_8625_nonzero__Reals__inverse,axiom,
% 0.22/0.55 ! [A: real] :
% 0.22/0.55 ( ( member_real @ A @ real_V470468836141973256s_real )
% 0.22/0.55 => ( ( A != zero_zero_real )
% 0.22/0.55 => ( member_real @ ( inverse_inverse_real @ A ) @ real_V470468836141973256s_real ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % nonzero_Reals_inverse
% 0.22/0.55 thf(fact_8626_nonzero__Reals__inverse,axiom,
% 0.22/0.55 ! [A: complex] :
% 0.22/0.55 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 0.22/0.55 => ( ( A != zero_zero_complex )
% 0.22/0.55 => ( member_complex @ ( invers8013647133539491842omplex @ A ) @ real_V2521375963428798218omplex ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % nonzero_Reals_inverse
% 0.22/0.55 thf(fact_8627_Re__complex__div__eq__0,axiom,
% 0.22/0.55 ! [A: complex,B: complex] :
% 0.22/0.55 ( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.22/0.55 = zero_zero_real )
% 0.22/0.55 = ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 0.22/0.55 = zero_zero_real ) ) ).
% 0.22/0.55
% 0.22/0.55 % Re_complex_div_eq_0
% 0.22/0.55 thf(fact_8628_Im__complex__div__eq__0,axiom,
% 0.22/0.55 ! [A: complex,B: complex] :
% 0.22/0.55 ( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.22/0.55 = zero_zero_real )
% 0.22/0.55 = ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 0.22/0.55 = zero_zero_real ) ) ).
% 0.22/0.55
% 0.22/0.55 % Im_complex_div_eq_0
% 0.22/0.55 thf(fact_8629_Re__complex__div__gt__0,axiom,
% 0.22/0.55 ! [A: complex,B: complex] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.22/0.55 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Re_complex_div_gt_0
% 0.22/0.55 thf(fact_8630_Re__complex__div__lt__0,axiom,
% 0.22/0.55 ! [A: complex,B: complex] :
% 0.22/0.55 ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.22/0.55 = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.22/0.55
% 0.22/0.55 % Re_complex_div_lt_0
% 0.22/0.55 thf(fact_8631_Re__complex__div__ge__0,axiom,
% 0.22/0.55 ! [A: complex,B: complex] :
% 0.22/0.55 ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.22/0.55 = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Re_complex_div_ge_0
% 0.22/0.55 thf(fact_8632_Re__complex__div__le__0,axiom,
% 0.22/0.55 ! [A: complex,B: complex] :
% 0.22/0.55 ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.22/0.55 = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.22/0.55
% 0.22/0.55 % Re_complex_div_le_0
% 0.22/0.55 thf(fact_8633_Im__complex__div__gt__0,axiom,
% 0.22/0.55 ! [A: complex,B: complex] :
% 0.22/0.55 ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.22/0.55 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Im_complex_div_gt_0
% 0.22/0.55 thf(fact_8634_Im__complex__div__lt__0,axiom,
% 0.22/0.55 ! [A: complex,B: complex] :
% 0.22/0.55 ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.22/0.55 = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.22/0.55
% 0.22/0.55 % Im_complex_div_lt_0
% 0.22/0.55 thf(fact_8635_Im__complex__div__ge__0,axiom,
% 0.22/0.55 ! [A: complex,B: complex] :
% 0.22/0.55 ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.22/0.55 = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % Im_complex_div_ge_0
% 0.22/0.55 thf(fact_8636_Im__complex__div__le__0,axiom,
% 0.22/0.55 ! [A: complex,B: complex] :
% 0.22/0.55 ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.22/0.55 = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.22/0.55
% 0.22/0.55 % Im_complex_div_le_0
% 0.22/0.55 thf(fact_8637_complex__mod__mult__cnj,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 0.22/0.55 = ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_mod_mult_cnj
% 0.22/0.55 thf(fact_8638_complex__div__gt__0,axiom,
% 0.22/0.55 ! [A: complex,B: complex] :
% 0.22/0.55 ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.22/0.55 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
% 0.22/0.55 & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.22/0.55 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_div_gt_0
% 0.22/0.55 thf(fact_8639_complex__norm__square,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.55 = ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_norm_square
% 0.22/0.55 thf(fact_8640_complex__add__cnj,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
% 0.22/0.55 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_add_cnj
% 0.22/0.55 thf(fact_8641_complex__div__cnj,axiom,
% 0.22/0.55 ( divide1717551699836669952omplex
% 0.22/0.55 = ( ^ [A3: complex,B2: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ ( cnj @ B2 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_div_cnj
% 0.22/0.55 thf(fact_8642_cnj__add__mult__eq__Re,axiom,
% 0.22/0.55 ! [Z: complex,W: complex] :
% 0.22/0.55 ( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
% 0.22/0.55 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % cnj_add_mult_eq_Re
% 0.22/0.55 thf(fact_8643_complex__mult__cnj,axiom,
% 0.22/0.55 ! [Z: complex] :
% 0.22/0.55 ( ( times_times_complex @ Z @ ( cnj @ Z ) )
% 0.22/0.55 = ( 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 ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % complex_mult_cnj
% 0.22/0.55 thf(fact_8644_or__int__rec,axiom,
% 0.22/0.55 ( bit_se1409905431419307370or_int
% 0.22/0.55 = ( ^ [K3: int,L3: int] :
% 0.22/0.55 ( plus_plus_int
% 0.22/0.55 @ ( zero_n2684676970156552555ol_int
% 0.22/0.55 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.22/0.55 | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
% 0.22/0.55 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_int_rec
% 0.22/0.55 thf(fact_8645_drop__bit__numeral__minus__bit1,axiom,
% 0.22/0.55 ! [L: num,K: num] :
% 0.22/0.55 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.22/0.55 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_numeral_minus_bit1
% 0.22/0.55 thf(fact_8646_or__numerals_I7_J,axiom,
% 0.22/0.55 ! [X: num,Y4: num] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y4 ) ) )
% 0.22/0.55 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_numerals(7)
% 0.22/0.55 thf(fact_8647_or__numerals_I7_J,axiom,
% 0.22/0.55 ! [X: num,Y4: num] :
% 0.22/0.55 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 0.22/0.55 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_numerals(7)
% 0.22/0.55 thf(fact_8648_or__numerals_I6_J,axiom,
% 0.22/0.55 ! [X: num,Y4: num] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y4 ) ) )
% 0.22/0.55 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_numerals(6)
% 0.22/0.55 thf(fact_8649_or__numerals_I6_J,axiom,
% 0.22/0.55 ! [X: num,Y4: num] :
% 0.22/0.55 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 0.22/0.55 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_numerals(6)
% 0.22/0.55 thf(fact_8650_or_Oright__idem,axiom,
% 0.22/0.55 ! [A: int,B: int] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ ( bit_se1409905431419307370or_int @ A @ B ) @ B )
% 0.22/0.55 = ( bit_se1409905431419307370or_int @ A @ B ) ) ).
% 0.22/0.55
% 0.22/0.55 % or.right_idem
% 0.22/0.55 thf(fact_8651_or_Oright__idem,axiom,
% 0.22/0.55 ! [A: nat,B: nat] :
% 0.22/0.55 ( ( bit_se1412395901928357646or_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ B )
% 0.22/0.55 = ( bit_se1412395901928357646or_nat @ A @ B ) ) ).
% 0.22/0.55
% 0.22/0.55 % or.right_idem
% 0.22/0.55 thf(fact_8652_or_Oleft__idem,axiom,
% 0.22/0.55 ! [A: int,B: int] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 0.22/0.55 = ( bit_se1409905431419307370or_int @ A @ B ) ) ).
% 0.22/0.55
% 0.22/0.55 % or.left_idem
% 0.22/0.55 thf(fact_8653_or_Oleft__idem,axiom,
% 0.22/0.55 ! [A: nat,B: nat] :
% 0.22/0.55 ( ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 0.22/0.55 = ( bit_se1412395901928357646or_nat @ A @ B ) ) ).
% 0.22/0.55
% 0.22/0.55 % or.left_idem
% 0.22/0.55 thf(fact_8654_or_Oidem,axiom,
% 0.22/0.55 ! [A: int] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ A @ A )
% 0.22/0.55 = A ) ).
% 0.22/0.55
% 0.22/0.55 % or.idem
% 0.22/0.55 thf(fact_8655_or_Oidem,axiom,
% 0.22/0.55 ! [A: nat] :
% 0.22/0.55 ( ( bit_se1412395901928357646or_nat @ A @ A )
% 0.22/0.55 = A ) ).
% 0.22/0.55
% 0.22/0.55 % or.idem
% 0.22/0.55 thf(fact_8656_or_Oleft__neutral,axiom,
% 0.22/0.55 ! [A: int] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ zero_zero_int @ A )
% 0.22/0.55 = A ) ).
% 0.22/0.55
% 0.22/0.55 % or.left_neutral
% 0.22/0.55 thf(fact_8657_or_Oleft__neutral,axiom,
% 0.22/0.55 ! [A: nat] :
% 0.22/0.55 ( ( bit_se1412395901928357646or_nat @ zero_zero_nat @ A )
% 0.22/0.55 = A ) ).
% 0.22/0.55
% 0.22/0.55 % or.left_neutral
% 0.22/0.55 thf(fact_8658_or_Oright__neutral,axiom,
% 0.22/0.55 ! [A: int] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ A @ zero_zero_int )
% 0.22/0.55 = A ) ).
% 0.22/0.55
% 0.22/0.55 % or.right_neutral
% 0.22/0.55 thf(fact_8659_or_Oright__neutral,axiom,
% 0.22/0.55 ! [A: nat] :
% 0.22/0.55 ( ( bit_se1412395901928357646or_nat @ A @ zero_zero_nat )
% 0.22/0.55 = A ) ).
% 0.22/0.55
% 0.22/0.55 % or.right_neutral
% 0.22/0.55 thf(fact_8660_drop__bit__of__0,axiom,
% 0.22/0.55 ! [N: nat] :
% 0.22/0.55 ( ( bit_se8568078237143864401it_int @ N @ zero_zero_int )
% 0.22/0.55 = zero_zero_int ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_of_0
% 0.22/0.55 thf(fact_8661_drop__bit__of__0,axiom,
% 0.22/0.55 ! [N: nat] :
% 0.22/0.55 ( ( bit_se8570568707652914677it_nat @ N @ zero_zero_nat )
% 0.22/0.55 = zero_zero_nat ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_of_0
% 0.22/0.55 thf(fact_8662_drop__bit__drop__bit,axiom,
% 0.22/0.55 ! [M: nat,N: nat,A: int] :
% 0.22/0.55 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se8568078237143864401it_int @ N @ A ) )
% 0.22/0.55 = ( bit_se8568078237143864401it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_drop_bit
% 0.22/0.55 thf(fact_8663_drop__bit__drop__bit,axiom,
% 0.22/0.55 ! [M: nat,N: nat,A: nat] :
% 0.22/0.55 ( ( bit_se8570568707652914677it_nat @ M @ ( bit_se8570568707652914677it_nat @ N @ A ) )
% 0.22/0.55 = ( bit_se8570568707652914677it_nat @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_drop_bit
% 0.22/0.55 thf(fact_8664_take__bit__or,axiom,
% 0.22/0.55 ! [N: nat,A: int,B: int] :
% 0.22/0.55 ( ( bit_se2923211474154528505it_int @ N @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 0.22/0.55 = ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % take_bit_or
% 0.22/0.55 thf(fact_8665_take__bit__or,axiom,
% 0.22/0.55 ! [N: nat,A: nat,B: nat] :
% 0.22/0.55 ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 0.22/0.55 = ( bit_se1412395901928357646or_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % take_bit_or
% 0.22/0.55 thf(fact_8666_push__bit__or,axiom,
% 0.22/0.55 ! [N: nat,A: int,B: int] :
% 0.22/0.55 ( ( bit_se545348938243370406it_int @ N @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 0.22/0.55 = ( bit_se1409905431419307370or_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_or
% 0.22/0.55 thf(fact_8667_push__bit__or,axiom,
% 0.22/0.55 ! [N: nat,A: nat,B: nat] :
% 0.22/0.55 ( ( bit_se547839408752420682it_nat @ N @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 0.22/0.55 = ( bit_se1412395901928357646or_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % push_bit_or
% 0.22/0.55 thf(fact_8668_drop__bit__and,axiom,
% 0.22/0.55 ! [N: nat,A: int,B: int] :
% 0.22/0.55 ( ( bit_se8568078237143864401it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 0.22/0.55 = ( bit_se725231765392027082nd_int @ ( bit_se8568078237143864401it_int @ N @ A ) @ ( bit_se8568078237143864401it_int @ N @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_and
% 0.22/0.55 thf(fact_8669_drop__bit__and,axiom,
% 0.22/0.55 ! [N: nat,A: nat,B: nat] :
% 0.22/0.55 ( ( bit_se8570568707652914677it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 0.22/0.55 = ( bit_se727722235901077358nd_nat @ ( bit_se8570568707652914677it_nat @ N @ A ) @ ( bit_se8570568707652914677it_nat @ N @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_and
% 0.22/0.55 thf(fact_8670_drop__bit__or,axiom,
% 0.22/0.55 ! [N: nat,A: int,B: int] :
% 0.22/0.55 ( ( bit_se8568078237143864401it_int @ N @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 0.22/0.55 = ( bit_se1409905431419307370or_int @ ( bit_se8568078237143864401it_int @ N @ A ) @ ( bit_se8568078237143864401it_int @ N @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_or
% 0.22/0.55 thf(fact_8671_drop__bit__or,axiom,
% 0.22/0.55 ! [N: nat,A: nat,B: nat] :
% 0.22/0.55 ( ( bit_se8570568707652914677it_nat @ N @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 0.22/0.55 = ( bit_se1412395901928357646or_nat @ ( bit_se8570568707652914677it_nat @ N @ A ) @ ( bit_se8570568707652914677it_nat @ N @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_or
% 0.22/0.55 thf(fact_8672_drop__bit__xor,axiom,
% 0.22/0.55 ! [N: nat,A: int,B: int] :
% 0.22/0.55 ( ( bit_se8568078237143864401it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B ) )
% 0.22/0.55 = ( bit_se6526347334894502574or_int @ ( bit_se8568078237143864401it_int @ N @ A ) @ ( bit_se8568078237143864401it_int @ N @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_xor
% 0.22/0.55 thf(fact_8673_drop__bit__xor,axiom,
% 0.22/0.55 ! [N: nat,A: nat,B: nat] :
% 0.22/0.55 ( ( bit_se8570568707652914677it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B ) )
% 0.22/0.55 = ( bit_se6528837805403552850or_nat @ ( bit_se8570568707652914677it_nat @ N @ A ) @ ( bit_se8570568707652914677it_nat @ N @ B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_xor
% 0.22/0.55 thf(fact_8674_bit_Odisj__one__right,axiom,
% 0.22/0.55 ! [X: code_integer] :
% 0.22/0.55 ( ( bit_se1080825931792720795nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.55 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.disj_one_right
% 0.22/0.55 thf(fact_8675_bit_Odisj__one__right,axiom,
% 0.22/0.55 ! [X: int] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.55 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.disj_one_right
% 0.22/0.55 thf(fact_8676_bit_Odisj__one__left,axiom,
% 0.22/0.55 ! [X: code_integer] :
% 0.22/0.55 ( ( bit_se1080825931792720795nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 0.22/0.55 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.disj_one_left
% 0.22/0.55 thf(fact_8677_bit_Odisj__one__left,axiom,
% 0.22/0.55 ! [X: int] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 0.22/0.55 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.disj_one_left
% 0.22/0.55 thf(fact_8678_drop__bit__of__bool,axiom,
% 0.22/0.55 ! [N: nat,B: $o] :
% 0.22/0.55 ( ( bit_se3928097537394005634nteger @ N @ ( zero_n356916108424825756nteger @ B ) )
% 0.22/0.55 = ( zero_n356916108424825756nteger
% 0.22/0.55 @ ( ( N = zero_zero_nat )
% 0.22/0.55 & B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_of_bool
% 0.22/0.55 thf(fact_8679_drop__bit__of__bool,axiom,
% 0.22/0.55 ! [N: nat,B: $o] :
% 0.22/0.55 ( ( bit_se8568078237143864401it_int @ N @ ( zero_n2684676970156552555ol_int @ B ) )
% 0.22/0.55 = ( zero_n2684676970156552555ol_int
% 0.22/0.55 @ ( ( N = zero_zero_nat )
% 0.22/0.55 & B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_of_bool
% 0.22/0.55 thf(fact_8680_drop__bit__of__bool,axiom,
% 0.22/0.55 ! [N: nat,B: $o] :
% 0.22/0.55 ( ( bit_se8570568707652914677it_nat @ N @ ( zero_n2687167440665602831ol_nat @ B ) )
% 0.22/0.55 = ( zero_n2687167440665602831ol_nat
% 0.22/0.55 @ ( ( N = zero_zero_nat )
% 0.22/0.55 & B ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_of_bool
% 0.22/0.55 thf(fact_8681_or__nonnegative__int__iff,axiom,
% 0.22/0.55 ! [K: int,L: int] :
% 0.22/0.55 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
% 0.22/0.55 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.22/0.55 & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_nonnegative_int_iff
% 0.22/0.55 thf(fact_8682_or__negative__int__iff,axiom,
% 0.22/0.55 ! [K: int,L: int] :
% 0.22/0.55 ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
% 0.22/0.55 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.22/0.55 | ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_negative_int_iff
% 0.22/0.55 thf(fact_8683_bit_Ode__Morgan__disj,axiom,
% 0.22/0.55 ! [X: int,Y4: int] :
% 0.22/0.55 ( ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ X @ Y4 ) )
% 0.22/0.55 = ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X ) @ ( bit_ri7919022796975470100ot_int @ Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.de_Morgan_disj
% 0.22/0.55 thf(fact_8684_bit_Ode__Morgan__conj,axiom,
% 0.22/0.55 ! [X: int,Y4: int] :
% 0.22/0.55 ( ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ X @ Y4 ) )
% 0.22/0.55 = ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ ( bit_ri7919022796975470100ot_int @ Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.de_Morgan_conj
% 0.22/0.55 thf(fact_8685_drop__bit__nonnegative__int__iff,axiom,
% 0.22/0.55 ! [N: nat,K: int] :
% 0.22/0.55 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
% 0.22/0.55 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_nonnegative_int_iff
% 0.22/0.55 thf(fact_8686_drop__bit__negative__int__iff,axiom,
% 0.22/0.55 ! [N: nat,K: int] :
% 0.22/0.55 ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
% 0.22/0.55 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_negative_int_iff
% 0.22/0.55 thf(fact_8687_drop__bit__minus__one,axiom,
% 0.22/0.55 ! [N: nat] :
% 0.22/0.55 ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.55 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_minus_one
% 0.22/0.55 thf(fact_8688_or__nat__numerals_I2_J,axiom,
% 0.22/0.55 ! [Y4: num] :
% 0.22/0.55 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 0.22/0.55 = ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_nat_numerals(2)
% 0.22/0.55 thf(fact_8689_or__nat__numerals_I4_J,axiom,
% 0.22/0.55 ! [X: num] :
% 0.22/0.55 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.22/0.55 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_nat_numerals(4)
% 0.22/0.55 thf(fact_8690_or__numerals_I2_J,axiom,
% 0.22/0.55 ! [Y4: num] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y4 ) ) )
% 0.22/0.55 = ( numeral_numeral_int @ ( bit1 @ Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_numerals(2)
% 0.22/0.55 thf(fact_8691_or__numerals_I2_J,axiom,
% 0.22/0.55 ! [Y4: num] :
% 0.22/0.55 ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 0.22/0.55 = ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_numerals(2)
% 0.22/0.55 thf(fact_8692_or__numerals_I8_J,axiom,
% 0.22/0.55 ! [X: num] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
% 0.22/0.55 = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_numerals(8)
% 0.22/0.55 thf(fact_8693_or__numerals_I8_J,axiom,
% 0.22/0.55 ! [X: num] :
% 0.22/0.55 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
% 0.22/0.55 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_numerals(8)
% 0.22/0.55 thf(fact_8694_drop__bit__Suc__bit0,axiom,
% 0.22/0.55 ! [N: nat,K: num] :
% 0.22/0.55 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.22/0.55 = ( bit_se8568078237143864401it_int @ N @ ( numeral_numeral_int @ K ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_Suc_bit0
% 0.22/0.55 thf(fact_8695_drop__bit__Suc__bit0,axiom,
% 0.22/0.55 ! [N: nat,K: num] :
% 0.22/0.55 ( ( bit_se8570568707652914677it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.22/0.55 = ( bit_se8570568707652914677it_nat @ N @ ( numeral_numeral_nat @ K ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_Suc_bit0
% 0.22/0.55 thf(fact_8696_drop__bit__Suc__bit1,axiom,
% 0.22/0.55 ! [N: nat,K: num] :
% 0.22/0.55 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.22/0.55 = ( bit_se8568078237143864401it_int @ N @ ( numeral_numeral_int @ K ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_Suc_bit1
% 0.22/0.55 thf(fact_8697_drop__bit__Suc__bit1,axiom,
% 0.22/0.55 ! [N: nat,K: num] :
% 0.22/0.55 ( ( bit_se8570568707652914677it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.22/0.55 = ( bit_se8570568707652914677it_nat @ N @ ( numeral_numeral_nat @ K ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_Suc_bit1
% 0.22/0.55 thf(fact_8698_bit_Odisj__cancel__left,axiom,
% 0.22/0.55 ! [X: code_integer] :
% 0.22/0.55 ( ( bit_se1080825931792720795nteger @ ( bit_ri7632146776885996613nteger @ X ) @ X )
% 0.22/0.55 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.disj_cancel_left
% 0.22/0.55 thf(fact_8699_bit_Odisj__cancel__left,axiom,
% 0.22/0.55 ! [X: int] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
% 0.22/0.55 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.disj_cancel_left
% 0.22/0.55 thf(fact_8700_bit_Odisj__cancel__right,axiom,
% 0.22/0.55 ! [X: code_integer] :
% 0.22/0.55 ( ( bit_se1080825931792720795nteger @ X @ ( bit_ri7632146776885996613nteger @ X ) )
% 0.22/0.55 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.disj_cancel_right
% 0.22/0.55 thf(fact_8701_bit_Odisj__cancel__right,axiom,
% 0.22/0.55 ! [X: int] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
% 0.22/0.55 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.22/0.55
% 0.22/0.55 % bit.disj_cancel_right
% 0.22/0.55 thf(fact_8702_drop__bit__of__1,axiom,
% 0.22/0.55 ! [N: nat] :
% 0.22/0.55 ( ( bit_se3928097537394005634nteger @ N @ one_one_Code_integer )
% 0.22/0.55 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_of_1
% 0.22/0.55 thf(fact_8703_drop__bit__of__1,axiom,
% 0.22/0.55 ! [N: nat] :
% 0.22/0.55 ( ( bit_se8568078237143864401it_int @ N @ one_one_int )
% 0.22/0.55 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_of_1
% 0.22/0.55 thf(fact_8704_drop__bit__of__1,axiom,
% 0.22/0.55 ! [N: nat] :
% 0.22/0.55 ( ( bit_se8570568707652914677it_nat @ N @ one_one_nat )
% 0.22/0.55 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_of_1
% 0.22/0.55 thf(fact_8705_or__nat__numerals_I1_J,axiom,
% 0.22/0.55 ! [Y4: num] :
% 0.22/0.55 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 0.22/0.55 = ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_nat_numerals(1)
% 0.22/0.55 thf(fact_8706_or__nat__numerals_I3_J,axiom,
% 0.22/0.55 ! [X: num] :
% 0.22/0.55 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.22/0.55 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_nat_numerals(3)
% 0.22/0.55 thf(fact_8707_or__numerals_I3_J,axiom,
% 0.22/0.55 ! [X: num,Y4: num] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y4 ) ) )
% 0.22/0.55 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_numerals(3)
% 0.22/0.55 thf(fact_8708_or__numerals_I3_J,axiom,
% 0.22/0.55 ! [X: num,Y4: num] :
% 0.22/0.55 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 0.22/0.55 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_numerals(3)
% 0.22/0.55 thf(fact_8709_or__numerals_I1_J,axiom,
% 0.22/0.55 ! [Y4: num] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y4 ) ) )
% 0.22/0.55 = ( numeral_numeral_int @ ( bit1 @ Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_numerals(1)
% 0.22/0.55 thf(fact_8710_or__numerals_I1_J,axiom,
% 0.22/0.55 ! [Y4: num] :
% 0.22/0.55 ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 0.22/0.55 = ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_numerals(1)
% 0.22/0.55 thf(fact_8711_or__numerals_I5_J,axiom,
% 0.22/0.55 ! [X: num] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
% 0.22/0.55 = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_numerals(5)
% 0.22/0.55 thf(fact_8712_or__numerals_I5_J,axiom,
% 0.22/0.55 ! [X: num] :
% 0.22/0.55 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
% 0.22/0.55 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_numerals(5)
% 0.22/0.55 thf(fact_8713_drop__bit__numeral__bit0,axiom,
% 0.22/0.55 ! [L: num,K: num] :
% 0.22/0.55 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.22/0.55 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_numeral_bit0
% 0.22/0.55 thf(fact_8714_drop__bit__numeral__bit0,axiom,
% 0.22/0.55 ! [L: num,K: num] :
% 0.22/0.55 ( ( bit_se8570568707652914677it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.22/0.55 = ( bit_se8570568707652914677it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_numeral_bit0
% 0.22/0.55 thf(fact_8715_drop__bit__numeral__bit1,axiom,
% 0.22/0.55 ! [L: num,K: num] :
% 0.22/0.55 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.22/0.55 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_numeral_bit1
% 0.22/0.55 thf(fact_8716_drop__bit__numeral__bit1,axiom,
% 0.22/0.55 ! [L: num,K: num] :
% 0.22/0.55 ( ( bit_se8570568707652914677it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.22/0.55 = ( bit_se8570568707652914677it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_numeral_bit1
% 0.22/0.55 thf(fact_8717_or__minus__numerals_I6_J,axiom,
% 0.22/0.55 ! [N: num] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 0.22/0.55 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_minus_numerals(6)
% 0.22/0.55 thf(fact_8718_or__minus__numerals_I2_J,axiom,
% 0.22/0.55 ! [N: num] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.22/0.55 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_minus_numerals(2)
% 0.22/0.55 thf(fact_8719_drop__bit__Suc__minus__bit0,axiom,
% 0.22/0.55 ! [N: nat,K: num] :
% 0.22/0.55 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.22/0.55 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_Suc_minus_bit0
% 0.22/0.55 thf(fact_8720_drop__bit__numeral__minus__bit0,axiom,
% 0.22/0.55 ! [L: num,K: num] :
% 0.22/0.55 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.22/0.55 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_numeral_minus_bit0
% 0.22/0.55 thf(fact_8721_drop__bit__Suc__minus__bit1,axiom,
% 0.22/0.55 ! [N: nat,K: num] :
% 0.22/0.55 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.22/0.55 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % drop_bit_Suc_minus_bit1
% 0.22/0.55 thf(fact_8722_and__minus__minus__numerals,axiom,
% 0.22/0.55 ! [M: num,N: num] :
% 0.22/0.55 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.55 = ( 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 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % and_minus_minus_numerals
% 0.22/0.55 thf(fact_8723_or__minus__minus__numerals,axiom,
% 0.22/0.55 ! [M: num,N: num] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.55 = ( 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 ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_minus_minus_numerals
% 0.22/0.55 thf(fact_8724_or__numerals_I4_J,axiom,
% 0.22/0.55 ! [X: num,Y4: num] :
% 0.22/0.55 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y4 ) ) )
% 0.22/0.55 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ) ).
% 0.22/0.55
% 0.22/0.55 % or_numerals(4)
% 0.22/0.55 thf(fact_8725_or__numerals_I4_J,axiom,
% 0.22/0.56 ! [X: num,Y4: num] :
% 0.22/0.56 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 0.22/0.56 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_numerals(4)
% 0.22/0.56 thf(fact_8726_of__int__or__eq,axiom,
% 0.22/0.56 ! [K: int,L: int] :
% 0.22/0.56 ( ( ring_1_of_int_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
% 0.22/0.56 = ( bit_se1409905431419307370or_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % of_int_or_eq
% 0.22/0.56 thf(fact_8727_or_Oleft__commute,axiom,
% 0.22/0.56 ! [B: int,A: int,C: int] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ B @ ( bit_se1409905431419307370or_int @ A @ C ) )
% 0.22/0.56 = ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ B @ C ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or.left_commute
% 0.22/0.56 thf(fact_8728_or_Oleft__commute,axiom,
% 0.22/0.56 ! [B: nat,A: nat,C: nat] :
% 0.22/0.56 ( ( bit_se1412395901928357646or_nat @ B @ ( bit_se1412395901928357646or_nat @ A @ C ) )
% 0.22/0.56 = ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ B @ C ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or.left_commute
% 0.22/0.56 thf(fact_8729_or_Ocommute,axiom,
% 0.22/0.56 ( bit_se1409905431419307370or_int
% 0.22/0.56 = ( ^ [A3: int,B2: int] : ( bit_se1409905431419307370or_int @ B2 @ A3 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or.commute
% 0.22/0.56 thf(fact_8730_or_Ocommute,axiom,
% 0.22/0.56 ( bit_se1412395901928357646or_nat
% 0.22/0.56 = ( ^ [A3: nat,B2: nat] : ( bit_se1412395901928357646or_nat @ B2 @ A3 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or.commute
% 0.22/0.56 thf(fact_8731_or_Oassoc,axiom,
% 0.22/0.56 ! [A: int,B: int,C: int] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ ( bit_se1409905431419307370or_int @ A @ B ) @ C )
% 0.22/0.56 = ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ B @ C ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or.assoc
% 0.22/0.56 thf(fact_8732_or_Oassoc,axiom,
% 0.22/0.56 ! [A: nat,B: nat,C: nat] :
% 0.22/0.56 ( ( bit_se1412395901928357646or_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ C )
% 0.22/0.56 = ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ B @ C ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or.assoc
% 0.22/0.56 thf(fact_8733_bit_Oconj__disj__distrib,axiom,
% 0.22/0.56 ! [X: int,Y4: int,Z: int] :
% 0.22/0.56 ( ( bit_se725231765392027082nd_int @ X @ ( bit_se1409905431419307370or_int @ Y4 @ Z ) )
% 0.22/0.56 = ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ X @ Y4 ) @ ( bit_se725231765392027082nd_int @ X @ Z ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % bit.conj_disj_distrib
% 0.22/0.56 thf(fact_8734_bit_Odisj__conj__distrib,axiom,
% 0.22/0.56 ! [X: int,Y4: int,Z: int] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ X @ ( bit_se725231765392027082nd_int @ Y4 @ Z ) )
% 0.22/0.56 = ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ X @ Y4 ) @ ( bit_se1409905431419307370or_int @ X @ Z ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % bit.disj_conj_distrib
% 0.22/0.56 thf(fact_8735_bit_Oconj__disj__distrib2,axiom,
% 0.22/0.56 ! [Y4: int,Z: int,X: int] :
% 0.22/0.56 ( ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ Y4 @ Z ) @ X )
% 0.22/0.56 = ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ Y4 @ X ) @ ( bit_se725231765392027082nd_int @ Z @ X ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % bit.conj_disj_distrib2
% 0.22/0.56 thf(fact_8736_bit_Odisj__conj__distrib2,axiom,
% 0.22/0.56 ! [Y4: int,Z: int,X: int] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ Y4 @ Z ) @ X )
% 0.22/0.56 = ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ Y4 @ X ) @ ( bit_se1409905431419307370or_int @ Z @ X ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % bit.disj_conj_distrib2
% 0.22/0.56 thf(fact_8737_bit_Odisj__zero__right,axiom,
% 0.22/0.56 ! [X: int] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ X @ zero_zero_int )
% 0.22/0.56 = X ) ).
% 0.22/0.56
% 0.22/0.56 % bit.disj_zero_right
% 0.22/0.56 thf(fact_8738_or__eq__0__iff,axiom,
% 0.22/0.56 ! [A: int,B: int] :
% 0.22/0.56 ( ( ( bit_se1409905431419307370or_int @ A @ B )
% 0.22/0.56 = zero_zero_int )
% 0.22/0.56 = ( ( A = zero_zero_int )
% 0.22/0.56 & ( B = zero_zero_int ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_eq_0_iff
% 0.22/0.56 thf(fact_8739_or__eq__0__iff,axiom,
% 0.22/0.56 ! [A: nat,B: nat] :
% 0.22/0.56 ( ( ( bit_se1412395901928357646or_nat @ A @ B )
% 0.22/0.56 = zero_zero_nat )
% 0.22/0.56 = ( ( A = zero_zero_nat )
% 0.22/0.56 & ( B = zero_zero_nat ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_eq_0_iff
% 0.22/0.56 thf(fact_8740_bit__or__int__iff,axiom,
% 0.22/0.56 ! [K: int,L: int,N: nat] :
% 0.22/0.56 ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ N )
% 0.22/0.56 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.22/0.56 | ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % bit_or_int_iff
% 0.22/0.56 thf(fact_8741_bit__or__iff,axiom,
% 0.22/0.56 ! [A: int,B: int,N: nat] :
% 0.22/0.56 ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ A @ B ) @ N )
% 0.22/0.56 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.22/0.56 | ( bit_se1146084159140164899it_int @ B @ N ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % bit_or_iff
% 0.22/0.56 thf(fact_8742_bit__or__iff,axiom,
% 0.22/0.56 ! [A: nat,B: nat,N: nat] :
% 0.22/0.56 ( ( bit_se1148574629649215175it_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ N )
% 0.22/0.56 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.22/0.56 | ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % bit_or_iff
% 0.22/0.56 thf(fact_8743_or__nat__def,axiom,
% 0.22/0.56 ( bit_se1412395901928357646or_nat
% 0.22/0.56 = ( ^ [M5: nat,N4: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_nat_def
% 0.22/0.56 thf(fact_8744_of__nat__drop__bit,axiom,
% 0.22/0.56 ! [M: nat,N: nat] :
% 0.22/0.56 ( ( semiri1314217659103216013at_int @ ( bit_se8570568707652914677it_nat @ M @ N ) )
% 0.22/0.56 = ( bit_se8568078237143864401it_int @ M @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % of_nat_drop_bit
% 0.22/0.56 thf(fact_8745_of__nat__drop__bit,axiom,
% 0.22/0.56 ! [M: nat,N: nat] :
% 0.22/0.56 ( ( semiri1316708129612266289at_nat @ ( bit_se8570568707652914677it_nat @ M @ N ) )
% 0.22/0.56 = ( bit_se8570568707652914677it_nat @ M @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % of_nat_drop_bit
% 0.22/0.56 thf(fact_8746_drop__bit__of__nat,axiom,
% 0.22/0.56 ! [N: nat,M: nat] :
% 0.22/0.56 ( ( bit_se8568078237143864401it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 0.22/0.56 = ( semiri1314217659103216013at_int @ ( bit_se8570568707652914677it_nat @ N @ M ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % drop_bit_of_nat
% 0.22/0.56 thf(fact_8747_drop__bit__of__nat,axiom,
% 0.22/0.56 ! [N: nat,M: nat] :
% 0.22/0.56 ( ( bit_se8570568707652914677it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 0.22/0.56 = ( semiri1316708129612266289at_nat @ ( bit_se8570568707652914677it_nat @ N @ M ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % drop_bit_of_nat
% 0.22/0.56 thf(fact_8748_of__nat__or__eq,axiom,
% 0.22/0.56 ! [M: nat,N: nat] :
% 0.22/0.56 ( ( semiri1314217659103216013at_int @ ( bit_se1412395901928357646or_nat @ M @ N ) )
% 0.22/0.56 = ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % of_nat_or_eq
% 0.22/0.56 thf(fact_8749_of__nat__or__eq,axiom,
% 0.22/0.56 ! [M: nat,N: nat] :
% 0.22/0.56 ( ( semiri1316708129612266289at_nat @ ( bit_se1412395901928357646or_nat @ M @ N ) )
% 0.22/0.56 = ( bit_se1412395901928357646or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % of_nat_or_eq
% 0.22/0.56 thf(fact_8750_or__greater__eq,axiom,
% 0.22/0.56 ! [L: int,K: int] :
% 0.22/0.56 ( ( ord_less_eq_int @ zero_zero_int @ L )
% 0.22/0.56 => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_greater_eq
% 0.22/0.56 thf(fact_8751_OR__lower,axiom,
% 0.22/0.56 ! [X: int,Y4: int] :
% 0.22/0.56 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.22/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.22/0.56 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y4 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % OR_lower
% 0.22/0.56 thf(fact_8752_disjunctive__add,axiom,
% 0.22/0.56 ! [A: int,B: int] :
% 0.22/0.56 ( ! [N2: nat] :
% 0.22/0.56 ( ~ ( bit_se1146084159140164899it_int @ A @ N2 )
% 0.22/0.56 | ~ ( bit_se1146084159140164899it_int @ B @ N2 ) )
% 0.22/0.56 => ( ( plus_plus_int @ A @ B )
% 0.22/0.56 = ( bit_se1409905431419307370or_int @ A @ B ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % disjunctive_add
% 0.22/0.56 thf(fact_8753_disjunctive__add,axiom,
% 0.22/0.56 ! [A: nat,B: nat] :
% 0.22/0.56 ( ! [N2: nat] :
% 0.22/0.56 ( ~ ( bit_se1148574629649215175it_nat @ A @ N2 )
% 0.22/0.56 | ~ ( bit_se1148574629649215175it_nat @ B @ N2 ) )
% 0.22/0.56 => ( ( plus_plus_nat @ A @ B )
% 0.22/0.56 = ( bit_se1412395901928357646or_nat @ A @ B ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % disjunctive_add
% 0.22/0.56 thf(fact_8754_plus__and__or,axiom,
% 0.22/0.56 ! [X: int,Y4: int] :
% 0.22/0.56 ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y4 ) @ ( bit_se1409905431419307370or_int @ X @ Y4 ) )
% 0.22/0.56 = ( plus_plus_int @ X @ Y4 ) ) ).
% 0.22/0.56
% 0.22/0.56 % plus_and_or
% 0.22/0.56 thf(fact_8755_or__eq__not__not__and,axiom,
% 0.22/0.56 ( bit_se1409905431419307370or_int
% 0.22/0.56 = ( ^ [A3: int,B2: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ A3 ) @ ( bit_ri7919022796975470100ot_int @ B2 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_eq_not_not_and
% 0.22/0.56 thf(fact_8756_and__eq__not__not__or,axiom,
% 0.22/0.56 ( bit_se725231765392027082nd_int
% 0.22/0.56 = ( ^ [A3: int,B2: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ A3 ) @ ( bit_ri7919022796975470100ot_int @ B2 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % and_eq_not_not_or
% 0.22/0.56 thf(fact_8757_or__int__def,axiom,
% 0.22/0.56 ( bit_se1409905431419307370or_int
% 0.22/0.56 = ( ^ [K3: int,L3: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ ( bit_ri7919022796975470100ot_int @ L3 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_int_def
% 0.22/0.56 thf(fact_8758_set__bit__nat__def,axiom,
% 0.22/0.56 ( bit_se7882103937844011126it_nat
% 0.22/0.56 = ( ^ [M5: nat,N4: nat] : ( bit_se1412395901928357646or_nat @ N4 @ ( bit_se547839408752420682it_nat @ M5 @ one_one_nat ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % set_bit_nat_def
% 0.22/0.56 thf(fact_8759_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
% 0.22/0.56 ! [N: nat,A: int] :
% 0.22/0.56 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.22/0.56 = A )
% 0.22/0.56 = ( ( bit_se8568078237143864401it_int @ N @ A )
% 0.22/0.56 = zero_zero_int ) ) ).
% 0.22/0.56
% 0.22/0.56 % take_bit_eq_self_iff_drop_bit_eq_0
% 0.22/0.56 thf(fact_8760_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
% 0.22/0.56 ! [N: nat,A: nat] :
% 0.22/0.56 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.22/0.56 = A )
% 0.22/0.56 = ( ( bit_se8570568707652914677it_nat @ N @ A )
% 0.22/0.56 = zero_zero_nat ) ) ).
% 0.22/0.56
% 0.22/0.56 % take_bit_eq_self_iff_drop_bit_eq_0
% 0.22/0.56 thf(fact_8761_drop__bit__push__bit__int,axiom,
% 0.22/0.56 ! [M: nat,N: nat,K: int] :
% 0.22/0.56 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N @ K ) )
% 0.22/0.56 = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M ) @ K ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % drop_bit_push_bit_int
% 0.22/0.56 thf(fact_8762_take__bit__drop__bit,axiom,
% 0.22/0.56 ! [M: nat,N: nat,A: int] :
% 0.22/0.56 ( ( bit_se2923211474154528505it_int @ M @ ( bit_se8568078237143864401it_int @ N @ A ) )
% 0.22/0.56 = ( bit_se8568078237143864401it_int @ N @ ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % take_bit_drop_bit
% 0.22/0.56 thf(fact_8763_take__bit__drop__bit,axiom,
% 0.22/0.56 ! [M: nat,N: nat,A: nat] :
% 0.22/0.56 ( ( bit_se2925701944663578781it_nat @ M @ ( bit_se8570568707652914677it_nat @ N @ A ) )
% 0.22/0.56 = ( bit_se8570568707652914677it_nat @ N @ ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N ) @ A ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % take_bit_drop_bit
% 0.22/0.56 thf(fact_8764_drop__bit__take__bit,axiom,
% 0.22/0.56 ! [M: nat,N: nat,A: int] :
% 0.22/0.56 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.22/0.56 = ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N @ M ) @ ( bit_se8568078237143864401it_int @ M @ A ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % drop_bit_take_bit
% 0.22/0.56 thf(fact_8765_drop__bit__take__bit,axiom,
% 0.22/0.56 ! [M: nat,N: nat,A: nat] :
% 0.22/0.56 ( ( bit_se8570568707652914677it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 0.22/0.56 = ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N @ M ) @ ( bit_se8570568707652914677it_nat @ M @ A ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % drop_bit_take_bit
% 0.22/0.56 thf(fact_8766_or__not__numerals_I1_J,axiom,
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.22/0.56 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_not_numerals(1)
% 0.22/0.56 thf(fact_8767_bit_Oxor__def2,axiom,
% 0.22/0.56 ( bit_se6526347334894502574or_int
% 0.22/0.56 = ( ^ [X3: int,Y6: int] : ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ X3 @ Y6 ) @ ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ X3 ) @ ( bit_ri7919022796975470100ot_int @ Y6 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % bit.xor_def2
% 0.22/0.56 thf(fact_8768_bit_Oxor__def,axiom,
% 0.22/0.56 ( bit_se6526347334894502574or_int
% 0.22/0.56 = ( ^ [X3: int,Y6: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ X3 @ ( bit_ri7919022796975470100ot_int @ Y6 ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X3 ) @ Y6 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % bit.xor_def
% 0.22/0.56 thf(fact_8769_set__bit__eq__or,axiom,
% 0.22/0.56 ( bit_se2793503036327961859nteger
% 0.22/0.56 = ( ^ [N4: nat,A3: code_integer] : ( bit_se1080825931792720795nteger @ A3 @ ( bit_se7788150548672797655nteger @ N4 @ one_one_Code_integer ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % set_bit_eq_or
% 0.22/0.56 thf(fact_8770_set__bit__eq__or,axiom,
% 0.22/0.56 ( bit_se7879613467334960850it_int
% 0.22/0.56 = ( ^ [N4: nat,A3: int] : ( bit_se1409905431419307370or_int @ A3 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % set_bit_eq_or
% 0.22/0.56 thf(fact_8771_set__bit__eq__or,axiom,
% 0.22/0.56 ( bit_se7882103937844011126it_nat
% 0.22/0.56 = ( ^ [N4: nat,A3: nat] : ( bit_se1412395901928357646or_nat @ A3 @ ( bit_se547839408752420682it_nat @ N4 @ one_one_nat ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % set_bit_eq_or
% 0.22/0.56 thf(fact_8772_xor__int__def,axiom,
% 0.22/0.56 ( bit_se6526347334894502574or_int
% 0.22/0.56 = ( ^ [K3: int,L3: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ L3 ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ L3 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % xor_int_def
% 0.22/0.56 thf(fact_8773_div__push__bit__of__1__eq__drop__bit,axiom,
% 0.22/0.56 ! [A: int,N: nat] :
% 0.22/0.56 ( ( divide_divide_int @ A @ ( bit_se545348938243370406it_int @ N @ one_one_int ) )
% 0.22/0.56 = ( bit_se8568078237143864401it_int @ N @ A ) ) ).
% 0.22/0.56
% 0.22/0.56 % div_push_bit_of_1_eq_drop_bit
% 0.22/0.56 thf(fact_8774_div__push__bit__of__1__eq__drop__bit,axiom,
% 0.22/0.56 ! [A: nat,N: nat] :
% 0.22/0.56 ( ( divide_divide_nat @ A @ ( bit_se547839408752420682it_nat @ N @ one_one_nat ) )
% 0.22/0.56 = ( bit_se8570568707652914677it_nat @ N @ A ) ) ).
% 0.22/0.56
% 0.22/0.56 % div_push_bit_of_1_eq_drop_bit
% 0.22/0.56 thf(fact_8775_concat__bit__def,axiom,
% 0.22/0.56 ( bit_concat_bit
% 0.22/0.56 = ( ^ [N4: nat,K3: int,L3: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N4 @ K3 ) @ ( bit_se545348938243370406it_int @ N4 @ L3 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % concat_bit_def
% 0.22/0.56 thf(fact_8776_bit__iff__and__drop__bit__eq__1,axiom,
% 0.22/0.56 ( bit_se1146084159140164899it_int
% 0.22/0.56 = ( ^ [A3: int,N4: nat] :
% 0.22/0.56 ( ( bit_se725231765392027082nd_int @ ( bit_se8568078237143864401it_int @ N4 @ A3 ) @ one_one_int )
% 0.22/0.56 = one_one_int ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % bit_iff_and_drop_bit_eq_1
% 0.22/0.56 thf(fact_8777_bit__iff__and__drop__bit__eq__1,axiom,
% 0.22/0.56 ( bit_se1148574629649215175it_nat
% 0.22/0.56 = ( ^ [A3: nat,N4: nat] :
% 0.22/0.56 ( ( bit_se727722235901077358nd_nat @ ( bit_se8570568707652914677it_nat @ N4 @ A3 ) @ one_one_nat )
% 0.22/0.56 = one_one_nat ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % bit_iff_and_drop_bit_eq_1
% 0.22/0.56 thf(fact_8778_set__bit__int__def,axiom,
% 0.22/0.56 ( bit_se7879613467334960850it_int
% 0.22/0.56 = ( ^ [N4: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % set_bit_int_def
% 0.22/0.56 thf(fact_8779_bits__ident,axiom,
% 0.22/0.56 ! [N: nat,A: int] :
% 0.22/0.56 ( ( plus_plus_int @ ( bit_se545348938243370406it_int @ N @ ( bit_se8568078237143864401it_int @ N @ A ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.22/0.56 = A ) ).
% 0.22/0.56
% 0.22/0.56 % bits_ident
% 0.22/0.56 thf(fact_8780_bits__ident,axiom,
% 0.22/0.56 ! [N: nat,A: nat] :
% 0.22/0.56 ( ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N @ ( bit_se8570568707652914677it_nat @ N @ A ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 0.22/0.56 = A ) ).
% 0.22/0.56
% 0.22/0.56 % bits_ident
% 0.22/0.56 thf(fact_8781_even__or__iff,axiom,
% 0.22/0.56 ! [A: code_integer,B: code_integer] :
% 0.22/0.56 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1080825931792720795nteger @ A @ B ) )
% 0.22/0.56 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.22/0.56 & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % even_or_iff
% 0.22/0.56 thf(fact_8782_even__or__iff,axiom,
% 0.22/0.56 ! [A: int,B: int] :
% 0.22/0.56 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 0.22/0.56 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.22/0.56 & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % even_or_iff
% 0.22/0.56 thf(fact_8783_even__or__iff,axiom,
% 0.22/0.56 ! [A: nat,B: nat] :
% 0.22/0.56 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 0.22/0.56 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.22/0.56 & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % even_or_iff
% 0.22/0.56 thf(fact_8784_bit_Ocomplement__unique,axiom,
% 0.22/0.56 ! [A: code_integer,X: code_integer,Y4: code_integer] :
% 0.22/0.56 ( ( ( bit_se3949692690581998587nteger @ A @ X )
% 0.22/0.56 = zero_z3403309356797280102nteger )
% 0.22/0.56 => ( ( ( bit_se1080825931792720795nteger @ A @ X )
% 0.22/0.56 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.56 => ( ( ( bit_se3949692690581998587nteger @ A @ Y4 )
% 0.22/0.56 = zero_z3403309356797280102nteger )
% 0.22/0.56 => ( ( ( bit_se1080825931792720795nteger @ A @ Y4 )
% 0.22/0.56 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.56 => ( X = Y4 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % bit.complement_unique
% 0.22/0.56 thf(fact_8785_bit_Ocomplement__unique,axiom,
% 0.22/0.56 ! [A: int,X: int,Y4: int] :
% 0.22/0.56 ( ( ( bit_se725231765392027082nd_int @ A @ X )
% 0.22/0.56 = zero_zero_int )
% 0.22/0.56 => ( ( ( bit_se1409905431419307370or_int @ A @ X )
% 0.22/0.56 = ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.56 => ( ( ( bit_se725231765392027082nd_int @ A @ Y4 )
% 0.22/0.56 = zero_zero_int )
% 0.22/0.56 => ( ( ( bit_se1409905431419307370or_int @ A @ Y4 )
% 0.22/0.56 = ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.56 => ( X = Y4 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % bit.complement_unique
% 0.22/0.56 thf(fact_8786_or__not__numerals_I4_J,axiom,
% 0.22/0.56 ! [M: num] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.22/0.56 = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_not_numerals(4)
% 0.22/0.56 thf(fact_8787_or__not__numerals_I2_J,axiom,
% 0.22/0.56 ! [N: num] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.22/0.56 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_not_numerals(2)
% 0.22/0.56 thf(fact_8788_stable__imp__drop__bit__eq,axiom,
% 0.22/0.56 ! [A: int,N: nat] :
% 0.22/0.56 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.56 = A )
% 0.22/0.56 => ( ( bit_se8568078237143864401it_int @ N @ A )
% 0.22/0.56 = A ) ) ).
% 0.22/0.56
% 0.22/0.56 % stable_imp_drop_bit_eq
% 0.22/0.56 thf(fact_8789_stable__imp__drop__bit__eq,axiom,
% 0.22/0.56 ! [A: nat,N: nat] :
% 0.22/0.56 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.56 = A )
% 0.22/0.56 => ( ( bit_se8570568707652914677it_nat @ N @ A )
% 0.22/0.56 = A ) ) ).
% 0.22/0.56
% 0.22/0.56 % stable_imp_drop_bit_eq
% 0.22/0.56 thf(fact_8790_drop__bit__half,axiom,
% 0.22/0.56 ! [N: nat,A: int] :
% 0.22/0.56 ( ( bit_se8568078237143864401it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.22/0.56 = ( divide_divide_int @ ( bit_se8568078237143864401it_int @ N @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % drop_bit_half
% 0.22/0.56 thf(fact_8791_drop__bit__half,axiom,
% 0.22/0.56 ! [N: nat,A: nat] :
% 0.22/0.56 ( ( bit_se8570568707652914677it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.56 = ( divide_divide_nat @ ( bit_se8570568707652914677it_nat @ N @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % drop_bit_half
% 0.22/0.56 thf(fact_8792_bit_Ocompl__unique,axiom,
% 0.22/0.56 ! [X: code_integer,Y4: code_integer] :
% 0.22/0.56 ( ( ( bit_se3949692690581998587nteger @ X @ Y4 )
% 0.22/0.56 = zero_z3403309356797280102nteger )
% 0.22/0.56 => ( ( ( bit_se1080825931792720795nteger @ X @ Y4 )
% 0.22/0.56 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.22/0.56 => ( ( bit_ri7632146776885996613nteger @ X )
% 0.22/0.56 = Y4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % bit.compl_unique
% 0.22/0.56 thf(fact_8793_bit_Ocompl__unique,axiom,
% 0.22/0.56 ! [X: int,Y4: int] :
% 0.22/0.56 ( ( ( bit_se725231765392027082nd_int @ X @ Y4 )
% 0.22/0.56 = zero_zero_int )
% 0.22/0.56 => ( ( ( bit_se1409905431419307370or_int @ X @ Y4 )
% 0.22/0.56 = ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.56 => ( ( bit_ri7919022796975470100ot_int @ X )
% 0.22/0.56 = Y4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % bit.compl_unique
% 0.22/0.56 thf(fact_8794_or__not__numerals_I3_J,axiom,
% 0.22/0.56 ! [N: num] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.22/0.56 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_not_numerals(3)
% 0.22/0.56 thf(fact_8795_or__not__numerals_I7_J,axiom,
% 0.22/0.56 ! [M: num] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.22/0.56 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_not_numerals(7)
% 0.22/0.56 thf(fact_8796_drop__bit__int__def,axiom,
% 0.22/0.56 ( bit_se8568078237143864401it_int
% 0.22/0.56 = ( ^ [N4: nat,K3: int] : ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % drop_bit_int_def
% 0.22/0.56 thf(fact_8797_signed__take__bit__eq__if__negative,axiom,
% 0.22/0.56 ! [A: int,N: nat] :
% 0.22/0.56 ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.22/0.56 => ( ( bit_ri631733984087533419it_int @ N @ A )
% 0.22/0.56 = ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % signed_take_bit_eq_if_negative
% 0.22/0.56 thf(fact_8798_drop__bit__Suc,axiom,
% 0.22/0.56 ! [N: nat,A: int] :
% 0.22/0.56 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ A )
% 0.22/0.56 = ( bit_se8568078237143864401it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % drop_bit_Suc
% 0.22/0.56 thf(fact_8799_drop__bit__Suc,axiom,
% 0.22/0.56 ! [N: nat,A: nat] :
% 0.22/0.56 ( ( bit_se8570568707652914677it_nat @ ( suc @ N ) @ A )
% 0.22/0.56 = ( bit_se8570568707652914677it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % drop_bit_Suc
% 0.22/0.56 thf(fact_8800_drop__bit__eq__div,axiom,
% 0.22/0.56 ( bit_se8568078237143864401it_int
% 0.22/0.56 = ( ^ [N4: nat,A3: int] : ( divide_divide_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % drop_bit_eq_div
% 0.22/0.56 thf(fact_8801_drop__bit__eq__div,axiom,
% 0.22/0.56 ( bit_se8570568707652914677it_nat
% 0.22/0.56 = ( ^ [N4: nat,A3: nat] : ( divide_divide_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % drop_bit_eq_div
% 0.22/0.56 thf(fact_8802_even__drop__bit__iff__not__bit,axiom,
% 0.22/0.56 ! [N: nat,A: code_integer] :
% 0.22/0.56 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3928097537394005634nteger @ N @ A ) )
% 0.22/0.56 = ( ~ ( bit_se9216721137139052372nteger @ A @ N ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % even_drop_bit_iff_not_bit
% 0.22/0.56 thf(fact_8803_even__drop__bit__iff__not__bit,axiom,
% 0.22/0.56 ! [N: nat,A: int] :
% 0.22/0.56 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se8568078237143864401it_int @ N @ A ) )
% 0.22/0.56 = ( ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % even_drop_bit_iff_not_bit
% 0.22/0.56 thf(fact_8804_even__drop__bit__iff__not__bit,axiom,
% 0.22/0.56 ! [N: nat,A: nat] :
% 0.22/0.56 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se8570568707652914677it_nat @ N @ A ) )
% 0.22/0.56 = ( ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % even_drop_bit_iff_not_bit
% 0.22/0.56 thf(fact_8805_bit__iff__odd__drop__bit,axiom,
% 0.22/0.56 ( bit_se9216721137139052372nteger
% 0.22/0.56 = ( ^ [A3: code_integer,N4: nat] :
% 0.22/0.56 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3928097537394005634nteger @ N4 @ A3 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % bit_iff_odd_drop_bit
% 0.22/0.56 thf(fact_8806_bit__iff__odd__drop__bit,axiom,
% 0.22/0.56 ( bit_se1146084159140164899it_int
% 0.22/0.56 = ( ^ [A3: int,N4: nat] :
% 0.22/0.56 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se8568078237143864401it_int @ N4 @ A3 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % bit_iff_odd_drop_bit
% 0.22/0.56 thf(fact_8807_bit__iff__odd__drop__bit,axiom,
% 0.22/0.56 ( bit_se1148574629649215175it_nat
% 0.22/0.56 = ( ^ [A3: nat,N4: nat] :
% 0.22/0.56 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se8570568707652914677it_nat @ N4 @ A3 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % bit_iff_odd_drop_bit
% 0.22/0.56 thf(fact_8808_mask__Suc__exp,axiom,
% 0.22/0.56 ! [N: nat] :
% 0.22/0.56 ( ( bit_se2002935070580805687sk_nat @ ( suc @ N ) )
% 0.22/0.56 = ( bit_se1412395901928357646or_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % mask_Suc_exp
% 0.22/0.56 thf(fact_8809_mask__Suc__exp,axiom,
% 0.22/0.56 ! [N: nat] :
% 0.22/0.56 ( ( bit_se2000444600071755411sk_int @ ( suc @ N ) )
% 0.22/0.56 = ( bit_se1409905431419307370or_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % mask_Suc_exp
% 0.22/0.56 thf(fact_8810_or__not__numerals_I6_J,axiom,
% 0.22/0.56 ! [M: num,N: num] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.22/0.56 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_not_numerals(6)
% 0.22/0.56 thf(fact_8811_or__one__eq,axiom,
% 0.22/0.56 ! [A: code_integer] :
% 0.22/0.56 ( ( bit_se1080825931792720795nteger @ A @ one_one_Code_integer )
% 0.22/0.56 = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_one_eq
% 0.22/0.56 thf(fact_8812_or__one__eq,axiom,
% 0.22/0.56 ! [A: int] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ A @ one_one_int )
% 0.22/0.56 = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_one_eq
% 0.22/0.56 thf(fact_8813_or__one__eq,axiom,
% 0.22/0.56 ! [A: nat] :
% 0.22/0.56 ( ( bit_se1412395901928357646or_nat @ A @ one_one_nat )
% 0.22/0.56 = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_one_eq
% 0.22/0.56 thf(fact_8814_one__or__eq,axiom,
% 0.22/0.56 ! [A: code_integer] :
% 0.22/0.56 ( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ A )
% 0.22/0.56 = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % one_or_eq
% 0.22/0.56 thf(fact_8815_one__or__eq,axiom,
% 0.22/0.56 ! [A: int] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ one_one_int @ A )
% 0.22/0.56 = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % one_or_eq
% 0.22/0.56 thf(fact_8816_one__or__eq,axiom,
% 0.22/0.56 ! [A: nat] :
% 0.22/0.56 ( ( bit_se1412395901928357646or_nat @ one_one_nat @ A )
% 0.22/0.56 = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % one_or_eq
% 0.22/0.56 thf(fact_8817_mask__Suc__double,axiom,
% 0.22/0.56 ! [N: nat] :
% 0.22/0.56 ( ( bit_se2002935070580805687sk_nat @ ( suc @ N ) )
% 0.22/0.56 = ( bit_se1412395901928357646or_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % mask_Suc_double
% 0.22/0.56 thf(fact_8818_mask__Suc__double,axiom,
% 0.22/0.56 ! [N: nat] :
% 0.22/0.56 ( ( bit_se2000444600071755411sk_int @ ( suc @ N ) )
% 0.22/0.56 = ( bit_se1409905431419307370or_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % mask_Suc_double
% 0.22/0.56 thf(fact_8819_OR__upper,axiom,
% 0.22/0.56 ! [X: int,N: nat,Y4: int] :
% 0.22/0.56 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.22/0.56 => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.56 => ( ( ord_less_int @ Y4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.22/0.56 => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y4 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % OR_upper
% 0.22/0.56 thf(fact_8820_or__not__numerals_I5_J,axiom,
% 0.22/0.56 ! [M: num,N: num] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.22/0.56 = ( 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 ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_not_numerals(5)
% 0.22/0.56 thf(fact_8821_slice__eq__mask,axiom,
% 0.22/0.56 ! [N: nat,M: nat,A: int] :
% 0.22/0.56 ( ( bit_se545348938243370406it_int @ N @ ( bit_se2923211474154528505it_int @ M @ ( bit_se8568078237143864401it_int @ N @ A ) ) )
% 0.22/0.56 = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ ( bit_se2000444600071755411sk_int @ ( plus_plus_nat @ M @ N ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % slice_eq_mask
% 0.22/0.56 thf(fact_8822_signed__take__bit__def,axiom,
% 0.22/0.56 ( bit_ri6519982836138164636nteger
% 0.22/0.56 = ( ^ [N4: nat,A3: code_integer] : ( bit_se1080825931792720795nteger @ ( bit_se1745604003318907178nteger @ N4 @ A3 ) @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( bit_se9216721137139052372nteger @ A3 @ N4 ) ) @ ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N4 ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % signed_take_bit_def
% 0.22/0.56 thf(fact_8823_signed__take__bit__def,axiom,
% 0.22/0.56 ( bit_ri631733984087533419it_int
% 0.22/0.56 = ( ^ [N4: nat,A3: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N4 @ A3 ) @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ A3 @ N4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N4 ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % signed_take_bit_def
% 0.22/0.56 thf(fact_8824_or__Suc__0__eq,axiom,
% 0.22/0.56 ! [N: nat] :
% 0.22/0.56 ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.22/0.56 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_Suc_0_eq
% 0.22/0.56 thf(fact_8825_Suc__0__or__eq,axiom,
% 0.22/0.56 ! [N: nat] :
% 0.22/0.56 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.22/0.56 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % Suc_0_or_eq
% 0.22/0.56 thf(fact_8826_drop__bit__rec,axiom,
% 0.22/0.56 ( bit_se8568078237143864401it_int
% 0.22/0.56 = ( ^ [N4: nat,A3: int] : ( if_int @ ( N4 = zero_zero_nat ) @ A3 @ ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % drop_bit_rec
% 0.22/0.56 thf(fact_8827_drop__bit__rec,axiom,
% 0.22/0.56 ( bit_se8570568707652914677it_nat
% 0.22/0.56 = ( ^ [N4: nat,A3: nat] : ( if_nat @ ( N4 = zero_zero_nat ) @ A3 @ ( bit_se8570568707652914677it_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % drop_bit_rec
% 0.22/0.56 thf(fact_8828_or__not__numerals_I9_J,axiom,
% 0.22/0.56 ! [M: num,N: num] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.22/0.56 = ( 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 ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_not_numerals(9)
% 0.22/0.56 thf(fact_8829_or__not__numerals_I8_J,axiom,
% 0.22/0.56 ! [M: num,N: num] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.22/0.56 = ( 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 ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_not_numerals(8)
% 0.22/0.56 thf(fact_8830_or__nat__rec,axiom,
% 0.22/0.56 ( bit_se1412395901928357646or_nat
% 0.22/0.56 = ( ^ [M5: nat,N4: nat] :
% 0.22/0.56 ( plus_plus_nat
% 0.22/0.56 @ ( zero_n2687167440665602831ol_nat
% 0.22/0.56 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 )
% 0.22/0.56 | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
% 0.22/0.56 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_nat_rec
% 0.22/0.56 thf(fact_8831_or__int__unfold,axiom,
% 0.22/0.56 ( bit_se1409905431419307370or_int
% 0.22/0.56 = ( ^ [K3: int,L3: int] :
% 0.22/0.56 ( if_int
% 0.22/0.56 @ ( ( K3
% 0.22/0.56 = ( uminus_uminus_int @ one_one_int ) )
% 0.22/0.56 | ( L3
% 0.22/0.56 = ( uminus_uminus_int @ one_one_int ) ) )
% 0.22/0.56 @ ( uminus_uminus_int @ one_one_int )
% 0.22/0.56 @ ( if_int @ ( K3 = zero_zero_int ) @ L3 @ ( if_int @ ( L3 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_int_unfold
% 0.22/0.56 thf(fact_8832_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
% 0.22/0.56 boolea2445317508997433345nteger @ bit_se3949692690581998587nteger @ bit_se1080825931792720795nteger @ bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ bit_se3222712562003087583nteger ).
% 0.22/0.56
% 0.22/0.56 % bit.abstract_boolean_algebra_sym_diff_axioms
% 0.22/0.56 thf(fact_8833_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
% 0.22/0.56 boolea8527374999097803216ff_int @ bit_se725231765392027082nd_int @ bit_se1409905431419307370or_int @ bit_ri7919022796975470100ot_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) @ bit_se6526347334894502574or_int ).
% 0.22/0.56
% 0.22/0.56 % bit.abstract_boolean_algebra_sym_diff_axioms
% 0.22/0.56 thf(fact_8834_or__minus__numerals_I8_J,axiom,
% 0.22/0.56 ! [N: num,M: num] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.22/0.56 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_minus_numerals(8)
% 0.22/0.56 thf(fact_8835_or__minus__numerals_I4_J,axiom,
% 0.22/0.56 ! [M: num,N: num] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.22/0.56 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_minus_numerals(4)
% 0.22/0.56 thf(fact_8836_max_Obounded__iff,axiom,
% 0.22/0.56 ! [B: extended_enat,C: extended_enat,A: extended_enat] :
% 0.22/0.56 ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
% 0.22/0.56 = ( ( ord_le2932123472753598470d_enat @ B @ A )
% 0.22/0.56 & ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.bounded_iff
% 0.22/0.56 thf(fact_8837_max_Obounded__iff,axiom,
% 0.22/0.56 ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.22/0.56 ( ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B @ C ) @ A )
% 0.22/0.56 = ( ( ord_le3102999989581377725nteger @ B @ A )
% 0.22/0.56 & ( ord_le3102999989581377725nteger @ C @ A ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.bounded_iff
% 0.22/0.56 thf(fact_8838_max_Obounded__iff,axiom,
% 0.22/0.56 ! [B: rat,C: rat,A: rat] :
% 0.22/0.56 ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 0.22/0.56 = ( ( ord_less_eq_rat @ B @ A )
% 0.22/0.56 & ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.bounded_iff
% 0.22/0.56 thf(fact_8839_max_Obounded__iff,axiom,
% 0.22/0.56 ! [B: num,C: num,A: num] :
% 0.22/0.56 ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 0.22/0.56 = ( ( ord_less_eq_num @ B @ A )
% 0.22/0.56 & ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.bounded_iff
% 0.22/0.56 thf(fact_8840_max_Obounded__iff,axiom,
% 0.22/0.56 ! [B: nat,C: nat,A: nat] :
% 0.22/0.56 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 0.22/0.56 = ( ( ord_less_eq_nat @ B @ A )
% 0.22/0.56 & ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.bounded_iff
% 0.22/0.56 thf(fact_8841_max_Obounded__iff,axiom,
% 0.22/0.56 ! [B: int,C: int,A: int] :
% 0.22/0.56 ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 0.22/0.56 = ( ( ord_less_eq_int @ B @ A )
% 0.22/0.56 & ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.bounded_iff
% 0.22/0.56 thf(fact_8842_max_Obounded__iff,axiom,
% 0.22/0.56 ! [B: real,C: real,A: real] :
% 0.22/0.56 ( ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A )
% 0.22/0.56 = ( ( ord_less_eq_real @ B @ A )
% 0.22/0.56 & ( ord_less_eq_real @ C @ A ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.bounded_iff
% 0.22/0.56 thf(fact_8843_max_Oabsorb2,axiom,
% 0.22/0.56 ! [A: extended_enat,B: extended_enat] :
% 0.22/0.56 ( ( ord_le2932123472753598470d_enat @ A @ B )
% 0.22/0.56 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 0.22/0.56 = B ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.absorb2
% 0.22/0.56 thf(fact_8844_max_Oabsorb2,axiom,
% 0.22/0.56 ! [A: code_integer,B: code_integer] :
% 0.22/0.56 ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.22/0.56 => ( ( ord_max_Code_integer @ A @ B )
% 0.22/0.56 = B ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.absorb2
% 0.22/0.56 thf(fact_8845_max_Oabsorb2,axiom,
% 0.22/0.56 ! [A: rat,B: rat] :
% 0.22/0.56 ( ( ord_less_eq_rat @ A @ B )
% 0.22/0.56 => ( ( ord_max_rat @ A @ B )
% 0.22/0.56 = B ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.absorb2
% 0.22/0.56 thf(fact_8846_max_Oabsorb2,axiom,
% 0.22/0.56 ! [A: num,B: num] :
% 0.22/0.56 ( ( ord_less_eq_num @ A @ B )
% 0.22/0.56 => ( ( ord_max_num @ A @ B )
% 0.22/0.56 = B ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.absorb2
% 0.22/0.56 thf(fact_8847_max_Oabsorb2,axiom,
% 0.22/0.56 ! [A: nat,B: nat] :
% 0.22/0.56 ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.56 => ( ( ord_max_nat @ A @ B )
% 0.22/0.56 = B ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.absorb2
% 0.22/0.56 thf(fact_8848_max_Oabsorb2,axiom,
% 0.22/0.56 ! [A: int,B: int] :
% 0.22/0.56 ( ( ord_less_eq_int @ A @ B )
% 0.22/0.56 => ( ( ord_max_int @ A @ B )
% 0.22/0.56 = B ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.absorb2
% 0.22/0.56 thf(fact_8849_max_Oabsorb2,axiom,
% 0.22/0.56 ! [A: real,B: real] :
% 0.22/0.56 ( ( ord_less_eq_real @ A @ B )
% 0.22/0.56 => ( ( ord_max_real @ A @ B )
% 0.22/0.56 = B ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.absorb2
% 0.22/0.56 thf(fact_8850_max_Oabsorb1,axiom,
% 0.22/0.56 ! [B: extended_enat,A: extended_enat] :
% 0.22/0.56 ( ( ord_le2932123472753598470d_enat @ B @ A )
% 0.22/0.56 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 0.22/0.56 = A ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.absorb1
% 0.22/0.56 thf(fact_8851_max_Oabsorb1,axiom,
% 0.22/0.56 ! [B: code_integer,A: code_integer] :
% 0.22/0.56 ( ( ord_le3102999989581377725nteger @ B @ A )
% 0.22/0.56 => ( ( ord_max_Code_integer @ A @ B )
% 0.22/0.56 = A ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.absorb1
% 0.22/0.56 thf(fact_8852_max_Oabsorb1,axiom,
% 0.22/0.56 ! [B: rat,A: rat] :
% 0.22/0.56 ( ( ord_less_eq_rat @ B @ A )
% 0.22/0.56 => ( ( ord_max_rat @ A @ B )
% 0.22/0.56 = A ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.absorb1
% 0.22/0.56 thf(fact_8853_max_Oabsorb1,axiom,
% 0.22/0.56 ! [B: num,A: num] :
% 0.22/0.56 ( ( ord_less_eq_num @ B @ A )
% 0.22/0.56 => ( ( ord_max_num @ A @ B )
% 0.22/0.56 = A ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.absorb1
% 0.22/0.56 thf(fact_8854_max_Oabsorb1,axiom,
% 0.22/0.56 ! [B: nat,A: nat] :
% 0.22/0.56 ( ( ord_less_eq_nat @ B @ A )
% 0.22/0.56 => ( ( ord_max_nat @ A @ B )
% 0.22/0.56 = A ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.absorb1
% 0.22/0.56 thf(fact_8855_max_Oabsorb1,axiom,
% 0.22/0.56 ! [B: int,A: int] :
% 0.22/0.56 ( ( ord_less_eq_int @ B @ A )
% 0.22/0.56 => ( ( ord_max_int @ A @ B )
% 0.22/0.56 = A ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.absorb1
% 0.22/0.56 thf(fact_8856_max_Oabsorb1,axiom,
% 0.22/0.56 ! [B: real,A: real] :
% 0.22/0.56 ( ( ord_less_eq_real @ B @ A )
% 0.22/0.56 => ( ( ord_max_real @ A @ B )
% 0.22/0.56 = A ) ) ).
% 0.22/0.56
% 0.22/0.56 % max.absorb1
% 0.22/0.56 thf(fact_8857_of__bool__or__iff,axiom,
% 0.22/0.56 ! [P: $o,Q: $o] :
% 0.22/0.56 ( ( zero_n2687167440665602831ol_nat
% 0.22/0.56 @ ( P
% 0.22/0.56 | Q ) )
% 0.22/0.56 = ( ord_max_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % of_bool_or_iff
% 0.22/0.56 thf(fact_8858_of__bool__or__iff,axiom,
% 0.22/0.56 ! [P: $o,Q: $o] :
% 0.22/0.56 ( ( zero_n2684676970156552555ol_int
% 0.22/0.56 @ ( P
% 0.22/0.56 | Q ) )
% 0.22/0.56 = ( ord_max_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % of_bool_or_iff
% 0.22/0.56 thf(fact_8859_of__bool__or__iff,axiom,
% 0.22/0.56 ! [P: $o,Q: $o] :
% 0.22/0.56 ( ( zero_n356916108424825756nteger
% 0.22/0.56 @ ( P
% 0.22/0.56 | Q ) )
% 0.22/0.56 = ( ord_max_Code_integer @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % of_bool_or_iff
% 0.22/0.56 thf(fact_8860_max__number__of_I1_J,axiom,
% 0.22/0.56 ! [U: num,V: num] :
% 0.22/0.56 ( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.22/0.56 => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.22/0.56 = ( numera1916890842035813515d_enat @ V ) ) )
% 0.22/0.56 & ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.22/0.56 => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.22/0.56 = ( numera1916890842035813515d_enat @ U ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_number_of(1)
% 0.22/0.56 thf(fact_8861_max__number__of_I1_J,axiom,
% 0.22/0.56 ! [U: num,V: num] :
% 0.22/0.56 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
% 0.22/0.56 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
% 0.22/0.56 = ( numera6620942414471956472nteger @ V ) ) )
% 0.22/0.56 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
% 0.22/0.56 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
% 0.22/0.56 = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_number_of(1)
% 0.22/0.56 thf(fact_8862_max__number__of_I1_J,axiom,
% 0.22/0.56 ! [U: num,V: num] :
% 0.22/0.56 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.22/0.56 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.22/0.56 = ( numeral_numeral_rat @ V ) ) )
% 0.22/0.56 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.22/0.56 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.22/0.56 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_number_of(1)
% 0.22/0.56 thf(fact_8863_max__number__of_I1_J,axiom,
% 0.22/0.56 ! [U: num,V: num] :
% 0.22/0.56 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.22/0.56 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.22/0.56 = ( numeral_numeral_nat @ V ) ) )
% 0.22/0.56 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.22/0.56 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.22/0.56 = ( numeral_numeral_nat @ U ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_number_of(1)
% 0.22/0.56 thf(fact_8864_max__number__of_I1_J,axiom,
% 0.22/0.56 ! [U: num,V: num] :
% 0.22/0.56 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.22/0.56 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.22/0.56 = ( numeral_numeral_int @ V ) ) )
% 0.22/0.56 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.22/0.56 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.22/0.56 = ( numeral_numeral_int @ U ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_number_of(1)
% 0.22/0.56 thf(fact_8865_max__number__of_I1_J,axiom,
% 0.22/0.56 ! [U: num,V: num] :
% 0.22/0.56 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.22/0.56 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.22/0.56 = ( numeral_numeral_real @ V ) ) )
% 0.22/0.56 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.22/0.56 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.22/0.56 = ( numeral_numeral_real @ U ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_number_of(1)
% 0.22/0.56 thf(fact_8866_max__0__1_I3_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X ) )
% 0.22/0.56 = ( numera1916890842035813515d_enat @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(3)
% 0.22/0.56 thf(fact_8867_max__0__1_I3_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ X ) )
% 0.22/0.56 = ( numera6620942414471956472nteger @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(3)
% 0.22/0.56 thf(fact_8868_max__0__1_I3_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
% 0.22/0.56 = ( numeral_numeral_real @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(3)
% 0.22/0.56 thf(fact_8869_max__0__1_I3_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
% 0.22/0.56 = ( numeral_numeral_rat @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(3)
% 0.22/0.56 thf(fact_8870_max__0__1_I3_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
% 0.22/0.56 = ( numeral_numeral_nat @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(3)
% 0.22/0.56 thf(fact_8871_max__0__1_I3_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
% 0.22/0.56 = ( numeral_numeral_int @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(3)
% 0.22/0.56 thf(fact_8872_max__0__1_I4_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ zero_z5237406670263579293d_enat )
% 0.22/0.56 = ( numera1916890842035813515d_enat @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(4)
% 0.22/0.56 thf(fact_8873_max__0__1_I4_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X ) @ zero_z3403309356797280102nteger )
% 0.22/0.56 = ( numera6620942414471956472nteger @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(4)
% 0.22/0.56 thf(fact_8874_max__0__1_I4_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
% 0.22/0.56 = ( numeral_numeral_real @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(4)
% 0.22/0.56 thf(fact_8875_max__0__1_I4_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
% 0.22/0.56 = ( numeral_numeral_rat @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(4)
% 0.22/0.56 thf(fact_8876_max__0__1_I4_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
% 0.22/0.56 = ( numeral_numeral_nat @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(4)
% 0.22/0.56 thf(fact_8877_max__0__1_I4_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
% 0.22/0.56 = ( numeral_numeral_int @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(4)
% 0.22/0.56 thf(fact_8878_max__0__1_I2_J,axiom,
% 0.22/0.56 ( ( ord_max_real @ one_one_real @ zero_zero_real )
% 0.22/0.56 = one_one_real ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(2)
% 0.22/0.56 thf(fact_8879_max__0__1_I2_J,axiom,
% 0.22/0.56 ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
% 0.22/0.56 = one_one_rat ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(2)
% 0.22/0.56 thf(fact_8880_max__0__1_I2_J,axiom,
% 0.22/0.56 ( ( ord_max_int @ one_one_int @ zero_zero_int )
% 0.22/0.56 = one_one_int ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(2)
% 0.22/0.56 thf(fact_8881_max__0__1_I2_J,axiom,
% 0.22/0.56 ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
% 0.22/0.56 = one_one_nat ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(2)
% 0.22/0.56 thf(fact_8882_max__0__1_I2_J,axiom,
% 0.22/0.56 ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
% 0.22/0.56 = one_on7984719198319812577d_enat ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(2)
% 0.22/0.56 thf(fact_8883_max__0__1_I2_J,axiom,
% 0.22/0.56 ( ( ord_max_Code_integer @ one_one_Code_integer @ zero_z3403309356797280102nteger )
% 0.22/0.56 = one_one_Code_integer ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(2)
% 0.22/0.56 thf(fact_8884_max__0__1_I1_J,axiom,
% 0.22/0.56 ( ( ord_max_real @ zero_zero_real @ one_one_real )
% 0.22/0.56 = one_one_real ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(1)
% 0.22/0.56 thf(fact_8885_max__0__1_I1_J,axiom,
% 0.22/0.56 ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
% 0.22/0.56 = one_one_rat ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(1)
% 0.22/0.56 thf(fact_8886_max__0__1_I1_J,axiom,
% 0.22/0.56 ( ( ord_max_int @ zero_zero_int @ one_one_int )
% 0.22/0.56 = one_one_int ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(1)
% 0.22/0.56 thf(fact_8887_max__0__1_I1_J,axiom,
% 0.22/0.56 ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
% 0.22/0.56 = one_one_nat ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(1)
% 0.22/0.56 thf(fact_8888_max__0__1_I1_J,axiom,
% 0.22/0.56 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
% 0.22/0.56 = one_on7984719198319812577d_enat ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(1)
% 0.22/0.56 thf(fact_8889_max__0__1_I1_J,axiom,
% 0.22/0.56 ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer )
% 0.22/0.56 = one_one_Code_integer ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(1)
% 0.22/0.56 thf(fact_8890_max__0__1_I5_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
% 0.22/0.56 = ( numera1916890842035813515d_enat @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(5)
% 0.22/0.56 thf(fact_8891_max__0__1_I5_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_Code_integer @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X ) )
% 0.22/0.56 = ( numera6620942414471956472nteger @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(5)
% 0.22/0.56 thf(fact_8892_max__0__1_I5_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X ) )
% 0.22/0.56 = ( numeral_numeral_real @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(5)
% 0.22/0.56 thf(fact_8893_max__0__1_I5_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
% 0.22/0.56 = ( numeral_numeral_rat @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(5)
% 0.22/0.56 thf(fact_8894_max__0__1_I5_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
% 0.22/0.56 = ( numeral_numeral_nat @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(5)
% 0.22/0.56 thf(fact_8895_max__0__1_I5_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
% 0.22/0.56 = ( numeral_numeral_int @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(5)
% 0.22/0.56 thf(fact_8896_max__0__1_I6_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat )
% 0.22/0.56 = ( numera1916890842035813515d_enat @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(6)
% 0.22/0.56 thf(fact_8897_max__0__1_I6_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer )
% 0.22/0.56 = ( numera6620942414471956472nteger @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(6)
% 0.22/0.56 thf(fact_8898_max__0__1_I6_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ one_one_real )
% 0.22/0.56 = ( numeral_numeral_real @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(6)
% 0.22/0.56 thf(fact_8899_max__0__1_I6_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
% 0.22/0.56 = ( numeral_numeral_rat @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(6)
% 0.22/0.56 thf(fact_8900_max__0__1_I6_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
% 0.22/0.56 = ( numeral_numeral_nat @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(6)
% 0.22/0.56 thf(fact_8901_max__0__1_I6_J,axiom,
% 0.22/0.56 ! [X: num] :
% 0.22/0.56 ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
% 0.22/0.56 = ( numeral_numeral_int @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_0_1(6)
% 0.22/0.56 thf(fact_8902_max__number__of_I4_J,axiom,
% 0.22/0.56 ! [U: num,V: num] :
% 0.22/0.56 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.22/0.56 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.22/0.56 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 0.22/0.56 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.22/0.56 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.22/0.56 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_number_of(4)
% 0.22/0.56 thf(fact_8903_max__number__of_I4_J,axiom,
% 0.22/0.56 ! [U: num,V: num] :
% 0.22/0.56 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.22/0.56 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.22/0.56 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 0.22/0.56 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.22/0.56 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.22/0.56 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_number_of(4)
% 0.22/0.56 thf(fact_8904_max__number__of_I4_J,axiom,
% 0.22/0.56 ! [U: num,V: num] :
% 0.22/0.56 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.22/0.56 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.22/0.56 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 0.22/0.56 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.22/0.56 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.22/0.56 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_number_of(4)
% 0.22/0.56 thf(fact_8905_max__number__of_I4_J,axiom,
% 0.22/0.56 ! [U: num,V: num] :
% 0.22/0.56 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.22/0.56 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.22/0.56 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 0.22/0.56 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.22/0.56 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.22/0.56 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_number_of(4)
% 0.22/0.56 thf(fact_8906_max__number__of_I3_J,axiom,
% 0.22/0.56 ! [U: num,V: num] :
% 0.22/0.56 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.22/0.56 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.22/0.56 = ( numera6620942414471956472nteger @ V ) ) )
% 0.22/0.56 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.22/0.56 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.22/0.56 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_number_of(3)
% 0.22/0.56 thf(fact_8907_max__number__of_I3_J,axiom,
% 0.22/0.56 ! [U: num,V: num] :
% 0.22/0.56 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.22/0.56 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.22/0.56 = ( numeral_numeral_rat @ V ) ) )
% 0.22/0.56 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.22/0.56 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.22/0.56 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_number_of(3)
% 0.22/0.56 thf(fact_8908_max__number__of_I3_J,axiom,
% 0.22/0.56 ! [U: num,V: num] :
% 0.22/0.56 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.22/0.56 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.22/0.56 = ( numeral_numeral_int @ V ) ) )
% 0.22/0.56 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.22/0.56 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.22/0.56 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_number_of(3)
% 0.22/0.56 thf(fact_8909_max__number__of_I3_J,axiom,
% 0.22/0.56 ! [U: num,V: num] :
% 0.22/0.56 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.22/0.56 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.22/0.56 = ( numeral_numeral_real @ V ) ) )
% 0.22/0.56 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.22/0.56 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.22/0.56 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_number_of(3)
% 0.22/0.56 thf(fact_8910_max__number__of_I2_J,axiom,
% 0.22/0.56 ! [U: num,V: num] :
% 0.22/0.56 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.22/0.56 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.22/0.56 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 0.22/0.56 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.22/0.56 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.22/0.56 = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_number_of(2)
% 0.22/0.56 thf(fact_8911_max__number__of_I2_J,axiom,
% 0.22/0.56 ! [U: num,V: num] :
% 0.22/0.56 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.22/0.56 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.22/0.56 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 0.22/0.56 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.22/0.56 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.22/0.56 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_number_of(2)
% 0.22/0.56 thf(fact_8912_max__number__of_I2_J,axiom,
% 0.22/0.56 ! [U: num,V: num] :
% 0.22/0.56 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.22/0.56 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.22/0.56 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 0.22/0.56 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.22/0.56 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.22/0.56 = ( numeral_numeral_int @ U ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_number_of(2)
% 0.22/0.56 thf(fact_8913_max__number__of_I2_J,axiom,
% 0.22/0.56 ! [U: num,V: num] :
% 0.22/0.56 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.22/0.56 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.22/0.56 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 0.22/0.56 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.22/0.56 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.22/0.56 = ( numeral_numeral_real @ U ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_number_of(2)
% 0.22/0.56 thf(fact_8914_drop__bit__of__Suc__0,axiom,
% 0.22/0.56 ! [N: nat] :
% 0.22/0.56 ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.22/0.56 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % drop_bit_of_Suc_0
% 0.22/0.56 thf(fact_8915_of__nat__max,axiom,
% 0.22/0.56 ! [X: nat,Y4: nat] :
% 0.22/0.56 ( ( semiri4216267220026989637d_enat @ ( ord_max_nat @ X @ Y4 ) )
% 0.22/0.56 = ( ord_ma741700101516333627d_enat @ ( semiri4216267220026989637d_enat @ X ) @ ( semiri4216267220026989637d_enat @ Y4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % of_nat_max
% 0.22/0.56 thf(fact_8916_of__nat__max,axiom,
% 0.22/0.56 ! [X: nat,Y4: nat] :
% 0.22/0.56 ( ( semiri4939895301339042750nteger @ ( ord_max_nat @ X @ Y4 ) )
% 0.22/0.56 = ( ord_max_Code_integer @ ( semiri4939895301339042750nteger @ X ) @ ( semiri4939895301339042750nteger @ Y4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % of_nat_max
% 0.22/0.56 thf(fact_8917_of__nat__max,axiom,
% 0.22/0.56 ! [X: nat,Y4: nat] :
% 0.22/0.56 ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y4 ) )
% 0.22/0.56 = ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % of_nat_max
% 0.22/0.56 thf(fact_8918_of__nat__max,axiom,
% 0.22/0.56 ! [X: nat,Y4: nat] :
% 0.22/0.56 ( ( semiri681578069525770553at_rat @ ( ord_max_nat @ X @ Y4 ) )
% 0.22/0.56 = ( ord_max_rat @ ( semiri681578069525770553at_rat @ X ) @ ( semiri681578069525770553at_rat @ Y4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % of_nat_max
% 0.22/0.56 thf(fact_8919_of__nat__max,axiom,
% 0.22/0.56 ! [X: nat,Y4: nat] :
% 0.22/0.56 ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y4 ) )
% 0.22/0.56 = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % of_nat_max
% 0.22/0.56 thf(fact_8920_of__nat__max,axiom,
% 0.22/0.56 ! [X: nat,Y4: nat] :
% 0.22/0.56 ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y4 ) )
% 0.22/0.56 = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % of_nat_max
% 0.22/0.56 thf(fact_8921_max__diff__distrib__left,axiom,
% 0.22/0.56 ! [X: code_integer,Y4: code_integer,Z: code_integer] :
% 0.22/0.56 ( ( minus_8373710615458151222nteger @ ( ord_max_Code_integer @ X @ Y4 ) @ Z )
% 0.22/0.56 = ( ord_max_Code_integer @ ( minus_8373710615458151222nteger @ X @ Z ) @ ( minus_8373710615458151222nteger @ Y4 @ Z ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_diff_distrib_left
% 0.22/0.56 thf(fact_8922_max__diff__distrib__left,axiom,
% 0.22/0.56 ! [X: real,Y4: real,Z: real] :
% 0.22/0.56 ( ( minus_minus_real @ ( ord_max_real @ X @ Y4 ) @ Z )
% 0.22/0.56 = ( ord_max_real @ ( minus_minus_real @ X @ Z ) @ ( minus_minus_real @ Y4 @ Z ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_diff_distrib_left
% 0.22/0.56 thf(fact_8923_max__diff__distrib__left,axiom,
% 0.22/0.56 ! [X: rat,Y4: rat,Z: rat] :
% 0.22/0.56 ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y4 ) @ Z )
% 0.22/0.56 = ( ord_max_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y4 @ Z ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_diff_distrib_left
% 0.22/0.56 thf(fact_8924_max__diff__distrib__left,axiom,
% 0.22/0.56 ! [X: int,Y4: int,Z: int] :
% 0.22/0.56 ( ( minus_minus_int @ ( ord_max_int @ X @ Y4 ) @ Z )
% 0.22/0.56 = ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y4 @ Z ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_diff_distrib_left
% 0.22/0.56 thf(fact_8925_max__add__distrib__right,axiom,
% 0.22/0.56 ! [X: real,Y4: real,Z: real] :
% 0.22/0.56 ( ( plus_plus_real @ X @ ( ord_max_real @ Y4 @ Z ) )
% 0.22/0.56 = ( ord_max_real @ ( plus_plus_real @ X @ Y4 ) @ ( plus_plus_real @ X @ Z ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_add_distrib_right
% 0.22/0.56 thf(fact_8926_max__add__distrib__right,axiom,
% 0.22/0.56 ! [X: rat,Y4: rat,Z: rat] :
% 0.22/0.56 ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y4 @ Z ) )
% 0.22/0.56 = ( ord_max_rat @ ( plus_plus_rat @ X @ Y4 ) @ ( plus_plus_rat @ X @ Z ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_add_distrib_right
% 0.22/0.56 thf(fact_8927_max__add__distrib__right,axiom,
% 0.22/0.56 ! [X: int,Y4: int,Z: int] :
% 0.22/0.56 ( ( plus_plus_int @ X @ ( ord_max_int @ Y4 @ Z ) )
% 0.22/0.56 = ( ord_max_int @ ( plus_plus_int @ X @ Y4 ) @ ( plus_plus_int @ X @ Z ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_add_distrib_right
% 0.22/0.56 thf(fact_8928_max__add__distrib__right,axiom,
% 0.22/0.56 ! [X: nat,Y4: nat,Z: nat] :
% 0.22/0.56 ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y4 @ Z ) )
% 0.22/0.56 = ( ord_max_nat @ ( plus_plus_nat @ X @ Y4 ) @ ( plus_plus_nat @ X @ Z ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_add_distrib_right
% 0.22/0.56 thf(fact_8929_max__add__distrib__right,axiom,
% 0.22/0.56 ! [X: code_integer,Y4: code_integer,Z: code_integer] :
% 0.22/0.56 ( ( plus_p5714425477246183910nteger @ X @ ( ord_max_Code_integer @ Y4 @ Z ) )
% 0.22/0.56 = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Y4 ) @ ( plus_p5714425477246183910nteger @ X @ Z ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_add_distrib_right
% 0.22/0.56 thf(fact_8930_max__add__distrib__left,axiom,
% 0.22/0.56 ! [X: real,Y4: real,Z: real] :
% 0.22/0.56 ( ( plus_plus_real @ ( ord_max_real @ X @ Y4 ) @ Z )
% 0.22/0.56 = ( ord_max_real @ ( plus_plus_real @ X @ Z ) @ ( plus_plus_real @ Y4 @ Z ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_add_distrib_left
% 0.22/0.56 thf(fact_8931_max__add__distrib__left,axiom,
% 0.22/0.56 ! [X: rat,Y4: rat,Z: rat] :
% 0.22/0.56 ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y4 ) @ Z )
% 0.22/0.56 = ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y4 @ Z ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_add_distrib_left
% 0.22/0.56 thf(fact_8932_max__add__distrib__left,axiom,
% 0.22/0.56 ! [X: int,Y4: int,Z: int] :
% 0.22/0.56 ( ( plus_plus_int @ ( ord_max_int @ X @ Y4 ) @ Z )
% 0.22/0.56 = ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y4 @ Z ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_add_distrib_left
% 0.22/0.56 thf(fact_8933_max__add__distrib__left,axiom,
% 0.22/0.56 ! [X: nat,Y4: nat,Z: nat] :
% 0.22/0.56 ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y4 ) @ Z )
% 0.22/0.56 = ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y4 @ Z ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_add_distrib_left
% 0.22/0.56 thf(fact_8934_max__add__distrib__left,axiom,
% 0.22/0.56 ! [X: code_integer,Y4: code_integer,Z: code_integer] :
% 0.22/0.56 ( ( plus_p5714425477246183910nteger @ ( ord_max_Code_integer @ X @ Y4 ) @ Z )
% 0.22/0.56 = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Z ) @ ( plus_p5714425477246183910nteger @ Y4 @ Z ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_add_distrib_left
% 0.22/0.56 thf(fact_8935_max__absorb2,axiom,
% 0.22/0.56 ! [X: extended_enat,Y4: extended_enat] :
% 0.22/0.56 ( ( ord_le2932123472753598470d_enat @ X @ Y4 )
% 0.22/0.56 => ( ( ord_ma741700101516333627d_enat @ X @ Y4 )
% 0.22/0.56 = Y4 ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_absorb2
% 0.22/0.56 thf(fact_8936_max__absorb2,axiom,
% 0.22/0.56 ! [X: code_integer,Y4: code_integer] :
% 0.22/0.56 ( ( ord_le3102999989581377725nteger @ X @ Y4 )
% 0.22/0.56 => ( ( ord_max_Code_integer @ X @ Y4 )
% 0.22/0.56 = Y4 ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_absorb2
% 0.22/0.56 thf(fact_8937_max__absorb2,axiom,
% 0.22/0.56 ! [X: set_int,Y4: set_int] :
% 0.22/0.56 ( ( ord_less_eq_set_int @ X @ Y4 )
% 0.22/0.56 => ( ( ord_max_set_int @ X @ Y4 )
% 0.22/0.56 = Y4 ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_absorb2
% 0.22/0.56 thf(fact_8938_max__absorb2,axiom,
% 0.22/0.56 ! [X: rat,Y4: rat] :
% 0.22/0.56 ( ( ord_less_eq_rat @ X @ Y4 )
% 0.22/0.56 => ( ( ord_max_rat @ X @ Y4 )
% 0.22/0.56 = Y4 ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_absorb2
% 0.22/0.56 thf(fact_8939_max__absorb2,axiom,
% 0.22/0.56 ! [X: num,Y4: num] :
% 0.22/0.56 ( ( ord_less_eq_num @ X @ Y4 )
% 0.22/0.56 => ( ( ord_max_num @ X @ Y4 )
% 0.22/0.56 = Y4 ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_absorb2
% 0.22/0.56 thf(fact_8940_max__absorb2,axiom,
% 0.22/0.56 ! [X: nat,Y4: nat] :
% 0.22/0.56 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.22/0.56 => ( ( ord_max_nat @ X @ Y4 )
% 0.22/0.56 = Y4 ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_absorb2
% 0.22/0.56 thf(fact_8941_max__absorb2,axiom,
% 0.22/0.56 ! [X: int,Y4: int] :
% 0.22/0.56 ( ( ord_less_eq_int @ X @ Y4 )
% 0.22/0.56 => ( ( ord_max_int @ X @ Y4 )
% 0.22/0.56 = Y4 ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_absorb2
% 0.22/0.56 thf(fact_8942_max__absorb2,axiom,
% 0.22/0.56 ! [X: real,Y4: real] :
% 0.22/0.56 ( ( ord_less_eq_real @ X @ Y4 )
% 0.22/0.56 => ( ( ord_max_real @ X @ Y4 )
% 0.22/0.56 = Y4 ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_absorb2
% 0.22/0.56 thf(fact_8943_max__absorb1,axiom,
% 0.22/0.56 ! [Y4: extended_enat,X: extended_enat] :
% 0.22/0.56 ( ( ord_le2932123472753598470d_enat @ Y4 @ X )
% 0.22/0.56 => ( ( ord_ma741700101516333627d_enat @ X @ Y4 )
% 0.22/0.56 = X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_absorb1
% 0.22/0.56 thf(fact_8944_max__absorb1,axiom,
% 0.22/0.56 ! [Y4: code_integer,X: code_integer] :
% 0.22/0.56 ( ( ord_le3102999989581377725nteger @ Y4 @ X )
% 0.22/0.56 => ( ( ord_max_Code_integer @ X @ Y4 )
% 0.22/0.56 = X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_absorb1
% 0.22/0.56 thf(fact_8945_max__absorb1,axiom,
% 0.22/0.56 ! [Y4: set_int,X: set_int] :
% 0.22/0.56 ( ( ord_less_eq_set_int @ Y4 @ X )
% 0.22/0.56 => ( ( ord_max_set_int @ X @ Y4 )
% 0.22/0.56 = X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_absorb1
% 0.22/0.56 thf(fact_8946_max__absorb1,axiom,
% 0.22/0.56 ! [Y4: rat,X: rat] :
% 0.22/0.56 ( ( ord_less_eq_rat @ Y4 @ X )
% 0.22/0.56 => ( ( ord_max_rat @ X @ Y4 )
% 0.22/0.56 = X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_absorb1
% 0.22/0.56 thf(fact_8947_max__absorb1,axiom,
% 0.22/0.56 ! [Y4: num,X: num] :
% 0.22/0.56 ( ( ord_less_eq_num @ Y4 @ X )
% 0.22/0.56 => ( ( ord_max_num @ X @ Y4 )
% 0.22/0.56 = X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_absorb1
% 0.22/0.56 thf(fact_8948_max__absorb1,axiom,
% 0.22/0.56 ! [Y4: nat,X: nat] :
% 0.22/0.56 ( ( ord_less_eq_nat @ Y4 @ X )
% 0.22/0.56 => ( ( ord_max_nat @ X @ Y4 )
% 0.22/0.56 = X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_absorb1
% 0.22/0.56 thf(fact_8949_max__absorb1,axiom,
% 0.22/0.56 ! [Y4: int,X: int] :
% 0.22/0.56 ( ( ord_less_eq_int @ Y4 @ X )
% 0.22/0.56 => ( ( ord_max_int @ X @ Y4 )
% 0.22/0.56 = X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_absorb1
% 0.22/0.56 thf(fact_8950_max__absorb1,axiom,
% 0.22/0.56 ! [Y4: real,X: real] :
% 0.22/0.56 ( ( ord_less_eq_real @ Y4 @ X )
% 0.22/0.56 => ( ( ord_max_real @ X @ Y4 )
% 0.22/0.56 = X ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_absorb1
% 0.22/0.56 thf(fact_8951_max__def,axiom,
% 0.22/0.56 ( ord_max_nat
% 0.22/0.56 = ( ^ [A3: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_def
% 0.22/0.56 thf(fact_8952_max__def,axiom,
% 0.22/0.56 ( ord_max_int
% 0.22/0.56 = ( ^ [A3: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_def
% 0.22/0.56 thf(fact_8953_max__def,axiom,
% 0.22/0.56 ( ord_max_real
% 0.22/0.56 = ( ^ [A3: real,B2: real] : ( if_real @ ( ord_less_eq_real @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_def
% 0.22/0.56 thf(fact_8954_or__not__num__neg_Osimps_I1_J,axiom,
% 0.22/0.56 ( ( bit_or_not_num_neg @ one @ one )
% 0.22/0.56 = one ) ).
% 0.22/0.56
% 0.22/0.56 % or_not_num_neg.simps(1)
% 0.22/0.56 thf(fact_8955_or__not__num__neg_Osimps_I4_J,axiom,
% 0.22/0.56 ! [N: num] :
% 0.22/0.56 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
% 0.22/0.56 = ( bit0 @ one ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_not_num_neg.simps(4)
% 0.22/0.56 thf(fact_8956_drop__bit__nat__eq,axiom,
% 0.22/0.56 ! [N: nat,K: int] :
% 0.22/0.56 ( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K ) )
% 0.22/0.56 = ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % drop_bit_nat_eq
% 0.22/0.56 thf(fact_8957_or__not__num__neg_Osimps_I6_J,axiom,
% 0.22/0.56 ! [N: num,M: num] :
% 0.22/0.56 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
% 0.22/0.56 = ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_not_num_neg.simps(6)
% 0.22/0.56 thf(fact_8958_or__not__num__neg_Osimps_I7_J,axiom,
% 0.22/0.56 ! [N: num] :
% 0.22/0.56 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
% 0.22/0.56 = one ) ).
% 0.22/0.56
% 0.22/0.56 % or_not_num_neg.simps(7)
% 0.22/0.56 thf(fact_8959_or__not__num__neg_Osimps_I3_J,axiom,
% 0.22/0.56 ! [M: num] :
% 0.22/0.56 ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
% 0.22/0.56 = ( bit1 @ M ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_not_num_neg.simps(3)
% 0.22/0.56 thf(fact_8960_or__not__num__neg_Osimps_I2_J,axiom,
% 0.22/0.56 ! [M: num] :
% 0.22/0.56 ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
% 0.22/0.56 = ( bit1 @ M ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_not_num_neg.simps(2)
% 0.22/0.56 thf(fact_8961_drop__bit__nat__def,axiom,
% 0.22/0.56 ( bit_se8570568707652914677it_nat
% 0.22/0.56 = ( ^ [N4: nat,M5: nat] : ( divide_divide_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % drop_bit_nat_def
% 0.22/0.56 thf(fact_8962_int__numeral__or__not__num__neg,axiom,
% 0.22/0.56 ! [M: num,N: num] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.56 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % int_numeral_or_not_num_neg
% 0.22/0.56 thf(fact_8963_int__numeral__not__or__num__neg,axiom,
% 0.22/0.56 ! [M: num,N: num] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.22/0.56 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % int_numeral_not_or_num_neg
% 0.22/0.56 thf(fact_8964_numeral__or__not__num__eq,axiom,
% 0.22/0.56 ! [M: num,N: num] :
% 0.22/0.56 ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
% 0.22/0.56 = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % numeral_or_not_num_eq
% 0.22/0.56 thf(fact_8965_or__minus__numerals_I1_J,axiom,
% 0.22/0.56 ! [N: num] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.22/0.56 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_minus_numerals(1)
% 0.22/0.56 thf(fact_8966_or__minus__numerals_I5_J,axiom,
% 0.22/0.56 ! [N: num] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 0.22/0.56 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_minus_numerals(5)
% 0.22/0.56 thf(fact_8967_or__nat__unfold,axiom,
% 0.22/0.56 ( bit_se1412395901928357646or_nat
% 0.22/0.56 = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M5 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_nat_unfold
% 0.22/0.56 thf(fact_8968_or__minus__numerals_I7_J,axiom,
% 0.22/0.56 ! [N: num,M: num] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.22/0.56 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_minus_numerals(7)
% 0.22/0.56 thf(fact_8969_max__enat__simps_I2_J,axiom,
% 0.22/0.56 ! [Q2: extended_enat] :
% 0.22/0.56 ( ( ord_ma741700101516333627d_enat @ Q2 @ zero_z5237406670263579293d_enat )
% 0.22/0.56 = Q2 ) ).
% 0.22/0.56
% 0.22/0.56 % max_enat_simps(2)
% 0.22/0.56 thf(fact_8970_max__enat__simps_I3_J,axiom,
% 0.22/0.56 ! [Q2: extended_enat] :
% 0.22/0.56 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q2 )
% 0.22/0.56 = Q2 ) ).
% 0.22/0.56
% 0.22/0.56 % max_enat_simps(3)
% 0.22/0.56 thf(fact_8971_max__Suc__Suc,axiom,
% 0.22/0.56 ! [M: nat,N: nat] :
% 0.22/0.56 ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.22/0.56 = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_Suc_Suc
% 0.22/0.56 thf(fact_8972_max__0R,axiom,
% 0.22/0.56 ! [N: nat] :
% 0.22/0.56 ( ( ord_max_nat @ N @ zero_zero_nat )
% 0.22/0.56 = N ) ).
% 0.22/0.56
% 0.22/0.56 % max_0R
% 0.22/0.56 thf(fact_8973_max__0L,axiom,
% 0.22/0.56 ! [N: nat] :
% 0.22/0.56 ( ( ord_max_nat @ zero_zero_nat @ N )
% 0.22/0.56 = N ) ).
% 0.22/0.56
% 0.22/0.56 % max_0L
% 0.22/0.56 thf(fact_8974_max__nat_Oright__neutral,axiom,
% 0.22/0.56 ! [A: nat] :
% 0.22/0.56 ( ( ord_max_nat @ A @ zero_zero_nat )
% 0.22/0.56 = A ) ).
% 0.22/0.56
% 0.22/0.56 % max_nat.right_neutral
% 0.22/0.56 thf(fact_8975_max__nat_Oneutr__eq__iff,axiom,
% 0.22/0.56 ! [A: nat,B: nat] :
% 0.22/0.56 ( ( zero_zero_nat
% 0.22/0.56 = ( ord_max_nat @ A @ B ) )
% 0.22/0.56 = ( ( A = zero_zero_nat )
% 0.22/0.56 & ( B = zero_zero_nat ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_nat.neutr_eq_iff
% 0.22/0.56 thf(fact_8976_max__nat_Oleft__neutral,axiom,
% 0.22/0.56 ! [A: nat] :
% 0.22/0.56 ( ( ord_max_nat @ zero_zero_nat @ A )
% 0.22/0.56 = A ) ).
% 0.22/0.56
% 0.22/0.56 % max_nat.left_neutral
% 0.22/0.56 thf(fact_8977_max__nat_Oeq__neutr__iff,axiom,
% 0.22/0.56 ! [A: nat,B: nat] :
% 0.22/0.56 ( ( ( ord_max_nat @ A @ B )
% 0.22/0.56 = zero_zero_nat )
% 0.22/0.56 = ( ( A = zero_zero_nat )
% 0.22/0.56 & ( B = zero_zero_nat ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_nat.eq_neutr_iff
% 0.22/0.56 thf(fact_8978_pred__numeral__simps_I2_J,axiom,
% 0.22/0.56 ! [K: num] :
% 0.22/0.56 ( ( pred_numeral @ ( bit0 @ K ) )
% 0.22/0.56 = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % pred_numeral_simps(2)
% 0.22/0.56 thf(fact_8979_max__numeral__Suc,axiom,
% 0.22/0.56 ! [K: num,N: nat] :
% 0.22/0.56 ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.22/0.56 = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_numeral_Suc
% 0.22/0.56 thf(fact_8980_max__Suc__numeral,axiom,
% 0.22/0.56 ! [N: nat,K: num] :
% 0.22/0.56 ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.22/0.56 = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % max_Suc_numeral
% 0.22/0.56 thf(fact_8981_or__minus__numerals_I3_J,axiom,
% 0.22/0.56 ! [M: num,N: num] :
% 0.22/0.56 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.22/0.56 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_minus_numerals(3)
% 0.22/0.56 thf(fact_8982_nat__mult__max__right,axiom,
% 0.22/0.56 ! [M: nat,N: nat,Q2: nat] :
% 0.22/0.56 ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 0.22/0.56 = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % nat_mult_max_right
% 0.22/0.56 thf(fact_8983_nat__mult__max__left,axiom,
% 0.22/0.56 ! [M: nat,N: nat,Q2: nat] :
% 0.22/0.56 ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 0.22/0.56 = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % nat_mult_max_left
% 0.22/0.56 thf(fact_8984_nat__add__max__right,axiom,
% 0.22/0.56 ! [M: nat,N: nat,Q2: nat] :
% 0.22/0.56 ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 0.22/0.56 = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % nat_add_max_right
% 0.22/0.56 thf(fact_8985_nat__add__max__left,axiom,
% 0.22/0.56 ! [M: nat,N: nat,Q2: nat] :
% 0.22/0.56 ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 0.22/0.56 = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % nat_add_max_left
% 0.22/0.56 thf(fact_8986_semiring__norm_I26_J,axiom,
% 0.22/0.56 ( ( bitM @ one )
% 0.22/0.56 = one ) ).
% 0.22/0.56
% 0.22/0.56 % semiring_norm(26)
% 0.22/0.56 thf(fact_8987_nat__minus__add__max,axiom,
% 0.22/0.56 ! [N: nat,M: nat] :
% 0.22/0.56 ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 0.22/0.56 = ( ord_max_nat @ N @ M ) ) ).
% 0.22/0.56
% 0.22/0.56 % nat_minus_add_max
% 0.22/0.56 thf(fact_8988_semiring__norm_I27_J,axiom,
% 0.22/0.56 ! [N: num] :
% 0.22/0.56 ( ( bitM @ ( bit0 @ N ) )
% 0.22/0.56 = ( bit1 @ ( bitM @ N ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % semiring_norm(27)
% 0.22/0.56 thf(fact_8989_semiring__norm_I28_J,axiom,
% 0.22/0.56 ! [N: num] :
% 0.22/0.56 ( ( bitM @ ( bit1 @ N ) )
% 0.22/0.56 = ( bit1 @ ( bit0 @ N ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % semiring_norm(28)
% 0.22/0.56 thf(fact_8990_or__not__num__neg_Osimps_I5_J,axiom,
% 0.22/0.56 ! [N: num,M: num] :
% 0.22/0.56 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
% 0.22/0.56 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_not_num_neg.simps(5)
% 0.22/0.56 thf(fact_8991_inc__BitM__eq,axiom,
% 0.22/0.56 ! [N: num] :
% 0.22/0.56 ( ( inc @ ( bitM @ N ) )
% 0.22/0.56 = ( bit0 @ N ) ) ).
% 0.22/0.56
% 0.22/0.56 % inc_BitM_eq
% 0.22/0.56 thf(fact_8992_or__not__num__neg_Osimps_I9_J,axiom,
% 0.22/0.56 ! [N: num,M: num] :
% 0.22/0.56 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
% 0.22/0.56 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_not_num_neg.simps(9)
% 0.22/0.56 thf(fact_8993_BitM__inc__eq,axiom,
% 0.22/0.56 ! [N: num] :
% 0.22/0.56 ( ( bitM @ ( inc @ N ) )
% 0.22/0.56 = ( bit1 @ N ) ) ).
% 0.22/0.56
% 0.22/0.56 % BitM_inc_eq
% 0.22/0.56 thf(fact_8994_eval__nat__numeral_I2_J,axiom,
% 0.22/0.56 ! [N: num] :
% 0.22/0.56 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 0.22/0.56 = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % eval_nat_numeral(2)
% 0.22/0.56 thf(fact_8995_one__plus__BitM,axiom,
% 0.22/0.56 ! [N: num] :
% 0.22/0.56 ( ( plus_plus_num @ one @ ( bitM @ N ) )
% 0.22/0.56 = ( bit0 @ N ) ) ).
% 0.22/0.56
% 0.22/0.56 % one_plus_BitM
% 0.22/0.56 thf(fact_8996_BitM__plus__one,axiom,
% 0.22/0.56 ! [N: num] :
% 0.22/0.56 ( ( plus_plus_num @ ( bitM @ N ) @ one )
% 0.22/0.56 = ( bit0 @ N ) ) ).
% 0.22/0.56
% 0.22/0.56 % BitM_plus_one
% 0.22/0.56 thf(fact_8997_or__not__num__neg_Osimps_I8_J,axiom,
% 0.22/0.56 ! [N: num,M: num] :
% 0.22/0.56 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
% 0.22/0.56 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_not_num_neg.simps(8)
% 0.22/0.56 thf(fact_8998_or__not__num__neg_Oelims,axiom,
% 0.22/0.56 ! [X: num,Xa: num,Y4: num] :
% 0.22/0.56 ( ( ( bit_or_not_num_neg @ X @ Xa )
% 0.22/0.56 = Y4 )
% 0.22/0.56 => ( ( ( X = one )
% 0.22/0.56 => ( ( Xa = one )
% 0.22/0.56 => ( Y4 != one ) ) )
% 0.22/0.56 => ( ( ( X = one )
% 0.22/0.56 => ! [M4: num] :
% 0.22/0.56 ( ( Xa
% 0.22/0.56 = ( bit0 @ M4 ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 != ( bit1 @ M4 ) ) ) )
% 0.22/0.56 => ( ( ( X = one )
% 0.22/0.56 => ! [M4: num] :
% 0.22/0.56 ( ( Xa
% 0.22/0.56 = ( bit1 @ M4 ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 != ( bit1 @ M4 ) ) ) )
% 0.22/0.56 => ( ( ? [N2: num] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( bit0 @ N2 ) )
% 0.22/0.56 => ( ( Xa = one )
% 0.22/0.56 => ( Y4
% 0.22/0.56 != ( bit0 @ one ) ) ) )
% 0.22/0.56 => ( ! [N2: num] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( bit0 @ N2 ) )
% 0.22/0.56 => ! [M4: num] :
% 0.22/0.56 ( ( Xa
% 0.22/0.56 = ( bit0 @ M4 ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
% 0.22/0.56 => ( ! [N2: num] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( bit0 @ N2 ) )
% 0.22/0.56 => ! [M4: num] :
% 0.22/0.56 ( ( Xa
% 0.22/0.56 = ( bit1 @ M4 ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 != ( bit0 @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
% 0.22/0.56 => ( ( ? [N2: num] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( bit1 @ N2 ) )
% 0.22/0.56 => ( ( Xa = one )
% 0.22/0.56 => ( Y4 != one ) ) )
% 0.22/0.56 => ( ! [N2: num] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( bit1 @ N2 ) )
% 0.22/0.56 => ! [M4: num] :
% 0.22/0.56 ( ( Xa
% 0.22/0.56 = ( bit0 @ M4 ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
% 0.22/0.56 => ~ ! [N2: num] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( bit1 @ N2 ) )
% 0.22/0.56 => ! [M4: num] :
% 0.22/0.56 ( ( Xa
% 0.22/0.56 = ( bit1 @ M4 ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % or_not_num_neg.elims
% 0.22/0.56 thf(fact_8999_root__powr__inverse,axiom,
% 0.22/0.56 ! [N: nat,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.56 => ( ( root @ N @ X )
% 0.22/0.56 = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % root_powr_inverse
% 0.22/0.56 thf(fact_9000_xor__minus__numerals_I2_J,axiom,
% 0.22/0.56 ! [K: int,N: num] :
% 0.22/0.56 ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.56 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % xor_minus_numerals(2)
% 0.22/0.56 thf(fact_9001_xor__minus__numerals_I1_J,axiom,
% 0.22/0.56 ! [N: num,K: int] :
% 0.22/0.56 ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
% 0.22/0.56 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % xor_minus_numerals(1)
% 0.22/0.56 thf(fact_9002_real__root__zero,axiom,
% 0.22/0.56 ! [N: nat] :
% 0.22/0.56 ( ( root @ N @ zero_zero_real )
% 0.22/0.56 = zero_zero_real ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_zero
% 0.22/0.56 thf(fact_9003_real__root__Suc__0,axiom,
% 0.22/0.56 ! [X: real] :
% 0.22/0.56 ( ( root @ ( suc @ zero_zero_nat ) @ X )
% 0.22/0.56 = X ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_Suc_0
% 0.22/0.56 thf(fact_9004_root__0,axiom,
% 0.22/0.56 ! [X: real] :
% 0.22/0.56 ( ( root @ zero_zero_nat @ X )
% 0.22/0.56 = zero_zero_real ) ).
% 0.22/0.56
% 0.22/0.56 % root_0
% 0.22/0.56 thf(fact_9005_real__root__eq__iff,axiom,
% 0.22/0.56 ! [N: nat,X: real,Y4: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ( root @ N @ X )
% 0.22/0.56 = ( root @ N @ Y4 ) )
% 0.22/0.56 = ( X = Y4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_eq_iff
% 0.22/0.56 thf(fact_9006_real__root__eq__0__iff,axiom,
% 0.22/0.56 ! [N: nat,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ( root @ N @ X )
% 0.22/0.56 = zero_zero_real )
% 0.22/0.56 = ( X = zero_zero_real ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_eq_0_iff
% 0.22/0.56 thf(fact_9007_real__root__less__iff,axiom,
% 0.22/0.56 ! [N: nat,X: real,Y4: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y4 ) )
% 0.22/0.56 = ( ord_less_real @ X @ Y4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_less_iff
% 0.22/0.56 thf(fact_9008_real__root__le__iff,axiom,
% 0.22/0.56 ! [N: nat,X: real,Y4: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y4 ) )
% 0.22/0.56 = ( ord_less_eq_real @ X @ Y4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_le_iff
% 0.22/0.56 thf(fact_9009_real__root__one,axiom,
% 0.22/0.56 ! [N: nat] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( root @ N @ one_one_real )
% 0.22/0.56 = one_one_real ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_one
% 0.22/0.56 thf(fact_9010_real__root__eq__1__iff,axiom,
% 0.22/0.56 ! [N: nat,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ( root @ N @ X )
% 0.22/0.56 = one_one_real )
% 0.22/0.56 = ( X = one_one_real ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_eq_1_iff
% 0.22/0.56 thf(fact_9011_real__root__gt__0__iff,axiom,
% 0.22/0.56 ! [N: nat,Y4: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y4 ) )
% 0.22/0.56 = ( ord_less_real @ zero_zero_real @ Y4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_gt_0_iff
% 0.22/0.56 thf(fact_9012_real__root__lt__0__iff,axiom,
% 0.22/0.56 ! [N: nat,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
% 0.22/0.56 = ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_lt_0_iff
% 0.22/0.56 thf(fact_9013_real__root__ge__0__iff,axiom,
% 0.22/0.56 ! [N: nat,Y4: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y4 ) )
% 0.22/0.56 = ( ord_less_eq_real @ zero_zero_real @ Y4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_ge_0_iff
% 0.22/0.56 thf(fact_9014_real__root__le__0__iff,axiom,
% 0.22/0.56 ! [N: nat,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
% 0.22/0.56 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_le_0_iff
% 0.22/0.56 thf(fact_9015_real__root__gt__1__iff,axiom,
% 0.22/0.56 ! [N: nat,Y4: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y4 ) )
% 0.22/0.56 = ( ord_less_real @ one_one_real @ Y4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_gt_1_iff
% 0.22/0.56 thf(fact_9016_real__root__lt__1__iff,axiom,
% 0.22/0.56 ! [N: nat,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
% 0.22/0.56 = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_lt_1_iff
% 0.22/0.56 thf(fact_9017_real__root__ge__1__iff,axiom,
% 0.22/0.56 ! [N: nat,Y4: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y4 ) )
% 0.22/0.56 = ( ord_less_eq_real @ one_one_real @ Y4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_ge_1_iff
% 0.22/0.56 thf(fact_9018_real__root__le__1__iff,axiom,
% 0.22/0.56 ! [N: nat,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
% 0.22/0.56 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_le_1_iff
% 0.22/0.56 thf(fact_9019_real__root__pow__pos2,axiom,
% 0.22/0.56 ! [N: nat,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.56 => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 0.22/0.56 = X ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_pow_pos2
% 0.22/0.56 thf(fact_9020_real__root__inverse,axiom,
% 0.22/0.56 ! [N: nat,X: real] :
% 0.22/0.56 ( ( root @ N @ ( inverse_inverse_real @ X ) )
% 0.22/0.56 = ( inverse_inverse_real @ ( root @ N @ X ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_inverse
% 0.22/0.56 thf(fact_9021_real__root__mult__exp,axiom,
% 0.22/0.56 ! [M: nat,N: nat,X: real] :
% 0.22/0.56 ( ( root @ ( times_times_nat @ M @ N ) @ X )
% 0.22/0.56 = ( root @ M @ ( root @ N @ X ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_mult_exp
% 0.22/0.56 thf(fact_9022_real__root__mult,axiom,
% 0.22/0.56 ! [N: nat,X: real,Y4: real] :
% 0.22/0.56 ( ( root @ N @ ( times_times_real @ X @ Y4 ) )
% 0.22/0.56 = ( times_times_real @ ( root @ N @ X ) @ ( root @ N @ Y4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_mult
% 0.22/0.56 thf(fact_9023_real__root__minus,axiom,
% 0.22/0.56 ! [N: nat,X: real] :
% 0.22/0.56 ( ( root @ N @ ( uminus_uminus_real @ X ) )
% 0.22/0.56 = ( uminus_uminus_real @ ( root @ N @ X ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_minus
% 0.22/0.56 thf(fact_9024_real__root__commute,axiom,
% 0.22/0.56 ! [M: nat,N: nat,X: real] :
% 0.22/0.56 ( ( root @ M @ ( root @ N @ X ) )
% 0.22/0.56 = ( root @ N @ ( root @ M @ X ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_commute
% 0.22/0.56 thf(fact_9025_real__root__divide,axiom,
% 0.22/0.56 ! [N: nat,X: real,Y4: real] :
% 0.22/0.56 ( ( root @ N @ ( divide_divide_real @ X @ Y4 ) )
% 0.22/0.56 = ( divide_divide_real @ ( root @ N @ X ) @ ( root @ N @ Y4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_divide
% 0.22/0.56 thf(fact_9026_real__root__pos__pos__le,axiom,
% 0.22/0.56 ! [X: real,N: nat] :
% 0.22/0.56 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.56 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_pos_pos_le
% 0.22/0.56 thf(fact_9027_real__root__less__mono,axiom,
% 0.22/0.56 ! [N: nat,X: real,Y4: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_real @ X @ Y4 )
% 0.22/0.56 => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y4 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_less_mono
% 0.22/0.56 thf(fact_9028_real__root__le__mono,axiom,
% 0.22/0.56 ! [N: nat,X: real,Y4: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_eq_real @ X @ Y4 )
% 0.22/0.56 => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y4 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_le_mono
% 0.22/0.56 thf(fact_9029_real__root__power,axiom,
% 0.22/0.56 ! [N: nat,X: real,K: nat] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( root @ N @ ( power_power_real @ X @ K ) )
% 0.22/0.56 = ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_power
% 0.22/0.56 thf(fact_9030_real__root__abs,axiom,
% 0.22/0.56 ! [N: nat,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( root @ N @ ( abs_abs_real @ X ) )
% 0.22/0.56 = ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_abs
% 0.22/0.56 thf(fact_9031_sgn__root,axiom,
% 0.22/0.56 ! [N: nat,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( sgn_sgn_real @ ( root @ N @ X ) )
% 0.22/0.56 = ( sgn_sgn_real @ X ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % sgn_root
% 0.22/0.56 thf(fact_9032_real__root__gt__zero,axiom,
% 0.22/0.56 ! [N: nat,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.56 => ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_gt_zero
% 0.22/0.56 thf(fact_9033_real__root__strict__decreasing,axiom,
% 0.22/0.56 ! [N: nat,N5: nat,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_nat @ N @ N5 )
% 0.22/0.56 => ( ( ord_less_real @ one_one_real @ X )
% 0.22/0.56 => ( ord_less_real @ ( root @ N5 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_strict_decreasing
% 0.22/0.56 thf(fact_9034_sqrt__def,axiom,
% 0.22/0.56 ( sqrt
% 0.22/0.56 = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % sqrt_def
% 0.22/0.56 thf(fact_9035_root__abs__power,axiom,
% 0.22/0.56 ! [N: nat,Y4: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y4 @ N ) ) )
% 0.22/0.56 = ( abs_abs_real @ Y4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % root_abs_power
% 0.22/0.56 thf(fact_9036_real__root__pos__pos,axiom,
% 0.22/0.56 ! [N: nat,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.56 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_pos_pos
% 0.22/0.56 thf(fact_9037_real__root__strict__increasing,axiom,
% 0.22/0.56 ! [N: nat,N5: nat,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_nat @ N @ N5 )
% 0.22/0.56 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.56 => ( ( ord_less_real @ X @ one_one_real )
% 0.22/0.56 => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_strict_increasing
% 0.22/0.56 thf(fact_9038_real__root__decreasing,axiom,
% 0.22/0.56 ! [N: nat,N5: nat,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.22/0.56 => ( ( ord_less_eq_real @ one_one_real @ X )
% 0.22/0.56 => ( ord_less_eq_real @ ( root @ N5 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_decreasing
% 0.22/0.56 thf(fact_9039_real__root__pow__pos,axiom,
% 0.22/0.56 ! [N: nat,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.56 => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 0.22/0.56 = X ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_pow_pos
% 0.22/0.56 thf(fact_9040_real__root__pos__unique,axiom,
% 0.22/0.56 ! [N: nat,Y4: real,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.22/0.56 => ( ( ( power_power_real @ Y4 @ N )
% 0.22/0.56 = X )
% 0.22/0.56 => ( ( root @ N @ X )
% 0.22/0.56 = Y4 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_pos_unique
% 0.22/0.56 thf(fact_9041_real__root__power__cancel,axiom,
% 0.22/0.56 ! [N: nat,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.56 => ( ( root @ N @ ( power_power_real @ X @ N ) )
% 0.22/0.56 = X ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_power_cancel
% 0.22/0.56 thf(fact_9042_odd__real__root__pow,axiom,
% 0.22/0.56 ! [N: nat,X: real] :
% 0.22/0.56 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.56 => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 0.22/0.56 = X ) ) ).
% 0.22/0.56
% 0.22/0.56 % odd_real_root_pow
% 0.22/0.56 thf(fact_9043_odd__real__root__unique,axiom,
% 0.22/0.56 ! [N: nat,Y4: real,X: real] :
% 0.22/0.56 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.56 => ( ( ( power_power_real @ Y4 @ N )
% 0.22/0.56 = X )
% 0.22/0.56 => ( ( root @ N @ X )
% 0.22/0.56 = Y4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % odd_real_root_unique
% 0.22/0.56 thf(fact_9044_odd__real__root__power__cancel,axiom,
% 0.22/0.56 ! [N: nat,X: real] :
% 0.22/0.56 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.56 => ( ( root @ N @ ( power_power_real @ X @ N ) )
% 0.22/0.56 = X ) ) ).
% 0.22/0.56
% 0.22/0.56 % odd_real_root_power_cancel
% 0.22/0.56 thf(fact_9045_real__root__increasing,axiom,
% 0.22/0.56 ! [N: nat,N5: nat,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.22/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.56 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.22/0.56 => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % real_root_increasing
% 0.22/0.56 thf(fact_9046_sub__BitM__One__eq,axiom,
% 0.22/0.56 ! [N: num] :
% 0.22/0.56 ( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
% 0.22/0.56 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % sub_BitM_One_eq
% 0.22/0.56 thf(fact_9047_root__sgn__power,axiom,
% 0.22/0.56 ! [N: nat,Y4: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y4 ) @ ( power_power_real @ ( abs_abs_real @ Y4 ) @ N ) ) )
% 0.22/0.56 = Y4 ) ) ).
% 0.22/0.56
% 0.22/0.56 % root_sgn_power
% 0.22/0.56 thf(fact_9048_sgn__power__root,axiom,
% 0.22/0.56 ! [N: nat,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
% 0.22/0.56 = X ) ) ).
% 0.22/0.56
% 0.22/0.56 % sgn_power_root
% 0.22/0.56 thf(fact_9049_ln__root,axiom,
% 0.22/0.56 ! [N: nat,B: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.22/0.56 => ( ( ln_ln_real @ ( root @ N @ B ) )
% 0.22/0.56 = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % ln_root
% 0.22/0.56 thf(fact_9050_log__root,axiom,
% 0.22/0.56 ! [N: nat,A: real,B: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.22/0.56 => ( ( log @ B @ ( root @ N @ A ) )
% 0.22/0.56 = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % log_root
% 0.22/0.56 thf(fact_9051_log__base__root,axiom,
% 0.22/0.56 ! [N: nat,B: real,X: real] :
% 0.22/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.22/0.56 => ( ( log @ ( root @ N @ B ) @ X )
% 0.22/0.56 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % log_base_root
% 0.22/0.56 thf(fact_9052_split__root,axiom,
% 0.22/0.56 ! [P: real > $o,N: nat,X: real] :
% 0.22/0.56 ( ( P @ ( root @ N @ X ) )
% 0.22/0.56 = ( ( ( N = zero_zero_nat )
% 0.22/0.56 => ( P @ zero_zero_real ) )
% 0.22/0.56 & ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ! [Y6: real] :
% 0.22/0.56 ( ( ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N ) )
% 0.22/0.56 = X )
% 0.22/0.56 => ( P @ Y6 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % split_root
% 0.22/0.56 thf(fact_9053_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
% 0.22/0.56 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.22/0.56 ( ( vEBT_VEBT_cnt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
% 0.22/0.56 = ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList ) @ zero_zero_real ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.cnt.simps(2)
% 0.22/0.56 thf(fact_9054_VEBT_Oinject_I1_J,axiom,
% 0.22/0.56 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
% 0.22/0.56 ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 0.22/0.56 = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
% 0.22/0.56 = ( ( X11 = Y11 )
% 0.22/0.56 & ( X12 = Y12 )
% 0.22/0.56 & ( X13 = Y13 )
% 0.22/0.56 & ( X14 = Y14 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT.inject(1)
% 0.22/0.56 thf(fact_9055_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
% 0.22/0.56 ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
% 0.22/0.56 ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.naive_member.simps(2)
% 0.22/0.56 thf(fact_9056_both__member__options__ding,axiom,
% 0.22/0.56 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
% 0.22/0.56 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
% 0.22/0.56 => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.22/0.56 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.56 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % both_member_options_ding
% 0.22/0.56 thf(fact_9057_VEBT__internal_Ospace_Osimps_I2_J,axiom,
% 0.22/0.56 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.22/0.56 ( ( vEBT_VEBT_space @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
% 0.22/0.56 = ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList ) @ zero_zero_nat ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.space.simps(2)
% 0.22/0.56 thf(fact_9058_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
% 0.22/0.56 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.22/0.56 ( ( vEBT_VEBT_space2 @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
% 0.22/0.56 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList ) @ zero_zero_nat ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.space'.simps(2)
% 0.22/0.56 thf(fact_9059_both__member__options__def,axiom,
% 0.22/0.56 ( vEBT_V8194947554948674370ptions
% 0.22/0.56 = ( ^ [T2: vEBT_VEBT,X3: nat] :
% 0.22/0.56 ( ( vEBT_V5719532721284313246member @ T2 @ X3 )
% 0.22/0.56 | ( vEBT_VEBT_membermima @ T2 @ X3 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % both_member_options_def
% 0.22/0.56 thf(fact_9060_VEBT_Oinject_I2_J,axiom,
% 0.22/0.56 ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
% 0.22/0.56 ( ( ( vEBT_Leaf @ X21 @ X222 )
% 0.22/0.56 = ( vEBT_Leaf @ Y21 @ Y22 ) )
% 0.22/0.56 = ( ( X21 = Y21 )
% 0.22/0.56 & ( X222 = Y22 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT.inject(2)
% 0.22/0.56 thf(fact_9061_VEBT__internal_Ospace_H_Ocases,axiom,
% 0.22/0.56 ! [X: vEBT_VEBT] :
% 0.22/0.56 ( ! [A4: $o,B3: $o] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( vEBT_Leaf @ A4 @ B3 ) )
% 0.22/0.56 => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.space'.cases
% 0.22/0.56 thf(fact_9062_VEBT_Oexhaust,axiom,
% 0.22/0.56 ! [Y4: vEBT_VEBT] :
% 0.22/0.56 ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
% 0.22/0.56 ( Y4
% 0.22/0.56 != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
% 0.22/0.56 => ~ ! [X212: $o,X223: $o] :
% 0.22/0.56 ( Y4
% 0.22/0.56 != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT.exhaust
% 0.22/0.56 thf(fact_9063_VEBT_Odistinct_I1_J,axiom,
% 0.22/0.56 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
% 0.22/0.56 ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 0.22/0.56 != ( vEBT_Leaf @ X21 @ X222 ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT.distinct(1)
% 0.22/0.56 thf(fact_9064_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
% 0.22/0.56 ! [Uu: $o,Uv: $o,Uw: nat] :
% 0.22/0.56 ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.membermima.simps(1)
% 0.22/0.56 thf(fact_9065_VEBT_Osize_I4_J,axiom,
% 0.22/0.56 ! [X21: $o,X222: $o] :
% 0.22/0.56 ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 0.22/0.56 = zero_zero_nat ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT.size(4)
% 0.22/0.56 thf(fact_9066_vebt__buildup_Osimps_I1_J,axiom,
% 0.22/0.56 ( ( vEBT_vebt_buildup @ zero_zero_nat )
% 0.22/0.56 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 0.22/0.56
% 0.22/0.56 % vebt_buildup.simps(1)
% 0.22/0.56 thf(fact_9067_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 0.22/0.56 ! [Uu: $o,Uv: $o,D: nat] :
% 0.22/0.56 ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
% 0.22/0.56 = ( D = one_one_nat ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.valid'.simps(1)
% 0.22/0.56 thf(fact_9068_VEBT__internal_Ocnt_Osimps_I1_J,axiom,
% 0.22/0.56 ! [A: $o,B: $o] :
% 0.22/0.56 ( ( vEBT_VEBT_cnt @ ( vEBT_Leaf @ A @ B ) )
% 0.22/0.56 = one_one_real ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.cnt.simps(1)
% 0.22/0.56 thf(fact_9069_invar__vebt_Ointros_I1_J,axiom,
% 0.22/0.56 ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% 0.22/0.56
% 0.22/0.56 % invar_vebt.intros(1)
% 0.22/0.56 thf(fact_9070_vebt__buildup_Osimps_I2_J,axiom,
% 0.22/0.56 ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
% 0.22/0.56 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 0.22/0.56
% 0.22/0.56 % vebt_buildup.simps(2)
% 0.22/0.56 thf(fact_9071_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 0.22/0.56 ! [A: $o,B: $o,X: nat] :
% 0.22/0.56 ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
% 0.22/0.56 = ( ( ( X = zero_zero_nat )
% 0.22/0.56 => A )
% 0.22/0.56 & ( ( X != zero_zero_nat )
% 0.22/0.56 => ( ( ( X = one_one_nat )
% 0.22/0.56 => B )
% 0.22/0.56 & ( X = one_one_nat ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.naive_member.simps(1)
% 0.22/0.56 thf(fact_9072_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
% 0.22/0.56 ! [A: $o,B: $o] :
% 0.22/0.56 ( ( vEBT_VEBT_space2 @ ( vEBT_Leaf @ A @ B ) )
% 0.22/0.56 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.space'.simps(1)
% 0.22/0.56 thf(fact_9073_VEBT__internal_Ospace_Osimps_I1_J,axiom,
% 0.22/0.56 ! [A: $o,B: $o] :
% 0.22/0.56 ( ( vEBT_VEBT_space @ ( vEBT_Leaf @ A @ B ) )
% 0.22/0.56 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.space.simps(1)
% 0.22/0.56 thf(fact_9074_VEBT__internal_Ospace_H_Oelims,axiom,
% 0.22/0.56 ! [X: vEBT_VEBT,Y4: nat] :
% 0.22/0.56 ( ( ( vEBT_VEBT_space2 @ X )
% 0.22/0.56 = Y4 )
% 0.22/0.56 => ( ( ? [A4: $o,B3: $o] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( vEBT_Leaf @ A4 @ B3 ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.22/0.56 => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 != ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList2 ) @ zero_zero_nat ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.space'.elims
% 0.22/0.56 thf(fact_9075_VEBT__internal_Ospace_Oelims,axiom,
% 0.22/0.56 ! [X: vEBT_VEBT,Y4: nat] :
% 0.22/0.56 ( ( ( vEBT_VEBT_space @ X )
% 0.22/0.56 = Y4 )
% 0.22/0.56 => ( ( ? [A4: $o,B3: $o] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( vEBT_Leaf @ A4 @ B3 ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
% 0.22/0.56 => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 != ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList2 ) @ zero_zero_nat ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.space.elims
% 0.22/0.56 thf(fact_9076_VEBT__internal_Ocnt_Oelims,axiom,
% 0.22/0.56 ! [X: vEBT_VEBT,Y4: real] :
% 0.22/0.56 ( ( ( vEBT_VEBT_cnt @ X )
% 0.22/0.56 = Y4 )
% 0.22/0.56 => ( ( ? [A4: $o,B3: $o] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( vEBT_Leaf @ A4 @ B3 ) )
% 0.22/0.56 => ( Y4 != one_one_real ) )
% 0.22/0.56 => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 != ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary2 ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList2 ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.cnt.elims
% 0.22/0.56 thf(fact_9077_in__children__def,axiom,
% 0.22/0.56 ( vEBT_V5917875025757280293ildren
% 0.22/0.56 = ( ^ [N4: nat,TreeList3: list_VEBT_VEBT,X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X3 @ N4 ) ) @ ( vEBT_VEBT_low @ X3 @ N4 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % in_children_def
% 0.22/0.56 thf(fact_9078_invar__vebt_Ointros_I3_J,axiom,
% 0.22/0.56 ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.22/0.56 ( ! [X4: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.22/0.56 => ( vEBT_invar_vebt @ X4 @ N ) )
% 0.22/0.56 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.22/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.22/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.22/0.56 => ( ( M
% 0.22/0.56 = ( suc @ N ) )
% 0.22/0.56 => ( ( Deg
% 0.22/0.56 = ( plus_plus_nat @ N @ M ) )
% 0.22/0.56 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 0.22/0.56 => ( ! [X4: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.22/0.56 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
% 0.22/0.56 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % invar_vebt.intros(3)
% 0.22/0.56 thf(fact_9079_VEBT__internal_Ospace_Opelims,axiom,
% 0.22/0.56 ! [X: vEBT_VEBT,Y4: nat] :
% 0.22/0.56 ( ( ( vEBT_VEBT_space @ X )
% 0.22/0.56 = Y4 )
% 0.22/0.56 => ( ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ X )
% 0.22/0.56 => ( ! [A4: $o,B3: $o] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Leaf @ A4 @ B3 ) )
% 0.22/0.56 => ( ( Y4
% 0.22/0.56 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.22/0.56 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
% 0.22/0.56 => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.22/0.56 => ( ( Y4
% 0.22/0.56 = ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList2 ) @ zero_zero_nat ) ) )
% 0.22/0.56 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.space.pelims
% 0.22/0.56 thf(fact_9080_invar__vebt_Ointros_I2_J,axiom,
% 0.22/0.56 ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.22/0.56 ( ! [X4: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.22/0.56 => ( vEBT_invar_vebt @ X4 @ N ) )
% 0.22/0.56 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.22/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.22/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.22/0.56 => ( ( M = N )
% 0.22/0.56 => ( ( Deg
% 0.22/0.56 = ( plus_plus_nat @ N @ M ) )
% 0.22/0.56 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 0.22/0.56 => ( ! [X4: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.22/0.56 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
% 0.22/0.56 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % invar_vebt.intros(2)
% 0.22/0.56 thf(fact_9081_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
% 0.22/0.56 ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
% 0.22/0.56 ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.membermima.simps(2)
% 0.22/0.56 thf(fact_9082_VEBT__internal_Ospace_H_Opelims,axiom,
% 0.22/0.56 ! [X: vEBT_VEBT,Y4: nat] :
% 0.22/0.56 ( ( ( vEBT_VEBT_space2 @ X )
% 0.22/0.56 = Y4 )
% 0.22/0.56 => ( ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ X )
% 0.22/0.56 => ( ! [A4: $o,B3: $o] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Leaf @ A4 @ B3 ) )
% 0.22/0.56 => ( ( Y4
% 0.22/0.56 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.22/0.56 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
% 0.22/0.56 => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.22/0.56 => ( ( Y4
% 0.22/0.56 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList2 ) @ zero_zero_nat ) ) )
% 0.22/0.56 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.space'.pelims
% 0.22/0.56 thf(fact_9083_VEBT__internal_Ocnt_Opelims,axiom,
% 0.22/0.56 ! [X: vEBT_VEBT,Y4: real] :
% 0.22/0.56 ( ( ( vEBT_VEBT_cnt @ X )
% 0.22/0.56 = Y4 )
% 0.22/0.56 => ( ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ X )
% 0.22/0.56 => ( ! [A4: $o,B3: $o] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Leaf @ A4 @ B3 ) )
% 0.22/0.56 => ( ( Y4 = one_one_real )
% 0.22/0.56 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
% 0.22/0.56 => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.22/0.56 => ( ( Y4
% 0.22/0.56 = ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary2 ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList2 ) @ zero_zero_real ) ) )
% 0.22/0.56 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.cnt.pelims
% 0.22/0.56 thf(fact_9084_mi__ma__2__deg,axiom,
% 0.22/0.56 ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.22/0.56 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 0.22/0.56 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.22/0.56 & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % mi_ma_2_deg
% 0.22/0.56 thf(fact_9085_both__member__options__from__complete__tree__to__child,axiom,
% 0.22/0.56 ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 0.22/0.56 ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 0.22/0.56 => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.22/0.56 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.56 | ( X = Mi )
% 0.22/0.56 | ( X = Ma ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % both_member_options_from_complete_tree_to_child
% 0.22/0.56 thf(fact_9086_both__member__options__from__chilf__to__complete__tree,axiom,
% 0.22/0.56 ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.22/0.56 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.22/0.56 => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 0.22/0.56 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.56 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % both_member_options_from_chilf_to_complete_tree
% 0.22/0.56 thf(fact_9087_prod__decode__aux_Ocases,axiom,
% 0.22/0.56 ! [X: product_prod_nat_nat] :
% 0.22/0.56 ~ ! [K2: nat,M4: nat] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( product_Pair_nat_nat @ K2 @ M4 ) ) ).
% 0.22/0.56
% 0.22/0.56 % prod_decode_aux.cases
% 0.22/0.56 thf(fact_9088_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
% 0.22/0.56 ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
% 0.22/0.56 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va2 @ Vb ) @ X )
% 0.22/0.56 = ( ( X = Mi )
% 0.22/0.56 | ( X = Ma ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.membermima.simps(3)
% 0.22/0.56 thf(fact_9089_invar__vebt_Ointros_I4_J,axiom,
% 0.22/0.56 ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 0.22/0.56 ( ! [X4: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.22/0.56 => ( vEBT_invar_vebt @ X4 @ N ) )
% 0.22/0.56 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.22/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.22/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.22/0.56 => ( ( M = N )
% 0.22/0.56 => ( ( Deg
% 0.22/0.56 = ( plus_plus_nat @ N @ M ) )
% 0.22/0.56 => ( ! [I3: nat] :
% 0.22/0.56 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.22/0.56 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
% 0.22/0.56 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 0.22/0.56 => ( ( ( Mi = Ma )
% 0.22/0.56 => ! [X4: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.22/0.56 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
% 0.22/0.56 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.22/0.56 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.22/0.56 => ( ( ( Mi != Ma )
% 0.22/0.56 => ! [I3: nat] :
% 0.22/0.56 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.22/0.56 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 0.22/0.56 = I3 )
% 0.22/0.56 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 0.22/0.56 & ! [X4: nat] :
% 0.22/0.56 ( ( ( ( vEBT_VEBT_high @ X4 @ N )
% 0.22/0.56 = I3 )
% 0.22/0.56 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
% 0.22/0.56 => ( ( ord_less_nat @ Mi @ X4 )
% 0.22/0.56 & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 0.22/0.56 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % invar_vebt.intros(4)
% 0.22/0.56 thf(fact_9090_invar__vebt_Ointros_I5_J,axiom,
% 0.22/0.56 ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 0.22/0.56 ( ! [X4: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.22/0.56 => ( vEBT_invar_vebt @ X4 @ N ) )
% 0.22/0.56 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.22/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.22/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.22/0.56 => ( ( M
% 0.22/0.56 = ( suc @ N ) )
% 0.22/0.56 => ( ( Deg
% 0.22/0.56 = ( plus_plus_nat @ N @ M ) )
% 0.22/0.56 => ( ! [I3: nat] :
% 0.22/0.56 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.22/0.56 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
% 0.22/0.56 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 0.22/0.56 => ( ( ( Mi = Ma )
% 0.22/0.56 => ! [X4: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.22/0.56 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
% 0.22/0.56 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.22/0.56 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.22/0.56 => ( ( ( Mi != Ma )
% 0.22/0.56 => ! [I3: nat] :
% 0.22/0.56 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.22/0.56 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 0.22/0.56 = I3 )
% 0.22/0.56 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 0.22/0.56 & ! [X4: nat] :
% 0.22/0.56 ( ( ( ( vEBT_VEBT_high @ X4 @ N )
% 0.22/0.56 = I3 )
% 0.22/0.56 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
% 0.22/0.56 => ( ( ord_less_nat @ Mi @ X4 )
% 0.22/0.56 & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 0.22/0.56 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % invar_vebt.intros(5)
% 0.22/0.56 thf(fact_9091_invar__vebt_Ocases,axiom,
% 0.22/0.56 ! [A1: vEBT_VEBT,A22: nat] :
% 0.22/0.56 ( ( vEBT_invar_vebt @ A1 @ A22 )
% 0.22/0.56 => ( ( ? [A4: $o,B3: $o] :
% 0.22/0.56 ( A1
% 0.22/0.56 = ( vEBT_Leaf @ A4 @ B3 ) )
% 0.22/0.56 => ( A22
% 0.22/0.56 != ( suc @ zero_zero_nat ) ) )
% 0.22/0.56 => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 0.22/0.56 ( ( A1
% 0.22/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.22/0.56 => ( ( A22 = Deg2 )
% 0.22/0.56 => ( ! [X5: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.56 => ( vEBT_invar_vebt @ X5 @ N2 ) )
% 0.22/0.56 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.22/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.22/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.22/0.56 => ( ( M4 = N2 )
% 0.22/0.56 => ( ( Deg2
% 0.22/0.56 = ( plus_plus_nat @ N2 @ M4 ) )
% 0.22/0.56 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 0.22/0.56 => ~ ! [X5: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.56 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
% 0.22/0.56 => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 0.22/0.56 ( ( A1
% 0.22/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.22/0.56 => ( ( A22 = Deg2 )
% 0.22/0.56 => ( ! [X5: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.56 => ( vEBT_invar_vebt @ X5 @ N2 ) )
% 0.22/0.56 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.22/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.22/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.22/0.56 => ( ( M4
% 0.22/0.56 = ( suc @ N2 ) )
% 0.22/0.56 => ( ( Deg2
% 0.22/0.56 = ( plus_plus_nat @ N2 @ M4 ) )
% 0.22/0.56 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 0.22/0.56 => ~ ! [X5: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.56 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
% 0.22/0.56 => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 0.22/0.56 ( ( A1
% 0.22/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.22/0.56 => ( ( A22 = Deg2 )
% 0.22/0.56 => ( ! [X5: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.56 => ( vEBT_invar_vebt @ X5 @ N2 ) )
% 0.22/0.56 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.22/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.22/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.22/0.56 => ( ( M4 = N2 )
% 0.22/0.56 => ( ( Deg2
% 0.22/0.56 = ( plus_plus_nat @ N2 @ M4 ) )
% 0.22/0.56 => ( ! [I4: nat] :
% 0.22/0.56 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.22/0.56 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
% 0.22/0.56 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.22/0.56 => ( ( ( Mi2 = Ma2 )
% 0.22/0.56 => ! [X5: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.56 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 0.22/0.56 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.22/0.56 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.22/0.56 => ~ ( ( Mi2 != Ma2 )
% 0.22/0.56 => ! [I4: nat] :
% 0.22/0.56 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.22/0.56 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 0.22/0.56 = I4 )
% 0.22/0.56 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 0.22/0.56 & ! [X5: nat] :
% 0.22/0.56 ( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
% 0.22/0.56 = I4 )
% 0.22/0.56 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
% 0.22/0.56 => ( ( ord_less_nat @ Mi2 @ X5 )
% 0.22/0.56 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.22/0.56 => ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 0.22/0.56 ( ( A1
% 0.22/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.22/0.56 => ( ( A22 = Deg2 )
% 0.22/0.56 => ( ! [X5: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.56 => ( vEBT_invar_vebt @ X5 @ N2 ) )
% 0.22/0.56 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.22/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.22/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.22/0.56 => ( ( M4
% 0.22/0.56 = ( suc @ N2 ) )
% 0.22/0.56 => ( ( Deg2
% 0.22/0.56 = ( plus_plus_nat @ N2 @ M4 ) )
% 0.22/0.56 => ( ! [I4: nat] :
% 0.22/0.56 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.22/0.56 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
% 0.22/0.56 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.22/0.56 => ( ( ( Mi2 = Ma2 )
% 0.22/0.56 => ! [X5: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.56 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 0.22/0.56 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.22/0.56 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.22/0.56 => ~ ( ( Mi2 != Ma2 )
% 0.22/0.56 => ! [I4: nat] :
% 0.22/0.56 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.22/0.56 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 0.22/0.56 = I4 )
% 0.22/0.56 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 0.22/0.56 & ! [X5: nat] :
% 0.22/0.56 ( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
% 0.22/0.56 = I4 )
% 0.22/0.56 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
% 0.22/0.56 => ( ( ord_less_nat @ Mi2 @ X5 )
% 0.22/0.56 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % invar_vebt.cases
% 0.22/0.56 thf(fact_9092_invar__vebt_Osimps,axiom,
% 0.22/0.56 ( vEBT_invar_vebt
% 0.22/0.56 = ( ^ [A12: vEBT_VEBT,A23: nat] :
% 0.22/0.56 ( ( ? [A3: $o,B2: $o] :
% 0.22/0.56 ( A12
% 0.22/0.56 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.22/0.56 & ( A23
% 0.22/0.56 = ( suc @ zero_zero_nat ) ) )
% 0.22/0.56 | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
% 0.22/0.56 ( ( A12
% 0.22/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList3 @ Summary3 ) )
% 0.22/0.56 & ! [X3: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.22/0.56 => ( vEBT_invar_vebt @ X3 @ N4 ) )
% 0.22/0.56 & ( vEBT_invar_vebt @ Summary3 @ N4 )
% 0.22/0.56 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.22/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
% 0.22/0.56 & ( A23
% 0.22/0.56 = ( plus_plus_nat @ N4 @ N4 ) )
% 0.22/0.56 & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
% 0.22/0.56 & ! [X3: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.22/0.56 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.22/0.56 | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
% 0.22/0.56 ( ( A12
% 0.22/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList3 @ Summary3 ) )
% 0.22/0.56 & ! [X3: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.22/0.56 => ( vEBT_invar_vebt @ X3 @ N4 ) )
% 0.22/0.56 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N4 ) )
% 0.22/0.56 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.22/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
% 0.22/0.56 & ( A23
% 0.22/0.56 = ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) )
% 0.22/0.56 & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
% 0.22/0.56 & ! [X3: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.22/0.56 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.22/0.56 | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.22/0.56 ( ( A12
% 0.22/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
% 0.22/0.56 & ! [X3: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.22/0.56 => ( vEBT_invar_vebt @ X3 @ N4 ) )
% 0.22/0.56 & ( vEBT_invar_vebt @ Summary3 @ N4 )
% 0.22/0.56 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.22/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
% 0.22/0.56 & ( A23
% 0.22/0.56 = ( plus_plus_nat @ N4 @ N4 ) )
% 0.22/0.56 & ! [I2: nat] :
% 0.22/0.56 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
% 0.22/0.56 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X8 ) )
% 0.22/0.56 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
% 0.22/0.56 & ( ( Mi3 = Ma3 )
% 0.22/0.56 => ! [X3: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.22/0.56 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.22/0.56 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.22/0.56 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 0.22/0.56 & ( ( Mi3 != Ma3 )
% 0.22/0.56 => ! [I2: nat] :
% 0.22/0.56 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
% 0.22/0.56 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N4 )
% 0.22/0.56 = I2 )
% 0.22/0.56 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
% 0.22/0.56 & ! [X3: nat] :
% 0.22/0.56 ( ( ( ( vEBT_VEBT_high @ X3 @ N4 )
% 0.22/0.56 = I2 )
% 0.22/0.56 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N4 ) ) )
% 0.22/0.56 => ( ( ord_less_nat @ Mi3 @ X3 )
% 0.22/0.56 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) )
% 0.22/0.56 | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.22/0.56 ( ( A12
% 0.22/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
% 0.22/0.56 & ! [X3: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.22/0.56 => ( vEBT_invar_vebt @ X3 @ N4 ) )
% 0.22/0.56 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N4 ) )
% 0.22/0.56 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.22/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
% 0.22/0.56 & ( A23
% 0.22/0.56 = ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) )
% 0.22/0.56 & ! [I2: nat] :
% 0.22/0.56 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
% 0.22/0.56 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X8 ) )
% 0.22/0.56 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
% 0.22/0.56 & ( ( Mi3 = Ma3 )
% 0.22/0.56 => ! [X3: vEBT_VEBT] :
% 0.22/0.56 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.22/0.56 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.22/0.56 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.22/0.56 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 0.22/0.56 & ( ( Mi3 != Ma3 )
% 0.22/0.56 => ! [I2: nat] :
% 0.22/0.56 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
% 0.22/0.56 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N4 )
% 0.22/0.56 = I2 )
% 0.22/0.56 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
% 0.22/0.56 & ! [X3: nat] :
% 0.22/0.56 ( ( ( ( vEBT_VEBT_high @ X3 @ N4 )
% 0.22/0.56 = I2 )
% 0.22/0.56 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N4 ) ) )
% 0.22/0.56 => ( ( ord_less_nat @ Mi3 @ X3 )
% 0.22/0.56 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % invar_vebt.simps
% 0.22/0.56 thf(fact_9093_VEBT__internal_Ovalid_H_Ocases,axiom,
% 0.22/0.56 ! [X: produc9072475918466114483BT_nat] :
% 0.22/0.56 ( ! [Uu2: $o,Uv2: $o,D2: nat] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D2 ) )
% 0.22/0.56 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Deg3 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.valid'.cases
% 0.22/0.56 thf(fact_9094_xor__num_Ocases,axiom,
% 0.22/0.56 ! [X: product_prod_num_num] :
% 0.22/0.56 ( ( X
% 0.22/0.56 != ( product_Pair_num_num @ one @ one ) )
% 0.22/0.56 => ( ! [N2: num] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) )
% 0.22/0.56 => ( ! [N2: num] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) )
% 0.22/0.56 => ( ! [M4: num] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) )
% 0.22/0.56 => ( ! [M4: num,N2: num] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) )
% 0.22/0.56 => ( ! [M4: num,N2: num] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) )
% 0.22/0.56 => ( ! [M4: num] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) )
% 0.22/0.56 => ( ! [M4: num,N2: num] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) )
% 0.22/0.56 => ~ ! [M4: num,N2: num] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % xor_num.cases
% 0.22/0.56 thf(fact_9095_VEBT__internal_Onaive__member_Ocases,axiom,
% 0.22/0.56 ! [X: produc9072475918466114483BT_nat] :
% 0.22/0.56 ( ! [A4: $o,B3: $o,X4: nat] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ X4 ) )
% 0.22/0.56 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
% 0.22/0.56 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) @ X4 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.naive_member.cases
% 0.22/0.56 thf(fact_9096_VEBT__internal_Omembermima_Ocases,axiom,
% 0.22/0.56 ! [X: produc9072475918466114483BT_nat] :
% 0.22/0.56 ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
% 0.22/0.56 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
% 0.22/0.56 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X4 ) )
% 0.22/0.56 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X4: nat] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ X4 ) )
% 0.22/0.56 => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X4: nat] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ X4 ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.membermima.cases
% 0.22/0.56 thf(fact_9097_neg__eucl__rel__int__mult__2,axiom,
% 0.22/0.56 ! [B: int,A: int,Q2: int,R2: int] :
% 0.22/0.56 ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.22/0.56 => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.22/0.56 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % neg_eucl_rel_int_mult_2
% 0.22/0.56 thf(fact_9098_Divides_Oadjust__div__eq,axiom,
% 0.22/0.56 ! [Q2: int,R2: int] :
% 0.22/0.56 ( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.22/0.56 = ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % Divides.adjust_div_eq
% 0.22/0.56 thf(fact_9099_numeral__div__minus__numeral,axiom,
% 0.22/0.56 ! [M: num,N: num] :
% 0.22/0.56 ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.56 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % numeral_div_minus_numeral
% 0.22/0.56 thf(fact_9100_minus__numeral__div__numeral,axiom,
% 0.22/0.56 ! [M: num,N: num] :
% 0.22/0.56 ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.22/0.56 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % minus_numeral_div_numeral
% 0.22/0.56 thf(fact_9101_one__div__minus__numeral,axiom,
% 0.22/0.56 ! [N: num] :
% 0.22/0.56 ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.56 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % one_div_minus_numeral
% 0.22/0.56 thf(fact_9102_minus__one__div__numeral,axiom,
% 0.22/0.56 ! [N: num] :
% 0.22/0.56 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.22/0.56 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % minus_one_div_numeral
% 0.22/0.56 thf(fact_9103_divmod__BitM__2__eq,axiom,
% 0.22/0.56 ! [M: num] :
% 0.22/0.56 ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 0.22/0.56 = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 0.22/0.56
% 0.22/0.56 % divmod_BitM_2_eq
% 0.22/0.56 thf(fact_9104_unique__quotient,axiom,
% 0.22/0.56 ! [A: int,B: int,Q2: int,R2: int,Q5: int,R4: int] :
% 0.22/0.56 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.22/0.56 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R4 ) )
% 0.22/0.56 => ( Q2 = Q5 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % unique_quotient
% 0.22/0.56 thf(fact_9105_unique__remainder,axiom,
% 0.22/0.56 ! [A: int,B: int,Q2: int,R2: int,Q5: int,R4: int] :
% 0.22/0.56 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.22/0.56 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R4 ) )
% 0.22/0.56 => ( R2 = R4 ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % unique_remainder
% 0.22/0.56 thf(fact_9106_eucl__rel__int__by0,axiom,
% 0.22/0.56 ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).
% 0.22/0.56
% 0.22/0.56 % eucl_rel_int_by0
% 0.22/0.56 thf(fact_9107_div__int__unique,axiom,
% 0.22/0.56 ! [K: int,L: int,Q2: int,R2: int] :
% 0.22/0.56 ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.22/0.56 => ( ( divide_divide_int @ K @ L )
% 0.22/0.56 = Q2 ) ) ).
% 0.22/0.56
% 0.22/0.56 % div_int_unique
% 0.22/0.56 thf(fact_9108_mod__int__unique,axiom,
% 0.22/0.56 ! [K: int,L: int,Q2: int,R2: int] :
% 0.22/0.56 ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.22/0.56 => ( ( modulo_modulo_int @ K @ L )
% 0.22/0.56 = R2 ) ) ).
% 0.22/0.56
% 0.22/0.56 % mod_int_unique
% 0.22/0.56 thf(fact_9109_eucl__rel__int__dividesI,axiom,
% 0.22/0.56 ! [L: int,K: int,Q2: int] :
% 0.22/0.56 ( ( L != zero_zero_int )
% 0.22/0.56 => ( ( K
% 0.22/0.56 = ( times_times_int @ Q2 @ L ) )
% 0.22/0.56 => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % eucl_rel_int_dividesI
% 0.22/0.56 thf(fact_9110_eucl__rel__int,axiom,
% 0.22/0.56 ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % eucl_rel_int
% 0.22/0.56 thf(fact_9111_divmod__int__def,axiom,
% 0.22/0.56 ( unique5052692396658037445od_int
% 0.22/0.56 = ( ^ [M5: num,N4: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N4 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N4 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % divmod_int_def
% 0.22/0.56 thf(fact_9112_divmod_H__nat__def,axiom,
% 0.22/0.56 ( unique5055182867167087721od_nat
% 0.22/0.56 = ( ^ [M5: num,N4: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N4 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N4 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % divmod'_nat_def
% 0.22/0.56 thf(fact_9113_zminus1__lemma,axiom,
% 0.22/0.56 ! [A: int,B: int,Q2: int,R2: int] :
% 0.22/0.56 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.22/0.56 => ( ( B != zero_zero_int )
% 0.22/0.56 => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( 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 @ B @ R2 ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % zminus1_lemma
% 0.22/0.56 thf(fact_9114_eucl__rel__int__iff,axiom,
% 0.22/0.56 ! [K: int,L: int,Q2: int,R2: int] :
% 0.22/0.56 ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.22/0.56 = ( ( K
% 0.22/0.56 = ( plus_plus_int @ ( times_times_int @ L @ Q2 ) @ R2 ) )
% 0.22/0.56 & ( ( ord_less_int @ zero_zero_int @ L )
% 0.22/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.22/0.56 & ( ord_less_int @ R2 @ L ) ) )
% 0.22/0.56 & ( ~ ( ord_less_int @ zero_zero_int @ L )
% 0.22/0.56 => ( ( ( ord_less_int @ L @ zero_zero_int )
% 0.22/0.56 => ( ( ord_less_int @ L @ R2 )
% 0.22/0.56 & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
% 0.22/0.56 & ( ~ ( ord_less_int @ L @ zero_zero_int )
% 0.22/0.56 => ( Q2 = zero_zero_int ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % eucl_rel_int_iff
% 0.22/0.56 thf(fact_9115_eucl__rel__int__remainderI,axiom,
% 0.22/0.56 ! [R2: int,L: int,K: int,Q2: int] :
% 0.22/0.56 ( ( ( sgn_sgn_int @ R2 )
% 0.22/0.56 = ( sgn_sgn_int @ L ) )
% 0.22/0.56 => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
% 0.22/0.56 => ( ( K
% 0.22/0.56 = ( plus_plus_int @ ( times_times_int @ Q2 @ L ) @ R2 ) )
% 0.22/0.56 => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R2 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % eucl_rel_int_remainderI
% 0.22/0.56 thf(fact_9116_eucl__rel__int_Osimps,axiom,
% 0.22/0.56 ( eucl_rel_int
% 0.22/0.56 = ( ^ [A12: int,A23: int,A32: product_prod_int_int] :
% 0.22/0.56 ( ? [K3: int] :
% 0.22/0.56 ( ( A12 = K3 )
% 0.22/0.56 & ( A23 = zero_zero_int )
% 0.22/0.56 & ( A32
% 0.22/0.56 = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
% 0.22/0.56 | ? [L3: int,K3: int,Q3: int] :
% 0.22/0.56 ( ( A12 = K3 )
% 0.22/0.56 & ( A23 = L3 )
% 0.22/0.56 & ( A32
% 0.22/0.56 = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
% 0.22/0.56 & ( L3 != zero_zero_int )
% 0.22/0.56 & ( K3
% 0.22/0.56 = ( times_times_int @ Q3 @ L3 ) ) )
% 0.22/0.56 | ? [R5: int,L3: int,K3: int,Q3: int] :
% 0.22/0.56 ( ( A12 = K3 )
% 0.22/0.56 & ( A23 = L3 )
% 0.22/0.56 & ( A32
% 0.22/0.56 = ( product_Pair_int_int @ Q3 @ R5 ) )
% 0.22/0.56 & ( ( sgn_sgn_int @ R5 )
% 0.22/0.56 = ( sgn_sgn_int @ L3 ) )
% 0.22/0.56 & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L3 ) )
% 0.22/0.56 & ( K3
% 0.22/0.56 = ( plus_plus_int @ ( times_times_int @ Q3 @ L3 ) @ R5 ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % eucl_rel_int.simps
% 0.22/0.56 thf(fact_9117_eucl__rel__int_Ocases,axiom,
% 0.22/0.56 ! [A1: int,A22: int,A33: product_prod_int_int] :
% 0.22/0.56 ( ( eucl_rel_int @ A1 @ A22 @ A33 )
% 0.22/0.56 => ( ( ( A22 = zero_zero_int )
% 0.22/0.56 => ( A33
% 0.22/0.56 != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
% 0.22/0.56 => ( ! [Q4: int] :
% 0.22/0.56 ( ( A33
% 0.22/0.56 = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
% 0.22/0.56 => ( ( A22 != zero_zero_int )
% 0.22/0.56 => ( A1
% 0.22/0.56 != ( times_times_int @ Q4 @ A22 ) ) ) )
% 0.22/0.56 => ~ ! [R3: int,Q4: int] :
% 0.22/0.56 ( ( A33
% 0.22/0.56 = ( product_Pair_int_int @ Q4 @ R3 ) )
% 0.22/0.56 => ( ( ( sgn_sgn_int @ R3 )
% 0.22/0.56 = ( sgn_sgn_int @ A22 ) )
% 0.22/0.56 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A22 ) )
% 0.22/0.56 => ( A1
% 0.22/0.56 != ( plus_plus_int @ ( times_times_int @ Q4 @ A22 ) @ R3 ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % eucl_rel_int.cases
% 0.22/0.56 thf(fact_9118_pos__eucl__rel__int__mult__2,axiom,
% 0.22/0.56 ! [B: int,A: int,Q2: int,R2: int] :
% 0.22/0.56 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.22/0.56 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 0.22/0.56 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % pos_eucl_rel_int_mult_2
% 0.22/0.56 thf(fact_9119_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 0.22/0.56 ! [X: vEBT_VEBT,Xa: nat] :
% 0.22/0.56 ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
% 0.22/0.56 => ( ! [Uu2: $o,Uv2: $o] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.22/0.56 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.22/0.56 => ( ! [Mi2: nat,Ma2: nat] :
% 0.22/0.56 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.22/0.56 => ( ( Xa = Mi2 )
% 0.22/0.56 | ( Xa = Ma2 ) ) )
% 0.22/0.56 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT] :
% 0.22/0.56 ( ? [Vc: vEBT_VEBT] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
% 0.22/0.56 => ( ( Xa = Mi2 )
% 0.22/0.56 | ( Xa = Ma2 )
% 0.22/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.22/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.22/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
% 0.22/0.56 => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT] :
% 0.22/0.56 ( ? [Vd: vEBT_VEBT] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
% 0.22/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.22/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.22/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.membermima.elims(3)
% 0.22/0.56 thf(fact_9120_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 0.22/0.56 ! [X: vEBT_VEBT,Xa: nat,Y4: $o] :
% 0.22/0.56 ( ( ( vEBT_VEBT_membermima @ X @ Xa )
% 0.22/0.56 = Y4 )
% 0.22/0.56 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.22/0.56 => Y4 )
% 0.22/0.56 => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.22/0.56 => Y4 )
% 0.22/0.56 => ( ! [Mi2: nat,Ma2: nat] :
% 0.22/0.56 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 = ( ~ ( ( Xa = Mi2 )
% 0.22/0.56 | ( Xa = Ma2 ) ) ) ) )
% 0.22/0.56 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT] :
% 0.22/0.56 ( ? [Vc: vEBT_VEBT] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 = ( ~ ( ( Xa = Mi2 )
% 0.22/0.56 | ( Xa = Ma2 )
% 0.22/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.22/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.22/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) )
% 0.22/0.56 => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT] :
% 0.22/0.56 ( ? [Vd: vEBT_VEBT] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.22/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.22/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.membermima.elims(1)
% 0.22/0.56 thf(fact_9121_prod__decode__aux_Osimps,axiom,
% 0.22/0.56 ( nat_prod_decode_aux
% 0.22/0.56 = ( ^ [K3: nat,M5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M5 @ K3 ) @ ( product_Pair_nat_nat @ M5 @ ( minus_minus_nat @ K3 @ M5 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M5 @ ( suc @ K3 ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % prod_decode_aux.simps
% 0.22/0.56 thf(fact_9122_prod__decode__aux_Oelims,axiom,
% 0.22/0.56 ! [X: nat,Xa: nat,Y4: product_prod_nat_nat] :
% 0.22/0.56 ( ( ( nat_prod_decode_aux @ X @ Xa )
% 0.22/0.56 = Y4 )
% 0.22/0.56 => ( ( ( ord_less_eq_nat @ Xa @ X )
% 0.22/0.56 => ( Y4
% 0.22/0.56 = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
% 0.22/0.56 & ( ~ ( ord_less_eq_nat @ Xa @ X )
% 0.22/0.56 => ( Y4
% 0.22/0.56 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % prod_decode_aux.elims
% 0.22/0.56 thf(fact_9123_foldr__same,axiom,
% 0.22/0.56 ! [Xs: list_real,Y4: real] :
% 0.22/0.56 ( ! [X4: real,Y: real] :
% 0.22/0.56 ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
% 0.22/0.56 => ( ( member_real @ Y @ ( set_real2 @ Xs ) )
% 0.22/0.56 => ( X4 = Y ) ) )
% 0.22/0.56 => ( ! [X4: real] :
% 0.22/0.56 ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
% 0.22/0.56 => ( X4 = Y4 ) )
% 0.22/0.56 => ( ( foldr_real_real @ plus_plus_real @ Xs @ zero_zero_real )
% 0.22/0.56 = ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_real @ Xs ) ) @ Y4 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % foldr_same
% 0.22/0.56 thf(fact_9124_foldr__mono,axiom,
% 0.22/0.56 ! [Xs: list_nat,Ys: list_nat,C: nat,D: nat] :
% 0.22/0.56 ( ( ( size_size_list_nat @ Xs )
% 0.22/0.56 = ( size_size_list_nat @ Ys ) )
% 0.22/0.56 => ( ! [I3: nat] :
% 0.22/0.56 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 0.22/0.56 => ( ord_less_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) )
% 0.22/0.56 => ( ( ord_less_eq_nat @ C @ D )
% 0.22/0.56 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( foldr_nat_nat @ plus_plus_nat @ Xs @ C ) @ ( size_size_list_nat @ Ys ) ) @ ( foldr_nat_nat @ plus_plus_nat @ Ys @ D ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % foldr_mono
% 0.22/0.56 thf(fact_9125_foldr__zero,axiom,
% 0.22/0.56 ! [Xs: list_nat,D: nat] :
% 0.22/0.56 ( ! [I3: nat] :
% 0.22/0.56 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 0.22/0.56 => ( ord_less_nat @ zero_zero_nat @ ( nth_nat @ Xs @ I3 ) ) )
% 0.22/0.56 => ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( minus_minus_nat @ ( foldr_nat_nat @ plus_plus_nat @ Xs @ D ) @ D ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % foldr_zero
% 0.22/0.56 thf(fact_9126_nat__leq__as__int,axiom,
% 0.22/0.56 ( ord_less_eq_nat
% 0.22/0.56 = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % nat_leq_as_int
% 0.22/0.56 thf(fact_9127_nat__less__as__int,axiom,
% 0.22/0.56 ( ord_less_nat
% 0.22/0.56 = ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % nat_less_as_int
% 0.22/0.56 thf(fact_9128_nat__plus__as__int,axiom,
% 0.22/0.56 ( plus_plus_nat
% 0.22/0.56 = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % nat_plus_as_int
% 0.22/0.56 thf(fact_9129_nat__times__as__int,axiom,
% 0.22/0.56 ( times_times_nat
% 0.22/0.56 = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % nat_times_as_int
% 0.22/0.56 thf(fact_9130_nat__minus__as__int,axiom,
% 0.22/0.56 ( minus_minus_nat
% 0.22/0.56 = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % nat_minus_as_int
% 0.22/0.56 thf(fact_9131_nat__div__as__int,axiom,
% 0.22/0.56 ( divide_divide_nat
% 0.22/0.56 = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % nat_div_as_int
% 0.22/0.56 thf(fact_9132_nat__mod__as__int,axiom,
% 0.22/0.56 ( modulo_modulo_nat
% 0.22/0.56 = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % nat_mod_as_int
% 0.22/0.56 thf(fact_9133_diff__nat__eq__if,axiom,
% 0.22/0.56 ! [Z6: int,Z: int] :
% 0.22/0.56 ( ( ( ord_less_int @ Z6 @ zero_zero_int )
% 0.22/0.56 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
% 0.22/0.56 = ( nat2 @ Z ) ) )
% 0.22/0.56 & ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
% 0.22/0.56 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
% 0.22/0.56 = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z6 ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % diff_nat_eq_if
% 0.22/0.56 thf(fact_9134_set__decode__def,axiom,
% 0.22/0.56 ( nat_set_decode
% 0.22/0.56 = ( ^ [X3: nat] :
% 0.22/0.56 ( collect_nat
% 0.22/0.56 @ ^ [N4: nat] :
% 0.22/0.56 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % set_decode_def
% 0.22/0.56 thf(fact_9135_vebt__buildup_Osimps_I3_J,axiom,
% 0.22/0.56 ! [Va2: nat] :
% 0.22/0.56 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.22/0.56 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
% 0.22/0.56 = ( 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 ) ) ) ) ) ) )
% 0.22/0.56 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.22/0.56 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
% 0.22/0.56 = ( 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 ) ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % vebt_buildup.simps(3)
% 0.22/0.56 thf(fact_9136_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
% 0.22/0.56 ! [Va2: nat] :
% 0.22/0.56 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.22/0.56 => ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va2 ) ) )
% 0.22/0.56 = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.22/0.56 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.22/0.56 => ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va2 ) ) )
% 0.22/0.56 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
% 0.22/0.56 thf(fact_9137_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
% 0.22/0.56 ! [Va2: nat] :
% 0.22/0.56 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.22/0.56 => ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va2 ) ) )
% 0.22/0.56 = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
% 0.22/0.56 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.22/0.56 => ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va2 ) ) )
% 0.22/0.56 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
% 0.22/0.56 thf(fact_9138_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 0.22/0.56 ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
% 0.22/0.56 ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S2 ) @ X )
% 0.22/0.56 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.22/0.56 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.22/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.naive_member.simps(3)
% 0.22/0.56 thf(fact_9139_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
% 0.22/0.56 ! [X: nat,Y4: nat] :
% 0.22/0.56 ( ( ( vEBT_V8646137997579335489_i_l_d @ X )
% 0.22/0.56 = Y4 )
% 0.22/0.56 => ( ( ( X = zero_zero_nat )
% 0.22/0.56 => ( Y4
% 0.22/0.56 != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.22/0.56 => ( ( ( X
% 0.22/0.56 = ( suc @ zero_zero_nat ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.22/0.56 => ~ ! [Va: nat] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( suc @ ( suc @ Va ) ) )
% 0.22/0.56 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.22/0.56 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
% 0.22/0.56 thf(fact_9140_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
% 0.22/0.56 ! [X: nat,Y4: nat] :
% 0.22/0.56 ( ( ( vEBT_V8346862874174094_d_u_p @ X )
% 0.22/0.56 = Y4 )
% 0.22/0.56 => ( ( ( X = zero_zero_nat )
% 0.22/0.56 => ( Y4
% 0.22/0.56 != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
% 0.22/0.56 => ( ( ( X
% 0.22/0.56 = ( suc @ zero_zero_nat ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
% 0.22/0.56 => ~ ! [Va: nat] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( suc @ ( suc @ Va ) ) )
% 0.22/0.56 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
% 0.22/0.56 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
% 0.22/0.56 thf(fact_9141_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 0.22/0.56 ! [V: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
% 0.22/0.56 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd2 ) @ X )
% 0.22/0.56 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.22/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.22/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.membermima.simps(5)
% 0.22/0.56 thf(fact_9142_vebt__buildup_Oelims,axiom,
% 0.22/0.56 ! [X: nat,Y4: vEBT_VEBT] :
% 0.22/0.56 ( ( ( vEBT_vebt_buildup @ X )
% 0.22/0.56 = Y4 )
% 0.22/0.56 => ( ( ( X = zero_zero_nat )
% 0.22/0.56 => ( Y4
% 0.22/0.56 != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.22/0.56 => ( ( ( X
% 0.22/0.56 = ( suc @ zero_zero_nat ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.22/0.56 => ~ ! [Va: nat] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( suc @ ( suc @ Va ) ) )
% 0.22/0.56 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 = ( 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 ) ) ) ) ) ) )
% 0.22/0.56 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % vebt_buildup.elims
% 0.22/0.56 thf(fact_9143_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 0.22/0.56 ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X: nat] :
% 0.22/0.56 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc2 ) @ X )
% 0.22/0.56 = ( ( X = Mi )
% 0.22/0.56 | ( X = Ma )
% 0.22/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.22/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.22/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.membermima.simps(4)
% 0.22/0.56 thf(fact_9144_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 0.22/0.56 ! [X: vEBT_VEBT,Xa: nat,Y4: $o] :
% 0.22/0.56 ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.22/0.56 = Y4 )
% 0.22/0.56 => ( ! [A4: $o,B3: $o] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Leaf @ A4 @ B3 ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 = ( ~ ( ( ( Xa = zero_zero_nat )
% 0.22/0.56 => A4 )
% 0.22/0.56 & ( ( Xa != zero_zero_nat )
% 0.22/0.56 => ( ( ( Xa = one_one_nat )
% 0.22/0.56 => B3 )
% 0.22/0.56 & ( Xa = one_one_nat ) ) ) ) ) ) )
% 0.22/0.56 => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.22/0.56 => Y4 )
% 0.22/0.56 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT] :
% 0.22/0.56 ( ? [S3: vEBT_VEBT] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
% 0.22/0.56 => ( Y4
% 0.22/0.56 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.22/0.56 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.22/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.naive_member.elims(1)
% 0.22/0.56 thf(fact_9145_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 0.22/0.56 ! [X: vEBT_VEBT,Xa: nat] :
% 0.22/0.56 ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.22/0.56 => ( ! [A4: $o,B3: $o] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Leaf @ A4 @ B3 ) )
% 0.22/0.56 => ~ ( ( ( Xa = zero_zero_nat )
% 0.22/0.56 => A4 )
% 0.22/0.56 & ( ( Xa != zero_zero_nat )
% 0.22/0.56 => ( ( ( Xa = one_one_nat )
% 0.22/0.56 => B3 )
% 0.22/0.56 & ( Xa = one_one_nat ) ) ) ) )
% 0.22/0.56 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT] :
% 0.22/0.56 ( ? [S3: vEBT_VEBT] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
% 0.22/0.56 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.22/0.56 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.22/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.naive_member.elims(2)
% 0.22/0.56 thf(fact_9146_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 0.22/0.56 ! [X: vEBT_VEBT,Xa: nat] :
% 0.22/0.56 ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.22/0.56 => ( ! [A4: $o,B3: $o] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Leaf @ A4 @ B3 ) )
% 0.22/0.56 => ( ( ( Xa = zero_zero_nat )
% 0.22/0.56 => A4 )
% 0.22/0.56 & ( ( Xa != zero_zero_nat )
% 0.22/0.56 => ( ( ( Xa = one_one_nat )
% 0.22/0.56 => B3 )
% 0.22/0.56 & ( Xa = one_one_nat ) ) ) ) )
% 0.22/0.56 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.22/0.56 ( X
% 0.22/0.56 != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.22/0.56 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT] :
% 0.22/0.56 ( ? [S3: vEBT_VEBT] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
% 0.22/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.22/0.56 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.22/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.naive_member.elims(3)
% 0.22/0.56 thf(fact_9147_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 0.22/0.56 ! [X: vEBT_VEBT,Xa: nat] :
% 0.22/0.56 ( ( vEBT_VEBT_membermima @ X @ Xa )
% 0.22/0.56 => ( ! [Mi2: nat,Ma2: nat] :
% 0.22/0.56 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.22/0.56 => ~ ( ( Xa = Mi2 )
% 0.22/0.56 | ( Xa = Ma2 ) ) )
% 0.22/0.56 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT] :
% 0.22/0.56 ( ? [Vc: vEBT_VEBT] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
% 0.22/0.56 => ~ ( ( Xa = Mi2 )
% 0.22/0.56 | ( Xa = Ma2 )
% 0.22/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.22/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.22/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
% 0.22/0.56 => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT] :
% 0.22/0.56 ( ? [Vd: vEBT_VEBT] :
% 0.22/0.56 ( X
% 0.22/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
% 0.22/0.56 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.22/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.22/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.membermima.elims(2)
% 0.22/0.56 thf(fact_9148_monoseq__arctan__series,axiom,
% 0.22/0.56 ! [X: real] :
% 0.22/0.56 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.22/0.56 => ( topolo6980174941875973593q_real
% 0.22/0.56 @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % monoseq_arctan_series
% 0.22/0.56 thf(fact_9149_ln__series,axiom,
% 0.22/0.56 ! [X: real] :
% 0.22/0.56 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.56 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.22/0.56 => ( ( ln_ln_real @ X )
% 0.22/0.56 = ( suminf_real
% 0.22/0.56 @ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N4 ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % ln_series
% 0.22/0.56 thf(fact_9150_monoseq__realpow,axiom,
% 0.22/0.56 ! [X: real] :
% 0.22/0.56 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.56 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.22/0.56 => ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % monoseq_realpow
% 0.22/0.56 thf(fact_9151_pi__series,axiom,
% 0.22/0.56 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.22/0.56 = ( suminf_real
% 0.22/0.56 @ ^ [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 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % pi_series
% 0.22/0.56 thf(fact_9152_arctan__series,axiom,
% 0.22/0.56 ! [X: real] :
% 0.22/0.56 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.22/0.56 => ( ( arctan @ X )
% 0.22/0.56 = ( suminf_real
% 0.22/0.56 @ ^ [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 @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % arctan_series
% 0.22/0.56 thf(fact_9153_bij__betw__nth__root__unity,axiom,
% 0.22/0.56 ! [C: complex,N: nat] :
% 0.22/0.56 ( ( C != zero_zero_complex )
% 0.22/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.56 => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 0.22/0.56 @ ( collect_complex
% 0.22/0.56 @ ^ [Z5: complex] :
% 0.22/0.56 ( ( power_power_complex @ Z5 @ N )
% 0.22/0.56 = one_one_complex ) )
% 0.22/0.56 @ ( collect_complex
% 0.22/0.56 @ ^ [Z5: complex] :
% 0.22/0.56 ( ( power_power_complex @ Z5 @ N )
% 0.22/0.56 = C ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % bij_betw_nth_root_unity
% 0.22/0.56 thf(fact_9154_summable__arctan__series,axiom,
% 0.22/0.56 ! [X: real] :
% 0.22/0.56 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.22/0.56 => ( summable_real
% 0.22/0.56 @ ^ [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 @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % summable_arctan_series
% 0.22/0.56 thf(fact_9155_summable__rabs__comparison__test,axiom,
% 0.22/0.56 ! [F: nat > real,G: nat > real] :
% 0.22/0.56 ( ? [N7: nat] :
% 0.22/0.56 ! [N2: nat] :
% 0.22/0.56 ( ( ord_less_eq_nat @ N7 @ N2 )
% 0.22/0.56 => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
% 0.22/0.56 => ( ( summable_real @ G )
% 0.22/0.56 => ( summable_real
% 0.22/0.56 @ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % summable_rabs_comparison_test
% 0.22/0.56 thf(fact_9156_summable__rabs,axiom,
% 0.22/0.56 ! [F: nat > real] :
% 0.22/0.56 ( ( summable_real
% 0.22/0.56 @ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) )
% 0.22/0.56 => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
% 0.22/0.56 @ ( suminf_real
% 0.22/0.56 @ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % summable_rabs
% 0.22/0.56 thf(fact_9157_summable__power__series,axiom,
% 0.22/0.56 ! [F: nat > real,Z: real] :
% 0.22/0.56 ( ! [I3: nat] : ( ord_less_eq_real @ ( F @ I3 ) @ one_one_real )
% 0.22/0.56 => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 0.22/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ Z )
% 0.22/0.56 => ( ( ord_less_real @ Z @ one_one_real )
% 0.22/0.56 => ( summable_real
% 0.22/0.56 @ ^ [I2: nat] : ( times_times_real @ ( F @ I2 ) @ ( power_power_real @ Z @ I2 ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % summable_power_series
% 0.22/0.56 thf(fact_9158_sin__paired,axiom,
% 0.22/0.56 ! [X: real] :
% 0.22/0.56 ( sums_real
% 0.22/0.56 @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
% 0.22/0.56 @ ( sin_real @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % sin_paired
% 0.22/0.56 thf(fact_9159_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 0.22/0.56 ! [X: vEBT_VEBT,Xa: nat] :
% 0.22/0.56 ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
% 0.22/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.22/0.56 => ( ! [Uu2: $o,Uv2: $o] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.22/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) )
% 0.22/0.56 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.22/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) )
% 0.22/0.56 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.22/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) )
% 0.22/0.56 => ( ( Xa = Mi2 )
% 0.22/0.56 | ( Xa = Ma2 ) ) ) )
% 0.22/0.56 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
% 0.22/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ Xa ) )
% 0.22/0.56 => ( ( Xa = Mi2 )
% 0.22/0.56 | ( Xa = Ma2 )
% 0.22/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.22/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.22/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
% 0.22/0.56 => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
% 0.22/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ Xa ) )
% 0.22/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.22/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.22/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.membermima.pelims(3)
% 0.22/0.56 thf(fact_9160_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 0.22/0.56 ! [X: vEBT_VEBT,Xa: nat,Y4: $o] :
% 0.22/0.56 ( ( ( vEBT_VEBT_membermima @ X @ Xa )
% 0.22/0.56 = Y4 )
% 0.22/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.22/0.56 => ( ! [Uu2: $o,Uv2: $o] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.22/0.56 => ( ~ Y4
% 0.22/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
% 0.22/0.56 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.22/0.56 => ( ~ Y4
% 0.22/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) ) )
% 0.22/0.56 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.22/0.56 => ( ( Y4
% 0.22/0.56 = ( ( Xa = Mi2 )
% 0.22/0.56 | ( Xa = Ma2 ) ) )
% 0.22/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) ) ) )
% 0.22/0.56 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
% 0.22/0.56 => ( ( Y4
% 0.22/0.56 = ( ( Xa = Mi2 )
% 0.22/0.56 | ( Xa = Ma2 )
% 0.22/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.22/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.22/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
% 0.22/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ Xa ) ) ) )
% 0.22/0.56 => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
% 0.22/0.56 => ( ( Y4
% 0.22/0.56 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.22/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.22/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
% 0.22/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % VEBT_internal.membermima.pelims(1)
% 0.22/0.56 thf(fact_9161_power__half__series,axiom,
% 0.22/0.56 ( sums_real
% 0.22/0.56 @ ^ [N4: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N4 ) )
% 0.22/0.56 @ one_one_real ) ).
% 0.22/0.56
% 0.22/0.56 % power_half_series
% 0.22/0.56 thf(fact_9162_sums__if_H,axiom,
% 0.22/0.56 ! [G: nat > real,X: real] :
% 0.22/0.56 ( ( sums_real @ G @ X )
% 0.22/0.56 => ( sums_real
% 0.22/0.56 @ ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.56 @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % sums_if'
% 0.22/0.56 thf(fact_9163_sums__if,axiom,
% 0.22/0.56 ! [G: nat > real,X: real,F: nat > real,Y4: real] :
% 0.22/0.56 ( ( sums_real @ G @ X )
% 0.22/0.56 => ( ( sums_real @ F @ Y4 )
% 0.22/0.56 => ( sums_real
% 0.22/0.56 @ ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( F @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.56 @ ( plus_plus_real @ X @ Y4 ) ) ) ) ).
% 0.22/0.56
% 0.22/0.56 % sums_if
% 0.22/0.56 thf(fact_9164_cos__paired,axiom,
% 0.22/0.56 ! [X: real] :
% 0.22/0.56 ( sums_real
% 0.22/0.56 @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
% 0.22/0.56 @ ( cos_real @ X ) ) ).
% 0.22/0.56
% 0.22/0.56 % cos_paired
% 0.22/0.56 thf(fact_9165_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 0.22/0.56 ! [X: vEBT_VEBT,Xa: nat] :
% 0.22/0.56 ( ( vEBT_VEBT_membermima @ X @ Xa )
% 0.22/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.22/0.56 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.22/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) )
% 0.22/0.56 => ~ ( ( Xa = Mi2 )
% 0.22/0.56 | ( Xa = Ma2 ) ) ) )
% 0.22/0.56 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
% 0.22/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ Xa ) )
% 0.22/0.56 => ~ ( ( Xa = Mi2 )
% 0.22/0.56 | ( Xa = Ma2 )
% 0.22/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.22/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.22/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
% 0.22/0.56 => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
% 0.22/0.56 ( ( X
% 0.22/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
% 0.22/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ Xa ) )
% 0.22/0.56 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.22/0.57 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.22/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % VEBT_internal.membermima.pelims(2)
% 0.22/0.57 thf(fact_9166_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 0.22/0.57 ! [X: vEBT_VEBT,Xa: nat] :
% 0.22/0.57 ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.22/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.22/0.57 => ( ! [A4: $o,B3: $o] :
% 0.22/0.57 ( ( X
% 0.22/0.57 = ( vEBT_Leaf @ A4 @ B3 ) )
% 0.22/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) )
% 0.22/0.57 => ( ( ( Xa = zero_zero_nat )
% 0.22/0.57 => A4 )
% 0.22/0.57 & ( ( Xa != zero_zero_nat )
% 0.22/0.57 => ( ( ( Xa = one_one_nat )
% 0.22/0.57 => B3 )
% 0.22/0.57 & ( Xa = one_one_nat ) ) ) ) ) )
% 0.22/0.57 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.22/0.57 ( ( X
% 0.22/0.57 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.22/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) )
% 0.22/0.57 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.22/0.57 ( ( X
% 0.22/0.57 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
% 0.22/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) @ Xa ) )
% 0.22/0.57 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.22/0.57 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.22/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % VEBT_internal.naive_member.pelims(3)
% 0.22/0.57 thf(fact_9167_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 0.22/0.57 ! [X: vEBT_VEBT,Xa: nat] :
% 0.22/0.57 ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.22/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.22/0.57 => ( ! [A4: $o,B3: $o] :
% 0.22/0.57 ( ( X
% 0.22/0.57 = ( vEBT_Leaf @ A4 @ B3 ) )
% 0.22/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) )
% 0.22/0.57 => ~ ( ( ( Xa = zero_zero_nat )
% 0.22/0.57 => A4 )
% 0.22/0.57 & ( ( Xa != zero_zero_nat )
% 0.22/0.57 => ( ( ( Xa = one_one_nat )
% 0.22/0.57 => B3 )
% 0.22/0.57 & ( Xa = one_one_nat ) ) ) ) ) )
% 0.22/0.57 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.22/0.57 ( ( X
% 0.22/0.57 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
% 0.22/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) @ Xa ) )
% 0.22/0.57 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.22/0.57 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.22/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % VEBT_internal.naive_member.pelims(2)
% 0.22/0.57 thf(fact_9168_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 0.22/0.57 ! [X: vEBT_VEBT,Xa: nat,Y4: $o] :
% 0.22/0.57 ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.22/0.57 = Y4 )
% 0.22/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.22/0.57 => ( ! [A4: $o,B3: $o] :
% 0.22/0.57 ( ( X
% 0.22/0.57 = ( vEBT_Leaf @ A4 @ B3 ) )
% 0.22/0.57 => ( ( Y4
% 0.22/0.57 = ( ( ( Xa = zero_zero_nat )
% 0.22/0.57 => A4 )
% 0.22/0.57 & ( ( Xa != zero_zero_nat )
% 0.22/0.57 => ( ( ( Xa = one_one_nat )
% 0.22/0.57 => B3 )
% 0.22/0.57 & ( Xa = one_one_nat ) ) ) ) )
% 0.22/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
% 0.22/0.57 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.22/0.57 ( ( X
% 0.22/0.57 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.22/0.57 => ( ~ Y4
% 0.22/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) ) )
% 0.22/0.57 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.22/0.57 ( ( X
% 0.22/0.57 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
% 0.22/0.57 => ( ( Y4
% 0.22/0.57 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.22/0.57 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.22/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
% 0.22/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) @ Xa ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % VEBT_internal.naive_member.pelims(1)
% 0.22/0.57 thf(fact_9169_Arg__def,axiom,
% 0.22/0.57 ( arg
% 0.22/0.57 = ( ^ [Z5: complex] :
% 0.22/0.57 ( if_real @ ( Z5 = zero_zero_complex ) @ zero_zero_real
% 0.22/0.57 @ ( fChoice_real
% 0.22/0.57 @ ^ [A3: real] :
% 0.22/0.57 ( ( ( sgn_sgn_complex @ Z5 )
% 0.22/0.57 = ( cis @ A3 ) )
% 0.22/0.57 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A3 )
% 0.22/0.57 & ( ord_less_eq_real @ A3 @ pi ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Arg_def
% 0.22/0.57 thf(fact_9170_set__vebt__def,axiom,
% 0.22/0.57 ( vEBT_set_vebt
% 0.22/0.57 = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % set_vebt_def
% 0.22/0.57 thf(fact_9171_sum__choose__lower,axiom,
% 0.22/0.57 ! [R2: nat,N: nat] :
% 0.22/0.57 ( ( groups3542108847815614940at_nat
% 0.22/0.57 @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K3 ) @ K3 )
% 0.22/0.57 @ ( set_ord_atMost_nat @ N ) )
% 0.22/0.57 = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N ) ) @ N ) ) ).
% 0.22/0.57
% 0.22/0.57 % sum_choose_lower
% 0.22/0.57 thf(fact_9172_sum__choose__upper,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( groups3542108847815614940at_nat
% 0.22/0.57 @ ^ [K3: nat] : ( binomial @ K3 @ M )
% 0.22/0.57 @ ( set_ord_atMost_nat @ N ) )
% 0.22/0.57 = ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % sum_choose_upper
% 0.22/0.57 thf(fact_9173_sum__choose__diagonal,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.57 => ( ( groups3542108847815614940at_nat
% 0.22/0.57 @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
% 0.22/0.57 @ ( set_ord_atMost_nat @ M ) )
% 0.22/0.57 = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % sum_choose_diagonal
% 0.22/0.57 thf(fact_9174_vandermonde,axiom,
% 0.22/0.57 ! [M: nat,N: nat,R2: nat] :
% 0.22/0.57 ( ( groups3542108847815614940at_nat
% 0.22/0.57 @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus_nat @ R2 @ K3 ) ) )
% 0.22/0.57 @ ( set_ord_atMost_nat @ R2 ) )
% 0.22/0.57 = ( binomial @ ( plus_plus_nat @ M @ N ) @ R2 ) ) ).
% 0.22/0.57
% 0.22/0.57 % vandermonde
% 0.22/0.57 thf(fact_9175_choose__row__sum,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
% 0.22/0.57 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.22/0.57
% 0.22/0.57 % choose_row_sum
% 0.22/0.57 thf(fact_9176_binomial,axiom,
% 0.22/0.57 ! [A: nat,B: nat,N: nat] :
% 0.22/0.57 ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
% 0.22/0.57 = ( groups3542108847815614940at_nat
% 0.22/0.57 @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
% 0.22/0.57 @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % binomial
% 0.22/0.57 thf(fact_9177_polynomial__product__nat,axiom,
% 0.22/0.57 ! [M: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
% 0.22/0.57 ( ! [I3: nat] :
% 0.22/0.57 ( ( ord_less_nat @ M @ I3 )
% 0.22/0.57 => ( ( A @ I3 )
% 0.22/0.57 = zero_zero_nat ) )
% 0.22/0.57 => ( ! [J2: nat] :
% 0.22/0.57 ( ( ord_less_nat @ N @ J2 )
% 0.22/0.57 => ( ( B @ J2 )
% 0.22/0.57 = zero_zero_nat ) )
% 0.22/0.57 => ( ( times_times_nat
% 0.22/0.57 @ ( groups3542108847815614940at_nat
% 0.22/0.57 @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( power_power_nat @ X @ I2 ) )
% 0.22/0.57 @ ( set_ord_atMost_nat @ M ) )
% 0.22/0.57 @ ( groups3542108847815614940at_nat
% 0.22/0.57 @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
% 0.22/0.57 @ ( set_ord_atMost_nat @ N ) ) )
% 0.22/0.57 = ( groups3542108847815614940at_nat
% 0.22/0.57 @ ^ [R5: nat] :
% 0.22/0.57 ( times_times_nat
% 0.22/0.57 @ ( groups3542108847815614940at_nat
% 0.22/0.57 @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 0.22/0.57 @ ( set_ord_atMost_nat @ R5 ) )
% 0.22/0.57 @ ( power_power_nat @ X @ R5 ) )
% 0.22/0.57 @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % polynomial_product_nat
% 0.22/0.57 thf(fact_9178_choose__square__sum,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( groups3542108847815614940at_nat
% 0.22/0.57 @ ^ [K3: nat] : ( power_power_nat @ ( binomial @ N @ K3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.57 @ ( set_ord_atMost_nat @ N ) )
% 0.22/0.57 = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 0.22/0.57
% 0.22/0.57 % choose_square_sum
% 0.22/0.57 thf(fact_9179_binomial__r__part__sum,axiom,
% 0.22/0.57 ! [M: nat] :
% 0.22/0.57 ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
% 0.22/0.57 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % binomial_r_part_sum
% 0.22/0.57 thf(fact_9180_choose__linear__sum,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( groups3542108847815614940at_nat
% 0.22/0.57 @ ^ [I2: nat] : ( times_times_nat @ I2 @ ( binomial @ N @ I2 ) )
% 0.22/0.57 @ ( set_ord_atMost_nat @ N ) )
% 0.22/0.57 = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % choose_linear_sum
% 0.22/0.57 thf(fact_9181_mask__eq__sum__exp__nat,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
% 0.22/0.57 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.22/0.57 @ ( collect_nat
% 0.22/0.57 @ ^ [Q3: nat] : ( ord_less_nat @ Q3 @ N ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % mask_eq_sum_exp_nat
% 0.22/0.57 thf(fact_9182_bounded__Max__nat,axiom,
% 0.22/0.57 ! [P: nat > $o,X: nat,M10: nat] :
% 0.22/0.57 ( ( P @ X )
% 0.22/0.57 => ( ! [X4: nat] :
% 0.22/0.57 ( ( P @ X4 )
% 0.22/0.57 => ( ord_less_eq_nat @ X4 @ M10 ) )
% 0.22/0.57 => ~ ! [M4: nat] :
% 0.22/0.57 ( ( P @ M4 )
% 0.22/0.57 => ~ ! [X5: nat] :
% 0.22/0.57 ( ( P @ X5 )
% 0.22/0.57 => ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bounded_Max_nat
% 0.22/0.57 thf(fact_9183_sum__nth__roots,axiom,
% 0.22/0.57 ! [N: nat,C: complex] :
% 0.22/0.57 ( ( ord_less_nat @ one_one_nat @ N )
% 0.22/0.57 => ( ( groups7754918857620584856omplex
% 0.22/0.57 @ ^ [X3: complex] : X3
% 0.22/0.57 @ ( collect_complex
% 0.22/0.57 @ ^ [Z5: complex] :
% 0.22/0.57 ( ( power_power_complex @ Z5 @ N )
% 0.22/0.57 = C ) ) )
% 0.22/0.57 = zero_zero_complex ) ) ).
% 0.22/0.57
% 0.22/0.57 % sum_nth_roots
% 0.22/0.57 thf(fact_9184_sum__roots__unity,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( ord_less_nat @ one_one_nat @ N )
% 0.22/0.57 => ( ( groups7754918857620584856omplex
% 0.22/0.57 @ ^ [X3: complex] : X3
% 0.22/0.57 @ ( collect_complex
% 0.22/0.57 @ ^ [Z5: complex] :
% 0.22/0.57 ( ( power_power_complex @ Z5 @ N )
% 0.22/0.57 = one_one_complex ) ) )
% 0.22/0.57 = zero_zero_complex ) ) ).
% 0.22/0.57
% 0.22/0.57 % sum_roots_unity
% 0.22/0.57 thf(fact_9185_Maclaurin__minus__cos__expansion,axiom,
% 0.22/0.57 ! [N: nat,X: real] :
% 0.22/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.57 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.22/0.57 => ? [T3: real] :
% 0.22/0.57 ( ( ord_less_real @ X @ T3 )
% 0.22/0.57 & ( ord_less_real @ T3 @ zero_zero_real )
% 0.22/0.57 & ( ( cos_real @ X )
% 0.22/0.57 = ( plus_plus_real
% 0.22/0.57 @ ( groups6591440286371151544t_real
% 0.22/0.57 @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
% 0.22/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.57 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Maclaurin_minus_cos_expansion
% 0.22/0.57 thf(fact_9186_Maclaurin__cos__expansion2,axiom,
% 0.22/0.57 ! [X: real,N: nat] :
% 0.22/0.57 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.57 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.57 => ? [T3: real] :
% 0.22/0.57 ( ( ord_less_real @ zero_zero_real @ T3 )
% 0.22/0.57 & ( ord_less_real @ T3 @ X )
% 0.22/0.57 & ( ( cos_real @ X )
% 0.22/0.57 = ( plus_plus_real
% 0.22/0.57 @ ( groups6591440286371151544t_real
% 0.22/0.57 @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
% 0.22/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.57 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Maclaurin_cos_expansion2
% 0.22/0.57 thf(fact_9187_Maclaurin__sin__expansion3,axiom,
% 0.22/0.57 ! [N: nat,X: real] :
% 0.22/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.57 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.57 => ? [T3: real] :
% 0.22/0.57 ( ( ord_less_real @ zero_zero_real @ T3 )
% 0.22/0.57 & ( ord_less_real @ T3 @ X )
% 0.22/0.57 & ( ( sin_real @ X )
% 0.22/0.57 = ( plus_plus_real
% 0.22/0.57 @ ( groups6591440286371151544t_real
% 0.22/0.57 @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
% 0.22/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.57 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Maclaurin_sin_expansion3
% 0.22/0.57 thf(fact_9188_Maclaurin__sin__expansion4,axiom,
% 0.22/0.57 ! [X: real,N: nat] :
% 0.22/0.57 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.57 => ? [T3: real] :
% 0.22/0.57 ( ( ord_less_real @ zero_zero_real @ T3 )
% 0.22/0.57 & ( ord_less_eq_real @ T3 @ X )
% 0.22/0.57 & ( ( sin_real @ X )
% 0.22/0.57 = ( plus_plus_real
% 0.22/0.57 @ ( groups6591440286371151544t_real
% 0.22/0.57 @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
% 0.22/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.57 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Maclaurin_sin_expansion4
% 0.22/0.57 thf(fact_9189_sumr__cos__zero__one,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( groups6591440286371151544t_real
% 0.22/0.57 @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ zero_zero_real @ M5 ) )
% 0.22/0.57 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.22/0.57 = one_one_real ) ).
% 0.22/0.57
% 0.22/0.57 % sumr_cos_zero_one
% 0.22/0.57 thf(fact_9190_lessThan__Suc__atMost,axiom,
% 0.22/0.57 ! [K: nat] :
% 0.22/0.57 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 0.22/0.57 = ( set_ord_atMost_nat @ K ) ) ).
% 0.22/0.57
% 0.22/0.57 % lessThan_Suc_atMost
% 0.22/0.57 thf(fact_9191_Maclaurin__lemma,axiom,
% 0.22/0.57 ! [H: real,F: real > real,J: nat > real,N: nat] :
% 0.22/0.57 ( ( ord_less_real @ zero_zero_real @ H )
% 0.22/0.57 => ? [B8: real] :
% 0.22/0.57 ( ( F @ H )
% 0.22/0.57 = ( plus_plus_real
% 0.22/0.57 @ ( groups6591440286371151544t_real
% 0.22/0.57 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H @ M5 ) )
% 0.22/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.57 @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Maclaurin_lemma
% 0.22/0.57 thf(fact_9192_sum__split__even__odd,axiom,
% 0.22/0.57 ! [F: nat > real,G: nat > real,N: nat] :
% 0.22/0.57 ( ( groups6591440286371151544t_real
% 0.22/0.57 @ ^ [I2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ ( F @ I2 ) @ ( G @ I2 ) )
% 0.22/0.57 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.22/0.57 = ( plus_plus_real
% 0.22/0.57 @ ( groups6591440286371151544t_real
% 0.22/0.57 @ ^ [I2: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) )
% 0.22/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.57 @ ( groups6591440286371151544t_real
% 0.22/0.57 @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ one_one_nat ) )
% 0.22/0.57 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % sum_split_even_odd
% 0.22/0.57 thf(fact_9193_Maclaurin__exp__le,axiom,
% 0.22/0.57 ! [X: real,N: nat] :
% 0.22/0.57 ? [T3: real] :
% 0.22/0.57 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.22/0.57 & ( ( exp_real @ X )
% 0.22/0.57 = ( plus_plus_real
% 0.22/0.57 @ ( groups6591440286371151544t_real
% 0.22/0.57 @ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
% 0.22/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.57 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Maclaurin_exp_le
% 0.22/0.57 thf(fact_9194_Maclaurin__sin__bound,axiom,
% 0.22/0.57 ! [X: real,N: nat] :
% 0.22/0.57 ( ord_less_eq_real
% 0.22/0.57 @ ( abs_abs_real
% 0.22/0.57 @ ( minus_minus_real @ ( sin_real @ X )
% 0.22/0.57 @ ( groups6591440286371151544t_real
% 0.22/0.57 @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
% 0.22/0.57 @ ( set_ord_lessThan_nat @ N ) ) ) )
% 0.22/0.57 @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Maclaurin_sin_bound
% 0.22/0.57 thf(fact_9195_sum__pos__lt__pair,axiom,
% 0.22/0.57 ! [F: nat > real,K: nat] :
% 0.22/0.57 ( ( summable_real @ F )
% 0.22/0.57 => ( ! [D2: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D2 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D2 ) @ one_one_nat ) ) ) ) )
% 0.22/0.57 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % sum_pos_lt_pair
% 0.22/0.57 thf(fact_9196_Maclaurin__exp__lt,axiom,
% 0.22/0.57 ! [X: real,N: nat] :
% 0.22/0.57 ( ( X != zero_zero_real )
% 0.22/0.57 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.57 => ? [T3: real] :
% 0.22/0.57 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
% 0.22/0.57 & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.22/0.57 & ( ( exp_real @ X )
% 0.22/0.57 = ( plus_plus_real
% 0.22/0.57 @ ( groups6591440286371151544t_real
% 0.22/0.57 @ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
% 0.22/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.57 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Maclaurin_exp_lt
% 0.22/0.57 thf(fact_9197_Maclaurin__sin__expansion,axiom,
% 0.22/0.57 ! [X: real,N: nat] :
% 0.22/0.57 ? [T3: real] :
% 0.22/0.57 ( ( sin_real @ X )
% 0.22/0.57 = ( plus_plus_real
% 0.22/0.57 @ ( groups6591440286371151544t_real
% 0.22/0.57 @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
% 0.22/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.57 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Maclaurin_sin_expansion
% 0.22/0.57 thf(fact_9198_Maclaurin__sin__expansion2,axiom,
% 0.22/0.57 ! [X: real,N: nat] :
% 0.22/0.57 ? [T3: real] :
% 0.22/0.57 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.22/0.57 & ( ( sin_real @ X )
% 0.22/0.57 = ( plus_plus_real
% 0.22/0.57 @ ( groups6591440286371151544t_real
% 0.22/0.57 @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
% 0.22/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.57 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Maclaurin_sin_expansion2
% 0.22/0.57 thf(fact_9199_Maclaurin__cos__expansion,axiom,
% 0.22/0.57 ! [X: real,N: nat] :
% 0.22/0.57 ? [T3: real] :
% 0.22/0.57 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.22/0.57 & ( ( cos_real @ X )
% 0.22/0.57 = ( plus_plus_real
% 0.22/0.57 @ ( groups6591440286371151544t_real
% 0.22/0.57 @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
% 0.22/0.57 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.57 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Maclaurin_cos_expansion
% 0.22/0.57 thf(fact_9200_bij__betw__roots__unity,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.57 => ( bij_betw_nat_complex
% 0.22/0.57 @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.22/0.57 @ ( set_ord_lessThan_nat @ N )
% 0.22/0.57 @ ( collect_complex
% 0.22/0.57 @ ^ [Z5: complex] :
% 0.22/0.57 ( ( power_power_complex @ Z5 @ N )
% 0.22/0.57 = one_one_complex ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bij_betw_roots_unity
% 0.22/0.57 thf(fact_9201_all__nat__less,axiom,
% 0.22/0.57 ! [N: nat,P: nat > $o] :
% 0.22/0.57 ( ( ! [M5: nat] :
% 0.22/0.57 ( ( ord_less_eq_nat @ M5 @ N )
% 0.22/0.57 => ( P @ M5 ) ) )
% 0.22/0.57 = ( ! [X3: nat] :
% 0.22/0.57 ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.22/0.57 => ( P @ X3 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % all_nat_less
% 0.22/0.57 thf(fact_9202_ex__nat__less,axiom,
% 0.22/0.57 ! [N: nat,P: nat > $o] :
% 0.22/0.57 ( ( ? [M5: nat] :
% 0.22/0.57 ( ( ord_less_eq_nat @ M5 @ N )
% 0.22/0.57 & ( P @ M5 ) ) )
% 0.22/0.57 = ( ? [X3: nat] :
% 0.22/0.57 ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.22/0.57 & ( P @ X3 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % ex_nat_less
% 0.22/0.57 thf(fact_9203_atMost__atLeast0,axiom,
% 0.22/0.57 ( set_ord_atMost_nat
% 0.22/0.57 = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% 0.22/0.57
% 0.22/0.57 % atMost_atLeast0
% 0.22/0.57 thf(fact_9204_gauss__sum__nat,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( groups3542108847815614940at_nat
% 0.22/0.57 @ ^ [X3: nat] : X3
% 0.22/0.57 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.22/0.57 = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gauss_sum_nat
% 0.22/0.57 thf(fact_9205_arith__series__nat,axiom,
% 0.22/0.57 ! [A: nat,D: nat,N: nat] :
% 0.22/0.57 ( ( groups3542108847815614940at_nat
% 0.22/0.57 @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I2 @ D ) )
% 0.22/0.57 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.22/0.57 = ( 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 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % arith_series_nat
% 0.22/0.57 thf(fact_9206_Sum__Icc__nat,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( groups3542108847815614940at_nat
% 0.22/0.57 @ ^ [X3: nat] : X3
% 0.22/0.57 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.22/0.57 = ( 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 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Sum_Icc_nat
% 0.22/0.57 thf(fact_9207_bset_I1_J,axiom,
% 0.22/0.57 ! [D3: int,B4: set_int,P: int > $o,Q: int > $o] :
% 0.22/0.57 ( ! [X4: int] :
% 0.22/0.57 ( ! [Xa2: int] :
% 0.22/0.57 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb: int] :
% 0.22/0.57 ( ( member_int @ Xb @ B4 )
% 0.22/0.57 => ( X4
% 0.22/0.57 != ( plus_plus_int @ Xb @ Xa2 ) ) ) )
% 0.22/0.57 => ( ( P @ X4 )
% 0.22/0.57 => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 0.22/0.57 => ( ! [X4: int] :
% 0.22/0.57 ( ! [Xa2: int] :
% 0.22/0.57 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb: int] :
% 0.22/0.57 ( ( member_int @ Xb @ B4 )
% 0.22/0.57 => ( X4
% 0.22/0.57 != ( plus_plus_int @ Xb @ Xa2 ) ) ) )
% 0.22/0.57 => ( ( Q @ X4 )
% 0.22/0.57 => ( Q @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ B4 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ( ( P @ X5 )
% 0.22/0.57 & ( Q @ X5 ) )
% 0.22/0.57 => ( ( P @ ( minus_minus_int @ X5 @ D3 ) )
% 0.22/0.57 & ( Q @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bset(1)
% 0.22/0.57 thf(fact_9208_bset_I2_J,axiom,
% 0.22/0.57 ! [D3: int,B4: set_int,P: int > $o,Q: int > $o] :
% 0.22/0.57 ( ! [X4: int] :
% 0.22/0.57 ( ! [Xa2: int] :
% 0.22/0.57 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb: int] :
% 0.22/0.57 ( ( member_int @ Xb @ B4 )
% 0.22/0.57 => ( X4
% 0.22/0.57 != ( plus_plus_int @ Xb @ Xa2 ) ) ) )
% 0.22/0.57 => ( ( P @ X4 )
% 0.22/0.57 => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 0.22/0.57 => ( ! [X4: int] :
% 0.22/0.57 ( ! [Xa2: int] :
% 0.22/0.57 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb: int] :
% 0.22/0.57 ( ( member_int @ Xb @ B4 )
% 0.22/0.57 => ( X4
% 0.22/0.57 != ( plus_plus_int @ Xb @ Xa2 ) ) ) )
% 0.22/0.57 => ( ( Q @ X4 )
% 0.22/0.57 => ( Q @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ B4 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ( ( P @ X5 )
% 0.22/0.57 | ( Q @ X5 ) )
% 0.22/0.57 => ( ( P @ ( minus_minus_int @ X5 @ D3 ) )
% 0.22/0.57 | ( Q @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bset(2)
% 0.22/0.57 thf(fact_9209_aset_I1_J,axiom,
% 0.22/0.57 ! [D3: int,A2: set_int,P: int > $o,Q: int > $o] :
% 0.22/0.57 ( ! [X4: int] :
% 0.22/0.57 ( ! [Xa2: int] :
% 0.22/0.57 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb: int] :
% 0.22/0.57 ( ( member_int @ Xb @ A2 )
% 0.22/0.57 => ( X4
% 0.22/0.57 != ( minus_minus_int @ Xb @ Xa2 ) ) ) )
% 0.22/0.57 => ( ( P @ X4 )
% 0.22/0.57 => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 0.22/0.57 => ( ! [X4: int] :
% 0.22/0.57 ( ! [Xa2: int] :
% 0.22/0.57 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb: int] :
% 0.22/0.57 ( ( member_int @ Xb @ A2 )
% 0.22/0.57 => ( X4
% 0.22/0.57 != ( minus_minus_int @ Xb @ Xa2 ) ) ) )
% 0.22/0.57 => ( ( Q @ X4 )
% 0.22/0.57 => ( Q @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ A2 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ( ( P @ X5 )
% 0.22/0.57 & ( Q @ X5 ) )
% 0.22/0.57 => ( ( P @ ( plus_plus_int @ X5 @ D3 ) )
% 0.22/0.57 & ( Q @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % aset(1)
% 0.22/0.57 thf(fact_9210_aset_I2_J,axiom,
% 0.22/0.57 ! [D3: int,A2: set_int,P: int > $o,Q: int > $o] :
% 0.22/0.57 ( ! [X4: int] :
% 0.22/0.57 ( ! [Xa2: int] :
% 0.22/0.57 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb: int] :
% 0.22/0.57 ( ( member_int @ Xb @ A2 )
% 0.22/0.57 => ( X4
% 0.22/0.57 != ( minus_minus_int @ Xb @ Xa2 ) ) ) )
% 0.22/0.57 => ( ( P @ X4 )
% 0.22/0.57 => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 0.22/0.57 => ( ! [X4: int] :
% 0.22/0.57 ( ! [Xa2: int] :
% 0.22/0.57 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb: int] :
% 0.22/0.57 ( ( member_int @ Xb @ A2 )
% 0.22/0.57 => ( X4
% 0.22/0.57 != ( minus_minus_int @ Xb @ Xa2 ) ) ) )
% 0.22/0.57 => ( ( Q @ X4 )
% 0.22/0.57 => ( Q @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ A2 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ( ( P @ X5 )
% 0.22/0.57 | ( Q @ X5 ) )
% 0.22/0.57 => ( ( P @ ( plus_plus_int @ X5 @ D3 ) )
% 0.22/0.57 | ( Q @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % aset(2)
% 0.22/0.57 thf(fact_9211_bset_I9_J,axiom,
% 0.22/0.57 ! [D: int,D3: int,B4: set_int,T: int] :
% 0.22/0.57 ( ( dvd_dvd_int @ D @ D3 )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ B4 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.22/0.57 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bset(9)
% 0.22/0.57 thf(fact_9212_bset_I10_J,axiom,
% 0.22/0.57 ! [D: int,D3: int,B4: set_int,T: int] :
% 0.22/0.57 ( ( dvd_dvd_int @ D @ D3 )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ B4 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.22/0.57 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bset(10)
% 0.22/0.57 thf(fact_9213_aset_I9_J,axiom,
% 0.22/0.57 ! [D: int,D3: int,A2: set_int,T: int] :
% 0.22/0.57 ( ( dvd_dvd_int @ D @ D3 )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ A2 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.22/0.57 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % aset(9)
% 0.22/0.57 thf(fact_9214_aset_I10_J,axiom,
% 0.22/0.57 ! [D: int,D3: int,A2: set_int,T: int] :
% 0.22/0.57 ( ( dvd_dvd_int @ D @ D3 )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ A2 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.22/0.57 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % aset(10)
% 0.22/0.57 thf(fact_9215_periodic__finite__ex,axiom,
% 0.22/0.57 ! [D: int,P: int > $o] :
% 0.22/0.57 ( ( ord_less_int @ zero_zero_int @ D )
% 0.22/0.57 => ( ! [X4: int,K2: int] :
% 0.22/0.57 ( ( P @ X4 )
% 0.22/0.57 = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 0.22/0.57 => ( ( ? [X8: int] : ( P @ X8 ) )
% 0.22/0.57 = ( ? [X3: int] :
% 0.22/0.57 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 0.22/0.57 & ( P @ X3 ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % periodic_finite_ex
% 0.22/0.57 thf(fact_9216_bset_I3_J,axiom,
% 0.22/0.57 ! [D3: int,T: int,B4: set_int] :
% 0.22/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.22/0.57 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B4 )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ B4 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ( X5 = T )
% 0.22/0.57 => ( ( minus_minus_int @ X5 @ D3 )
% 0.22/0.57 = T ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bset(3)
% 0.22/0.57 thf(fact_9217_bset_I4_J,axiom,
% 0.22/0.57 ! [D3: int,T: int,B4: set_int] :
% 0.22/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.22/0.57 => ( ( member_int @ T @ B4 )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ B4 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ( X5 != T )
% 0.22/0.57 => ( ( minus_minus_int @ X5 @ D3 )
% 0.22/0.57 != T ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bset(4)
% 0.22/0.57 thf(fact_9218_bset_I5_J,axiom,
% 0.22/0.57 ! [D3: int,B4: set_int,T: int] :
% 0.22/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ B4 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ( ord_less_int @ X5 @ T )
% 0.22/0.57 => ( ord_less_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bset(5)
% 0.22/0.57 thf(fact_9219_bset_I7_J,axiom,
% 0.22/0.57 ! [D3: int,T: int,B4: set_int] :
% 0.22/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.22/0.57 => ( ( member_int @ T @ B4 )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ B4 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ( ord_less_int @ T @ X5 )
% 0.22/0.57 => ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bset(7)
% 0.22/0.57 thf(fact_9220_aset_I3_J,axiom,
% 0.22/0.57 ! [D3: int,T: int,A2: set_int] :
% 0.22/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.22/0.57 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ A2 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ( X5 = T )
% 0.22/0.57 => ( ( plus_plus_int @ X5 @ D3 )
% 0.22/0.57 = T ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % aset(3)
% 0.22/0.57 thf(fact_9221_aset_I4_J,axiom,
% 0.22/0.57 ! [D3: int,T: int,A2: set_int] :
% 0.22/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.22/0.57 => ( ( member_int @ T @ A2 )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ A2 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ( X5 != T )
% 0.22/0.57 => ( ( plus_plus_int @ X5 @ D3 )
% 0.22/0.57 != T ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % aset(4)
% 0.22/0.57 thf(fact_9222_aset_I5_J,axiom,
% 0.22/0.57 ! [D3: int,T: int,A2: set_int] :
% 0.22/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.22/0.57 => ( ( member_int @ T @ A2 )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ A2 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ( ord_less_int @ X5 @ T )
% 0.22/0.57 => ( ord_less_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % aset(5)
% 0.22/0.57 thf(fact_9223_aset_I7_J,axiom,
% 0.22/0.57 ! [D3: int,A2: set_int,T: int] :
% 0.22/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ A2 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ( ord_less_int @ T @ X5 )
% 0.22/0.57 => ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % aset(7)
% 0.22/0.57 thf(fact_9224_fact__eq__fact__times,axiom,
% 0.22/0.57 ! [N: nat,M: nat] :
% 0.22/0.57 ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.57 => ( ( semiri1408675320244567234ct_nat @ M )
% 0.22/0.57 = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
% 0.22/0.57 @ ( groups708209901874060359at_nat
% 0.22/0.57 @ ^ [X3: nat] : X3
% 0.22/0.57 @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % fact_eq_fact_times
% 0.22/0.57 thf(fact_9225_bset_I6_J,axiom,
% 0.22/0.57 ! [D3: int,B4: set_int,T: int] :
% 0.22/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ B4 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ( ord_less_eq_int @ X5 @ T )
% 0.22/0.57 => ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bset(6)
% 0.22/0.57 thf(fact_9226_bset_I8_J,axiom,
% 0.22/0.57 ! [D3: int,T: int,B4: set_int] :
% 0.22/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.22/0.57 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B4 )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ B4 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ( ord_less_eq_int @ T @ X5 )
% 0.22/0.57 => ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bset(8)
% 0.22/0.57 thf(fact_9227_aset_I6_J,axiom,
% 0.22/0.57 ! [D3: int,T: int,A2: set_int] :
% 0.22/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.22/0.57 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ A2 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ( ord_less_eq_int @ X5 @ T )
% 0.22/0.57 => ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % aset(6)
% 0.22/0.57 thf(fact_9228_aset_I8_J,axiom,
% 0.22/0.57 ! [D3: int,A2: set_int,T: int] :
% 0.22/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.22/0.57 => ! [X5: int] :
% 0.22/0.57 ( ! [Xa3: int] :
% 0.22/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb2: int] :
% 0.22/0.57 ( ( member_int @ Xb2 @ A2 )
% 0.22/0.57 => ( X5
% 0.22/0.57 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.22/0.57 => ( ( ord_less_eq_int @ T @ X5 )
% 0.22/0.57 => ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % aset(8)
% 0.22/0.57 thf(fact_9229_cppi,axiom,
% 0.22/0.57 ! [D3: int,P: int > $o,P4: int > $o,A2: set_int] :
% 0.22/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.22/0.57 => ( ? [Z3: int] :
% 0.22/0.57 ! [X4: int] :
% 0.22/0.57 ( ( ord_less_int @ Z3 @ X4 )
% 0.22/0.57 => ( ( P @ X4 )
% 0.22/0.57 = ( P4 @ X4 ) ) )
% 0.22/0.57 => ( ! [X4: int] :
% 0.22/0.57 ( ! [Xa2: int] :
% 0.22/0.57 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb: int] :
% 0.22/0.57 ( ( member_int @ Xb @ A2 )
% 0.22/0.57 => ( X4
% 0.22/0.57 != ( minus_minus_int @ Xb @ Xa2 ) ) ) )
% 0.22/0.57 => ( ( P @ X4 )
% 0.22/0.57 => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 0.22/0.57 => ( ! [X4: int,K2: int] :
% 0.22/0.57 ( ( P4 @ X4 )
% 0.22/0.57 = ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 0.22/0.57 => ( ( ? [X8: int] : ( P @ X8 ) )
% 0.22/0.57 = ( ? [X3: int] :
% 0.22/0.57 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 & ( P4 @ X3 ) )
% 0.22/0.57 | ? [X3: int] :
% 0.22/0.57 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 & ? [Y6: int] :
% 0.22/0.57 ( ( member_int @ Y6 @ A2 )
% 0.22/0.57 & ( P @ ( minus_minus_int @ Y6 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % cppi
% 0.22/0.57 thf(fact_9230_cpmi,axiom,
% 0.22/0.57 ! [D3: int,P: int > $o,P4: int > $o,B4: set_int] :
% 0.22/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.22/0.57 => ( ? [Z3: int] :
% 0.22/0.57 ! [X4: int] :
% 0.22/0.57 ( ( ord_less_int @ X4 @ Z3 )
% 0.22/0.57 => ( ( P @ X4 )
% 0.22/0.57 = ( P4 @ X4 ) ) )
% 0.22/0.57 => ( ! [X4: int] :
% 0.22/0.57 ( ! [Xa2: int] :
% 0.22/0.57 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 => ! [Xb: int] :
% 0.22/0.57 ( ( member_int @ Xb @ B4 )
% 0.22/0.57 => ( X4
% 0.22/0.57 != ( plus_plus_int @ Xb @ Xa2 ) ) ) )
% 0.22/0.57 => ( ( P @ X4 )
% 0.22/0.57 => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 0.22/0.57 => ( ! [X4: int,K2: int] :
% 0.22/0.57 ( ( P4 @ X4 )
% 0.22/0.57 = ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 0.22/0.57 => ( ( ? [X8: int] : ( P @ X8 ) )
% 0.22/0.57 = ( ? [X3: int] :
% 0.22/0.57 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 & ( P4 @ X3 ) )
% 0.22/0.57 | ? [X3: int] :
% 0.22/0.57 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.22/0.57 & ? [Y6: int] :
% 0.22/0.57 ( ( member_int @ Y6 @ B4 )
% 0.22/0.57 & ( P @ ( plus_plus_int @ Y6 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % cpmi
% 0.22/0.57 thf(fact_9231_fact__div__fact,axiom,
% 0.22/0.57 ! [N: nat,M: nat] :
% 0.22/0.57 ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.57 => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.22/0.57 = ( groups708209901874060359at_nat
% 0.22/0.57 @ ^ [X3: nat] : X3
% 0.22/0.57 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % fact_div_fact
% 0.22/0.57 thf(fact_9232_divmod__step__nat__def,axiom,
% 0.22/0.57 ( unique5026877609467782581ep_nat
% 0.22/0.57 = ( ^ [L3: num] :
% 0.22/0.57 ( produc2626176000494625587at_nat
% 0.22/0.57 @ ^ [Q3: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L3 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L3 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R5 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % divmod_step_nat_def
% 0.22/0.57 thf(fact_9233_divmod__step__int__def,axiom,
% 0.22/0.57 ( unique5024387138958732305ep_int
% 0.22/0.57 = ( ^ [L3: num] :
% 0.22/0.57 ( produc4245557441103728435nt_int
% 0.22/0.57 @ ^ [Q3: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L3 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L3 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R5 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % divmod_step_int_def
% 0.22/0.57 thf(fact_9234_Sum__Icc__int,axiom,
% 0.22/0.57 ! [M: int,N: int] :
% 0.22/0.57 ( ( ord_less_eq_int @ M @ N )
% 0.22/0.57 => ( ( groups4538972089207619220nt_int
% 0.22/0.57 @ ^ [X3: int] : X3
% 0.22/0.57 @ ( set_or1266510415728281911st_int @ M @ N ) )
% 0.22/0.57 = ( 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 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Sum_Icc_int
% 0.22/0.57 thf(fact_9235_divmod__nat__if,axiom,
% 0.22/0.57 ( divmod_nat
% 0.22/0.57 = ( ^ [M5: nat,N4: nat] :
% 0.22/0.57 ( if_Pro6206227464963214023at_nat
% 0.22/0.57 @ ( ( N4 = zero_zero_nat )
% 0.22/0.57 | ( ord_less_nat @ M5 @ N4 ) )
% 0.22/0.57 @ ( product_Pair_nat_nat @ zero_zero_nat @ M5 )
% 0.22/0.57 @ ( produc2626176000494625587at_nat
% 0.22/0.57 @ ^ [Q3: nat] : ( product_Pair_nat_nat @ ( suc @ Q3 ) )
% 0.22/0.57 @ ( divmod_nat @ ( minus_minus_nat @ M5 @ N4 ) @ N4 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % divmod_nat_if
% 0.22/0.57 thf(fact_9236_vebt__buildup_Opelims,axiom,
% 0.22/0.57 ! [X: nat,Y4: vEBT_VEBT] :
% 0.22/0.57 ( ( ( vEBT_vebt_buildup @ X )
% 0.22/0.57 = Y4 )
% 0.22/0.57 => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
% 0.22/0.57 => ( ( ( X = zero_zero_nat )
% 0.22/0.57 => ( ( Y4
% 0.22/0.57 = ( vEBT_Leaf @ $false @ $false ) )
% 0.22/0.57 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 0.22/0.57 => ( ( ( X
% 0.22/0.57 = ( suc @ zero_zero_nat ) )
% 0.22/0.57 => ( ( Y4
% 0.22/0.57 = ( vEBT_Leaf @ $false @ $false ) )
% 0.22/0.57 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 0.22/0.57 => ~ ! [Va: nat] :
% 0.22/0.57 ( ( X
% 0.22/0.57 = ( suc @ ( suc @ Va ) ) )
% 0.22/0.57 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.22/0.57 => ( Y4
% 0.22/0.57 = ( 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 ) ) ) ) ) ) )
% 0.22/0.57 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.22/0.57 => ( Y4
% 0.22/0.57 = ( 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 ) ) ) ) ) ) ) ) )
% 0.22/0.57 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % vebt_buildup.pelims
% 0.22/0.57 thf(fact_9237_arctan__def,axiom,
% 0.22/0.57 ( arctan
% 0.22/0.57 = ( ^ [Y6: real] :
% 0.22/0.57 ( the_real
% 0.22/0.57 @ ^ [X3: real] :
% 0.22/0.57 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.22/0.57 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.57 & ( ( tan_real @ X3 )
% 0.22/0.57 = Y6 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % arctan_def
% 0.22/0.57 thf(fact_9238_prod__int__eq,axiom,
% 0.22/0.57 ! [I: nat,J: nat] :
% 0.22/0.57 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 0.22/0.57 = ( groups1705073143266064639nt_int
% 0.22/0.57 @ ^ [X3: int] : X3
% 0.22/0.57 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % prod_int_eq
% 0.22/0.57 thf(fact_9239_prod__int__plus__eq,axiom,
% 0.22/0.57 ! [I: nat,J: nat] :
% 0.22/0.57 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
% 0.22/0.57 = ( groups1705073143266064639nt_int
% 0.22/0.57 @ ^ [X3: int] : X3
% 0.22/0.57 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % prod_int_plus_eq
% 0.22/0.57 thf(fact_9240_ln__neg__is__const,axiom,
% 0.22/0.57 ! [X: real] :
% 0.22/0.57 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.22/0.57 => ( ( ln_ln_real @ X )
% 0.22/0.57 = ( the_real
% 0.22/0.57 @ ^ [X3: real] : $false ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % ln_neg_is_const
% 0.22/0.57 thf(fact_9241_arccos__def,axiom,
% 0.22/0.57 ( arccos
% 0.22/0.57 = ( ^ [Y6: real] :
% 0.22/0.57 ( the_real
% 0.22/0.57 @ ^ [X3: real] :
% 0.22/0.57 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.22/0.57 & ( ord_less_eq_real @ X3 @ pi )
% 0.22/0.57 & ( ( cos_real @ X3 )
% 0.22/0.57 = Y6 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % arccos_def
% 0.22/0.57 thf(fact_9242_divmod__nat__def,axiom,
% 0.22/0.57 ( divmod_nat
% 0.22/0.57 = ( ^ [M5: nat,N4: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M5 @ N4 ) @ ( modulo_modulo_nat @ M5 @ N4 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % divmod_nat_def
% 0.22/0.57 thf(fact_9243_pi__half,axiom,
% 0.22/0.57 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.22/0.57 = ( the_real
% 0.22/0.57 @ ^ [X3: real] :
% 0.22/0.57 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.22/0.57 & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.22/0.57 & ( ( cos_real @ X3 )
% 0.22/0.57 = zero_zero_real ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % pi_half
% 0.22/0.57 thf(fact_9244_pi__def,axiom,
% 0.22/0.57 ( pi
% 0.22/0.57 = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.22/0.57 @ ( the_real
% 0.22/0.57 @ ^ [X3: real] :
% 0.22/0.57 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.22/0.57 & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.22/0.57 & ( ( cos_real @ X3 )
% 0.22/0.57 = zero_zero_real ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % pi_def
% 0.22/0.57 thf(fact_9245_arcsin__def,axiom,
% 0.22/0.57 ( arcsin
% 0.22/0.57 = ( ^ [Y6: real] :
% 0.22/0.57 ( the_real
% 0.22/0.57 @ ^ [X3: real] :
% 0.22/0.57 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.22/0.57 & ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.57 & ( ( sin_real @ X3 )
% 0.22/0.57 = Y6 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % arcsin_def
% 0.22/0.57 thf(fact_9246_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
% 0.22/0.57 ! [X: nat,Y4: nat] :
% 0.22/0.57 ( ( ( vEBT_V8346862874174094_d_u_p @ X )
% 0.22/0.57 = Y4 )
% 0.22/0.57 => ( ( accp_nat @ vEBT_V1247956027447740395_p_rel @ X )
% 0.22/0.57 => ( ( ( X = zero_zero_nat )
% 0.22/0.57 => ( ( Y4
% 0.22/0.57 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.22/0.57 => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ zero_zero_nat ) ) )
% 0.22/0.57 => ( ( ( X
% 0.22/0.57 = ( suc @ zero_zero_nat ) )
% 0.22/0.57 => ( ( Y4
% 0.22/0.57 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.22/0.57 => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ zero_zero_nat ) ) ) )
% 0.22/0.57 => ~ ! [Va: nat] :
% 0.22/0.57 ( ( X
% 0.22/0.57 = ( suc @ ( suc @ Va ) ) )
% 0.22/0.57 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.22/0.57 => ( Y4
% 0.22/0.57 = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
% 0.22/0.57 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.22/0.57 => ( Y4
% 0.22/0.57 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
% 0.22/0.57 => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
% 0.22/0.57 thf(fact_9247_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
% 0.22/0.57 ! [X: nat,Y4: nat] :
% 0.22/0.57 ( ( ( vEBT_V8646137997579335489_i_l_d @ X )
% 0.22/0.57 = Y4 )
% 0.22/0.57 => ( ( accp_nat @ vEBT_V5144397997797733112_d_rel @ X )
% 0.22/0.57 => ( ( ( X = zero_zero_nat )
% 0.22/0.57 => ( ( Y4
% 0.22/0.57 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.22/0.57 => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ zero_zero_nat ) ) )
% 0.22/0.57 => ( ( ( X
% 0.22/0.57 = ( suc @ zero_zero_nat ) )
% 0.22/0.57 => ( ( Y4
% 0.22/0.57 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.22/0.57 => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ zero_zero_nat ) ) ) )
% 0.22/0.57 => ~ ! [Va: nat] :
% 0.22/0.57 ( ( X
% 0.22/0.57 = ( suc @ ( suc @ Va ) ) )
% 0.22/0.57 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.22/0.57 => ( Y4
% 0.22/0.57 = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.22/0.57 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.22/0.57 => ( Y4
% 0.22/0.57 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.22/0.57 => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
% 0.22/0.57 thf(fact_9248_set__encode__def,axiom,
% 0.22/0.57 ( nat_set_encode
% 0.22/0.57 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % set_encode_def
% 0.22/0.57 thf(fact_9249_set__decode__inverse,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( nat_set_encode @ ( nat_set_decode @ N ) )
% 0.22/0.57 = N ) ).
% 0.22/0.57
% 0.22/0.57 % set_decode_inverse
% 0.22/0.57 thf(fact_9250_VEBT_Osize_I3_J,axiom,
% 0.22/0.57 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.22/0.57 ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.22/0.57 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % VEBT.size(3)
% 0.22/0.57 thf(fact_9251_Sum__Ico__nat,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( groups3542108847815614940at_nat
% 0.22/0.57 @ ^ [X3: nat] : X3
% 0.22/0.57 @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 0.22/0.57 = ( 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 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Sum_Ico_nat
% 0.22/0.57 thf(fact_9252_sum__power2,axiom,
% 0.22/0.57 ! [K: nat] :
% 0.22/0.57 ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 0.22/0.57 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 0.22/0.57
% 0.22/0.57 % sum_power2
% 0.22/0.57 thf(fact_9253_ex__nat__less__eq,axiom,
% 0.22/0.57 ! [N: nat,P: nat > $o] :
% 0.22/0.57 ( ( ? [M5: nat] :
% 0.22/0.57 ( ( ord_less_nat @ M5 @ N )
% 0.22/0.57 & ( P @ M5 ) ) )
% 0.22/0.57 = ( ? [X3: nat] :
% 0.22/0.57 ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.22/0.57 & ( P @ X3 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % ex_nat_less_eq
% 0.22/0.57 thf(fact_9254_all__nat__less__eq,axiom,
% 0.22/0.57 ! [N: nat,P: nat > $o] :
% 0.22/0.57 ( ( ! [M5: nat] :
% 0.22/0.57 ( ( ord_less_nat @ M5 @ N )
% 0.22/0.57 => ( P @ M5 ) ) )
% 0.22/0.57 = ( ! [X3: nat] :
% 0.22/0.57 ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.22/0.57 => ( P @ X3 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % all_nat_less_eq
% 0.22/0.57 thf(fact_9255_atLeastLessThanSuc__atLeastAtMost,axiom,
% 0.22/0.57 ! [L: nat,U: nat] :
% 0.22/0.57 ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
% 0.22/0.57 = ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% 0.22/0.57
% 0.22/0.57 % atLeastLessThanSuc_atLeastAtMost
% 0.22/0.57 thf(fact_9256_lessThan__atLeast0,axiom,
% 0.22/0.57 ( set_ord_lessThan_nat
% 0.22/0.57 = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% 0.22/0.57
% 0.22/0.57 % lessThan_atLeast0
% 0.22/0.57 thf(fact_9257_prod__Suc__Suc__fact,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.22/0.57 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.22/0.57
% 0.22/0.57 % prod_Suc_Suc_fact
% 0.22/0.57 thf(fact_9258_prod__Suc__fact,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.22/0.57 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.22/0.57
% 0.22/0.57 % prod_Suc_fact
% 0.22/0.57 thf(fact_9259_Chebyshev__sum__upper__nat,axiom,
% 0.22/0.57 ! [N: nat,A: nat > nat,B: nat > nat] :
% 0.22/0.57 ( ! [I3: nat,J2: nat] :
% 0.22/0.57 ( ( ord_less_eq_nat @ I3 @ J2 )
% 0.22/0.57 => ( ( ord_less_nat @ J2 @ N )
% 0.22/0.57 => ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
% 0.22/0.57 => ( ! [I3: nat,J2: nat] :
% 0.22/0.57 ( ( ord_less_eq_nat @ I3 @ J2 )
% 0.22/0.57 => ( ( ord_less_nat @ J2 @ N )
% 0.22/0.57 => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I3 ) ) ) )
% 0.22/0.57 => ( ord_less_eq_nat
% 0.22/0.57 @ ( times_times_nat @ N
% 0.22/0.57 @ ( groups3542108847815614940at_nat
% 0.22/0.57 @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( B @ I2 ) )
% 0.22/0.57 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 0.22/0.57 @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Chebyshev_sum_upper_nat
% 0.22/0.57 thf(fact_9260_VEBT_Osize__gen_I1_J,axiom,
% 0.22/0.57 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.22/0.57 ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.22/0.57 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % VEBT.size_gen(1)
% 0.22/0.57 thf(fact_9261_VEBT_Osize__gen_I2_J,axiom,
% 0.22/0.57 ! [X21: $o,X222: $o] :
% 0.22/0.57 ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 0.22/0.57 = zero_zero_nat ) ).
% 0.22/0.57
% 0.22/0.57 % VEBT.size_gen(2)
% 0.22/0.57 thf(fact_9262_int__of__nat__def,axiom,
% 0.22/0.57 code_T6385005292777649522of_nat = semiri1314217659103216013at_int ).
% 0.22/0.57
% 0.22/0.57 % int_of_nat_def
% 0.22/0.57 thf(fact_9263_prod__decode__aux_Opelims,axiom,
% 0.22/0.57 ! [X: nat,Xa: nat,Y4: product_prod_nat_nat] :
% 0.22/0.57 ( ( ( nat_prod_decode_aux @ X @ Xa )
% 0.22/0.57 = Y4 )
% 0.22/0.57 => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 0.22/0.57 => ~ ( ( ( ( ord_less_eq_nat @ Xa @ X )
% 0.22/0.57 => ( Y4
% 0.22/0.57 = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
% 0.22/0.57 & ( ~ ( ord_less_eq_nat @ Xa @ X )
% 0.22/0.57 => ( Y4
% 0.22/0.57 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) )
% 0.22/0.57 => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % prod_decode_aux.pelims
% 0.22/0.57 thf(fact_9264_or__not__num__neg_Opelims,axiom,
% 0.22/0.57 ! [X: num,Xa: num,Y4: num] :
% 0.22/0.57 ( ( ( bit_or_not_num_neg @ X @ Xa )
% 0.22/0.57 = Y4 )
% 0.22/0.57 => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 0.22/0.57 => ( ( ( X = one )
% 0.22/0.57 => ( ( Xa = one )
% 0.22/0.57 => ( ( Y4 = one )
% 0.22/0.57 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.22/0.57 => ( ( ( X = one )
% 0.22/0.57 => ! [M4: num] :
% 0.22/0.57 ( ( Xa
% 0.22/0.57 = ( bit0 @ M4 ) )
% 0.22/0.57 => ( ( Y4
% 0.22/0.57 = ( bit1 @ M4 ) )
% 0.22/0.57 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M4 ) ) ) ) ) )
% 0.22/0.57 => ( ( ( X = one )
% 0.22/0.57 => ! [M4: num] :
% 0.22/0.57 ( ( Xa
% 0.22/0.57 = ( bit1 @ M4 ) )
% 0.22/0.57 => ( ( Y4
% 0.22/0.57 = ( bit1 @ M4 ) )
% 0.22/0.57 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M4 ) ) ) ) ) )
% 0.22/0.57 => ( ! [N2: num] :
% 0.22/0.57 ( ( X
% 0.22/0.57 = ( bit0 @ N2 ) )
% 0.22/0.57 => ( ( Xa = one )
% 0.22/0.57 => ( ( Y4
% 0.22/0.57 = ( bit0 @ one ) )
% 0.22/0.57 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ one ) ) ) ) )
% 0.22/0.57 => ( ! [N2: num] :
% 0.22/0.57 ( ( X
% 0.22/0.57 = ( bit0 @ N2 ) )
% 0.22/0.57 => ! [M4: num] :
% 0.22/0.57 ( ( Xa
% 0.22/0.57 = ( bit0 @ M4 ) )
% 0.22/0.57 => ( ( Y4
% 0.22/0.57 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
% 0.22/0.57 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 0.22/0.57 => ( ! [N2: num] :
% 0.22/0.57 ( ( X
% 0.22/0.57 = ( bit0 @ N2 ) )
% 0.22/0.57 => ! [M4: num] :
% 0.22/0.57 ( ( Xa
% 0.22/0.57 = ( bit1 @ M4 ) )
% 0.22/0.57 => ( ( Y4
% 0.22/0.57 = ( bit0 @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
% 0.22/0.57 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ ( bit1 @ M4 ) ) ) ) ) )
% 0.22/0.57 => ( ! [N2: num] :
% 0.22/0.57 ( ( X
% 0.22/0.57 = ( bit1 @ N2 ) )
% 0.22/0.57 => ( ( Xa = one )
% 0.22/0.57 => ( ( Y4 = one )
% 0.22/0.57 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ one ) ) ) ) )
% 0.22/0.57 => ( ! [N2: num] :
% 0.22/0.57 ( ( X
% 0.22/0.57 = ( bit1 @ N2 ) )
% 0.22/0.57 => ! [M4: num] :
% 0.22/0.57 ( ( Xa
% 0.22/0.57 = ( bit0 @ M4 ) )
% 0.22/0.57 => ( ( Y4
% 0.22/0.57 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
% 0.22/0.57 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 0.22/0.57 => ~ ! [N2: num] :
% 0.22/0.57 ( ( X
% 0.22/0.57 = ( bit1 @ N2 ) )
% 0.22/0.57 => ! [M4: num] :
% 0.22/0.57 ( ( Xa
% 0.22/0.57 = ( bit1 @ M4 ) )
% 0.22/0.57 => ( ( Y4
% 0.22/0.57 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
% 0.22/0.57 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ ( bit1 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % or_not_num_neg.pelims
% 0.22/0.57 thf(fact_9265_int__ge__less__than__def,axiom,
% 0.22/0.57 ( int_ge_less_than
% 0.22/0.57 = ( ^ [D4: int] :
% 0.22/0.57 ( collec213857154873943460nt_int
% 0.22/0.57 @ ( produc4947309494688390418_int_o
% 0.22/0.57 @ ^ [Z7: int,Z5: int] :
% 0.22/0.57 ( ( ord_less_eq_int @ D4 @ Z7 )
% 0.22/0.57 & ( ord_less_int @ Z7 @ Z5 ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % int_ge_less_than_def
% 0.22/0.57 thf(fact_9266_int__ge__less__than2__def,axiom,
% 0.22/0.57 ( int_ge_less_than2
% 0.22/0.57 = ( ^ [D4: int] :
% 0.22/0.57 ( collec213857154873943460nt_int
% 0.22/0.57 @ ( produc4947309494688390418_int_o
% 0.22/0.57 @ ^ [Z7: int,Z5: int] :
% 0.22/0.57 ( ( ord_less_eq_int @ D4 @ Z5 )
% 0.22/0.57 & ( ord_less_int @ Z7 @ Z5 ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % int_ge_less_than2_def
% 0.22/0.57 thf(fact_9267_upto_Opinduct,axiom,
% 0.22/0.57 ! [A0: int,A1: int,P: int > int > $o] :
% 0.22/0.57 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
% 0.22/0.57 => ( ! [I3: int,J2: int] :
% 0.22/0.57 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
% 0.22/0.57 => ( ( ( ord_less_eq_int @ I3 @ J2 )
% 0.22/0.57 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
% 0.22/0.57 => ( P @ I3 @ J2 ) ) )
% 0.22/0.57 => ( P @ A0 @ A1 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % upto.pinduct
% 0.22/0.57 thf(fact_9268_Code__Target__Int_Opositive__def,axiom,
% 0.22/0.57 code_Target_positive = numeral_numeral_int ).
% 0.22/0.57
% 0.22/0.57 % Code_Target_Int.positive_def
% 0.22/0.57 thf(fact_9269_divmod__step__integer__def,axiom,
% 0.22/0.57 ( unique4921790084139445826nteger
% 0.22/0.57 = ( ^ [L3: num] :
% 0.22/0.57 ( produc6916734918728496179nteger
% 0.22/0.57 @ ^ [Q3: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L3 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L3 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ R5 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % divmod_step_integer_def
% 0.22/0.57 thf(fact_9270_less__eq__integer__code_I1_J,axiom,
% 0.22/0.57 ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).
% 0.22/0.57
% 0.22/0.57 % less_eq_integer_code(1)
% 0.22/0.57 thf(fact_9271_plus__integer__code_I1_J,axiom,
% 0.22/0.57 ! [K: code_integer] :
% 0.22/0.57 ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
% 0.22/0.57 = K ) ).
% 0.22/0.57
% 0.22/0.57 % plus_integer_code(1)
% 0.22/0.57 thf(fact_9272_plus__integer__code_I2_J,axiom,
% 0.22/0.57 ! [L: code_integer] :
% 0.22/0.57 ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L )
% 0.22/0.57 = L ) ).
% 0.22/0.57
% 0.22/0.57 % plus_integer_code(2)
% 0.22/0.57 thf(fact_9273_times__integer__code_I2_J,axiom,
% 0.22/0.57 ! [L: code_integer] :
% 0.22/0.57 ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
% 0.22/0.57 = zero_z3403309356797280102nteger ) ).
% 0.22/0.57
% 0.22/0.57 % times_integer_code(2)
% 0.22/0.57 thf(fact_9274_times__integer__code_I1_J,axiom,
% 0.22/0.57 ! [K: code_integer] :
% 0.22/0.57 ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
% 0.22/0.57 = zero_z3403309356797280102nteger ) ).
% 0.22/0.57
% 0.22/0.57 % times_integer_code(1)
% 0.22/0.57 thf(fact_9275_minus__integer__code_I2_J,axiom,
% 0.22/0.57 ! [L: code_integer] :
% 0.22/0.57 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
% 0.22/0.57 = ( uminus1351360451143612070nteger @ L ) ) ).
% 0.22/0.57
% 0.22/0.57 % minus_integer_code(2)
% 0.22/0.57 thf(fact_9276_minus__integer__code_I1_J,axiom,
% 0.22/0.57 ! [K: code_integer] :
% 0.22/0.57 ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
% 0.22/0.57 = K ) ).
% 0.22/0.57
% 0.22/0.57 % minus_integer_code(1)
% 0.22/0.57 thf(fact_9277_sgn__integer__code,axiom,
% 0.22/0.57 ( sgn_sgn_Code_integer
% 0.22/0.57 = ( ^ [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 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % sgn_integer_code
% 0.22/0.57 thf(fact_9278_nat_Odisc__eq__case_I2_J,axiom,
% 0.22/0.57 ! [Nat: nat] :
% 0.22/0.57 ( ( Nat != zero_zero_nat )
% 0.22/0.57 = ( case_nat_o @ $false
% 0.22/0.57 @ ^ [Uu3: nat] : $true
% 0.22/0.57 @ Nat ) ) ).
% 0.22/0.57
% 0.22/0.57 % nat.disc_eq_case(2)
% 0.22/0.57 thf(fact_9279_nat_Odisc__eq__case_I1_J,axiom,
% 0.22/0.57 ! [Nat: nat] :
% 0.22/0.57 ( ( Nat = zero_zero_nat )
% 0.22/0.57 = ( case_nat_o @ $true
% 0.22/0.57 @ ^ [Uu3: nat] : $false
% 0.22/0.57 @ Nat ) ) ).
% 0.22/0.57
% 0.22/0.57 % nat.disc_eq_case(1)
% 0.22/0.57 thf(fact_9280_zero__natural_Orsp,axiom,
% 0.22/0.57 zero_zero_nat = zero_zero_nat ).
% 0.22/0.57
% 0.22/0.57 % zero_natural.rsp
% 0.22/0.57 thf(fact_9281_zero__integer_Orsp,axiom,
% 0.22/0.57 zero_zero_int = zero_zero_int ).
% 0.22/0.57
% 0.22/0.57 % zero_integer.rsp
% 0.22/0.57 thf(fact_9282_less__eq__nat_Osimps_I2_J,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.22/0.57 = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 0.22/0.57
% 0.22/0.57 % less_eq_nat.simps(2)
% 0.22/0.57 thf(fact_9283_max__Suc2,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( ord_max_nat @ M @ ( suc @ N ) )
% 0.22/0.57 = ( case_nat_nat @ ( suc @ N )
% 0.22/0.57 @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ M6 @ N ) )
% 0.22/0.57 @ M ) ) ).
% 0.22/0.57
% 0.22/0.57 % max_Suc2
% 0.22/0.57 thf(fact_9284_max__Suc1,axiom,
% 0.22/0.57 ! [N: nat,M: nat] :
% 0.22/0.57 ( ( ord_max_nat @ ( suc @ N ) @ M )
% 0.22/0.57 = ( case_nat_nat @ ( suc @ N )
% 0.22/0.57 @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ N @ M6 ) )
% 0.22/0.57 @ M ) ) ).
% 0.22/0.57
% 0.22/0.57 % max_Suc1
% 0.22/0.57 thf(fact_9285_diff__Suc,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.22/0.57 = ( case_nat_nat @ zero_zero_nat
% 0.22/0.57 @ ^ [K3: nat] : K3
% 0.22/0.57 @ ( minus_minus_nat @ M @ N ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % diff_Suc
% 0.22/0.57 thf(fact_9286_integer__of__int__code,axiom,
% 0.22/0.57 ( code_integer_of_int
% 0.22/0.57 = ( ^ [K3: int] :
% 0.22/0.57 ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
% 0.22/0.57 @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 0.22/0.57 @ ( if_Code_integer
% 0.22/0.57 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.22/0.57 = zero_zero_int )
% 0.22/0.57 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.22/0.57 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % integer_of_int_code
% 0.22/0.57 thf(fact_9287_flip__bit__integer_Oabs__eq,axiom,
% 0.22/0.57 ! [Xa: nat,X: int] :
% 0.22/0.57 ( ( bit_se1345352211410354436nteger @ Xa @ ( code_integer_of_int @ X ) )
% 0.22/0.57 = ( code_integer_of_int @ ( bit_se2159334234014336723it_int @ Xa @ X ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % flip_bit_integer.abs_eq
% 0.22/0.57 thf(fact_9288_zero__integer__def,axiom,
% 0.22/0.57 ( zero_z3403309356797280102nteger
% 0.22/0.57 = ( code_integer_of_int @ zero_zero_int ) ) ).
% 0.22/0.57
% 0.22/0.57 % zero_integer_def
% 0.22/0.57 thf(fact_9289_unset__bit__integer_Oabs__eq,axiom,
% 0.22/0.57 ! [Xa: nat,X: int] :
% 0.22/0.57 ( ( bit_se8260200283734997820nteger @ Xa @ ( code_integer_of_int @ X ) )
% 0.22/0.57 = ( code_integer_of_int @ ( bit_se4203085406695923979it_int @ Xa @ X ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % unset_bit_integer.abs_eq
% 0.22/0.57 thf(fact_9290_divide__integer_Oabs__eq,axiom,
% 0.22/0.57 ! [Xa: int,X: int] :
% 0.22/0.57 ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.22/0.57 = ( code_integer_of_int @ ( divide_divide_int @ Xa @ X ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % divide_integer.abs_eq
% 0.22/0.57 thf(fact_9291_set__bit__integer_Oabs__eq,axiom,
% 0.22/0.57 ! [Xa: nat,X: int] :
% 0.22/0.57 ( ( bit_se2793503036327961859nteger @ Xa @ ( code_integer_of_int @ X ) )
% 0.22/0.57 = ( code_integer_of_int @ ( bit_se7879613467334960850it_int @ Xa @ X ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % set_bit_integer.abs_eq
% 0.22/0.57 thf(fact_9292_uminus__integer__code_I1_J,axiom,
% 0.22/0.57 ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 0.22/0.57 = zero_z3403309356797280102nteger ) ).
% 0.22/0.57
% 0.22/0.57 % uminus_integer_code(1)
% 0.22/0.57 thf(fact_9293_less__integer__code_I1_J,axiom,
% 0.22/0.57 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% 0.22/0.57
% 0.22/0.57 % less_integer_code(1)
% 0.22/0.57 thf(fact_9294_abs__integer__code,axiom,
% 0.22/0.57 ( abs_abs_Code_integer
% 0.22/0.57 = ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % abs_integer_code
% 0.22/0.57 thf(fact_9295_times__integer_Oabs__eq,axiom,
% 0.22/0.57 ! [Xa: int,X: int] :
% 0.22/0.57 ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.22/0.57 = ( code_integer_of_int @ ( times_times_int @ Xa @ X ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % times_integer.abs_eq
% 0.22/0.57 thf(fact_9296_less__eq__integer_Oabs__eq,axiom,
% 0.22/0.57 ! [Xa: int,X: int] :
% 0.22/0.57 ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.22/0.57 = ( ord_less_eq_int @ Xa @ X ) ) ).
% 0.22/0.57
% 0.22/0.57 % less_eq_integer.abs_eq
% 0.22/0.57 thf(fact_9297_minus__integer_Oabs__eq,axiom,
% 0.22/0.57 ! [Xa: int,X: int] :
% 0.22/0.57 ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.22/0.57 = ( code_integer_of_int @ ( minus_minus_int @ Xa @ X ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % minus_integer.abs_eq
% 0.22/0.57 thf(fact_9298_floor__real__def,axiom,
% 0.22/0.57 ( archim6058952711729229775r_real
% 0.22/0.57 = ( ^ [X3: real] :
% 0.22/0.57 ( the_int
% 0.22/0.57 @ ^ [Z5: int] :
% 0.22/0.57 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z5 ) @ X3 )
% 0.22/0.57 & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % floor_real_def
% 0.22/0.57 thf(fact_9299_floor__rat__def,axiom,
% 0.22/0.57 ( archim3151403230148437115or_rat
% 0.22/0.57 = ( ^ [X3: rat] :
% 0.22/0.57 ( the_int
% 0.22/0.57 @ ^ [Z5: int] :
% 0.22/0.57 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z5 ) @ X3 )
% 0.22/0.57 & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % floor_rat_def
% 0.22/0.57 thf(fact_9300_integer__of__num_I3_J,axiom,
% 0.22/0.57 ! [N: num] :
% 0.22/0.57 ( ( code_integer_of_num @ ( bit1 @ N ) )
% 0.22/0.57 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).
% 0.22/0.57
% 0.22/0.57 % integer_of_num(3)
% 0.22/0.57 thf(fact_9301_less__eq__rat__def,axiom,
% 0.22/0.57 ( ord_less_eq_rat
% 0.22/0.57 = ( ^ [X3: rat,Y6: rat] :
% 0.22/0.57 ( ( ord_less_rat @ X3 @ Y6 )
% 0.22/0.57 | ( X3 = Y6 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % less_eq_rat_def
% 0.22/0.57 thf(fact_9302_obtain__pos__sum,axiom,
% 0.22/0.57 ! [R2: rat] :
% 0.22/0.57 ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 0.22/0.57 => ~ ! [S3: rat] :
% 0.22/0.57 ( ( ord_less_rat @ zero_zero_rat @ S3 )
% 0.22/0.57 => ! [T3: rat] :
% 0.22/0.57 ( ( ord_less_rat @ zero_zero_rat @ T3 )
% 0.22/0.57 => ( R2
% 0.22/0.57 != ( plus_plus_rat @ S3 @ T3 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % obtain_pos_sum
% 0.22/0.57 thf(fact_9303_sgn__rat__def,axiom,
% 0.22/0.57 ( sgn_sgn_rat
% 0.22/0.57 = ( ^ [A3: rat] : ( if_rat @ ( A3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % sgn_rat_def
% 0.22/0.57 thf(fact_9304_abs__rat__def,axiom,
% 0.22/0.57 ( abs_abs_rat
% 0.22/0.57 = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % abs_rat_def
% 0.22/0.57 thf(fact_9305_integer__of__num__triv_I1_J,axiom,
% 0.22/0.57 ( ( code_integer_of_num @ one )
% 0.22/0.57 = one_one_Code_integer ) ).
% 0.22/0.57
% 0.22/0.57 % integer_of_num_triv(1)
% 0.22/0.57 thf(fact_9306_integer__of__num_I2_J,axiom,
% 0.22/0.57 ! [N: num] :
% 0.22/0.57 ( ( code_integer_of_num @ ( bit0 @ N ) )
% 0.22/0.57 = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % integer_of_num(2)
% 0.22/0.57 thf(fact_9307_integer__of__num__triv_I2_J,axiom,
% 0.22/0.57 ( ( code_integer_of_num @ ( bit0 @ one ) )
% 0.22/0.57 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % integer_of_num_triv(2)
% 0.22/0.57 thf(fact_9308_pred__def,axiom,
% 0.22/0.57 ( pred
% 0.22/0.57 = ( case_nat_nat @ zero_zero_nat
% 0.22/0.57 @ ^ [X23: nat] : X23 ) ) ).
% 0.22/0.57
% 0.22/0.57 % pred_def
% 0.22/0.57 thf(fact_9309_rat__inverse__code,axiom,
% 0.22/0.57 ! [P5: rat] :
% 0.22/0.57 ( ( quotient_of @ ( inverse_inverse_rat @ P5 ) )
% 0.22/0.57 = ( produc4245557441103728435nt_int
% 0.22/0.57 @ ^ [A3: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( A3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A3 ) @ B2 ) @ ( abs_abs_int @ A3 ) ) )
% 0.22/0.57 @ ( quotient_of @ P5 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % rat_inverse_code
% 0.22/0.57 thf(fact_9310_normalize__negative,axiom,
% 0.22/0.57 ! [Q2: int,P5: int] :
% 0.22/0.57 ( ( ord_less_int @ Q2 @ zero_zero_int )
% 0.22/0.57 => ( ( normalize @ ( product_Pair_int_int @ P5 @ Q2 ) )
% 0.22/0.57 = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P5 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % normalize_negative
% 0.22/0.57 thf(fact_9311_bit__cut__integer__def,axiom,
% 0.22/0.57 ( code_bit_cut_integer
% 0.22/0.57 = ( ^ [K3: code_integer] :
% 0.22/0.57 ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.22/0.57 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bit_cut_integer_def
% 0.22/0.57 thf(fact_9312_quotient__of__number_I3_J,axiom,
% 0.22/0.57 ! [K: num] :
% 0.22/0.57 ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
% 0.22/0.57 = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% 0.22/0.57
% 0.22/0.57 % quotient_of_number(3)
% 0.22/0.57 thf(fact_9313_normalize__denom__zero,axiom,
% 0.22/0.57 ! [P5: int] :
% 0.22/0.57 ( ( normalize @ ( product_Pair_int_int @ P5 @ zero_zero_int ) )
% 0.22/0.57 = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 0.22/0.57
% 0.22/0.57 % normalize_denom_zero
% 0.22/0.57 thf(fact_9314_rat__zero__code,axiom,
% 0.22/0.57 ( ( quotient_of @ zero_zero_rat )
% 0.22/0.57 = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 0.22/0.57
% 0.22/0.57 % rat_zero_code
% 0.22/0.57 thf(fact_9315_quotient__of__number_I5_J,axiom,
% 0.22/0.57 ! [K: num] :
% 0.22/0.57 ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.22/0.57 = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 0.22/0.57
% 0.22/0.57 % quotient_of_number(5)
% 0.22/0.57 thf(fact_9316_diff__rat__def,axiom,
% 0.22/0.57 ( minus_minus_rat
% 0.22/0.57 = ( ^ [Q3: rat,R5: rat] : ( plus_plus_rat @ Q3 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % diff_rat_def
% 0.22/0.57 thf(fact_9317_rat__times__code,axiom,
% 0.22/0.57 ! [P5: rat,Q2: rat] :
% 0.22/0.57 ( ( quotient_of @ ( times_times_rat @ P5 @ Q2 ) )
% 0.22/0.57 = ( produc4245557441103728435nt_int
% 0.22/0.57 @ ^ [A3: int,C5: int] :
% 0.22/0.57 ( produc4245557441103728435nt_int
% 0.22/0.57 @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ B2 ) @ ( times_times_int @ C5 @ D4 ) ) )
% 0.22/0.57 @ ( quotient_of @ Q2 ) )
% 0.22/0.57 @ ( quotient_of @ P5 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % rat_times_code
% 0.22/0.57 thf(fact_9318_rat__divide__code,axiom,
% 0.22/0.57 ! [P5: rat,Q2: rat] :
% 0.22/0.57 ( ( quotient_of @ ( divide_divide_rat @ P5 @ Q2 ) )
% 0.22/0.57 = ( produc4245557441103728435nt_int
% 0.22/0.57 @ ^ [A3: int,C5: int] :
% 0.22/0.57 ( produc4245557441103728435nt_int
% 0.22/0.57 @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ D4 ) @ ( times_times_int @ C5 @ B2 ) ) )
% 0.22/0.57 @ ( quotient_of @ Q2 ) )
% 0.22/0.57 @ ( quotient_of @ P5 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % rat_divide_code
% 0.22/0.57 thf(fact_9319_rat__plus__code,axiom,
% 0.22/0.57 ! [P5: rat,Q2: rat] :
% 0.22/0.57 ( ( quotient_of @ ( plus_plus_rat @ P5 @ Q2 ) )
% 0.22/0.57 = ( produc4245557441103728435nt_int
% 0.22/0.57 @ ^ [A3: int,C5: int] :
% 0.22/0.57 ( produc4245557441103728435nt_int
% 0.22/0.57 @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A3 @ D4 ) @ ( times_times_int @ B2 @ C5 ) ) @ ( times_times_int @ C5 @ D4 ) ) )
% 0.22/0.57 @ ( quotient_of @ Q2 ) )
% 0.22/0.57 @ ( quotient_of @ P5 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % rat_plus_code
% 0.22/0.57 thf(fact_9320_rat__minus__code,axiom,
% 0.22/0.57 ! [P5: rat,Q2: rat] :
% 0.22/0.57 ( ( quotient_of @ ( minus_minus_rat @ P5 @ Q2 ) )
% 0.22/0.57 = ( produc4245557441103728435nt_int
% 0.22/0.57 @ ^ [A3: int,C5: int] :
% 0.22/0.57 ( produc4245557441103728435nt_int
% 0.22/0.57 @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A3 @ D4 ) @ ( times_times_int @ B2 @ C5 ) ) @ ( times_times_int @ C5 @ D4 ) ) )
% 0.22/0.57 @ ( quotient_of @ Q2 ) )
% 0.22/0.57 @ ( quotient_of @ P5 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % rat_minus_code
% 0.22/0.57 thf(fact_9321_quotient__of__denom__pos,axiom,
% 0.22/0.57 ! [R2: rat,P5: int,Q2: int] :
% 0.22/0.57 ( ( ( quotient_of @ R2 )
% 0.22/0.57 = ( product_Pair_int_int @ P5 @ Q2 ) )
% 0.22/0.57 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 0.22/0.57
% 0.22/0.57 % quotient_of_denom_pos
% 0.22/0.57 thf(fact_9322_normalize__denom__pos,axiom,
% 0.22/0.57 ! [R2: product_prod_int_int,P5: int,Q2: int] :
% 0.22/0.57 ( ( ( normalize @ R2 )
% 0.22/0.57 = ( product_Pair_int_int @ P5 @ Q2 ) )
% 0.22/0.57 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 0.22/0.57
% 0.22/0.57 % normalize_denom_pos
% 0.22/0.57 thf(fact_9323_normalize__crossproduct,axiom,
% 0.22/0.57 ! [Q2: int,S2: int,P5: int,R2: int] :
% 0.22/0.57 ( ( Q2 != zero_zero_int )
% 0.22/0.57 => ( ( S2 != zero_zero_int )
% 0.22/0.57 => ( ( ( normalize @ ( product_Pair_int_int @ P5 @ Q2 ) )
% 0.22/0.57 = ( normalize @ ( product_Pair_int_int @ R2 @ S2 ) ) )
% 0.22/0.57 => ( ( times_times_int @ P5 @ S2 )
% 0.22/0.57 = ( times_times_int @ R2 @ Q2 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % normalize_crossproduct
% 0.22/0.57 thf(fact_9324_rat__less__code,axiom,
% 0.22/0.57 ( ord_less_rat
% 0.22/0.57 = ( ^ [P6: rat,Q3: rat] :
% 0.22/0.57 ( produc4947309494688390418_int_o
% 0.22/0.57 @ ^ [A3: int,C5: int] :
% 0.22/0.57 ( produc4947309494688390418_int_o
% 0.22/0.57 @ ^ [B2: int,D4: int] : ( ord_less_int @ ( times_times_int @ A3 @ D4 ) @ ( times_times_int @ C5 @ B2 ) )
% 0.22/0.57 @ ( quotient_of @ Q3 ) )
% 0.22/0.57 @ ( quotient_of @ P6 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % rat_less_code
% 0.22/0.57 thf(fact_9325_rat__less__eq__code,axiom,
% 0.22/0.57 ( ord_less_eq_rat
% 0.22/0.57 = ( ^ [P6: rat,Q3: rat] :
% 0.22/0.57 ( produc4947309494688390418_int_o
% 0.22/0.57 @ ^ [A3: int,C5: int] :
% 0.22/0.57 ( produc4947309494688390418_int_o
% 0.22/0.57 @ ^ [B2: int,D4: int] : ( ord_less_eq_int @ ( times_times_int @ A3 @ D4 ) @ ( times_times_int @ C5 @ B2 ) )
% 0.22/0.57 @ ( quotient_of @ Q3 ) )
% 0.22/0.57 @ ( quotient_of @ P6 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % rat_less_eq_code
% 0.22/0.57 thf(fact_9326_bit__cut__integer__code,axiom,
% 0.22/0.57 ( code_bit_cut_integer
% 0.22/0.57 = ( ^ [K3: code_integer] :
% 0.22/0.57 ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 0.22/0.57 @ ( produc9125791028180074456eger_o
% 0.22/0.57 @ ^ [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 ) )
% 0.22/0.57 @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bit_cut_integer_code
% 0.22/0.57 thf(fact_9327_bezw__0,axiom,
% 0.22/0.57 ! [X: nat] :
% 0.22/0.57 ( ( bezw @ X @ zero_zero_nat )
% 0.22/0.57 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% 0.22/0.57
% 0.22/0.57 % bezw_0
% 0.22/0.57 thf(fact_9328_set__encode__inverse,axiom,
% 0.22/0.57 ! [A2: set_nat] :
% 0.22/0.57 ( ( finite_finite_nat @ A2 )
% 0.22/0.57 => ( ( nat_set_decode @ ( nat_set_encode @ A2 ) )
% 0.22/0.57 = A2 ) ) ).
% 0.22/0.57
% 0.22/0.57 % set_encode_inverse
% 0.22/0.57 thf(fact_9329_finite__set__decode,axiom,
% 0.22/0.57 ! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).
% 0.22/0.57
% 0.22/0.57 % finite_set_decode
% 0.22/0.57 thf(fact_9330_set__encode__eq,axiom,
% 0.22/0.57 ! [A2: set_nat,B4: set_nat] :
% 0.22/0.57 ( ( finite_finite_nat @ A2 )
% 0.22/0.57 => ( ( finite_finite_nat @ B4 )
% 0.22/0.57 => ( ( ( nat_set_encode @ A2 )
% 0.22/0.57 = ( nat_set_encode @ B4 ) )
% 0.22/0.57 = ( A2 = B4 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % set_encode_eq
% 0.22/0.57 thf(fact_9331_finite__nat__set__iff__bounded__le,axiom,
% 0.22/0.57 ( finite_finite_nat
% 0.22/0.57 = ( ^ [N8: set_nat] :
% 0.22/0.57 ? [M5: nat] :
% 0.22/0.57 ! [X3: nat] :
% 0.22/0.57 ( ( member_nat @ X3 @ N8 )
% 0.22/0.57 => ( ord_less_eq_nat @ X3 @ M5 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % finite_nat_set_iff_bounded_le
% 0.22/0.57 thf(fact_9332_finite__less__ub,axiom,
% 0.22/0.57 ! [F: nat > nat,U: nat] :
% 0.22/0.57 ( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
% 0.22/0.57 => ( finite_finite_nat
% 0.22/0.57 @ ( collect_nat
% 0.22/0.57 @ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % finite_less_ub
% 0.22/0.57 thf(fact_9333_set__encode__inf,axiom,
% 0.22/0.57 ! [A2: set_nat] :
% 0.22/0.57 ( ~ ( finite_finite_nat @ A2 )
% 0.22/0.57 => ( ( nat_set_encode @ A2 )
% 0.22/0.57 = zero_zero_nat ) ) ).
% 0.22/0.57
% 0.22/0.57 % set_encode_inf
% 0.22/0.57 thf(fact_9334_finite__divisors__nat,axiom,
% 0.22/0.57 ! [M: nat] :
% 0.22/0.57 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.22/0.57 => ( finite_finite_nat
% 0.22/0.57 @ ( collect_nat
% 0.22/0.57 @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % finite_divisors_nat
% 0.22/0.57 thf(fact_9335_subset__eq__atLeast0__atMost__finite,axiom,
% 0.22/0.57 ! [N5: set_nat,N: nat] :
% 0.22/0.57 ( ( ord_less_eq_set_nat @ N5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.22/0.57 => ( finite_finite_nat @ N5 ) ) ).
% 0.22/0.57
% 0.22/0.57 % subset_eq_atLeast0_atMost_finite
% 0.22/0.57 thf(fact_9336_subset__eq__atLeast0__lessThan__finite,axiom,
% 0.22/0.57 ! [N5: set_nat,N: nat] :
% 0.22/0.57 ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.22/0.57 => ( finite_finite_nat @ N5 ) ) ).
% 0.22/0.57
% 0.22/0.57 % subset_eq_atLeast0_lessThan_finite
% 0.22/0.57 thf(fact_9337_divmod__abs__code_I6_J,axiom,
% 0.22/0.57 ! [J: code_integer] :
% 0.22/0.57 ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
% 0.22/0.57 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).
% 0.22/0.57
% 0.22/0.57 % divmod_abs_code(6)
% 0.22/0.57 thf(fact_9338_divmod__abs__code_I5_J,axiom,
% 0.22/0.57 ! [J: code_integer] :
% 0.22/0.57 ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
% 0.22/0.57 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % divmod_abs_code(5)
% 0.22/0.57 thf(fact_9339_even__set__encode__iff,axiom,
% 0.22/0.57 ! [A2: set_nat] :
% 0.22/0.57 ( ( finite_finite_nat @ A2 )
% 0.22/0.57 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
% 0.22/0.57 = ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % even_set_encode_iff
% 0.22/0.57 thf(fact_9340_finite__Collect__le__nat,axiom,
% 0.22/0.57 ! [K: nat] :
% 0.22/0.57 ( finite_finite_nat
% 0.22/0.57 @ ( collect_nat
% 0.22/0.57 @ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % finite_Collect_le_nat
% 0.22/0.57 thf(fact_9341_finite__interval__int1,axiom,
% 0.22/0.57 ! [A: int,B: int] :
% 0.22/0.57 ( finite_finite_int
% 0.22/0.57 @ ( collect_int
% 0.22/0.57 @ ^ [I2: int] :
% 0.22/0.57 ( ( ord_less_eq_int @ A @ I2 )
% 0.22/0.57 & ( ord_less_eq_int @ I2 @ B ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % finite_interval_int1
% 0.22/0.57 thf(fact_9342_finite__interval__int4,axiom,
% 0.22/0.57 ! [A: int,B: int] :
% 0.22/0.57 ( finite_finite_int
% 0.22/0.57 @ ( collect_int
% 0.22/0.57 @ ^ [I2: int] :
% 0.22/0.57 ( ( ord_less_int @ A @ I2 )
% 0.22/0.57 & ( ord_less_int @ I2 @ B ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % finite_interval_int4
% 0.22/0.57 thf(fact_9343_finite__interval__int2,axiom,
% 0.22/0.57 ! [A: int,B: int] :
% 0.22/0.57 ( finite_finite_int
% 0.22/0.57 @ ( collect_int
% 0.22/0.57 @ ^ [I2: int] :
% 0.22/0.57 ( ( ord_less_eq_int @ A @ I2 )
% 0.22/0.57 & ( ord_less_int @ I2 @ B ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % finite_interval_int2
% 0.22/0.57 thf(fact_9344_finite__interval__int3,axiom,
% 0.22/0.57 ! [A: int,B: int] :
% 0.22/0.57 ( finite_finite_int
% 0.22/0.57 @ ( collect_int
% 0.22/0.57 @ ^ [I2: int] :
% 0.22/0.57 ( ( ord_less_int @ A @ I2 )
% 0.22/0.57 & ( ord_less_eq_int @ I2 @ B ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % finite_interval_int3
% 0.22/0.57 thf(fact_9345_finite__nth__roots,axiom,
% 0.22/0.57 ! [N: nat,C: complex] :
% 0.22/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.57 => ( finite3207457112153483333omplex
% 0.22/0.57 @ ( collect_complex
% 0.22/0.57 @ ^ [Z5: complex] :
% 0.22/0.57 ( ( power_power_complex @ Z5 @ N )
% 0.22/0.57 = C ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % finite_nth_roots
% 0.22/0.57 thf(fact_9346_finite__atLeastZeroLessThan__int,axiom,
% 0.22/0.57 ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% 0.22/0.57
% 0.22/0.57 % finite_atLeastZeroLessThan_int
% 0.22/0.57 thf(fact_9347_finite__divisors__int,axiom,
% 0.22/0.57 ! [I: int] :
% 0.22/0.57 ( ( I != zero_zero_int )
% 0.22/0.57 => ( finite_finite_int
% 0.22/0.57 @ ( collect_int
% 0.22/0.57 @ ^ [D4: int] : ( dvd_dvd_int @ D4 @ I ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % finite_divisors_int
% 0.22/0.57 thf(fact_9348_finite__nat__iff__bounded__le,axiom,
% 0.22/0.57 ( finite_finite_nat
% 0.22/0.57 = ( ^ [S5: set_nat] :
% 0.22/0.57 ? [K3: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % finite_nat_iff_bounded_le
% 0.22/0.57 thf(fact_9349_finite__nat__bounded,axiom,
% 0.22/0.57 ! [S: set_nat] :
% 0.22/0.57 ( ( finite_finite_nat @ S )
% 0.22/0.57 => ? [K2: nat] : ( ord_less_eq_set_nat @ S @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % finite_nat_bounded
% 0.22/0.57 thf(fact_9350_finite__nat__iff__bounded,axiom,
% 0.22/0.57 ( finite_finite_nat
% 0.22/0.57 = ( ^ [S5: set_nat] :
% 0.22/0.57 ? [K3: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % finite_nat_iff_bounded
% 0.22/0.57 thf(fact_9351_infinite__int__iff__unbounded__le,axiom,
% 0.22/0.57 ! [S: set_int] :
% 0.22/0.57 ( ( ~ ( finite_finite_int @ S ) )
% 0.22/0.57 = ( ! [M5: int] :
% 0.22/0.57 ? [N4: int] :
% 0.22/0.57 ( ( ord_less_eq_int @ M5 @ ( abs_abs_int @ N4 ) )
% 0.22/0.57 & ( member_int @ N4 @ S ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % infinite_int_iff_unbounded_le
% 0.22/0.57 thf(fact_9352_infinite__nat__iff__unbounded__le,axiom,
% 0.22/0.57 ! [S: set_nat] :
% 0.22/0.57 ( ( ~ ( finite_finite_nat @ S ) )
% 0.22/0.57 = ( ! [M5: nat] :
% 0.22/0.57 ? [N4: nat] :
% 0.22/0.57 ( ( ord_less_eq_nat @ M5 @ N4 )
% 0.22/0.57 & ( member_nat @ N4 @ S ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % infinite_nat_iff_unbounded_le
% 0.22/0.57 thf(fact_9353_Frct__code__post_I5_J,axiom,
% 0.22/0.57 ! [K: num] :
% 0.22/0.57 ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 0.22/0.57 = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Frct_code_post(5)
% 0.22/0.57 thf(fact_9354_set__encode__insert,axiom,
% 0.22/0.57 ! [A2: set_nat,N: nat] :
% 0.22/0.57 ( ( finite_finite_nat @ A2 )
% 0.22/0.57 => ( ~ ( member_nat @ N @ A2 )
% 0.22/0.57 => ( ( nat_set_encode @ ( insert_nat @ N @ A2 ) )
% 0.22/0.57 = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % set_encode_insert
% 0.22/0.57 thf(fact_9355_lessThan__Suc,axiom,
% 0.22/0.57 ! [K: nat] :
% 0.22/0.57 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 0.22/0.57 = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % lessThan_Suc
% 0.22/0.57 thf(fact_9356_atMost__Suc,axiom,
% 0.22/0.57 ! [K: nat] :
% 0.22/0.57 ( ( set_ord_atMost_nat @ ( suc @ K ) )
% 0.22/0.57 = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % atMost_Suc
% 0.22/0.57 thf(fact_9357_atLeast0__atMost__Suc,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.22/0.57 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % atLeast0_atMost_Suc
% 0.22/0.57 thf(fact_9358_atLeast0__lessThan__Suc,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.22/0.57 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % atLeast0_lessThan_Suc
% 0.22/0.57 thf(fact_9359_atLeastAtMost__insertL,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.57 => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.22/0.57 = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % atLeastAtMost_insertL
% 0.22/0.57 thf(fact_9360_atLeastAtMostSuc__conv,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.22/0.57 => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
% 0.22/0.57 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % atLeastAtMostSuc_conv
% 0.22/0.57 thf(fact_9361_Icc__eq__insert__lb__nat,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.57 => ( ( set_or1269000886237332187st_nat @ M @ N )
% 0.22/0.57 = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Icc_eq_insert_lb_nat
% 0.22/0.57 thf(fact_9362_lessThan__nat__numeral,axiom,
% 0.22/0.57 ! [K: num] :
% 0.22/0.57 ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
% 0.22/0.57 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % lessThan_nat_numeral
% 0.22/0.57 thf(fact_9363_atMost__nat__numeral,axiom,
% 0.22/0.57 ! [K: num] :
% 0.22/0.57 ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
% 0.22/0.57 = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % atMost_nat_numeral
% 0.22/0.57 thf(fact_9364_Frct__code__post_I2_J,axiom,
% 0.22/0.57 ! [A: int] :
% 0.22/0.57 ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
% 0.22/0.57 = zero_zero_rat ) ).
% 0.22/0.57
% 0.22/0.57 % Frct_code_post(2)
% 0.22/0.57 thf(fact_9365_Frct__code__post_I1_J,axiom,
% 0.22/0.57 ! [A: int] :
% 0.22/0.57 ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
% 0.22/0.57 = zero_zero_rat ) ).
% 0.22/0.57
% 0.22/0.57 % Frct_code_post(1)
% 0.22/0.57 thf(fact_9366_set__decode__plus__power__2,axiom,
% 0.22/0.57 ! [N: nat,Z: nat] :
% 0.22/0.57 ( ~ ( member_nat @ N @ ( nat_set_decode @ Z ) )
% 0.22/0.57 => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z ) )
% 0.22/0.57 = ( insert_nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % set_decode_plus_power_2
% 0.22/0.57 thf(fact_9367_Frct__code__post_I4_J,axiom,
% 0.22/0.57 ! [K: num] :
% 0.22/0.57 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
% 0.22/0.57 = ( numeral_numeral_rat @ K ) ) ).
% 0.22/0.57
% 0.22/0.57 % Frct_code_post(4)
% 0.22/0.57 thf(fact_9368_Frct__code__post_I6_J,axiom,
% 0.22/0.57 ! [K: num,L: num] :
% 0.22/0.57 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
% 0.22/0.57 = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Frct_code_post(6)
% 0.22/0.57 thf(fact_9369_divmod__integer__code,axiom,
% 0.22/0.57 ( code_divmod_integer
% 0.22/0.57 = ( ^ [K3: code_integer,L3: code_integer] :
% 0.22/0.57 ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.22/0.57 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L3 )
% 0.22/0.57 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L3 )
% 0.22/0.57 @ ( produc6916734918728496179nteger
% 0.22/0.57 @ ^ [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 @ L3 @ S4 ) ) )
% 0.22/0.57 @ ( code_divmod_abs @ K3 @ L3 ) ) )
% 0.22/0.57 @ ( if_Pro6119634080678213985nteger @ ( L3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 0.22/0.57 @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 0.22/0.57 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L3 )
% 0.22/0.57 @ ( produc6916734918728496179nteger
% 0.22/0.57 @ ^ [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 @ L3 ) @ S4 ) ) )
% 0.22/0.57 @ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % divmod_integer_code
% 0.22/0.57 thf(fact_9370_card__atMost,axiom,
% 0.22/0.57 ! [U: nat] :
% 0.22/0.57 ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
% 0.22/0.57 = ( suc @ U ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_atMost
% 0.22/0.57 thf(fact_9371_card__atLeastLessThan,axiom,
% 0.22/0.57 ! [L: nat,U: nat] :
% 0.22/0.57 ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
% 0.22/0.57 = ( minus_minus_nat @ U @ L ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_atLeastLessThan
% 0.22/0.57 thf(fact_9372_card__Collect__le__nat,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( finite_card_nat
% 0.22/0.57 @ ( collect_nat
% 0.22/0.57 @ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N ) ) )
% 0.22/0.57 = ( suc @ N ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_Collect_le_nat
% 0.22/0.57 thf(fact_9373_card__atLeastAtMost,axiom,
% 0.22/0.57 ! [L: nat,U: nat] :
% 0.22/0.57 ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 0.22/0.57 = ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_atLeastAtMost
% 0.22/0.57 thf(fact_9374_card__atLeastLessThan__int,axiom,
% 0.22/0.57 ! [L: int,U: int] :
% 0.22/0.57 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L @ U ) )
% 0.22/0.57 = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_atLeastLessThan_int
% 0.22/0.57 thf(fact_9375_card__atLeastAtMost__int,axiom,
% 0.22/0.57 ! [L: int,U: int] :
% 0.22/0.57 ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
% 0.22/0.57 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_atLeastAtMost_int
% 0.22/0.57 thf(fact_9376_card__length__sum__list__rec,axiom,
% 0.22/0.57 ! [M: nat,N5: nat] :
% 0.22/0.57 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.22/0.57 => ( ( finite_card_list_nat
% 0.22/0.57 @ ( collect_list_nat
% 0.22/0.57 @ ^ [L3: list_nat] :
% 0.22/0.57 ( ( ( size_size_list_nat @ L3 )
% 0.22/0.57 = M )
% 0.22/0.57 & ( ( groups4561878855575611511st_nat @ L3 )
% 0.22/0.57 = N5 ) ) ) )
% 0.22/0.57 = ( plus_plus_nat
% 0.22/0.57 @ ( finite_card_list_nat
% 0.22/0.57 @ ( collect_list_nat
% 0.22/0.57 @ ^ [L3: list_nat] :
% 0.22/0.57 ( ( ( size_size_list_nat @ L3 )
% 0.22/0.57 = ( minus_minus_nat @ M @ one_one_nat ) )
% 0.22/0.57 & ( ( groups4561878855575611511st_nat @ L3 )
% 0.22/0.57 = N5 ) ) ) )
% 0.22/0.57 @ ( finite_card_list_nat
% 0.22/0.57 @ ( collect_list_nat
% 0.22/0.57 @ ^ [L3: list_nat] :
% 0.22/0.57 ( ( ( size_size_list_nat @ L3 )
% 0.22/0.57 = M )
% 0.22/0.57 & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L3 ) @ one_one_nat )
% 0.22/0.57 = N5 ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_length_sum_list_rec
% 0.22/0.57 thf(fact_9377_card__length__sum__list,axiom,
% 0.22/0.57 ! [M: nat,N5: nat] :
% 0.22/0.57 ( ( finite_card_list_nat
% 0.22/0.57 @ ( collect_list_nat
% 0.22/0.57 @ ^ [L3: list_nat] :
% 0.22/0.57 ( ( ( size_size_list_nat @ L3 )
% 0.22/0.57 = M )
% 0.22/0.57 & ( ( groups4561878855575611511st_nat @ L3 )
% 0.22/0.57 = N5 ) ) ) )
% 0.22/0.57 = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N5 @ M ) @ one_one_nat ) @ N5 ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_length_sum_list
% 0.22/0.57 thf(fact_9378_card__less__Suc2,axiom,
% 0.22/0.57 ! [M10: set_nat,I: nat] :
% 0.22/0.57 ( ~ ( member_nat @ zero_zero_nat @ M10 )
% 0.22/0.57 => ( ( finite_card_nat
% 0.22/0.57 @ ( collect_nat
% 0.22/0.57 @ ^ [K3: nat] :
% 0.22/0.57 ( ( member_nat @ ( suc @ K3 ) @ M10 )
% 0.22/0.57 & ( ord_less_nat @ K3 @ I ) ) ) )
% 0.22/0.57 = ( finite_card_nat
% 0.22/0.57 @ ( collect_nat
% 0.22/0.57 @ ^ [K3: nat] :
% 0.22/0.57 ( ( member_nat @ K3 @ M10 )
% 0.22/0.57 & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_less_Suc2
% 0.22/0.57 thf(fact_9379_card__less__Suc,axiom,
% 0.22/0.57 ! [M10: set_nat,I: nat] :
% 0.22/0.57 ( ( member_nat @ zero_zero_nat @ M10 )
% 0.22/0.57 => ( ( suc
% 0.22/0.57 @ ( finite_card_nat
% 0.22/0.57 @ ( collect_nat
% 0.22/0.57 @ ^ [K3: nat] :
% 0.22/0.57 ( ( member_nat @ ( suc @ K3 ) @ M10 )
% 0.22/0.57 & ( ord_less_nat @ K3 @ I ) ) ) ) )
% 0.22/0.57 = ( finite_card_nat
% 0.22/0.57 @ ( collect_nat
% 0.22/0.57 @ ^ [K3: nat] :
% 0.22/0.57 ( ( member_nat @ K3 @ M10 )
% 0.22/0.57 & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_less_Suc
% 0.22/0.57 thf(fact_9380_card__less,axiom,
% 0.22/0.57 ! [M10: set_nat,I: nat] :
% 0.22/0.57 ( ( member_nat @ zero_zero_nat @ M10 )
% 0.22/0.57 => ( ( finite_card_nat
% 0.22/0.57 @ ( collect_nat
% 0.22/0.57 @ ^ [K3: nat] :
% 0.22/0.57 ( ( member_nat @ K3 @ M10 )
% 0.22/0.57 & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
% 0.22/0.57 != zero_zero_nat ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_less
% 0.22/0.57 thf(fact_9381_card__atLeastZeroLessThan__int,axiom,
% 0.22/0.57 ! [U: int] :
% 0.22/0.57 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
% 0.22/0.57 = ( nat2 @ U ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_atLeastZeroLessThan_int
% 0.22/0.57 thf(fact_9382_subset__card__intvl__is__intvl,axiom,
% 0.22/0.57 ! [A2: set_nat,K: nat] :
% 0.22/0.57 ( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
% 0.22/0.57 => ( A2
% 0.22/0.57 = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % subset_card_intvl_is_intvl
% 0.22/0.57 thf(fact_9383_atLeastAtMostPlus1__int__conv,axiom,
% 0.22/0.57 ! [M: int,N: int] :
% 0.22/0.57 ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 0.22/0.57 => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 0.22/0.57 = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % atLeastAtMostPlus1_int_conv
% 0.22/0.57 thf(fact_9384_subset__eq__atLeast0__lessThan__card,axiom,
% 0.22/0.57 ! [N5: set_nat,N: nat] :
% 0.22/0.57 ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.22/0.57 => ( ord_less_eq_nat @ ( finite_card_nat @ N5 ) @ N ) ) ).
% 0.22/0.57
% 0.22/0.57 % subset_eq_atLeast0_lessThan_card
% 0.22/0.57 thf(fact_9385_card__sum__le__nat__sum,axiom,
% 0.22/0.57 ! [S: set_nat] :
% 0.22/0.57 ( ord_less_eq_nat
% 0.22/0.57 @ ( groups3542108847815614940at_nat
% 0.22/0.57 @ ^ [X3: nat] : X3
% 0.22/0.57 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S ) ) )
% 0.22/0.57 @ ( groups3542108847815614940at_nat
% 0.22/0.57 @ ^ [X3: nat] : X3
% 0.22/0.57 @ S ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_sum_le_nat_sum
% 0.22/0.57 thf(fact_9386_card__nth__roots,axiom,
% 0.22/0.57 ! [C: complex,N: nat] :
% 0.22/0.57 ( ( C != zero_zero_complex )
% 0.22/0.57 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.57 => ( ( finite_card_complex
% 0.22/0.57 @ ( collect_complex
% 0.22/0.57 @ ^ [Z5: complex] :
% 0.22/0.57 ( ( power_power_complex @ Z5 @ N )
% 0.22/0.57 = C ) ) )
% 0.22/0.57 = N ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_nth_roots
% 0.22/0.57 thf(fact_9387_card__roots__unity__eq,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.57 => ( ( finite_card_complex
% 0.22/0.57 @ ( collect_complex
% 0.22/0.57 @ ^ [Z5: complex] :
% 0.22/0.57 ( ( power_power_complex @ Z5 @ N )
% 0.22/0.57 = one_one_complex ) ) )
% 0.22/0.57 = N ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_roots_unity_eq
% 0.22/0.57 thf(fact_9388_and__int_Osimps,axiom,
% 0.22/0.57 ( bit_se725231765392027082nd_int
% 0.22/0.57 = ( ^ [K3: int,L3: int] :
% 0.22/0.57 ( if_int
% 0.22/0.57 @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.22/0.57 & ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.22/0.57 @ ( uminus_uminus_int
% 0.22/0.57 @ ( zero_n2684676970156552555ol_int
% 0.22/0.57 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.22/0.57 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) ) )
% 0.22/0.57 @ ( plus_plus_int
% 0.22/0.57 @ ( zero_n2684676970156552555ol_int
% 0.22/0.57 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.22/0.57 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
% 0.22/0.57 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % and_int.simps
% 0.22/0.57 thf(fact_9389_and__int_Oelims,axiom,
% 0.22/0.57 ! [X: int,Xa: int,Y4: int] :
% 0.22/0.57 ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 0.22/0.57 = Y4 )
% 0.22/0.57 => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.22/0.57 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.22/0.57 => ( Y4
% 0.22/0.57 = ( uminus_uminus_int
% 0.22/0.57 @ ( zero_n2684676970156552555ol_int
% 0.22/0.57 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.22/0.57 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 0.22/0.57 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.22/0.57 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.22/0.57 => ( Y4
% 0.22/0.57 = ( plus_plus_int
% 0.22/0.57 @ ( zero_n2684676970156552555ol_int
% 0.22/0.57 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.22/0.57 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 0.22/0.57 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % and_int.elims
% 0.22/0.57 thf(fact_9390_and__int_Opsimps,axiom,
% 0.22/0.57 ! [K: int,L: int] :
% 0.22/0.57 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
% 0.22/0.57 => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.22/0.57 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.22/0.57 => ( ( bit_se725231765392027082nd_int @ K @ L )
% 0.22/0.57 = ( uminus_uminus_int
% 0.22/0.57 @ ( zero_n2684676970156552555ol_int
% 0.22/0.57 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.22/0.57 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
% 0.22/0.57 & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.22/0.57 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.22/0.57 => ( ( bit_se725231765392027082nd_int @ K @ L )
% 0.22/0.57 = ( plus_plus_int
% 0.22/0.57 @ ( zero_n2684676970156552555ol_int
% 0.22/0.57 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.22/0.57 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 0.22/0.57 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % and_int.psimps
% 0.22/0.57 thf(fact_9391_and__int_Opelims,axiom,
% 0.22/0.57 ! [X: int,Xa: int,Y4: int] :
% 0.22/0.57 ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 0.22/0.57 = Y4 )
% 0.22/0.57 => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) )
% 0.22/0.57 => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.22/0.57 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.22/0.57 => ( Y4
% 0.22/0.57 = ( uminus_uminus_int
% 0.22/0.57 @ ( zero_n2684676970156552555ol_int
% 0.22/0.57 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.22/0.57 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 0.22/0.57 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.22/0.57 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.22/0.57 => ( Y4
% 0.22/0.57 = ( plus_plus_int
% 0.22/0.57 @ ( zero_n2684676970156552555ol_int
% 0.22/0.57 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.22/0.57 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 0.22/0.57 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.22/0.57 => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % and_int.pelims
% 0.22/0.57 thf(fact_9392_and__int_Opinduct,axiom,
% 0.22/0.57 ! [A0: int,A1: int,P: int > int > $o] :
% 0.22/0.57 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
% 0.22/0.57 => ( ! [K2: int,L2: int] :
% 0.22/0.57 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L2 ) )
% 0.22/0.57 => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.22/0.57 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.22/0.57 => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.22/0.57 => ( P @ K2 @ L2 ) ) )
% 0.22/0.57 => ( P @ A0 @ A1 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % and_int.pinduct
% 0.22/0.57 thf(fact_9393_lessThan__0,axiom,
% 0.22/0.57 ( ( set_ord_lessThan_nat @ zero_zero_nat )
% 0.22/0.57 = bot_bot_set_nat ) ).
% 0.22/0.57
% 0.22/0.57 % lessThan_0
% 0.22/0.57 thf(fact_9394_set__decode__zero,axiom,
% 0.22/0.57 ( ( nat_set_decode @ zero_zero_nat )
% 0.22/0.57 = bot_bot_set_nat ) ).
% 0.22/0.57
% 0.22/0.57 % set_decode_zero
% 0.22/0.57 thf(fact_9395_set__encode__empty,axiom,
% 0.22/0.57 ( ( nat_set_encode @ bot_bot_set_nat )
% 0.22/0.57 = zero_zero_nat ) ).
% 0.22/0.57
% 0.22/0.57 % set_encode_empty
% 0.22/0.57 thf(fact_9396_atLeastLessThan__singleton,axiom,
% 0.22/0.57 ! [M: nat] :
% 0.22/0.57 ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
% 0.22/0.57 = ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% 0.22/0.57
% 0.22/0.57 % atLeastLessThan_singleton
% 0.22/0.57 thf(fact_9397_atMost__0,axiom,
% 0.22/0.57 ( ( set_ord_atMost_nat @ zero_zero_nat )
% 0.22/0.57 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% 0.22/0.57
% 0.22/0.57 % atMost_0
% 0.22/0.57 thf(fact_9398_bot__enat__def,axiom,
% 0.22/0.57 bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% 0.22/0.57
% 0.22/0.57 % bot_enat_def
% 0.22/0.57 thf(fact_9399_bot__nat__def,axiom,
% 0.22/0.57 bot_bot_nat = zero_zero_nat ).
% 0.22/0.57
% 0.22/0.57 % bot_nat_def
% 0.22/0.57 thf(fact_9400_atLeastLessThan0,axiom,
% 0.22/0.57 ! [M: nat] :
% 0.22/0.57 ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
% 0.22/0.57 = bot_bot_set_nat ) ).
% 0.22/0.57
% 0.22/0.57 % atLeastLessThan0
% 0.22/0.57 thf(fact_9401_lessThan__empty__iff,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( ( set_ord_lessThan_nat @ N )
% 0.22/0.57 = bot_bot_set_nat )
% 0.22/0.57 = ( N = zero_zero_nat ) ) ).
% 0.22/0.57
% 0.22/0.57 % lessThan_empty_iff
% 0.22/0.57 thf(fact_9402_atLeastLessThanSuc,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.57 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 0.22/0.57 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
% 0.22/0.57 & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.22/0.57 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 0.22/0.57 = bot_bot_set_nat ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % atLeastLessThanSuc
% 0.22/0.57 thf(fact_9403_atLeast1__lessThan__eq__remove0,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.22/0.57 = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % atLeast1_lessThan_eq_remove0
% 0.22/0.57 thf(fact_9404_atLeast1__atMost__eq__remove0,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.22/0.57 = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % atLeast1_atMost_eq_remove0
% 0.22/0.57 thf(fact_9405_atLeastLessThan__nat__numeral,axiom,
% 0.22/0.57 ! [M: nat,K: num] :
% 0.22/0.57 ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.22/0.57 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.22/0.57 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 0.22/0.57 & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.22/0.57 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.22/0.57 = bot_bot_set_nat ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % atLeastLessThan_nat_numeral
% 0.22/0.57 thf(fact_9406_length__upt,axiom,
% 0.22/0.57 ! [I: nat,J: nat] :
% 0.22/0.57 ( ( size_size_list_nat @ ( upt @ I @ J ) )
% 0.22/0.57 = ( minus_minus_nat @ J @ I ) ) ).
% 0.22/0.57
% 0.22/0.57 % length_upt
% 0.22/0.57 thf(fact_9407_sum__list__upt,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.57 => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
% 0.22/0.57 = ( groups3542108847815614940at_nat
% 0.22/0.57 @ ^ [X3: nat] : X3
% 0.22/0.57 @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % sum_list_upt
% 0.22/0.57 thf(fact_9408_map__Suc__upt,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
% 0.22/0.57 = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % map_Suc_upt
% 0.22/0.57 thf(fact_9409_map__add__upt,axiom,
% 0.22/0.57 ! [N: nat,M: nat] :
% 0.22/0.57 ( ( map_nat_nat
% 0.22/0.57 @ ^ [I2: nat] : ( plus_plus_nat @ I2 @ N )
% 0.22/0.57 @ ( upt @ zero_zero_nat @ M ) )
% 0.22/0.57 = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % map_add_upt
% 0.22/0.57 thf(fact_9410_map__decr__upt,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( map_nat_nat
% 0.22/0.57 @ ^ [N4: nat] : ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) )
% 0.22/0.57 @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.22/0.57 = ( upt @ M @ N ) ) ).
% 0.22/0.57
% 0.22/0.57 % map_decr_upt
% 0.22/0.57 thf(fact_9411_atLeastAtMost__upt,axiom,
% 0.22/0.57 ( set_or1269000886237332187st_nat
% 0.22/0.57 = ( ^ [N4: nat,M5: nat] : ( set_nat2 @ ( upt @ N4 @ ( suc @ M5 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % atLeastAtMost_upt
% 0.22/0.57 thf(fact_9412_atLeast__upt,axiom,
% 0.22/0.57 ( set_ord_lessThan_nat
% 0.22/0.57 = ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N4 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % atLeast_upt
% 0.22/0.57 thf(fact_9413_atMost__upto,axiom,
% 0.22/0.57 ( set_ord_atMost_nat
% 0.22/0.57 = ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N4 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % atMost_upto
% 0.22/0.57 thf(fact_9414_sorted__upt,axiom,
% 0.22/0.57 ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% 0.22/0.57
% 0.22/0.57 % sorted_upt
% 0.22/0.57 thf(fact_9415_sorted__wrt__less__idx,axiom,
% 0.22/0.57 ! [Ns: list_nat,I: nat] :
% 0.22/0.57 ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
% 0.22/0.57 => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
% 0.22/0.57 => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % sorted_wrt_less_idx
% 0.22/0.57 thf(fact_9416_binomial__def,axiom,
% 0.22/0.57 ( binomial
% 0.22/0.57 = ( ^ [N4: nat,K3: nat] :
% 0.22/0.57 ( finite_card_set_nat
% 0.22/0.57 @ ( collect_set_nat
% 0.22/0.57 @ ^ [K7: set_nat] :
% 0.22/0.57 ( ( member_set_nat @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) )
% 0.22/0.57 & ( ( finite_card_nat @ K7 )
% 0.22/0.57 = K3 ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % binomial_def
% 0.22/0.57 thf(fact_9417_Suc__0__div__numeral,axiom,
% 0.22/0.57 ! [K: num] :
% 0.22/0.57 ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.22/0.57 = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Suc_0_div_numeral
% 0.22/0.57 thf(fact_9418_Suc__0__mod__numeral,axiom,
% 0.22/0.57 ! [K: num] :
% 0.22/0.57 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.22/0.57 = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Suc_0_mod_numeral
% 0.22/0.57 thf(fact_9419_fst__divmod__nat,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
% 0.22/0.57 = ( divide_divide_nat @ M @ N ) ) ).
% 0.22/0.57
% 0.22/0.57 % fst_divmod_nat
% 0.22/0.57 thf(fact_9420_snd__divmod__nat,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( product_snd_nat_nat @ ( divmod_nat @ M @ N ) )
% 0.22/0.57 = ( modulo_modulo_nat @ M @ N ) ) ).
% 0.22/0.57
% 0.22/0.57 % snd_divmod_nat
% 0.22/0.57 thf(fact_9421_quotient__of__denom__pos_H,axiom,
% 0.22/0.57 ! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % quotient_of_denom_pos'
% 0.22/0.57 thf(fact_9422_bezw__non__0,axiom,
% 0.22/0.57 ! [Y4: nat,X: nat] :
% 0.22/0.57 ( ( ord_less_nat @ zero_zero_nat @ Y4 )
% 0.22/0.57 => ( ( bezw @ X @ Y4 )
% 0.22/0.57 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X @ Y4 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X @ Y4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X @ Y4 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y4 ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bezw_non_0
% 0.22/0.57 thf(fact_9423_bezw_Osimps,axiom,
% 0.22/0.57 ( bezw
% 0.22/0.57 = ( ^ [X3: nat,Y6: nat] : ( if_Pro3027730157355071871nt_int @ ( Y6 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X3 @ Y6 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X3 @ Y6 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X3 @ Y6 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Y6 ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bezw.simps
% 0.22/0.57 thf(fact_9424_bezw_Oelims,axiom,
% 0.22/0.57 ! [X: nat,Xa: nat,Y4: product_prod_int_int] :
% 0.22/0.57 ( ( ( bezw @ X @ Xa )
% 0.22/0.57 = Y4 )
% 0.22/0.57 => ( ( ( Xa = zero_zero_nat )
% 0.22/0.57 => ( Y4
% 0.22/0.57 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.22/0.57 & ( ( Xa != zero_zero_nat )
% 0.22/0.57 => ( Y4
% 0.22/0.57 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bezw.elims
% 0.22/0.57 thf(fact_9425_bezw_Opelims,axiom,
% 0.22/0.57 ! [X: nat,Xa: nat,Y4: product_prod_int_int] :
% 0.22/0.57 ( ( ( bezw @ X @ Xa )
% 0.22/0.57 = Y4 )
% 0.22/0.57 => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 0.22/0.57 => ~ ( ( ( ( Xa = zero_zero_nat )
% 0.22/0.57 => ( Y4
% 0.22/0.57 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.22/0.57 & ( ( Xa != zero_zero_nat )
% 0.22/0.57 => ( Y4
% 0.22/0.57 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) )
% 0.22/0.57 => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bezw.pelims
% 0.22/0.57 thf(fact_9426_one__mod__minus__numeral,axiom,
% 0.22/0.57 ! [N: num] :
% 0.22/0.57 ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.57 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % one_mod_minus_numeral
% 0.22/0.57 thf(fact_9427_minus__one__mod__numeral,axiom,
% 0.22/0.57 ! [N: num] :
% 0.22/0.57 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.22/0.57 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % minus_one_mod_numeral
% 0.22/0.57 thf(fact_9428_minus__numeral__mod__numeral,axiom,
% 0.22/0.57 ! [M: num,N: num] :
% 0.22/0.57 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.22/0.57 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % minus_numeral_mod_numeral
% 0.22/0.57 thf(fact_9429_numeral__mod__minus__numeral,axiom,
% 0.22/0.57 ! [M: num,N: num] :
% 0.22/0.57 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.57 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % numeral_mod_minus_numeral
% 0.22/0.57 thf(fact_9430_Divides_Oadjust__mod__def,axiom,
% 0.22/0.57 ( adjust_mod
% 0.22/0.57 = ( ^ [L3: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L3 @ R5 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Divides.adjust_mod_def
% 0.22/0.57 thf(fact_9431_normalize__def,axiom,
% 0.22/0.57 ( normalize
% 0.22/0.57 = ( ^ [P6: product_prod_int_int] :
% 0.22/0.57 ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P6 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) )
% 0.22/0.57 @ ( if_Pro3027730157355071871nt_int
% 0.22/0.57 @ ( ( product_snd_int_int @ P6 )
% 0.22/0.57 = zero_zero_int )
% 0.22/0.57 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.22/0.57 @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % normalize_def
% 0.22/0.57 thf(fact_9432_gcd__pos__int,axiom,
% 0.22/0.57 ! [M: int,N: int] :
% 0.22/0.57 ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
% 0.22/0.57 = ( ( M != zero_zero_int )
% 0.22/0.57 | ( N != zero_zero_int ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_pos_int
% 0.22/0.57 thf(fact_9433_gcd__neg__numeral__1__int,axiom,
% 0.22/0.57 ! [N: num,X: int] :
% 0.22/0.57 ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ X )
% 0.22/0.57 = ( gcd_gcd_int @ ( numeral_numeral_int @ N ) @ X ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_neg_numeral_1_int
% 0.22/0.57 thf(fact_9434_gcd__neg__numeral__2__int,axiom,
% 0.22/0.57 ! [X: int,N: num] :
% 0.22/0.57 ( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.57 = ( gcd_gcd_int @ X @ ( numeral_numeral_int @ N ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_neg_numeral_2_int
% 0.22/0.57 thf(fact_9435_gcd__0__int,axiom,
% 0.22/0.57 ! [X: int] :
% 0.22/0.57 ( ( gcd_gcd_int @ X @ zero_zero_int )
% 0.22/0.57 = ( abs_abs_int @ X ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_0_int
% 0.22/0.57 thf(fact_9436_gcd__0__left__int,axiom,
% 0.22/0.57 ! [X: int] :
% 0.22/0.57 ( ( gcd_gcd_int @ zero_zero_int @ X )
% 0.22/0.57 = ( abs_abs_int @ X ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_0_left_int
% 0.22/0.57 thf(fact_9437_gcd__proj2__if__dvd__int,axiom,
% 0.22/0.57 ! [Y4: int,X: int] :
% 0.22/0.57 ( ( dvd_dvd_int @ Y4 @ X )
% 0.22/0.57 => ( ( gcd_gcd_int @ X @ Y4 )
% 0.22/0.57 = ( abs_abs_int @ Y4 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_proj2_if_dvd_int
% 0.22/0.57 thf(fact_9438_gcd__proj1__if__dvd__int,axiom,
% 0.22/0.57 ! [X: int,Y4: int] :
% 0.22/0.57 ( ( dvd_dvd_int @ X @ Y4 )
% 0.22/0.57 => ( ( gcd_gcd_int @ X @ Y4 )
% 0.22/0.57 = ( abs_abs_int @ X ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_proj1_if_dvd_int
% 0.22/0.57 thf(fact_9439_gcd__ge__0__int,axiom,
% 0.22/0.57 ! [X: int,Y4: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y4 ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_ge_0_int
% 0.22/0.57 thf(fact_9440_bezout__int,axiom,
% 0.22/0.57 ! [X: int,Y4: int] :
% 0.22/0.57 ? [U2: int,V3: int] :
% 0.22/0.57 ( ( plus_plus_int @ ( times_times_int @ U2 @ X ) @ ( times_times_int @ V3 @ Y4 ) )
% 0.22/0.57 = ( gcd_gcd_int @ X @ Y4 ) ) ).
% 0.22/0.57
% 0.22/0.57 % bezout_int
% 0.22/0.57 thf(fact_9441_gcd__mult__distrib__int,axiom,
% 0.22/0.57 ! [K: int,M: int,N: int] :
% 0.22/0.57 ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
% 0.22/0.57 = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_mult_distrib_int
% 0.22/0.57 thf(fact_9442_gcd__le1__int,axiom,
% 0.22/0.57 ! [A: int,B: int] :
% 0.22/0.57 ( ( ord_less_int @ zero_zero_int @ A )
% 0.22/0.57 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_le1_int
% 0.22/0.57 thf(fact_9443_gcd__le2__int,axiom,
% 0.22/0.57 ! [B: int,A: int] :
% 0.22/0.57 ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.57 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_le2_int
% 0.22/0.57 thf(fact_9444_gcd__cases__int,axiom,
% 0.22/0.57 ! [X: int,Y4: int,P: int > $o] :
% 0.22/0.57 ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.22/0.57 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.22/0.57 => ( P @ ( gcd_gcd_int @ X @ Y4 ) ) ) )
% 0.22/0.57 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.22/0.57 => ( ( ord_less_eq_int @ Y4 @ zero_zero_int )
% 0.22/0.57 => ( P @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y4 ) ) ) ) )
% 0.22/0.57 => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 0.22/0.57 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 0.22/0.57 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y4 ) ) ) )
% 0.22/0.57 => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 0.22/0.57 => ( ( ord_less_eq_int @ Y4 @ zero_zero_int )
% 0.22/0.57 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y4 ) ) ) ) )
% 0.22/0.57 => ( P @ ( gcd_gcd_int @ X @ Y4 ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_cases_int
% 0.22/0.57 thf(fact_9445_gcd__unique__int,axiom,
% 0.22/0.57 ! [D: int,A: int,B: int] :
% 0.22/0.57 ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 0.22/0.57 & ( dvd_dvd_int @ D @ A )
% 0.22/0.57 & ( dvd_dvd_int @ D @ B )
% 0.22/0.57 & ! [E3: int] :
% 0.22/0.57 ( ( ( dvd_dvd_int @ E3 @ A )
% 0.22/0.57 & ( dvd_dvd_int @ E3 @ B ) )
% 0.22/0.57 => ( dvd_dvd_int @ E3 @ D ) ) )
% 0.22/0.57 = ( D
% 0.22/0.57 = ( gcd_gcd_int @ A @ B ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_unique_int
% 0.22/0.57 thf(fact_9446_gcd__non__0__int,axiom,
% 0.22/0.57 ! [Y4: int,X: int] :
% 0.22/0.57 ( ( ord_less_int @ zero_zero_int @ Y4 )
% 0.22/0.57 => ( ( gcd_gcd_int @ X @ Y4 )
% 0.22/0.57 = ( gcd_gcd_int @ Y4 @ ( modulo_modulo_int @ X @ Y4 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_non_0_int
% 0.22/0.57 thf(fact_9447_gcd__code__int,axiom,
% 0.22/0.57 ( gcd_gcd_int
% 0.22/0.57 = ( ^ [K3: int,L3: int] : ( abs_abs_int @ ( if_int @ ( L3 = zero_zero_int ) @ K3 @ ( gcd_gcd_int @ L3 @ ( modulo_modulo_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L3 ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_code_int
% 0.22/0.57 thf(fact_9448_gcd__0__left__nat,axiom,
% 0.22/0.57 ! [X: nat] :
% 0.22/0.57 ( ( gcd_gcd_nat @ zero_zero_nat @ X )
% 0.22/0.57 = X ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_0_left_nat
% 0.22/0.57 thf(fact_9449_gcd__0__nat,axiom,
% 0.22/0.57 ! [X: nat] :
% 0.22/0.57 ( ( gcd_gcd_nat @ X @ zero_zero_nat )
% 0.22/0.57 = X ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_0_nat
% 0.22/0.57 thf(fact_9450_gcd__nat_Oright__neutral,axiom,
% 0.22/0.57 ! [A: nat] :
% 0.22/0.57 ( ( gcd_gcd_nat @ A @ zero_zero_nat )
% 0.22/0.57 = A ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.right_neutral
% 0.22/0.57 thf(fact_9451_gcd__nat_Oneutr__eq__iff,axiom,
% 0.22/0.57 ! [A: nat,B: nat] :
% 0.22/0.57 ( ( zero_zero_nat
% 0.22/0.57 = ( gcd_gcd_nat @ A @ B ) )
% 0.22/0.57 = ( ( A = zero_zero_nat )
% 0.22/0.57 & ( B = zero_zero_nat ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.neutr_eq_iff
% 0.22/0.57 thf(fact_9452_gcd__nat_Oleft__neutral,axiom,
% 0.22/0.57 ! [A: nat] :
% 0.22/0.57 ( ( gcd_gcd_nat @ zero_zero_nat @ A )
% 0.22/0.57 = A ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.left_neutral
% 0.22/0.57 thf(fact_9453_gcd__nat_Oeq__neutr__iff,axiom,
% 0.22/0.57 ! [A: nat,B: nat] :
% 0.22/0.57 ( ( ( gcd_gcd_nat @ A @ B )
% 0.22/0.57 = zero_zero_nat )
% 0.22/0.57 = ( ( A = zero_zero_nat )
% 0.22/0.57 & ( B = zero_zero_nat ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.eq_neutr_iff
% 0.22/0.57 thf(fact_9454_gcd__nat_Oabsorb1,axiom,
% 0.22/0.57 ! [A: nat,B: nat] :
% 0.22/0.57 ( ( dvd_dvd_nat @ A @ B )
% 0.22/0.57 => ( ( gcd_gcd_nat @ A @ B )
% 0.22/0.57 = A ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.absorb1
% 0.22/0.57 thf(fact_9455_gcd__nat_Oabsorb2,axiom,
% 0.22/0.57 ! [B: nat,A: nat] :
% 0.22/0.57 ( ( dvd_dvd_nat @ B @ A )
% 0.22/0.57 => ( ( gcd_gcd_nat @ A @ B )
% 0.22/0.57 = B ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.absorb2
% 0.22/0.57 thf(fact_9456_gcd__nat_Obounded__iff,axiom,
% 0.22/0.57 ! [A: nat,B: nat,C: nat] :
% 0.22/0.57 ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
% 0.22/0.57 = ( ( dvd_dvd_nat @ A @ B )
% 0.22/0.57 & ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.bounded_iff
% 0.22/0.57 thf(fact_9457_gcd__proj1__if__dvd__nat,axiom,
% 0.22/0.57 ! [X: nat,Y4: nat] :
% 0.22/0.57 ( ( dvd_dvd_nat @ X @ Y4 )
% 0.22/0.57 => ( ( gcd_gcd_nat @ X @ Y4 )
% 0.22/0.57 = X ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_proj1_if_dvd_nat
% 0.22/0.57 thf(fact_9458_gcd__proj2__if__dvd__nat,axiom,
% 0.22/0.57 ! [Y4: nat,X: nat] :
% 0.22/0.57 ( ( dvd_dvd_nat @ Y4 @ X )
% 0.22/0.57 => ( ( gcd_gcd_nat @ X @ Y4 )
% 0.22/0.57 = Y4 ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_proj2_if_dvd_nat
% 0.22/0.57 thf(fact_9459_gcd__Suc__0,axiom,
% 0.22/0.57 ! [M: nat] :
% 0.22/0.57 ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.22/0.57 = ( suc @ zero_zero_nat ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_Suc_0
% 0.22/0.57 thf(fact_9460_gcd__pos__nat,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
% 0.22/0.57 = ( ( M != zero_zero_nat )
% 0.22/0.57 | ( N != zero_zero_nat ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_pos_nat
% 0.22/0.57 thf(fact_9461_gcd__int__int__eq,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.22/0.57 = ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_int_int_eq
% 0.22/0.57 thf(fact_9462_gcd__nat__abs__right__eq,axiom,
% 0.22/0.57 ! [N: nat,K: int] :
% 0.22/0.57 ( ( gcd_gcd_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 0.22/0.57 = ( nat2 @ ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat_abs_right_eq
% 0.22/0.57 thf(fact_9463_gcd__nat__abs__left__eq,axiom,
% 0.22/0.57 ! [K: int,N: nat] :
% 0.22/0.57 ( ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 0.22/0.57 = ( nat2 @ ( gcd_gcd_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat_abs_left_eq
% 0.22/0.57 thf(fact_9464_gcd__mult__distrib__nat,axiom,
% 0.22/0.57 ! [K: nat,M: nat,N: nat] :
% 0.22/0.57 ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
% 0.22/0.57 = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_mult_distrib_nat
% 0.22/0.57 thf(fact_9465_gcd__nat_Omono,axiom,
% 0.22/0.57 ! [A: nat,C: nat,B: nat,D: nat] :
% 0.22/0.57 ( ( dvd_dvd_nat @ A @ C )
% 0.22/0.57 => ( ( dvd_dvd_nat @ B @ D )
% 0.22/0.57 => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ ( gcd_gcd_nat @ C @ D ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.mono
% 0.22/0.57 thf(fact_9466_gcd__nat_OorderE,axiom,
% 0.22/0.57 ! [A: nat,B: nat] :
% 0.22/0.57 ( ( dvd_dvd_nat @ A @ B )
% 0.22/0.57 => ( A
% 0.22/0.57 = ( gcd_gcd_nat @ A @ B ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.orderE
% 0.22/0.57 thf(fact_9467_gcd__nat_OorderI,axiom,
% 0.22/0.57 ! [A: nat,B: nat] :
% 0.22/0.57 ( ( A
% 0.22/0.57 = ( gcd_gcd_nat @ A @ B ) )
% 0.22/0.57 => ( dvd_dvd_nat @ A @ B ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.orderI
% 0.22/0.57 thf(fact_9468_gcd__nat_Oabsorb3,axiom,
% 0.22/0.57 ! [A: nat,B: nat] :
% 0.22/0.57 ( ( ( dvd_dvd_nat @ A @ B )
% 0.22/0.57 & ( A != B ) )
% 0.22/0.57 => ( ( gcd_gcd_nat @ A @ B )
% 0.22/0.57 = A ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.absorb3
% 0.22/0.57 thf(fact_9469_gcd__nat_Oabsorb4,axiom,
% 0.22/0.57 ! [B: nat,A: nat] :
% 0.22/0.57 ( ( ( dvd_dvd_nat @ B @ A )
% 0.22/0.57 & ( B != A ) )
% 0.22/0.57 => ( ( gcd_gcd_nat @ A @ B )
% 0.22/0.57 = B ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.absorb4
% 0.22/0.57 thf(fact_9470_gcd__nat_OboundedE,axiom,
% 0.22/0.57 ! [A: nat,B: nat,C: nat] :
% 0.22/0.57 ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
% 0.22/0.57 => ~ ( ( dvd_dvd_nat @ A @ B )
% 0.22/0.57 => ~ ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.boundedE
% 0.22/0.57 thf(fact_9471_gcd__nat_OboundedI,axiom,
% 0.22/0.57 ! [A: nat,B: nat,C: nat] :
% 0.22/0.57 ( ( dvd_dvd_nat @ A @ B )
% 0.22/0.57 => ( ( dvd_dvd_nat @ A @ C )
% 0.22/0.57 => ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.boundedI
% 0.22/0.57 thf(fact_9472_gcd__nat_Oorder__iff,axiom,
% 0.22/0.57 ( dvd_dvd_nat
% 0.22/0.57 = ( ^ [A3: nat,B2: nat] :
% 0.22/0.57 ( A3
% 0.22/0.57 = ( gcd_gcd_nat @ A3 @ B2 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.order_iff
% 0.22/0.57 thf(fact_9473_gcd__nat_Ocobounded1,axiom,
% 0.22/0.57 ! [A: nat,B: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.cobounded1
% 0.22/0.57 thf(fact_9474_gcd__nat_Ocobounded2,axiom,
% 0.22/0.57 ! [A: nat,B: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.cobounded2
% 0.22/0.57 thf(fact_9475_gcd__nat_Oabsorb__iff1,axiom,
% 0.22/0.57 ( dvd_dvd_nat
% 0.22/0.57 = ( ^ [A3: nat,B2: nat] :
% 0.22/0.57 ( ( gcd_gcd_nat @ A3 @ B2 )
% 0.22/0.57 = A3 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.absorb_iff1
% 0.22/0.57 thf(fact_9476_gcd__nat_Oabsorb__iff2,axiom,
% 0.22/0.57 ( dvd_dvd_nat
% 0.22/0.57 = ( ^ [B2: nat,A3: nat] :
% 0.22/0.57 ( ( gcd_gcd_nat @ A3 @ B2 )
% 0.22/0.57 = B2 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.absorb_iff2
% 0.22/0.57 thf(fact_9477_gcd__nat_OcoboundedI1,axiom,
% 0.22/0.57 ! [A: nat,C: nat,B: nat] :
% 0.22/0.57 ( ( dvd_dvd_nat @ A @ C )
% 0.22/0.57 => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.coboundedI1
% 0.22/0.57 thf(fact_9478_gcd__nat_OcoboundedI2,axiom,
% 0.22/0.57 ! [B: nat,C: nat,A: nat] :
% 0.22/0.57 ( ( dvd_dvd_nat @ B @ C )
% 0.22/0.57 => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.coboundedI2
% 0.22/0.57 thf(fact_9479_gcd__nat_Ostrict__boundedE,axiom,
% 0.22/0.57 ! [A: nat,B: nat,C: nat] :
% 0.22/0.57 ( ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
% 0.22/0.57 & ( A
% 0.22/0.57 != ( gcd_gcd_nat @ B @ C ) ) )
% 0.22/0.57 => ~ ( ( ( dvd_dvd_nat @ A @ B )
% 0.22/0.57 & ( A != B ) )
% 0.22/0.57 => ~ ( ( dvd_dvd_nat @ A @ C )
% 0.22/0.57 & ( A != C ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.strict_boundedE
% 0.22/0.57 thf(fact_9480_gcd__nat_Ostrict__order__iff,axiom,
% 0.22/0.57 ! [A: nat,B: nat] :
% 0.22/0.57 ( ( ( dvd_dvd_nat @ A @ B )
% 0.22/0.57 & ( A != B ) )
% 0.22/0.57 = ( ( A
% 0.22/0.57 = ( gcd_gcd_nat @ A @ B ) )
% 0.22/0.57 & ( A != B ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.strict_order_iff
% 0.22/0.57 thf(fact_9481_gcd__nat_Ostrict__coboundedI1,axiom,
% 0.22/0.57 ! [A: nat,C: nat,B: nat] :
% 0.22/0.57 ( ( ( dvd_dvd_nat @ A @ C )
% 0.22/0.57 & ( A != C ) )
% 0.22/0.57 => ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
% 0.22/0.57 & ( ( gcd_gcd_nat @ A @ B )
% 0.22/0.57 != C ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.strict_coboundedI1
% 0.22/0.57 thf(fact_9482_gcd__nat_Ostrict__coboundedI2,axiom,
% 0.22/0.57 ! [B: nat,C: nat,A: nat] :
% 0.22/0.57 ( ( ( dvd_dvd_nat @ B @ C )
% 0.22/0.57 & ( B != C ) )
% 0.22/0.57 => ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
% 0.22/0.57 & ( ( gcd_gcd_nat @ A @ B )
% 0.22/0.57 != C ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.strict_coboundedI2
% 0.22/0.57 thf(fact_9483_gcd__unique__nat,axiom,
% 0.22/0.57 ! [D: nat,A: nat,B: nat] :
% 0.22/0.57 ( ( ( dvd_dvd_nat @ D @ A )
% 0.22/0.57 & ( dvd_dvd_nat @ D @ B )
% 0.22/0.57 & ! [E3: nat] :
% 0.22/0.57 ( ( ( dvd_dvd_nat @ E3 @ A )
% 0.22/0.57 & ( dvd_dvd_nat @ E3 @ B ) )
% 0.22/0.57 => ( dvd_dvd_nat @ E3 @ D ) ) )
% 0.22/0.57 = ( D
% 0.22/0.57 = ( gcd_gcd_nat @ A @ B ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_unique_nat
% 0.22/0.57 thf(fact_9484_gcd__le2__nat,axiom,
% 0.22/0.57 ! [B: nat,A: nat] :
% 0.22/0.57 ( ( B != zero_zero_nat )
% 0.22/0.57 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_le2_nat
% 0.22/0.57 thf(fact_9485_gcd__le1__nat,axiom,
% 0.22/0.57 ! [A: nat,B: nat] :
% 0.22/0.57 ( ( A != zero_zero_nat )
% 0.22/0.57 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_le1_nat
% 0.22/0.57 thf(fact_9486_gcd__diff1__nat,axiom,
% 0.22/0.57 ! [N: nat,M: nat] :
% 0.22/0.57 ( ( ord_less_eq_nat @ N @ M )
% 0.22/0.57 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 0.22/0.57 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_diff1_nat
% 0.22/0.57 thf(fact_9487_gcd__diff2__nat,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.22/0.57 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
% 0.22/0.57 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_diff2_nat
% 0.22/0.57 thf(fact_9488_gcd__nat_Oelims,axiom,
% 0.22/0.57 ! [X: nat,Xa: nat,Y4: nat] :
% 0.22/0.57 ( ( ( gcd_gcd_nat @ X @ Xa )
% 0.22/0.57 = Y4 )
% 0.22/0.57 => ( ( ( Xa = zero_zero_nat )
% 0.22/0.57 => ( Y4 = X ) )
% 0.22/0.57 & ( ( Xa != zero_zero_nat )
% 0.22/0.57 => ( Y4
% 0.22/0.57 = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.elims
% 0.22/0.57 thf(fact_9489_gcd__nat_Osimps,axiom,
% 0.22/0.57 ( gcd_gcd_nat
% 0.22/0.57 = ( ^ [X3: nat,Y6: nat] : ( if_nat @ ( Y6 = zero_zero_nat ) @ X3 @ ( gcd_gcd_nat @ Y6 @ ( modulo_modulo_nat @ X3 @ Y6 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.simps
% 0.22/0.57 thf(fact_9490_gcd__non__0__nat,axiom,
% 0.22/0.57 ! [Y4: nat,X: nat] :
% 0.22/0.57 ( ( Y4 != zero_zero_nat )
% 0.22/0.57 => ( ( gcd_gcd_nat @ X @ Y4 )
% 0.22/0.57 = ( gcd_gcd_nat @ Y4 @ ( modulo_modulo_nat @ X @ Y4 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_non_0_nat
% 0.22/0.57 thf(fact_9491_gcd__code__integer,axiom,
% 0.22/0.57 ( gcd_gcd_Code_integer
% 0.22/0.57 = ( ^ [K3: code_integer,L3: code_integer] : ( abs_abs_Code_integer @ ( if_Code_integer @ ( L3 = zero_z3403309356797280102nteger ) @ K3 @ ( gcd_gcd_Code_integer @ L3 @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L3 ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_code_integer
% 0.22/0.57 thf(fact_9492_bezout__nat,axiom,
% 0.22/0.57 ! [A: nat,B: nat] :
% 0.22/0.57 ( ( A != zero_zero_nat )
% 0.22/0.57 => ? [X4: nat,Y: nat] :
% 0.22/0.57 ( ( times_times_nat @ A @ X4 )
% 0.22/0.57 = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bezout_nat
% 0.22/0.57 thf(fact_9493_bezout__gcd__nat_H,axiom,
% 0.22/0.57 ! [B: nat,A: nat] :
% 0.22/0.57 ? [X4: nat,Y: nat] :
% 0.22/0.57 ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y ) @ ( times_times_nat @ A @ X4 ) )
% 0.22/0.57 & ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y ) )
% 0.22/0.57 = ( gcd_gcd_nat @ A @ B ) ) )
% 0.22/0.57 | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y ) @ ( times_times_nat @ B @ X4 ) )
% 0.22/0.57 & ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y ) )
% 0.22/0.57 = ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bezout_gcd_nat'
% 0.22/0.57 thf(fact_9494_gcd__int__def,axiom,
% 0.22/0.57 ( gcd_gcd_int
% 0.22/0.57 = ( ^ [X3: int,Y6: int] : ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ X3 ) ) @ ( nat2 @ ( abs_abs_int @ Y6 ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_int_def
% 0.22/0.57 thf(fact_9495_bezw__aux,axiom,
% 0.22/0.57 ! [X: nat,Y4: nat] :
% 0.22/0.57 ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y4 ) )
% 0.22/0.57 = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y4 ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y4 ) ) @ ( semiri1314217659103216013at_int @ Y4 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % bezw_aux
% 0.22/0.57 thf(fact_9496_nat__descend__induct,axiom,
% 0.22/0.57 ! [N: nat,P: nat > $o,M: nat] :
% 0.22/0.57 ( ! [K2: nat] :
% 0.22/0.57 ( ( ord_less_nat @ N @ K2 )
% 0.22/0.57 => ( P @ K2 ) )
% 0.22/0.57 => ( ! [K2: nat] :
% 0.22/0.57 ( ( ord_less_eq_nat @ K2 @ N )
% 0.22/0.57 => ( ! [I4: nat] :
% 0.22/0.57 ( ( ord_less_nat @ K2 @ I4 )
% 0.22/0.57 => ( P @ I4 ) )
% 0.22/0.57 => ( P @ K2 ) ) )
% 0.22/0.57 => ( P @ M ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % nat_descend_induct
% 0.22/0.57 thf(fact_9497_gcd__nat_Opelims,axiom,
% 0.22/0.57 ! [X: nat,Xa: nat,Y4: nat] :
% 0.22/0.57 ( ( ( gcd_gcd_nat @ X @ Xa )
% 0.22/0.57 = Y4 )
% 0.22/0.57 => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 0.22/0.57 => ~ ( ( ( ( Xa = zero_zero_nat )
% 0.22/0.57 => ( Y4 = X ) )
% 0.22/0.57 & ( ( Xa != zero_zero_nat )
% 0.22/0.57 => ( Y4
% 0.22/0.57 = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) ) )
% 0.22/0.57 => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.pelims
% 0.22/0.57 thf(fact_9498_Sup__nat__empty,axiom,
% 0.22/0.57 ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
% 0.22/0.57 = zero_zero_nat ) ).
% 0.22/0.57
% 0.22/0.57 % Sup_nat_empty
% 0.22/0.57 thf(fact_9499_divmod__integer__eq__cases,axiom,
% 0.22/0.57 ( code_divmod_integer
% 0.22/0.57 = ( ^ [K3: code_integer,L3: code_integer] :
% 0.22/0.57 ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.22/0.57 @ ( if_Pro6119634080678213985nteger @ ( L3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 0.22/0.57 @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L3
% 0.22/0.57 @ ( if_Pro6119634080678213985nteger
% 0.22/0.57 @ ( ( sgn_sgn_Code_integer @ K3 )
% 0.22/0.57 = ( sgn_sgn_Code_integer @ L3 ) )
% 0.22/0.57 @ ( code_divmod_abs @ K3 @ L3 )
% 0.22/0.57 @ ( produc6916734918728496179nteger
% 0.22/0.57 @ ^ [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 @ L3 ) @ S4 ) ) )
% 0.22/0.57 @ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % divmod_integer_eq_cases
% 0.22/0.57 thf(fact_9500_card__greaterThanLessThan__int,axiom,
% 0.22/0.57 ! [L: int,U: int] :
% 0.22/0.57 ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
% 0.22/0.57 = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_greaterThanLessThan_int
% 0.22/0.57 thf(fact_9501_Suc__funpow,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( compow_nat_nat @ N @ suc )
% 0.22/0.57 = ( plus_plus_nat @ N ) ) ).
% 0.22/0.57
% 0.22/0.57 % Suc_funpow
% 0.22/0.57 thf(fact_9502_card__greaterThanLessThan,axiom,
% 0.22/0.57 ! [L: nat,U: nat] :
% 0.22/0.57 ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
% 0.22/0.57 = ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_greaterThanLessThan
% 0.22/0.57 thf(fact_9503_card_Ocomp__fun__commute__on,axiom,
% 0.22/0.57 ( ( comp_nat_nat_nat @ suc @ suc )
% 0.22/0.57 = ( comp_nat_nat_nat @ suc @ suc ) ) ).
% 0.22/0.57
% 0.22/0.57 % card.comp_fun_commute_on
% 0.22/0.57 thf(fact_9504_atLeastSucLessThan__greaterThanLessThan,axiom,
% 0.22/0.57 ! [L: nat,U: nat] :
% 0.22/0.57 ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
% 0.22/0.57 = ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% 0.22/0.57
% 0.22/0.57 % atLeastSucLessThan_greaterThanLessThan
% 0.22/0.57 thf(fact_9505_greaterThanLessThan__upt,axiom,
% 0.22/0.57 ( set_or5834768355832116004an_nat
% 0.22/0.57 = ( ^ [N4: nat,M5: nat] : ( set_nat2 @ ( upt @ ( suc @ N4 ) @ M5 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % greaterThanLessThan_upt
% 0.22/0.57 thf(fact_9506_Code__Target__Int_Onegative__def,axiom,
% 0.22/0.57 ( code_Target_negative
% 0.22/0.57 = ( comp_int_int_num @ uminus_uminus_int @ numeral_numeral_int ) ) ).
% 0.22/0.57
% 0.22/0.57 % Code_Target_Int.negative_def
% 0.22/0.57 thf(fact_9507_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 0.22/0.57 ! [N: nat,J: nat,I: nat] :
% 0.22/0.57 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
% 0.22/0.57 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
% 0.22/0.57 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % nth_sorted_list_of_set_greaterThanLessThan
% 0.22/0.57 thf(fact_9508_nat__of__integer__non__positive,axiom,
% 0.22/0.57 ! [K: code_integer] :
% 0.22/0.57 ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
% 0.22/0.57 => ( ( code_nat_of_integer @ K )
% 0.22/0.57 = zero_zero_nat ) ) ).
% 0.22/0.57
% 0.22/0.57 % nat_of_integer_non_positive
% 0.22/0.57 thf(fact_9509_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.22/0.57 ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 0.22/0.57 @ ^ [X3: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X3 )
% 0.22/0.57 @ ^ [X3: nat,Y6: nat] : ( ord_less_nat @ Y6 @ X3 ) ) ).
% 0.22/0.57
% 0.22/0.57 % max_nat.semilattice_neutr_order_axioms
% 0.22/0.57 thf(fact_9510_of__nat__of__integer,axiom,
% 0.22/0.57 ! [K: code_integer] :
% 0.22/0.57 ( ( semiri4939895301339042750nteger @ ( code_nat_of_integer @ K ) )
% 0.22/0.57 = ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ K ) ) ).
% 0.22/0.57
% 0.22/0.57 % of_nat_of_integer
% 0.22/0.57 thf(fact_9511_nat__of__integer__code__post_I1_J,axiom,
% 0.22/0.57 ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
% 0.22/0.57 = zero_zero_nat ) ).
% 0.22/0.57
% 0.22/0.57 % nat_of_integer_code_post(1)
% 0.22/0.57 thf(fact_9512_nat__of__integer__code__post_I3_J,axiom,
% 0.22/0.57 ! [K: num] :
% 0.22/0.57 ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 0.22/0.57 = ( numeral_numeral_nat @ K ) ) ).
% 0.22/0.57
% 0.22/0.57 % nat_of_integer_code_post(3)
% 0.22/0.57 thf(fact_9513_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.22/0.57 ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
% 0.22/0.57 @ ^ [M5: nat,N4: nat] :
% 0.22/0.57 ( ( dvd_dvd_nat @ M5 @ N4 )
% 0.22/0.57 & ( M5 != N4 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % gcd_nat.semilattice_neutr_order_axioms
% 0.22/0.57 thf(fact_9514_nat__of__integer__code,axiom,
% 0.22/0.57 ( code_nat_of_integer
% 0.22/0.57 = ( ^ [K3: code_integer] :
% 0.22/0.57 ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 0.22/0.57 @ ( produc1555791787009142072er_nat
% 0.22/0.57 @ ^ [L3: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ one_one_nat ) )
% 0.22/0.57 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % nat_of_integer_code
% 0.22/0.57 thf(fact_9515_int__of__integer__code,axiom,
% 0.22/0.57 ( code_int_of_integer
% 0.22/0.57 = ( ^ [K3: code_integer] :
% 0.22/0.57 ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
% 0.22/0.57 @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 0.22/0.57 @ ( produc1553301316500091796er_int
% 0.22/0.57 @ ^ [L3: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ one_one_int ) )
% 0.22/0.57 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % int_of_integer_code
% 0.22/0.57 thf(fact_9516_times__int_Oabs__eq,axiom,
% 0.22/0.57 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.22/0.57 ( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.22/0.57 = ( abs_Integ
% 0.22/0.57 @ ( produc27273713700761075at_nat
% 0.22/0.57 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.57 ( produc2626176000494625587at_nat
% 0.22/0.57 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) )
% 0.22/0.57 @ Xa
% 0.22/0.57 @ X ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % times_int.abs_eq
% 0.22/0.57 thf(fact_9517_zero__integer_Orep__eq,axiom,
% 0.22/0.57 ( ( code_int_of_integer @ zero_z3403309356797280102nteger )
% 0.22/0.57 = zero_zero_int ) ).
% 0.22/0.57
% 0.22/0.57 % zero_integer.rep_eq
% 0.22/0.57 thf(fact_9518_int__of__integer__numeral,axiom,
% 0.22/0.57 ! [K: num] :
% 0.22/0.57 ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 0.22/0.57 = ( numeral_numeral_int @ K ) ) ).
% 0.22/0.57
% 0.22/0.57 % int_of_integer_numeral
% 0.22/0.57 thf(fact_9519_times__integer_Orep__eq,axiom,
% 0.22/0.57 ! [X: code_integer,Xa: code_integer] :
% 0.22/0.57 ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X @ Xa ) )
% 0.22/0.57 = ( times_times_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % times_integer.rep_eq
% 0.22/0.57 thf(fact_9520_minus__integer_Orep__eq,axiom,
% 0.22/0.57 ! [X: code_integer,Xa: code_integer] :
% 0.22/0.57 ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X @ Xa ) )
% 0.22/0.57 = ( minus_minus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % minus_integer.rep_eq
% 0.22/0.57 thf(fact_9521_divide__integer_Orep__eq,axiom,
% 0.22/0.57 ! [X: code_integer,Xa: code_integer] :
% 0.22/0.57 ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X @ Xa ) )
% 0.22/0.57 = ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % divide_integer.rep_eq
% 0.22/0.57 thf(fact_9522_int__of__integer__of__nat,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( code_int_of_integer @ ( semiri4939895301339042750nteger @ N ) )
% 0.22/0.57 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.22/0.57
% 0.22/0.57 % int_of_integer_of_nat
% 0.22/0.57 thf(fact_9523_eq__Abs__Integ,axiom,
% 0.22/0.57 ! [Z: int] :
% 0.22/0.57 ~ ! [X4: nat,Y: nat] :
% 0.22/0.57 ( Z
% 0.22/0.57 != ( abs_Integ @ ( product_Pair_nat_nat @ X4 @ Y ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % eq_Abs_Integ
% 0.22/0.57 thf(fact_9524_int_Oabs__induct,axiom,
% 0.22/0.57 ! [P: int > $o,X: int] :
% 0.22/0.57 ( ! [Y: product_prod_nat_nat] : ( P @ ( abs_Integ @ Y ) )
% 0.22/0.57 => ( P @ X ) ) ).
% 0.22/0.57
% 0.22/0.57 % int.abs_induct
% 0.22/0.57 thf(fact_9525_less__eq__integer_Orep__eq,axiom,
% 0.22/0.57 ( ord_le3102999989581377725nteger
% 0.22/0.57 = ( ^ [X3: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % less_eq_integer.rep_eq
% 0.22/0.57 thf(fact_9526_integer__less__eq__iff,axiom,
% 0.22/0.57 ( ord_le3102999989581377725nteger
% 0.22/0.57 = ( ^ [K3: code_integer,L3: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % integer_less_eq_iff
% 0.22/0.57 thf(fact_9527_nat_Oabs__eq,axiom,
% 0.22/0.57 ! [X: product_prod_nat_nat] :
% 0.22/0.57 ( ( nat2 @ ( abs_Integ @ X ) )
% 0.22/0.57 = ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).
% 0.22/0.57
% 0.22/0.57 % nat.abs_eq
% 0.22/0.57 thf(fact_9528_unset__bit__integer_Orep__eq,axiom,
% 0.22/0.57 ! [X: nat,Xa: code_integer] :
% 0.22/0.57 ( ( code_int_of_integer @ ( bit_se8260200283734997820nteger @ X @ Xa ) )
% 0.22/0.57 = ( bit_se4203085406695923979it_int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % unset_bit_integer.rep_eq
% 0.22/0.57 thf(fact_9529_set__bit__integer_Orep__eq,axiom,
% 0.22/0.57 ! [X: nat,Xa: code_integer] :
% 0.22/0.57 ( ( code_int_of_integer @ ( bit_se2793503036327961859nteger @ X @ Xa ) )
% 0.22/0.57 = ( bit_se7879613467334960850it_int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % set_bit_integer.rep_eq
% 0.22/0.57 thf(fact_9530_flip__bit__integer_Orep__eq,axiom,
% 0.22/0.57 ! [X: nat,Xa: code_integer] :
% 0.22/0.57 ( ( code_int_of_integer @ ( bit_se1345352211410354436nteger @ X @ Xa ) )
% 0.22/0.57 = ( bit_se2159334234014336723it_int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % flip_bit_integer.rep_eq
% 0.22/0.57 thf(fact_9531_zero__int__def,axiom,
% 0.22/0.57 ( zero_zero_int
% 0.22/0.57 = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % zero_int_def
% 0.22/0.57 thf(fact_9532_int__def,axiom,
% 0.22/0.57 ( semiri1314217659103216013at_int
% 0.22/0.57 = ( ^ [N4: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N4 @ zero_zero_nat ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % int_def
% 0.22/0.57 thf(fact_9533_uminus__int_Oabs__eq,axiom,
% 0.22/0.57 ! [X: product_prod_nat_nat] :
% 0.22/0.57 ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
% 0.22/0.57 = ( abs_Integ
% 0.22/0.57 @ ( produc2626176000494625587at_nat
% 0.22/0.57 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 )
% 0.22/0.57 @ X ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % uminus_int.abs_eq
% 0.22/0.57 thf(fact_9534_one__int__def,axiom,
% 0.22/0.57 ( one_one_int
% 0.22/0.57 = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % one_int_def
% 0.22/0.57 thf(fact_9535_less__int_Oabs__eq,axiom,
% 0.22/0.57 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.22/0.57 ( ( ord_less_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.22/0.57 = ( produc8739625826339149834_nat_o
% 0.22/0.57 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.57 ( produc6081775807080527818_nat_o
% 0.22/0.57 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) )
% 0.22/0.57 @ Xa
% 0.22/0.57 @ X ) ) ).
% 0.22/0.57
% 0.22/0.57 % less_int.abs_eq
% 0.22/0.57 thf(fact_9536_less__eq__int_Oabs__eq,axiom,
% 0.22/0.57 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.22/0.57 ( ( ord_less_eq_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.22/0.57 = ( produc8739625826339149834_nat_o
% 0.22/0.57 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.57 ( produc6081775807080527818_nat_o
% 0.22/0.57 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) )
% 0.22/0.57 @ Xa
% 0.22/0.57 @ X ) ) ).
% 0.22/0.57
% 0.22/0.57 % less_eq_int.abs_eq
% 0.22/0.57 thf(fact_9537_plus__int_Oabs__eq,axiom,
% 0.22/0.57 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.22/0.57 ( ( plus_plus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.22/0.57 = ( abs_Integ
% 0.22/0.57 @ ( produc27273713700761075at_nat
% 0.22/0.57 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.57 ( produc2626176000494625587at_nat
% 0.22/0.57 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) )
% 0.22/0.57 @ Xa
% 0.22/0.57 @ X ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % plus_int.abs_eq
% 0.22/0.57 thf(fact_9538_minus__int_Oabs__eq,axiom,
% 0.22/0.57 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.22/0.57 ( ( minus_minus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.22/0.57 = ( abs_Integ
% 0.22/0.57 @ ( produc27273713700761075at_nat
% 0.22/0.57 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.57 ( produc2626176000494625587at_nat
% 0.22/0.57 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) )
% 0.22/0.57 @ Xa
% 0.22/0.57 @ X ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % minus_int.abs_eq
% 0.22/0.57 thf(fact_9539_Gcd__remove0__nat,axiom,
% 0.22/0.57 ! [M10: set_nat] :
% 0.22/0.57 ( ( finite_finite_nat @ M10 )
% 0.22/0.57 => ( ( gcd_Gcd_nat @ M10 )
% 0.22/0.57 = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M10 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Gcd_remove0_nat
% 0.22/0.57 thf(fact_9540_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 0.22/0.57 ! [N: nat,J: nat,I: nat] :
% 0.22/0.57 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
% 0.22/0.57 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
% 0.22/0.57 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % nth_sorted_list_of_set_greaterThanAtMost
% 0.22/0.57 thf(fact_9541_card__greaterThanAtMost,axiom,
% 0.22/0.57 ! [L: nat,U: nat] :
% 0.22/0.57 ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
% 0.22/0.57 = ( minus_minus_nat @ U @ L ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_greaterThanAtMost
% 0.22/0.57 thf(fact_9542_Gcd__dvd__nat,axiom,
% 0.22/0.57 ! [A: nat,A2: set_nat] :
% 0.22/0.57 ( ( member_nat @ A @ A2 )
% 0.22/0.57 => ( dvd_dvd_nat @ ( gcd_Gcd_nat @ A2 ) @ A ) ) ).
% 0.22/0.57
% 0.22/0.57 % Gcd_dvd_nat
% 0.22/0.57 thf(fact_9543_Gcd__greatest__nat,axiom,
% 0.22/0.57 ! [A2: set_nat,A: nat] :
% 0.22/0.57 ( ! [B3: nat] :
% 0.22/0.57 ( ( member_nat @ B3 @ A2 )
% 0.22/0.57 => ( dvd_dvd_nat @ A @ B3 ) )
% 0.22/0.57 => ( dvd_dvd_nat @ A @ ( gcd_Gcd_nat @ A2 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Gcd_greatest_nat
% 0.22/0.57 thf(fact_9544_atLeastSucAtMost__greaterThanAtMost,axiom,
% 0.22/0.57 ! [L: nat,U: nat] :
% 0.22/0.57 ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
% 0.22/0.57 = ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% 0.22/0.57
% 0.22/0.57 % atLeastSucAtMost_greaterThanAtMost
% 0.22/0.57 thf(fact_9545_greaterThanAtMost__upt,axiom,
% 0.22/0.57 ( set_or6659071591806873216st_nat
% 0.22/0.57 = ( ^ [N4: nat,M5: nat] : ( set_nat2 @ ( upt @ ( suc @ N4 ) @ ( suc @ M5 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % greaterThanAtMost_upt
% 0.22/0.57 thf(fact_9546_less__eq__int_Orep__eq,axiom,
% 0.22/0.57 ( ord_less_eq_int
% 0.22/0.57 = ( ^ [X3: int,Xa4: int] :
% 0.22/0.57 ( produc8739625826339149834_nat_o
% 0.22/0.57 @ ^ [Y6: nat,Z5: nat] :
% 0.22/0.57 ( produc6081775807080527818_nat_o
% 0.22/0.57 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y6 @ V4 ) @ ( plus_plus_nat @ U3 @ Z5 ) ) )
% 0.22/0.57 @ ( rep_Integ @ X3 )
% 0.22/0.57 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % less_eq_int.rep_eq
% 0.22/0.57 thf(fact_9547_less__int_Orep__eq,axiom,
% 0.22/0.57 ( ord_less_int
% 0.22/0.57 = ( ^ [X3: int,Xa4: int] :
% 0.22/0.57 ( produc8739625826339149834_nat_o
% 0.22/0.57 @ ^ [Y6: nat,Z5: nat] :
% 0.22/0.57 ( produc6081775807080527818_nat_o
% 0.22/0.57 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y6 @ V4 ) @ ( plus_plus_nat @ U3 @ Z5 ) ) )
% 0.22/0.57 @ ( rep_Integ @ X3 )
% 0.22/0.57 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % less_int.rep_eq
% 0.22/0.57 thf(fact_9548_card__greaterThanAtMost__int,axiom,
% 0.22/0.57 ! [L: int,U: int] :
% 0.22/0.57 ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L @ U ) )
% 0.22/0.57 = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_greaterThanAtMost_int
% 0.22/0.57 thf(fact_9549_Gcd__dvd__int,axiom,
% 0.22/0.57 ! [A: int,A2: set_int] :
% 0.22/0.57 ( ( member_int @ A @ A2 )
% 0.22/0.57 => ( dvd_dvd_int @ ( gcd_Gcd_int @ A2 ) @ A ) ) ).
% 0.22/0.57
% 0.22/0.57 % Gcd_dvd_int
% 0.22/0.57 thf(fact_9550_Gcd__greatest__int,axiom,
% 0.22/0.57 ! [A2: set_int,A: int] :
% 0.22/0.57 ( ! [B3: int] :
% 0.22/0.57 ( ( member_int @ B3 @ A2 )
% 0.22/0.57 => ( dvd_dvd_int @ A @ B3 ) )
% 0.22/0.57 => ( dvd_dvd_int @ A @ ( gcd_Gcd_int @ A2 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Gcd_greatest_int
% 0.22/0.57 thf(fact_9551_Gcd__int__greater__eq__0,axiom,
% 0.22/0.57 ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).
% 0.22/0.57
% 0.22/0.57 % Gcd_int_greater_eq_0
% 0.22/0.57 thf(fact_9552_nat_Orep__eq,axiom,
% 0.22/0.57 ( nat2
% 0.22/0.57 = ( ^ [X3: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X3 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % nat.rep_eq
% 0.22/0.57 thf(fact_9553_uminus__int__def,axiom,
% 0.22/0.57 ( uminus_uminus_int
% 0.22/0.57 = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
% 0.22/0.57 @ ( produc2626176000494625587at_nat
% 0.22/0.57 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % uminus_int_def
% 0.22/0.57 thf(fact_9554_prod__encode__def,axiom,
% 0.22/0.57 ( nat_prod_encode
% 0.22/0.57 = ( produc6842872674320459806at_nat
% 0.22/0.57 @ ^ [M5: nat,N4: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M5 @ N4 ) ) @ M5 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % prod_encode_def
% 0.22/0.57 thf(fact_9555_num__of__nat_Osimps_I2_J,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.57 => ( ( num_of_nat @ ( suc @ N ) )
% 0.22/0.57 = ( inc @ ( num_of_nat @ N ) ) ) )
% 0.22/0.57 & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.57 => ( ( num_of_nat @ ( suc @ N ) )
% 0.22/0.57 = one ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % num_of_nat.simps(2)
% 0.22/0.57 thf(fact_9556_prod__encode__eq,axiom,
% 0.22/0.57 ! [X: product_prod_nat_nat,Y4: product_prod_nat_nat] :
% 0.22/0.57 ( ( ( nat_prod_encode @ X )
% 0.22/0.57 = ( nat_prod_encode @ Y4 ) )
% 0.22/0.57 = ( X = Y4 ) ) ).
% 0.22/0.57
% 0.22/0.57 % prod_encode_eq
% 0.22/0.57 thf(fact_9557_num__of__nat__numeral__eq,axiom,
% 0.22/0.57 ! [Q2: num] :
% 0.22/0.57 ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
% 0.22/0.57 = Q2 ) ).
% 0.22/0.57
% 0.22/0.57 % num_of_nat_numeral_eq
% 0.22/0.57 thf(fact_9558_num__of__nat_Osimps_I1_J,axiom,
% 0.22/0.57 ( ( num_of_nat @ zero_zero_nat )
% 0.22/0.57 = one ) ).
% 0.22/0.57
% 0.22/0.57 % num_of_nat.simps(1)
% 0.22/0.57 thf(fact_9559_le__prod__encode__1,axiom,
% 0.22/0.57 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % le_prod_encode_1
% 0.22/0.57 thf(fact_9560_le__prod__encode__2,axiom,
% 0.22/0.57 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % le_prod_encode_2
% 0.22/0.57 thf(fact_9561_numeral__num__of__nat,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.57 => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
% 0.22/0.57 = N ) ) ).
% 0.22/0.57
% 0.22/0.57 % numeral_num_of_nat
% 0.22/0.57 thf(fact_9562_num__of__nat__One,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( ord_less_eq_nat @ N @ one_one_nat )
% 0.22/0.57 => ( ( num_of_nat @ N )
% 0.22/0.57 = one ) ) ).
% 0.22/0.57
% 0.22/0.57 % num_of_nat_One
% 0.22/0.57 thf(fact_9563_num__of__nat__double,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.57 => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
% 0.22/0.57 = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % num_of_nat_double
% 0.22/0.57 thf(fact_9564_num__of__nat__plus__distrib,axiom,
% 0.22/0.57 ! [M: nat,N: nat] :
% 0.22/0.57 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.22/0.57 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.57 => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
% 0.22/0.57 = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % num_of_nat_plus_distrib
% 0.22/0.57 thf(fact_9565_prod__encode__prod__decode__aux,axiom,
% 0.22/0.57 ! [K: nat,M: nat] :
% 0.22/0.57 ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
% 0.22/0.57 = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).
% 0.22/0.57
% 0.22/0.57 % prod_encode_prod_decode_aux
% 0.22/0.57 thf(fact_9566_times__int__def,axiom,
% 0.22/0.57 ( times_times_int
% 0.22/0.57 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.22/0.57 @ ( produc27273713700761075at_nat
% 0.22/0.57 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.57 ( produc2626176000494625587at_nat
% 0.22/0.57 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % times_int_def
% 0.22/0.57 thf(fact_9567_minus__int__def,axiom,
% 0.22/0.57 ( minus_minus_int
% 0.22/0.57 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.22/0.57 @ ( produc27273713700761075at_nat
% 0.22/0.57 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.57 ( produc2626176000494625587at_nat
% 0.22/0.57 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % minus_int_def
% 0.22/0.57 thf(fact_9568_plus__int__def,axiom,
% 0.22/0.57 ( plus_plus_int
% 0.22/0.57 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.22/0.57 @ ( produc27273713700761075at_nat
% 0.22/0.57 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.57 ( produc2626176000494625587at_nat
% 0.22/0.57 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % plus_int_def
% 0.22/0.57 thf(fact_9569_pred__nat__def,axiom,
% 0.22/0.57 ( pred_nat
% 0.22/0.57 = ( collec3392354462482085612at_nat
% 0.22/0.57 @ ( produc6081775807080527818_nat_o
% 0.22/0.57 @ ^ [M5: nat,N4: nat] :
% 0.22/0.57 ( N4
% 0.22/0.57 = ( suc @ M5 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % pred_nat_def
% 0.22/0.57 thf(fact_9570_pow_Osimps_I3_J,axiom,
% 0.22/0.57 ! [X: num,Y4: num] :
% 0.22/0.57 ( ( pow @ X @ ( bit1 @ Y4 ) )
% 0.22/0.57 = ( times_times_num @ ( sqr @ ( pow @ X @ Y4 ) ) @ X ) ) ).
% 0.22/0.57
% 0.22/0.57 % pow.simps(3)
% 0.22/0.57 thf(fact_9571_image__minus__const__atLeastLessThan__nat,axiom,
% 0.22/0.57 ! [C: nat,Y4: nat,X: nat] :
% 0.22/0.57 ( ( ( ord_less_nat @ C @ Y4 )
% 0.22/0.57 => ( ( image_nat_nat
% 0.22/0.57 @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
% 0.22/0.57 @ ( set_or4665077453230672383an_nat @ X @ Y4 ) )
% 0.22/0.57 = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y4 @ C ) ) ) )
% 0.22/0.57 & ( ~ ( ord_less_nat @ C @ Y4 )
% 0.22/0.57 => ( ( ( ord_less_nat @ X @ Y4 )
% 0.22/0.57 => ( ( image_nat_nat
% 0.22/0.57 @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
% 0.22/0.57 @ ( set_or4665077453230672383an_nat @ X @ Y4 ) )
% 0.22/0.57 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 0.22/0.57 & ( ~ ( ord_less_nat @ X @ Y4 )
% 0.22/0.57 => ( ( image_nat_nat
% 0.22/0.57 @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
% 0.22/0.57 @ ( set_or4665077453230672383an_nat @ X @ Y4 ) )
% 0.22/0.57 = bot_bot_set_nat ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % image_minus_const_atLeastLessThan_nat
% 0.22/0.57 thf(fact_9572_bij__betw__Suc,axiom,
% 0.22/0.57 ! [M10: set_nat,N5: set_nat] :
% 0.22/0.57 ( ( bij_betw_nat_nat @ suc @ M10 @ N5 )
% 0.22/0.57 = ( ( image_nat_nat @ suc @ M10 )
% 0.22/0.57 = N5 ) ) ).
% 0.22/0.57
% 0.22/0.57 % bij_betw_Suc
% 0.22/0.57 thf(fact_9573_image__Suc__atLeastAtMost,axiom,
% 0.22/0.57 ! [I: nat,J: nat] :
% 0.22/0.57 ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 0.22/0.57 = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % image_Suc_atLeastAtMost
% 0.22/0.57 thf(fact_9574_image__Suc__atLeastLessThan,axiom,
% 0.22/0.57 ! [I: nat,J: nat] :
% 0.22/0.57 ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
% 0.22/0.57 = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % image_Suc_atLeastLessThan
% 0.22/0.57 thf(fact_9575_sqr_Osimps_I2_J,axiom,
% 0.22/0.57 ! [N: num] :
% 0.22/0.57 ( ( sqr @ ( bit0 @ N ) )
% 0.22/0.57 = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % sqr.simps(2)
% 0.22/0.57 thf(fact_9576_sqr_Osimps_I1_J,axiom,
% 0.22/0.57 ( ( sqr @ one )
% 0.22/0.57 = one ) ).
% 0.22/0.57
% 0.22/0.57 % sqr.simps(1)
% 0.22/0.57 thf(fact_9577_zero__notin__Suc__image,axiom,
% 0.22/0.57 ! [A2: set_nat] :
% 0.22/0.57 ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% 0.22/0.57
% 0.22/0.57 % zero_notin_Suc_image
% 0.22/0.57 thf(fact_9578_sqr__conv__mult,axiom,
% 0.22/0.57 ( sqr
% 0.22/0.57 = ( ^ [X3: num] : ( times_times_num @ X3 @ X3 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % sqr_conv_mult
% 0.22/0.57 thf(fact_9579_image__Suc__lessThan,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.57 = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% 0.22/0.57
% 0.22/0.57 % image_Suc_lessThan
% 0.22/0.57 thf(fact_9580_image__Suc__atMost,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
% 0.22/0.57 = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % image_Suc_atMost
% 0.22/0.57 thf(fact_9581_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.22/0.57 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % atLeast0_atMost_Suc_eq_insert_0
% 0.22/0.57 thf(fact_9582_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.22/0.57 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % atLeast0_lessThan_Suc_eq_insert_0
% 0.22/0.57 thf(fact_9583_lessThan__Suc__eq__insert__0,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( set_ord_lessThan_nat @ ( suc @ N ) )
% 0.22/0.57 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % lessThan_Suc_eq_insert_0
% 0.22/0.57 thf(fact_9584_atMost__Suc__eq__insert__0,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( set_ord_atMost_nat @ ( suc @ N ) )
% 0.22/0.57 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % atMost_Suc_eq_insert_0
% 0.22/0.57 thf(fact_9585_pow_Osimps_I2_J,axiom,
% 0.22/0.57 ! [X: num,Y4: num] :
% 0.22/0.57 ( ( pow @ X @ ( bit0 @ Y4 ) )
% 0.22/0.57 = ( sqr @ ( pow @ X @ Y4 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % pow.simps(2)
% 0.22/0.57 thf(fact_9586_sqr_Osimps_I3_J,axiom,
% 0.22/0.57 ! [N: num] :
% 0.22/0.57 ( ( sqr @ ( bit1 @ N ) )
% 0.22/0.57 = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % sqr.simps(3)
% 0.22/0.57 thf(fact_9587_Gcd__int__eq,axiom,
% 0.22/0.57 ! [N5: set_nat] :
% 0.22/0.57 ( ( gcd_Gcd_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N5 ) )
% 0.22/0.57 = ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ N5 ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Gcd_int_eq
% 0.22/0.57 thf(fact_9588_Inf__real__def,axiom,
% 0.22/0.57 ( comple4887499456419720421f_real
% 0.22/0.57 = ( ^ [X8: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X8 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Inf_real_def
% 0.22/0.57 thf(fact_9589_finite__int__iff__bounded,axiom,
% 0.22/0.57 ( finite_finite_int
% 0.22/0.57 = ( ^ [S5: set_int] :
% 0.22/0.57 ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S5 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % finite_int_iff_bounded
% 0.22/0.57 thf(fact_9590_finite__int__iff__bounded__le,axiom,
% 0.22/0.57 ( finite_finite_int
% 0.22/0.57 = ( ^ [S5: set_int] :
% 0.22/0.57 ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S5 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % finite_int_iff_bounded_le
% 0.22/0.57 thf(fact_9591_image__int__atLeastAtMost,axiom,
% 0.22/0.57 ! [A: nat,B: nat] :
% 0.22/0.57 ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.22/0.57 = ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % image_int_atLeastAtMost
% 0.22/0.57 thf(fact_9592_image__int__atLeastLessThan,axiom,
% 0.22/0.57 ! [A: nat,B: nat] :
% 0.22/0.57 ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B ) )
% 0.22/0.57 = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % image_int_atLeastLessThan
% 0.22/0.57 thf(fact_9593_image__add__int__atLeastLessThan,axiom,
% 0.22/0.57 ! [L: int,U: int] :
% 0.22/0.57 ( ( image_int_int
% 0.22/0.57 @ ^ [X3: int] : ( plus_plus_int @ X3 @ L )
% 0.22/0.57 @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
% 0.22/0.57 = ( set_or4662586982721622107an_int @ L @ U ) ) ).
% 0.22/0.57
% 0.22/0.57 % image_add_int_atLeastLessThan
% 0.22/0.57 thf(fact_9594_Gcd__int__def,axiom,
% 0.22/0.57 ( gcd_Gcd_int
% 0.22/0.57 = ( ^ [K7: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K7 ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % Gcd_int_def
% 0.22/0.57 thf(fact_9595_image__atLeastZeroLessThan__int,axiom,
% 0.22/0.57 ! [U: int] :
% 0.22/0.57 ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.22/0.57 => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
% 0.22/0.57 = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % image_atLeastZeroLessThan_int
% 0.22/0.57 thf(fact_9596_range__mod,axiom,
% 0.22/0.57 ! [N: nat] :
% 0.22/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.57 => ( ( image_nat_nat
% 0.22/0.57 @ ^ [M5: nat] : ( modulo_modulo_nat @ M5 @ N )
% 0.22/0.57 @ top_top_set_nat )
% 0.22/0.57 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % range_mod
% 0.22/0.57 thf(fact_9597_suminf__eq__SUP__real,axiom,
% 0.22/0.57 ! [X9: nat > real] :
% 0.22/0.57 ( ( summable_real @ X9 )
% 0.22/0.57 => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X9 @ I3 ) )
% 0.22/0.57 => ( ( suminf_real @ X9 )
% 0.22/0.57 = ( comple1385675409528146559p_real
% 0.22/0.57 @ ( image_nat_real
% 0.22/0.57 @ ^ [I2: nat] : ( groups6591440286371151544t_real @ X9 @ ( set_ord_lessThan_nat @ I2 ) )
% 0.22/0.57 @ top_top_set_nat ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % suminf_eq_SUP_real
% 0.22/0.57 thf(fact_9598_UNIV__nat__eq,axiom,
% 0.22/0.57 ( top_top_set_nat
% 0.22/0.57 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % UNIV_nat_eq
% 0.22/0.57 thf(fact_9599_measure__function__int,axiom,
% 0.22/0.57 fun_is_measure_int @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) ).
% 0.22/0.57
% 0.22/0.57 % measure_function_int
% 0.22/0.57 thf(fact_9600_card__UNIV__bool,axiom,
% 0.22/0.57 ( ( finite_card_o @ top_top_set_o )
% 0.22/0.57 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_UNIV_bool
% 0.22/0.57 thf(fact_9601_range__mult,axiom,
% 0.22/0.57 ! [A: real] :
% 0.22/0.57 ( ( ( A = zero_zero_real )
% 0.22/0.57 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.22/0.57 = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 0.22/0.57 & ( ( A != zero_zero_real )
% 0.22/0.57 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.22/0.57 = top_top_set_real ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % range_mult
% 0.22/0.57 thf(fact_9602_infinite__UNIV__int,axiom,
% 0.22/0.57 ~ ( finite_finite_int @ top_top_set_int ) ).
% 0.22/0.57
% 0.22/0.57 % infinite_UNIV_int
% 0.22/0.57 thf(fact_9603_bij__prod__encode,axiom,
% 0.22/0.57 bij_be5333170631980326235at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat @ top_top_set_nat ).
% 0.22/0.57
% 0.22/0.57 % bij_prod_encode
% 0.22/0.57 thf(fact_9604_surj__prod__encode,axiom,
% 0.22/0.57 ( ( image_2486076414777270412at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat )
% 0.22/0.57 = top_top_set_nat ) ).
% 0.22/0.57
% 0.22/0.57 % surj_prod_encode
% 0.22/0.57 thf(fact_9605_int__in__range__abs,axiom,
% 0.22/0.57 ! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).
% 0.22/0.57
% 0.22/0.57 % int_in_range_abs
% 0.22/0.57 thf(fact_9606_root__def,axiom,
% 0.22/0.57 ( root
% 0.22/0.57 = ( ^ [N4: nat,X3: real] :
% 0.22/0.57 ( if_real @ ( N4 = zero_zero_nat ) @ zero_zero_real
% 0.22/0.57 @ ( the_in5290026491893676941l_real @ top_top_set_real
% 0.22/0.57 @ ^ [Y6: real] : ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N4 ) )
% 0.22/0.57 @ X3 ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % root_def
% 0.22/0.57 thf(fact_9607_card__UNIV__char,axiom,
% 0.22/0.57 ( ( finite_card_char @ top_top_set_char )
% 0.22/0.57 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.57
% 0.22/0.57 % card_UNIV_char
% 0.22/0.57 thf(fact_9608_UNIV__bool,axiom,
% 0.22/0.57 ( top_top_set_o
% 0.22/0.57 = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % UNIV_bool
% 0.22/0.58 thf(fact_9609_UNIV__char__of__nat,axiom,
% 0.22/0.58 ( top_top_set_char
% 0.22/0.58 = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % UNIV_char_of_nat
% 0.22/0.58 thf(fact_9610_char_Osize_I2_J,axiom,
% 0.22/0.58 ! [X1: $o,X2: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
% 0.22/0.58 ( ( size_size_char @ ( char2 @ X1 @ X2 @ X33 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
% 0.22/0.58 = zero_zero_nat ) ).
% 0.22/0.58
% 0.22/0.58 % char.size(2)
% 0.22/0.58 thf(fact_9611_nat__of__char__less__256,axiom,
% 0.22/0.58 ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % nat_of_char_less_256
% 0.22/0.58 thf(fact_9612_range__nat__of__char,axiom,
% 0.22/0.58 ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
% 0.22/0.58 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % range_nat_of_char
% 0.22/0.58 thf(fact_9613_integer__of__char__code,axiom,
% 0.22/0.58 ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 0.22/0.58 ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 0.22/0.58 = ( 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 ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % integer_of_char_code
% 0.22/0.58 thf(fact_9614_char_Osize__gen,axiom,
% 0.22/0.58 ! [X1: $o,X2: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
% 0.22/0.58 ( ( size_char @ ( char2 @ X1 @ X2 @ X33 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
% 0.22/0.58 = zero_zero_nat ) ).
% 0.22/0.58
% 0.22/0.58 % char.size_gen
% 0.22/0.58 thf(fact_9615_String_Ochar__of__ascii__of,axiom,
% 0.22/0.58 ! [C: char] :
% 0.22/0.58 ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
% 0.22/0.58 = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % String.char_of_ascii_of
% 0.22/0.58 thf(fact_9616_DERIV__real__root__generic,axiom,
% 0.22/0.58 ! [N: nat,X: real,D3: real] :
% 0.22/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.58 => ( ( X != zero_zero_real )
% 0.22/0.58 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.58 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.58 => ( D3
% 0.22/0.58 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 0.22/0.58 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.58 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.22/0.58 => ( D3
% 0.22/0.58 = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 0.22/0.58 => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.58 => ( D3
% 0.22/0.58 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_real_root_generic
% 0.22/0.58 thf(fact_9617_has__real__derivative__neg__dec__right,axiom,
% 0.22/0.58 ! [F: real > real,L: real,X: real,S: set_real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S ) )
% 0.22/0.58 => ( ( ord_less_real @ L @ zero_zero_real )
% 0.22/0.58 => ? [D2: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.22/0.58 & ! [H2: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.22/0.58 => ( ( member_real @ ( plus_plus_real @ X @ H2 ) @ S )
% 0.22/0.58 => ( ( ord_less_real @ H2 @ D2 )
% 0.22/0.58 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % has_real_derivative_neg_dec_right
% 0.22/0.58 thf(fact_9618_has__real__derivative__pos__inc__right,axiom,
% 0.22/0.58 ! [F: real > real,L: real,X: real,S: set_real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S ) )
% 0.22/0.58 => ( ( ord_less_real @ zero_zero_real @ L )
% 0.22/0.58 => ? [D2: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.22/0.58 & ! [H2: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.22/0.58 => ( ( member_real @ ( plus_plus_real @ X @ H2 ) @ S )
% 0.22/0.58 => ( ( ord_less_real @ H2 @ D2 )
% 0.22/0.58 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H2 ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % has_real_derivative_pos_inc_right
% 0.22/0.58 thf(fact_9619_has__real__derivative__pos__inc__left,axiom,
% 0.22/0.58 ! [F: real > real,L: real,X: real,S: set_real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S ) )
% 0.22/0.58 => ( ( ord_less_real @ zero_zero_real @ L )
% 0.22/0.58 => ? [D2: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.22/0.58 & ! [H2: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.22/0.58 => ( ( member_real @ ( minus_minus_real @ X @ H2 ) @ S )
% 0.22/0.58 => ( ( ord_less_real @ H2 @ D2 )
% 0.22/0.58 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % has_real_derivative_pos_inc_left
% 0.22/0.58 thf(fact_9620_has__real__derivative__neg__dec__left,axiom,
% 0.22/0.58 ! [F: real > real,L: real,X: real,S: set_real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S ) )
% 0.22/0.58 => ( ( ord_less_real @ L @ zero_zero_real )
% 0.22/0.58 => ? [D2: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.22/0.58 & ! [H2: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.22/0.58 => ( ( member_real @ ( minus_minus_real @ X @ H2 ) @ S )
% 0.22/0.58 => ( ( ord_less_real @ H2 @ D2 )
% 0.22/0.58 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H2 ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % has_real_derivative_neg_dec_left
% 0.22/0.58 thf(fact_9621_deriv__nonneg__imp__mono,axiom,
% 0.22/0.58 ! [A: real,B: real,G: real > real,G2: real > real] :
% 0.22/0.58 ( ! [X4: real] :
% 0.22/0.58 ( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 0.22/0.58 => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X4 ) ) )
% 0.22/0.58 => ( ( ord_less_eq_real @ A @ B )
% 0.22/0.58 => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % deriv_nonneg_imp_mono
% 0.22/0.58 thf(fact_9622_DERIV__nonneg__imp__nondecreasing,axiom,
% 0.22/0.58 ! [A: real,B: real,F: real > real] :
% 0.22/0.58 ( ( ord_less_eq_real @ A @ B )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ A @ X4 )
% 0.22/0.58 => ( ( ord_less_eq_real @ X4 @ B )
% 0.22/0.58 => ? [Y3: real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.22/0.58 & ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ) )
% 0.22/0.58 => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_nonneg_imp_nondecreasing
% 0.22/0.58 thf(fact_9623_DERIV__nonpos__imp__nonincreasing,axiom,
% 0.22/0.58 ! [A: real,B: real,F: real > real] :
% 0.22/0.58 ( ( ord_less_eq_real @ A @ B )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ A @ X4 )
% 0.22/0.58 => ( ( ord_less_eq_real @ X4 @ B )
% 0.22/0.58 => ? [Y3: real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.22/0.58 & ( ord_less_eq_real @ Y3 @ zero_zero_real ) ) ) )
% 0.22/0.58 => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_nonpos_imp_nonincreasing
% 0.22/0.58 thf(fact_9624_DERIV__pos__imp__increasing,axiom,
% 0.22/0.58 ! [A: real,B: real,F: real > real] :
% 0.22/0.58 ( ( ord_less_real @ A @ B )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ A @ X4 )
% 0.22/0.58 => ( ( ord_less_eq_real @ X4 @ B )
% 0.22/0.58 => ? [Y3: real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.22/0.58 & ( ord_less_real @ zero_zero_real @ Y3 ) ) ) )
% 0.22/0.58 => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_pos_imp_increasing
% 0.22/0.58 thf(fact_9625_DERIV__neg__imp__decreasing,axiom,
% 0.22/0.58 ! [A: real,B: real,F: real > real] :
% 0.22/0.58 ( ( ord_less_real @ A @ B )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ A @ X4 )
% 0.22/0.58 => ( ( ord_less_eq_real @ X4 @ B )
% 0.22/0.58 => ? [Y3: real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.22/0.58 & ( ord_less_real @ Y3 @ zero_zero_real ) ) ) )
% 0.22/0.58 => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_neg_imp_decreasing
% 0.22/0.58 thf(fact_9626_DERIV__isconst__all,axiom,
% 0.22/0.58 ! [F: real > real,X: real,Y4: real] :
% 0.22/0.58 ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.22/0.58 => ( ( F @ X )
% 0.22/0.58 = ( F @ Y4 ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_isconst_all
% 0.22/0.58 thf(fact_9627_DERIV__pos__inc__right,axiom,
% 0.22/0.58 ! [F: real > real,L: real,X: real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( ( ord_less_real @ zero_zero_real @ L )
% 0.22/0.58 => ? [D2: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.22/0.58 & ! [H2: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.22/0.58 => ( ( ord_less_real @ H2 @ D2 )
% 0.22/0.58 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H2 ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_pos_inc_right
% 0.22/0.58 thf(fact_9628_DERIV__neg__dec__right,axiom,
% 0.22/0.58 ! [F: real > real,L: real,X: real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( ( ord_less_real @ L @ zero_zero_real )
% 0.22/0.58 => ? [D2: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.22/0.58 & ! [H2: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.22/0.58 => ( ( ord_less_real @ H2 @ D2 )
% 0.22/0.58 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_neg_dec_right
% 0.22/0.58 thf(fact_9629_DERIV__const__ratio__const2,axiom,
% 0.22/0.58 ! [A: real,B: real,F: real > real,K: real] :
% 0.22/0.58 ( ( A != B )
% 0.22/0.58 => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.22/0.58 => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
% 0.22/0.58 = K ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_const_ratio_const2
% 0.22/0.58 thf(fact_9630_DERIV__const__ratio__const,axiom,
% 0.22/0.58 ! [A: real,B: real,F: real > real,K: real] :
% 0.22/0.58 ( ( A != B )
% 0.22/0.58 => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.22/0.58 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.22/0.58 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ K ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_const_ratio_const
% 0.22/0.58 thf(fact_9631_DERIV__pos__inc__left,axiom,
% 0.22/0.58 ! [F: real > real,L: real,X: real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( ( ord_less_real @ zero_zero_real @ L )
% 0.22/0.58 => ? [D2: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.22/0.58 & ! [H2: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.22/0.58 => ( ( ord_less_real @ H2 @ D2 )
% 0.22/0.58 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_pos_inc_left
% 0.22/0.58 thf(fact_9632_DERIV__neg__dec__left,axiom,
% 0.22/0.58 ! [F: real > real,L: real,X: real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( ( ord_less_real @ L @ zero_zero_real )
% 0.22/0.58 => ? [D2: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.22/0.58 & ! [H2: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.22/0.58 => ( ( ord_less_real @ H2 @ D2 )
% 0.22/0.58 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H2 ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_neg_dec_left
% 0.22/0.58 thf(fact_9633_DERIV__isconst3,axiom,
% 0.22/0.58 ! [A: real,B: real,X: real,Y4: real,F: real > real] :
% 0.22/0.58 ( ( ord_less_real @ A @ B )
% 0.22/0.58 => ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.22/0.58 => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.22/0.58 => ( ( F @ X )
% 0.22/0.58 = ( F @ Y4 ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_isconst3
% 0.22/0.58 thf(fact_9634_MVT2,axiom,
% 0.22/0.58 ! [A: real,B: real,F: real > real,F2: real > real] :
% 0.22/0.58 ( ( ord_less_real @ A @ B )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ A @ X4 )
% 0.22/0.58 => ( ( ord_less_eq_real @ X4 @ B )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ F @ ( F2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.22/0.58 => ? [Z2: real] :
% 0.22/0.58 ( ( ord_less_real @ A @ Z2 )
% 0.22/0.58 & ( ord_less_real @ Z2 @ B )
% 0.22/0.58 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.22/0.58 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F2 @ Z2 ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % MVT2
% 0.22/0.58 thf(fact_9635_DERIV__local__const,axiom,
% 0.22/0.58 ! [F: real > real,L: real,X: real,D: real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.22/0.58 => ( ! [Y: real] :
% 0.22/0.58 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) @ D )
% 0.22/0.58 => ( ( F @ X )
% 0.22/0.58 = ( F @ Y ) ) )
% 0.22/0.58 => ( L = zero_zero_real ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_local_const
% 0.22/0.58 thf(fact_9636_DERIV__ln,axiom,
% 0.22/0.58 ! [X: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_ln
% 0.22/0.58 thf(fact_9637_DERIV__const__average,axiom,
% 0.22/0.58 ! [A: real,B: real,V: real > real,K: real] :
% 0.22/0.58 ( ( A != B )
% 0.22/0.58 => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.22/0.58 => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.22/0.58 = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_const_average
% 0.22/0.58 thf(fact_9638_DERIV__local__min,axiom,
% 0.22/0.58 ! [F: real > real,L: real,X: real,D: real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.22/0.58 => ( ! [Y: real] :
% 0.22/0.58 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) @ D )
% 0.22/0.58 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
% 0.22/0.58 => ( L = zero_zero_real ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_local_min
% 0.22/0.58 thf(fact_9639_DERIV__local__max,axiom,
% 0.22/0.58 ! [F: real > real,L: real,X: real,D: real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.22/0.58 => ( ! [Y: real] :
% 0.22/0.58 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) @ D )
% 0.22/0.58 => ( ord_less_eq_real @ ( F @ Y ) @ ( F @ X ) ) )
% 0.22/0.58 => ( L = zero_zero_real ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_local_max
% 0.22/0.58 thf(fact_9640_DERIV__ln__divide,axiom,
% 0.22/0.58 ! [X: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_ln_divide
% 0.22/0.58 thf(fact_9641_DERIV__pow,axiom,
% 0.22/0.58 ! [N: nat,X: real,S2: set_real] :
% 0.22/0.58 ( has_fi5821293074295781190e_real
% 0.22/0.58 @ ^ [X3: real] : ( power_power_real @ X3 @ N )
% 0.22/0.58 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ S2 ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_pow
% 0.22/0.58 thf(fact_9642_DERIV__fun__pow,axiom,
% 0.22/0.58 ! [G: real > real,M: real,X: real,N: nat] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( has_fi5821293074295781190e_real
% 0.22/0.58 @ ^ [X3: real] : ( power_power_real @ ( G @ X3 ) @ N )
% 0.22/0.58 @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_fun_pow
% 0.22/0.58 thf(fact_9643_has__real__derivative__powr,axiom,
% 0.22/0.58 ! [Z: real,R2: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.22/0.58 => ( has_fi5821293074295781190e_real
% 0.22/0.58 @ ^ [Z5: real] : ( powr_real @ Z5 @ R2 )
% 0.22/0.58 @ ( times_times_real @ R2 @ ( powr_real @ Z @ ( minus_minus_real @ R2 @ one_one_real ) ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % has_real_derivative_powr
% 0.22/0.58 thf(fact_9644_DERIV__series_H,axiom,
% 0.22/0.58 ! [F: real > nat > real,F2: real > nat > real,X0: real,A: real,B: real,L4: nat > real] :
% 0.22/0.58 ( ! [N2: nat] :
% 0.22/0.58 ( has_fi5821293074295781190e_real
% 0.22/0.58 @ ^ [X3: real] : ( F @ X3 @ N2 )
% 0.22/0.58 @ ( F2 @ X0 @ N2 )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.22/0.58 => ( summable_real @ ( F @ X4 ) ) )
% 0.22/0.58 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.22/0.58 => ( ( summable_real @ ( F2 @ X0 ) )
% 0.22/0.58 => ( ( summable_real @ L4 )
% 0.22/0.58 => ( ! [N2: nat,X4: real,Y: real] :
% 0.22/0.58 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.22/0.58 => ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.22/0.58 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X4 @ N2 ) @ ( F @ Y @ N2 ) ) ) @ ( times_times_real @ ( L4 @ N2 ) @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y ) ) ) ) ) )
% 0.22/0.58 => ( has_fi5821293074295781190e_real
% 0.22/0.58 @ ^ [X3: real] : ( suminf_real @ ( F @ X3 ) )
% 0.22/0.58 @ ( suminf_real @ ( F2 @ X0 ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_series'
% 0.22/0.58 thf(fact_9645_DERIV__log,axiom,
% 0.22/0.58 ! [X: real,B: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_log
% 0.22/0.58 thf(fact_9646_DERIV__fun__powr,axiom,
% 0.22/0.58 ! [G: real > real,M: real,X: real,R2: real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 0.22/0.58 => ( has_fi5821293074295781190e_real
% 0.22/0.58 @ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ R2 )
% 0.22/0.58 @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_fun_powr
% 0.22/0.58 thf(fact_9647_DERIV__powr,axiom,
% 0.22/0.58 ! [G: real > real,M: real,X: real,F: real > real,R2: real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 0.22/0.58 => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( has_fi5821293074295781190e_real
% 0.22/0.58 @ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ ( F @ X3 ) )
% 0.22/0.58 @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_powr
% 0.22/0.58 thf(fact_9648_artanh__real__has__field__derivative,axiom,
% 0.22/0.58 ! [X: real,A2: set_real] :
% 0.22/0.58 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % artanh_real_has_field_derivative
% 0.22/0.58 thf(fact_9649_DERIV__real__sqrt,axiom,
% 0.22/0.58 ! [X: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_real_sqrt
% 0.22/0.58 thf(fact_9650_DERIV__arctan,axiom,
% 0.22/0.58 ! [X: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_arctan
% 0.22/0.58 thf(fact_9651_arsinh__real__has__field__derivative,axiom,
% 0.22/0.58 ! [X: real,A2: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ).
% 0.22/0.58
% 0.22/0.58 % arsinh_real_has_field_derivative
% 0.22/0.58 thf(fact_9652_DERIV__real__sqrt__generic,axiom,
% 0.22/0.58 ! [X: real,D3: real] :
% 0.22/0.58 ( ( X != zero_zero_real )
% 0.22/0.58 => ( ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.58 => ( D3
% 0.22/0.58 = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.22/0.58 => ( ( ( ord_less_real @ X @ zero_zero_real )
% 0.22/0.58 => ( D3
% 0.22/0.58 = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ sqrt @ D3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_real_sqrt_generic
% 0.22/0.58 thf(fact_9653_arcosh__real__has__field__derivative,axiom,
% 0.22/0.58 ! [X: real,A2: set_real] :
% 0.22/0.58 ( ( ord_less_real @ one_one_real @ X )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % arcosh_real_has_field_derivative
% 0.22/0.58 thf(fact_9654_DERIV__power__series_H,axiom,
% 0.22/0.58 ! [R: real,F: nat > real,X0: real] :
% 0.22/0.58 ( ! [X4: real] :
% 0.22/0.58 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 0.22/0.58 => ( summable_real
% 0.22/0.58 @ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( F @ N4 ) @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ ( power_power_real @ X4 @ N4 ) ) ) )
% 0.22/0.58 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 0.22/0.58 => ( ( ord_less_real @ zero_zero_real @ R )
% 0.22/0.58 => ( has_fi5821293074295781190e_real
% 0.22/0.58 @ ^ [X3: real] :
% 0.22/0.58 ( suminf_real
% 0.22/0.58 @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ X3 @ ( suc @ N4 ) ) ) )
% 0.22/0.58 @ ( suminf_real
% 0.22/0.58 @ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( F @ N4 ) @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ ( power_power_real @ X0 @ N4 ) ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_power_series'
% 0.22/0.58 thf(fact_9655_DERIV__real__root,axiom,
% 0.22/0.58 ! [N: nat,X: real] :
% 0.22/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.58 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_real_root
% 0.22/0.58 thf(fact_9656_DERIV__arccos,axiom,
% 0.22/0.58 ! [X: real] :
% 0.22/0.58 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.22/0.58 => ( ( ord_less_real @ X @ one_one_real )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_arccos
% 0.22/0.58 thf(fact_9657_DERIV__arcsin,axiom,
% 0.22/0.58 ! [X: real] :
% 0.22/0.58 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.22/0.58 => ( ( ord_less_real @ X @ one_one_real )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_arcsin
% 0.22/0.58 thf(fact_9658_Maclaurin__all__le__objl,axiom,
% 0.22/0.58 ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 0.22/0.58 ( ( ( ( Diff @ zero_zero_nat )
% 0.22/0.58 = F )
% 0.22/0.58 & ! [M4: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.22/0.58 => ? [T3: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.22/0.58 & ( ( F @ X )
% 0.22/0.58 = ( plus_plus_real
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.58 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Maclaurin_all_le_objl
% 0.22/0.58 thf(fact_9659_Maclaurin__all__le,axiom,
% 0.22/0.58 ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 0.22/0.58 ( ( ( Diff @ zero_zero_nat )
% 0.22/0.58 = F )
% 0.22/0.58 => ( ! [M4: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.22/0.58 => ? [T3: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.22/0.58 & ( ( F @ X )
% 0.22/0.58 = ( plus_plus_real
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.58 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Maclaurin_all_le
% 0.22/0.58 thf(fact_9660_DERIV__odd__real__root,axiom,
% 0.22/0.58 ! [N: nat,X: real] :
% 0.22/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.58 => ( ( X != zero_zero_real )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_odd_real_root
% 0.22/0.58 thf(fact_9661_Maclaurin__minus,axiom,
% 0.22/0.58 ! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
% 0.22/0.58 ( ( ord_less_real @ H @ zero_zero_real )
% 0.22/0.58 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.58 => ( ( ( Diff @ zero_zero_nat )
% 0.22/0.58 = F )
% 0.22/0.58 => ( ! [M4: nat,T3: real] :
% 0.22/0.58 ( ( ( ord_less_nat @ M4 @ N )
% 0.22/0.58 & ( ord_less_eq_real @ H @ T3 )
% 0.22/0.58 & ( ord_less_eq_real @ T3 @ zero_zero_real ) )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.22/0.58 => ? [T3: real] :
% 0.22/0.58 ( ( ord_less_real @ H @ T3 )
% 0.22/0.58 & ( ord_less_real @ T3 @ zero_zero_real )
% 0.22/0.58 & ( ( F @ H )
% 0.22/0.58 = ( plus_plus_real
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H @ M5 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.58 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Maclaurin_minus
% 0.22/0.58 thf(fact_9662_Maclaurin2,axiom,
% 0.22/0.58 ! [H: real,Diff: nat > real > real,F: real > real,N: nat] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ H )
% 0.22/0.58 => ( ( ( Diff @ zero_zero_nat )
% 0.22/0.58 = F )
% 0.22/0.58 => ( ! [M4: nat,T3: real] :
% 0.22/0.58 ( ( ( ord_less_nat @ M4 @ N )
% 0.22/0.58 & ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.22/0.58 & ( ord_less_eq_real @ T3 @ H ) )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.22/0.58 => ? [T3: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ T3 )
% 0.22/0.58 & ( ord_less_eq_real @ T3 @ H )
% 0.22/0.58 & ( ( F @ H )
% 0.22/0.58 = ( plus_plus_real
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H @ M5 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.58 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Maclaurin2
% 0.22/0.58 thf(fact_9663_Maclaurin,axiom,
% 0.22/0.58 ! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ H )
% 0.22/0.58 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.58 => ( ( ( Diff @ zero_zero_nat )
% 0.22/0.58 = F )
% 0.22/0.58 => ( ! [M4: nat,T3: real] :
% 0.22/0.58 ( ( ( ord_less_nat @ M4 @ N )
% 0.22/0.58 & ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.22/0.58 & ( ord_less_eq_real @ T3 @ H ) )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.22/0.58 => ? [T3: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ T3 )
% 0.22/0.58 & ( ord_less_real @ T3 @ H )
% 0.22/0.58 & ( ( F @ H )
% 0.22/0.58 = ( plus_plus_real
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H @ M5 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.58 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Maclaurin
% 0.22/0.58 thf(fact_9664_Maclaurin__all__lt,axiom,
% 0.22/0.58 ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 0.22/0.58 ( ( ( Diff @ zero_zero_nat )
% 0.22/0.58 = F )
% 0.22/0.58 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.58 => ( ( X != zero_zero_real )
% 0.22/0.58 => ( ! [M4: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.22/0.58 => ? [T3: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
% 0.22/0.58 & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.22/0.58 & ( ( F @ X )
% 0.22/0.58 = ( plus_plus_real
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.58 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Maclaurin_all_lt
% 0.22/0.58 thf(fact_9665_Maclaurin__bi__le,axiom,
% 0.22/0.58 ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 0.22/0.58 ( ( ( Diff @ zero_zero_nat )
% 0.22/0.58 = F )
% 0.22/0.58 => ( ! [M4: nat,T3: real] :
% 0.22/0.58 ( ( ( ord_less_nat @ M4 @ N )
% 0.22/0.58 & ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) ) )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.22/0.58 => ? [T3: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.22/0.58 & ( ( F @ X )
% 0.22/0.58 = ( plus_plus_real
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.58 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Maclaurin_bi_le
% 0.22/0.58 thf(fact_9666_Taylor__down,axiom,
% 0.22/0.58 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 0.22/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.58 => ( ( ( Diff @ zero_zero_nat )
% 0.22/0.58 = F )
% 0.22/0.58 => ( ! [M4: nat,T3: real] :
% 0.22/0.58 ( ( ( ord_less_nat @ M4 @ N )
% 0.22/0.58 & ( ord_less_eq_real @ A @ T3 )
% 0.22/0.58 & ( ord_less_eq_real @ T3 @ B ) )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.22/0.58 => ( ( ord_less_real @ A @ C )
% 0.22/0.58 => ( ( ord_less_eq_real @ C @ B )
% 0.22/0.58 => ? [T3: real] :
% 0.22/0.58 ( ( ord_less_real @ A @ T3 )
% 0.22/0.58 & ( ord_less_real @ T3 @ C )
% 0.22/0.58 & ( ( F @ A )
% 0.22/0.58 = ( plus_plus_real
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M5 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.58 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Taylor_down
% 0.22/0.58 thf(fact_9667_Taylor__up,axiom,
% 0.22/0.58 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 0.22/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.58 => ( ( ( Diff @ zero_zero_nat )
% 0.22/0.58 = F )
% 0.22/0.58 => ( ! [M4: nat,T3: real] :
% 0.22/0.58 ( ( ( ord_less_nat @ M4 @ N )
% 0.22/0.58 & ( ord_less_eq_real @ A @ T3 )
% 0.22/0.58 & ( ord_less_eq_real @ T3 @ B ) )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.22/0.58 => ( ( ord_less_eq_real @ A @ C )
% 0.22/0.58 => ( ( ord_less_real @ C @ B )
% 0.22/0.58 => ? [T3: real] :
% 0.22/0.58 ( ( ord_less_real @ C @ T3 )
% 0.22/0.58 & ( ord_less_real @ T3 @ B )
% 0.22/0.58 & ( ( F @ B )
% 0.22/0.58 = ( plus_plus_real
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M5 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.58 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Taylor_up
% 0.22/0.58 thf(fact_9668_Taylor,axiom,
% 0.22/0.58 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
% 0.22/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.58 => ( ( ( Diff @ zero_zero_nat )
% 0.22/0.58 = F )
% 0.22/0.58 => ( ! [M4: nat,T3: real] :
% 0.22/0.58 ( ( ( ord_less_nat @ M4 @ N )
% 0.22/0.58 & ( ord_less_eq_real @ A @ T3 )
% 0.22/0.58 & ( ord_less_eq_real @ T3 @ B ) )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.22/0.58 => ( ( ord_less_eq_real @ A @ C )
% 0.22/0.58 => ( ( ord_less_eq_real @ C @ B )
% 0.22/0.58 => ( ( ord_less_eq_real @ A @ X )
% 0.22/0.58 => ( ( ord_less_eq_real @ X @ B )
% 0.22/0.58 => ( ( X != C )
% 0.22/0.58 => ? [T3: real] :
% 0.22/0.58 ( ( ( ord_less_real @ X @ C )
% 0.22/0.58 => ( ( ord_less_real @ X @ T3 )
% 0.22/0.58 & ( ord_less_real @ T3 @ C ) ) )
% 0.22/0.58 & ( ~ ( ord_less_real @ X @ C )
% 0.22/0.58 => ( ( ord_less_real @ C @ T3 )
% 0.22/0.58 & ( ord_less_real @ T3 @ X ) ) )
% 0.22/0.58 & ( ( F @ X )
% 0.22/0.58 = ( plus_plus_real
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M5 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ N ) )
% 0.22/0.58 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Taylor
% 0.22/0.58 thf(fact_9669_Maclaurin__lemma2,axiom,
% 0.22/0.58 ! [N: nat,H: real,Diff: nat > real > real,K: nat,B4: real] :
% 0.22/0.58 ( ! [M4: nat,T3: real] :
% 0.22/0.58 ( ( ( ord_less_nat @ M4 @ N )
% 0.22/0.58 & ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.22/0.58 & ( ord_less_eq_real @ T3 @ H ) )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.22/0.58 => ( ( N
% 0.22/0.58 = ( suc @ K ) )
% 0.22/0.58 => ! [M2: nat,T4: real] :
% 0.22/0.58 ( ( ( ord_less_nat @ M2 @ N )
% 0.22/0.58 & ( ord_less_eq_real @ zero_zero_real @ T4 )
% 0.22/0.58 & ( ord_less_eq_real @ T4 @ H ) )
% 0.22/0.58 => ( has_fi5821293074295781190e_real
% 0.22/0.58 @ ^ [U3: real] :
% 0.22/0.58 ( minus_minus_real @ ( Diff @ M2 @ U3 )
% 0.22/0.58 @ ( plus_plus_real
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ U3 @ P6 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M2 ) ) )
% 0.22/0.58 @ ( times_times_real @ B4 @ ( divide_divide_real @ ( power_power_real @ U3 @ ( minus_minus_nat @ N @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) )
% 0.22/0.58 @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T4 )
% 0.22/0.58 @ ( plus_plus_real
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ T4 @ P6 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) )
% 0.22/0.58 @ ( times_times_real @ B4 @ ( divide_divide_real @ ( power_power_real @ T4 @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) ) ) ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Maclaurin_lemma2
% 0.22/0.58 thf(fact_9670_DERIV__arctan__series,axiom,
% 0.22/0.58 ! [X: real] :
% 0.22/0.58 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.22/0.58 => ( has_fi5821293074295781190e_real
% 0.22/0.58 @ ^ [X10: real] :
% 0.22/0.58 ( suminf_real
% 0.22/0.58 @ ^ [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 @ X10 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 0.22/0.58 @ ( suminf_real
% 0.22/0.58 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_arctan_series
% 0.22/0.58 thf(fact_9671_DERIV__even__real__root,axiom,
% 0.22/0.58 ! [N: nat,X: real] :
% 0.22/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.58 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.58 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_even_real_root
% 0.22/0.58 thf(fact_9672_isCont__Lb__Ub,axiom,
% 0.22/0.58 ! [A: real,B: real,F: real > real] :
% 0.22/0.58 ( ( ord_less_eq_real @ A @ B )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( ( ord_less_eq_real @ A @ X4 )
% 0.22/0.58 & ( ord_less_eq_real @ X4 @ B ) )
% 0.22/0.58 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
% 0.22/0.58 => ? [L5: real,M8: real] :
% 0.22/0.58 ( ! [X5: real] :
% 0.22/0.58 ( ( ( ord_less_eq_real @ A @ X5 )
% 0.22/0.58 & ( ord_less_eq_real @ X5 @ B ) )
% 0.22/0.58 => ( ( ord_less_eq_real @ L5 @ ( F @ X5 ) )
% 0.22/0.58 & ( ord_less_eq_real @ ( F @ X5 ) @ M8 ) ) )
% 0.22/0.58 & ! [Y3: real] :
% 0.22/0.58 ( ( ( ord_less_eq_real @ L5 @ Y3 )
% 0.22/0.58 & ( ord_less_eq_real @ Y3 @ M8 ) )
% 0.22/0.58 => ? [X4: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ A @ X4 )
% 0.22/0.58 & ( ord_less_eq_real @ X4 @ B )
% 0.22/0.58 & ( ( F @ X4 )
% 0.22/0.58 = Y3 ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % isCont_Lb_Ub
% 0.22/0.58 thf(fact_9673_LIM__fun__less__zero,axiom,
% 0.22/0.58 ! [F: real > real,L: real,C: real] :
% 0.22/0.58 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.22/0.58 => ( ( ord_less_real @ L @ zero_zero_real )
% 0.22/0.58 => ? [R3: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.22/0.58 & ! [X5: real] :
% 0.22/0.58 ( ( ( X5 != C )
% 0.22/0.58 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
% 0.22/0.58 => ( ord_less_real @ ( F @ X5 ) @ zero_zero_real ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % LIM_fun_less_zero
% 0.22/0.58 thf(fact_9674_LIM__fun__not__zero,axiom,
% 0.22/0.58 ! [F: real > real,L: real,C: real] :
% 0.22/0.58 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.22/0.58 => ( ( L != zero_zero_real )
% 0.22/0.58 => ? [R3: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.22/0.58 & ! [X5: real] :
% 0.22/0.58 ( ( ( X5 != C )
% 0.22/0.58 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
% 0.22/0.58 => ( ( F @ X5 )
% 0.22/0.58 != zero_zero_real ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % LIM_fun_not_zero
% 0.22/0.58 thf(fact_9675_LIM__fun__gt__zero,axiom,
% 0.22/0.58 ! [F: real > real,L: real,C: real] :
% 0.22/0.58 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.22/0.58 => ( ( ord_less_real @ zero_zero_real @ L )
% 0.22/0.58 => ? [R3: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.22/0.58 & ! [X5: real] :
% 0.22/0.58 ( ( ( X5 != C )
% 0.22/0.58 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
% 0.22/0.58 => ( ord_less_real @ zero_zero_real @ ( F @ X5 ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % LIM_fun_gt_zero
% 0.22/0.58 thf(fact_9676_isCont__real__sqrt,axiom,
% 0.22/0.58 ! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ sqrt ) ).
% 0.22/0.58
% 0.22/0.58 % isCont_real_sqrt
% 0.22/0.58 thf(fact_9677_isCont__real__root,axiom,
% 0.22/0.58 ! [X: real,N: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( root @ N ) ) ).
% 0.22/0.58
% 0.22/0.58 % isCont_real_root
% 0.22/0.58 thf(fact_9678_isCont__inverse__function2,axiom,
% 0.22/0.58 ! [A: real,X: real,B: real,G: real > real,F: real > real] :
% 0.22/0.58 ( ( ord_less_real @ A @ X )
% 0.22/0.58 => ( ( ord_less_real @ X @ B )
% 0.22/0.58 => ( ! [Z2: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ A @ Z2 )
% 0.22/0.58 => ( ( ord_less_eq_real @ Z2 @ B )
% 0.22/0.58 => ( ( G @ ( F @ Z2 ) )
% 0.22/0.58 = Z2 ) ) )
% 0.22/0.58 => ( ! [Z2: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ A @ Z2 )
% 0.22/0.58 => ( ( ord_less_eq_real @ Z2 @ B )
% 0.22/0.58 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
% 0.22/0.58 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % isCont_inverse_function2
% 0.22/0.58 thf(fact_9679_isCont__ln,axiom,
% 0.22/0.58 ! [X: real] :
% 0.22/0.58 ( ( X != zero_zero_real )
% 0.22/0.58 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ln_ln_real ) ) ).
% 0.22/0.58
% 0.22/0.58 % isCont_ln
% 0.22/0.58 thf(fact_9680_LIM__cos__div__sin,axiom,
% 0.22/0.58 ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( cos_real @ X3 ) @ ( sin_real @ X3 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).
% 0.22/0.58
% 0.22/0.58 % LIM_cos_div_sin
% 0.22/0.58 thf(fact_9681_DERIV__inverse__function,axiom,
% 0.22/0.58 ! [F: real > real,D3: real,G: real > real,X: real,A: real,B: real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ D3 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
% 0.22/0.58 => ( ( D3 != zero_zero_real )
% 0.22/0.58 => ( ( ord_less_real @ A @ X )
% 0.22/0.58 => ( ( ord_less_real @ X @ B )
% 0.22/0.58 => ( ! [Y: real] :
% 0.22/0.58 ( ( ord_less_real @ A @ Y )
% 0.22/0.58 => ( ( ord_less_real @ Y @ B )
% 0.22/0.58 => ( ( F @ ( G @ Y ) )
% 0.22/0.58 = Y ) ) )
% 0.22/0.58 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D3 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_inverse_function
% 0.22/0.58 thf(fact_9682_LIM__less__bound,axiom,
% 0.22/0.58 ! [B: real,X: real,F: real > real] :
% 0.22/0.58 ( ( ord_less_real @ B @ X )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ B @ X ) )
% 0.22/0.58 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.22/0.58 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
% 0.22/0.58 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % LIM_less_bound
% 0.22/0.58 thf(fact_9683_isCont__inverse__function,axiom,
% 0.22/0.58 ! [D: real,X: real,G: real > real,F: real > real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ D )
% 0.22/0.58 => ( ! [Z2: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X ) ) @ D )
% 0.22/0.58 => ( ( G @ ( F @ Z2 ) )
% 0.22/0.58 = Z2 ) )
% 0.22/0.58 => ( ! [Z2: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X ) ) @ D )
% 0.22/0.58 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) )
% 0.22/0.58 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % isCont_inverse_function
% 0.22/0.58 thf(fact_9684_GMVT_H,axiom,
% 0.22/0.58 ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F2: real > real] :
% 0.22/0.58 ( ( ord_less_real @ A @ B )
% 0.22/0.58 => ( ! [Z2: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ A @ Z2 )
% 0.22/0.58 => ( ( ord_less_eq_real @ Z2 @ B )
% 0.22/0.58 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
% 0.22/0.58 => ( ! [Z2: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ A @ Z2 )
% 0.22/0.58 => ( ( ord_less_eq_real @ Z2 @ B )
% 0.22/0.58 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ G ) ) )
% 0.22/0.58 => ( ! [Z2: real] :
% 0.22/0.58 ( ( ord_less_real @ A @ Z2 )
% 0.22/0.58 => ( ( ord_less_real @ Z2 @ B )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
% 0.22/0.58 => ( ! [Z2: real] :
% 0.22/0.58 ( ( ord_less_real @ A @ Z2 )
% 0.22/0.58 => ( ( ord_less_real @ Z2 @ B )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ F @ ( F2 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
% 0.22/0.58 => ? [C3: real] :
% 0.22/0.58 ( ( ord_less_real @ A @ C3 )
% 0.22/0.58 & ( ord_less_real @ C3 @ B )
% 0.22/0.58 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C3 ) )
% 0.22/0.58 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F2 @ C3 ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % GMVT'
% 0.22/0.58 thf(fact_9685_summable__Leibniz_I3_J,axiom,
% 0.22/0.58 ! [A: nat > real] :
% 0.22/0.58 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.22/0.58 => ( ( topolo6980174941875973593q_real @ A )
% 0.22/0.58 => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 0.22/0.58 => ! [N6: nat] :
% 0.22/0.58 ( member_real
% 0.22/0.58 @ ( suminf_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
% 0.22/0.58 @ ( set_or1222579329274155063t_real
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) )
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % summable_Leibniz(3)
% 0.22/0.58 thf(fact_9686_summable__Leibniz_I2_J,axiom,
% 0.22/0.58 ! [A: nat > real] :
% 0.22/0.58 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.22/0.58 => ( ( topolo6980174941875973593q_real @ A )
% 0.22/0.58 => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 0.22/0.58 => ! [N6: nat] :
% 0.22/0.58 ( member_real
% 0.22/0.58 @ ( suminf_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
% 0.22/0.58 @ ( set_or1222579329274155063t_real
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % summable_Leibniz(2)
% 0.22/0.58 thf(fact_9687_filterlim__Suc,axiom,
% 0.22/0.58 filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% 0.22/0.58
% 0.22/0.58 % filterlim_Suc
% 0.22/0.58 thf(fact_9688_mult__nat__right__at__top,axiom,
% 0.22/0.58 ! [C: nat] :
% 0.22/0.58 ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.22/0.58 => ( filterlim_nat_nat
% 0.22/0.58 @ ^ [X3: nat] : ( times_times_nat @ X3 @ C )
% 0.22/0.58 @ at_top_nat
% 0.22/0.58 @ at_top_nat ) ) ).
% 0.22/0.58
% 0.22/0.58 % mult_nat_right_at_top
% 0.22/0.58 thf(fact_9689_mult__nat__left__at__top,axiom,
% 0.22/0.58 ! [C: nat] :
% 0.22/0.58 ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.22/0.58 => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 0.22/0.58
% 0.22/0.58 % mult_nat_left_at_top
% 0.22/0.58 thf(fact_9690_monoseq__convergent,axiom,
% 0.22/0.58 ! [X9: nat > real,B4: real] :
% 0.22/0.58 ( ( topolo6980174941875973593q_real @ X9 )
% 0.22/0.58 => ( ! [I3: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X9 @ I3 ) ) @ B4 )
% 0.22/0.58 => ~ ! [L5: real] :
% 0.22/0.58 ~ ( filterlim_nat_real @ X9 @ ( topolo2815343760600316023s_real @ L5 ) @ at_top_nat ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % monoseq_convergent
% 0.22/0.58 thf(fact_9691_LIMSEQ__root,axiom,
% 0.22/0.58 ( filterlim_nat_real
% 0.22/0.58 @ ^ [N4: nat] : ( root @ N4 @ ( semiri5074537144036343181t_real @ N4 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.22/0.58 @ at_top_nat ) ).
% 0.22/0.58
% 0.22/0.58 % LIMSEQ_root
% 0.22/0.58 thf(fact_9692_nested__sequence__unique,axiom,
% 0.22/0.58 ! [F: nat > real,G: nat > real] :
% 0.22/0.58 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.22/0.58 => ( ! [N2: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N2 ) ) @ ( G @ N2 ) )
% 0.22/0.58 => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.22/0.58 => ( ( filterlim_nat_real
% 0.22/0.58 @ ^ [N4: nat] : ( minus_minus_real @ ( F @ N4 ) @ ( G @ N4 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.22/0.58 @ at_top_nat )
% 0.22/0.58 => ? [L2: real] :
% 0.22/0.58 ( ! [N6: nat] : ( ord_less_eq_real @ ( F @ N6 ) @ L2 )
% 0.22/0.58 & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat )
% 0.22/0.58 & ! [N6: nat] : ( ord_less_eq_real @ L2 @ ( G @ N6 ) )
% 0.22/0.58 & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % nested_sequence_unique
% 0.22/0.58 thf(fact_9693_LIMSEQ__inverse__zero,axiom,
% 0.22/0.58 ! [X9: nat > real] :
% 0.22/0.58 ( ! [R3: real] :
% 0.22/0.58 ? [N7: nat] :
% 0.22/0.58 ! [N2: nat] :
% 0.22/0.58 ( ( ord_less_eq_nat @ N7 @ N2 )
% 0.22/0.58 => ( ord_less_real @ R3 @ ( X9 @ N2 ) ) )
% 0.22/0.58 => ( filterlim_nat_real
% 0.22/0.58 @ ^ [N4: nat] : ( inverse_inverse_real @ ( X9 @ N4 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.22/0.58 @ at_top_nat ) ) ).
% 0.22/0.58
% 0.22/0.58 % LIMSEQ_inverse_zero
% 0.22/0.58 thf(fact_9694_lim__inverse__n_H,axiom,
% 0.22/0.58 ( filterlim_nat_real
% 0.22/0.58 @ ^ [N4: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N4 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.22/0.58 @ at_top_nat ) ).
% 0.22/0.58
% 0.22/0.58 % lim_inverse_n'
% 0.22/0.58 thf(fact_9695_LIMSEQ__root__const,axiom,
% 0.22/0.58 ! [C: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ C )
% 0.22/0.58 => ( filterlim_nat_real
% 0.22/0.58 @ ^ [N4: nat] : ( root @ N4 @ C )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.22/0.58 @ at_top_nat ) ) ).
% 0.22/0.58
% 0.22/0.58 % LIMSEQ_root_const
% 0.22/0.58 thf(fact_9696_LIMSEQ__inverse__real__of__nat,axiom,
% 0.22/0.58 ( filterlim_nat_real
% 0.22/0.58 @ ^ [N4: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.22/0.58 @ at_top_nat ) ).
% 0.22/0.58
% 0.22/0.58 % LIMSEQ_inverse_real_of_nat
% 0.22/0.58 thf(fact_9697_LIMSEQ__inverse__real__of__nat__add,axiom,
% 0.22/0.58 ! [R2: real] :
% 0.22/0.58 ( filterlim_nat_real
% 0.22/0.58 @ ^ [N4: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ R2 )
% 0.22/0.58 @ at_top_nat ) ).
% 0.22/0.58
% 0.22/0.58 % LIMSEQ_inverse_real_of_nat_add
% 0.22/0.58 thf(fact_9698_increasing__LIMSEQ,axiom,
% 0.22/0.58 ! [F: nat > real,L: real] :
% 0.22/0.58 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.22/0.58 => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ L )
% 0.22/0.58 => ( ! [E: real] :
% 0.22/0.58 ( ( ord_less_real @ zero_zero_real @ E )
% 0.22/0.58 => ? [N6: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N6 ) @ E ) ) )
% 0.22/0.58 => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % increasing_LIMSEQ
% 0.22/0.58 thf(fact_9699_LIMSEQ__realpow__zero,axiom,
% 0.22/0.58 ! [X: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.58 => ( ( ord_less_real @ X @ one_one_real )
% 0.22/0.58 => ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % LIMSEQ_realpow_zero
% 0.22/0.58 thf(fact_9700_LIMSEQ__divide__realpow__zero,axiom,
% 0.22/0.58 ! [X: real,A: real] :
% 0.22/0.58 ( ( ord_less_real @ one_one_real @ X )
% 0.22/0.58 => ( filterlim_nat_real
% 0.22/0.58 @ ^ [N4: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N4 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.22/0.58 @ at_top_nat ) ) ).
% 0.22/0.58
% 0.22/0.58 % LIMSEQ_divide_realpow_zero
% 0.22/0.58 thf(fact_9701_LIMSEQ__abs__realpow__zero,axiom,
% 0.22/0.58 ! [C: real] :
% 0.22/0.58 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.22/0.58 => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.22/0.58
% 0.22/0.58 % LIMSEQ_abs_realpow_zero
% 0.22/0.58 thf(fact_9702_LIMSEQ__abs__realpow__zero2,axiom,
% 0.22/0.58 ! [C: real] :
% 0.22/0.58 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.22/0.58 => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.22/0.58
% 0.22/0.58 % LIMSEQ_abs_realpow_zero2
% 0.22/0.58 thf(fact_9703_LIMSEQ__inverse__realpow__zero,axiom,
% 0.22/0.58 ! [X: real] :
% 0.22/0.58 ( ( ord_less_real @ one_one_real @ X )
% 0.22/0.58 => ( filterlim_nat_real
% 0.22/0.58 @ ^ [N4: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N4 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.22/0.58 @ at_top_nat ) ) ).
% 0.22/0.58
% 0.22/0.58 % LIMSEQ_inverse_realpow_zero
% 0.22/0.58 thf(fact_9704_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
% 0.22/0.58 ! [R2: real] :
% 0.22/0.58 ( filterlim_nat_real
% 0.22/0.58 @ ^ [N4: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ R2 )
% 0.22/0.58 @ at_top_nat ) ).
% 0.22/0.58
% 0.22/0.58 % LIMSEQ_inverse_real_of_nat_add_minus
% 0.22/0.58 thf(fact_9705_tendsto__exp__limit__sequentially,axiom,
% 0.22/0.58 ! [X: real] :
% 0.22/0.58 ( filterlim_nat_real
% 0.22/0.58 @ ^ [N4: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ) @ N4 )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 0.22/0.58 @ at_top_nat ) ).
% 0.22/0.58
% 0.22/0.58 % tendsto_exp_limit_sequentially
% 0.22/0.58 thf(fact_9706_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
% 0.22/0.58 ! [R2: real] :
% 0.22/0.58 ( filterlim_nat_real
% 0.22/0.58 @ ^ [N4: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) ) ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ R2 )
% 0.22/0.58 @ at_top_nat ) ).
% 0.22/0.58
% 0.22/0.58 % LIMSEQ_inverse_real_of_nat_add_minus_mult
% 0.22/0.58 thf(fact_9707_summable__Leibniz_I1_J,axiom,
% 0.22/0.58 ! [A: nat > real] :
% 0.22/0.58 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.22/0.58 => ( ( topolo6980174941875973593q_real @ A )
% 0.22/0.58 => ( summable_real
% 0.22/0.58 @ ^ [N4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( A @ N4 ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % summable_Leibniz(1)
% 0.22/0.58 thf(fact_9708_summable,axiom,
% 0.22/0.58 ! [A: nat > real] :
% 0.22/0.58 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.22/0.58 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.22/0.58 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.22/0.58 => ( summable_real
% 0.22/0.58 @ ^ [N4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( A @ N4 ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % summable
% 0.22/0.58 thf(fact_9709_cos__diff__limit__1,axiom,
% 0.22/0.58 ! [Theta: nat > real,Theta2: real] :
% 0.22/0.58 ( ( filterlim_nat_real
% 0.22/0.58 @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.22/0.58 @ at_top_nat )
% 0.22/0.58 => ~ ! [K2: nat > int] :
% 0.22/0.58 ~ ( filterlim_nat_real
% 0.22/0.58 @ ^ [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 ) ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ Theta2 )
% 0.22/0.58 @ at_top_nat ) ) ).
% 0.22/0.58
% 0.22/0.58 % cos_diff_limit_1
% 0.22/0.58 thf(fact_9710_cos__limit__1,axiom,
% 0.22/0.58 ! [Theta: nat > real] :
% 0.22/0.58 ( ( filterlim_nat_real
% 0.22/0.58 @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.22/0.58 @ at_top_nat )
% 0.22/0.58 => ? [K2: nat > int] :
% 0.22/0.58 ( filterlim_nat_real
% 0.22/0.58 @ ^ [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 ) ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.22/0.58 @ at_top_nat ) ) ).
% 0.22/0.58
% 0.22/0.58 % cos_limit_1
% 0.22/0.58 thf(fact_9711_summable__Leibniz_I4_J,axiom,
% 0.22/0.58 ! [A: nat > real] :
% 0.22/0.58 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.22/0.58 => ( ( topolo6980174941875973593q_real @ A )
% 0.22/0.58 => ( filterlim_nat_real
% 0.22/0.58 @ ^ [N4: nat] :
% 0.22/0.58 ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real
% 0.22/0.58 @ ( suminf_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 0.22/0.58 @ at_top_nat ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % summable_Leibniz(4)
% 0.22/0.58 thf(fact_9712_zeroseq__arctan__series,axiom,
% 0.22/0.58 ! [X: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.22/0.58 => ( filterlim_nat_real
% 0.22/0.58 @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.22/0.58 @ at_top_nat ) ) ).
% 0.22/0.58
% 0.22/0.58 % zeroseq_arctan_series
% 0.22/0.58 thf(fact_9713_summable__Leibniz_H_I2_J,axiom,
% 0.22/0.58 ! [A: nat > real,N: nat] :
% 0.22/0.58 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.22/0.58 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.22/0.58 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.22/0.58 => ( ord_less_eq_real
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.22/0.58 @ ( suminf_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % summable_Leibniz'(2)
% 0.22/0.58 thf(fact_9714_summable__Leibniz_H_I3_J,axiom,
% 0.22/0.58 ! [A: nat > real] :
% 0.22/0.58 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.22/0.58 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.22/0.58 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.22/0.58 => ( filterlim_nat_real
% 0.22/0.58 @ ^ [N4: nat] :
% 0.22/0.58 ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real
% 0.22/0.58 @ ( suminf_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 0.22/0.58 @ at_top_nat ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % summable_Leibniz'(3)
% 0.22/0.58 thf(fact_9715_sums__alternating__upper__lower,axiom,
% 0.22/0.58 ! [A: nat > real] :
% 0.22/0.58 ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.22/0.58 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.22/0.58 => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.22/0.58 => ? [L2: real] :
% 0.22/0.58 ( ! [N6: nat] :
% 0.22/0.58 ( ord_less_eq_real
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
% 0.22/0.58 @ L2 )
% 0.22/0.58 & ( filterlim_nat_real
% 0.22/0.58 @ ^ [N4: nat] :
% 0.22/0.58 ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ L2 )
% 0.22/0.58 @ at_top_nat )
% 0.22/0.58 & ! [N6: nat] :
% 0.22/0.58 ( ord_less_eq_real @ L2
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) )
% 0.22/0.58 & ( filterlim_nat_real
% 0.22/0.58 @ ^ [N4: nat] :
% 0.22/0.58 ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ L2 )
% 0.22/0.58 @ at_top_nat ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % sums_alternating_upper_lower
% 0.22/0.58 thf(fact_9716_summable__Leibniz_I5_J,axiom,
% 0.22/0.58 ! [A: nat > real] :
% 0.22/0.58 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.22/0.58 => ( ( topolo6980174941875973593q_real @ A )
% 0.22/0.58 => ( filterlim_nat_real
% 0.22/0.58 @ ^ [N4: nat] :
% 0.22/0.58 ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real
% 0.22/0.58 @ ( suminf_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 0.22/0.58 @ at_top_nat ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % summable_Leibniz(5)
% 0.22/0.58 thf(fact_9717_summable__Leibniz_H_I5_J,axiom,
% 0.22/0.58 ! [A: nat > real] :
% 0.22/0.58 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.22/0.58 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.22/0.58 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.22/0.58 => ( filterlim_nat_real
% 0.22/0.58 @ ^ [N4: nat] :
% 0.22/0.58 ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real
% 0.22/0.58 @ ( suminf_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 0.22/0.58 @ at_top_nat ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % summable_Leibniz'(5)
% 0.22/0.58 thf(fact_9718_summable__Leibniz_H_I4_J,axiom,
% 0.22/0.58 ! [A: nat > real,N: nat] :
% 0.22/0.58 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.22/0.58 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.22/0.58 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.22/0.58 => ( ord_less_eq_real
% 0.22/0.58 @ ( suminf_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
% 0.22/0.58 @ ( groups6591440286371151544t_real
% 0.22/0.58 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.22/0.58 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % summable_Leibniz'(4)
% 0.22/0.58 thf(fact_9719_real__bounded__linear,axiom,
% 0.22/0.58 ( real_V5970128139526366754l_real
% 0.22/0.58 = ( ^ [F3: real > real] :
% 0.22/0.58 ? [C5: real] :
% 0.22/0.58 ( F3
% 0.22/0.58 = ( ^ [X3: real] : ( times_times_real @ X3 @ C5 ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % real_bounded_linear
% 0.22/0.58 thf(fact_9720_tendsto__arctan__at__bot,axiom,
% 0.22/0.58 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).
% 0.22/0.58
% 0.22/0.58 % tendsto_arctan_at_bot
% 0.22/0.58 thf(fact_9721_dist__real__def,axiom,
% 0.22/0.58 ( real_V975177566351809787t_real
% 0.22/0.58 = ( ^ [X3: real,Y6: real] : ( abs_abs_real @ ( minus_minus_real @ X3 @ Y6 ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % dist_real_def
% 0.22/0.58 thf(fact_9722_dist__complex__def,axiom,
% 0.22/0.58 ( real_V3694042436643373181omplex
% 0.22/0.58 = ( ^ [X3: complex,Y6: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y6 ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % dist_complex_def
% 0.22/0.58 thf(fact_9723_exp__at__bot,axiom,
% 0.22/0.58 filterlim_real_real @ exp_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_bot_real ).
% 0.22/0.58
% 0.22/0.58 % exp_at_bot
% 0.22/0.58 thf(fact_9724_filterlim__inverse__at__bot__neg,axiom,
% 0.22/0.58 filterlim_real_real @ inverse_inverse_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5984915006950818249n_real @ zero_zero_real ) ) ).
% 0.22/0.58
% 0.22/0.58 % filterlim_inverse_at_bot_neg
% 0.22/0.58 thf(fact_9725_DERIV__pos__imp__increasing__at__bot,axiom,
% 0.22/0.58 ! [B: real,F: real > real,Flim: real] :
% 0.22/0.58 ( ! [X4: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ X4 @ B )
% 0.22/0.58 => ? [Y3: real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.22/0.58 & ( ord_less_real @ zero_zero_real @ Y3 ) ) )
% 0.22/0.58 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 0.22/0.58 => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_pos_imp_increasing_at_bot
% 0.22/0.58 thf(fact_9726_filterlim__pow__at__bot__odd,axiom,
% 0.22/0.58 ! [N: nat,F: real > real,F4: filter_real] :
% 0.22/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.58 => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
% 0.22/0.58 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.58 => ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N )
% 0.22/0.58 @ at_bot_real
% 0.22/0.58 @ F4 ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % filterlim_pow_at_bot_odd
% 0.22/0.58 thf(fact_9727_tendsto__exp__limit__at__right,axiom,
% 0.22/0.58 ! [X: real] :
% 0.22/0.58 ( filterlim_real_real
% 0.22/0.58 @ ^ [Y6: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y6 ) ) @ ( divide_divide_real @ one_one_real @ Y6 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % tendsto_exp_limit_at_right
% 0.22/0.58 thf(fact_9728_ln__at__0,axiom,
% 0.22/0.58 filterlim_real_real @ ln_ln_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).
% 0.22/0.58
% 0.22/0.58 % ln_at_0
% 0.22/0.58 thf(fact_9729_tendsto__arcosh__at__left__1,axiom,
% 0.22/0.58 filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).
% 0.22/0.58
% 0.22/0.58 % tendsto_arcosh_at_left_1
% 0.22/0.58 thf(fact_9730_filterlim__tan__at__right,axiom,
% 0.22/0.58 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 ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % filterlim_tan_at_right
% 0.22/0.58 thf(fact_9731_atLeast__0,axiom,
% 0.22/0.58 ( ( set_ord_atLeast_nat @ zero_zero_nat )
% 0.22/0.58 = top_top_set_nat ) ).
% 0.22/0.58
% 0.22/0.58 % atLeast_0
% 0.22/0.58 thf(fact_9732_atLeast__Suc__greaterThan,axiom,
% 0.22/0.58 ! [K: nat] :
% 0.22/0.58 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 0.22/0.58 = ( set_or1210151606488870762an_nat @ K ) ) ).
% 0.22/0.58
% 0.22/0.58 % atLeast_Suc_greaterThan
% 0.22/0.58 thf(fact_9733_greaterThan__0,axiom,
% 0.22/0.58 ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
% 0.22/0.58 = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% 0.22/0.58
% 0.22/0.58 % greaterThan_0
% 0.22/0.58 thf(fact_9734_greaterThan__Suc,axiom,
% 0.22/0.58 ! [K: nat] :
% 0.22/0.58 ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 0.22/0.58 = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % greaterThan_Suc
% 0.22/0.58 thf(fact_9735_atLeast__Suc,axiom,
% 0.22/0.58 ! [K: nat] :
% 0.22/0.58 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 0.22/0.58 = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % atLeast_Suc
% 0.22/0.58 thf(fact_9736_filterlim__pow__at__bot__even,axiom,
% 0.22/0.58 ! [N: nat,F: real > real,F4: filter_real] :
% 0.22/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.58 => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
% 0.22/0.58 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.22/0.58 => ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N )
% 0.22/0.58 @ at_top_real
% 0.22/0.58 @ F4 ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % filterlim_pow_at_bot_even
% 0.22/0.58 thf(fact_9737_filterlim__tan__at__left,axiom,
% 0.22/0.58 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 ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % filterlim_tan_at_left
% 0.22/0.58 thf(fact_9738_sqrt__at__top,axiom,
% 0.22/0.58 filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).
% 0.22/0.58
% 0.22/0.58 % sqrt_at_top
% 0.22/0.58 thf(fact_9739_filterlim__real__sequentially,axiom,
% 0.22/0.58 filterlim_nat_real @ semiri5074537144036343181t_real @ at_top_real @ at_top_nat ).
% 0.22/0.58
% 0.22/0.58 % filterlim_real_sequentially
% 0.22/0.58 thf(fact_9740_ln__x__over__x__tendsto__0,axiom,
% 0.22/0.58 ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( ln_ln_real @ X3 ) @ X3 )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.22/0.58 @ at_top_real ) ).
% 0.22/0.58
% 0.22/0.58 % ln_x_over_x_tendsto_0
% 0.22/0.58 thf(fact_9741_filterlim__inverse__at__right__top,axiom,
% 0.22/0.58 filterlim_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) @ at_top_real ).
% 0.22/0.58
% 0.22/0.58 % filterlim_inverse_at_right_top
% 0.22/0.58 thf(fact_9742_filterlim__inverse__at__top__right,axiom,
% 0.22/0.58 filterlim_real_real @ inverse_inverse_real @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).
% 0.22/0.58
% 0.22/0.58 % filterlim_inverse_at_top_right
% 0.22/0.58 thf(fact_9743_tendsto__power__div__exp__0,axiom,
% 0.22/0.58 ! [K: nat] :
% 0.22/0.58 ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( power_power_real @ X3 @ K ) @ ( exp_real @ X3 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.22/0.58 @ at_top_real ) ).
% 0.22/0.58
% 0.22/0.58 % tendsto_power_div_exp_0
% 0.22/0.58 thf(fact_9744_tendsto__exp__limit__at__top,axiom,
% 0.22/0.58 ! [X: real] :
% 0.22/0.58 ( filterlim_real_real
% 0.22/0.58 @ ^ [Y6: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y6 ) ) @ Y6 )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 0.22/0.58 @ at_top_real ) ).
% 0.22/0.58
% 0.22/0.58 % tendsto_exp_limit_at_top
% 0.22/0.58 thf(fact_9745_DERIV__neg__imp__decreasing__at__top,axiom,
% 0.22/0.58 ! [B: real,F: real > real,Flim: real] :
% 0.22/0.58 ( ! [X4: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ B @ X4 )
% 0.22/0.58 => ? [Y3: real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.22/0.58 & ( ord_less_real @ Y3 @ zero_zero_real ) ) )
% 0.22/0.58 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 0.22/0.58 => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_neg_imp_decreasing_at_top
% 0.22/0.58 thf(fact_9746_tendsto__arctan__at__top,axiom,
% 0.22/0.58 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).
% 0.22/0.58
% 0.22/0.58 % tendsto_arctan_at_top
% 0.22/0.58 thf(fact_9747_lhopital__left__at__top,axiom,
% 0.22/0.58 ! [G: real > real,X: real,G2: real > real,F: real > real,F2: real > real,Y4: real] :
% 0.22/0.58 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] :
% 0.22/0.58 ( ( G2 @ X3 )
% 0.22/0.58 != zero_zero_real )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.22/0.58 => ( ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ Y4 )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.22/0.58 => ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ Y4 )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % lhopital_left_at_top
% 0.22/0.58 thf(fact_9748_at__top__le__at__infinity,axiom,
% 0.22/0.58 ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).
% 0.22/0.58
% 0.22/0.58 % at_top_le_at_infinity
% 0.22/0.58 thf(fact_9749_at__bot__le__at__infinity,axiom,
% 0.22/0.58 ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).
% 0.22/0.58
% 0.22/0.58 % at_bot_le_at_infinity
% 0.22/0.58 thf(fact_9750_filterlim__real__at__infinity__sequentially,axiom,
% 0.22/0.58 filterlim_nat_real @ semiri5074537144036343181t_real @ at_infinity_real @ at_top_nat ).
% 0.22/0.58
% 0.22/0.58 % filterlim_real_at_infinity_sequentially
% 0.22/0.58 thf(fact_9751_eventually__at__right__to__0,axiom,
% 0.22/0.58 ! [P: real > $o,A: real] :
% 0.22/0.58 ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.22/0.58 = ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( P @ ( plus_plus_real @ X3 @ A ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % eventually_at_right_to_0
% 0.22/0.58 thf(fact_9752_eventually__at__top__to__right,axiom,
% 0.22/0.58 ! [P: real > $o] :
% 0.22/0.58 ( ( eventually_real @ P @ at_top_real )
% 0.22/0.58 = ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( P @ ( inverse_inverse_real @ X3 ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % eventually_at_top_to_right
% 0.22/0.58 thf(fact_9753_eventually__at__right__to__top,axiom,
% 0.22/0.58 ! [P: real > $o] :
% 0.22/0.58 ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.22/0.58 = ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( P @ ( inverse_inverse_real @ X3 ) )
% 0.22/0.58 @ at_top_real ) ) ).
% 0.22/0.58
% 0.22/0.58 % eventually_at_right_to_top
% 0.22/0.58 thf(fact_9754_lhopital__at__top__at__top,axiom,
% 0.22/0.58 ! [F: real > real,A: real,G: real > real,F2: real > real,G2: real > real] :
% 0.22/0.58 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.22/0.58 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.22/0.58 => ( ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 0.22/0.58 @ at_top_real
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.22/0.58 => ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.22/0.58 @ at_top_real
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % lhopital_at_top_at_top
% 0.22/0.58 thf(fact_9755_lhopital,axiom,
% 0.22/0.58 ! [F: real > real,X: real,G: real > real,G2: real > real,F2: real > real,F4: filter_real] :
% 0.22/0.58 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] :
% 0.22/0.58 ( ( G @ X3 )
% 0.22/0.58 != zero_zero_real )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] :
% 0.22/0.58 ( ( G2 @ X3 )
% 0.22/0.58 != zero_zero_real )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 0.22/0.58 @ F4
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.22/0.58 @ F4
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % lhopital
% 0.22/0.58 thf(fact_9756_lhopital__right__at__top__at__top,axiom,
% 0.22/0.58 ! [F: real > real,A: real,G: real > real,F2: real > real,G2: real > real] :
% 0.22/0.58 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.22/0.58 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.22/0.58 => ( ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 0.22/0.58 @ at_top_real
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.22/0.58 => ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.22/0.58 @ at_top_real
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % lhopital_right_at_top_at_top
% 0.22/0.58 thf(fact_9757_lhopital__at__top__at__bot,axiom,
% 0.22/0.58 ! [F: real > real,A: real,G: real > real,F2: real > real,G2: real > real] :
% 0.22/0.58 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.22/0.58 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.22/0.58 => ( ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 0.22/0.58 @ at_bot_real
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.22/0.58 => ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.22/0.58 @ at_bot_real
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % lhopital_at_top_at_bot
% 0.22/0.58 thf(fact_9758_lhopital__left__at__top__at__top,axiom,
% 0.22/0.58 ! [F: real > real,A: real,G: real > real,F2: real > real,G2: real > real] :
% 0.22/0.58 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.22/0.58 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.22/0.58 => ( ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 0.22/0.58 @ at_top_real
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.22/0.58 => ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.22/0.58 @ at_top_real
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % lhopital_left_at_top_at_top
% 0.22/0.58 thf(fact_9759_lhospital__at__top__at__top,axiom,
% 0.22/0.58 ! [G: real > real,G2: real > real,F: real > real,F2: real > real,X: real] :
% 0.22/0.58 ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] :
% 0.22/0.58 ( ( G2 @ X3 )
% 0.22/0.58 != zero_zero_real )
% 0.22/0.58 @ at_top_real )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ at_top_real )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ at_top_real )
% 0.22/0.58 => ( ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ X )
% 0.22/0.58 @ at_top_real )
% 0.22/0.58 => ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ X )
% 0.22/0.58 @ at_top_real ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % lhospital_at_top_at_top
% 0.22/0.58 thf(fact_9760_lhopital__at__top,axiom,
% 0.22/0.58 ! [G: real > real,X: real,G2: real > real,F: real > real,F2: real > real,Y4: real] :
% 0.22/0.58 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] :
% 0.22/0.58 ( ( G2 @ X3 )
% 0.22/0.58 != zero_zero_real )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ Y4 )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.22/0.58 => ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ Y4 )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % lhopital_at_top
% 0.22/0.58 thf(fact_9761_lhopital__right__0,axiom,
% 0.22/0.58 ! [F0: real > real,G0: real > real,G2: real > real,F2: real > real,F4: filter_real] :
% 0.22/0.58 ( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.22/0.58 => ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] :
% 0.22/0.58 ( ( G0 @ X3 )
% 0.22/0.58 != zero_zero_real )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] :
% 0.22/0.58 ( ( G2 @ X3 )
% 0.22/0.58 != zero_zero_real )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.22/0.58 => ( ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 0.22/0.58 @ F4
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.22/0.58 => ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F0 @ X3 ) @ ( G0 @ X3 ) )
% 0.22/0.58 @ F4
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % lhopital_right_0
% 0.22/0.58 thf(fact_9762_lhopital__right,axiom,
% 0.22/0.58 ! [F: real > real,X: real,G: real > real,G2: real > real,F2: real > real,F4: filter_real] :
% 0.22/0.58 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.22/0.58 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] :
% 0.22/0.58 ( ( G @ X3 )
% 0.22/0.58 != zero_zero_real )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] :
% 0.22/0.58 ( ( G2 @ X3 )
% 0.22/0.58 != zero_zero_real )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.22/0.58 => ( ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 0.22/0.58 @ F4
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.22/0.58 => ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.22/0.58 @ F4
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % lhopital_right
% 0.22/0.58 thf(fact_9763_lhopital__left,axiom,
% 0.22/0.58 ! [F: real > real,X: real,G: real > real,G2: real > real,F2: real > real,F4: filter_real] :
% 0.22/0.58 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.22/0.58 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] :
% 0.22/0.58 ( ( G @ X3 )
% 0.22/0.58 != zero_zero_real )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] :
% 0.22/0.58 ( ( G2 @ X3 )
% 0.22/0.58 != zero_zero_real )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.22/0.58 => ( ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 0.22/0.58 @ F4
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.22/0.58 => ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.22/0.58 @ F4
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % lhopital_left
% 0.22/0.58 thf(fact_9764_lhopital__right__at__top__at__bot,axiom,
% 0.22/0.58 ! [F: real > real,A: real,G: real > real,F2: real > real,G2: real > real] :
% 0.22/0.58 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.22/0.58 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.22/0.58 => ( ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 0.22/0.58 @ at_bot_real
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.22/0.58 => ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.22/0.58 @ at_bot_real
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % lhopital_right_at_top_at_bot
% 0.22/0.58 thf(fact_9765_lhopital__left__at__top__at__bot,axiom,
% 0.22/0.58 ! [F: real > real,A: real,G: real > real,F2: real > real,G2: real > real] :
% 0.22/0.58 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.22/0.58 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.22/0.58 => ( ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 0.22/0.58 @ at_bot_real
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.22/0.58 => ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.22/0.58 @ at_bot_real
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % lhopital_left_at_top_at_bot
% 0.22/0.58 thf(fact_9766_lhopital__right__at__top,axiom,
% 0.22/0.58 ! [G: real > real,X: real,G2: real > real,F: real > real,F2: real > real,Y4: real] :
% 0.22/0.58 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] :
% 0.22/0.58 ( ( G2 @ X3 )
% 0.22/0.58 != zero_zero_real )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.22/0.58 => ( ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ Y4 )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.22/0.58 => ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ Y4 )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % lhopital_right_at_top
% 0.22/0.58 thf(fact_9767_lhopital__right__0__at__top,axiom,
% 0.22/0.58 ! [G: real > real,G2: real > real,F: real > real,F2: real > real,X: real] :
% 0.22/0.58 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] :
% 0.22/0.58 ( ( G2 @ X3 )
% 0.22/0.58 != zero_zero_real )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.22/0.58 => ( ( eventually_real
% 0.22/0.58 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.22/0.58 => ( ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ X )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.22/0.58 => ( filterlim_real_real
% 0.22/0.58 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.22/0.58 @ ( topolo2815343760600316023s_real @ X )
% 0.22/0.58 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % lhopital_right_0_at_top
% 0.22/0.58 thf(fact_9768_eventually__sequentially__Suc,axiom,
% 0.22/0.58 ! [P: nat > $o] :
% 0.22/0.58 ( ( eventually_nat
% 0.22/0.58 @ ^ [I2: nat] : ( P @ ( suc @ I2 ) )
% 0.22/0.58 @ at_top_nat )
% 0.22/0.58 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.22/0.58
% 0.22/0.58 % eventually_sequentially_Suc
% 0.22/0.58 thf(fact_9769_le__sequentially,axiom,
% 0.22/0.58 ! [F4: filter_nat] :
% 0.22/0.58 ( ( ord_le2510731241096832064er_nat @ F4 @ at_top_nat )
% 0.22/0.58 = ( ! [N8: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N8 ) @ F4 ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % le_sequentially
% 0.22/0.58 thf(fact_9770_eventually__sequentially,axiom,
% 0.22/0.58 ! [P: nat > $o] :
% 0.22/0.58 ( ( eventually_nat @ P @ at_top_nat )
% 0.22/0.58 = ( ? [N8: nat] :
% 0.22/0.58 ! [N4: nat] :
% 0.22/0.58 ( ( ord_less_eq_nat @ N8 @ N4 )
% 0.22/0.58 => ( P @ N4 ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % eventually_sequentially
% 0.22/0.58 thf(fact_9771_eventually__sequentiallyI,axiom,
% 0.22/0.58 ! [C: nat,P: nat > $o] :
% 0.22/0.58 ( ! [X4: nat] :
% 0.22/0.58 ( ( ord_less_eq_nat @ C @ X4 )
% 0.22/0.58 => ( P @ X4 ) )
% 0.22/0.58 => ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.22/0.58
% 0.22/0.58 % eventually_sequentiallyI
% 0.22/0.58 thf(fact_9772_filterlim__int__sequentially,axiom,
% 0.22/0.58 filterlim_nat_int @ semiri1314217659103216013at_int @ at_top_int @ at_top_nat ).
% 0.22/0.58
% 0.22/0.58 % filterlim_int_sequentially
% 0.22/0.58 thf(fact_9773_Bseq__eq__bounded,axiom,
% 0.22/0.58 ! [F: nat > real,A: real,B: real] :
% 0.22/0.58 ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A @ B ) )
% 0.22/0.58 => ( bfun_nat_real @ F @ at_top_nat ) ) ).
% 0.22/0.58
% 0.22/0.58 % Bseq_eq_bounded
% 0.22/0.58 thf(fact_9774_Bseq__realpow,axiom,
% 0.22/0.58 ! [X: real] :
% 0.22/0.58 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.58 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.22/0.58 => ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Bseq_realpow
% 0.22/0.58 thf(fact_9775_GreatestI__nat,axiom,
% 0.22/0.58 ! [P: nat > $o,K: nat,B: nat] :
% 0.22/0.58 ( ( P @ K )
% 0.22/0.58 => ( ! [Y: nat] :
% 0.22/0.58 ( ( P @ Y )
% 0.22/0.58 => ( ord_less_eq_nat @ Y @ B ) )
% 0.22/0.58 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % GreatestI_nat
% 0.22/0.58 thf(fact_9776_Greatest__le__nat,axiom,
% 0.22/0.58 ! [P: nat > $o,K: nat,B: nat] :
% 0.22/0.58 ( ( P @ K )
% 0.22/0.58 => ( ! [Y: nat] :
% 0.22/0.58 ( ( P @ Y )
% 0.22/0.58 => ( ord_less_eq_nat @ Y @ B ) )
% 0.22/0.58 => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Greatest_le_nat
% 0.22/0.58 thf(fact_9777_GreatestI__ex__nat,axiom,
% 0.22/0.58 ! [P: nat > $o,B: nat] :
% 0.22/0.58 ( ? [X_12: nat] : ( P @ X_12 )
% 0.22/0.58 => ( ! [Y: nat] :
% 0.22/0.58 ( ( P @ Y )
% 0.22/0.58 => ( ord_less_eq_nat @ Y @ B ) )
% 0.22/0.58 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % GreatestI_ex_nat
% 0.22/0.58 thf(fact_9778_GMVT,axiom,
% 0.22/0.58 ! [A: real,B: real,F: real > real,G: real > real] :
% 0.22/0.58 ( ( ord_less_real @ A @ B )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( ( ord_less_eq_real @ A @ X4 )
% 0.22/0.58 & ( ord_less_eq_real @ X4 @ B ) )
% 0.22/0.58 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( ( ord_less_real @ A @ X4 )
% 0.22/0.58 & ( ord_less_real @ X4 @ B ) )
% 0.22/0.58 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( ( ord_less_eq_real @ A @ X4 )
% 0.22/0.58 & ( ord_less_eq_real @ X4 @ B ) )
% 0.22/0.58 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G ) )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( ( ord_less_real @ A @ X4 )
% 0.22/0.58 & ( ord_less_real @ X4 @ B ) )
% 0.22/0.58 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.22/0.58 => ? [G_c: real,F_c: real,C3: real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 0.22/0.58 & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 0.22/0.58 & ( ord_less_real @ A @ C3 )
% 0.22/0.58 & ( ord_less_real @ C3 @ B )
% 0.22/0.58 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
% 0.22/0.58 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % GMVT
% 0.22/0.58 thf(fact_9779_MVT,axiom,
% 0.22/0.58 ! [A: real,B: real,F: real > real] :
% 0.22/0.58 ( ( ord_less_real @ A @ B )
% 0.22/0.58 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( ord_less_real @ A @ X4 )
% 0.22/0.58 => ( ( ord_less_real @ X4 @ B )
% 0.22/0.58 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.22/0.58 => ? [L2: real,Z2: real] :
% 0.22/0.58 ( ( ord_less_real @ A @ Z2 )
% 0.22/0.58 & ( ord_less_real @ Z2 @ B )
% 0.22/0.58 & ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) )
% 0.22/0.58 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.22/0.58 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L2 ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % MVT
% 0.22/0.58 thf(fact_9780_continuous__on__arcosh_H,axiom,
% 0.22/0.58 ! [A2: set_real,F: real > real] :
% 0.22/0.58 ( ( topolo5044208981011980120l_real @ A2 @ F )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( member_real @ X4 @ A2 )
% 0.22/0.58 => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 0.22/0.58 => ( topolo5044208981011980120l_real @ A2
% 0.22/0.58 @ ^ [X3: real] : ( arcosh_real @ ( F @ X3 ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % continuous_on_arcosh'
% 0.22/0.58 thf(fact_9781_continuous__image__closed__interval,axiom,
% 0.22/0.58 ! [A: real,B: real,F: real > real] :
% 0.22/0.58 ( ( ord_less_eq_real @ A @ B )
% 0.22/0.58 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.22/0.58 => ? [C3: real,D2: real] :
% 0.22/0.58 ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
% 0.22/0.58 = ( set_or1222579329274155063t_real @ C3 @ D2 ) )
% 0.22/0.58 & ( ord_less_eq_real @ C3 @ D2 ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % continuous_image_closed_interval
% 0.22/0.58 thf(fact_9782_continuous__on__arcosh,axiom,
% 0.22/0.58 ! [A2: set_real] :
% 0.22/0.58 ( ( ord_less_eq_set_real @ A2 @ ( set_ord_atLeast_real @ one_one_real ) )
% 0.22/0.58 => ( topolo5044208981011980120l_real @ A2 @ arcosh_real ) ) ).
% 0.22/0.58
% 0.22/0.58 % continuous_on_arcosh
% 0.22/0.58 thf(fact_9783_Rolle__deriv,axiom,
% 0.22/0.58 ! [A: real,B: real,F: real > real,F2: real > real > real] :
% 0.22/0.58 ( ( ord_less_real @ A @ B )
% 0.22/0.58 => ( ( ( F @ A )
% 0.22/0.58 = ( F @ B ) )
% 0.22/0.58 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( ord_less_real @ A @ X4 )
% 0.22/0.58 => ( ( ord_less_real @ X4 @ B )
% 0.22/0.58 => ( has_de1759254742604945161l_real @ F @ ( F2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.22/0.58 => ? [Z2: real] :
% 0.22/0.58 ( ( ord_less_real @ A @ Z2 )
% 0.22/0.58 & ( ord_less_real @ Z2 @ B )
% 0.22/0.58 & ( ( F2 @ Z2 )
% 0.22/0.58 = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Rolle_deriv
% 0.22/0.58 thf(fact_9784_mvt,axiom,
% 0.22/0.58 ! [A: real,B: real,F: real > real,F2: real > real > real] :
% 0.22/0.58 ( ( ord_less_real @ A @ B )
% 0.22/0.58 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( ord_less_real @ A @ X4 )
% 0.22/0.58 => ( ( ord_less_real @ X4 @ B )
% 0.22/0.58 => ( has_de1759254742604945161l_real @ F @ ( F2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.22/0.58 => ~ ! [Xi: real] :
% 0.22/0.58 ( ( ord_less_real @ A @ Xi )
% 0.22/0.58 => ( ( ord_less_real @ Xi @ B )
% 0.22/0.58 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.22/0.58 != ( F2 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % mvt
% 0.22/0.58 thf(fact_9785_DERIV__pos__imp__increasing__open,axiom,
% 0.22/0.58 ! [A: real,B: real,F: real > real] :
% 0.22/0.58 ( ( ord_less_real @ A @ B )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( ord_less_real @ A @ X4 )
% 0.22/0.58 => ( ( ord_less_real @ X4 @ B )
% 0.22/0.58 => ? [Y3: real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.22/0.58 & ( ord_less_real @ zero_zero_real @ Y3 ) ) ) )
% 0.22/0.58 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.22/0.58 => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_pos_imp_increasing_open
% 0.22/0.58 thf(fact_9786_DERIV__neg__imp__decreasing__open,axiom,
% 0.22/0.58 ! [A: real,B: real,F: real > real] :
% 0.22/0.58 ( ( ord_less_real @ A @ B )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( ord_less_real @ A @ X4 )
% 0.22/0.58 => ( ( ord_less_real @ X4 @ B )
% 0.22/0.58 => ? [Y3: real] :
% 0.22/0.58 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.22/0.58 & ( ord_less_real @ Y3 @ zero_zero_real ) ) ) )
% 0.22/0.58 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.22/0.58 => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_neg_imp_decreasing_open
% 0.22/0.58 thf(fact_9787_DERIV__isconst__end,axiom,
% 0.22/0.58 ! [A: real,B: real,F: real > real] :
% 0.22/0.58 ( ( ord_less_real @ A @ B )
% 0.22/0.58 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( ord_less_real @ A @ X4 )
% 0.22/0.58 => ( ( ord_less_real @ X4 @ B )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.22/0.58 => ( ( F @ B )
% 0.22/0.58 = ( F @ A ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_isconst_end
% 0.22/0.58 thf(fact_9788_continuous__on__artanh,axiom,
% 0.22/0.58 ! [A2: set_real] :
% 0.22/0.58 ( ( ord_less_eq_set_real @ A2 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
% 0.22/0.58 => ( topolo5044208981011980120l_real @ A2 @ artanh_real ) ) ).
% 0.22/0.58
% 0.22/0.58 % continuous_on_artanh
% 0.22/0.58 thf(fact_9789_DERIV__isconst2,axiom,
% 0.22/0.58 ! [A: real,B: real,F: real > real,X: real] :
% 0.22/0.58 ( ( ord_less_real @ A @ B )
% 0.22/0.58 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( ord_less_real @ A @ X4 )
% 0.22/0.58 => ( ( ord_less_real @ X4 @ B )
% 0.22/0.58 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.22/0.58 => ( ( ord_less_eq_real @ A @ X )
% 0.22/0.58 => ( ( ord_less_eq_real @ X @ B )
% 0.22/0.58 => ( ( F @ X )
% 0.22/0.58 = ( F @ A ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % DERIV_isconst2
% 0.22/0.58 thf(fact_9790_Rolle,axiom,
% 0.22/0.58 ! [A: real,B: real,F: real > real] :
% 0.22/0.58 ( ( ord_less_real @ A @ B )
% 0.22/0.58 => ( ( ( F @ A )
% 0.22/0.58 = ( F @ B ) )
% 0.22/0.58 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.22/0.58 => ( ! [X4: real] :
% 0.22/0.58 ( ( ord_less_real @ A @ X4 )
% 0.22/0.58 => ( ( ord_less_real @ X4 @ B )
% 0.22/0.58 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.22/0.58 => ? [Z2: real] :
% 0.22/0.58 ( ( ord_less_real @ A @ Z2 )
% 0.22/0.58 & ( ord_less_real @ Z2 @ B )
% 0.22/0.58 & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Rolle
% 0.22/0.58 thf(fact_9791_surj__int__encode,axiom,
% 0.22/0.58 ( ( image_int_nat @ nat_int_encode @ top_top_set_int )
% 0.22/0.58 = top_top_set_nat ) ).
% 0.22/0.58
% 0.22/0.58 % surj_int_encode
% 0.22/0.58 thf(fact_9792_int__encode__eq,axiom,
% 0.22/0.58 ! [X: int,Y4: int] :
% 0.22/0.58 ( ( ( nat_int_encode @ X )
% 0.22/0.58 = ( nat_int_encode @ Y4 ) )
% 0.22/0.58 = ( X = Y4 ) ) ).
% 0.22/0.58
% 0.22/0.58 % int_encode_eq
% 0.22/0.58 thf(fact_9793_bij__int__encode,axiom,
% 0.22/0.58 bij_betw_int_nat @ nat_int_encode @ top_top_set_int @ top_top_set_nat ).
% 0.22/0.58
% 0.22/0.58 % bij_int_encode
% 0.22/0.58 thf(fact_9794_uniformity__real__def,axiom,
% 0.22/0.58 ( topolo1511823702728130853y_real
% 0.22/0.58 = ( comple2936214249959783750l_real
% 0.22/0.58 @ ( image_2178119161166701260l_real
% 0.22/0.58 @ ^ [E3: real] :
% 0.22/0.58 ( princi6114159922880469582l_real
% 0.22/0.58 @ ( collec3799799289383736868l_real
% 0.22/0.58 @ ( produc5414030515140494994real_o
% 0.22/0.58 @ ^ [X3: real,Y6: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X3 @ Y6 ) @ E3 ) ) ) )
% 0.22/0.58 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % uniformity_real_def
% 0.22/0.58 thf(fact_9795_uniformity__complex__def,axiom,
% 0.22/0.58 ( topolo896644834953643431omplex
% 0.22/0.58 = ( comple8358262395181532106omplex
% 0.22/0.58 @ ( image_5971271580939081552omplex
% 0.22/0.58 @ ^ [E3: real] :
% 0.22/0.58 ( princi3496590319149328850omplex
% 0.22/0.58 @ ( collec8663557070575231912omplex
% 0.22/0.58 @ ( produc6771430404735790350plex_o
% 0.22/0.58 @ ^ [X3: complex,Y6: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X3 @ Y6 ) @ E3 ) ) ) )
% 0.22/0.58 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % uniformity_complex_def
% 0.22/0.58 thf(fact_9796_range__abs__Nats,axiom,
% 0.22/0.58 ( ( image_int_int @ abs_abs_int @ top_top_set_int )
% 0.22/0.58 = semiring_1_Nats_int ) ).
% 0.22/0.58
% 0.22/0.58 % range_abs_Nats
% 0.22/0.58 thf(fact_9797_inj__sgn__power,axiom,
% 0.22/0.58 ! [N: nat] :
% 0.22/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.58 => ( inj_on_real_real
% 0.22/0.58 @ ^ [Y6: real] : ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N ) )
% 0.22/0.58 @ top_top_set_real ) ) ).
% 0.22/0.58
% 0.22/0.58 % inj_sgn_power
% 0.22/0.58 thf(fact_9798_surj__int__decode,axiom,
% 0.22/0.58 ( ( image_nat_int @ nat_int_decode @ top_top_set_nat )
% 0.22/0.58 = top_top_set_int ) ).
% 0.22/0.58
% 0.22/0.58 % surj_int_decode
% 0.22/0.58 thf(fact_9799_int__decode__inverse,axiom,
% 0.22/0.58 ! [N: nat] :
% 0.22/0.58 ( ( nat_int_encode @ ( nat_int_decode @ N ) )
% 0.22/0.58 = N ) ).
% 0.22/0.58
% 0.22/0.58 % int_decode_inverse
% 0.22/0.58 thf(fact_9800_int__encode__inverse,axiom,
% 0.22/0.58 ! [X: int] :
% 0.22/0.58 ( ( nat_int_decode @ ( nat_int_encode @ X ) )
% 0.22/0.58 = X ) ).
% 0.22/0.58
% 0.22/0.58 % int_encode_inverse
% 0.22/0.58 thf(fact_9801_int__decode__eq,axiom,
% 0.22/0.58 ! [X: nat,Y4: nat] :
% 0.22/0.58 ( ( ( nat_int_decode @ X )
% 0.22/0.58 = ( nat_int_decode @ Y4 ) )
% 0.22/0.58 = ( X = Y4 ) ) ).
% 0.22/0.58
% 0.22/0.58 % int_decode_eq
% 0.22/0.58 thf(fact_9802_bij__int__decode,axiom,
% 0.22/0.58 bij_betw_nat_int @ nat_int_decode @ top_top_set_nat @ top_top_set_int ).
% 0.22/0.58
% 0.22/0.58 % bij_int_decode
% 0.22/0.58 thf(fact_9803_log__inj,axiom,
% 0.22/0.58 ! [B: real] :
% 0.22/0.58 ( ( ord_less_real @ one_one_real @ B )
% 0.22/0.58 => ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % log_inj
% 0.22/0.58 thf(fact_9804_complex__is__Nat__iff,axiom,
% 0.22/0.58 ! [Z: complex] :
% 0.22/0.58 ( ( member_complex @ Z @ semiri3842193898606819883omplex )
% 0.22/0.58 = ( ( ( im @ Z )
% 0.22/0.58 = zero_zero_real )
% 0.22/0.58 & ? [I2: nat] :
% 0.22/0.58 ( ( re @ Z )
% 0.22/0.58 = ( semiri5074537144036343181t_real @ I2 ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % complex_is_Nat_iff
% 0.22/0.58 thf(fact_9805_inj__int__encode,axiom,
% 0.22/0.58 ! [A2: set_int] : ( inj_on_int_nat @ nat_int_encode @ A2 ) ).
% 0.22/0.58
% 0.22/0.58 % inj_int_encode
% 0.22/0.58 thf(fact_9806_inj__on__set__encode,axiom,
% 0.22/0.58 inj_on_set_nat_nat @ nat_set_encode @ ( collect_set_nat @ finite_finite_nat ) ).
% 0.22/0.58
% 0.22/0.58 % inj_on_set_encode
% 0.22/0.58 thf(fact_9807_inj__prod__encode,axiom,
% 0.22/0.58 ! [A2: set_Pr1261947904930325089at_nat] : ( inj_on2178005380612969504at_nat @ nat_prod_encode @ A2 ) ).
% 0.22/0.58
% 0.22/0.58 % inj_prod_encode
% 0.22/0.58 thf(fact_9808_inj__Suc,axiom,
% 0.22/0.58 ! [N5: set_nat] : ( inj_on_nat_nat @ suc @ N5 ) ).
% 0.22/0.58
% 0.22/0.58 % inj_Suc
% 0.22/0.58 thf(fact_9809_inj__int__decode,axiom,
% 0.22/0.58 ! [A2: set_nat] : ( inj_on_nat_int @ nat_int_decode @ A2 ) ).
% 0.22/0.58
% 0.22/0.58 % inj_int_decode
% 0.22/0.58 thf(fact_9810_inj__on__diff__nat,axiom,
% 0.22/0.58 ! [N5: set_nat,K: nat] :
% 0.22/0.58 ( ! [N2: nat] :
% 0.22/0.58 ( ( member_nat @ N2 @ N5 )
% 0.22/0.58 => ( ord_less_eq_nat @ K @ N2 ) )
% 0.22/0.58 => ( inj_on_nat_nat
% 0.22/0.58 @ ^ [N4: nat] : ( minus_minus_nat @ N4 @ K )
% 0.22/0.58 @ N5 ) ) ).
% 0.22/0.58
% 0.22/0.58 % inj_on_diff_nat
% 0.22/0.58 thf(fact_9811_summable__reindex,axiom,
% 0.22/0.58 ! [F: nat > real,G: nat > nat] :
% 0.22/0.58 ( ( summable_real @ F )
% 0.22/0.58 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.22/0.58 => ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 0.22/0.58 => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % summable_reindex
% 0.22/0.58 thf(fact_9812_suminf__reindex__mono,axiom,
% 0.22/0.58 ! [F: nat > real,G: nat > nat] :
% 0.22/0.58 ( ( summable_real @ F )
% 0.22/0.58 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.22/0.58 => ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 0.22/0.58 => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % suminf_reindex_mono
% 0.22/0.58 thf(fact_9813_inj__on__char__of__nat,axiom,
% 0.22/0.58 inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % inj_on_char_of_nat
% 0.22/0.58 thf(fact_9814_suminf__reindex,axiom,
% 0.22/0.58 ! [F: nat > real,G: nat > nat] :
% 0.22/0.58 ( ( summable_real @ F )
% 0.22/0.58 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.22/0.58 => ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 0.22/0.58 => ( ! [X4: nat] :
% 0.22/0.58 ( ~ ( member_nat @ X4 @ ( image_nat_nat @ G @ top_top_set_nat ) )
% 0.22/0.58 => ( ( F @ X4 )
% 0.22/0.58 = zero_zero_real ) )
% 0.22/0.58 => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
% 0.22/0.58 = ( suminf_real @ F ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % suminf_reindex
% 0.22/0.58 thf(fact_9815_powr__real__of__int_H,axiom,
% 0.22/0.58 ! [X: real,N: int] :
% 0.22/0.58 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.58 => ( ( ( X != zero_zero_real )
% 0.22/0.58 | ( ord_less_int @ zero_zero_int @ N ) )
% 0.22/0.58 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 0.22/0.58 = ( power_int_real @ X @ N ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % powr_real_of_int'
% 0.22/0.58 thf(fact_9816_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 0.22/0.58 ! [X: vEBT_VEBT,Xa: nat] :
% 0.22/0.58 ( ~ ( vEBT_VEBT_valid @ X @ Xa )
% 0.22/0.58 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.22/0.58 ( X
% 0.22/0.58 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.22/0.58 => ( Xa = one_one_nat ) )
% 0.22/0.58 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.22/0.58 => ( ( Deg2 = Xa )
% 0.22/0.58 & ! [X4: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.58 => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.58 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.58 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.22/0.58 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.22/0.58 & ( case_o184042715313410164at_nat
% 0.22/0.58 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 0.22/0.58 & ! [X3: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.58 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.22/0.58 @ ( produc6081775807080527818_nat_o
% 0.22/0.58 @ ^ [Mi3: nat,Ma3: nat] :
% 0.22/0.58 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.22/0.58 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.22/0.58 & ! [I2: nat] :
% 0.22/0.58 ( ( 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 ) ) ) ) ) )
% 0.22/0.58 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 0.22/0.58 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.22/0.58 & ( ( Mi3 = Ma3 )
% 0.22/0.58 => ! [X3: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.58 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.22/0.58 & ( ( Mi3 != Ma3 )
% 0.22/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 0.22/0.58 & ! [X3: nat] :
% 0.22/0.58 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.22/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
% 0.22/0.58 => ( ( ord_less_nat @ Mi3 @ X3 )
% 0.22/0.58 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 0.22/0.58 @ Mima ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % VEBT_internal.valid'.elims(3)
% 0.22/0.58 thf(fact_9817_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 0.22/0.58 ! [X: vEBT_VEBT,Xa: nat] :
% 0.22/0.58 ( ( vEBT_VEBT_valid @ X @ Xa )
% 0.22/0.58 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.22/0.58 ( X
% 0.22/0.58 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.22/0.58 => ( Xa != one_one_nat ) )
% 0.22/0.58 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.22/0.58 => ~ ( ( Deg2 = Xa )
% 0.22/0.58 & ! [X5: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.58 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.58 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.58 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.22/0.58 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.22/0.58 & ( case_o184042715313410164at_nat
% 0.22/0.58 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 0.22/0.58 & ! [X3: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.58 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.22/0.58 @ ( produc6081775807080527818_nat_o
% 0.22/0.58 @ ^ [Mi3: nat,Ma3: nat] :
% 0.22/0.58 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.22/0.58 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.22/0.58 & ! [I2: nat] :
% 0.22/0.58 ( ( 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 ) ) ) ) ) )
% 0.22/0.58 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 0.22/0.58 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.22/0.58 & ( ( Mi3 = Ma3 )
% 0.22/0.58 => ! [X3: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.58 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.22/0.58 & ( ( Mi3 != Ma3 )
% 0.22/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 0.22/0.58 & ! [X3: nat] :
% 0.22/0.58 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.22/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
% 0.22/0.58 => ( ( ord_less_nat @ Mi3 @ X3 )
% 0.22/0.58 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 0.22/0.58 @ Mima ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % VEBT_internal.valid'.elims(2)
% 0.22/0.58 thf(fact_9818_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 0.22/0.58 ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
% 0.22/0.58 ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg4 )
% 0.22/0.58 = ( ( Deg = Deg4 )
% 0.22/0.58 & ! [X3: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.22/0.58 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.58 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.58 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.22/0.58 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.22/0.58 & ( case_o184042715313410164at_nat
% 0.22/0.58 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
% 0.22/0.58 & ! [X3: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.22/0.58 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.22/0.58 @ ( produc6081775807080527818_nat_o
% 0.22/0.58 @ ^ [Mi3: nat,Ma3: nat] :
% 0.22/0.58 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.22/0.58 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.22/0.58 & ! [I2: nat] :
% 0.22/0.58 ( ( 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 ) ) ) ) ) )
% 0.22/0.58 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X8 ) )
% 0.22/0.58 = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 0.22/0.58 & ( ( Mi3 = Ma3 )
% 0.22/0.58 => ! [X3: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.22/0.58 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.22/0.58 & ( ( Mi3 != Ma3 )
% 0.22/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
% 0.22/0.58 & ! [X3: nat] :
% 0.22/0.58 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.22/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X3 )
% 0.22/0.58 => ( ( ord_less_nat @ Mi3 @ X3 )
% 0.22/0.58 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 0.22/0.58 @ Mima2 ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % VEBT_internal.valid'.simps(2)
% 0.22/0.58 thf(fact_9819_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 0.22/0.58 ! [X: vEBT_VEBT,Xa: nat,Y4: $o] :
% 0.22/0.58 ( ( ( vEBT_VEBT_valid @ X @ Xa )
% 0.22/0.58 = Y4 )
% 0.22/0.58 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.22/0.58 ( X
% 0.22/0.58 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.22/0.58 => ( Y4
% 0.22/0.58 = ( Xa != one_one_nat ) ) )
% 0.22/0.58 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.22/0.58 => ( Y4
% 0.22/0.58 = ( ~ ( ( Deg2 = Xa )
% 0.22/0.58 & ! [X3: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.58 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.58 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.58 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.22/0.58 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.22/0.58 & ( case_o184042715313410164at_nat
% 0.22/0.58 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 0.22/0.58 & ! [X3: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.58 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.22/0.58 @ ( produc6081775807080527818_nat_o
% 0.22/0.58 @ ^ [Mi3: nat,Ma3: nat] :
% 0.22/0.58 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.22/0.58 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.22/0.58 & ! [I2: nat] :
% 0.22/0.58 ( ( 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 ) ) ) ) ) )
% 0.22/0.58 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 0.22/0.58 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.22/0.58 & ( ( Mi3 = Ma3 )
% 0.22/0.58 => ! [X3: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.58 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.22/0.58 & ( ( Mi3 != Ma3 )
% 0.22/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 0.22/0.58 & ! [X3: nat] :
% 0.22/0.58 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.22/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
% 0.22/0.58 => ( ( ord_less_nat @ Mi3 @ X3 )
% 0.22/0.58 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 0.22/0.58 @ Mima ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % VEBT_internal.valid'.elims(1)
% 0.22/0.58 thf(fact_9820_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 0.22/0.58 ! [X: vEBT_VEBT,Xa: nat] :
% 0.22/0.58 ( ~ ( vEBT_VEBT_valid @ X @ Xa )
% 0.22/0.58 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.22/0.58 => ( ! [Uu2: $o,Uv2: $o] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.22/0.58 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
% 0.22/0.58 => ( Xa = one_one_nat ) ) )
% 0.22/0.58 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.22/0.58 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) )
% 0.22/0.58 => ( ( Deg2 = Xa )
% 0.22/0.58 & ! [X4: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.58 => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.58 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.58 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.22/0.58 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.22/0.58 & ( case_o184042715313410164at_nat
% 0.22/0.58 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 0.22/0.58 & ! [X3: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.58 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.22/0.58 @ ( produc6081775807080527818_nat_o
% 0.22/0.58 @ ^ [Mi3: nat,Ma3: nat] :
% 0.22/0.58 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.22/0.58 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.22/0.58 & ! [I2: nat] :
% 0.22/0.58 ( ( 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 ) ) ) ) ) )
% 0.22/0.58 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 0.22/0.58 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.22/0.58 & ( ( Mi3 = Ma3 )
% 0.22/0.58 => ! [X3: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.58 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.22/0.58 & ( ( Mi3 != Ma3 )
% 0.22/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 0.22/0.58 & ! [X3: nat] :
% 0.22/0.58 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.22/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
% 0.22/0.58 => ( ( ord_less_nat @ Mi3 @ X3 )
% 0.22/0.58 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 0.22/0.58 @ Mima ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % VEBT_internal.valid'.pelims(3)
% 0.22/0.58 thf(fact_9821_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 0.22/0.58 ! [X: vEBT_VEBT,Xa: nat] :
% 0.22/0.58 ( ( vEBT_VEBT_valid @ X @ Xa )
% 0.22/0.58 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.22/0.58 => ( ! [Uu2: $o,Uv2: $o] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.22/0.58 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
% 0.22/0.58 => ( Xa != one_one_nat ) ) )
% 0.22/0.58 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.22/0.58 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) )
% 0.22/0.58 => ~ ( ( Deg2 = Xa )
% 0.22/0.58 & ! [X5: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.58 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.58 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.58 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.22/0.58 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.22/0.58 & ( case_o184042715313410164at_nat
% 0.22/0.58 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 0.22/0.58 & ! [X3: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.58 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.22/0.58 @ ( produc6081775807080527818_nat_o
% 0.22/0.58 @ ^ [Mi3: nat,Ma3: nat] :
% 0.22/0.58 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.22/0.58 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.22/0.58 & ! [I2: nat] :
% 0.22/0.58 ( ( 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 ) ) ) ) ) )
% 0.22/0.58 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 0.22/0.58 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.22/0.58 & ( ( Mi3 = Ma3 )
% 0.22/0.58 => ! [X3: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.58 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.22/0.58 & ( ( Mi3 != Ma3 )
% 0.22/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 0.22/0.58 & ! [X3: nat] :
% 0.22/0.58 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.22/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
% 0.22/0.58 => ( ( ord_less_nat @ Mi3 @ X3 )
% 0.22/0.58 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 0.22/0.58 @ Mima ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % VEBT_internal.valid'.pelims(2)
% 0.22/0.58 thf(fact_9822_Sup__int__def,axiom,
% 0.22/0.58 ( complete_Sup_Sup_int
% 0.22/0.58 = ( ^ [X8: set_int] :
% 0.22/0.58 ( the_int
% 0.22/0.58 @ ^ [X3: int] :
% 0.22/0.58 ( ( member_int @ X3 @ X8 )
% 0.22/0.58 & ! [Y6: int] :
% 0.22/0.58 ( ( member_int @ Y6 @ X8 )
% 0.22/0.58 => ( ord_less_eq_int @ Y6 @ X3 ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Sup_int_def
% 0.22/0.58 thf(fact_9823_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 0.22/0.58 ! [X: vEBT_VEBT,Xa: nat,Y4: $o] :
% 0.22/0.58 ( ( ( vEBT_VEBT_valid @ X @ Xa )
% 0.22/0.58 = Y4 )
% 0.22/0.58 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.22/0.58 => ( ! [Uu2: $o,Uv2: $o] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( Xa = one_one_nat ) )
% 0.22/0.58 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
% 0.22/0.58 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( ( Deg2 = Xa )
% 0.22/0.58 & ! [X3: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.58 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.58 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.22/0.58 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.22/0.58 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.22/0.58 & ( case_o184042715313410164at_nat
% 0.22/0.58 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 0.22/0.58 & ! [X3: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.58 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.22/0.58 @ ( produc6081775807080527818_nat_o
% 0.22/0.58 @ ^ [Mi3: nat,Ma3: nat] :
% 0.22/0.58 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.22/0.58 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.22/0.58 & ! [I2: nat] :
% 0.22/0.58 ( ( 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 ) ) ) ) ) )
% 0.22/0.58 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 0.22/0.58 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.22/0.58 & ( ( Mi3 = Ma3 )
% 0.22/0.58 => ! [X3: vEBT_VEBT] :
% 0.22/0.58 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.22/0.58 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.22/0.58 & ( ( Mi3 != Ma3 )
% 0.22/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 0.22/0.58 & ! [X3: nat] :
% 0.22/0.58 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.22/0.58 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
% 0.22/0.58 => ( ( ord_less_nat @ Mi3 @ X3 )
% 0.22/0.58 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 0.22/0.58 @ Mima ) ) )
% 0.22/0.58 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % VEBT_internal.valid'.pelims(1)
% 0.22/0.58 thf(fact_9824_Rats__eq__int__div__nat,axiom,
% 0.22/0.58 ( field_5140801741446780682s_real
% 0.22/0.58 = ( collect_real
% 0.22/0.58 @ ^ [Uu3: real] :
% 0.22/0.58 ? [I2: int,N4: nat] :
% 0.22/0.58 ( ( Uu3
% 0.22/0.58 = ( divide_divide_real @ ( ring_1_of_int_real @ I2 ) @ ( semiri5074537144036343181t_real @ N4 ) ) )
% 0.22/0.58 & ( N4 != zero_zero_nat ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Rats_eq_int_div_nat
% 0.22/0.58 thf(fact_9825_Rats__abs__iff,axiom,
% 0.22/0.58 ! [X: real] :
% 0.22/0.58 ( ( member_real @ ( abs_abs_real @ X ) @ field_5140801741446780682s_real )
% 0.22/0.58 = ( member_real @ X @ field_5140801741446780682s_real ) ) ).
% 0.22/0.58
% 0.22/0.58 % Rats_abs_iff
% 0.22/0.58 thf(fact_9826_Rats__dense__in__real,axiom,
% 0.22/0.58 ! [X: real,Y4: real] :
% 0.22/0.58 ( ( ord_less_real @ X @ Y4 )
% 0.22/0.58 => ? [X4: real] :
% 0.22/0.58 ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 0.22/0.58 & ( ord_less_real @ X @ X4 )
% 0.22/0.58 & ( ord_less_real @ X4 @ Y4 ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Rats_dense_in_real
% 0.22/0.58 thf(fact_9827_Rats__no__bot__less,axiom,
% 0.22/0.58 ! [X: real] :
% 0.22/0.58 ? [X4: real] :
% 0.22/0.58 ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 0.22/0.58 & ( ord_less_real @ X4 @ X ) ) ).
% 0.22/0.58
% 0.22/0.58 % Rats_no_bot_less
% 0.22/0.58 thf(fact_9828_Rats__no__top__le,axiom,
% 0.22/0.58 ! [X: real] :
% 0.22/0.58 ? [X4: real] :
% 0.22/0.58 ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 0.22/0.58 & ( ord_less_eq_real @ X @ X4 ) ) ).
% 0.22/0.58
% 0.22/0.58 % Rats_no_top_le
% 0.22/0.58 thf(fact_9829_Rats__eq__int__div__int,axiom,
% 0.22/0.58 ( field_5140801741446780682s_real
% 0.22/0.58 = ( collect_real
% 0.22/0.58 @ ^ [Uu3: real] :
% 0.22/0.58 ? [I2: int,J3: int] :
% 0.22/0.58 ( ( Uu3
% 0.22/0.58 = ( divide_divide_real @ ( ring_1_of_int_real @ I2 ) @ ( ring_1_of_int_real @ J3 ) ) )
% 0.22/0.58 & ( J3 != zero_zero_int ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Rats_eq_int_div_int
% 0.22/0.58 thf(fact_9830_rat__floor__lemma,axiom,
% 0.22/0.58 ! [A: int,B: int] :
% 0.22/0.58 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
% 0.22/0.58 & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % rat_floor_lemma
% 0.22/0.58 thf(fact_9831_mult__rat,axiom,
% 0.22/0.58 ! [A: int,B: int,C: int,D: int] :
% 0.22/0.58 ( ( times_times_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.22/0.58 = ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % mult_rat
% 0.22/0.58 thf(fact_9832_divide__rat,axiom,
% 0.22/0.58 ! [A: int,B: int,C: int,D: int] :
% 0.22/0.58 ( ( divide_divide_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.22/0.58 = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % divide_rat
% 0.22/0.58 thf(fact_9833_floor__Fract,axiom,
% 0.22/0.58 ! [A: int,B: int] :
% 0.22/0.58 ( ( archim3151403230148437115or_rat @ ( fract @ A @ B ) )
% 0.22/0.58 = ( divide_divide_int @ A @ B ) ) ).
% 0.22/0.58
% 0.22/0.58 % floor_Fract
% 0.22/0.58 thf(fact_9834_less__rat,axiom,
% 0.22/0.58 ! [B: int,D: int,A: int,C: int] :
% 0.22/0.58 ( ( B != zero_zero_int )
% 0.22/0.58 => ( ( D != zero_zero_int )
% 0.22/0.58 => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.22/0.58 = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % less_rat
% 0.22/0.58 thf(fact_9835_add__rat,axiom,
% 0.22/0.58 ! [B: int,D: int,A: int,C: int] :
% 0.22/0.58 ( ( B != zero_zero_int )
% 0.22/0.58 => ( ( D != zero_zero_int )
% 0.22/0.58 => ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.22/0.58 = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % add_rat
% 0.22/0.58 thf(fact_9836_le__rat,axiom,
% 0.22/0.58 ! [B: int,D: int,A: int,C: int] :
% 0.22/0.58 ( ( B != zero_zero_int )
% 0.22/0.58 => ( ( D != zero_zero_int )
% 0.22/0.58 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.22/0.58 = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % le_rat
% 0.22/0.58 thf(fact_9837_diff__rat,axiom,
% 0.22/0.58 ! [B: int,D: int,A: int,C: int] :
% 0.22/0.58 ( ( B != zero_zero_int )
% 0.22/0.58 => ( ( D != zero_zero_int )
% 0.22/0.58 => ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.22/0.58 = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % diff_rat
% 0.22/0.58 thf(fact_9838_sgn__rat,axiom,
% 0.22/0.58 ! [A: int,B: int] :
% 0.22/0.58 ( ( sgn_sgn_rat @ ( fract @ A @ B ) )
% 0.22/0.58 = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % sgn_rat
% 0.22/0.58 thf(fact_9839_Rat__induct__pos,axiom,
% 0.22/0.58 ! [P: rat > $o,Q2: rat] :
% 0.22/0.58 ( ! [A4: int,B3: int] :
% 0.22/0.58 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.22/0.58 => ( P @ ( fract @ A4 @ B3 ) ) )
% 0.22/0.58 => ( P @ Q2 ) ) ).
% 0.22/0.58
% 0.22/0.58 % Rat_induct_pos
% 0.22/0.58 thf(fact_9840_Fract__coprime,axiom,
% 0.22/0.58 ! [A: int,B: int] :
% 0.22/0.58 ( ( fract @ ( divide_divide_int @ A @ ( gcd_gcd_int @ A @ B ) ) @ ( divide_divide_int @ B @ ( gcd_gcd_int @ A @ B ) ) )
% 0.22/0.58 = ( fract @ A @ B ) ) ).
% 0.22/0.58
% 0.22/0.58 % Fract_coprime
% 0.22/0.58 thf(fact_9841_rat__number__collapse_I1_J,axiom,
% 0.22/0.58 ! [K: int] :
% 0.22/0.58 ( ( fract @ zero_zero_int @ K )
% 0.22/0.58 = zero_zero_rat ) ).
% 0.22/0.58
% 0.22/0.58 % rat_number_collapse(1)
% 0.22/0.58 thf(fact_9842_rat__number__collapse_I6_J,axiom,
% 0.22/0.58 ! [K: int] :
% 0.22/0.58 ( ( fract @ K @ zero_zero_int )
% 0.22/0.58 = zero_zero_rat ) ).
% 0.22/0.58
% 0.22/0.58 % rat_number_collapse(6)
% 0.22/0.58 thf(fact_9843_eq__rat_I2_J,axiom,
% 0.22/0.58 ! [A: int] :
% 0.22/0.58 ( ( fract @ A @ zero_zero_int )
% 0.22/0.58 = ( fract @ zero_zero_int @ one_one_int ) ) ).
% 0.22/0.58
% 0.22/0.58 % eq_rat(2)
% 0.22/0.58 thf(fact_9844_mult__rat__cancel,axiom,
% 0.22/0.58 ! [C: int,A: int,B: int] :
% 0.22/0.58 ( ( C != zero_zero_int )
% 0.22/0.58 => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.22/0.58 = ( fract @ A @ B ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % mult_rat_cancel
% 0.22/0.58 thf(fact_9845_eq__rat_I1_J,axiom,
% 0.22/0.58 ! [B: int,D: int,A: int,C: int] :
% 0.22/0.58 ( ( B != zero_zero_int )
% 0.22/0.58 => ( ( D != zero_zero_int )
% 0.22/0.58 => ( ( ( fract @ A @ B )
% 0.22/0.58 = ( fract @ C @ D ) )
% 0.22/0.58 = ( ( times_times_int @ A @ D )
% 0.22/0.58 = ( times_times_int @ C @ B ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % eq_rat(1)
% 0.22/0.58 thf(fact_9846_eq__rat_I3_J,axiom,
% 0.22/0.58 ! [A: int,C: int] :
% 0.22/0.58 ( ( fract @ zero_zero_int @ A )
% 0.22/0.58 = ( fract @ zero_zero_int @ C ) ) ).
% 0.22/0.58
% 0.22/0.58 % eq_rat(3)
% 0.22/0.58 thf(fact_9847_Fract__of__nat__eq,axiom,
% 0.22/0.58 ! [K: nat] :
% 0.22/0.58 ( ( fract @ ( semiri1314217659103216013at_int @ K ) @ one_one_int )
% 0.22/0.58 = ( semiri681578069525770553at_rat @ K ) ) ).
% 0.22/0.58
% 0.22/0.58 % Fract_of_nat_eq
% 0.22/0.58 thf(fact_9848_Zero__rat__def,axiom,
% 0.22/0.58 ( zero_zero_rat
% 0.22/0.58 = ( fract @ zero_zero_int @ one_one_int ) ) ).
% 0.22/0.58
% 0.22/0.58 % Zero_rat_def
% 0.22/0.58 thf(fact_9849_atLeastLessThan__add__Un,axiom,
% 0.22/0.58 ! [I: nat,J: nat,K: nat] :
% 0.22/0.58 ( ( ord_less_eq_nat @ I @ J )
% 0.22/0.58 => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
% 0.22/0.58 = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % atLeastLessThan_add_Un
% 0.22/0.58 thf(fact_9850_rat__number__collapse_I3_J,axiom,
% 0.22/0.58 ! [W: num] :
% 0.22/0.58 ( ( fract @ ( numeral_numeral_int @ W ) @ one_one_int )
% 0.22/0.58 = ( numeral_numeral_rat @ W ) ) ).
% 0.22/0.58
% 0.22/0.58 % rat_number_collapse(3)
% 0.22/0.58 thf(fact_9851_rat__number__expand_I3_J,axiom,
% 0.22/0.58 ( numeral_numeral_rat
% 0.22/0.58 = ( ^ [K3: num] : ( fract @ ( numeral_numeral_int @ K3 ) @ one_one_int ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % rat_number_expand(3)
% 0.22/0.58 thf(fact_9852_zero__less__Fract__iff,axiom,
% 0.22/0.58 ! [B: int,A: int] :
% 0.22/0.58 ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.58 => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 0.22/0.58 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % zero_less_Fract_iff
% 0.22/0.58 thf(fact_9853_Fract__less__zero__iff,axiom,
% 0.22/0.58 ! [B: int,A: int] :
% 0.22/0.58 ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.58 => ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 0.22/0.58 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Fract_less_zero_iff
% 0.22/0.58 thf(fact_9854_Fract__less__one__iff,axiom,
% 0.22/0.58 ! [B: int,A: int] :
% 0.22/0.58 ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.58 => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
% 0.22/0.58 = ( ord_less_int @ A @ B ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Fract_less_one_iff
% 0.22/0.58 thf(fact_9855_one__less__Fract__iff,axiom,
% 0.22/0.58 ! [B: int,A: int] :
% 0.22/0.58 ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.58 => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
% 0.22/0.58 = ( ord_less_int @ B @ A ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % one_less_Fract_iff
% 0.22/0.58 thf(fact_9856_Fract__add__one,axiom,
% 0.22/0.58 ! [N: int,M: int] :
% 0.22/0.58 ( ( N != zero_zero_int )
% 0.22/0.58 => ( ( fract @ ( plus_plus_int @ M @ N ) @ N )
% 0.22/0.58 = ( plus_plus_rat @ ( fract @ M @ N ) @ one_one_rat ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Fract_add_one
% 0.22/0.58 thf(fact_9857_Fract__le__zero__iff,axiom,
% 0.22/0.58 ! [B: int,A: int] :
% 0.22/0.58 ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.58 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 0.22/0.58 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Fract_le_zero_iff
% 0.22/0.58 thf(fact_9858_zero__le__Fract__iff,axiom,
% 0.22/0.58 ! [B: int,A: int] :
% 0.22/0.58 ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.58 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 0.22/0.58 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % zero_le_Fract_iff
% 0.22/0.58 thf(fact_9859_one__le__Fract__iff,axiom,
% 0.22/0.58 ! [B: int,A: int] :
% 0.22/0.58 ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.58 => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
% 0.22/0.58 = ( ord_less_eq_int @ B @ A ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % one_le_Fract_iff
% 0.22/0.58 thf(fact_9860_Fract__le__one__iff,axiom,
% 0.22/0.58 ! [B: int,A: int] :
% 0.22/0.58 ( ( ord_less_int @ zero_zero_int @ B )
% 0.22/0.58 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
% 0.22/0.58 = ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Fract_le_one_iff
% 0.22/0.58 thf(fact_9861_rat__number__expand_I5_J,axiom,
% 0.22/0.58 ! [K: num] :
% 0.22/0.58 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) )
% 0.22/0.58 = ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 0.22/0.58
% 0.22/0.58 % rat_number_expand(5)
% 0.22/0.58 thf(fact_9862_rat__number__collapse_I4_J,axiom,
% 0.22/0.58 ! [W: num] :
% 0.22/0.58 ( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ one_one_int )
% 0.22/0.58 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % rat_number_collapse(4)
% 0.22/0.58 thf(fact_9863_sup__int__def,axiom,
% 0.22/0.58 sup_sup_int = ord_max_int ).
% 0.22/0.58
% 0.22/0.58 % sup_int_def
% 0.22/0.58 thf(fact_9864_sup__enat__def,axiom,
% 0.22/0.58 sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).
% 0.22/0.58
% 0.22/0.58 % sup_enat_def
% 0.22/0.58 thf(fact_9865_sup__nat__def,axiom,
% 0.22/0.58 sup_sup_nat = ord_max_nat ).
% 0.22/0.58
% 0.22/0.58 % sup_nat_def
% 0.22/0.58 thf(fact_9866_pos__deriv__imp__strict__mono,axiom,
% 0.22/0.58 ! [F: real > real,F2: real > real] :
% 0.22/0.58 ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.22/0.58 => ( ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( F2 @ X4 ) )
% 0.22/0.58 => ( order_7092887310737990675l_real @ F ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % pos_deriv_imp_strict_mono
% 0.22/0.58 thf(fact_9867_take__bit__numeral__minus__numeral__int,axiom,
% 0.22/0.58 ! [M: num,N: num] :
% 0.22/0.58 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.58 = ( case_option_int_num @ zero_zero_int
% 0.22/0.58 @ ^ [Q3: 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 @ Q3 ) ) )
% 0.22/0.58 @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % take_bit_numeral_minus_numeral_int
% 0.22/0.58 thf(fact_9868_of__nat__eq__id,axiom,
% 0.22/0.58 semiri1316708129612266289at_nat = id_nat ).
% 0.22/0.58
% 0.22/0.58 % of_nat_eq_id
% 0.22/0.58 thf(fact_9869_take__bit__num__simps_I1_J,axiom,
% 0.22/0.58 ! [M: num] :
% 0.22/0.58 ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 0.22/0.58 = none_num ) ).
% 0.22/0.58
% 0.22/0.58 % take_bit_num_simps(1)
% 0.22/0.58 thf(fact_9870_take__bit__num__simps_I2_J,axiom,
% 0.22/0.58 ! [N: nat] :
% 0.22/0.58 ( ( bit_take_bit_num @ ( suc @ N ) @ one )
% 0.22/0.58 = ( some_num @ one ) ) ).
% 0.22/0.58
% 0.22/0.58 % take_bit_num_simps(2)
% 0.22/0.58 thf(fact_9871_take__bit__num__simps_I5_J,axiom,
% 0.22/0.58 ! [R2: num] :
% 0.22/0.58 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ one )
% 0.22/0.58 = ( some_num @ one ) ) ).
% 0.22/0.58
% 0.22/0.58 % take_bit_num_simps(5)
% 0.22/0.58 thf(fact_9872_strict__mono__imp__increasing,axiom,
% 0.22/0.58 ! [F: nat > nat,N: nat] :
% 0.22/0.58 ( ( order_5726023648592871131at_nat @ F )
% 0.22/0.58 => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % strict_mono_imp_increasing
% 0.22/0.58 thf(fact_9873_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 0.22/0.58 ! [N: nat] :
% 0.22/0.58 ( ( bit_take_bit_num @ N @ one )
% 0.22/0.58 = ( case_nat_option_num @ none_num
% 0.22/0.58 @ ^ [N4: nat] : ( some_num @ one )
% 0.22/0.58 @ N ) ) ).
% 0.22/0.58
% 0.22/0.58 % Code_Abstract_Nat.take_bit_num_code(1)
% 0.22/0.58 thf(fact_9874_less__int__def,axiom,
% 0.22/0.58 ( ord_less_int
% 0.22/0.58 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 0.22/0.58 @ ( produc8739625826339149834_nat_o
% 0.22/0.58 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.58 ( produc6081775807080527818_nat_o
% 0.22/0.58 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % less_int_def
% 0.22/0.58 thf(fact_9875_nat__def,axiom,
% 0.22/0.58 ( nat2
% 0.22/0.58 = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % nat_def
% 0.22/0.58 thf(fact_9876_less__eq__int__def,axiom,
% 0.22/0.58 ( ord_less_eq_int
% 0.22/0.58 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 0.22/0.58 @ ( produc8739625826339149834_nat_o
% 0.22/0.58 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.58 ( produc6081775807080527818_nat_o
% 0.22/0.58 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % less_eq_int_def
% 0.22/0.58 thf(fact_9877_take__bit__num__def,axiom,
% 0.22/0.58 ( bit_take_bit_num
% 0.22/0.58 = ( ^ [N4: nat,M5: num] :
% 0.22/0.58 ( if_option_num
% 0.22/0.58 @ ( ( bit_se2925701944663578781it_nat @ N4 @ ( numeral_numeral_nat @ M5 ) )
% 0.22/0.58 = zero_zero_nat )
% 0.22/0.58 @ none_num
% 0.22/0.58 @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N4 @ ( numeral_numeral_nat @ M5 ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % take_bit_num_def
% 0.22/0.58 thf(fact_9878_and__minus__numerals_I3_J,axiom,
% 0.22/0.58 ! [M: num,N: num] :
% 0.22/0.58 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.22/0.58 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_minus_numerals(3)
% 0.22/0.58 thf(fact_9879_and__minus__numerals_I7_J,axiom,
% 0.22/0.58 ! [N: num,M: num] :
% 0.22/0.58 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.22/0.58 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_minus_numerals(7)
% 0.22/0.58 thf(fact_9880_take__bit__num__simps_I4_J,axiom,
% 0.22/0.58 ! [N: nat,M: num] :
% 0.22/0.58 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
% 0.22/0.58 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % take_bit_num_simps(4)
% 0.22/0.58 thf(fact_9881_take__bit__num__simps_I3_J,axiom,
% 0.22/0.58 ! [N: nat,M: num] :
% 0.22/0.58 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
% 0.22/0.58 = ( case_o6005452278849405969um_num @ none_num
% 0.22/0.58 @ ^ [Q3: num] : ( some_num @ ( bit0 @ Q3 ) )
% 0.22/0.58 @ ( bit_take_bit_num @ N @ M ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % take_bit_num_simps(3)
% 0.22/0.58 thf(fact_9882_take__bit__num__simps_I7_J,axiom,
% 0.22/0.58 ! [R2: num,M: num] :
% 0.22/0.58 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M ) )
% 0.22/0.58 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % take_bit_num_simps(7)
% 0.22/0.58 thf(fact_9883_take__bit__num__simps_I6_J,axiom,
% 0.22/0.58 ! [R2: num,M: num] :
% 0.22/0.58 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
% 0.22/0.58 = ( case_o6005452278849405969um_num @ none_num
% 0.22/0.58 @ ^ [Q3: num] : ( some_num @ ( bit0 @ Q3 ) )
% 0.22/0.58 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % take_bit_num_simps(6)
% 0.22/0.58 thf(fact_9884_and__minus__numerals_I8_J,axiom,
% 0.22/0.58 ! [N: num,M: num] :
% 0.22/0.58 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.22/0.58 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_minus_numerals(8)
% 0.22/0.58 thf(fact_9885_and__minus__numerals_I4_J,axiom,
% 0.22/0.58 ! [M: num,N: num] :
% 0.22/0.58 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.22/0.58 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_minus_numerals(4)
% 0.22/0.58 thf(fact_9886_and__not__num_Osimps_I8_J,axiom,
% 0.22/0.58 ! [M: num,N: num] :
% 0.22/0.58 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.22/0.58 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.22/0.58 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 0.22/0.58 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_not_num.simps(8)
% 0.22/0.58 thf(fact_9887_and__not__num_Osimps_I1_J,axiom,
% 0.22/0.58 ( ( bit_and_not_num @ one @ one )
% 0.22/0.58 = none_num ) ).
% 0.22/0.58
% 0.22/0.58 % and_not_num.simps(1)
% 0.22/0.58 thf(fact_9888_and__not__num_Osimps_I3_J,axiom,
% 0.22/0.58 ! [N: num] :
% 0.22/0.58 ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
% 0.22/0.58 = none_num ) ).
% 0.22/0.58
% 0.22/0.58 % and_not_num.simps(3)
% 0.22/0.58 thf(fact_9889_and__not__num_Osimps_I4_J,axiom,
% 0.22/0.58 ! [M: num] :
% 0.22/0.58 ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
% 0.22/0.58 = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_not_num.simps(4)
% 0.22/0.58 thf(fact_9890_and__not__num_Osimps_I2_J,axiom,
% 0.22/0.58 ! [N: num] :
% 0.22/0.58 ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
% 0.22/0.58 = ( some_num @ one ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_not_num.simps(2)
% 0.22/0.58 thf(fact_9891_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 0.22/0.58 ! [N: nat,M: num] :
% 0.22/0.58 ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
% 0.22/0.58 = ( case_nat_option_num @ none_num
% 0.22/0.58 @ ^ [N4: nat] :
% 0.22/0.58 ( case_o6005452278849405969um_num @ none_num
% 0.22/0.58 @ ^ [Q3: num] : ( some_num @ ( bit0 @ Q3 ) )
% 0.22/0.58 @ ( bit_take_bit_num @ N4 @ M ) )
% 0.22/0.58 @ N ) ) ).
% 0.22/0.58
% 0.22/0.58 % Code_Abstract_Nat.take_bit_num_code(2)
% 0.22/0.58 thf(fact_9892_and__not__num_Osimps_I7_J,axiom,
% 0.22/0.58 ! [M: num] :
% 0.22/0.58 ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
% 0.22/0.58 = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_not_num.simps(7)
% 0.22/0.58 thf(fact_9893_and__not__num__eq__Some__iff,axiom,
% 0.22/0.58 ! [M: num,N: num,Q2: num] :
% 0.22/0.58 ( ( ( bit_and_not_num @ M @ N )
% 0.22/0.58 = ( some_num @ Q2 ) )
% 0.22/0.58 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.58 = ( numeral_numeral_int @ Q2 ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_not_num_eq_Some_iff
% 0.22/0.58 thf(fact_9894_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 0.22/0.58 ! [N: nat,M: num] :
% 0.22/0.58 ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
% 0.22/0.58 = ( case_nat_option_num @ none_num
% 0.22/0.58 @ ^ [N4: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N4 @ M ) ) )
% 0.22/0.58 @ N ) ) ).
% 0.22/0.58
% 0.22/0.58 % Code_Abstract_Nat.take_bit_num_code(3)
% 0.22/0.58 thf(fact_9895_and__not__num__eq__None__iff,axiom,
% 0.22/0.58 ! [M: num,N: num] :
% 0.22/0.58 ( ( ( bit_and_not_num @ M @ N )
% 0.22/0.58 = none_num )
% 0.22/0.58 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.58 = zero_zero_int ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_not_num_eq_None_iff
% 0.22/0.58 thf(fact_9896_int__numeral__and__not__num,axiom,
% 0.22/0.58 ! [M: num,N: num] :
% 0.22/0.58 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.58 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % int_numeral_and_not_num
% 0.22/0.58 thf(fact_9897_int__numeral__not__and__num,axiom,
% 0.22/0.58 ! [M: num,N: num] :
% 0.22/0.58 ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.22/0.58 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % int_numeral_not_and_num
% 0.22/0.58 thf(fact_9898_positive__rat,axiom,
% 0.22/0.58 ! [A: int,B: int] :
% 0.22/0.58 ( ( positive @ ( fract @ A @ B ) )
% 0.22/0.58 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % positive_rat
% 0.22/0.58 thf(fact_9899_Rat_Opositive__zero,axiom,
% 0.22/0.58 ~ ( positive @ zero_zero_rat ) ).
% 0.22/0.58
% 0.22/0.58 % Rat.positive_zero
% 0.22/0.58 thf(fact_9900_Rat_Opositive__minus,axiom,
% 0.22/0.58 ! [X: rat] :
% 0.22/0.58 ( ~ ( positive @ X )
% 0.22/0.58 => ( ( X != zero_zero_rat )
% 0.22/0.58 => ( positive @ ( uminus_uminus_rat @ X ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Rat.positive_minus
% 0.22/0.58 thf(fact_9901_less__rat__def,axiom,
% 0.22/0.58 ( ord_less_rat
% 0.22/0.58 = ( ^ [X3: rat,Y6: rat] : ( positive @ ( minus_minus_rat @ Y6 @ X3 ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % less_rat_def
% 0.22/0.58 thf(fact_9902_Rat_Opositive_Orep__eq,axiom,
% 0.22/0.58 ( positive
% 0.22/0.58 = ( ^ [X3: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X3 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X3 ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Rat.positive.rep_eq
% 0.22/0.58 thf(fact_9903_and__not__num_Oelims,axiom,
% 0.22/0.58 ! [X: num,Xa: num,Y4: option_num] :
% 0.22/0.58 ( ( ( bit_and_not_num @ X @ Xa )
% 0.22/0.58 = Y4 )
% 0.22/0.58 => ( ( ( X = one )
% 0.22/0.58 => ( ( Xa = one )
% 0.22/0.58 => ( Y4 != none_num ) ) )
% 0.22/0.58 => ( ( ( X = one )
% 0.22/0.58 => ( ? [N2: num] :
% 0.22/0.58 ( Xa
% 0.22/0.58 = ( bit0 @ N2 ) )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( some_num @ one ) ) ) )
% 0.22/0.58 => ( ( ( X = one )
% 0.22/0.58 => ( ? [N2: num] :
% 0.22/0.58 ( Xa
% 0.22/0.58 = ( bit1 @ N2 ) )
% 0.22/0.58 => ( Y4 != none_num ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit0 @ M4 ) )
% 0.22/0.58 => ( ( Xa = one )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit0 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit0 @ N2 ) )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit0 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit1 @ N2 ) )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit1 @ M4 ) )
% 0.22/0.58 => ( ( Xa = one )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit1 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit0 @ N2 ) )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.22/0.58 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 0.22/0.58 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) )
% 0.22/0.58 => ~ ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit1 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit1 @ N2 ) )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_not_num.elims
% 0.22/0.58 thf(fact_9904_and__not__num_Osimps_I5_J,axiom,
% 0.22/0.58 ! [M: num,N: num] :
% 0.22/0.58 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.22/0.58 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_not_num.simps(5)
% 0.22/0.58 thf(fact_9905_and__not__num_Osimps_I9_J,axiom,
% 0.22/0.58 ! [M: num,N: num] :
% 0.22/0.58 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.22/0.58 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_not_num.simps(9)
% 0.22/0.58 thf(fact_9906_and__not__num_Osimps_I6_J,axiom,
% 0.22/0.58 ! [M: num,N: num] :
% 0.22/0.58 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.22/0.58 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_not_num.simps(6)
% 0.22/0.58 thf(fact_9907_and__not__num_Opelims,axiom,
% 0.22/0.58 ! [X: num,Xa: num,Y4: option_num] :
% 0.22/0.58 ( ( ( bit_and_not_num @ X @ Xa )
% 0.22/0.58 = Y4 )
% 0.22/0.58 => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 0.22/0.58 => ( ( ( X = one )
% 0.22/0.58 => ( ( Xa = one )
% 0.22/0.58 => ( ( Y4 = none_num )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.22/0.58 => ( ( ( X = one )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit0 @ N2 ) )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( some_num @ one ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 0.22/0.58 => ( ( ( X = one )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit1 @ N2 ) )
% 0.22/0.58 => ( ( Y4 = none_num )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit0 @ M4 ) )
% 0.22/0.58 => ( ( Xa = one )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( some_num @ ( bit0 @ M4 ) ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit0 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit0 @ N2 ) )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit0 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit1 @ N2 ) )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit1 @ M4 ) )
% 0.22/0.58 => ( ( Xa = one )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( some_num @ ( bit0 @ M4 ) ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit1 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit0 @ N2 ) )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.22/0.58 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 0.22/0.58 @ ( bit_and_not_num @ M4 @ N2 ) ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.22/0.58 => ~ ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit1 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit1 @ N2 ) )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_not_num.pelims
% 0.22/0.58 thf(fact_9908_Rat_Opositive__def,axiom,
% 0.22/0.58 ( positive
% 0.22/0.58 = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
% 0.22/0.58 @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Rat.positive_def
% 0.22/0.58 thf(fact_9909_and__num_Oelims,axiom,
% 0.22/0.58 ! [X: num,Xa: num,Y4: option_num] :
% 0.22/0.58 ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 0.22/0.58 = Y4 )
% 0.22/0.58 => ( ( ( X = one )
% 0.22/0.58 => ( ( Xa = one )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( some_num @ one ) ) ) )
% 0.22/0.58 => ( ( ( X = one )
% 0.22/0.58 => ( ? [N2: num] :
% 0.22/0.58 ( Xa
% 0.22/0.58 = ( bit0 @ N2 ) )
% 0.22/0.58 => ( Y4 != none_num ) ) )
% 0.22/0.58 => ( ( ( X = one )
% 0.22/0.58 => ( ? [N2: num] :
% 0.22/0.58 ( Xa
% 0.22/0.58 = ( bit1 @ N2 ) )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( some_num @ one ) ) ) )
% 0.22/0.58 => ( ( ? [M4: num] :
% 0.22/0.58 ( X
% 0.22/0.58 = ( bit0 @ M4 ) )
% 0.22/0.58 => ( ( Xa = one )
% 0.22/0.58 => ( Y4 != none_num ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit0 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit0 @ N2 ) )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit0 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit1 @ N2 ) )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) )
% 0.22/0.58 => ( ( ? [M4: num] :
% 0.22/0.58 ( X
% 0.22/0.58 = ( bit1 @ M4 ) )
% 0.22/0.58 => ( ( Xa = one )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( some_num @ one ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit1 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit0 @ N2 ) )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) )
% 0.22/0.58 => ~ ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit1 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit1 @ N2 ) )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.22/0.58 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 0.22/0.58 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_num.elims
% 0.22/0.58 thf(fact_9910_xor__num_Oelims,axiom,
% 0.22/0.58 ! [X: num,Xa: num,Y4: option_num] :
% 0.22/0.58 ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 0.22/0.58 = Y4 )
% 0.22/0.58 => ( ( ( X = one )
% 0.22/0.58 => ( ( Xa = one )
% 0.22/0.58 => ( Y4 != none_num ) ) )
% 0.22/0.58 => ( ( ( X = one )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit0 @ N2 ) )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( some_num @ ( bit1 @ N2 ) ) ) ) )
% 0.22/0.58 => ( ( ( X = one )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit1 @ N2 ) )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( some_num @ ( bit0 @ N2 ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit0 @ M4 ) )
% 0.22/0.58 => ( ( Xa = one )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( some_num @ ( bit1 @ M4 ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit0 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit0 @ N2 ) )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit0 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit1 @ N2 ) )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit1 @ M4 ) )
% 0.22/0.58 => ( ( Xa = one )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit1 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit0 @ N2 ) )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) ) )
% 0.22/0.58 => ~ ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit1 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit1 @ N2 ) )
% 0.22/0.58 => ( Y4
% 0.22/0.58 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % xor_num.elims
% 0.22/0.58 thf(fact_9911_and__num_Osimps_I1_J,axiom,
% 0.22/0.58 ( ( bit_un7362597486090784418nd_num @ one @ one )
% 0.22/0.58 = ( some_num @ one ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_num.simps(1)
% 0.22/0.58 thf(fact_9912_xor__num_Osimps_I1_J,axiom,
% 0.22/0.58 ( ( bit_un2480387367778600638or_num @ one @ one )
% 0.22/0.58 = none_num ) ).
% 0.22/0.58
% 0.22/0.58 % xor_num.simps(1)
% 0.22/0.58 thf(fact_9913_xor__num_Osimps_I5_J,axiom,
% 0.22/0.58 ! [M: num,N: num] :
% 0.22/0.58 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.22/0.58 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % xor_num.simps(5)
% 0.22/0.58 thf(fact_9914_and__num_Osimps_I5_J,axiom,
% 0.22/0.58 ! [M: num,N: num] :
% 0.22/0.58 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.22/0.58 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_num.simps(5)
% 0.22/0.58 thf(fact_9915_and__num_Osimps_I7_J,axiom,
% 0.22/0.58 ! [M: num] :
% 0.22/0.58 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
% 0.22/0.58 = ( some_num @ one ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_num.simps(7)
% 0.22/0.58 thf(fact_9916_and__num_Osimps_I3_J,axiom,
% 0.22/0.58 ! [N: num] :
% 0.22/0.58 ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
% 0.22/0.58 = ( some_num @ one ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_num.simps(3)
% 0.22/0.58 thf(fact_9917_and__num_Osimps_I4_J,axiom,
% 0.22/0.58 ! [M: num] :
% 0.22/0.58 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 0.22/0.58 = none_num ) ).
% 0.22/0.58
% 0.22/0.58 % and_num.simps(4)
% 0.22/0.58 thf(fact_9918_and__num_Osimps_I2_J,axiom,
% 0.22/0.58 ! [N: num] :
% 0.22/0.58 ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
% 0.22/0.58 = none_num ) ).
% 0.22/0.58
% 0.22/0.58 % and_num.simps(2)
% 0.22/0.58 thf(fact_9919_and__num_Osimps_I8_J,axiom,
% 0.22/0.58 ! [M: num,N: num] :
% 0.22/0.58 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.22/0.58 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_num.simps(8)
% 0.22/0.58 thf(fact_9920_and__num_Osimps_I6_J,axiom,
% 0.22/0.58 ! [M: num,N: num] :
% 0.22/0.58 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.22/0.58 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_num.simps(6)
% 0.22/0.58 thf(fact_9921_xor__num_Osimps_I9_J,axiom,
% 0.22/0.58 ! [M: num,N: num] :
% 0.22/0.58 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.22/0.58 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % xor_num.simps(9)
% 0.22/0.58 thf(fact_9922_xor__num_Osimps_I2_J,axiom,
% 0.22/0.58 ! [N: num] :
% 0.22/0.58 ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
% 0.22/0.58 = ( some_num @ ( bit1 @ N ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % xor_num.simps(2)
% 0.22/0.58 thf(fact_9923_xor__num_Osimps_I3_J,axiom,
% 0.22/0.58 ! [N: num] :
% 0.22/0.58 ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
% 0.22/0.58 = ( some_num @ ( bit0 @ N ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % xor_num.simps(3)
% 0.22/0.58 thf(fact_9924_xor__num_Osimps_I4_J,axiom,
% 0.22/0.58 ! [M: num] :
% 0.22/0.58 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
% 0.22/0.58 = ( some_num @ ( bit1 @ M ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % xor_num.simps(4)
% 0.22/0.58 thf(fact_9925_xor__num_Osimps_I7_J,axiom,
% 0.22/0.58 ! [M: num] :
% 0.22/0.58 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
% 0.22/0.58 = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % xor_num.simps(7)
% 0.22/0.58 thf(fact_9926_and__num_Osimps_I9_J,axiom,
% 0.22/0.58 ! [M: num,N: num] :
% 0.22/0.58 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.22/0.58 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.22/0.58 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 0.22/0.58 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_num.simps(9)
% 0.22/0.58 thf(fact_9927_xor__num_Osimps_I8_J,axiom,
% 0.22/0.58 ! [M: num,N: num] :
% 0.22/0.58 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.22/0.58 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % xor_num.simps(8)
% 0.22/0.58 thf(fact_9928_xor__num_Osimps_I6_J,axiom,
% 0.22/0.58 ! [M: num,N: num] :
% 0.22/0.58 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.22/0.58 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % xor_num.simps(6)
% 0.22/0.58 thf(fact_9929_and__num_Opelims,axiom,
% 0.22/0.58 ! [X: num,Xa: num,Y4: option_num] :
% 0.22/0.58 ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 0.22/0.58 = Y4 )
% 0.22/0.58 => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 0.22/0.58 => ( ( ( X = one )
% 0.22/0.58 => ( ( Xa = one )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( some_num @ one ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.22/0.58 => ( ( ( X = one )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit0 @ N2 ) )
% 0.22/0.58 => ( ( Y4 = none_num )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 0.22/0.58 => ( ( ( X = one )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit1 @ N2 ) )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( some_num @ one ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit0 @ M4 ) )
% 0.22/0.58 => ( ( Xa = one )
% 0.22/0.58 => ( ( Y4 = none_num )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit0 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit0 @ N2 ) )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit0 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit1 @ N2 ) )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit1 @ M4 ) )
% 0.22/0.58 => ( ( Xa = one )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( some_num @ one ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit1 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit0 @ N2 ) )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.22/0.58 => ~ ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit1 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit1 @ N2 ) )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.22/0.58 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 0.22/0.58 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % and_num.pelims
% 0.22/0.58 thf(fact_9930_xor__num_Opelims,axiom,
% 0.22/0.58 ! [X: num,Xa: num,Y4: option_num] :
% 0.22/0.58 ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 0.22/0.58 = Y4 )
% 0.22/0.58 => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 0.22/0.58 => ( ( ( X = one )
% 0.22/0.58 => ( ( Xa = one )
% 0.22/0.58 => ( ( Y4 = none_num )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.22/0.58 => ( ( ( X = one )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit0 @ N2 ) )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( some_num @ ( bit1 @ N2 ) ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 0.22/0.58 => ( ( ( X = one )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit1 @ N2 ) )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( some_num @ ( bit0 @ N2 ) ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit0 @ M4 ) )
% 0.22/0.58 => ( ( Xa = one )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( some_num @ ( bit1 @ M4 ) ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit0 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit0 @ N2 ) )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit0 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit1 @ N2 ) )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit1 @ M4 ) )
% 0.22/0.58 => ( ( Xa = one )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( some_num @ ( bit0 @ M4 ) ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 0.22/0.58 => ( ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit1 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit0 @ N2 ) )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.22/0.58 => ~ ! [M4: num] :
% 0.22/0.58 ( ( X
% 0.22/0.58 = ( bit1 @ M4 ) )
% 0.22/0.58 => ! [N2: num] :
% 0.22/0.58 ( ( Xa
% 0.22/0.58 = ( bit1 @ N2 ) )
% 0.22/0.58 => ( ( Y4
% 0.22/0.58 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) )
% 0.22/0.58 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % xor_num.pelims
% 0.22/0.58 thf(fact_9931_and__num__rel__dict,axiom,
% 0.22/0.58 bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).
% 0.22/0.58
% 0.22/0.58 % and_num_rel_dict
% 0.22/0.58 thf(fact_9932_xor__num__rel__dict,axiom,
% 0.22/0.58 bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).
% 0.22/0.58
% 0.22/0.58 % xor_num_rel_dict
% 0.22/0.58 thf(fact_9933_xor__num__dict,axiom,
% 0.22/0.58 bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).
% 0.22/0.58
% 0.22/0.58 % xor_num_dict
% 0.22/0.58 thf(fact_9934_and__num__dict,axiom,
% 0.22/0.58 bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).
% 0.22/0.58
% 0.22/0.58 % and_num_dict
% 0.22/0.58 thf(fact_9935_Bit__Operations_Otake__bit__num__code,axiom,
% 0.22/0.58 ( bit_take_bit_num
% 0.22/0.58 = ( ^ [N4: nat,M5: num] :
% 0.22/0.58 ( produc478579273971653890on_num
% 0.22/0.58 @ ^ [A3: nat,X3: num] :
% 0.22/0.58 ( case_nat_option_num @ none_num
% 0.22/0.58 @ ^ [O: nat] :
% 0.22/0.58 ( case_num_option_num @ ( some_num @ one )
% 0.22/0.58 @ ^ [P6: num] :
% 0.22/0.58 ( case_o6005452278849405969um_num @ none_num
% 0.22/0.58 @ ^ [Q3: num] : ( some_num @ ( bit0 @ Q3 ) )
% 0.22/0.58 @ ( bit_take_bit_num @ O @ P6 ) )
% 0.22/0.58 @ ^ [P6: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) )
% 0.22/0.58 @ X3 )
% 0.22/0.58 @ A3 )
% 0.22/0.58 @ ( product_Pair_nat_num @ N4 @ M5 ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Bit_Operations.take_bit_num_code
% 0.22/0.58 thf(fact_9936_of__rat__dense,axiom,
% 0.22/0.58 ! [X: real,Y4: real] :
% 0.22/0.58 ( ( ord_less_real @ X @ Y4 )
% 0.22/0.58 => ? [Q4: rat] :
% 0.22/0.58 ( ( ord_less_real @ X @ ( field_7254667332652039916t_real @ Q4 ) )
% 0.22/0.58 & ( ord_less_real @ ( field_7254667332652039916t_real @ Q4 ) @ Y4 ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % of_rat_dense
% 0.22/0.58 thf(fact_9937_plus__rat__def,axiom,
% 0.22/0.58 ( plus_plus_rat
% 0.22/0.58 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 0.22/0.58 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % plus_rat_def
% 0.22/0.58 thf(fact_9938_inverse__rat__def,axiom,
% 0.22/0.58 ( inverse_inverse_rat
% 0.22/0.58 = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
% 0.22/0.58 @ ^ [X3: product_prod_int_int] :
% 0.22/0.58 ( if_Pro3027730157355071871nt_int
% 0.22/0.58 @ ( ( product_fst_int_int @ X3 )
% 0.22/0.58 = zero_zero_int )
% 0.22/0.58 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.22/0.58 @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % inverse_rat_def
% 0.22/0.58 thf(fact_9939_Fract_Oabs__eq,axiom,
% 0.22/0.58 ( fract
% 0.22/0.58 = ( ^ [Xa4: int,X3: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X3 ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % Fract.abs_eq
% 0.22/0.58 thf(fact_9940_zero__rat__def,axiom,
% 0.22/0.58 ( zero_zero_rat
% 0.22/0.58 = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % zero_rat_def
% 0.22/0.58 thf(fact_9941_times__rat__def,axiom,
% 0.22/0.58 ( times_times_rat
% 0.22/0.58 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 0.22/0.58 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % times_rat_def
% 0.22/0.58 thf(fact_9942_plus__rat_Oabs__eq,axiom,
% 0.22/0.58 ! [Xa: product_prod_int_int,X: product_prod_int_int] :
% 0.22/0.58 ( ( ratrel @ Xa @ Xa )
% 0.22/0.58 => ( ( ratrel @ X @ X )
% 0.22/0.58 => ( ( plus_plus_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 0.22/0.58 = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% 0.22/0.58
% 0.22/0.58 % plus_rat.abs_eq
% 0.22/0.58 thf(fact_9943_inverse__rat_Oabs__eq,axiom,
% 0.22/0.58 ! [X: product_prod_int_int] :
% 0.22/0.58 ( ( ratrel @ X @ X )
% 0.22/0.58 => ( ( inverse_inverse_rat @ ( abs_Rat @ X ) )
% 0.22/0.58 = ( abs_Rat
% 0.22/0.59 @ ( if_Pro3027730157355071871nt_int
% 0.22/0.59 @ ( ( product_fst_int_int @ X )
% 0.22/0.59 = zero_zero_int )
% 0.22/0.59 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.22/0.59 @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % inverse_rat.abs_eq
% 0.22/0.59 thf(fact_9944_ratrel__iff,axiom,
% 0.22/0.59 ( ratrel
% 0.22/0.59 = ( ^ [X3: product_prod_int_int,Y6: product_prod_int_int] :
% 0.22/0.59 ( ( ( product_snd_int_int @ X3 )
% 0.22/0.59 != zero_zero_int )
% 0.22/0.59 & ( ( product_snd_int_int @ Y6 )
% 0.22/0.59 != zero_zero_int )
% 0.22/0.59 & ( ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) )
% 0.22/0.59 = ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % ratrel_iff
% 0.22/0.59 thf(fact_9945_zero__rat_Orsp,axiom,
% 0.22/0.59 ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).
% 0.22/0.59
% 0.22/0.59 % zero_rat.rsp
% 0.22/0.59 thf(fact_9946_ratrel__def,axiom,
% 0.22/0.59 ( ratrel
% 0.22/0.59 = ( ^ [X3: product_prod_int_int,Y6: product_prod_int_int] :
% 0.22/0.59 ( ( ( product_snd_int_int @ X3 )
% 0.22/0.59 != zero_zero_int )
% 0.22/0.59 & ( ( product_snd_int_int @ Y6 )
% 0.22/0.59 != zero_zero_int )
% 0.22/0.59 & ( ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) )
% 0.22/0.59 = ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % ratrel_def
% 0.22/0.59 thf(fact_9947_times__rat_Oabs__eq,axiom,
% 0.22/0.59 ! [Xa: product_prod_int_int,X: product_prod_int_int] :
% 0.22/0.59 ( ( ratrel @ Xa @ Xa )
% 0.22/0.59 => ( ( ratrel @ X @ X )
% 0.22/0.59 => ( ( times_times_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 0.22/0.59 = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_fst_int_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % times_rat.abs_eq
% 0.22/0.59 thf(fact_9948_Rat_Opositive_Oabs__eq,axiom,
% 0.22/0.59 ! [X: product_prod_int_int] :
% 0.22/0.59 ( ( ratrel @ X @ X )
% 0.22/0.59 => ( ( positive @ ( abs_Rat @ X ) )
% 0.22/0.59 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Rat.positive.abs_eq
% 0.22/0.59 thf(fact_9949_num__of__integer__code,axiom,
% 0.22/0.59 ( code_num_of_integer
% 0.22/0.59 = ( ^ [K3: code_integer] :
% 0.22/0.59 ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
% 0.22/0.59 @ ( produc7336495610019696514er_num
% 0.22/0.59 @ ^ [L3: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ one ) )
% 0.22/0.59 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % num_of_integer_code
% 0.22/0.59 thf(fact_9950_take__upt,axiom,
% 0.22/0.59 ! [I: nat,M: nat,N: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
% 0.22/0.59 => ( ( take_nat @ M @ ( upt @ I @ N ) )
% 0.22/0.59 = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % take_upt
% 0.22/0.59 thf(fact_9951_min__Suc__Suc,axiom,
% 0.22/0.59 ! [M: nat,N: nat] :
% 0.22/0.59 ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.22/0.59 = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % min_Suc_Suc
% 0.22/0.59 thf(fact_9952_min__0R,axiom,
% 0.22/0.59 ! [N: nat] :
% 0.22/0.59 ( ( ord_min_nat @ N @ zero_zero_nat )
% 0.22/0.59 = zero_zero_nat ) ).
% 0.22/0.59
% 0.22/0.59 % min_0R
% 0.22/0.59 thf(fact_9953_min__0L,axiom,
% 0.22/0.59 ! [N: nat] :
% 0.22/0.59 ( ( ord_min_nat @ zero_zero_nat @ N )
% 0.22/0.59 = zero_zero_nat ) ).
% 0.22/0.59
% 0.22/0.59 % min_0L
% 0.22/0.59 thf(fact_9954_min__Suc__numeral,axiom,
% 0.22/0.59 ! [N: nat,K: num] :
% 0.22/0.59 ( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.22/0.59 = ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % min_Suc_numeral
% 0.22/0.59 thf(fact_9955_min__numeral__Suc,axiom,
% 0.22/0.59 ! [K: num,N: nat] :
% 0.22/0.59 ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.22/0.59 = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % min_numeral_Suc
% 0.22/0.59 thf(fact_9956_inf__nat__def,axiom,
% 0.22/0.59 inf_inf_nat = ord_min_nat ).
% 0.22/0.59
% 0.22/0.59 % inf_nat_def
% 0.22/0.59 thf(fact_9957_concat__bit__assoc__sym,axiom,
% 0.22/0.59 ! [M: nat,N: nat,K: int,L: int,R2: int] :
% 0.22/0.59 ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L ) @ R2 )
% 0.22/0.59 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N ) @ L @ R2 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % concat_bit_assoc_sym
% 0.22/0.59 thf(fact_9958_min__diff,axiom,
% 0.22/0.59 ! [M: nat,I: nat,N: nat] :
% 0.22/0.59 ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
% 0.22/0.59 = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% 0.22/0.59
% 0.22/0.59 % min_diff
% 0.22/0.59 thf(fact_9959_nat__mult__min__right,axiom,
% 0.22/0.59 ! [M: nat,N: nat,Q2: nat] :
% 0.22/0.59 ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q2 ) )
% 0.22/0.59 = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % nat_mult_min_right
% 0.22/0.59 thf(fact_9960_nat__mult__min__left,axiom,
% 0.22/0.59 ! [M: nat,N: nat,Q2: nat] :
% 0.22/0.59 ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q2 )
% 0.22/0.59 = ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % nat_mult_min_left
% 0.22/0.59 thf(fact_9961_take__bit__concat__bit__eq,axiom,
% 0.22/0.59 ! [M: nat,N: nat,K: int,L: int] :
% 0.22/0.59 ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N @ K @ L ) )
% 0.22/0.59 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ L ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % take_bit_concat_bit_eq
% 0.22/0.59 thf(fact_9962_min__Suc2,axiom,
% 0.22/0.59 ! [M: nat,N: nat] :
% 0.22/0.59 ( ( ord_min_nat @ M @ ( suc @ N ) )
% 0.22/0.59 = ( case_nat_nat @ zero_zero_nat
% 0.22/0.59 @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ M6 @ N ) )
% 0.22/0.59 @ M ) ) ).
% 0.22/0.59
% 0.22/0.59 % min_Suc2
% 0.22/0.59 thf(fact_9963_min__Suc1,axiom,
% 0.22/0.59 ! [N: nat,M: nat] :
% 0.22/0.59 ( ( ord_min_nat @ ( suc @ N ) @ M )
% 0.22/0.59 = ( case_nat_nat @ zero_zero_nat
% 0.22/0.59 @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ N @ M6 ) )
% 0.22/0.59 @ M ) ) ).
% 0.22/0.59
% 0.22/0.59 % min_Suc1
% 0.22/0.59 thf(fact_9964_min__enat__simps_I2_J,axiom,
% 0.22/0.59 ! [Q2: extended_enat] :
% 0.22/0.59 ( ( ord_mi8085742599997312461d_enat @ Q2 @ zero_z5237406670263579293d_enat )
% 0.22/0.59 = zero_z5237406670263579293d_enat ) ).
% 0.22/0.59
% 0.22/0.59 % min_enat_simps(2)
% 0.22/0.59 thf(fact_9965_min__enat__simps_I3_J,axiom,
% 0.22/0.59 ! [Q2: extended_enat] :
% 0.22/0.59 ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q2 )
% 0.22/0.59 = zero_z5237406670263579293d_enat ) ).
% 0.22/0.59
% 0.22/0.59 % min_enat_simps(3)
% 0.22/0.59 thf(fact_9966_inf__int__def,axiom,
% 0.22/0.59 inf_inf_int = ord_min_int ).
% 0.22/0.59
% 0.22/0.59 % inf_int_def
% 0.22/0.59 thf(fact_9967_Max__divisors__self__nat,axiom,
% 0.22/0.59 ! [N: nat] :
% 0.22/0.59 ( ( N != zero_zero_nat )
% 0.22/0.59 => ( ( lattic8265883725875713057ax_nat
% 0.22/0.59 @ ( collect_nat
% 0.22/0.59 @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ N ) ) )
% 0.22/0.59 = N ) ) ).
% 0.22/0.59
% 0.22/0.59 % Max_divisors_self_nat
% 0.22/0.59 thf(fact_9968_inf__enat__def,axiom,
% 0.22/0.59 inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).
% 0.22/0.59
% 0.22/0.59 % inf_enat_def
% 0.22/0.59 thf(fact_9969_card__le__Suc__Max,axiom,
% 0.22/0.59 ! [S: set_nat] :
% 0.22/0.59 ( ( finite_finite_nat @ S )
% 0.22/0.59 => ( ord_less_eq_nat @ ( finite_card_nat @ S ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % card_le_Suc_Max
% 0.22/0.59 thf(fact_9970_Sup__nat__def,axiom,
% 0.22/0.59 ( complete_Sup_Sup_nat
% 0.22/0.59 = ( ^ [X8: set_nat] : ( if_nat @ ( X8 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X8 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Sup_nat_def
% 0.22/0.59 thf(fact_9971_divide__nat__def,axiom,
% 0.22/0.59 ( divide_divide_nat
% 0.22/0.59 = ( ^ [M5: nat,N4: nat] :
% 0.22/0.59 ( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat
% 0.22/0.59 @ ( lattic8265883725875713057ax_nat
% 0.22/0.59 @ ( collect_nat
% 0.22/0.59 @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N4 ) @ M5 ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % divide_nat_def
% 0.22/0.59 thf(fact_9972_gcd__is__Max__divisors__nat,axiom,
% 0.22/0.59 ! [N: nat,M: nat] :
% 0.22/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.59 => ( ( gcd_gcd_nat @ M @ N )
% 0.22/0.59 = ( lattic8265883725875713057ax_nat
% 0.22/0.59 @ ( collect_nat
% 0.22/0.59 @ ^ [D4: nat] :
% 0.22/0.59 ( ( dvd_dvd_nat @ D4 @ M )
% 0.22/0.59 & ( dvd_dvd_nat @ D4 @ N ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % gcd_is_Max_divisors_nat
% 0.22/0.59 thf(fact_9973_Gcd__eq__Max,axiom,
% 0.22/0.59 ! [M10: set_nat] :
% 0.22/0.59 ( ( finite_finite_nat @ M10 )
% 0.22/0.59 => ( ( M10 != bot_bot_set_nat )
% 0.22/0.59 => ( ~ ( member_nat @ zero_zero_nat @ M10 )
% 0.22/0.59 => ( ( gcd_Gcd_nat @ M10 )
% 0.22/0.59 = ( lattic8265883725875713057ax_nat
% 0.22/0.59 @ ( comple7806235888213564991et_nat
% 0.22/0.59 @ ( image_nat_set_nat
% 0.22/0.59 @ ^ [M5: nat] :
% 0.22/0.59 ( collect_nat
% 0.22/0.59 @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M5 ) )
% 0.22/0.59 @ M10 ) ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Gcd_eq_Max
% 0.22/0.59 thf(fact_9974_Max__divisors__self__int,axiom,
% 0.22/0.59 ! [N: int] :
% 0.22/0.59 ( ( N != zero_zero_int )
% 0.22/0.59 => ( ( lattic8263393255366662781ax_int
% 0.22/0.59 @ ( collect_int
% 0.22/0.59 @ ^ [D4: int] : ( dvd_dvd_int @ D4 @ N ) ) )
% 0.22/0.59 = ( abs_abs_int @ N ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Max_divisors_self_int
% 0.22/0.59 thf(fact_9975_upt__conv__Cons__Cons,axiom,
% 0.22/0.59 ! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
% 0.22/0.59 ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
% 0.22/0.59 = ( upt @ M @ Q2 ) )
% 0.22/0.59 = ( ( cons_nat @ N @ Ns )
% 0.22/0.59 = ( upt @ ( suc @ M ) @ Q2 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upt_conv_Cons_Cons
% 0.22/0.59 thf(fact_9976_upt__conv__Cons,axiom,
% 0.22/0.59 ! [I: nat,J: nat] :
% 0.22/0.59 ( ( ord_less_nat @ I @ J )
% 0.22/0.59 => ( ( upt @ I @ J )
% 0.22/0.59 = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upt_conv_Cons
% 0.22/0.59 thf(fact_9977_gcd__is__Max__divisors__int,axiom,
% 0.22/0.59 ! [N: int,M: int] :
% 0.22/0.59 ( ( N != zero_zero_int )
% 0.22/0.59 => ( ( gcd_gcd_int @ M @ N )
% 0.22/0.59 = ( lattic8263393255366662781ax_int
% 0.22/0.59 @ ( collect_int
% 0.22/0.59 @ ^ [D4: int] :
% 0.22/0.59 ( ( dvd_dvd_int @ D4 @ M )
% 0.22/0.59 & ( dvd_dvd_int @ D4 @ N ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % gcd_is_Max_divisors_int
% 0.22/0.59 thf(fact_9978_sorted__list__of__set__greaterThanAtMost,axiom,
% 0.22/0.59 ! [I: nat,J: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
% 0.22/0.59 => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
% 0.22/0.59 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % sorted_list_of_set_greaterThanAtMost
% 0.22/0.59 thf(fact_9979_sorted__list__of__set__greaterThanLessThan,axiom,
% 0.22/0.59 ! [I: nat,J: nat] :
% 0.22/0.59 ( ( ord_less_nat @ ( suc @ I ) @ J )
% 0.22/0.59 => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
% 0.22/0.59 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % sorted_list_of_set_greaterThanLessThan
% 0.22/0.59 thf(fact_9980_upto__aux__rec,axiom,
% 0.22/0.59 ( upto_aux
% 0.22/0.59 = ( ^ [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 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upto_aux_rec
% 0.22/0.59 thf(fact_9981_list__encode_Osimps_I2_J,axiom,
% 0.22/0.59 ! [X: nat,Xs: list_nat] :
% 0.22/0.59 ( ( nat_list_encode @ ( cons_nat @ X @ Xs ) )
% 0.22/0.59 = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % list_encode.simps(2)
% 0.22/0.59 thf(fact_9982_upt__conv__Nil,axiom,
% 0.22/0.59 ! [J: nat,I: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ J @ I )
% 0.22/0.59 => ( ( upt @ I @ J )
% 0.22/0.59 = nil_nat ) ) ).
% 0.22/0.59
% 0.22/0.59 % upt_conv_Nil
% 0.22/0.59 thf(fact_9983_upt__eq__Nil__conv,axiom,
% 0.22/0.59 ! [I: nat,J: nat] :
% 0.22/0.59 ( ( ( upt @ I @ J )
% 0.22/0.59 = nil_nat )
% 0.22/0.59 = ( ( J = zero_zero_nat )
% 0.22/0.59 | ( ord_less_eq_nat @ J @ I ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upt_eq_Nil_conv
% 0.22/0.59 thf(fact_9984_sorted__list__of__set__lessThan__Suc,axiom,
% 0.22/0.59 ! [K: nat] :
% 0.22/0.59 ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
% 0.22/0.59 = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % sorted_list_of_set_lessThan_Suc
% 0.22/0.59 thf(fact_9985_sorted__list__of__set__atMost__Suc,axiom,
% 0.22/0.59 ! [K: nat] :
% 0.22/0.59 ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
% 0.22/0.59 = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % sorted_list_of_set_atMost_Suc
% 0.22/0.59 thf(fact_9986_upt__rec__numeral,axiom,
% 0.22/0.59 ! [M: num,N: num] :
% 0.22/0.59 ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.22/0.59 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.22/0.59 = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
% 0.22/0.59 & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.22/0.59 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.22/0.59 = nil_nat ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upt_rec_numeral
% 0.22/0.59 thf(fact_9987_upt__Suc,axiom,
% 0.22/0.59 ! [I: nat,J: nat] :
% 0.22/0.59 ( ( ( ord_less_eq_nat @ I @ J )
% 0.22/0.59 => ( ( upt @ I @ ( suc @ J ) )
% 0.22/0.59 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
% 0.22/0.59 & ( ~ ( ord_less_eq_nat @ I @ J )
% 0.22/0.59 => ( ( upt @ I @ ( suc @ J ) )
% 0.22/0.59 = nil_nat ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upt_Suc
% 0.22/0.59 thf(fact_9988_upt__Suc__append,axiom,
% 0.22/0.59 ! [I: nat,J: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ I @ J )
% 0.22/0.59 => ( ( upt @ I @ ( suc @ J ) )
% 0.22/0.59 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upt_Suc_append
% 0.22/0.59 thf(fact_9989_list__encode_Ocases,axiom,
% 0.22/0.59 ! [X: list_nat] :
% 0.22/0.59 ( ( X != nil_nat )
% 0.22/0.59 => ~ ! [X4: nat,Xs2: list_nat] :
% 0.22/0.59 ( X
% 0.22/0.59 != ( cons_nat @ X4 @ Xs2 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % list_encode.cases
% 0.22/0.59 thf(fact_9990_list__encode_Osimps_I1_J,axiom,
% 0.22/0.59 ( ( nat_list_encode @ nil_nat )
% 0.22/0.59 = zero_zero_nat ) ).
% 0.22/0.59
% 0.22/0.59 % list_encode.simps(1)
% 0.22/0.59 thf(fact_9991_upt__0,axiom,
% 0.22/0.59 ! [I: nat] :
% 0.22/0.59 ( ( upt @ I @ zero_zero_nat )
% 0.22/0.59 = nil_nat ) ).
% 0.22/0.59
% 0.22/0.59 % upt_0
% 0.22/0.59 thf(fact_9992_inj__list__encode,axiom,
% 0.22/0.59 ! [A2: set_list_nat] : ( inj_on_list_nat_nat @ nat_list_encode @ A2 ) ).
% 0.22/0.59
% 0.22/0.59 % inj_list_encode
% 0.22/0.59 thf(fact_9993_list__encode__eq,axiom,
% 0.22/0.59 ! [X: list_nat,Y4: list_nat] :
% 0.22/0.59 ( ( ( nat_list_encode @ X )
% 0.22/0.59 = ( nat_list_encode @ Y4 ) )
% 0.22/0.59 = ( X = Y4 ) ) ).
% 0.22/0.59
% 0.22/0.59 % list_encode_eq
% 0.22/0.59 thf(fact_9994_bij__list__encode,axiom,
% 0.22/0.59 bij_be8532844293280997160at_nat @ nat_list_encode @ top_top_set_list_nat @ top_top_set_nat ).
% 0.22/0.59
% 0.22/0.59 % bij_list_encode
% 0.22/0.59 thf(fact_9995_surj__list__encode,axiom,
% 0.22/0.59 ( ( image_list_nat_nat @ nat_list_encode @ top_top_set_list_nat )
% 0.22/0.59 = top_top_set_nat ) ).
% 0.22/0.59
% 0.22/0.59 % surj_list_encode
% 0.22/0.59 thf(fact_9996_upt__add__eq__append,axiom,
% 0.22/0.59 ! [I: nat,J: nat,K: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ I @ J )
% 0.22/0.59 => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
% 0.22/0.59 = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upt_add_eq_append
% 0.22/0.59 thf(fact_9997_list__encode_Oelims,axiom,
% 0.22/0.59 ! [X: list_nat,Y4: nat] :
% 0.22/0.59 ( ( ( nat_list_encode @ X )
% 0.22/0.59 = Y4 )
% 0.22/0.59 => ( ( ( X = nil_nat )
% 0.22/0.59 => ( Y4 != zero_zero_nat ) )
% 0.22/0.59 => ~ ! [X4: nat,Xs2: list_nat] :
% 0.22/0.59 ( ( X
% 0.22/0.59 = ( cons_nat @ X4 @ Xs2 ) )
% 0.22/0.59 => ( Y4
% 0.22/0.59 != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % list_encode.elims
% 0.22/0.59 thf(fact_9998_upt__rec,axiom,
% 0.22/0.59 ( upt
% 0.22/0.59 = ( ^ [I2: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J3 ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J3 ) ) @ nil_nat ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upt_rec
% 0.22/0.59 thf(fact_9999_list__encode_Opelims,axiom,
% 0.22/0.59 ! [X: list_nat,Y4: nat] :
% 0.22/0.59 ( ( ( nat_list_encode @ X )
% 0.22/0.59 = Y4 )
% 0.22/0.59 => ( ( accp_list_nat @ nat_list_encode_rel @ X )
% 0.22/0.59 => ( ( ( X = nil_nat )
% 0.22/0.59 => ( ( Y4 = zero_zero_nat )
% 0.22/0.59 => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
% 0.22/0.59 => ~ ! [X4: nat,Xs2: list_nat] :
% 0.22/0.59 ( ( X
% 0.22/0.59 = ( cons_nat @ X4 @ Xs2 ) )
% 0.22/0.59 => ( ( Y4
% 0.22/0.59 = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) )
% 0.22/0.59 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X4 @ Xs2 ) ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % list_encode.pelims
% 0.22/0.59 thf(fact_10000_upto_Opelims,axiom,
% 0.22/0.59 ! [X: int,Xa: int,Y4: list_int] :
% 0.22/0.59 ( ( ( upto @ X @ Xa )
% 0.22/0.59 = Y4 )
% 0.22/0.59 => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) )
% 0.22/0.59 => ~ ( ( ( ( ord_less_eq_int @ X @ Xa )
% 0.22/0.59 => ( Y4
% 0.22/0.59 = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
% 0.22/0.59 & ( ~ ( ord_less_eq_int @ X @ Xa )
% 0.22/0.59 => ( Y4 = nil_int ) ) )
% 0.22/0.59 => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upto.pelims
% 0.22/0.59 thf(fact_10001_nth__upto,axiom,
% 0.22/0.59 ! [I: int,K: nat,J: int] :
% 0.22/0.59 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 0.22/0.59 => ( ( nth_int @ ( upto @ I @ J ) @ K )
% 0.22/0.59 = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % nth_upto
% 0.22/0.59 thf(fact_10002_length__upto,axiom,
% 0.22/0.59 ! [I: int,J: int] :
% 0.22/0.59 ( ( size_size_list_int @ ( upto @ I @ J ) )
% 0.22/0.59 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % length_upto
% 0.22/0.59 thf(fact_10003_upto__rec__numeral_I1_J,axiom,
% 0.22/0.59 ! [M: num,N: num] :
% 0.22/0.59 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.22/0.59 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.22/0.59 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 0.22/0.59 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.22/0.59 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.22/0.59 = nil_int ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upto_rec_numeral(1)
% 0.22/0.59 thf(fact_10004_upto__rec__numeral_I4_J,axiom,
% 0.22/0.59 ! [M: num,N: num] :
% 0.22/0.59 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.59 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.59 = ( 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 ) ) ) ) ) )
% 0.22/0.59 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.59 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.59 = nil_int ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upto_rec_numeral(4)
% 0.22/0.59 thf(fact_10005_upto__rec__numeral_I3_J,axiom,
% 0.22/0.59 ! [M: num,N: num] :
% 0.22/0.59 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.22/0.59 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.22/0.59 = ( 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 ) ) ) ) )
% 0.22/0.59 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.22/0.59 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.22/0.59 = nil_int ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upto_rec_numeral(3)
% 0.22/0.59 thf(fact_10006_upto__rec__numeral_I2_J,axiom,
% 0.22/0.59 ! [M: num,N: num] :
% 0.22/0.59 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.59 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.59 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 0.22/0.59 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.59 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.22/0.59 = nil_int ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upto_rec_numeral(2)
% 0.22/0.59 thf(fact_10007_sorted__upto,axiom,
% 0.22/0.59 ! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).
% 0.22/0.59
% 0.22/0.59 % sorted_upto
% 0.22/0.59 thf(fact_10008_upto__split2,axiom,
% 0.22/0.59 ! [I: int,J: int,K: int] :
% 0.22/0.59 ( ( ord_less_eq_int @ I @ J )
% 0.22/0.59 => ( ( ord_less_eq_int @ J @ K )
% 0.22/0.59 => ( ( upto @ I @ K )
% 0.22/0.59 = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upto_split2
% 0.22/0.59 thf(fact_10009_upto__split1,axiom,
% 0.22/0.59 ! [I: int,J: int,K: int] :
% 0.22/0.59 ( ( ord_less_eq_int @ I @ J )
% 0.22/0.59 => ( ( ord_less_eq_int @ J @ K )
% 0.22/0.59 => ( ( upto @ I @ K )
% 0.22/0.59 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upto_split1
% 0.22/0.59 thf(fact_10010_atLeastLessThan__upto,axiom,
% 0.22/0.59 ( set_or4662586982721622107an_int
% 0.22/0.59 = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % atLeastLessThan_upto
% 0.22/0.59 thf(fact_10011_upto_Osimps,axiom,
% 0.22/0.59 ( upto
% 0.22/0.59 = ( ^ [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 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upto.simps
% 0.22/0.59 thf(fact_10012_upto_Oelims,axiom,
% 0.22/0.59 ! [X: int,Xa: int,Y4: list_int] :
% 0.22/0.59 ( ( ( upto @ X @ Xa )
% 0.22/0.59 = Y4 )
% 0.22/0.59 => ( ( ( ord_less_eq_int @ X @ Xa )
% 0.22/0.59 => ( Y4
% 0.22/0.59 = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
% 0.22/0.59 & ( ~ ( ord_less_eq_int @ X @ Xa )
% 0.22/0.59 => ( Y4 = nil_int ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upto.elims
% 0.22/0.59 thf(fact_10013_upto__rec1,axiom,
% 0.22/0.59 ! [I: int,J: int] :
% 0.22/0.59 ( ( ord_less_eq_int @ I @ J )
% 0.22/0.59 => ( ( upto @ I @ J )
% 0.22/0.59 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upto_rec1
% 0.22/0.59 thf(fact_10014_upto__rec2,axiom,
% 0.22/0.59 ! [I: int,J: int] :
% 0.22/0.59 ( ( ord_less_eq_int @ I @ J )
% 0.22/0.59 => ( ( upto @ I @ J )
% 0.22/0.59 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upto_rec2
% 0.22/0.59 thf(fact_10015_greaterThanLessThan__upto,axiom,
% 0.22/0.59 ( set_or5832277885323065728an_int
% 0.22/0.59 = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % greaterThanLessThan_upto
% 0.22/0.59 thf(fact_10016_upto__split3,axiom,
% 0.22/0.59 ! [I: int,J: int,K: int] :
% 0.22/0.59 ( ( ord_less_eq_int @ I @ J )
% 0.22/0.59 => ( ( ord_less_eq_int @ J @ K )
% 0.22/0.59 => ( ( upto @ I @ K )
% 0.22/0.59 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upto_split3
% 0.22/0.59 thf(fact_10017_upto_Opsimps,axiom,
% 0.22/0.59 ! [I: int,J: int] :
% 0.22/0.59 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
% 0.22/0.59 => ( ( ( ord_less_eq_int @ I @ J )
% 0.22/0.59 => ( ( upto @ I @ J )
% 0.22/0.59 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
% 0.22/0.59 & ( ~ ( ord_less_eq_int @ I @ J )
% 0.22/0.59 => ( ( upto @ I @ J )
% 0.22/0.59 = nil_int ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % upto.psimps
% 0.22/0.59 thf(fact_10018_tl__upt,axiom,
% 0.22/0.59 ! [M: nat,N: nat] :
% 0.22/0.59 ( ( tl_nat @ ( upt @ M @ N ) )
% 0.22/0.59 = ( upt @ ( suc @ M ) @ N ) ) ).
% 0.22/0.59
% 0.22/0.59 % tl_upt
% 0.22/0.59 thf(fact_10019_quotient__of__def,axiom,
% 0.22/0.59 ( quotient_of
% 0.22/0.59 = ( ^ [X3: rat] :
% 0.22/0.59 ( the_Pr4378521158711661632nt_int
% 0.22/0.59 @ ^ [Pair: product_prod_int_int] :
% 0.22/0.59 ( ( X3
% 0.22/0.59 = ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
% 0.22/0.59 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
% 0.22/0.59 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % quotient_of_def
% 0.22/0.59 thf(fact_10020_coprime__abs__left__iff,axiom,
% 0.22/0.59 ! [K: int,L: int] :
% 0.22/0.59 ( ( algebr932160517623751201me_int @ ( abs_abs_int @ K ) @ L )
% 0.22/0.59 = ( algebr932160517623751201me_int @ K @ L ) ) ).
% 0.22/0.59
% 0.22/0.59 % coprime_abs_left_iff
% 0.22/0.59 thf(fact_10021_coprime__abs__right__iff,axiom,
% 0.22/0.59 ! [K: int,L: int] :
% 0.22/0.59 ( ( algebr932160517623751201me_int @ K @ ( abs_abs_int @ L ) )
% 0.22/0.59 = ( algebr932160517623751201me_int @ K @ L ) ) ).
% 0.22/0.59
% 0.22/0.59 % coprime_abs_right_iff
% 0.22/0.59 thf(fact_10022_normalize__stable,axiom,
% 0.22/0.59 ! [Q2: int,P5: int] :
% 0.22/0.59 ( ( ord_less_int @ zero_zero_int @ Q2 )
% 0.22/0.59 => ( ( algebr932160517623751201me_int @ P5 @ Q2 )
% 0.22/0.59 => ( ( normalize @ ( product_Pair_int_int @ P5 @ Q2 ) )
% 0.22/0.59 = ( product_Pair_int_int @ P5 @ Q2 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % normalize_stable
% 0.22/0.59 thf(fact_10023_mono__Suc,axiom,
% 0.22/0.59 order_mono_nat_nat @ suc ).
% 0.22/0.59
% 0.22/0.59 % mono_Suc
% 0.22/0.59 thf(fact_10024_coprime__crossproduct__int,axiom,
% 0.22/0.59 ! [A: int,D: int,B: int,C: int] :
% 0.22/0.59 ( ( algebr932160517623751201me_int @ A @ D )
% 0.22/0.59 => ( ( algebr932160517623751201me_int @ B @ C )
% 0.22/0.59 => ( ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ C ) )
% 0.22/0.59 = ( times_times_int @ ( abs_abs_int @ B ) @ ( abs_abs_int @ D ) ) )
% 0.22/0.59 = ( ( ( abs_abs_int @ A )
% 0.22/0.59 = ( abs_abs_int @ B ) )
% 0.22/0.59 & ( ( abs_abs_int @ C )
% 0.22/0.59 = ( abs_abs_int @ D ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % coprime_crossproduct_int
% 0.22/0.59 thf(fact_10025_mono__times__nat,axiom,
% 0.22/0.59 ! [N: nat] :
% 0.22/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.59 => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % mono_times_nat
% 0.22/0.59 thf(fact_10026_coprime__common__divisor__int,axiom,
% 0.22/0.59 ! [A: int,B: int,X: int] :
% 0.22/0.59 ( ( algebr932160517623751201me_int @ A @ B )
% 0.22/0.59 => ( ( dvd_dvd_int @ X @ A )
% 0.22/0.59 => ( ( dvd_dvd_int @ X @ B )
% 0.22/0.59 => ( ( abs_abs_int @ X )
% 0.22/0.59 = one_one_int ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % coprime_common_divisor_int
% 0.22/0.59 thf(fact_10027_Rat__cases,axiom,
% 0.22/0.59 ! [Q2: rat] :
% 0.22/0.59 ~ ! [A4: int,B3: int] :
% 0.22/0.59 ( ( Q2
% 0.22/0.59 = ( fract @ A4 @ B3 ) )
% 0.22/0.59 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.22/0.59 => ~ ( algebr932160517623751201me_int @ A4 @ B3 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Rat_cases
% 0.22/0.59 thf(fact_10028_Rat__induct,axiom,
% 0.22/0.59 ! [P: rat > $o,Q2: rat] :
% 0.22/0.59 ( ! [A4: int,B3: int] :
% 0.22/0.59 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.22/0.59 => ( ( algebr932160517623751201me_int @ A4 @ B3 )
% 0.22/0.59 => ( P @ ( fract @ A4 @ B3 ) ) ) )
% 0.22/0.59 => ( P @ Q2 ) ) ).
% 0.22/0.59
% 0.22/0.59 % Rat_induct
% 0.22/0.59 thf(fact_10029_Rat__cases__nonzero,axiom,
% 0.22/0.59 ! [Q2: rat] :
% 0.22/0.59 ( ! [A4: int,B3: int] :
% 0.22/0.59 ( ( Q2
% 0.22/0.59 = ( fract @ A4 @ B3 ) )
% 0.22/0.59 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.22/0.59 => ( ( A4 != zero_zero_int )
% 0.22/0.59 => ~ ( algebr932160517623751201me_int @ A4 @ B3 ) ) ) )
% 0.22/0.59 => ( Q2 = zero_zero_rat ) ) ).
% 0.22/0.59
% 0.22/0.59 % Rat_cases_nonzero
% 0.22/0.59 thf(fact_10030_quotient__of__unique,axiom,
% 0.22/0.59 ! [R2: rat] :
% 0.22/0.59 ? [X4: product_prod_int_int] :
% 0.22/0.59 ( ( R2
% 0.22/0.59 = ( fract @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) )
% 0.22/0.59 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X4 ) )
% 0.22/0.59 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) )
% 0.22/0.59 & ! [Y3: product_prod_int_int] :
% 0.22/0.59 ( ( ( R2
% 0.22/0.59 = ( fract @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ Y3 ) ) )
% 0.22/0.59 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y3 ) )
% 0.22/0.59 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ Y3 ) ) )
% 0.22/0.59 => ( Y3 = X4 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % quotient_of_unique
% 0.22/0.59 thf(fact_10031_mono__ge2__power__minus__self,axiom,
% 0.22/0.59 ! [K: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.22/0.59 => ( order_mono_nat_nat
% 0.22/0.59 @ ^ [M5: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M5 ) @ M5 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % mono_ge2_power_minus_self
% 0.22/0.59 thf(fact_10032_tendsto__at__topI__sequentially__real,axiom,
% 0.22/0.59 ! [F: real > real,Y4: real] :
% 0.22/0.59 ( ( order_mono_real_real @ F )
% 0.22/0.59 => ( ( filterlim_nat_real
% 0.22/0.59 @ ^ [N4: nat] : ( F @ ( semiri5074537144036343181t_real @ N4 ) )
% 0.22/0.59 @ ( topolo2815343760600316023s_real @ Y4 )
% 0.22/0.59 @ at_top_nat )
% 0.22/0.59 => ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Y4 ) @ at_top_real ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % tendsto_at_topI_sequentially_real
% 0.22/0.59 thf(fact_10033_coprime__int__iff,axiom,
% 0.22/0.59 ! [M: nat,N: nat] :
% 0.22/0.59 ( ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.22/0.59 = ( algebr934650988132801477me_nat @ M @ N ) ) ).
% 0.22/0.59
% 0.22/0.59 % coprime_int_iff
% 0.22/0.59 thf(fact_10034_coprime__nat__abs__left__iff,axiom,
% 0.22/0.59 ! [K: int,N: nat] :
% 0.22/0.59 ( ( algebr934650988132801477me_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 0.22/0.59 = ( algebr932160517623751201me_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % coprime_nat_abs_left_iff
% 0.22/0.59 thf(fact_10035_coprime__nat__abs__right__iff,axiom,
% 0.22/0.59 ! [N: nat,K: int] :
% 0.22/0.59 ( ( algebr934650988132801477me_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 0.22/0.59 = ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).
% 0.22/0.59
% 0.22/0.59 % coprime_nat_abs_right_iff
% 0.22/0.59 thf(fact_10036_coprime__Suc__0__right,axiom,
% 0.22/0.59 ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).
% 0.22/0.59
% 0.22/0.59 % coprime_Suc_0_right
% 0.22/0.59 thf(fact_10037_coprime__Suc__0__left,axiom,
% 0.22/0.59 ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).
% 0.22/0.59
% 0.22/0.59 % coprime_Suc_0_left
% 0.22/0.59 thf(fact_10038_coprime__crossproduct__nat,axiom,
% 0.22/0.59 ! [A: nat,D: nat,B: nat,C: nat] :
% 0.22/0.59 ( ( algebr934650988132801477me_nat @ A @ D )
% 0.22/0.59 => ( ( algebr934650988132801477me_nat @ B @ C )
% 0.22/0.59 => ( ( ( times_times_nat @ A @ C )
% 0.22/0.59 = ( times_times_nat @ B @ D ) )
% 0.22/0.59 = ( ( A = B )
% 0.22/0.59 & ( C = D ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % coprime_crossproduct_nat
% 0.22/0.59 thf(fact_10039_coprime__Suc__left__nat,axiom,
% 0.22/0.59 ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ N ) @ N ) ).
% 0.22/0.59
% 0.22/0.59 % coprime_Suc_left_nat
% 0.22/0.59 thf(fact_10040_coprime__Suc__right__nat,axiom,
% 0.22/0.59 ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ N ) ) ).
% 0.22/0.59
% 0.22/0.59 % coprime_Suc_right_nat
% 0.22/0.59 thf(fact_10041_coprime__common__divisor__nat,axiom,
% 0.22/0.59 ! [A: nat,B: nat,X: nat] :
% 0.22/0.59 ( ( algebr934650988132801477me_nat @ A @ B )
% 0.22/0.59 => ( ( dvd_dvd_nat @ X @ A )
% 0.22/0.59 => ( ( dvd_dvd_nat @ X @ B )
% 0.22/0.59 => ( X = one_one_nat ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % coprime_common_divisor_nat
% 0.22/0.59 thf(fact_10042_incseq__bounded,axiom,
% 0.22/0.59 ! [X9: nat > real,B4: real] :
% 0.22/0.59 ( ( order_mono_nat_real @ X9 )
% 0.22/0.59 => ( ! [I3: nat] : ( ord_less_eq_real @ ( X9 @ I3 ) @ B4 )
% 0.22/0.59 => ( bfun_nat_real @ X9 @ at_top_nat ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % incseq_bounded
% 0.22/0.59 thf(fact_10043_coprime__diff__one__right__nat,axiom,
% 0.22/0.59 ! [N: nat] :
% 0.22/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.59 => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % coprime_diff_one_right_nat
% 0.22/0.59 thf(fact_10044_coprime__diff__one__left__nat,axiom,
% 0.22/0.59 ! [N: nat] :
% 0.22/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/0.59 => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
% 0.22/0.59
% 0.22/0.59 % coprime_diff_one_left_nat
% 0.22/0.59 thf(fact_10045_incseq__convergent,axiom,
% 0.22/0.59 ! [X9: nat > real,B4: real] :
% 0.22/0.59 ( ( order_mono_nat_real @ X9 )
% 0.22/0.59 => ( ! [I3: nat] : ( ord_less_eq_real @ ( X9 @ I3 ) @ B4 )
% 0.22/0.59 => ~ ! [L5: real] :
% 0.22/0.59 ( ( filterlim_nat_real @ X9 @ ( topolo2815343760600316023s_real @ L5 ) @ at_top_nat )
% 0.22/0.59 => ~ ! [I4: nat] : ( ord_less_eq_real @ ( X9 @ I4 ) @ L5 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % incseq_convergent
% 0.22/0.59 thf(fact_10046_Rats__abs__nat__div__natE,axiom,
% 0.22/0.59 ! [X: real] :
% 0.22/0.59 ( ( member_real @ X @ field_5140801741446780682s_real )
% 0.22/0.59 => ~ ! [M4: nat,N2: nat] :
% 0.22/0.59 ( ( N2 != zero_zero_nat )
% 0.22/0.59 => ( ( ( abs_abs_real @ X )
% 0.22/0.59 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M4 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.22/0.59 => ~ ( algebr934650988132801477me_nat @ M4 @ N2 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Rats_abs_nat_div_natE
% 0.22/0.59 thf(fact_10047_nonneg__incseq__Bseq__subseq__iff,axiom,
% 0.22/0.59 ! [F: nat > real,G: nat > nat] :
% 0.22/0.59 ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 0.22/0.59 => ( ( order_mono_nat_real @ F )
% 0.22/0.59 => ( ( order_5726023648592871131at_nat @ G )
% 0.22/0.59 => ( ( bfun_nat_real
% 0.22/0.59 @ ^ [X3: nat] : ( F @ ( G @ X3 ) )
% 0.22/0.59 @ at_top_nat )
% 0.22/0.59 = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % nonneg_incseq_Bseq_subseq_iff
% 0.22/0.59 thf(fact_10048_Rat_Opositive_Orsp,axiom,
% 0.22/0.59 ( bNF_re8699439704749558557nt_o_o @ ratrel
% 0.22/0.59 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
% 0.22/0.59 @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Rat.positive.rsp
% 0.22/0.59 thf(fact_10049_vanishes__mult__bounded,axiom,
% 0.22/0.59 ! [X9: nat > rat,Y7: nat > rat] :
% 0.22/0.59 ( ? [A7: rat] :
% 0.22/0.59 ( ( ord_less_rat @ zero_zero_rat @ A7 )
% 0.22/0.59 & ! [N2: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N2 ) ) @ A7 ) )
% 0.22/0.59 => ( ( vanishes @ Y7 )
% 0.22/0.59 => ( vanishes
% 0.22/0.59 @ ^ [N4: nat] : ( times_times_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % vanishes_mult_bounded
% 0.22/0.59 thf(fact_10050_vanishes__const,axiom,
% 0.22/0.59 ! [C: rat] :
% 0.22/0.59 ( ( vanishes
% 0.22/0.59 @ ^ [N4: nat] : C )
% 0.22/0.59 = ( C = zero_zero_rat ) ) ).
% 0.22/0.59
% 0.22/0.59 % vanishes_const
% 0.22/0.59 thf(fact_10051_vanishes__diff,axiom,
% 0.22/0.59 ! [X9: nat > rat,Y7: nat > rat] :
% 0.22/0.59 ( ( vanishes @ X9 )
% 0.22/0.59 => ( ( vanishes @ Y7 )
% 0.22/0.59 => ( vanishes
% 0.22/0.59 @ ^ [N4: nat] : ( minus_minus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % vanishes_diff
% 0.22/0.59 thf(fact_10052_vanishes__add,axiom,
% 0.22/0.59 ! [X9: nat > rat,Y7: nat > rat] :
% 0.22/0.59 ( ( vanishes @ X9 )
% 0.22/0.59 => ( ( vanishes @ Y7 )
% 0.22/0.59 => ( vanishes
% 0.22/0.59 @ ^ [N4: nat] : ( plus_plus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % vanishes_add
% 0.22/0.59 thf(fact_10053_vanishes__minus,axiom,
% 0.22/0.59 ! [X9: nat > rat] :
% 0.22/0.59 ( ( vanishes @ X9 )
% 0.22/0.59 => ( vanishes
% 0.22/0.59 @ ^ [N4: nat] : ( uminus_uminus_rat @ ( X9 @ N4 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % vanishes_minus
% 0.22/0.59 thf(fact_10054_vanishes__def,axiom,
% 0.22/0.59 ( vanishes
% 0.22/0.59 = ( ^ [X8: nat > rat] :
% 0.22/0.59 ! [R5: rat] :
% 0.22/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.22/0.59 => ? [K3: nat] :
% 0.22/0.59 ! [N4: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K3 @ N4 )
% 0.22/0.59 => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N4 ) ) @ R5 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % vanishes_def
% 0.22/0.59 thf(fact_10055_vanishesI,axiom,
% 0.22/0.59 ! [X9: nat > rat] :
% 0.22/0.59 ( ! [R3: rat] :
% 0.22/0.59 ( ( ord_less_rat @ zero_zero_rat @ R3 )
% 0.22/0.59 => ? [K4: nat] :
% 0.22/0.59 ! [N2: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K4 @ N2 )
% 0.22/0.59 => ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N2 ) ) @ R3 ) ) )
% 0.22/0.59 => ( vanishes @ X9 ) ) ).
% 0.22/0.59
% 0.22/0.59 % vanishesI
% 0.22/0.59 thf(fact_10056_vanishesD,axiom,
% 0.22/0.59 ! [X9: nat > rat,R2: rat] :
% 0.22/0.59 ( ( vanishes @ X9 )
% 0.22/0.59 => ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 0.22/0.59 => ? [K2: nat] :
% 0.22/0.59 ! [N6: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K2 @ N6 )
% 0.22/0.59 => ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N6 ) ) @ R2 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % vanishesD
% 0.22/0.59 thf(fact_10057_Fract_Orsp,axiom,
% 0.22/0.59 ( bNF_re157797125943740599nt_int
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.22/0.59 @ ( bNF_re6250860962936578807nt_int
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.22/0.59 @ ratrel )
% 0.22/0.59 @ ^ [A3: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B2 ) )
% 0.22/0.59 @ ^ [A3: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B2 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Fract.rsp
% 0.22/0.59 thf(fact_10058_integer__of__natural_Orsp,axiom,
% 0.22/0.59 ( bNF_re6650684261131312217nt_int
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.22/0.59 @ semiri1314217659103216013at_int
% 0.22/0.59 @ semiri1314217659103216013at_int ) ).
% 0.22/0.59
% 0.22/0.59 % integer_of_natural.rsp
% 0.22/0.59 thf(fact_10059_less__eq__integer_Orsp,axiom,
% 0.22/0.59 ( bNF_re3403563459893282935_int_o
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.22/0.59 @ ( bNF_re5089333283451836215nt_o_o
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.22/0.59 @ ord_less_eq_int
% 0.22/0.59 @ ord_less_eq_int ) ).
% 0.22/0.59
% 0.22/0.59 % less_eq_integer.rsp
% 0.22/0.59 thf(fact_10060_less__eq__natural_Orsp,axiom,
% 0.22/0.59 ( bNF_re578469030762574527_nat_o
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ( bNF_re4705727531993890431at_o_o
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.22/0.59 @ ord_less_eq_nat
% 0.22/0.59 @ ord_less_eq_nat ) ).
% 0.22/0.59
% 0.22/0.59 % less_eq_natural.rsp
% 0.22/0.59 thf(fact_10061_times__natural_Orsp,axiom,
% 0.22/0.59 ( bNF_re1345281282404953727at_nat
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ( bNF_re5653821019739307937at_nat
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.22/0.59 @ times_times_nat
% 0.22/0.59 @ times_times_nat ) ).
% 0.22/0.59
% 0.22/0.59 % times_natural.rsp
% 0.22/0.59 thf(fact_10062_times__integer_Orsp,axiom,
% 0.22/0.59 ( bNF_re711492959462206631nt_int
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.22/0.59 @ ( bNF_re4712519889275205905nt_int
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.22/0.59 @ times_times_int
% 0.22/0.59 @ times_times_int ) ).
% 0.22/0.59
% 0.22/0.59 % times_integer.rsp
% 0.22/0.59 thf(fact_10063_divide__integer_Orsp,axiom,
% 0.22/0.59 ( bNF_re711492959462206631nt_int
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.22/0.59 @ ( bNF_re4712519889275205905nt_int
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.22/0.59 @ divide_divide_int
% 0.22/0.59 @ divide_divide_int ) ).
% 0.22/0.59
% 0.22/0.59 % divide_integer.rsp
% 0.22/0.59 thf(fact_10064_divide__natural_Orsp,axiom,
% 0.22/0.59 ( bNF_re1345281282404953727at_nat
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ( bNF_re5653821019739307937at_nat
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.22/0.59 @ divide_divide_nat
% 0.22/0.59 @ divide_divide_nat ) ).
% 0.22/0.59
% 0.22/0.59 % divide_natural.rsp
% 0.22/0.59 thf(fact_10065_set__bit__natural_Orsp,axiom,
% 0.22/0.59 ( bNF_re1345281282404953727at_nat
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ( bNF_re5653821019739307937at_nat
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.22/0.59 @ bit_se7882103937844011126it_nat
% 0.22/0.59 @ bit_se7882103937844011126it_nat ) ).
% 0.22/0.59
% 0.22/0.59 % set_bit_natural.rsp
% 0.22/0.59 thf(fact_10066_set__bit__integer_Orsp,axiom,
% 0.22/0.59 ( bNF_re4785983289428654063nt_int
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ( bNF_re4712519889275205905nt_int
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.22/0.59 @ bit_se7879613467334960850it_int
% 0.22/0.59 @ bit_se7879613467334960850it_int ) ).
% 0.22/0.59
% 0.22/0.59 % set_bit_integer.rsp
% 0.22/0.59 thf(fact_10067_unset__bit__natural_Orsp,axiom,
% 0.22/0.59 ( bNF_re1345281282404953727at_nat
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ( bNF_re5653821019739307937at_nat
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.22/0.59 @ bit_se4205575877204974255it_nat
% 0.22/0.59 @ bit_se4205575877204974255it_nat ) ).
% 0.22/0.59
% 0.22/0.59 % unset_bit_natural.rsp
% 0.22/0.59 thf(fact_10068_unset__bit__integer_Orsp,axiom,
% 0.22/0.59 ( bNF_re4785983289428654063nt_int
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ( bNF_re4712519889275205905nt_int
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.22/0.59 @ bit_se4203085406695923979it_int
% 0.22/0.59 @ bit_se4203085406695923979it_int ) ).
% 0.22/0.59
% 0.22/0.59 % unset_bit_integer.rsp
% 0.22/0.59 thf(fact_10069_flip__bit__natural_Orsp,axiom,
% 0.22/0.59 ( bNF_re1345281282404953727at_nat
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ( bNF_re5653821019739307937at_nat
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.22/0.59 @ bit_se2161824704523386999it_nat
% 0.22/0.59 @ bit_se2161824704523386999it_nat ) ).
% 0.22/0.59
% 0.22/0.59 % flip_bit_natural.rsp
% 0.22/0.59 thf(fact_10070_flip__bit__integer_Orsp,axiom,
% 0.22/0.59 ( bNF_re4785983289428654063nt_int
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ( bNF_re4712519889275205905nt_int
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.22/0.59 @ bit_se2159334234014336723it_int
% 0.22/0.59 @ bit_se2159334234014336723it_int ) ).
% 0.22/0.59
% 0.22/0.59 % flip_bit_integer.rsp
% 0.22/0.59 thf(fact_10071_Suc_Orsp,axiom,
% 0.22/0.59 ( bNF_re5653821019739307937at_nat
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ suc
% 0.22/0.59 @ suc ) ).
% 0.22/0.59
% 0.22/0.59 % Suc.rsp
% 0.22/0.59 thf(fact_10072_minus__integer_Orsp,axiom,
% 0.22/0.59 ( bNF_re711492959462206631nt_int
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.22/0.59 @ ( bNF_re4712519889275205905nt_int
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.22/0.59 @ minus_minus_int
% 0.22/0.59 @ minus_minus_int ) ).
% 0.22/0.59
% 0.22/0.59 % minus_integer.rsp
% 0.22/0.59 thf(fact_10073_minus__natural_Orsp,axiom,
% 0.22/0.59 ( bNF_re1345281282404953727at_nat
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ( bNF_re5653821019739307937at_nat
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.22/0.59 @ minus_minus_nat
% 0.22/0.59 @ minus_minus_nat ) ).
% 0.22/0.59
% 0.22/0.59 % minus_natural.rsp
% 0.22/0.59 thf(fact_10074_sub_Orsp,axiom,
% 0.22/0.59 ( bNF_re8402795839162346335um_int
% 0.22/0.59 @ ^ [Y5: num,Z4: num] : ( Y5 = Z4 )
% 0.22/0.59 @ ( bNF_re1822329894187522285nt_int
% 0.22/0.59 @ ^ [Y5: num,Z4: num] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.22/0.59 @ ^ [M5: num,N4: num] : ( minus_minus_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N4 ) )
% 0.22/0.59 @ ^ [M5: num,N4: num] : ( minus_minus_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N4 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % sub.rsp
% 0.22/0.59 thf(fact_10075_times__rat_Orsp,axiom,
% 0.22/0.59 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 0.22/0.59 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) )
% 0.22/0.59 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % times_rat.rsp
% 0.22/0.59 thf(fact_10076_plus__rat_Orsp,axiom,
% 0.22/0.59 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 0.22/0.59 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) )
% 0.22/0.59 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % plus_rat.rsp
% 0.22/0.59 thf(fact_10077_inverse__rat_Orsp,axiom,
% 0.22/0.59 ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
% 0.22/0.59 @ ^ [X3: product_prod_int_int] :
% 0.22/0.59 ( if_Pro3027730157355071871nt_int
% 0.22/0.59 @ ( ( product_fst_int_int @ X3 )
% 0.22/0.59 = zero_zero_int )
% 0.22/0.59 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.22/0.59 @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
% 0.22/0.59 @ ^ [X3: product_prod_int_int] :
% 0.22/0.59 ( if_Pro3027730157355071871nt_int
% 0.22/0.59 @ ( ( product_fst_int_int @ X3 )
% 0.22/0.59 = zero_zero_int )
% 0.22/0.59 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.22/0.59 @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % inverse_rat.rsp
% 0.22/0.59 thf(fact_10078_plus__rat_Otransfer,axiom,
% 0.22/0.59 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 0.22/0.59 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) )
% 0.22/0.59 @ plus_plus_rat ) ).
% 0.22/0.59
% 0.22/0.59 % plus_rat.transfer
% 0.22/0.59 thf(fact_10079_inverse__rat_Otransfer,axiom,
% 0.22/0.59 ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
% 0.22/0.59 @ ^ [X3: product_prod_int_int] :
% 0.22/0.59 ( if_Pro3027730157355071871nt_int
% 0.22/0.59 @ ( ( product_fst_int_int @ X3 )
% 0.22/0.59 = zero_zero_int )
% 0.22/0.59 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.22/0.59 @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
% 0.22/0.59 @ inverse_inverse_rat ) ).
% 0.22/0.59
% 0.22/0.59 % inverse_rat.transfer
% 0.22/0.59 thf(fact_10080_zero__rat_Otransfer,axiom,
% 0.22/0.59 pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).
% 0.22/0.59
% 0.22/0.59 % zero_rat.transfer
% 0.22/0.59 thf(fact_10081_Fract_Otransfer,axiom,
% 0.22/0.59 ( bNF_re3461391660133120880nt_rat
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.22/0.59 @ ( bNF_re2214769303045360666nt_rat
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.22/0.59 @ pcr_rat )
% 0.22/0.59 @ ^ [A3: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B2 ) )
% 0.22/0.59 @ fract ) ).
% 0.22/0.59
% 0.22/0.59 % Fract.transfer
% 0.22/0.59 thf(fact_10082_times__rat_Otransfer,axiom,
% 0.22/0.59 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 0.22/0.59 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) )
% 0.22/0.59 @ times_times_rat ) ).
% 0.22/0.59
% 0.22/0.59 % times_rat.transfer
% 0.22/0.59 thf(fact_10083_Rat_Opositive_Otransfer,axiom,
% 0.22/0.59 ( bNF_re1494630372529172596at_o_o @ pcr_rat
% 0.22/0.59 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
% 0.22/0.59 @ positive ) ).
% 0.22/0.59
% 0.22/0.59 % Rat.positive.transfer
% 0.22/0.59 thf(fact_10084_times__int_Otransfer,axiom,
% 0.22/0.59 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 0.22/0.59 @ ( produc27273713700761075at_nat
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.59 ( produc2626176000494625587at_nat
% 0.22/0.59 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) )
% 0.22/0.59 @ times_times_int ) ).
% 0.22/0.59
% 0.22/0.59 % times_int.transfer
% 0.22/0.59 thf(fact_10085_zero__int_Otransfer,axiom,
% 0.22/0.59 pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).
% 0.22/0.59
% 0.22/0.59 % zero_int.transfer
% 0.22/0.59 thf(fact_10086_int__transfer,axiom,
% 0.22/0.59 ( bNF_re6830278522597306478at_int
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ pcr_int
% 0.22/0.59 @ ^ [N4: nat] : ( product_Pair_nat_nat @ N4 @ zero_zero_nat )
% 0.22/0.59 @ semiri1314217659103216013at_int ) ).
% 0.22/0.59
% 0.22/0.59 % int_transfer
% 0.22/0.59 thf(fact_10087_uminus__int_Otransfer,axiom,
% 0.22/0.59 ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
% 0.22/0.59 @ ( produc2626176000494625587at_nat
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 ) )
% 0.22/0.59 @ uminus_uminus_int ) ).
% 0.22/0.59
% 0.22/0.59 % uminus_int.transfer
% 0.22/0.59 thf(fact_10088_nat_Otransfer,axiom,
% 0.22/0.59 ( bNF_re4555766996558763186at_nat @ pcr_int
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 0.22/0.59 @ nat2 ) ).
% 0.22/0.59
% 0.22/0.59 % nat.transfer
% 0.22/0.59 thf(fact_10089_one__int_Otransfer,axiom,
% 0.22/0.59 pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% 0.22/0.59
% 0.22/0.59 % one_int.transfer
% 0.22/0.59 thf(fact_10090_less__int_Otransfer,axiom,
% 0.22/0.59 ( bNF_re717283939379294677_int_o @ pcr_int
% 0.22/0.59 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 0.22/0.59 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.22/0.59 @ ( produc8739625826339149834_nat_o
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.59 ( produc6081775807080527818_nat_o
% 0.22/0.59 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
% 0.22/0.59 @ ord_less_int ) ).
% 0.22/0.59
% 0.22/0.59 % less_int.transfer
% 0.22/0.59 thf(fact_10091_less__eq__int_Otransfer,axiom,
% 0.22/0.59 ( bNF_re717283939379294677_int_o @ pcr_int
% 0.22/0.59 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 0.22/0.59 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.22/0.59 @ ( produc8739625826339149834_nat_o
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.59 ( produc6081775807080527818_nat_o
% 0.22/0.59 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
% 0.22/0.59 @ ord_less_eq_int ) ).
% 0.22/0.59
% 0.22/0.59 % less_eq_int.transfer
% 0.22/0.59 thf(fact_10092_plus__int_Otransfer,axiom,
% 0.22/0.59 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 0.22/0.59 @ ( produc27273713700761075at_nat
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.59 ( produc2626176000494625587at_nat
% 0.22/0.59 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) )
% 0.22/0.59 @ plus_plus_int ) ).
% 0.22/0.59
% 0.22/0.59 % plus_int.transfer
% 0.22/0.59 thf(fact_10093_minus__int_Otransfer,axiom,
% 0.22/0.59 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 0.22/0.59 @ ( produc27273713700761075at_nat
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.59 ( produc2626176000494625587at_nat
% 0.22/0.59 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) )
% 0.22/0.59 @ minus_minus_int ) ).
% 0.22/0.59
% 0.22/0.59 % minus_int.transfer
% 0.22/0.59 thf(fact_10094_product__atMost__eq__Un,axiom,
% 0.22/0.59 ! [A2: set_nat,M: nat] :
% 0.22/0.59 ( ( produc457027306803732586at_nat @ A2
% 0.22/0.59 @ ^ [Uu3: nat] : ( set_ord_atMost_nat @ M ) )
% 0.22/0.59 = ( sup_su6327502436637775413at_nat
% 0.22/0.59 @ ( produc457027306803732586at_nat @ A2
% 0.22/0.59 @ ^ [I2: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ I2 ) ) )
% 0.22/0.59 @ ( produc457027306803732586at_nat @ A2
% 0.22/0.59 @ ^ [I2: nat] : ( set_or6659071591806873216st_nat @ ( minus_minus_nat @ M @ I2 ) @ M ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % product_atMost_eq_Un
% 0.22/0.59 thf(fact_10095_pairs__le__eq__Sigma,axiom,
% 0.22/0.59 ! [M: nat] :
% 0.22/0.59 ( ( collec3392354462482085612at_nat
% 0.22/0.59 @ ( produc6081775807080527818_nat_o
% 0.22/0.59 @ ^ [I2: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ J3 ) @ M ) ) )
% 0.22/0.59 = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
% 0.22/0.59 @ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R5 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % pairs_le_eq_Sigma
% 0.22/0.59 thf(fact_10096_inverse__real_Otransfer,axiom,
% 0.22/0.59 ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
% 0.22/0.59 @ ^ [X8: nat > rat] :
% 0.22/0.59 ( if_nat_rat @ ( vanishes @ X8 )
% 0.22/0.59 @ ^ [N4: nat] : zero_zero_rat
% 0.22/0.59 @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X8 @ N4 ) ) )
% 0.22/0.59 @ inverse_inverse_real ) ).
% 0.22/0.59
% 0.22/0.59 % inverse_real.transfer
% 0.22/0.59 thf(fact_10097_zero__real_Otransfer,axiom,
% 0.22/0.59 ( pcr_real
% 0.22/0.59 @ ^ [N4: nat] : zero_zero_rat
% 0.22/0.59 @ zero_zero_real ) ).
% 0.22/0.59
% 0.22/0.59 % zero_real.transfer
% 0.22/0.59 thf(fact_10098_one__real_Otransfer,axiom,
% 0.22/0.59 ( pcr_real
% 0.22/0.59 @ ^ [N4: nat] : one_one_rat
% 0.22/0.59 @ one_one_real ) ).
% 0.22/0.59
% 0.22/0.59 % one_real.transfer
% 0.22/0.59 thf(fact_10099_uminus__real_Otransfer,axiom,
% 0.22/0.59 ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
% 0.22/0.59 @ ^ [X8: nat > rat,N4: nat] : ( uminus_uminus_rat @ ( X8 @ N4 ) )
% 0.22/0.59 @ uminus_uminus_real ) ).
% 0.22/0.59
% 0.22/0.59 % uminus_real.transfer
% 0.22/0.59 thf(fact_10100_plus__real_Otransfer,axiom,
% 0.22/0.59 ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
% 0.22/0.59 @ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( plus_plus_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) )
% 0.22/0.59 @ plus_plus_real ) ).
% 0.22/0.59
% 0.22/0.59 % plus_real.transfer
% 0.22/0.59 thf(fact_10101_times__real_Otransfer,axiom,
% 0.22/0.59 ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
% 0.22/0.59 @ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) )
% 0.22/0.59 @ times_times_real ) ).
% 0.22/0.59
% 0.22/0.59 % times_real.transfer
% 0.22/0.59 thf(fact_10102_rat__floor__code,axiom,
% 0.22/0.59 ( archim3151403230148437115or_rat
% 0.22/0.59 = ( ^ [P6: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P6 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % rat_floor_code
% 0.22/0.59 thf(fact_10103_Divides_Oadjust__div__def,axiom,
% 0.22/0.59 ( adjust_div
% 0.22/0.59 = ( produc8211389475949308722nt_int
% 0.22/0.59 @ ^ [Q3: int,R5: int] : ( plus_plus_int @ Q3 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Divides.adjust_div_def
% 0.22/0.59 thf(fact_10104_Real_Opositive_Otransfer,axiom,
% 0.22/0.59 ( bNF_re4297313714947099218al_o_o @ pcr_real
% 0.22/0.59 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [X8: nat > rat] :
% 0.22/0.59 ? [R5: rat] :
% 0.22/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.22/0.59 & ? [K3: nat] :
% 0.22/0.59 ! [N4: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K3 @ N4 )
% 0.22/0.59 => ( ord_less_rat @ R5 @ ( X8 @ N4 ) ) ) )
% 0.22/0.59 @ positive2 ) ).
% 0.22/0.59
% 0.22/0.59 % Real.positive.transfer
% 0.22/0.59 thf(fact_10105_times__int_Orsp,axiom,
% 0.22/0.59 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 0.22/0.59 @ ( produc27273713700761075at_nat
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.59 ( produc2626176000494625587at_nat
% 0.22/0.59 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) )
% 0.22/0.59 @ ( produc27273713700761075at_nat
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.59 ( produc2626176000494625587at_nat
% 0.22/0.59 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % times_int.rsp
% 0.22/0.59 thf(fact_10106_intrel__iff,axiom,
% 0.22/0.59 ! [X: nat,Y4: nat,U: nat,V: nat] :
% 0.22/0.59 ( ( intrel @ ( product_Pair_nat_nat @ X @ Y4 ) @ ( product_Pair_nat_nat @ U @ V ) )
% 0.22/0.59 = ( ( plus_plus_nat @ X @ V )
% 0.22/0.59 = ( plus_plus_nat @ U @ Y4 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % intrel_iff
% 0.22/0.59 thf(fact_10107_uminus__int_Orsp,axiom,
% 0.22/0.59 ( bNF_re2241393799969408733at_nat @ intrel @ intrel
% 0.22/0.59 @ ( produc2626176000494625587at_nat
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 ) )
% 0.22/0.59 @ ( produc2626176000494625587at_nat
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % uminus_int.rsp
% 0.22/0.59 thf(fact_10108_zero__int_Orsp,axiom,
% 0.22/0.59 intrel @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.22/0.59
% 0.22/0.59 % zero_int.rsp
% 0.22/0.59 thf(fact_10109_int_Oabs__eq__iff,axiom,
% 0.22/0.59 ! [X: product_prod_nat_nat,Y4: product_prod_nat_nat] :
% 0.22/0.59 ( ( ( abs_Integ @ X )
% 0.22/0.59 = ( abs_Integ @ Y4 ) )
% 0.22/0.59 = ( intrel @ X @ Y4 ) ) ).
% 0.22/0.59
% 0.22/0.59 % int.abs_eq_iff
% 0.22/0.59 thf(fact_10110_Real_Opositive__mult,axiom,
% 0.22/0.59 ! [X: real,Y4: real] :
% 0.22/0.59 ( ( positive2 @ X )
% 0.22/0.59 => ( ( positive2 @ Y4 )
% 0.22/0.59 => ( positive2 @ ( times_times_real @ X @ Y4 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Real.positive_mult
% 0.22/0.59 thf(fact_10111_Real_Opositive__zero,axiom,
% 0.22/0.59 ~ ( positive2 @ zero_zero_real ) ).
% 0.22/0.59
% 0.22/0.59 % Real.positive_zero
% 0.22/0.59 thf(fact_10112_Real_Opositive__add,axiom,
% 0.22/0.59 ! [X: real,Y4: real] :
% 0.22/0.59 ( ( positive2 @ X )
% 0.22/0.59 => ( ( positive2 @ Y4 )
% 0.22/0.59 => ( positive2 @ ( plus_plus_real @ X @ Y4 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Real.positive_add
% 0.22/0.59 thf(fact_10113_Real_Opositive__minus,axiom,
% 0.22/0.59 ! [X: real] :
% 0.22/0.59 ( ~ ( positive2 @ X )
% 0.22/0.59 => ( ( X != zero_zero_real )
% 0.22/0.59 => ( positive2 @ ( uminus_uminus_real @ X ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Real.positive_minus
% 0.22/0.59 thf(fact_10114_nat_Orsp,axiom,
% 0.22/0.59 ( bNF_re8246922863344978751at_nat @ intrel
% 0.22/0.59 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.22/0.59 @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 0.22/0.59 @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).
% 0.22/0.59
% 0.22/0.59 % nat.rsp
% 0.22/0.59 thf(fact_10115_less__real__def,axiom,
% 0.22/0.59 ( ord_less_real
% 0.22/0.59 = ( ^ [X3: real,Y6: real] : ( positive2 @ ( minus_minus_real @ Y6 @ X3 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % less_real_def
% 0.22/0.59 thf(fact_10116_one__int_Orsp,axiom,
% 0.22/0.59 intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.22/0.59
% 0.22/0.59 % one_int.rsp
% 0.22/0.59 thf(fact_10117_intrel__def,axiom,
% 0.22/0.59 ( intrel
% 0.22/0.59 = ( produc8739625826339149834_nat_o
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.59 ( produc6081775807080527818_nat_o
% 0.22/0.59 @ ^ [U3: nat,V4: nat] :
% 0.22/0.59 ( ( plus_plus_nat @ X3 @ V4 )
% 0.22/0.59 = ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % intrel_def
% 0.22/0.59 thf(fact_10118_less__int_Orsp,axiom,
% 0.22/0.59 ( bNF_re4202695980764964119_nat_o @ intrel
% 0.22/0.59 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 0.22/0.59 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.22/0.59 @ ( produc8739625826339149834_nat_o
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.59 ( produc6081775807080527818_nat_o
% 0.22/0.59 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
% 0.22/0.59 @ ( produc8739625826339149834_nat_o
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.59 ( produc6081775807080527818_nat_o
% 0.22/0.59 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % less_int.rsp
% 0.22/0.59 thf(fact_10119_less__eq__int_Orsp,axiom,
% 0.22/0.59 ( bNF_re4202695980764964119_nat_o @ intrel
% 0.22/0.59 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 0.22/0.59 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.22/0.59 @ ( produc8739625826339149834_nat_o
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.59 ( produc6081775807080527818_nat_o
% 0.22/0.59 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
% 0.22/0.59 @ ( produc8739625826339149834_nat_o
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.59 ( produc6081775807080527818_nat_o
% 0.22/0.59 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % less_eq_int.rsp
% 0.22/0.59 thf(fact_10120_int_Orel__eq__transfer,axiom,
% 0.22/0.59 ( bNF_re717283939379294677_int_o @ pcr_int
% 0.22/0.59 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 0.22/0.59 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.22/0.59 @ intrel
% 0.22/0.59 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) ) ).
% 0.22/0.59
% 0.22/0.59 % int.rel_eq_transfer
% 0.22/0.59 thf(fact_10121_plus__int_Orsp,axiom,
% 0.22/0.59 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 0.22/0.59 @ ( produc27273713700761075at_nat
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.59 ( produc2626176000494625587at_nat
% 0.22/0.59 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) )
% 0.22/0.59 @ ( produc27273713700761075at_nat
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.59 ( produc2626176000494625587at_nat
% 0.22/0.59 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % plus_int.rsp
% 0.22/0.59 thf(fact_10122_minus__int_Orsp,axiom,
% 0.22/0.59 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 0.22/0.59 @ ( produc27273713700761075at_nat
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.59 ( produc2626176000494625587at_nat
% 0.22/0.59 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) )
% 0.22/0.59 @ ( produc27273713700761075at_nat
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] :
% 0.22/0.59 ( produc2626176000494625587at_nat
% 0.22/0.59 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % minus_int.rsp
% 0.22/0.59 thf(fact_10123_Real_Opositive_Orep__eq,axiom,
% 0.22/0.59 ( positive2
% 0.22/0.59 = ( ^ [X3: real] :
% 0.22/0.59 ? [R5: rat] :
% 0.22/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.22/0.59 & ? [K3: nat] :
% 0.22/0.59 ! [N4: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K3 @ N4 )
% 0.22/0.59 => ( ord_less_rat @ R5 @ ( rep_real @ X3 @ N4 ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Real.positive.rep_eq
% 0.22/0.59 thf(fact_10124_Real_Opositive_Orsp,axiom,
% 0.22/0.59 ( bNF_re728719798268516973at_o_o @ realrel
% 0.22/0.59 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 )
% 0.22/0.59 @ ^ [X8: nat > rat] :
% 0.22/0.59 ? [R5: rat] :
% 0.22/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.22/0.59 & ? [K3: nat] :
% 0.22/0.59 ! [N4: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K3 @ N4 )
% 0.22/0.59 => ( ord_less_rat @ R5 @ ( X8 @ N4 ) ) ) )
% 0.22/0.59 @ ^ [X8: nat > rat] :
% 0.22/0.59 ? [R5: rat] :
% 0.22/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.22/0.59 & ? [K3: nat] :
% 0.22/0.59 ! [N4: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K3 @ N4 )
% 0.22/0.59 => ( ord_less_rat @ R5 @ ( X8 @ N4 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Real.positive.rsp
% 0.22/0.59 thf(fact_10125_times__real_Orsp,axiom,
% 0.22/0.59 ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
% 0.22/0.59 @ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) )
% 0.22/0.59 @ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % times_real.rsp
% 0.22/0.59 thf(fact_10126_plus__real_Orsp,axiom,
% 0.22/0.59 ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
% 0.22/0.59 @ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( plus_plus_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) )
% 0.22/0.59 @ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( plus_plus_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % plus_real.rsp
% 0.22/0.59 thf(fact_10127_uminus__real_Orsp,axiom,
% 0.22/0.59 ( bNF_re895249473297799549at_rat @ realrel @ realrel
% 0.22/0.59 @ ^ [X8: nat > rat,N4: nat] : ( uminus_uminus_rat @ ( X8 @ N4 ) )
% 0.22/0.59 @ ^ [X8: nat > rat,N4: nat] : ( uminus_uminus_rat @ ( X8 @ N4 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % uminus_real.rsp
% 0.22/0.59 thf(fact_10128_zero__real_Orsp,axiom,
% 0.22/0.59 ( realrel
% 0.22/0.59 @ ^ [N4: nat] : zero_zero_rat
% 0.22/0.59 @ ^ [N4: nat] : zero_zero_rat ) ).
% 0.22/0.59
% 0.22/0.59 % zero_real.rsp
% 0.22/0.59 thf(fact_10129_one__real_Orsp,axiom,
% 0.22/0.59 ( realrel
% 0.22/0.59 @ ^ [N4: nat] : one_one_rat
% 0.22/0.59 @ ^ [N4: nat] : one_one_rat ) ).
% 0.22/0.59
% 0.22/0.59 % one_real.rsp
% 0.22/0.59 thf(fact_10130_real_Orel__eq__transfer,axiom,
% 0.22/0.59 ( bNF_re4521903465945308077real_o @ pcr_real
% 0.22/0.59 @ ( bNF_re4297313714947099218al_o_o @ pcr_real
% 0.22/0.59 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.22/0.59 @ realrel
% 0.22/0.59 @ ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) ) ).
% 0.22/0.59
% 0.22/0.59 % real.rel_eq_transfer
% 0.22/0.59 thf(fact_10131_inverse__real_Orsp,axiom,
% 0.22/0.59 ( bNF_re895249473297799549at_rat @ realrel @ realrel
% 0.22/0.59 @ ^ [X8: nat > rat] :
% 0.22/0.59 ( if_nat_rat @ ( vanishes @ X8 )
% 0.22/0.59 @ ^ [N4: nat] : zero_zero_rat
% 0.22/0.59 @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X8 @ N4 ) ) )
% 0.22/0.59 @ ^ [X8: nat > rat] :
% 0.22/0.59 ( if_nat_rat @ ( vanishes @ X8 )
% 0.22/0.59 @ ^ [N4: nat] : zero_zero_rat
% 0.22/0.59 @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X8 @ N4 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % inverse_real.rsp
% 0.22/0.59 thf(fact_10132_Real_Opositive__def,axiom,
% 0.22/0.59 ( positive2
% 0.22/0.59 = ( map_fu1856342031159181835at_o_o @ rep_real @ id_o
% 0.22/0.59 @ ^ [X8: nat > rat] :
% 0.22/0.59 ? [R5: rat] :
% 0.22/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.22/0.59 & ? [K3: nat] :
% 0.22/0.59 ! [N4: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K3 @ N4 )
% 0.22/0.59 => ( ord_less_rat @ R5 @ ( X8 @ N4 ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Real.positive_def
% 0.22/0.59 thf(fact_10133_inverse__real_Oabs__eq,axiom,
% 0.22/0.59 ! [X: nat > rat] :
% 0.22/0.59 ( ( realrel @ X @ X )
% 0.22/0.59 => ( ( inverse_inverse_real @ ( real2 @ X ) )
% 0.22/0.59 = ( real2
% 0.22/0.59 @ ( if_nat_rat @ ( vanishes @ X )
% 0.22/0.59 @ ^ [N4: nat] : zero_zero_rat
% 0.22/0.59 @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X @ N4 ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % inverse_real.abs_eq
% 0.22/0.59 thf(fact_10134_real_Oabs__induct,axiom,
% 0.22/0.59 ! [P: real > $o,X: real] :
% 0.22/0.59 ( ! [Y: nat > rat] :
% 0.22/0.59 ( ( realrel @ Y @ Y )
% 0.22/0.59 => ( P @ ( real2 @ Y ) ) )
% 0.22/0.59 => ( P @ X ) ) ).
% 0.22/0.59
% 0.22/0.59 % real.abs_induct
% 0.22/0.59 thf(fact_10135_of__int__Real,axiom,
% 0.22/0.59 ( ring_1_of_int_real
% 0.22/0.59 = ( ^ [X3: int] :
% 0.22/0.59 ( real2
% 0.22/0.59 @ ^ [N4: nat] : ( ring_1_of_int_rat @ X3 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % of_int_Real
% 0.22/0.59 thf(fact_10136_of__rat__Real,axiom,
% 0.22/0.59 ( field_7254667332652039916t_real
% 0.22/0.59 = ( ^ [X3: rat] :
% 0.22/0.59 ( real2
% 0.22/0.59 @ ^ [N4: nat] : X3 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % of_rat_Real
% 0.22/0.59 thf(fact_10137_one__real__def,axiom,
% 0.22/0.59 ( one_one_real
% 0.22/0.59 = ( real2
% 0.22/0.59 @ ^ [N4: nat] : one_one_rat ) ) ).
% 0.22/0.59
% 0.22/0.59 % one_real_def
% 0.22/0.59 thf(fact_10138_zero__real__def,axiom,
% 0.22/0.59 ( zero_zero_real
% 0.22/0.59 = ( real2
% 0.22/0.59 @ ^ [N4: nat] : zero_zero_rat ) ) ).
% 0.22/0.59
% 0.22/0.59 % zero_real_def
% 0.22/0.59 thf(fact_10139_of__nat__Real,axiom,
% 0.22/0.59 ( semiri5074537144036343181t_real
% 0.22/0.59 = ( ^ [X3: nat] :
% 0.22/0.59 ( real2
% 0.22/0.59 @ ^ [N4: nat] : ( semiri681578069525770553at_rat @ X3 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % of_nat_Real
% 0.22/0.59 thf(fact_10140_uminus__real_Oabs__eq,axiom,
% 0.22/0.59 ! [X: nat > rat] :
% 0.22/0.59 ( ( realrel @ X @ X )
% 0.22/0.59 => ( ( uminus_uminus_real @ ( real2 @ X ) )
% 0.22/0.59 = ( real2
% 0.22/0.59 @ ^ [N4: nat] : ( uminus_uminus_rat @ ( X @ N4 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % uminus_real.abs_eq
% 0.22/0.59 thf(fact_10141_plus__real_Oabs__eq,axiom,
% 0.22/0.59 ! [Xa: nat > rat,X: nat > rat] :
% 0.22/0.59 ( ( realrel @ Xa @ Xa )
% 0.22/0.59 => ( ( realrel @ X @ X )
% 0.22/0.59 => ( ( plus_plus_real @ ( real2 @ Xa ) @ ( real2 @ X ) )
% 0.22/0.59 = ( real2
% 0.22/0.59 @ ^ [N4: nat] : ( plus_plus_rat @ ( Xa @ N4 ) @ ( X @ N4 ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % plus_real.abs_eq
% 0.22/0.59 thf(fact_10142_times__real_Oabs__eq,axiom,
% 0.22/0.59 ! [Xa: nat > rat,X: nat > rat] :
% 0.22/0.59 ( ( realrel @ Xa @ Xa )
% 0.22/0.59 => ( ( realrel @ X @ X )
% 0.22/0.59 => ( ( times_times_real @ ( real2 @ Xa ) @ ( real2 @ X ) )
% 0.22/0.59 = ( real2
% 0.22/0.59 @ ^ [N4: nat] : ( times_times_rat @ ( Xa @ N4 ) @ ( X @ N4 ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % times_real.abs_eq
% 0.22/0.59 thf(fact_10143_Real_Opositive_Oabs__eq,axiom,
% 0.22/0.59 ! [X: nat > rat] :
% 0.22/0.59 ( ( realrel @ X @ X )
% 0.22/0.59 => ( ( positive2 @ ( real2 @ X ) )
% 0.22/0.59 = ( ? [R5: rat] :
% 0.22/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.22/0.59 & ? [K3: nat] :
% 0.22/0.59 ! [N4: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K3 @ N4 )
% 0.22/0.59 => ( ord_less_rat @ R5 @ ( X @ N4 ) ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Real.positive.abs_eq
% 0.22/0.59 thf(fact_10144_inverse__real__def,axiom,
% 0.22/0.59 ( inverse_inverse_real
% 0.22/0.59 = ( map_fu7146612038024189824t_real @ rep_real @ real2
% 0.22/0.59 @ ^ [X8: nat > rat] :
% 0.22/0.59 ( if_nat_rat @ ( vanishes @ X8 )
% 0.22/0.59 @ ^ [N4: nat] : zero_zero_rat
% 0.22/0.59 @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X8 @ N4 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % inverse_real_def
% 0.22/0.59 thf(fact_10145_le__Real,axiom,
% 0.22/0.59 ! [X9: nat > rat,Y7: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( ( cauchy @ Y7 )
% 0.22/0.59 => ( ( ord_less_eq_real @ ( real2 @ X9 ) @ ( real2 @ Y7 ) )
% 0.22/0.59 = ( ! [R5: rat] :
% 0.22/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.22/0.59 => ? [K3: nat] :
% 0.22/0.59 ! [N4: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K3 @ N4 )
% 0.22/0.59 => ( ord_less_eq_rat @ ( X9 @ N4 ) @ ( plus_plus_rat @ ( Y7 @ N4 ) @ R5 ) ) ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % le_Real
% 0.22/0.59 thf(fact_10146_cr__real__eq,axiom,
% 0.22/0.59 ( pcr_real
% 0.22/0.59 = ( ^ [X3: nat > rat,Y6: real] :
% 0.22/0.59 ( ( cauchy @ X3 )
% 0.22/0.59 & ( ( real2 @ X3 )
% 0.22/0.59 = Y6 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % cr_real_eq
% 0.22/0.59 thf(fact_10147_Real__induct,axiom,
% 0.22/0.59 ! [P: real > $o,X: real] :
% 0.22/0.59 ( ! [X15: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X15 )
% 0.22/0.59 => ( P @ ( real2 @ X15 ) ) )
% 0.22/0.59 => ( P @ X ) ) ).
% 0.22/0.59
% 0.22/0.59 % Real_induct
% 0.22/0.59 thf(fact_10148_cauchy__inverse,axiom,
% 0.22/0.59 ! [X9: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( ~ ( vanishes @ X9 )
% 0.22/0.59 => ( cauchy
% 0.22/0.59 @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X9 @ N4 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % cauchy_inverse
% 0.22/0.59 thf(fact_10149_cauchy__add,axiom,
% 0.22/0.59 ! [X9: nat > rat,Y7: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( ( cauchy @ Y7 )
% 0.22/0.59 => ( cauchy
% 0.22/0.59 @ ^ [N4: nat] : ( plus_plus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % cauchy_add
% 0.22/0.59 thf(fact_10150_cauchy__minus,axiom,
% 0.22/0.59 ! [X9: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( cauchy
% 0.22/0.59 @ ^ [N4: nat] : ( uminus_uminus_rat @ ( X9 @ N4 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % cauchy_minus
% 0.22/0.59 thf(fact_10151_cauchy__const,axiom,
% 0.22/0.59 ! [X: rat] :
% 0.22/0.59 ( cauchy
% 0.22/0.59 @ ^ [N4: nat] : X ) ).
% 0.22/0.59
% 0.22/0.59 % cauchy_const
% 0.22/0.59 thf(fact_10152_cauchy__mult,axiom,
% 0.22/0.59 ! [X9: nat > rat,Y7: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( ( cauchy @ Y7 )
% 0.22/0.59 => ( cauchy
% 0.22/0.59 @ ^ [N4: nat] : ( times_times_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % cauchy_mult
% 0.22/0.59 thf(fact_10153_cauchy__diff,axiom,
% 0.22/0.59 ! [X9: nat > rat,Y7: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( ( cauchy @ Y7 )
% 0.22/0.59 => ( cauchy
% 0.22/0.59 @ ^ [N4: nat] : ( minus_minus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % cauchy_diff
% 0.22/0.59 thf(fact_10154_realrel__refl,axiom,
% 0.22/0.59 ! [X9: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( realrel @ X9 @ X9 ) ) ).
% 0.22/0.59
% 0.22/0.59 % realrel_refl
% 0.22/0.59 thf(fact_10155_cauchy__imp__bounded,axiom,
% 0.22/0.59 ! [X9: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ? [B3: rat] :
% 0.22/0.59 ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 0.22/0.59 & ! [N6: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N6 ) ) @ B3 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % cauchy_imp_bounded
% 0.22/0.59 thf(fact_10156_less__RealD,axiom,
% 0.22/0.59 ! [Y7: nat > rat,X: real] :
% 0.22/0.59 ( ( cauchy @ Y7 )
% 0.22/0.59 => ( ( ord_less_real @ X @ ( real2 @ Y7 ) )
% 0.22/0.59 => ? [N2: nat] : ( ord_less_real @ X @ ( field_7254667332652039916t_real @ ( Y7 @ N2 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % less_RealD
% 0.22/0.59 thf(fact_10157_Real__leI,axiom,
% 0.22/0.59 ! [X9: nat > rat,Y4: real] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( ! [N2: nat] : ( ord_less_eq_real @ ( field_7254667332652039916t_real @ ( X9 @ N2 ) ) @ Y4 )
% 0.22/0.59 => ( ord_less_eq_real @ ( real2 @ X9 ) @ Y4 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Real_leI
% 0.22/0.59 thf(fact_10158_le__RealI,axiom,
% 0.22/0.59 ! [Y7: nat > rat,X: real] :
% 0.22/0.59 ( ( cauchy @ Y7 )
% 0.22/0.59 => ( ! [N2: nat] : ( ord_less_eq_real @ X @ ( field_7254667332652039916t_real @ ( Y7 @ N2 ) ) )
% 0.22/0.59 => ( ord_less_eq_real @ X @ ( real2 @ Y7 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % le_RealI
% 0.22/0.59 thf(fact_10159_minus__Real,axiom,
% 0.22/0.59 ! [X9: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( ( uminus_uminus_real @ ( real2 @ X9 ) )
% 0.22/0.59 = ( real2
% 0.22/0.59 @ ^ [N4: nat] : ( uminus_uminus_rat @ ( X9 @ N4 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % minus_Real
% 0.22/0.59 thf(fact_10160_add__Real,axiom,
% 0.22/0.59 ! [X9: nat > rat,Y7: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( ( cauchy @ Y7 )
% 0.22/0.59 => ( ( plus_plus_real @ ( real2 @ X9 ) @ ( real2 @ Y7 ) )
% 0.22/0.59 = ( real2
% 0.22/0.59 @ ^ [N4: nat] : ( plus_plus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % add_Real
% 0.22/0.59 thf(fact_10161_mult__Real,axiom,
% 0.22/0.59 ! [X9: nat > rat,Y7: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( ( cauchy @ Y7 )
% 0.22/0.59 => ( ( times_times_real @ ( real2 @ X9 ) @ ( real2 @ Y7 ) )
% 0.22/0.59 = ( real2
% 0.22/0.59 @ ^ [N4: nat] : ( times_times_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % mult_Real
% 0.22/0.59 thf(fact_10162_diff__Real,axiom,
% 0.22/0.59 ! [X9: nat > rat,Y7: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( ( cauchy @ Y7 )
% 0.22/0.59 => ( ( minus_minus_real @ ( real2 @ X9 ) @ ( real2 @ Y7 ) )
% 0.22/0.59 = ( real2
% 0.22/0.59 @ ^ [N4: nat] : ( minus_minus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % diff_Real
% 0.22/0.59 thf(fact_10163_realrelI,axiom,
% 0.22/0.59 ! [X9: nat > rat,Y7: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( ( cauchy @ Y7 )
% 0.22/0.59 => ( ( vanishes
% 0.22/0.59 @ ^ [N4: nat] : ( minus_minus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) )
% 0.22/0.59 => ( realrel @ X9 @ Y7 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % realrelI
% 0.22/0.59 thf(fact_10164_eq__Real,axiom,
% 0.22/0.59 ! [X9: nat > rat,Y7: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( ( cauchy @ Y7 )
% 0.22/0.59 => ( ( ( real2 @ X9 )
% 0.22/0.59 = ( real2 @ Y7 ) )
% 0.22/0.59 = ( vanishes
% 0.22/0.59 @ ^ [N4: nat] : ( minus_minus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % eq_Real
% 0.22/0.59 thf(fact_10165_vanishes__diff__inverse,axiom,
% 0.22/0.59 ! [X9: nat > rat,Y7: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( ~ ( vanishes @ X9 )
% 0.22/0.59 => ( ( cauchy @ Y7 )
% 0.22/0.59 => ( ~ ( vanishes @ Y7 )
% 0.22/0.59 => ( ( vanishes
% 0.22/0.59 @ ^ [N4: nat] : ( minus_minus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) )
% 0.22/0.59 => ( vanishes
% 0.22/0.59 @ ^ [N4: nat] : ( minus_minus_rat @ ( inverse_inverse_rat @ ( X9 @ N4 ) ) @ ( inverse_inverse_rat @ ( Y7 @ N4 ) ) ) ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % vanishes_diff_inverse
% 0.22/0.59 thf(fact_10166_realrel__def,axiom,
% 0.22/0.59 ( realrel
% 0.22/0.59 = ( ^ [X8: nat > rat,Y8: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X8 )
% 0.22/0.59 & ( cauchy @ Y8 )
% 0.22/0.59 & ( vanishes
% 0.22/0.59 @ ^ [N4: nat] : ( minus_minus_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % realrel_def
% 0.22/0.59 thf(fact_10167_cauchy__not__vanishes__cases,axiom,
% 0.22/0.59 ! [X9: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( ~ ( vanishes @ X9 )
% 0.22/0.59 => ? [B3: rat] :
% 0.22/0.59 ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 0.22/0.59 & ? [K2: nat] :
% 0.22/0.59 ( ! [N6: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K2 @ N6 )
% 0.22/0.59 => ( ord_less_rat @ B3 @ ( uminus_uminus_rat @ ( X9 @ N6 ) ) ) )
% 0.22/0.59 | ! [N6: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K2 @ N6 )
% 0.22/0.59 => ( ord_less_rat @ B3 @ ( X9 @ N6 ) ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % cauchy_not_vanishes_cases
% 0.22/0.59 thf(fact_10168_positive__Real,axiom,
% 0.22/0.59 ! [X9: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( ( positive2 @ ( real2 @ X9 ) )
% 0.22/0.59 = ( ? [R5: rat] :
% 0.22/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.22/0.59 & ? [K3: nat] :
% 0.22/0.59 ! [N4: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K3 @ N4 )
% 0.22/0.59 => ( ord_less_rat @ R5 @ ( X9 @ N4 ) ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % positive_Real
% 0.22/0.59 thf(fact_10169_uminus__real__def,axiom,
% 0.22/0.59 ( uminus_uminus_real
% 0.22/0.59 = ( map_fu7146612038024189824t_real @ rep_real @ real2
% 0.22/0.59 @ ^ [X8: nat > rat,N4: nat] : ( uminus_uminus_rat @ ( X8 @ N4 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % uminus_real_def
% 0.22/0.59 thf(fact_10170_cauchy__not__vanishes,axiom,
% 0.22/0.59 ! [X9: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( ~ ( vanishes @ X9 )
% 0.22/0.59 => ? [B3: rat] :
% 0.22/0.59 ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 0.22/0.59 & ? [K2: nat] :
% 0.22/0.59 ! [N6: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K2 @ N6 )
% 0.22/0.59 => ( ord_less_rat @ B3 @ ( abs_abs_rat @ ( X9 @ N6 ) ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % cauchy_not_vanishes
% 0.22/0.59 thf(fact_10171_cauchyD,axiom,
% 0.22/0.59 ! [X9: nat > rat,R2: rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 0.22/0.59 => ? [K2: nat] :
% 0.22/0.59 ! [M2: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K2 @ M2 )
% 0.22/0.59 => ! [N6: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K2 @ N6 )
% 0.22/0.59 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X9 @ M2 ) @ ( X9 @ N6 ) ) ) @ R2 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % cauchyD
% 0.22/0.59 thf(fact_10172_cauchyI,axiom,
% 0.22/0.59 ! [X9: nat > rat] :
% 0.22/0.59 ( ! [R3: rat] :
% 0.22/0.59 ( ( ord_less_rat @ zero_zero_rat @ R3 )
% 0.22/0.59 => ? [K4: nat] :
% 0.22/0.59 ! [M4: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K4 @ M4 )
% 0.22/0.59 => ! [N2: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K4 @ N2 )
% 0.22/0.59 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X9 @ M4 ) @ ( X9 @ N2 ) ) ) @ R3 ) ) ) )
% 0.22/0.59 => ( cauchy @ X9 ) ) ).
% 0.22/0.59
% 0.22/0.59 % cauchyI
% 0.22/0.59 thf(fact_10173_cauchy__def,axiom,
% 0.22/0.59 ( cauchy
% 0.22/0.59 = ( ^ [X8: nat > rat] :
% 0.22/0.59 ! [R5: rat] :
% 0.22/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.22/0.59 => ? [K3: nat] :
% 0.22/0.59 ! [M5: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K3 @ M5 )
% 0.22/0.59 => ! [N4: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K3 @ N4 )
% 0.22/0.59 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M5 ) @ ( X8 @ N4 ) ) ) @ R5 ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % cauchy_def
% 0.22/0.59 thf(fact_10174_inverse__Real,axiom,
% 0.22/0.59 ! [X9: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( ( ( vanishes @ X9 )
% 0.22/0.59 => ( ( inverse_inverse_real @ ( real2 @ X9 ) )
% 0.22/0.59 = zero_zero_real ) )
% 0.22/0.59 & ( ~ ( vanishes @ X9 )
% 0.22/0.59 => ( ( inverse_inverse_real @ ( real2 @ X9 ) )
% 0.22/0.59 = ( real2
% 0.22/0.59 @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X9 @ N4 ) ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % inverse_Real
% 0.22/0.59 thf(fact_10175_not__positive__Real,axiom,
% 0.22/0.59 ! [X9: nat > rat] :
% 0.22/0.59 ( ( cauchy @ X9 )
% 0.22/0.59 => ( ( ~ ( positive2 @ ( real2 @ X9 ) ) )
% 0.22/0.59 = ( ! [R5: rat] :
% 0.22/0.59 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.22/0.59 => ? [K3: nat] :
% 0.22/0.59 ! [N4: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ K3 @ N4 )
% 0.22/0.59 => ( ord_less_eq_rat @ ( X9 @ N4 ) @ R5 ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % not_positive_Real
% 0.22/0.59 thf(fact_10176_times__real__def,axiom,
% 0.22/0.59 ( times_times_real
% 0.22/0.59 = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
% 0.22/0.59 @ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % times_real_def
% 0.22/0.59 thf(fact_10177_plus__real__def,axiom,
% 0.22/0.59 ( plus_plus_real
% 0.22/0.59 = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
% 0.22/0.59 @ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( plus_plus_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % plus_real_def
% 0.22/0.59 thf(fact_10178_cr__real__def,axiom,
% 0.22/0.59 ( cr_real
% 0.22/0.59 = ( ^ [X3: nat > rat,Y6: real] :
% 0.22/0.59 ( ( realrel @ X3 @ X3 )
% 0.22/0.59 & ( ( real2 @ X3 )
% 0.22/0.59 = Y6 ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % cr_real_def
% 0.22/0.59 thf(fact_10179_int_Obi__total,axiom,
% 0.22/0.59 bi_tot896582865486249351at_int @ pcr_int ).
% 0.22/0.59
% 0.22/0.59 % int.bi_total
% 0.22/0.59 thf(fact_10180_real_Opcr__cr__eq,axiom,
% 0.22/0.59 pcr_real = cr_real ).
% 0.22/0.59
% 0.22/0.59 % real.pcr_cr_eq
% 0.22/0.59 thf(fact_10181_rcis__inverse,axiom,
% 0.22/0.59 ! [R2: real,A: real] :
% 0.22/0.59 ( ( invers8013647133539491842omplex @ ( rcis @ R2 @ A ) )
% 0.22/0.59 = ( rcis @ ( divide_divide_real @ one_one_real @ R2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % rcis_inverse
% 0.22/0.59 thf(fact_10182_rcis__zero__arg,axiom,
% 0.22/0.59 ! [R2: real] :
% 0.22/0.59 ( ( rcis @ R2 @ zero_zero_real )
% 0.22/0.59 = ( real_V4546457046886955230omplex @ R2 ) ) ).
% 0.22/0.59
% 0.22/0.59 % rcis_zero_arg
% 0.22/0.59 thf(fact_10183_rcis__eq__zero__iff,axiom,
% 0.22/0.59 ! [R2: real,A: real] :
% 0.22/0.59 ( ( ( rcis @ R2 @ A )
% 0.22/0.59 = zero_zero_complex )
% 0.22/0.59 = ( R2 = zero_zero_real ) ) ).
% 0.22/0.59
% 0.22/0.59 % rcis_eq_zero_iff
% 0.22/0.59 thf(fact_10184_rcis__zero__mod,axiom,
% 0.22/0.59 ! [A: real] :
% 0.22/0.59 ( ( rcis @ zero_zero_real @ A )
% 0.22/0.59 = zero_zero_complex ) ).
% 0.22/0.59
% 0.22/0.59 % rcis_zero_mod
% 0.22/0.59 thf(fact_10185_Re__rcis,axiom,
% 0.22/0.59 ! [R2: real,A: real] :
% 0.22/0.59 ( ( re @ ( rcis @ R2 @ A ) )
% 0.22/0.59 = ( times_times_real @ R2 @ ( cos_real @ A ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Re_rcis
% 0.22/0.59 thf(fact_10186_Im__rcis,axiom,
% 0.22/0.59 ! [R2: real,A: real] :
% 0.22/0.59 ( ( im @ ( rcis @ R2 @ A ) )
% 0.22/0.59 = ( times_times_real @ R2 @ ( sin_real @ A ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Im_rcis
% 0.22/0.59 thf(fact_10187_rcis__mult,axiom,
% 0.22/0.59 ! [R1: real,A: real,R22: real,B: real] :
% 0.22/0.59 ( ( times_times_complex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B ) )
% 0.22/0.59 = ( rcis @ ( times_times_real @ R1 @ R22 ) @ ( plus_plus_real @ A @ B ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % rcis_mult
% 0.22/0.59 thf(fact_10188_rcis__divide,axiom,
% 0.22/0.59 ! [R1: real,A: real,R22: real,B: real] :
% 0.22/0.59 ( ( divide1717551699836669952omplex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B ) )
% 0.22/0.59 = ( rcis @ ( divide_divide_real @ R1 @ R22 ) @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % rcis_divide
% 0.22/0.59 thf(fact_10189_DeMoivre2,axiom,
% 0.22/0.59 ! [R2: real,A: real,N: nat] :
% 0.22/0.59 ( ( power_power_complex @ ( rcis @ R2 @ A ) @ N )
% 0.22/0.59 = ( rcis @ ( power_power_real @ R2 @ N ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % DeMoivre2
% 0.22/0.59 thf(fact_10190_pred__nat__trancl__eq__le,axiom,
% 0.22/0.59 ! [M: nat,N: nat] :
% 0.22/0.59 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
% 0.22/0.59 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.22/0.59
% 0.22/0.59 % pred_nat_trancl_eq_le
% 0.22/0.59 thf(fact_10191_le__enumerate,axiom,
% 0.22/0.59 ! [S: set_nat,N: nat] :
% 0.22/0.59 ( ~ ( finite_finite_nat @ S )
% 0.22/0.59 => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % le_enumerate
% 0.22/0.59 thf(fact_10192_finite__le__enumerate,axiom,
% 0.22/0.59 ! [S: set_nat,N: nat] :
% 0.22/0.59 ( ( finite_finite_nat @ S )
% 0.22/0.59 => ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
% 0.22/0.59 => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % finite_le_enumerate
% 0.22/0.59 thf(fact_10193_Least__eq__0,axiom,
% 0.22/0.59 ! [P: nat > $o] :
% 0.22/0.59 ( ( P @ zero_zero_nat )
% 0.22/0.59 => ( ( ord_Least_nat @ P )
% 0.22/0.59 = zero_zero_nat ) ) ).
% 0.22/0.59
% 0.22/0.59 % Least_eq_0
% 0.22/0.59 thf(fact_10194_Least__Suc2,axiom,
% 0.22/0.59 ! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
% 0.22/0.59 ( ( P @ N )
% 0.22/0.59 => ( ( Q @ M )
% 0.22/0.59 => ( ~ ( P @ zero_zero_nat )
% 0.22/0.59 => ( ! [K2: nat] :
% 0.22/0.59 ( ( P @ ( suc @ K2 ) )
% 0.22/0.59 = ( Q @ K2 ) )
% 0.22/0.59 => ( ( ord_Least_nat @ P )
% 0.22/0.59 = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Least_Suc2
% 0.22/0.59 thf(fact_10195_Least__Suc,axiom,
% 0.22/0.59 ! [P: nat > $o,N: nat] :
% 0.22/0.59 ( ( P @ N )
% 0.22/0.59 => ( ~ ( P @ zero_zero_nat )
% 0.22/0.59 => ( ( ord_Least_nat @ P )
% 0.22/0.59 = ( suc
% 0.22/0.59 @ ( ord_Least_nat
% 0.22/0.59 @ ^ [M5: nat] : ( P @ ( suc @ M5 ) ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Least_Suc
% 0.22/0.59 thf(fact_10196_Sup__real__def,axiom,
% 0.22/0.59 ( comple1385675409528146559p_real
% 0.22/0.59 = ( ^ [X8: set_real] :
% 0.22/0.59 ( ord_Least_real
% 0.22/0.59 @ ^ [Z5: real] :
% 0.22/0.59 ! [X3: real] :
% 0.22/0.59 ( ( member_real @ X3 @ X8 )
% 0.22/0.59 => ( ord_less_eq_real @ X3 @ Z5 ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Sup_real_def
% 0.22/0.59 thf(fact_10197_last__upt,axiom,
% 0.22/0.59 ! [I: nat,J: nat] :
% 0.22/0.59 ( ( ord_less_nat @ I @ J )
% 0.22/0.59 => ( ( last_nat @ ( upt @ I @ J ) )
% 0.22/0.59 = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % last_upt
% 0.22/0.59 thf(fact_10198_vimage__Suc__insert__0,axiom,
% 0.22/0.59 ! [A2: set_nat] :
% 0.22/0.59 ( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A2 ) )
% 0.22/0.59 = ( vimage_nat_nat @ suc @ A2 ) ) ).
% 0.22/0.59
% 0.22/0.59 % vimage_Suc_insert_0
% 0.22/0.59 thf(fact_10199_vimage__Suc__insert__Suc,axiom,
% 0.22/0.59 ! [N: nat,A2: set_nat] :
% 0.22/0.59 ( ( vimage_nat_nat @ suc @ ( insert_nat @ ( suc @ N ) @ A2 ) )
% 0.22/0.59 = ( insert_nat @ N @ ( vimage_nat_nat @ suc @ A2 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % vimage_Suc_insert_Suc
% 0.22/0.59 thf(fact_10200_finite__vimage__Suc__iff,axiom,
% 0.22/0.59 ! [F4: set_nat] :
% 0.22/0.59 ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F4 ) )
% 0.22/0.59 = ( finite_finite_nat @ F4 ) ) ).
% 0.22/0.59
% 0.22/0.59 % finite_vimage_Suc_iff
% 0.22/0.59 thf(fact_10201_set__decode__div__2,axiom,
% 0.22/0.59 ! [X: nat] :
% 0.22/0.59 ( ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.22/0.59 = ( vimage_nat_nat @ suc @ ( nat_set_decode @ X ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % set_decode_div_2
% 0.22/0.59 thf(fact_10202_set__encode__vimage__Suc,axiom,
% 0.22/0.59 ! [A2: set_nat] :
% 0.22/0.59 ( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A2 ) )
% 0.22/0.59 = ( divide_divide_nat @ ( nat_set_encode @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % set_encode_vimage_Suc
% 0.22/0.59 thf(fact_10203_Bseq__monoseq__convergent_H__dec,axiom,
% 0.22/0.59 ! [F: nat > real,M10: nat] :
% 0.22/0.59 ( ( bfun_nat_real
% 0.22/0.59 @ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ M10 ) )
% 0.22/0.59 @ at_top_nat )
% 0.22/0.59 => ( ! [M4: nat,N2: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ M10 @ M4 )
% 0.22/0.59 => ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.22/0.59 => ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ M4 ) ) ) )
% 0.22/0.59 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Bseq_monoseq_convergent'_dec
% 0.22/0.59 thf(fact_10204_Bseq__mono__convergent,axiom,
% 0.22/0.59 ! [X9: nat > real] :
% 0.22/0.59 ( ( bfun_nat_real @ X9 @ at_top_nat )
% 0.22/0.59 => ( ! [M4: nat,N2: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.22/0.59 => ( ord_less_eq_real @ ( X9 @ M4 ) @ ( X9 @ N2 ) ) )
% 0.22/0.59 => ( topolo7531315842566124627t_real @ X9 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Bseq_mono_convergent
% 0.22/0.59 thf(fact_10205_convergent__realpow,axiom,
% 0.22/0.59 ! [X: real] :
% 0.22/0.59 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.22/0.59 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.22/0.59 => ( topolo7531315842566124627t_real @ ( power_power_real @ X ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % convergent_realpow
% 0.22/0.59 thf(fact_10206_Bseq__monoseq__convergent_H__inc,axiom,
% 0.22/0.59 ! [F: nat > real,M10: nat] :
% 0.22/0.59 ( ( bfun_nat_real
% 0.22/0.59 @ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ M10 ) )
% 0.22/0.59 @ at_top_nat )
% 0.22/0.59 => ( ! [M4: nat,N2: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ M10 @ M4 )
% 0.22/0.59 => ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.22/0.59 => ( ord_less_eq_real @ ( F @ M4 ) @ ( F @ N2 ) ) ) )
% 0.22/0.59 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % Bseq_monoseq_convergent'_inc
% 0.22/0.59 thf(fact_10207_pair__lessI2,axiom,
% 0.22/0.59 ! [A: nat,B: nat,S2: nat,T: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.59 => ( ( ord_less_nat @ S2 @ T )
% 0.22/0.59 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % pair_lessI2
% 0.22/0.59 thf(fact_10208_pair__leqI2,axiom,
% 0.22/0.59 ! [A: nat,B: nat,S2: nat,T: nat] :
% 0.22/0.59 ( ( ord_less_eq_nat @ A @ B )
% 0.22/0.59 => ( ( ord_less_eq_nat @ S2 @ T )
% 0.22/0.59 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % pair_leqI2
% 0.22/0.59 thf(fact_10209_filtermap__at__right__shift,axiom,
% 0.22/0.59 ! [D: real,A: real] :
% 0.22/0.59 ( ( filtermap_real_real
% 0.22/0.59 @ ^ [X3: real] : ( minus_minus_real @ X3 @ D )
% 0.22/0.59 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.22/0.59 = ( topolo2177554685111907308n_real @ ( minus_minus_real @ A @ D ) @ ( set_or5849166863359141190n_real @ ( minus_minus_real @ A @ D ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % filtermap_at_right_shift
% 0.22/0.59 thf(fact_10210_at__right__to__0,axiom,
% 0.22/0.59 ! [A: real] :
% 0.22/0.59 ( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
% 0.22/0.59 = ( filtermap_real_real
% 0.22/0.59 @ ^ [X3: real] : ( plus_plus_real @ X3 @ A )
% 0.22/0.59 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % at_right_to_0
% 0.22/0.59 thf(fact_10211_at__right__to__top,axiom,
% 0.22/0.59 ( ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) )
% 0.22/0.59 = ( filtermap_real_real @ inverse_inverse_real @ at_top_real ) ) ).
% 0.22/0.59
% 0.22/0.59 % at_right_to_top
% 0.22/0.59 thf(fact_10212_at__top__to__right,axiom,
% 0.22/0.59 ( at_top_real
% 0.22/0.59 = ( filtermap_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 % at_top_to_right
% 0.22/0.59 thf(fact_10213_filtermap__ln__at__right,axiom,
% 0.22/0.59 ( ( filtermap_real_real @ ln_ln_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.22/0.59 = at_bot_real ) ).
% 0.22/0.59
% 0.22/0.59 % filtermap_ln_at_right
% 0.22/0.59 thf(fact_10214_gcd__nat_Oordering__top__axioms,axiom,
% 0.22/0.59 ( ordering_top_nat @ dvd_dvd_nat
% 0.22/0.59 @ ^ [M5: nat,N4: nat] :
% 0.22/0.59 ( ( dvd_dvd_nat @ M5 @ N4 )
% 0.22/0.59 & ( M5 != N4 ) )
% 0.22/0.59 @ zero_zero_nat ) ).
% 0.22/0.59
% 0.22/0.59 % gcd_nat.ordering_top_axioms
% 0.22/0.59 thf(fact_10215_bot__nat__0_Oordering__top__axioms,axiom,
% 0.22/0.59 ( ordering_top_nat
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X3 )
% 0.22/0.59 @ ^ [X3: nat,Y6: nat] : ( ord_less_nat @ Y6 @ X3 )
% 0.22/0.59 @ zero_zero_nat ) ).
% 0.22/0.59
% 0.22/0.59 % bot_nat_0.ordering_top_axioms
% 0.22/0.59
% 0.22/0.59 % Helper facts (40)
% 0.22/0.59 thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 0.22/0.59 ! [X: int,Y4: int] :
% 0.22/0.59 ( ( if_int @ $false @ X @ Y4 )
% 0.22/0.59 = Y4 ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 0.22/0.59 ! [X: int,Y4: int] :
% 0.22/0.59 ( ( if_int @ $true @ X @ Y4 )
% 0.22/0.59 = X ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.22/0.59 ! [X: nat,Y4: nat] :
% 0.22/0.59 ( ( if_nat @ $false @ X @ Y4 )
% 0.22/0.59 = Y4 ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.22/0.59 ! [X: nat,Y4: nat] :
% 0.22/0.59 ( ( if_nat @ $true @ X @ Y4 )
% 0.22/0.59 = X ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
% 0.22/0.59 ! [X: num,Y4: num] :
% 0.22/0.59 ( ( if_num @ $false @ X @ Y4 )
% 0.22/0.59 = Y4 ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
% 0.22/0.59 ! [X: num,Y4: num] :
% 0.22/0.59 ( ( if_num @ $true @ X @ Y4 )
% 0.22/0.59 = X ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 0.22/0.59 ! [X: rat,Y4: rat] :
% 0.22/0.59 ( ( if_rat @ $false @ X @ Y4 )
% 0.22/0.59 = Y4 ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 0.22/0.59 ! [X: rat,Y4: rat] :
% 0.22/0.59 ( ( if_rat @ $true @ X @ Y4 )
% 0.22/0.59 = X ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 0.22/0.59 ! [X: real,Y4: real] :
% 0.22/0.59 ( ( if_real @ $false @ X @ Y4 )
% 0.22/0.59 = Y4 ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 0.22/0.59 ! [X: real,Y4: real] :
% 0.22/0.59 ( ( if_real @ $true @ X @ Y4 )
% 0.22/0.59 = X ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 0.22/0.59 ! [P: real > $o] :
% 0.22/0.59 ( ( P @ ( fChoice_real @ P ) )
% 0.22/0.59 = ( ? [X8: real] : ( P @ X8 ) ) ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.22/0.59 ! [X: complex,Y4: complex] :
% 0.22/0.59 ( ( if_complex @ $false @ X @ Y4 )
% 0.22/0.59 = Y4 ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.22/0.59 ! [X: complex,Y4: complex] :
% 0.22/0.59 ( ( if_complex @ $true @ X @ Y4 )
% 0.22/0.59 = X ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.22/0.59 ! [X: code_integer,Y4: code_integer] :
% 0.22/0.59 ( ( if_Code_integer @ $false @ X @ Y4 )
% 0.22/0.59 = Y4 ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.22/0.59 ! [X: code_integer,Y4: code_integer] :
% 0.22/0.59 ( ( if_Code_integer @ $true @ X @ Y4 )
% 0.22/0.59 = X ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.22/0.59 ! [X: list_int,Y4: list_int] :
% 0.22/0.59 ( ( if_list_int @ $false @ X @ Y4 )
% 0.22/0.59 = Y4 ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.22/0.59 ! [X: list_int,Y4: list_int] :
% 0.22/0.59 ( ( if_list_int @ $true @ X @ Y4 )
% 0.22/0.59 = X ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 0.22/0.59 ! [X: list_nat,Y4: list_nat] :
% 0.22/0.59 ( ( if_list_nat @ $false @ X @ Y4 )
% 0.22/0.59 = Y4 ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 0.22/0.59 ! [X: list_nat,Y4: list_nat] :
% 0.22/0.59 ( ( if_list_nat @ $true @ X @ Y4 )
% 0.22/0.59 = X ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.22/0.59 ! [X: int > int,Y4: int > int] :
% 0.22/0.59 ( ( if_int_int @ $false @ X @ Y4 )
% 0.22/0.59 = Y4 ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.22/0.59 ! [X: int > int,Y4: int > int] :
% 0.22/0.59 ( ( if_int_int @ $true @ X @ Y4 )
% 0.22/0.59 = X ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_2_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
% 0.22/0.59 ! [X: nat > rat,Y4: nat > rat] :
% 0.22/0.59 ( ( if_nat_rat @ $false @ X @ Y4 )
% 0.22/0.59 = Y4 ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_1_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
% 0.22/0.59 ! [X: nat > rat,Y4: nat > rat] :
% 0.22/0.59 ( ( if_nat_rat @ $true @ X @ Y4 )
% 0.22/0.59 = X ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.22/0.59 ! [X: option_num,Y4: option_num] :
% 0.22/0.59 ( ( if_option_num @ $false @ X @ Y4 )
% 0.22/0.59 = Y4 ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.22/0.59 ! [X: option_num,Y4: option_num] :
% 0.22/0.59 ( ( if_option_num @ $true @ X @ Y4 )
% 0.22/0.59 = X ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.22/0.59 ! [X: product_prod_int_int,Y4: product_prod_int_int] :
% 0.22/0.59 ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y4 )
% 0.22/0.59 = Y4 ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.22/0.59 ! [X: product_prod_int_int,Y4: product_prod_int_int] :
% 0.22/0.59 ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y4 )
% 0.22/0.59 = X ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.22/0.59 ! [X: product_prod_nat_nat,Y4: product_prod_nat_nat] :
% 0.22/0.59 ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y4 )
% 0.22/0.59 = Y4 ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.22/0.59 ! [X: product_prod_nat_nat,Y4: product_prod_nat_nat] :
% 0.22/0.59 ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y4 )
% 0.22/0.59 = X ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
% 0.22/0.59 ! [X: nat > int > int,Y4: nat > int > int] :
% 0.22/0.59 ( ( if_nat_int_int @ $false @ X @ Y4 )
% 0.22/0.59 = Y4 ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
% 0.22/0.59 ! [X: nat > int > int,Y4: nat > int > int] :
% 0.22/0.59 ( ( if_nat_int_int @ $true @ X @ Y4 )
% 0.22/0.59 = X ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
% 0.22/0.59 ! [X: nat > nat > nat,Y4: nat > nat > nat] :
% 0.22/0.59 ( ( if_nat_nat_nat @ $false @ X @ Y4 )
% 0.22/0.59 = Y4 ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
% 0.22/0.59 ! [X: nat > nat > nat,Y4: nat > nat > nat] :
% 0.22/0.59 ( ( if_nat_nat_nat @ $true @ X @ Y4 )
% 0.22/0.59 = X ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.22/0.59 ! [X: produc6271795597528267376eger_o,Y4: produc6271795597528267376eger_o] :
% 0.22/0.59 ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y4 )
% 0.22/0.59 = Y4 ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.22/0.59 ! [X: produc6271795597528267376eger_o,Y4: produc6271795597528267376eger_o] :
% 0.22/0.59 ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y4 )
% 0.22/0.59 = X ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.22/0.59 ! [X: produc8923325533196201883nteger,Y4: produc8923325533196201883nteger] :
% 0.22/0.59 ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y4 )
% 0.22/0.59 = Y4 ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.22/0.59 ! [X: produc8923325533196201883nteger,Y4: produc8923325533196201883nteger] :
% 0.22/0.59 ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y4 )
% 0.22/0.59 = X ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_3_1_If_001_062_It__Nat__Onat_M_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_T,axiom,
% 0.22/0.59 ! [P: $o] :
% 0.22/0.59 ( ( P = $true )
% 0.22/0.59 | ( P = $false ) ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_T,axiom,
% 0.22/0.59 ! [X: nat > code_integer > code_integer,Y4: nat > code_integer > code_integer] :
% 0.22/0.59 ( ( if_nat5617392847756311170nteger @ $false @ X @ Y4 )
% 0.22/0.59 = Y4 ) ).
% 0.22/0.59
% 0.22/0.59 thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_T,axiom,
% 0.22/0.59 ! [X: nat > code_integer > code_integer,Y4: nat > code_integer > code_integer] :
% 0.22/0.59 ( ( if_nat5617392847756311170nteger @ $true @ X @ Y4 )
% 0.22/0.59 = X ) ).
% 0.22/0.59
% 0.22/0.59 % Conjectures (1)
% 0.22/0.59 thf(conj_0,conjecture,
% 0.22/0.59 ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.22/0.59
% 0.22/0.59 %------------------------------------------------------------------------------
% 0.22/0.59 ------------------------
% 9.33/3.43 ---- Embedded in THF ---
% 9.33/3.43 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 9.33/3.43 %%% Generated on Fri Feb 27 14:50:12 EST 2026
% 9.33/3.43 %%% using '$$dhol' embedding, version 1.3.0.
% 9.33/3.43 %%% Logic specification used:
% 9.33/3.43 %%% thf(spec, logic, $$dhol).
% 9.33/3.43
% 9.33/3.43 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2DTF24167.p
% See solution above
% 9.33/3.47 ------------------------
% 12.76/6.88 ---- Cleaned THF ---
% 12.76/6.88 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,
% 12.76/6.88 set_Pr8693737435421807431at_nat: $tType ).
% 12.76/6.88
% 12.76/6.88 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,
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% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J_J,type,
% 12.76/6.88 set_fi4554929511873752355omplex: $tType ).
% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J_J,type,
% 12.76/6.88 set_fi7789364187291644575l_real: $tType ).
% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 12.76/6.88 filter6041513312241820739omplex: $tType ).
% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
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% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 12.76/6.88 option4927543243414619207at_nat: $tType ).
% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 12.76/6.88 set_Pr6218003697084177305l_real: $tType ).
% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 12.76/6.88 produc9072475918466114483BT_nat: $tType ).
% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 12.76/6.88 set_Pr1261947904930325089at_nat: $tType ).
% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 12.76/6.88 set_Pr958786334691620121nt_int: $tType ).
% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 12.76/6.88 produc4411394909380815293omplex: $tType ).
% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 12.76/6.88 produc6271795597528267376eger_o: $tType ).
% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 12.76/6.88 produc2422161461964618553l_real: $tType ).
% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 12.76/6.88 product_prod_num_num: $tType ).
% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 12.76/6.88 product_prod_nat_num: $tType ).
% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.76/6.88 product_prod_nat_nat: $tType ).
% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 12.76/6.88 product_prod_int_int: $tType ).
% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__List__Olist_It__List__Olist_It__Real__Oreal_J_J,type,
% 12.76/6.88 list_list_real: $tType ).
% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__Set__Oset_It__List__Olist_It__Real__Oreal_J_J,type,
% 12.76/6.88 set_list_real: $tType ).
% 12.76/6.88
% 12.76/6.88 thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
% 12.76/6.89 list_list_nat: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
% 12.76/6.89 list_list_int: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 12.76/6.89 list_VEBT_VEBT: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 12.76/6.89 set_list_nat: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
% 12.76/6.89 set_list_int: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 12.76/6.89 list_set_nat: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
% 12.76/6.89 set_VEBT_VEBT: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 12.76/6.89 set_set_nat: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
% 12.76/6.89 set_Code_integer: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__List__Olist_It__List__Olist_I_Eo_J_J,type,
% 12.76/6.89 list_list_o: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
% 12.76/6.89 list_complex: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
% 12.76/6.89 set_list_o: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
% 12.76/6.89 set_complex: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
% 12.76/6.89 filter_real: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
% 12.76/6.89 option_num: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
% 12.76/6.89 filter_nat: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Filter__Ofilter_It__Int__Oint_J,type,
% 12.76/6.89 filter_int: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
% 12.76/6.89 set_char: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
% 12.76/6.89 list_real: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 12.76/6.89 set_real: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
% 12.76/6.89 list_nat: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
% 12.76/6.89 list_int: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__VEBT____Definitions__OVEBT,type,
% 12.76/6.89 vEBT_VEBT: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
% 12.76/6.89 set_rat: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 12.76/6.89 set_nat: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
% 12.76/6.89 set_int: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Code____Numeral__Ointeger,type,
% 12.76/6.89 code_integer: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Extended____Nat__Oenat,type,
% 12.76/6.89 extended_enat: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__List__Olist_I_Eo_J,type,
% 12.76/6.89 list_o: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Complex__Ocomplex,type,
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% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Set__Oset_I_Eo_J,type,
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% 12.76/6.89 thf(ty_n_t__String__Ochar,type,
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% 12.76/6.89 thf(ty_n_t__Real__Oreal,type,
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% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Rat__Orat,type,
% 12.76/6.89 rat: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Num__Onum,type,
% 12.76/6.89 num: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Nat__Onat,type,
% 12.76/6.89 nat: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(ty_n_t__Int__Oint,type,
% 12.76/6.89 int: $tType ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
% 12.76/6.89 archim2889992004027027881ng_rat: rat > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
% 12.76/6.89 archim7802044766580827645g_real: real > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
% 12.76/6.89 archim3151403230148437115or_rat: rat > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
% 12.76/6.89 archimedean_frac_rat: rat > rat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
% 12.76/6.89 archim2898591450579166408c_real: real > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
% 12.76/6.89 archim8280529875227126926d_real: real > int ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 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 ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 bNF_re895249473297799549at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 bNF_re4695409256820837752l_real: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( real > real > real ) > $o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 bNF_re4521903465945308077real_o: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > $o ) > ( real > $o ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( real > real > $o ) > $o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 bNF_re3023117138289059399t_real: ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_Eo_001_Eo,type,
% 12.76/6.89 bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 12.76/6.89 bNF_re157797125943740599nt_int: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > product_prod_int_int ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Rat__Orat_J,type,
% 12.76/6.89 bNF_re3461391660133120880nt_rat: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > rat ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
% 12.76/6.89 bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
% 12.76/6.89 bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 12.76/6.89 bNF_re6250860962936578807nt_int: ( int > int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 12.76/6.89 bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 12.76/6.89 bNF_re4785983289428654063nt_int: ( nat > nat > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( nat > int > int ) > ( nat > int > int ) > $o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
% 12.76/6.89 bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint_001t__Int__Oint,type,
% 12.76/6.89 bNF_re6650684261131312217nt_int: ( nat > nat > $o ) > ( int > int > $o ) > ( nat > int ) > ( nat > int ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.76/6.89 bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 12.76/6.89 bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Int__Oint,type,
% 12.76/6.89 bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 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 ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 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 ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 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 ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 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 ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 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 ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 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 ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 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 ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 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 ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 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 ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 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 ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 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 ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 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 ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 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 ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Binomial_Obinomial,type,
% 12.76/6.89 binomial: nat > nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
% 12.76/6.89 gbinomial_complex: complex > nat > complex ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
% 12.76/6.89 gbinomial_int: int > nat > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
% 12.76/6.89 gbinomial_nat: nat > nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
% 12.76/6.89 gbinomial_rat: rat > nat > rat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
% 12.76/6.89 gbinomial_real: real > nat > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Bit__Operations_Oand__int__rel,type,
% 12.76/6.89 bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Bit__Operations_Oand__not__num,type,
% 12.76/6.89 bit_and_not_num: num > num > option_num ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
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% 12.76/6.89 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Code____Numeral__Ointeger,type,
% 12.76/6.89 semiri3624122377584611663nteger: nat > code_integer ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
% 12.76/6.89 semiri5044797733671781792omplex: nat > complex ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
% 12.76/6.89 semiri1406184849735516958ct_int: nat > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
% 12.76/6.89 semiri1408675320244567234ct_nat: nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
% 12.76/6.89 semiri773545260158071498ct_rat: nat > rat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
% 12.76/6.89 semiri2265585572941072030t_real: nat > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
% 12.76/6.89 invers8013647133539491842omplex: complex > complex ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
% 12.76/6.89 inverse_inverse_rat: rat > rat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
% 12.76/6.89 inverse_inverse_real: real > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
% 12.76/6.89 at_bot_real: filter_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Filter_Oat__top_001t__Int__Oint,type,
% 12.76/6.89 at_top_int: filter_int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
% 12.76/6.89 at_top_nat: filter_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
% 12.76/6.89 at_top_real: filter_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
% 12.76/6.89 eventually_nat: ( nat > $o ) > filter_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
% 12.76/6.89 eventually_real: ( real > $o ) > filter_real > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Int__Oint,type,
% 12.76/6.89 filterlim_nat_int: ( nat > int ) > filter_int > filter_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.76/6.89 filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
% 12.76/6.89 filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.76/6.89 filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Filter_Ofiltermap_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.76/6.89 filtermap_real_real: ( real > real ) > filter_real > filter_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 12.76/6.89 princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 12.76/6.89 princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Finite__Set_Ocard_001_Eo,type,
% 12.76/6.89 finite_card_o: set_o > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
% 12.76/6.89 finite_card_complex: set_complex > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
% 12.76/6.89 finite_card_int: set_int > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
% 12.76/6.89 finite_card_list_nat: set_list_nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
% 12.76/6.89 finite_card_nat: set_nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
% 12.76/6.89 finite_card_set_nat: set_set_nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Finite__Set_Ocard_001t__String__Ochar,type,
% 12.76/6.89 finite_card_char: set_char > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
% 12.76/6.89 finite3207457112153483333omplex: set_complex > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
% 12.76/6.89 finite_finite_int: set_int > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 12.76/6.89 finite_finite_nat: set_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 12.76/6.89 bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Obij__betw_001t__Int__Oint_001t__Nat__Onat,type,
% 12.76/6.89 bij_betw_int_nat: ( int > nat ) > set_int > set_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Obij__betw_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
% 12.76/6.89 bij_be8532844293280997160at_nat: ( list_nat > nat ) > set_list_nat > set_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 12.76/6.89 bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Int__Oint,type,
% 12.76/6.89 bij_betw_nat_int: ( nat > int ) > set_nat > set_int > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.76/6.89 bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Obij__betw_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 12.76/6.89 bij_be5333170631980326235at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > set_nat > $o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Num__Onum,type,
% 12.76/6.89 comp_int_int_num: ( int > int ) > ( num > int ) > num > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
% 12.76/6.89 comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.76/6.89 comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
% 12.76/6.89 comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Oid_001_Eo,type,
% 12.76/6.89 id_o: $o > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
% 12.76/6.89 id_nat: nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Nat__Onat,type,
% 12.76/6.89 inj_on_int_nat: ( int > nat ) > set_int > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
% 12.76/6.89 inj_on_list_nat_nat: ( list_nat > nat ) > set_list_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Int__Oint,type,
% 12.76/6.89 inj_on_nat_int: ( nat > int ) > set_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.76/6.89 inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
% 12.76/6.89 inj_on_nat_char: ( nat > char ) > set_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 12.76/6.89 inj_on2178005380612969504at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.76/6.89 inj_on_real_real: ( real > real ) > set_real > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
% 12.76/6.89 inj_on_set_nat_nat: ( set_nat > nat ) > set_set_nat > $o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 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 ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 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 ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 map_fu4826362097070443709at_o_o: ( int > product_prod_nat_nat ) > ( $o > $o ) > ( product_prod_nat_nat > $o ) > int > $o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 map_fu2345160673673942751at_nat: ( int > product_prod_nat_nat ) > ( nat > nat ) > ( product_prod_nat_nat > nat ) > int > nat ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 map_fu3667384564859982768at_int: ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int ) > ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 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 ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 map_fu898904425404107465nt_o_o: ( rat > product_prod_int_int ) > ( $o > $o ) > ( product_prod_int_int > $o ) > rat > $o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 map_fu5673905371560938248nt_rat: ( rat > product_prod_int_int ) > ( product_prod_int_int > rat ) > ( product_prod_int_int > product_prod_int_int ) > rat > rat ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 map_fu1532550112467129777l_real: ( real > nat > rat ) > ( ( ( nat > rat ) > nat > rat ) > real > real ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > real > real > real ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 map_fu7146612038024189824t_real: ( real > nat > rat ) > ( ( nat > rat ) > real ) > ( ( nat > rat ) > nat > rat ) > real > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Omap__fun_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_Eo_001_Eo,type,
% 12.76/6.89 map_fu1856342031159181835at_o_o: ( real > nat > rat ) > ( $o > $o ) > ( ( nat > rat ) > $o ) > real > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.76/6.89 the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun__Def_Ois__measure_001t__Int__Oint,type,
% 12.76/6.89 fun_is_measure_int: ( int > nat ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun__Def_Opair__leq,type,
% 12.76/6.89 fun_pair_leq: set_Pr8693737435421807431at_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Fun__Def_Opair__less,type,
% 12.76/6.89 fun_pair_less: set_Pr8693737435421807431at_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_GCD_OGcd__class_OGcd_001t__Int__Oint,type,
% 12.76/6.89 gcd_Gcd_int: set_int > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
% 12.76/6.89 gcd_Gcd_nat: set_nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_GCD_Obezw,type,
% 12.76/6.89 bezw: nat > nat > product_prod_int_int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_GCD_Obezw__rel,type,
% 12.76/6.89 bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Code____Numeral__Ointeger,type,
% 12.76/6.89 gcd_gcd_Code_integer: code_integer > code_integer > code_integer ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Int__Oint,type,
% 12.76/6.89 gcd_gcd_int: int > int > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Nat__Onat,type,
% 12.76/6.89 gcd_gcd_nat: nat > nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_GCD_Ogcd__nat__rel,type,
% 12.76/6.89 gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Groups_Oabs__class_Oabs_001t__Code____Numeral__Ointeger,type,
% 12.76/6.89 abs_abs_Code_integer: code_integer > code_integer ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex,type,
% 12.76/6.89 abs_abs_complex: complex > complex ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
% 12.76/6.89 abs_abs_int: int > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
% 12.76/6.89 abs_abs_rat: rat > rat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
% 12.76/6.89 abs_abs_real: real > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Ointeger,type,
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% 12.76/6.89
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
% 12.76/6.89 ring_1_Ints_real: set_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
% 12.76/6.89 ring_18347121197199848620nteger: int > code_integer ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
% 12.76/6.89 ring_17405671764205052669omplex: int > complex ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
% 12.76/6.89 ring_1_of_int_int: int > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
% 12.76/6.89 ring_1_of_int_rat: int > rat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
% 12.76/6.89 ring_1_of_int_real: int > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Extended____Nat__Oenat,type,
% 12.76/6.89 inf_in1870772243966228564d_enat: extended_enat > extended_enat > extended_enat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
% 12.76/6.89 inf_inf_int: int > int > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
% 12.76/6.89 inf_inf_nat: nat > nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
% 12.76/6.89 semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Lattices_Osup__class_Osup_001t__Extended____Nat__Oenat,type,
% 12.76/6.89 sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
% 12.76/6.89 sup_sup_int: int > int > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
% 12.76/6.89 sup_sup_nat: nat > nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
% 12.76/6.89 sup_sup_set_nat: set_nat > set_nat > set_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 12.76/6.89 sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
% 12.76/6.89 lattic8263393255366662781ax_int: set_int > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
% 12.76/6.89 lattic8265883725875713057ax_nat: set_nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
% 12.76/6.89 bfun_nat_real: ( nat > real ) > filter_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
% 12.76/6.89 at_infinity_real: filter_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Oappend_001t__Int__Oint,type,
% 12.76/6.89 append_int: list_int > list_int > list_int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Oappend_001t__Nat__Onat,type,
% 12.76/6.89 append_nat: list_nat > list_nat > list_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Oconcat_001_Eo,type,
% 12.76/6.89 concat_o: list_list_o > list_o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Oconcat_001t__Int__Oint,type,
% 12.76/6.89 concat_int: list_list_int > list_int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
% 12.76/6.89 concat_nat: list_list_nat > list_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Oconcat_001t__Real__Oreal,type,
% 12.76/6.89 concat_real: list_list_real > list_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Ofoldr_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.76/6.89 foldr_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.76/6.89 foldr_real_real: ( real > real > real ) > list_real > real > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olast_001t__Nat__Onat,type,
% 12.76/6.89 last_nat: list_nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
% 12.76/6.89 linord2614967742042102400et_nat: set_nat > list_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
% 12.76/6.89 cons_int: int > list_int > list_int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
% 12.76/6.89 cons_nat: nat > list_nat > list_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
% 12.76/6.89 nil_int: list_int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
% 12.76/6.89 nil_nat: list_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__Real__Oreal,type,
% 12.76/6.89 map_complex_real: ( complex > real ) > list_complex > list_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Real__Oreal,type,
% 12.76/6.89 map_int_real: ( int > real ) > list_int > list_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.76/6.89 map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Real__Oreal,type,
% 12.76/6.89 map_nat_real: ( nat > real ) > list_nat > list_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.76/6.89 map_real_real: ( real > real ) > list_real > list_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Nat__Onat_J_001t__Real__Oreal,type,
% 12.76/6.89 map_set_nat_real: ( set_nat > real ) > list_set_nat > list_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 12.76/6.89 map_VEBT_VEBT_nat: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > list_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
% 12.76/6.89 map_VEBT_VEBT_real: ( vEBT_VEBT > real ) > list_VEBT_VEBT > list_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Oset_001_Eo,type,
% 12.76/6.89 set_o2: list_o > set_o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
% 12.76/6.89 set_complex2: list_complex > set_complex ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
% 12.76/6.89 set_int2: list_int > set_int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Oset_001t__List__Olist_I_Eo_J,type,
% 12.76/6.89 set_list_o2: list_list_o > set_list_o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Int__Oint_J,type,
% 12.76/6.89 set_list_int2: list_list_int > set_list_int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
% 12.76/6.89 set_list_nat2: list_list_nat > set_list_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Real__Oreal_J,type,
% 12.76/6.89 set_list_real2: list_list_real > set_list_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
% 12.76/6.89 set_nat2: list_nat > set_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
% 12.76/6.89 set_real2: list_real > set_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
% 12.76/6.89 set_set_nat2: list_set_nat > set_set_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
% 12.76/6.89 set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
% 12.76/6.89 size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
% 12.76/6.89 tl_nat: list_nat > list_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Onth_001_Eo,type,
% 12.76/6.89 nth_o: list_o > nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
% 12.76/6.89 nth_complex: list_complex > nat > complex ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Onth_001t__Int__Oint,type,
% 12.76/6.89 nth_int: list_int > nat > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 12.76/6.89 nth_nat: list_nat > nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Onth_001t__Real__Oreal,type,
% 12.76/6.89 nth_real: list_real > nat > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
% 12.76/6.89 nth_set_nat: list_set_nat > nat > set_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
% 12.76/6.89 nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Oreplicate_001t__Complex__Ocomplex,type,
% 12.76/6.89 replicate_complex: nat > complex > list_complex ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
% 12.76/6.89 replicate_int: nat > int > list_int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
% 12.76/6.89 replicate_nat: nat > nat > list_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
% 12.76/6.89 replicate_real: nat > real > list_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
% 12.76/6.89 replicate_set_nat: nat > set_nat > list_set_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 12.76/6.89 replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
% 12.76/6.89 sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
% 12.76/6.89 sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Osubseqs_001_Eo,type,
% 12.76/6.89 subseqs_o: list_o > list_list_o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Osubseqs_001t__Int__Oint,type,
% 12.76/6.89 subseqs_int: list_int > list_list_int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
% 12.76/6.89 subseqs_nat: list_nat > list_list_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Osubseqs_001t__Real__Oreal,type,
% 12.76/6.89 subseqs_real: list_real > list_list_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Otake_001t__Nat__Onat,type,
% 12.76/6.89 take_nat: nat > list_nat > list_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Oupt,type,
% 12.76/6.89 upt: nat > nat > list_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Oupto,type,
% 12.76/6.89 upto: int > int > list_int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Oupto__aux,type,
% 12.76/6.89 upto_aux: int > int > list_int > list_int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_List_Oupto__rel,type,
% 12.76/6.89 upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_OSuc,type,
% 12.76/6.89 suc: nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.76/6.89 compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 12.76/6.89 case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 12.76/6.89 case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
% 12.76/6.89 case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Onat_Opred,type,
% 12.76/6.89 pred: nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Complex__Ocomplex,type,
% 12.76/6.89 semiri3842193898606819883omplex: set_complex ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
% 12.76/6.89 semiring_1_Nats_int: set_int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
% 12.76/6.89 semiri4939895301339042750nteger: nat > code_integer ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
% 12.76/6.89 semiri8010041392384452111omplex: nat > complex ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
% 12.76/6.89 semiri4216267220026989637d_enat: nat > extended_enat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 12.76/6.89 semiri1314217659103216013at_int: nat > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 12.76/6.89 semiri1316708129612266289at_nat: nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
% 12.76/6.89 semiri681578069525770553at_rat: nat > rat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 12.76/6.89 semiri5074537144036343181t_real: nat > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
% 12.76/6.89 size_size_list_o: list_o > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 12.76/6.89 size_s3451745648224563538omplex: list_complex > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
% 12.76/6.89 size_size_list_int: list_int > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
% 12.76/6.89 size_s2710708370519433104list_o: list_list_o > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
% 12.76/6.89 size_s533118279054570080st_int: list_list_int > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
% 12.76/6.89 size_s3023201423986296836st_nat: list_list_nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Real__Oreal_J_J,type,
% 12.76/6.89 size_s6660260683639930848t_real: list_list_real > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 12.76/6.89 size_size_list_nat: list_nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
% 12.76/6.89 size_size_list_real: list_real > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 12.76/6.89 size_s3254054031482475050et_nat: list_set_nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 12.76/6.89 size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
% 12.76/6.89 size_size_num: num > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
% 12.76/6.89 size_size_char: char > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
% 12.76/6.89 size_size_VEBT_VEBT: vEBT_VEBT > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat__Bijection_Oint__decode,type,
% 12.76/6.89 nat_int_decode: nat > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat__Bijection_Oint__encode,type,
% 12.76/6.89 nat_int_encode: int > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat__Bijection_Olist__encode,type,
% 12.76/6.89 nat_list_encode: list_nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
% 12.76/6.89 nat_list_encode_rel: list_nat > list_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 12.76/6.89 nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 12.76/6.89 nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 12.76/6.89 nat_prod_encode: product_prod_nat_nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat__Bijection_Oset__decode,type,
% 12.76/6.89 nat_set_decode: nat > set_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat__Bijection_Oset__encode,type,
% 12.76/6.89 nat_set_encode: set_nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Nat__Bijection_Otriangle,type,
% 12.76/6.89 nat_triangle: nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_NthRoot_Oroot,type,
% 12.76/6.89 root: nat > real > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_NthRoot_Osqrt,type,
% 12.76/6.89 sqrt: real > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_OBitM,type,
% 12.76/6.89 bitM: num > num ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Oinc,type,
% 12.76/6.89 inc: num > num ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
% 12.76/6.89 neg_nu8804712462038260780nteger: code_integer > code_integer ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
% 12.76/6.89 neg_nu7009210354673126013omplex: complex > complex ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
% 12.76/6.89 neg_numeral_dbl_int: int > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
% 12.76/6.89 neg_numeral_dbl_rat: rat > rat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
% 12.76/6.89 neg_numeral_dbl_real: real > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
% 12.76/6.89 neg_nu7757733837767384882nteger: code_integer > code_integer ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
% 12.76/6.89 neg_nu6511756317524482435omplex: complex > complex ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
% 12.76/6.89 neg_nu3811975205180677377ec_int: int > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
% 12.76/6.89 neg_nu3179335615603231917ec_rat: rat > rat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
% 12.76/6.89 neg_nu6075765906172075777c_real: real > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
% 12.76/6.89 neg_nu5831290666863070958nteger: code_integer > code_integer ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
% 12.76/6.89 neg_nu8557863876264182079omplex: complex > complex ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
% 12.76/6.89 neg_nu5851722552734809277nc_int: int > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
% 12.76/6.89 neg_nu5219082963157363817nc_rat: rat > rat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
% 12.76/6.89 neg_nu8295874005876285629c_real: real > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
% 12.76/6.89 neg_numeral_sub_int: num > num > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Onum_OBit0,type,
% 12.76/6.89 bit0: num > num ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Onum_OBit1,type,
% 12.76/6.89 bit1: num > num ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Onum_OOne,type,
% 12.76/6.89 one: num ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
% 12.76/6.89 case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Onum_Osize__num,type,
% 12.76/6.89 size_num: num > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Onum__of__nat,type,
% 12.76/6.89 num_of_nat: nat > num ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
% 12.76/6.89 numera6620942414471956472nteger: num > code_integer ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
% 12.76/6.89 numera6690914467698888265omplex: num > complex ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
% 12.76/6.89 numera1916890842035813515d_enat: num > extended_enat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
% 12.76/6.89 numeral_numeral_int: num > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
% 12.76/6.89 numeral_numeral_nat: num > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
% 12.76/6.89 numeral_numeral_rat: num > rat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
% 12.76/6.89 numeral_numeral_real: num > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Opow,type,
% 12.76/6.89 pow: num > num > num ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Opred__numeral,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Num_Osqr,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
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% 12.76/6.89 thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
% 12.76/6.89 some_num: num > option_num ).
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% 12.76/6.89 thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.76/6.89 some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
% 12.76/6.89
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
% 12.76/6.89 ord_Least_nat: ( nat > $o ) > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oord__class_OLeast_001t__Real__Oreal,type,
% 12.76/6.89 ord_Least_real: ( real > $o ) > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
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% 12.76/6.89 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
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% 12.76/6.89 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
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% 12.76/6.89 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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% 12.76/6.89 thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Real__Oreal,type,
% 12.76/6.89 order_mono_nat_real: ( nat > real ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oorder__class_Omono_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.76/6.89 order_mono_real_real: ( real > real ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.76/6.89 order_5726023648592871131at_nat: ( nat > nat ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
% 12.76/6.89 order_7092887310737990675l_real: ( real > real ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
% 12.76/6.89 ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
% 12.76/6.89 top_top_set_o: set_o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 12.76/6.89 top_top_set_list_nat: set_list_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
% 12.76/6.89 top_top_set_nat: set_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
% 12.76/6.89 top_top_set_real: set_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
% 12.76/6.89 power_power_complex: complex > nat > complex ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
% 12.76/6.89 power_power_int: int > nat > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
% 12.76/6.89 power_power_nat: nat > nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
% 12.76/6.89 power_power_rat: rat > nat > rat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
% 12.76/6.89 power_power_real: real > nat > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
% 12.76/6.89 produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 12.76/6.89 produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
% 12.76/6.89 product_Pair_int_int: int > int > product_prod_int_int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.76/6.89 product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
% 12.76/6.89 product_Pair_nat_num: nat > num > product_prod_nat_num ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
% 12.76/6.89 product_Pair_num_num: num > num > product_prod_num_num ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 12.76/6.89 produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.76/6.89 produc457027306803732586at_nat: set_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 12.76/6.89 produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
% 12.76/6.89 produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
% 12.76/6.89 produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
% 12.76/6.89 produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
% 12.76/6.89 produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
% 12.76/6.89 produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
% 12.76/6.89 produc8211389475949308722nt_int: ( int > int > int ) > product_prod_int_int > int ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
% 12.76/6.89 produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.76/6.89 produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
% 12.76/6.89 produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
% 12.76/6.89 produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
% 12.76/6.89 product_fst_int_int: product_prod_int_int > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.76/6.89 product_fst_nat_nat: product_prod_nat_nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
% 12.76/6.89 product_snd_int_int: product_prod_int_int > int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
% 12.76/6.89 product_snd_nat_nat: product_prod_nat_nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rat_OAbs__Rat,type,
% 12.76/6.89 abs_Rat: product_prod_int_int > rat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rat_OFract,type,
% 12.76/6.89 fract: int > int > rat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rat_OFrct,type,
% 12.76/6.89 frct: product_prod_int_int > rat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rat_ORep__Rat,type,
% 12.76/6.89 rep_Rat: rat > product_prod_int_int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
% 12.76/6.89 field_5140801741446780682s_real: set_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
% 12.76/6.89 field_7254667332652039916t_real: rat > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rat_Onormalize,type,
% 12.76/6.89 normalize: product_prod_int_int > product_prod_int_int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rat_Opcr__rat,type,
% 12.76/6.89 pcr_rat: product_prod_int_int > rat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rat_Opositive,type,
% 12.76/6.89 positive: rat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rat_Oquotient__of,type,
% 12.76/6.89 quotient_of: rat > product_prod_int_int ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rat_Oratrel,type,
% 12.76/6.89 ratrel: product_prod_int_int > product_prod_int_int > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Real_OReal,type,
% 12.76/6.89 real2: ( nat > rat ) > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Real_Ocauchy,type,
% 12.76/6.89 cauchy: ( nat > rat ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Real_Ocr__real,type,
% 12.76/6.89 cr_real: ( nat > rat ) > real > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Real_Opcr__real,type,
% 12.76/6.89 pcr_real: ( nat > rat ) > real > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Real_Opositive,type,
% 12.76/6.89 positive2: real > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Real_Orealrel,type,
% 12.76/6.89 realrel: ( nat > rat ) > ( nat > rat ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Real_Orep__real,type,
% 12.76/6.89 rep_real: real > nat > rat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Real_Ovanishes,type,
% 12.76/6.89 vanishes: ( nat > rat ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
% 12.76/6.89 real_V2521375963428798218omplex: set_complex ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Real__Vector__Spaces_OReals_001t__Real__Oreal,type,
% 12.76/6.89 real_V470468836141973256s_real: set_real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
% 12.76/6.89 real_V3694042436643373181omplex: complex > complex > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
% 12.76/6.89 algebr932160517623751201me_int: int > int > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
% 12.76/6.89 algebr934650988132801477me_nat: nat > nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
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% 12.76/6.89 thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
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% 12.76/6.89 thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
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% 12.76/6.89
% 12.76/6.89 thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
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% 12.76/6.89 thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
% 12.76/6.89 tanh_real: real > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Transfer_Obi__total_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 12.76/6.89 bi_tot896582865486249351at_int: ( product_prod_nat_nat > int > $o ) > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
% 12.76/6.89 transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
% 12.76/6.89 vEBT_Leaf: $o > $o > vEBT_VEBT ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
% 12.76/6.89 vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
% 12.76/6.89 vEBT_size_VEBT: vEBT_VEBT > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
% 12.76/6.89 vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
% 12.76/6.89 vEBT_VEBT_high: nat > nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
% 12.76/6.89 vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
% 12.76/6.89 vEBT_VEBT_low: nat > nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
% 12.76/6.89 vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
% 12.76/6.89 vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
% 12.76/6.89 vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
% 12.76/6.89 vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
% 12.76/6.89 vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
% 12.76/6.89 vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
% 12.76/6.89 vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Definitions_Oset__vebt,type,
% 12.76/6.89 vEBT_set_vebt: vEBT_VEBT > set_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
% 12.76/6.89 vEBT_vebt_buildup: nat > vEBT_VEBT ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
% 12.76/6.89 vEBT_v4011308405150292612up_rel: nat > nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
% 12.76/6.89 vEBT_V8646137997579335489_i_l_d: nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
% 12.76/6.89 vEBT_V8346862874174094_d_u_p: nat > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
% 12.76/6.89 vEBT_V1247956027447740395_p_rel: nat > nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
% 12.76/6.89 vEBT_V5144397997797733112_d_rel: nat > nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
% 12.76/6.89 vEBT_VEBT_cnt: vEBT_VEBT > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
% 12.76/6.89 vEBT_VEBT_cnt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
% 12.76/6.89 vEBT_VEBT_space: vEBT_VEBT > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
% 12.76/6.89 vEBT_VEBT_space2: vEBT_VEBT > nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
% 12.76/6.89 vEBT_VEBT_space_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
% 12.76/6.89 vEBT_VEBT_space_rel2: vEBT_VEBT > vEBT_VEBT > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
% 12.76/6.89 accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
% 12.76/6.89 accp_nat: ( nat > nat > $o ) > nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 12.76/6.89 accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.76/6.89 accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 12.76/6.89 accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 12.76/6.89 accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
% 12.76/6.89 accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_Wellfounded_Opred__nat,type,
% 12.76/6.89 pred_nat: set_Pr1261947904930325089at_nat ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_fChoice_001t__Real__Oreal,type,
% 12.76/6.89 fChoice_real: ( real > $o ) > real ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_member_001_Eo,type,
% 12.76/6.89 member_o: $o > set_o > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_member_001t__Code____Numeral__Ointeger,type,
% 12.76/6.89 member_Code_integer: code_integer > set_Code_integer > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_member_001t__Complex__Ocomplex,type,
% 12.76/6.89 member_complex: complex > set_complex > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_member_001t__Int__Oint,type,
% 12.76/6.89 member_int: int > set_int > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
% 12.76/6.89 member_list_o: list_o > set_list_o > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
% 12.76/6.89 member_list_int: list_int > set_list_int > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
% 12.76/6.89 member_list_nat: list_nat > set_list_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_member_001t__List__Olist_It__Real__Oreal_J,type,
% 12.76/6.89 member_list_real: list_real > set_list_real > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_member_001t__Nat__Onat,type,
% 12.76/6.89 member_nat: nat > set_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 12.76/6.89 member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
% 12.76/6.89
% 12.76/6.89 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,
% 12.76/6.89 member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_member_001t__Rat__Orat,type,
% 12.76/6.89 member_rat: rat > set_rat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_member_001t__Real__Oreal,type,
% 12.76/6.89 member_real: real > set_real > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 12.76/6.89 member_set_nat: set_nat > set_set_nat > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
% 12.76/6.89 member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
% 12.76/6.89
% 12.76/6.89 thf(sy_v_va____,type,
% 12.76/6.89 va: nat ).
% 12.76/6.89
% 12.76/6.89 thf(fact_0_False,axiom,
% 12.76/6.89 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_1__C3_OIH_C_I3_J,axiom,
% 12.76/6.89 ! [X: nat] :
% 12.76/6.89 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
% 12.76/6.89 => ( ( X
% 12.76/6.89 = ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.76/6.89 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ X ) ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ X ) ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_2__C3_OIH_C_I4_J,axiom,
% 12.76/6.89 ! [X: nat] :
% 12.76/6.89 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
% 12.76/6.89 => ( ( X
% 12.76/6.89 = ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.76/6.89 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ X ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ X ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_3__C3_OIH_C_I1_J,axiom,
% 12.76/6.89 ! [X: nat] :
% 12.76/6.89 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
% 12.76/6.89 => ( ( X
% 12.76/6.89 = ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.76/6.89 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ X ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ X ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_4_space__cnt,axiom,
% 12.76/6.89 ! [T: vEBT_VEBT] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_space2 @ T ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_cnt @ T ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_5_div2__Suc__Suc,axiom,
% 12.76/6.89 ! [M: nat] :
% 12.76/6.89 ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.76/6.89 = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_6_divide__le__eq__numeral1_I1_J,axiom,
% 12.76/6.89 ! [B: real,W: num,A: real] :
% 12.76/6.89 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 12.76/6.89 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_7_divide__le__eq__numeral1_I1_J,axiom,
% 12.76/6.89 ! [B: rat,W: num,A: rat] :
% 12.76/6.89 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 12.76/6.89 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_8_le__divide__eq__numeral1_I1_J,axiom,
% 12.76/6.89 ! [A: real,B: real,W: num] :
% 12.76/6.89 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 12.76/6.89 = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_9_le__divide__eq__numeral1_I1_J,axiom,
% 12.76/6.89 ! [A: rat,B: rat,W: num] :
% 12.76/6.89 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 12.76/6.89 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_10_of__nat__numeral,axiom,
% 12.76/6.89 ! [N: num] :
% 12.76/6.89 ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
% 12.76/6.89 = ( numera6690914467698888265omplex @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_11_of__nat__numeral,axiom,
% 12.76/6.89 ! [N: num] :
% 12.76/6.89 ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
% 12.76/6.89 = ( numeral_numeral_real @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_12_of__nat__numeral,axiom,
% 12.76/6.89 ! [N: num] :
% 12.76/6.89 ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
% 12.76/6.89 = ( numeral_numeral_rat @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_13_of__nat__numeral,axiom,
% 12.76/6.89 ! [N: num] :
% 12.76/6.89 ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
% 12.76/6.89 = ( numeral_numeral_nat @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_14_of__nat__numeral,axiom,
% 12.76/6.89 ! [N: num] :
% 12.76/6.89 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 12.76/6.89 = ( numeral_numeral_int @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_15_semiring__norm_I86_J,axiom,
% 12.76/6.89 ! [M: num] :
% 12.76/6.89 ( ( bit1 @ M )
% 12.76/6.89 != one ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_16_semiring__norm_I84_J,axiom,
% 12.76/6.89 ! [N: num] :
% 12.76/6.89 ( one
% 12.76/6.89 != ( bit1 @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_17_semiring__norm_I89_J,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( bit1 @ M )
% 12.76/6.89 != ( bit0 @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_18_semiring__norm_I88_J,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( bit0 @ M )
% 12.76/6.89 != ( bit1 @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_19_of__nat__mult,axiom,
% 12.76/6.89 ! [M: nat,N: nat] :
% 12.76/6.89 ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
% 12.76/6.89 = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_20_of__nat__mult,axiom,
% 12.76/6.89 ! [M: nat,N: nat] :
% 12.76/6.89 ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
% 12.76/6.89 = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_21_of__nat__mult,axiom,
% 12.76/6.89 ! [M: nat,N: nat] :
% 12.76/6.89 ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
% 12.76/6.89 = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_22_of__nat__mult,axiom,
% 12.76/6.89 ! [M: nat,N: nat] :
% 12.76/6.89 ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
% 12.76/6.89 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_23_of__nat__mult,axiom,
% 12.76/6.89 ! [M: nat,N: nat] :
% 12.76/6.89 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
% 12.76/6.89 = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_24_numeral__Bit1__div__2,axiom,
% 12.76/6.89 ! [N: num] :
% 12.76/6.89 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.76/6.89 = ( numeral_numeral_nat @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_25_numeral__Bit1__div__2,axiom,
% 12.76/6.89 ! [N: num] :
% 12.76/6.89 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.76/6.89 = ( numeral_numeral_int @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_26_of__nat__le__iff,axiom,
% 12.76/6.89 ! [M: nat,N: nat] :
% 12.76/6.89 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 12.76/6.89 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_27_of__nat__le__iff,axiom,
% 12.76/6.89 ! [M: nat,N: nat] :
% 12.76/6.89 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 12.76/6.89 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_28_of__nat__le__iff,axiom,
% 12.76/6.89 ! [M: nat,N: nat] :
% 12.76/6.89 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 12.76/6.89 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_29_of__nat__le__iff,axiom,
% 12.76/6.89 ! [M: nat,N: nat] :
% 12.76/6.89 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 12.76/6.89 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_30_numeral__eq__iff,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( ( numera6690914467698888265omplex @ M )
% 12.76/6.89 = ( numera6690914467698888265omplex @ N ) )
% 12.76/6.89 = ( M = N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_31_numeral__eq__iff,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( ( numeral_numeral_real @ M )
% 12.76/6.89 = ( numeral_numeral_real @ N ) )
% 12.76/6.89 = ( M = N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_32_numeral__eq__iff,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( ( numeral_numeral_rat @ M )
% 12.76/6.89 = ( numeral_numeral_rat @ N ) )
% 12.76/6.89 = ( M = N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_33_numeral__eq__iff,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( ( numeral_numeral_nat @ M )
% 12.76/6.89 = ( numeral_numeral_nat @ N ) )
% 12.76/6.89 = ( M = N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_34_numeral__eq__iff,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( ( numeral_numeral_int @ M )
% 12.76/6.89 = ( numeral_numeral_int @ N ) )
% 12.76/6.89 = ( M = N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_35_semiring__norm_I13_J,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.76/6.89 = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_36_semiring__norm_I71_J,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.76/6.89 = ( ord_less_eq_num @ M @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_37_semiring__norm_I87_J,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( ( bit0 @ M )
% 12.76/6.89 = ( bit0 @ N ) )
% 12.76/6.89 = ( M = N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_38_semiring__norm_I11_J,axiom,
% 12.76/6.89 ! [M: num] :
% 12.76/6.89 ( ( times_times_num @ M @ one )
% 12.76/6.89 = M ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_39_semiring__norm_I12_J,axiom,
% 12.76/6.89 ! [N: num] :
% 12.76/6.89 ( ( times_times_num @ one @ N )
% 12.76/6.89 = N ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_40_semiring__norm_I68_J,axiom,
% 12.76/6.89 ! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_41_old_Onat_Oinject,axiom,
% 12.76/6.89 ! [Nat: nat,Nat2: nat] :
% 12.76/6.89 ( ( ( suc @ Nat )
% 12.76/6.89 = ( suc @ Nat2 ) )
% 12.76/6.89 = ( Nat = Nat2 ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_42_nat_Oinject,axiom,
% 12.76/6.89 ! [X2: nat,Y2: nat] :
% 12.76/6.89 ( ( ( suc @ X2 )
% 12.76/6.89 = ( suc @ Y2 ) )
% 12.76/6.89 = ( X2 = Y2 ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_43_of__nat__eq__iff,axiom,
% 12.76/6.89 ! [M: nat,N: nat] :
% 12.76/6.89 ( ( ( semiri8010041392384452111omplex @ M )
% 12.76/6.89 = ( semiri8010041392384452111omplex @ N ) )
% 12.76/6.89 = ( M = N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_44_of__nat__eq__iff,axiom,
% 12.76/6.89 ! [M: nat,N: nat] :
% 12.76/6.89 ( ( ( semiri5074537144036343181t_real @ M )
% 12.76/6.89 = ( semiri5074537144036343181t_real @ N ) )
% 12.76/6.89 = ( M = N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_45_of__nat__eq__iff,axiom,
% 12.76/6.89 ! [M: nat,N: nat] :
% 12.76/6.89 ( ( ( semiri681578069525770553at_rat @ M )
% 12.76/6.89 = ( semiri681578069525770553at_rat @ N ) )
% 12.76/6.89 = ( M = N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_46_of__nat__eq__iff,axiom,
% 12.76/6.89 ! [M: nat,N: nat] :
% 12.76/6.89 ( ( ( semiri1316708129612266289at_nat @ M )
% 12.76/6.89 = ( semiri1316708129612266289at_nat @ N ) )
% 12.76/6.89 = ( M = N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_47_of__nat__eq__iff,axiom,
% 12.76/6.89 ! [M: nat,N: nat] :
% 12.76/6.89 ( ( ( semiri1314217659103216013at_int @ M )
% 12.76/6.89 = ( semiri1314217659103216013at_int @ N ) )
% 12.76/6.89 = ( M = N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_48_semiring__norm_I73_J,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.76/6.89 = ( ord_less_eq_num @ M @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_49_semiring__norm_I90_J,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( ( bit1 @ M )
% 12.76/6.89 = ( bit1 @ N ) )
% 12.76/6.89 = ( M = N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_50_numeral__le__iff,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 12.76/6.89 = ( ord_less_eq_num @ M @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_51_numeral__le__iff,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 12.76/6.89 = ( ord_less_eq_num @ M @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_52_numeral__le__iff,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 12.76/6.89 = ( ord_less_eq_num @ M @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_53_numeral__le__iff,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 12.76/6.89 = ( ord_less_eq_num @ M @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_54_mult__numeral__left__semiring__numeral,axiom,
% 12.76/6.89 ! [V: num,W: num,Z: complex] :
% 12.76/6.89 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 12.76/6.89 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_55_mult__numeral__left__semiring__numeral,axiom,
% 12.76/6.89 ! [V: num,W: num,Z: real] :
% 12.76/6.89 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 12.76/6.89 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_56_mult__numeral__left__semiring__numeral,axiom,
% 12.76/6.89 ! [V: num,W: num,Z: rat] :
% 12.76/6.89 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 12.76/6.89 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_57_mult__numeral__left__semiring__numeral,axiom,
% 12.76/6.89 ! [V: num,W: num,Z: nat] :
% 12.76/6.89 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 12.76/6.89 = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_58_mult__numeral__left__semiring__numeral,axiom,
% 12.76/6.89 ! [V: num,W: num,Z: int] :
% 12.76/6.89 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 12.76/6.89 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_59_numeral__times__numeral,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 12.76/6.89 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_60_numeral__times__numeral,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 12.76/6.89 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_61_numeral__times__numeral,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 12.76/6.89 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_62_numeral__times__numeral,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 12.76/6.89 = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_63_numeral__times__numeral,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 12.76/6.89 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_64_num__double,axiom,
% 12.76/6.89 ! [N: num] :
% 12.76/6.89 ( ( times_times_num @ ( bit0 @ one ) @ N )
% 12.76/6.89 = ( bit0 @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_65_semiring__norm_I69_J,axiom,
% 12.76/6.89 ! [M: num] :
% 12.76/6.89 ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_66_semiring__norm_I83_J,axiom,
% 12.76/6.89 ! [N: num] :
% 12.76/6.89 ( one
% 12.76/6.89 != ( bit0 @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_67_semiring__norm_I85_J,axiom,
% 12.76/6.89 ! [M: num] :
% 12.76/6.89 ( ( bit0 @ M )
% 12.76/6.89 != one ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_68_semiring__norm_I14_J,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.76/6.89 = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_69_semiring__norm_I15_J,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.76/6.89 = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_70_semiring__norm_I72_J,axiom,
% 12.76/6.89 ! [M: num,N: num] :
% 12.76/6.89 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.76/6.89 = ( ord_less_eq_num @ M @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_71_semiring__norm_I70_J,axiom,
% 12.76/6.89 ! [M: num] :
% 12.76/6.89 ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_72_Suc__le__mono,axiom,
% 12.76/6.89 ! [N: nat,M: nat] :
% 12.76/6.89 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 12.76/6.89 = ( ord_less_eq_nat @ N @ M ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_73_le__cube,axiom,
% 12.76/6.89 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_74_le__refl,axiom,
% 12.76/6.89 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_75_le__trans,axiom,
% 12.76/6.89 ! [I: nat,J: nat,K: nat] :
% 12.76/6.89 ( ( ord_less_eq_nat @ I @ J )
% 12.76/6.89 => ( ( ord_less_eq_nat @ J @ K )
% 12.76/6.89 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_76_mem__Collect__eq,axiom,
% 12.76/6.89 ! [A: complex,P: complex > $o] :
% 12.76/6.89 ( ( member_complex @ A @ ( collect_complex @ P ) )
% 12.76/6.89 = ( P @ A ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_77_mem__Collect__eq,axiom,
% 12.76/6.89 ! [A: real,P: real > $o] :
% 12.76/6.89 ( ( member_real @ A @ ( collect_real @ P ) )
% 12.76/6.89 = ( P @ A ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_78_mem__Collect__eq,axiom,
% 12.76/6.89 ! [A: list_nat,P: list_nat > $o] :
% 12.76/6.89 ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
% 12.76/6.89 = ( P @ A ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_79_mem__Collect__eq,axiom,
% 12.76/6.89 ! [A: set_nat,P: set_nat > $o] :
% 12.76/6.89 ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
% 12.76/6.89 = ( P @ A ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_80_mem__Collect__eq,axiom,
% 12.76/6.89 ! [A: nat,P: nat > $o] :
% 12.76/6.89 ( ( member_nat @ A @ ( collect_nat @ P ) )
% 12.76/6.89 = ( P @ A ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_81_mem__Collect__eq,axiom,
% 12.76/6.89 ! [A: int,P: int > $o] :
% 12.76/6.89 ( ( member_int @ A @ ( collect_int @ P ) )
% 12.76/6.89 = ( P @ A ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_82_Collect__mem__eq,axiom,
% 12.76/6.89 ! [A2: set_complex] :
% 12.76/6.89 ( ( collect_complex
% 12.76/6.89 @ ^ [X3: complex] : ( member_complex @ X3 @ A2 ) )
% 12.76/6.89 = A2 ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_83_Collect__mem__eq,axiom,
% 12.76/6.89 ! [A2: set_real] :
% 12.76/6.89 ( ( collect_real
% 12.76/6.89 @ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
% 12.76/6.89 = A2 ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_84_Collect__mem__eq,axiom,
% 12.76/6.89 ! [A2: set_list_nat] :
% 12.76/6.89 ( ( collect_list_nat
% 12.76/6.89 @ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A2 ) )
% 12.76/6.89 = A2 ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_85_Collect__mem__eq,axiom,
% 12.76/6.89 ! [A2: set_set_nat] :
% 12.76/6.89 ( ( collect_set_nat
% 12.76/6.89 @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
% 12.76/6.89 = A2 ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_86_Collect__mem__eq,axiom,
% 12.76/6.89 ! [A2: set_nat] :
% 12.76/6.89 ( ( collect_nat
% 12.76/6.89 @ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
% 12.76/6.89 = A2 ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_87_Collect__mem__eq,axiom,
% 12.76/6.89 ! [A2: set_int] :
% 12.76/6.89 ( ( collect_int
% 12.76/6.89 @ ^ [X3: int] : ( member_int @ X3 @ A2 ) )
% 12.76/6.89 = A2 ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_88_Collect__cong,axiom,
% 12.76/6.89 ! [P: real > $o,Q: real > $o] :
% 12.76/6.89 ( ! [X4: real] :
% 12.76/6.89 ( ( P @ X4 )
% 12.76/6.89 = ( Q @ X4 ) )
% 12.76/6.89 => ( ( collect_real @ P )
% 12.76/6.89 = ( collect_real @ Q ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_89_Collect__cong,axiom,
% 12.76/6.89 ! [P: list_nat > $o,Q: list_nat > $o] :
% 12.76/6.89 ( ! [X4: list_nat] :
% 12.76/6.89 ( ( P @ X4 )
% 12.76/6.89 = ( Q @ X4 ) )
% 12.76/6.89 => ( ( collect_list_nat @ P )
% 12.76/6.89 = ( collect_list_nat @ Q ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_90_Collect__cong,axiom,
% 12.76/6.89 ! [P: set_nat > $o,Q: set_nat > $o] :
% 12.76/6.89 ( ! [X4: set_nat] :
% 12.76/6.89 ( ( P @ X4 )
% 12.76/6.89 = ( Q @ X4 ) )
% 12.76/6.89 => ( ( collect_set_nat @ P )
% 12.76/6.89 = ( collect_set_nat @ Q ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_91_Collect__cong,axiom,
% 12.76/6.89 ! [P: nat > $o,Q: nat > $o] :
% 12.76/6.89 ( ! [X4: nat] :
% 12.76/6.89 ( ( P @ X4 )
% 12.76/6.89 = ( Q @ X4 ) )
% 12.76/6.89 => ( ( collect_nat @ P )
% 12.76/6.89 = ( collect_nat @ Q ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_92_Collect__cong,axiom,
% 12.76/6.89 ! [P: int > $o,Q: int > $o] :
% 12.76/6.89 ( ! [X4: int] :
% 12.76/6.89 ( ( P @ X4 )
% 12.76/6.89 = ( Q @ X4 ) )
% 12.76/6.89 => ( ( collect_int @ P )
% 12.76/6.89 = ( collect_int @ Q ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_93_eq__imp__le,axiom,
% 12.76/6.89 ! [M: nat,N: nat] :
% 12.76/6.89 ( ( M = N )
% 12.76/6.89 => ( ord_less_eq_nat @ M @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_94_le__square,axiom,
% 12.76/6.89 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_95_le__antisym,axiom,
% 12.76/6.89 ! [M: nat,N: nat] :
% 12.76/6.89 ( ( ord_less_eq_nat @ M @ N )
% 12.76/6.89 => ( ( ord_less_eq_nat @ N @ M )
% 12.76/6.89 => ( M = N ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_96_dvd__antisym,axiom,
% 12.76/6.89 ! [M: nat,N: nat] :
% 12.76/6.89 ( ( dvd_dvd_nat @ M @ N )
% 12.76/6.89 => ( ( dvd_dvd_nat @ N @ M )
% 12.76/6.89 => ( M = N ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_97_mult__le__mono,axiom,
% 12.76/6.89 ! [I: nat,J: nat,K: nat,L: nat] :
% 12.76/6.89 ( ( ord_less_eq_nat @ I @ J )
% 12.76/6.89 => ( ( ord_less_eq_nat @ K @ L )
% 12.76/6.89 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_98_mult__le__mono1,axiom,
% 12.76/6.89 ! [I: nat,J: nat,K: nat] :
% 12.76/6.89 ( ( ord_less_eq_nat @ I @ J )
% 12.76/6.89 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_99_mult__le__mono2,axiom,
% 12.76/6.89 ! [I: nat,J: nat,K: nat] :
% 12.76/6.89 ( ( ord_less_eq_nat @ I @ J )
% 12.76/6.89 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_100_nat__le__linear,axiom,
% 12.76/6.89 ! [M: nat,N: nat] :
% 12.76/6.89 ( ( ord_less_eq_nat @ M @ N )
% 12.76/6.89 | ( ord_less_eq_nat @ N @ M ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_101_Suc__mult__le__cancel1,axiom,
% 12.76/6.89 ! [K: nat,M: nat,N: nat] :
% 12.76/6.89 ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 12.76/6.89 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_102_Nat_Oex__has__greatest__nat,axiom,
% 12.76/6.89 ! [P: nat > $o,K: nat,B: nat] :
% 12.76/6.89 ( ( P @ K )
% 12.76/6.89 => ( ! [Y: nat] :
% 12.76/6.89 ( ( P @ Y )
% 12.76/6.89 => ( ord_less_eq_nat @ Y @ B ) )
% 12.76/6.89 => ? [X4: nat] :
% 12.76/6.89 ( ( P @ X4 )
% 12.76/6.89 & ! [Y3: nat] :
% 12.76/6.89 ( ( P @ Y3 )
% 12.76/6.89 => ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_103_div__times__less__eq__dividend,axiom,
% 12.76/6.89 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_104_times__div__less__eq__dividend,axiom,
% 12.76/6.89 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_105_le__num__One__iff,axiom,
% 12.76/6.89 ! [X: num] :
% 12.76/6.89 ( ( ord_less_eq_num @ X @ one )
% 12.76/6.89 = ( X = one ) ) ).
% 12.76/6.89
% 12.76/6.89 thf(fact_106_Suc__mult__cancel1,axiom,
% 12.76/6.89 ! [K: nat,M: nat,N: nat] :
% 12.76/6.89 ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 12.76/6.89 = ( times_times_nat @ ( suc @ K ) @ N ) )
% 12.76/6.90 = ( M = N ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_107_transitive__stepwise__le,axiom,
% 12.76/6.90 ! [M: nat,N: nat,R: nat > nat > $o] :
% 12.76/6.90 ( ( ord_less_eq_nat @ M @ N )
% 12.76/6.90 => ( ! [X4: nat] : ( R @ X4 @ X4 )
% 12.76/6.90 => ( ! [X4: nat,Y: nat,Z2: nat] :
% 12.76/6.90 ( ( R @ X4 @ Y )
% 12.76/6.90 => ( ( R @ Y @ Z2 )
% 12.76/6.90 => ( R @ X4 @ Z2 ) ) )
% 12.76/6.90 => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
% 12.76/6.90 => ( R @ M @ N ) ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_108_nat__induct__at__least,axiom,
% 12.76/6.90 ! [M: nat,N: nat,P: nat > $o] :
% 12.76/6.90 ( ( ord_less_eq_nat @ M @ N )
% 12.76/6.90 => ( ( P @ M )
% 12.76/6.90 => ( ! [N2: nat] :
% 12.76/6.90 ( ( ord_less_eq_nat @ M @ N2 )
% 12.76/6.90 => ( ( P @ N2 )
% 12.76/6.90 => ( P @ ( suc @ N2 ) ) ) )
% 12.76/6.90 => ( P @ N ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_109_full__nat__induct,axiom,
% 12.76/6.90 ! [P: nat > $o,N: nat] :
% 12.76/6.90 ( ! [N2: nat] :
% 12.76/6.90 ( ! [M2: nat] :
% 12.76/6.90 ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
% 12.76/6.90 => ( P @ M2 ) )
% 12.76/6.90 => ( P @ N2 ) )
% 12.76/6.90 => ( P @ N ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_110_not__less__eq__eq,axiom,
% 12.76/6.90 ! [M: nat,N: nat] :
% 12.76/6.90 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 12.76/6.90 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_111_Suc__n__not__le__n,axiom,
% 12.76/6.90 ! [N: nat] :
% 12.76/6.90 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_112_le__Suc__eq,axiom,
% 12.76/6.90 ! [M: nat,N: nat] :
% 12.76/6.90 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 12.76/6.90 = ( ( ord_less_eq_nat @ M @ N )
% 12.76/6.90 | ( M
% 12.76/6.90 = ( suc @ N ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_113_Suc__le__D,axiom,
% 12.76/6.90 ! [N: nat,M3: nat] :
% 12.76/6.90 ( ( ord_less_eq_nat @ ( suc @ N ) @ M3 )
% 12.76/6.90 => ? [M4: nat] :
% 12.76/6.90 ( M3
% 12.76/6.90 = ( suc @ M4 ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_114_le__SucI,axiom,
% 12.76/6.90 ! [M: nat,N: nat] :
% 12.76/6.90 ( ( ord_less_eq_nat @ M @ N )
% 12.76/6.90 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_115_le__SucE,axiom,
% 12.76/6.90 ! [M: nat,N: nat] :
% 12.76/6.90 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 12.76/6.90 => ( ~ ( ord_less_eq_nat @ M @ N )
% 12.76/6.90 => ( M
% 12.76/6.90 = ( suc @ N ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_116_Suc__leD,axiom,
% 12.76/6.90 ! [M: nat,N: nat] :
% 12.76/6.90 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 12.76/6.90 => ( ord_less_eq_nat @ M @ N ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_117_div__mult2__eq,axiom,
% 12.76/6.90 ! [M: nat,N: nat,Q2: nat] :
% 12.76/6.90 ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 12.76/6.90 = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_118_div__le__dividend,axiom,
% 12.76/6.90 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_119_div__le__mono,axiom,
% 12.76/6.90 ! [M: nat,N: nat,K: nat] :
% 12.76/6.90 ( ( ord_less_eq_nat @ M @ N )
% 12.76/6.90 => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_120_of__nat__dvd__iff,axiom,
% 12.76/6.90 ! [M: nat,N: nat] :
% 12.76/6.90 ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
% 12.76/6.90 = ( dvd_dvd_nat @ M @ N ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_121_of__nat__dvd__iff,axiom,
% 12.76/6.90 ! [M: nat,N: nat] :
% 12.76/6.90 ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 12.76/6.90 = ( dvd_dvd_nat @ M @ N ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_122_of__nat__dvd__iff,axiom,
% 12.76/6.90 ! [M: nat,N: nat] :
% 12.76/6.90 ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 12.76/6.90 = ( dvd_dvd_nat @ M @ N ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_123_div__mult2__numeral__eq,axiom,
% 12.76/6.90 ! [A: nat,K: num,L: num] :
% 12.76/6.90 ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
% 12.76/6.90 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_124_div__mult2__numeral__eq,axiom,
% 12.76/6.90 ! [A: int,K: num,L: num] :
% 12.76/6.90 ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
% 12.76/6.90 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_125_Suc__div__le__mono,axiom,
% 12.76/6.90 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_126_n__not__Suc__n,axiom,
% 12.76/6.90 ! [N: nat] :
% 12.76/6.90 ( N
% 12.76/6.90 != ( suc @ N ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_127_Suc__inject,axiom,
% 12.76/6.90 ! [X: nat,Y4: nat] :
% 12.76/6.90 ( ( ( suc @ X )
% 12.76/6.90 = ( suc @ Y4 ) )
% 12.76/6.90 => ( X = Y4 ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_128_lift__Suc__antimono__le,axiom,
% 12.76/6.90 ! [F: nat > set_int,N: nat,N3: nat] :
% 12.76/6.90 ( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 12.76/6.90 => ( ( ord_less_eq_nat @ N @ N3 )
% 12.76/6.90 => ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_129_lift__Suc__antimono__le,axiom,
% 12.76/6.90 ! [F: nat > rat,N: nat,N3: nat] :
% 12.76/6.90 ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 12.76/6.90 => ( ( ord_less_eq_nat @ N @ N3 )
% 12.76/6.90 => ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_130_lift__Suc__antimono__le,axiom,
% 12.76/6.90 ! [F: nat > num,N: nat,N3: nat] :
% 12.76/6.90 ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 12.76/6.90 => ( ( ord_less_eq_nat @ N @ N3 )
% 12.76/6.90 => ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_131_lift__Suc__antimono__le,axiom,
% 12.76/6.90 ! [F: nat > nat,N: nat,N3: nat] :
% 12.76/6.90 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 12.76/6.90 => ( ( ord_less_eq_nat @ N @ N3 )
% 12.76/6.90 => ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_132_lift__Suc__antimono__le,axiom,
% 12.76/6.90 ! [F: nat > int,N: nat,N3: nat] :
% 12.76/6.90 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 12.76/6.90 => ( ( ord_less_eq_nat @ N @ N3 )
% 12.76/6.90 => ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_133_lift__Suc__antimono__le,axiom,
% 12.76/6.90 ! [F: nat > real,N: nat,N3: nat] :
% 12.76/6.90 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 12.76/6.90 => ( ( ord_less_eq_nat @ N @ N3 )
% 12.76/6.90 => ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_134_lift__Suc__mono__le,axiom,
% 12.76/6.90 ! [F: nat > set_int,N: nat,N3: nat] :
% 12.76/6.90 ( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 12.76/6.90 => ( ( ord_less_eq_nat @ N @ N3 )
% 12.76/6.90 => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_135_lift__Suc__mono__le,axiom,
% 12.76/6.90 ! [F: nat > rat,N: nat,N3: nat] :
% 12.76/6.90 ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 12.76/6.90 => ( ( ord_less_eq_nat @ N @ N3 )
% 12.76/6.90 => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_136_lift__Suc__mono__le,axiom,
% 12.76/6.90 ! [F: nat > num,N: nat,N3: nat] :
% 12.76/6.90 ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 12.76/6.90 => ( ( ord_less_eq_nat @ N @ N3 )
% 12.76/6.90 => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_137_lift__Suc__mono__le,axiom,
% 12.76/6.90 ! [F: nat > nat,N: nat,N3: nat] :
% 12.76/6.90 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 12.76/6.90 => ( ( ord_less_eq_nat @ N @ N3 )
% 12.76/6.90 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_138_lift__Suc__mono__le,axiom,
% 12.76/6.90 ! [F: nat > int,N: nat,N3: nat] :
% 12.76/6.90 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 12.76/6.90 => ( ( ord_less_eq_nat @ N @ N3 )
% 12.76/6.90 => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_139_lift__Suc__mono__le,axiom,
% 12.76/6.90 ! [F: nat > real,N: nat,N3: nat] :
% 12.76/6.90 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 12.76/6.90 => ( ( ord_less_eq_nat @ N @ N3 )
% 12.76/6.90 => ( ord_less_eq_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_140_of__nat__mono,axiom,
% 12.76/6.90 ! [I: nat,J: nat] :
% 12.76/6.90 ( ( ord_less_eq_nat @ I @ J )
% 12.76/6.90 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_141_of__nat__mono,axiom,
% 12.76/6.90 ! [I: nat,J: nat] :
% 12.76/6.90 ( ( ord_less_eq_nat @ I @ J )
% 12.76/6.90 => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_142_of__nat__mono,axiom,
% 12.76/6.90 ! [I: nat,J: nat] :
% 12.76/6.90 ( ( ord_less_eq_nat @ I @ J )
% 12.76/6.90 => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_143_of__nat__mono,axiom,
% 12.76/6.90 ! [I: nat,J: nat] :
% 12.76/6.90 ( ( ord_less_eq_nat @ I @ J )
% 12.76/6.90 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_144_mult__of__nat__commute,axiom,
% 12.76/6.90 ! [X: nat,Y4: complex] :
% 12.76/6.90 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y4 )
% 12.76/6.90 = ( times_times_complex @ Y4 @ ( semiri8010041392384452111omplex @ X ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_145_mult__of__nat__commute,axiom,
% 12.76/6.90 ! [X: nat,Y4: real] :
% 12.76/6.90 ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y4 )
% 12.76/6.90 = ( times_times_real @ Y4 @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_146_mult__of__nat__commute,axiom,
% 12.76/6.90 ! [X: nat,Y4: rat] :
% 12.76/6.90 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y4 )
% 12.76/6.90 = ( times_times_rat @ Y4 @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_147_mult__of__nat__commute,axiom,
% 12.76/6.90 ! [X: nat,Y4: nat] :
% 12.76/6.90 ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y4 )
% 12.76/6.90 = ( times_times_nat @ Y4 @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_148_mult__of__nat__commute,axiom,
% 12.76/6.90 ! [X: nat,Y4: int] :
% 12.76/6.90 ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y4 )
% 12.76/6.90 = ( times_times_int @ Y4 @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_149_mult__numeral__1__right,axiom,
% 12.76/6.90 ! [A: complex] :
% 12.76/6.90 ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
% 12.76/6.90 = A ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_150_mult__numeral__1__right,axiom,
% 12.76/6.90 ! [A: real] :
% 12.76/6.90 ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
% 12.76/6.90 = A ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_151_mult__numeral__1__right,axiom,
% 12.76/6.90 ! [A: rat] :
% 12.76/6.90 ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
% 12.76/6.90 = A ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_152_mult__numeral__1__right,axiom,
% 12.76/6.90 ! [A: nat] :
% 12.76/6.90 ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
% 12.76/6.90 = A ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_153_mult__numeral__1__right,axiom,
% 12.76/6.90 ! [A: int] :
% 12.76/6.90 ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
% 12.76/6.90 = A ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_154_mult__numeral__1,axiom,
% 12.76/6.90 ! [A: complex] :
% 12.76/6.90 ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
% 12.76/6.90 = A ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_155_mult__numeral__1,axiom,
% 12.76/6.90 ! [A: real] :
% 12.76/6.90 ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
% 12.76/6.90 = A ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_156_mult__numeral__1,axiom,
% 12.76/6.90 ! [A: rat] :
% 12.76/6.90 ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
% 12.76/6.90 = A ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_157_mult__numeral__1,axiom,
% 12.76/6.90 ! [A: nat] :
% 12.76/6.90 ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
% 12.76/6.90 = A ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_158_mult__numeral__1,axiom,
% 12.76/6.90 ! [A: int] :
% 12.76/6.90 ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
% 12.76/6.90 = A ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_159_divide__numeral__1,axiom,
% 12.76/6.90 ! [A: complex] :
% 12.76/6.90 ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
% 12.76/6.90 = A ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_160_divide__numeral__1,axiom,
% 12.76/6.90 ! [A: real] :
% 12.76/6.90 ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
% 12.76/6.90 = A ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_161_divide__numeral__1,axiom,
% 12.76/6.90 ! [A: rat] :
% 12.76/6.90 ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
% 12.76/6.90 = A ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_162_num_Oexhaust,axiom,
% 12.76/6.90 ! [Y4: num] :
% 12.76/6.90 ( ( Y4 != one )
% 12.76/6.90 => ( ! [X22: num] :
% 12.76/6.90 ( Y4
% 12.76/6.90 != ( bit0 @ X22 ) )
% 12.76/6.90 => ~ ! [X32: num] :
% 12.76/6.90 ( Y4
% 12.76/6.90 != ( bit1 @ X32 ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_163_div__mult2__eq_H,axiom,
% 12.76/6.90 ! [A: nat,M: nat,N: nat] :
% 12.76/6.90 ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 12.76/6.90 = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_164_div__mult2__eq_H,axiom,
% 12.76/6.90 ! [A: int,M: nat,N: nat] :
% 12.76/6.90 ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 12.76/6.90 = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_165_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 12.76/6.90 ! [M: nat,N: nat] :
% 12.76/6.90 ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
% 12.76/6.90 = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_166_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 12.76/6.90 ! [M: nat,N: nat] :
% 12.76/6.90 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
% 12.76/6.90 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_167_numeral__Bit0__div__2,axiom,
% 12.76/6.90 ! [N: num] :
% 12.76/6.90 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.76/6.90 = ( numeral_numeral_nat @ N ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_168_numeral__Bit0__div__2,axiom,
% 12.76/6.90 ! [N: num] :
% 12.76/6.90 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.76/6.90 = ( numeral_numeral_int @ N ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_169_eval__nat__numeral_I3_J,axiom,
% 12.76/6.90 ! [N: num] :
% 12.76/6.90 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 12.76/6.90 = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_170_even__of__nat,axiom,
% 12.76/6.90 ! [N: nat] :
% 12.76/6.90 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
% 12.76/6.90 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_171_even__of__nat,axiom,
% 12.76/6.90 ! [N: nat] :
% 12.76/6.90 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
% 12.76/6.90 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_172_even__of__nat,axiom,
% 12.76/6.90 ! [N: nat] :
% 12.76/6.90 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
% 12.76/6.90 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.76/6.90
% 12.76/6.90 thf(fact_173_even__Suc__div__two,axiom,
% 12.76/6.90 ! [N: nat] :
% 12.76/6.90 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.76/6.90 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.76/6.90 = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_174_odd__Suc__div__two,axiom,
% 12.85/6.90 ! [N: nat] :
% 12.85/6.90 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.90 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.90 = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_175_numeral__le__real__of__nat__iff,axiom,
% 12.85/6.90 ! [N: num,M: nat] :
% 12.85/6.90 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
% 12.85/6.90 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_176_even__Suc__Suc__iff,axiom,
% 12.85/6.90 ! [N: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
% 12.85/6.90 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_177_even__Suc,axiom,
% 12.85/6.90 ! [N: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
% 12.85/6.90 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_178_even__mult__iff,axiom,
% 12.85/6.90 ! [A: code_integer,B: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
% 12.85/6.90 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.90 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_179_even__mult__iff,axiom,
% 12.85/6.90 ! [A: nat,B: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
% 12.85/6.90 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.90 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_180_even__mult__iff,axiom,
% 12.85/6.90 ! [A: int,B: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
% 12.85/6.90 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.90 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_181_dvd__mult__div__cancel,axiom,
% 12.85/6.90 ! [A: code_integer,B: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ A @ B )
% 12.85/6.90 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
% 12.85/6.90 = B ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_182_dvd__mult__div__cancel,axiom,
% 12.85/6.90 ! [A: nat,B: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.90 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
% 12.85/6.90 = B ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_183_dvd__mult__div__cancel,axiom,
% 12.85/6.90 ! [A: int,B: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ A @ B )
% 12.85/6.90 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
% 12.85/6.90 = B ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_184_dvd__div__mult__self,axiom,
% 12.85/6.90 ! [A: code_integer,B: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ A @ B )
% 12.85/6.90 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 12.85/6.90 = B ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_185_dvd__div__mult__self,axiom,
% 12.85/6.90 ! [A: nat,B: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.90 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 12.85/6.90 = B ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_186_dvd__div__mult__self,axiom,
% 12.85/6.90 ! [A: int,B: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ A @ B )
% 12.85/6.90 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 12.85/6.90 = B ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_187_even__two__times__div__two,axiom,
% 12.85/6.90 ! [A: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.90 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 12.85/6.90 = A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_188_even__two__times__div__two,axiom,
% 12.85/6.90 ! [A: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.90 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.90 = A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_189_even__two__times__div__two,axiom,
% 12.85/6.90 ! [A: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.90 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 12.85/6.90 = A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_190_div2__even__ext__nat,axiom,
% 12.85/6.90 ! [X: nat,Y4: nat] :
% 12.85/6.90 ( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.90 = ( divide_divide_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.90 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
% 12.85/6.90 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y4 ) )
% 12.85/6.90 => ( X = Y4 ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_191_enat__ord__number_I1_J,axiom,
% 12.85/6.90 ! [M: num,N: num] :
% 12.85/6.90 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 12.85/6.90 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_192_div__dvd__div,axiom,
% 12.85/6.90 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ A @ B )
% 12.85/6.90 => ( ( dvd_dvd_Code_integer @ A @ C )
% 12.85/6.90 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
% 12.85/6.90 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_193_div__dvd__div,axiom,
% 12.85/6.90 ! [A: nat,B: nat,C: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.90 => ( ( dvd_dvd_nat @ A @ C )
% 12.85/6.90 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
% 12.85/6.90 = ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_194_div__dvd__div,axiom,
% 12.85/6.90 ! [A: int,B: int,C: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ A @ B )
% 12.85/6.90 => ( ( dvd_dvd_int @ A @ C )
% 12.85/6.90 => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
% 12.85/6.90 = ( dvd_dvd_int @ B @ C ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_195_dvd__trans,axiom,
% 12.85/6.90 ! [A: nat,B: nat,C: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.90 => ( ( dvd_dvd_nat @ B @ C )
% 12.85/6.90 => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_196_dvd__trans,axiom,
% 12.85/6.90 ! [A: int,B: int,C: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ A @ B )
% 12.85/6.90 => ( ( dvd_dvd_int @ B @ C )
% 12.85/6.90 => ( dvd_dvd_int @ A @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_197_dvd__trans,axiom,
% 12.85/6.90 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ A @ B )
% 12.85/6.90 => ( ( dvd_dvd_Code_integer @ B @ C )
% 12.85/6.90 => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_198_dvd__refl,axiom,
% 12.85/6.90 ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_199_dvd__refl,axiom,
% 12.85/6.90 ! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_200_dvd__refl,axiom,
% 12.85/6.90 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_201_complete__real,axiom,
% 12.85/6.90 ! [S: set_real] :
% 12.85/6.90 ( ? [X5: real] : ( member_real @ X5 @ S )
% 12.85/6.90 => ( ? [Z3: real] :
% 12.85/6.90 ! [X4: real] :
% 12.85/6.90 ( ( member_real @ X4 @ S )
% 12.85/6.90 => ( ord_less_eq_real @ X4 @ Z3 ) )
% 12.85/6.90 => ? [Y: real] :
% 12.85/6.90 ( ! [X5: real] :
% 12.85/6.90 ( ( member_real @ X5 @ S )
% 12.85/6.90 => ( ord_less_eq_real @ X5 @ Y ) )
% 12.85/6.90 & ! [Z3: real] :
% 12.85/6.90 ( ! [X4: real] :
% 12.85/6.90 ( ( member_real @ X4 @ S )
% 12.85/6.90 => ( ord_less_eq_real @ X4 @ Z3 ) )
% 12.85/6.90 => ( ord_less_eq_real @ Y @ Z3 ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_202_dvdE,axiom,
% 12.85/6.90 ! [B: code_integer,A: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ B @ A )
% 12.85/6.90 => ~ ! [K2: code_integer] :
% 12.85/6.90 ( A
% 12.85/6.90 != ( times_3573771949741848930nteger @ B @ K2 ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_203_dvdE,axiom,
% 12.85/6.90 ! [B: real,A: real] :
% 12.85/6.90 ( ( dvd_dvd_real @ B @ A )
% 12.85/6.90 => ~ ! [K2: real] :
% 12.85/6.90 ( A
% 12.85/6.90 != ( times_times_real @ B @ K2 ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_204_dvdE,axiom,
% 12.85/6.90 ! [B: rat,A: rat] :
% 12.85/6.90 ( ( dvd_dvd_rat @ B @ A )
% 12.85/6.90 => ~ ! [K2: rat] :
% 12.85/6.90 ( A
% 12.85/6.90 != ( times_times_rat @ B @ K2 ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_205_dvdE,axiom,
% 12.85/6.90 ! [B: nat,A: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ B @ A )
% 12.85/6.90 => ~ ! [K2: nat] :
% 12.85/6.90 ( A
% 12.85/6.90 != ( times_times_nat @ B @ K2 ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_206_dvdE,axiom,
% 12.85/6.90 ! [B: int,A: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ B @ A )
% 12.85/6.90 => ~ ! [K2: int] :
% 12.85/6.90 ( A
% 12.85/6.90 != ( times_times_int @ B @ K2 ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_207_dvdI,axiom,
% 12.85/6.90 ! [A: code_integer,B: code_integer,K: code_integer] :
% 12.85/6.90 ( ( A
% 12.85/6.90 = ( times_3573771949741848930nteger @ B @ K ) )
% 12.85/6.90 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_208_dvdI,axiom,
% 12.85/6.90 ! [A: real,B: real,K: real] :
% 12.85/6.90 ( ( A
% 12.85/6.90 = ( times_times_real @ B @ K ) )
% 12.85/6.90 => ( dvd_dvd_real @ B @ A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_209_dvdI,axiom,
% 12.85/6.90 ! [A: rat,B: rat,K: rat] :
% 12.85/6.90 ( ( A
% 12.85/6.90 = ( times_times_rat @ B @ K ) )
% 12.85/6.90 => ( dvd_dvd_rat @ B @ A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_210_dvdI,axiom,
% 12.85/6.90 ! [A: nat,B: nat,K: nat] :
% 12.85/6.90 ( ( A
% 12.85/6.90 = ( times_times_nat @ B @ K ) )
% 12.85/6.90 => ( dvd_dvd_nat @ B @ A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_211_dvdI,axiom,
% 12.85/6.90 ! [A: int,B: int,K: int] :
% 12.85/6.90 ( ( A
% 12.85/6.90 = ( times_times_int @ B @ K ) )
% 12.85/6.90 => ( dvd_dvd_int @ B @ A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_212_dvd__def,axiom,
% 12.85/6.90 ( dvd_dvd_Code_integer
% 12.85/6.90 = ( ^ [B2: code_integer,A3: code_integer] :
% 12.85/6.90 ? [K3: code_integer] :
% 12.85/6.90 ( A3
% 12.85/6.90 = ( times_3573771949741848930nteger @ B2 @ K3 ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_213_dvd__def,axiom,
% 12.85/6.90 ( dvd_dvd_real
% 12.85/6.90 = ( ^ [B2: real,A3: real] :
% 12.85/6.90 ? [K3: real] :
% 12.85/6.90 ( A3
% 12.85/6.90 = ( times_times_real @ B2 @ K3 ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_214_dvd__def,axiom,
% 12.85/6.90 ( dvd_dvd_rat
% 12.85/6.90 = ( ^ [B2: rat,A3: rat] :
% 12.85/6.90 ? [K3: rat] :
% 12.85/6.90 ( A3
% 12.85/6.90 = ( times_times_rat @ B2 @ K3 ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_215_dvd__def,axiom,
% 12.85/6.90 ( dvd_dvd_nat
% 12.85/6.90 = ( ^ [B2: nat,A3: nat] :
% 12.85/6.90 ? [K3: nat] :
% 12.85/6.90 ( A3
% 12.85/6.90 = ( times_times_nat @ B2 @ K3 ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_216_dvd__def,axiom,
% 12.85/6.90 ( dvd_dvd_int
% 12.85/6.90 = ( ^ [B2: int,A3: int] :
% 12.85/6.90 ? [K3: int] :
% 12.85/6.90 ( A3
% 12.85/6.90 = ( times_times_int @ B2 @ K3 ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_217_dvd__mult,axiom,
% 12.85/6.90 ! [A: code_integer,C: code_integer,B: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ A @ C )
% 12.85/6.90 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_218_dvd__mult,axiom,
% 12.85/6.90 ! [A: real,C: real,B: real] :
% 12.85/6.90 ( ( dvd_dvd_real @ A @ C )
% 12.85/6.90 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_219_dvd__mult,axiom,
% 12.85/6.90 ! [A: rat,C: rat,B: rat] :
% 12.85/6.90 ( ( dvd_dvd_rat @ A @ C )
% 12.85/6.90 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_220_dvd__mult,axiom,
% 12.85/6.90 ! [A: nat,C: nat,B: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ A @ C )
% 12.85/6.90 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_221_dvd__mult,axiom,
% 12.85/6.90 ! [A: int,C: int,B: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ A @ C )
% 12.85/6.90 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_222_dvd__mult2,axiom,
% 12.85/6.90 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ A @ B )
% 12.85/6.90 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_223_dvd__mult2,axiom,
% 12.85/6.90 ! [A: real,B: real,C: real] :
% 12.85/6.90 ( ( dvd_dvd_real @ A @ B )
% 12.85/6.90 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_224_dvd__mult2,axiom,
% 12.85/6.90 ! [A: rat,B: rat,C: rat] :
% 12.85/6.90 ( ( dvd_dvd_rat @ A @ B )
% 12.85/6.90 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_225_dvd__mult2,axiom,
% 12.85/6.90 ! [A: nat,B: nat,C: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.90 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_226_dvd__mult2,axiom,
% 12.85/6.90 ! [A: int,B: int,C: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ A @ B )
% 12.85/6.90 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_227_dvd__mult__left,axiom,
% 12.85/6.90 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 12.85/6.90 => ( dvd_dvd_Code_integer @ A @ C ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_228_dvd__mult__left,axiom,
% 12.85/6.90 ! [A: real,B: real,C: real] :
% 12.85/6.90 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 12.85/6.90 => ( dvd_dvd_real @ A @ C ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_229_dvd__mult__left,axiom,
% 12.85/6.90 ! [A: rat,B: rat,C: rat] :
% 12.85/6.90 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 12.85/6.90 => ( dvd_dvd_rat @ A @ C ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_230_dvd__mult__left,axiom,
% 12.85/6.90 ! [A: nat,B: nat,C: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 12.85/6.90 => ( dvd_dvd_nat @ A @ C ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_231_dvd__mult__left,axiom,
% 12.85/6.90 ! [A: int,B: int,C: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 12.85/6.90 => ( dvd_dvd_int @ A @ C ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_232_dvd__triv__left,axiom,
% 12.85/6.90 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_233_dvd__triv__left,axiom,
% 12.85/6.90 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_234_dvd__triv__left,axiom,
% 12.85/6.90 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_235_dvd__triv__left,axiom,
% 12.85/6.90 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_236_dvd__triv__left,axiom,
% 12.85/6.90 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_237_mult__dvd__mono,axiom,
% 12.85/6.90 ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ A @ B )
% 12.85/6.90 => ( ( dvd_dvd_Code_integer @ C @ D )
% 12.85/6.90 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_238_mult__dvd__mono,axiom,
% 12.85/6.90 ! [A: real,B: real,C: real,D: real] :
% 12.85/6.90 ( ( dvd_dvd_real @ A @ B )
% 12.85/6.90 => ( ( dvd_dvd_real @ C @ D )
% 12.85/6.90 => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_239_mult__dvd__mono,axiom,
% 12.85/6.90 ! [A: rat,B: rat,C: rat,D: rat] :
% 12.85/6.90 ( ( dvd_dvd_rat @ A @ B )
% 12.85/6.90 => ( ( dvd_dvd_rat @ C @ D )
% 12.85/6.90 => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_240_mult__dvd__mono,axiom,
% 12.85/6.90 ! [A: nat,B: nat,C: nat,D: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.90 => ( ( dvd_dvd_nat @ C @ D )
% 12.85/6.90 => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_241_mult__dvd__mono,axiom,
% 12.85/6.90 ! [A: int,B: int,C: int,D: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ A @ B )
% 12.85/6.90 => ( ( dvd_dvd_int @ C @ D )
% 12.85/6.90 => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_242_dvd__mult__right,axiom,
% 12.85/6.90 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 12.85/6.90 => ( dvd_dvd_Code_integer @ B @ C ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_243_dvd__mult__right,axiom,
% 12.85/6.90 ! [A: real,B: real,C: real] :
% 12.85/6.90 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 12.85/6.90 => ( dvd_dvd_real @ B @ C ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_244_dvd__mult__right,axiom,
% 12.85/6.90 ! [A: rat,B: rat,C: rat] :
% 12.85/6.90 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 12.85/6.90 => ( dvd_dvd_rat @ B @ C ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_245_dvd__mult__right,axiom,
% 12.85/6.90 ! [A: nat,B: nat,C: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 12.85/6.90 => ( dvd_dvd_nat @ B @ C ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_246_dvd__mult__right,axiom,
% 12.85/6.90 ! [A: int,B: int,C: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 12.85/6.90 => ( dvd_dvd_int @ B @ C ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_247_dvd__triv__right,axiom,
% 12.85/6.90 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_248_dvd__triv__right,axiom,
% 12.85/6.90 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_249_dvd__triv__right,axiom,
% 12.85/6.90 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_250_dvd__triv__right,axiom,
% 12.85/6.90 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_251_dvd__triv__right,axiom,
% 12.85/6.90 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_252_dvd__div__eq__iff,axiom,
% 12.85/6.90 ! [C: code_integer,A: code_integer,B: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ C @ A )
% 12.85/6.90 => ( ( dvd_dvd_Code_integer @ C @ B )
% 12.85/6.90 => ( ( ( divide6298287555418463151nteger @ A @ C )
% 12.85/6.90 = ( divide6298287555418463151nteger @ B @ C ) )
% 12.85/6.90 = ( A = B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_253_dvd__div__eq__iff,axiom,
% 12.85/6.90 ! [C: complex,A: complex,B: complex] :
% 12.85/6.90 ( ( dvd_dvd_complex @ C @ A )
% 12.85/6.90 => ( ( dvd_dvd_complex @ C @ B )
% 12.85/6.90 => ( ( ( divide1717551699836669952omplex @ A @ C )
% 12.85/6.90 = ( divide1717551699836669952omplex @ B @ C ) )
% 12.85/6.90 = ( A = B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_254_dvd__div__eq__iff,axiom,
% 12.85/6.90 ! [C: real,A: real,B: real] :
% 12.85/6.90 ( ( dvd_dvd_real @ C @ A )
% 12.85/6.90 => ( ( dvd_dvd_real @ C @ B )
% 12.85/6.90 => ( ( ( divide_divide_real @ A @ C )
% 12.85/6.90 = ( divide_divide_real @ B @ C ) )
% 12.85/6.90 = ( A = B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_255_dvd__div__eq__iff,axiom,
% 12.85/6.90 ! [C: rat,A: rat,B: rat] :
% 12.85/6.90 ( ( dvd_dvd_rat @ C @ A )
% 12.85/6.90 => ( ( dvd_dvd_rat @ C @ B )
% 12.85/6.90 => ( ( ( divide_divide_rat @ A @ C )
% 12.85/6.90 = ( divide_divide_rat @ B @ C ) )
% 12.85/6.90 = ( A = B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_256_dvd__div__eq__iff,axiom,
% 12.85/6.90 ! [C: nat,A: nat,B: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ C @ A )
% 12.85/6.90 => ( ( dvd_dvd_nat @ C @ B )
% 12.85/6.90 => ( ( ( divide_divide_nat @ A @ C )
% 12.85/6.90 = ( divide_divide_nat @ B @ C ) )
% 12.85/6.90 = ( A = B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_257_dvd__div__eq__iff,axiom,
% 12.85/6.90 ! [C: int,A: int,B: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ C @ A )
% 12.85/6.90 => ( ( dvd_dvd_int @ C @ B )
% 12.85/6.90 => ( ( ( divide_divide_int @ A @ C )
% 12.85/6.90 = ( divide_divide_int @ B @ C ) )
% 12.85/6.90 = ( A = B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_258_dvd__div__eq__cancel,axiom,
% 12.85/6.90 ! [A: code_integer,C: code_integer,B: code_integer] :
% 12.85/6.90 ( ( ( divide6298287555418463151nteger @ A @ C )
% 12.85/6.90 = ( divide6298287555418463151nteger @ B @ C ) )
% 12.85/6.90 => ( ( dvd_dvd_Code_integer @ C @ A )
% 12.85/6.90 => ( ( dvd_dvd_Code_integer @ C @ B )
% 12.85/6.90 => ( A = B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_259_dvd__div__eq__cancel,axiom,
% 12.85/6.90 ! [A: complex,C: complex,B: complex] :
% 12.85/6.90 ( ( ( divide1717551699836669952omplex @ A @ C )
% 12.85/6.90 = ( divide1717551699836669952omplex @ B @ C ) )
% 12.85/6.90 => ( ( dvd_dvd_complex @ C @ A )
% 12.85/6.90 => ( ( dvd_dvd_complex @ C @ B )
% 12.85/6.90 => ( A = B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_260_dvd__div__eq__cancel,axiom,
% 12.85/6.90 ! [A: real,C: real,B: real] :
% 12.85/6.90 ( ( ( divide_divide_real @ A @ C )
% 12.85/6.90 = ( divide_divide_real @ B @ C ) )
% 12.85/6.90 => ( ( dvd_dvd_real @ C @ A )
% 12.85/6.90 => ( ( dvd_dvd_real @ C @ B )
% 12.85/6.90 => ( A = B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_261_dvd__div__eq__cancel,axiom,
% 12.85/6.90 ! [A: rat,C: rat,B: rat] :
% 12.85/6.90 ( ( ( divide_divide_rat @ A @ C )
% 12.85/6.90 = ( divide_divide_rat @ B @ C ) )
% 12.85/6.90 => ( ( dvd_dvd_rat @ C @ A )
% 12.85/6.90 => ( ( dvd_dvd_rat @ C @ B )
% 12.85/6.90 => ( A = B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_262_dvd__div__eq__cancel,axiom,
% 12.85/6.90 ! [A: nat,C: nat,B: nat] :
% 12.85/6.90 ( ( ( divide_divide_nat @ A @ C )
% 12.85/6.90 = ( divide_divide_nat @ B @ C ) )
% 12.85/6.90 => ( ( dvd_dvd_nat @ C @ A )
% 12.85/6.90 => ( ( dvd_dvd_nat @ C @ B )
% 12.85/6.90 => ( A = B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_263_dvd__div__eq__cancel,axiom,
% 12.85/6.90 ! [A: int,C: int,B: int] :
% 12.85/6.90 ( ( ( divide_divide_int @ A @ C )
% 12.85/6.90 = ( divide_divide_int @ B @ C ) )
% 12.85/6.90 => ( ( dvd_dvd_int @ C @ A )
% 12.85/6.90 => ( ( dvd_dvd_int @ C @ B )
% 12.85/6.90 => ( A = B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_264_div__div__div__same,axiom,
% 12.85/6.90 ! [D: code_integer,B: code_integer,A: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ D @ B )
% 12.85/6.90 => ( ( dvd_dvd_Code_integer @ B @ A )
% 12.85/6.90 => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
% 12.85/6.90 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_265_div__div__div__same,axiom,
% 12.85/6.90 ! [D: nat,B: nat,A: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ D @ B )
% 12.85/6.90 => ( ( dvd_dvd_nat @ B @ A )
% 12.85/6.90 => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
% 12.85/6.90 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_266_div__div__div__same,axiom,
% 12.85/6.90 ! [D: int,B: int,A: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ D @ B )
% 12.85/6.90 => ( ( dvd_dvd_int @ B @ A )
% 12.85/6.90 => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
% 12.85/6.90 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_267_dvd__div__mult,axiom,
% 12.85/6.90 ! [C: code_integer,B: code_integer,A: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ C @ B )
% 12.85/6.90 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
% 12.85/6.90 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_268_dvd__div__mult,axiom,
% 12.85/6.90 ! [C: nat,B: nat,A: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ C @ B )
% 12.85/6.90 => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
% 12.85/6.90 = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_269_dvd__div__mult,axiom,
% 12.85/6.90 ! [C: int,B: int,A: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ C @ B )
% 12.85/6.90 => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
% 12.85/6.90 = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_270_div__mult__swap,axiom,
% 12.85/6.90 ! [C: code_integer,B: code_integer,A: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ C @ B )
% 12.85/6.90 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 12.85/6.90 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_271_div__mult__swap,axiom,
% 12.85/6.90 ! [C: nat,B: nat,A: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ C @ B )
% 12.85/6.90 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 12.85/6.90 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_272_div__mult__swap,axiom,
% 12.85/6.90 ! [C: int,B: int,A: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ C @ B )
% 12.85/6.90 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 12.85/6.90 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_273_div__div__eq__right,axiom,
% 12.85/6.90 ! [C: code_integer,B: code_integer,A: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ C @ B )
% 12.85/6.90 => ( ( dvd_dvd_Code_integer @ B @ A )
% 12.85/6.90 => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 12.85/6.90 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_274_div__div__eq__right,axiom,
% 12.85/6.90 ! [C: nat,B: nat,A: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ C @ B )
% 12.85/6.90 => ( ( dvd_dvd_nat @ B @ A )
% 12.85/6.90 => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 12.85/6.90 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_275_div__div__eq__right,axiom,
% 12.85/6.90 ! [C: int,B: int,A: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ C @ B )
% 12.85/6.90 => ( ( dvd_dvd_int @ B @ A )
% 12.85/6.90 => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
% 12.85/6.90 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_276_dvd__div__mult2__eq,axiom,
% 12.85/6.90 ! [B: code_integer,C: code_integer,A: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
% 12.85/6.90 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 12.85/6.90 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_277_dvd__div__mult2__eq,axiom,
% 12.85/6.90 ! [B: nat,C: nat,A: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
% 12.85/6.90 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 12.85/6.90 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_278_dvd__div__mult2__eq,axiom,
% 12.85/6.90 ! [B: int,C: int,A: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
% 12.85/6.90 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 12.85/6.90 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_279_dvd__mult__imp__div,axiom,
% 12.85/6.90 ! [A: code_integer,C: code_integer,B: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
% 12.85/6.90 => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_280_dvd__mult__imp__div,axiom,
% 12.85/6.90 ! [A: nat,C: nat,B: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
% 12.85/6.90 => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_281_dvd__mult__imp__div,axiom,
% 12.85/6.90 ! [A: int,C: int,B: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
% 12.85/6.90 => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_282_div__mult__div__if__dvd,axiom,
% 12.85/6.90 ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ B @ A )
% 12.85/6.90 => ( ( dvd_dvd_Code_integer @ D @ C )
% 12.85/6.90 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
% 12.85/6.90 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_283_div__mult__div__if__dvd,axiom,
% 12.85/6.90 ! [B: nat,A: nat,D: nat,C: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ B @ A )
% 12.85/6.90 => ( ( dvd_dvd_nat @ D @ C )
% 12.85/6.90 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
% 12.85/6.90 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_284_div__mult__div__if__dvd,axiom,
% 12.85/6.90 ! [B: int,A: int,D: int,C: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ B @ A )
% 12.85/6.90 => ( ( dvd_dvd_int @ D @ C )
% 12.85/6.90 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
% 12.85/6.90 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_285_real__of__nat__div4,axiom,
% 12.85/6.90 ! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_286_real__of__nat__div,axiom,
% 12.85/6.90 ! [D: nat,N: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ D @ N )
% 12.85/6.90 => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
% 12.85/6.90 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_287_even__numeral,axiom,
% 12.85/6.90 ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_288_even__numeral,axiom,
% 12.85/6.90 ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_289_even__numeral,axiom,
% 12.85/6.90 ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_290_evenE,axiom,
% 12.85/6.90 ! [A: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.90 => ~ ! [B3: code_integer] :
% 12.85/6.90 ( A
% 12.85/6.90 != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_291_evenE,axiom,
% 12.85/6.90 ! [A: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.90 => ~ ! [B3: nat] :
% 12.85/6.90 ( A
% 12.85/6.90 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_292_evenE,axiom,
% 12.85/6.90 ! [A: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.90 => ~ ! [B3: int] :
% 12.85/6.90 ( A
% 12.85/6.90 != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_293_odd__numeral,axiom,
% 12.85/6.90 ! [N: num] :
% 12.85/6.90 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_294_odd__numeral,axiom,
% 12.85/6.90 ! [N: num] :
% 12.85/6.90 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_295_odd__numeral,axiom,
% 12.85/6.90 ! [N: num] :
% 12.85/6.90 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_296_Suc__double__not__eq__double,axiom,
% 12.85/6.90 ! [M: nat,N: nat] :
% 12.85/6.90 ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.85/6.90 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_297_double__not__eq__Suc__double,axiom,
% 12.85/6.90 ! [M: nat,N: nat] :
% 12.85/6.90 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 12.85/6.90 != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_298_zdiv__numeral__Bit1,axiom,
% 12.85/6.90 ! [V: num,W: num] :
% 12.85/6.90 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 12.85/6.90 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_299_bit__eq__rec,axiom,
% 12.85/6.90 ( ( ^ [Y5: code_integer,Z4: code_integer] : ( Y5 = Z4 ) )
% 12.85/6.90 = ( ^ [A3: code_integer,B2: code_integer] :
% 12.85/6.90 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
% 12.85/6.90 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) )
% 12.85/6.90 & ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.90 = ( divide6298287555418463151nteger @ B2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_300_bit__eq__rec,axiom,
% 12.85/6.90 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 12.85/6.90 = ( ^ [A3: nat,B2: nat] :
% 12.85/6.90 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
% 12.85/6.90 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
% 12.85/6.90 & ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.90 = ( divide_divide_nat @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_301_bit__eq__rec,axiom,
% 12.85/6.90 ( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 12.85/6.90 = ( ^ [A3: int,B2: int] :
% 12.85/6.90 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
% 12.85/6.90 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
% 12.85/6.90 & ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.90 = ( divide_divide_int @ B2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_302_times__divide__eq__left,axiom,
% 12.85/6.90 ! [B: complex,C: complex,A: complex] :
% 12.85/6.90 ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
% 12.85/6.90 = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_303_times__divide__eq__left,axiom,
% 12.85/6.90 ! [B: real,C: real,A: real] :
% 12.85/6.90 ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
% 12.85/6.90 = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_304_times__divide__eq__left,axiom,
% 12.85/6.90 ! [B: rat,C: rat,A: rat] :
% 12.85/6.90 ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 12.85/6.90 = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_305_divide__divide__eq__left,axiom,
% 12.85/6.90 ! [A: complex,B: complex,C: complex] :
% 12.85/6.90 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 12.85/6.90 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_306_divide__divide__eq__left,axiom,
% 12.85/6.90 ! [A: real,B: real,C: real] :
% 12.85/6.90 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 12.85/6.90 = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_307_divide__divide__eq__left,axiom,
% 12.85/6.90 ! [A: rat,B: rat,C: rat] :
% 12.85/6.90 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 12.85/6.90 = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_308_divide__divide__eq__right,axiom,
% 12.85/6.90 ! [A: complex,B: complex,C: complex] :
% 12.85/6.90 ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 12.85/6.90 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_309_divide__divide__eq__right,axiom,
% 12.85/6.90 ! [A: real,B: real,C: real] :
% 12.85/6.90 ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
% 12.85/6.90 = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_310_divide__divide__eq__right,axiom,
% 12.85/6.90 ! [A: rat,B: rat,C: rat] :
% 12.85/6.90 ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 12.85/6.90 = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_311_times__divide__eq__right,axiom,
% 12.85/6.90 ! [A: complex,B: complex,C: complex] :
% 12.85/6.90 ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 12.85/6.90 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_312_times__divide__eq__right,axiom,
% 12.85/6.90 ! [A: real,B: real,C: real] :
% 12.85/6.90 ( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
% 12.85/6.90 = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_313_times__divide__eq__right,axiom,
% 12.85/6.90 ! [A: rat,B: rat,C: rat] :
% 12.85/6.90 ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 12.85/6.90 = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_314_buildup__nothing__in__leaf,axiom,
% 12.85/6.90 ! [N: nat,X: nat] :
% 12.85/6.90 ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_315_triangle__def,axiom,
% 12.85/6.90 ( nat_triangle
% 12.85/6.90 = ( ^ [N4: nat] : ( divide_divide_nat @ ( times_times_nat @ N4 @ ( suc @ N4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_316_set__decode__Suc,axiom,
% 12.85/6.90 ! [N: nat,X: nat] :
% 12.85/6.90 ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
% 12.85/6.90 = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_317_int__dvd__int__iff,axiom,
% 12.85/6.90 ! [M: nat,N: nat] :
% 12.85/6.90 ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 12.85/6.90 = ( dvd_dvd_nat @ M @ N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_318_real__divide__square__eq,axiom,
% 12.85/6.90 ! [R2: real,A: real] :
% 12.85/6.90 ( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
% 12.85/6.90 = ( divide_divide_real @ A @ R2 ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_319_int__eq__iff__numeral,axiom,
% 12.85/6.90 ! [M: nat,V: num] :
% 12.85/6.90 ( ( ( semiri1314217659103216013at_int @ M )
% 12.85/6.90 = ( numeral_numeral_int @ V ) )
% 12.85/6.90 = ( M
% 12.85/6.90 = ( numeral_numeral_nat @ V ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_320_zdiv__numeral__Bit0,axiom,
% 12.85/6.90 ! [V: num,W: num] :
% 12.85/6.90 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 12.85/6.90 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_321_int__int__eq,axiom,
% 12.85/6.90 ! [M: nat,N: nat] :
% 12.85/6.90 ( ( ( semiri1314217659103216013at_int @ M )
% 12.85/6.90 = ( semiri1314217659103216013at_int @ N ) )
% 12.85/6.90 = ( M = N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_322_subset__decode__imp__le,axiom,
% 12.85/6.90 ! [M: nat,N: nat] :
% 12.85/6.90 ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 12.85/6.90 => ( ord_less_eq_nat @ M @ N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_323_zle__int,axiom,
% 12.85/6.90 ! [M: nat,N: nat] :
% 12.85/6.90 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 12.85/6.90 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_324_zdiv__int,axiom,
% 12.85/6.90 ! [A: nat,B: nat] :
% 12.85/6.90 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
% 12.85/6.90 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_325_divide__divide__eq__left_H,axiom,
% 12.85/6.90 ! [A: complex,B: complex,C: complex] :
% 12.85/6.90 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 12.85/6.90 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_326_divide__divide__eq__left_H,axiom,
% 12.85/6.90 ! [A: real,B: real,C: real] :
% 12.85/6.90 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 12.85/6.90 = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_327_divide__divide__eq__left_H,axiom,
% 12.85/6.90 ! [A: rat,B: rat,C: rat] :
% 12.85/6.90 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 12.85/6.90 = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_328_divide__divide__times__eq,axiom,
% 12.85/6.90 ! [X: complex,Y4: complex,Z: complex,W: complex] :
% 12.85/6.90 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y4 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 12.85/6.90 = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y4 @ Z ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_329_divide__divide__times__eq,axiom,
% 12.85/6.90 ! [X: real,Y4: real,Z: real,W: real] :
% 12.85/6.90 ( ( divide_divide_real @ ( divide_divide_real @ X @ Y4 ) @ ( divide_divide_real @ Z @ W ) )
% 12.85/6.90 = ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y4 @ Z ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_330_divide__divide__times__eq,axiom,
% 12.85/6.90 ! [X: rat,Y4: rat,Z: rat,W: rat] :
% 12.85/6.90 ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y4 ) @ ( divide_divide_rat @ Z @ W ) )
% 12.85/6.90 = ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y4 @ Z ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_331_times__divide__times__eq,axiom,
% 12.85/6.90 ! [X: complex,Y4: complex,Z: complex,W: complex] :
% 12.85/6.90 ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y4 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 12.85/6.90 = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y4 @ W ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_332_times__divide__times__eq,axiom,
% 12.85/6.90 ! [X: real,Y4: real,Z: real,W: real] :
% 12.85/6.90 ( ( times_times_real @ ( divide_divide_real @ X @ Y4 ) @ ( divide_divide_real @ Z @ W ) )
% 12.85/6.90 = ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y4 @ W ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_333_times__divide__times__eq,axiom,
% 12.85/6.90 ! [X: rat,Y4: rat,Z: rat,W: rat] :
% 12.85/6.90 ( ( times_times_rat @ ( divide_divide_rat @ X @ Y4 ) @ ( divide_divide_rat @ Z @ W ) )
% 12.85/6.90 = ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y4 @ W ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_334_int__ops_I3_J,axiom,
% 12.85/6.90 ! [N: num] :
% 12.85/6.90 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 12.85/6.90 = ( numeral_numeral_int @ N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_335_verit__eq__simplify_I9_J,axiom,
% 12.85/6.90 ! [X33: num,Y32: num] :
% 12.85/6.90 ( ( ( bit1 @ X33 )
% 12.85/6.90 = ( bit1 @ Y32 ) )
% 12.85/6.90 = ( X33 = Y32 ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_336_verit__eq__simplify_I8_J,axiom,
% 12.85/6.90 ! [X2: num,Y2: num] :
% 12.85/6.90 ( ( ( bit0 @ X2 )
% 12.85/6.90 = ( bit0 @ Y2 ) )
% 12.85/6.90 = ( X2 = Y2 ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_337_dual__order_Orefl,axiom,
% 12.85/6.90 ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_338_dual__order_Orefl,axiom,
% 12.85/6.90 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_339_dual__order_Orefl,axiom,
% 12.85/6.90 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_340_dual__order_Orefl,axiom,
% 12.85/6.90 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_341_dual__order_Orefl,axiom,
% 12.85/6.90 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_342_dual__order_Orefl,axiom,
% 12.85/6.90 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_343_order__refl,axiom,
% 12.85/6.90 ! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_344_order__refl,axiom,
% 12.85/6.90 ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_345_order__refl,axiom,
% 12.85/6.90 ! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_346_order__refl,axiom,
% 12.85/6.90 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_347_order__refl,axiom,
% 12.85/6.90 ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_348_order__refl,axiom,
% 12.85/6.90 ! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_349_cnt__non__neg,axiom,
% 12.85/6.90 ! [T: vEBT_VEBT] : ( ord_less_eq_real @ zero_zero_real @ ( vEBT_VEBT_cnt @ T ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_350_buildup__nothing__in__min__max,axiom,
% 12.85/6.90 ! [N: nat,X: nat] :
% 12.85/6.90 ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_351_set__decode__0,axiom,
% 12.85/6.90 ! [X: nat] :
% 12.85/6.90 ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
% 12.85/6.90 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_352_int__ops_I7_J,axiom,
% 12.85/6.90 ! [A: nat,B: nat] :
% 12.85/6.90 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
% 12.85/6.90 = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_353_nat__int__comparison_I3_J,axiom,
% 12.85/6.90 ( ord_less_eq_nat
% 12.85/6.90 = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_354_verit__eq__simplify_I12_J,axiom,
% 12.85/6.90 ! [X33: num] :
% 12.85/6.90 ( one
% 12.85/6.90 != ( bit1 @ X33 ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_355_verit__eq__simplify_I14_J,axiom,
% 12.85/6.90 ! [X2: num,X33: num] :
% 12.85/6.90 ( ( bit0 @ X2 )
% 12.85/6.90 != ( bit1 @ X33 ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_356_mult__zero__left,axiom,
% 12.85/6.90 ! [A: complex] :
% 12.85/6.90 ( ( times_times_complex @ zero_zero_complex @ A )
% 12.85/6.90 = zero_zero_complex ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_357_mult__zero__left,axiom,
% 12.85/6.90 ! [A: real] :
% 12.85/6.90 ( ( times_times_real @ zero_zero_real @ A )
% 12.85/6.90 = zero_zero_real ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_358_mult__zero__left,axiom,
% 12.85/6.90 ! [A: rat] :
% 12.85/6.90 ( ( times_times_rat @ zero_zero_rat @ A )
% 12.85/6.90 = zero_zero_rat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_359_mult__zero__left,axiom,
% 12.85/6.90 ! [A: nat] :
% 12.85/6.90 ( ( times_times_nat @ zero_zero_nat @ A )
% 12.85/6.90 = zero_zero_nat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_360_mult__zero__left,axiom,
% 12.85/6.90 ! [A: int] :
% 12.85/6.90 ( ( times_times_int @ zero_zero_int @ A )
% 12.85/6.90 = zero_zero_int ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_361_mult__zero__right,axiom,
% 12.85/6.90 ! [A: complex] :
% 12.85/6.90 ( ( times_times_complex @ A @ zero_zero_complex )
% 12.85/6.90 = zero_zero_complex ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_362_mult__zero__right,axiom,
% 12.85/6.90 ! [A: real] :
% 12.85/6.90 ( ( times_times_real @ A @ zero_zero_real )
% 12.85/6.90 = zero_zero_real ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_363_mult__zero__right,axiom,
% 12.85/6.90 ! [A: rat] :
% 12.85/6.90 ( ( times_times_rat @ A @ zero_zero_rat )
% 12.85/6.90 = zero_zero_rat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_364_mult__zero__right,axiom,
% 12.85/6.90 ! [A: nat] :
% 12.85/6.90 ( ( times_times_nat @ A @ zero_zero_nat )
% 12.85/6.90 = zero_zero_nat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_365_mult__zero__right,axiom,
% 12.85/6.90 ! [A: int] :
% 12.85/6.90 ( ( times_times_int @ A @ zero_zero_int )
% 12.85/6.90 = zero_zero_int ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_366_mult__eq__0__iff,axiom,
% 12.85/6.90 ! [A: complex,B: complex] :
% 12.85/6.90 ( ( ( times_times_complex @ A @ B )
% 12.85/6.90 = zero_zero_complex )
% 12.85/6.90 = ( ( A = zero_zero_complex )
% 12.85/6.90 | ( B = zero_zero_complex ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_367_mult__eq__0__iff,axiom,
% 12.85/6.90 ! [A: real,B: real] :
% 12.85/6.90 ( ( ( times_times_real @ A @ B )
% 12.85/6.90 = zero_zero_real )
% 12.85/6.90 = ( ( A = zero_zero_real )
% 12.85/6.90 | ( B = zero_zero_real ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_368_mult__eq__0__iff,axiom,
% 12.85/6.90 ! [A: rat,B: rat] :
% 12.85/6.90 ( ( ( times_times_rat @ A @ B )
% 12.85/6.90 = zero_zero_rat )
% 12.85/6.90 = ( ( A = zero_zero_rat )
% 12.85/6.90 | ( B = zero_zero_rat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_369_mult__eq__0__iff,axiom,
% 12.85/6.90 ! [A: nat,B: nat] :
% 12.85/6.90 ( ( ( times_times_nat @ A @ B )
% 12.85/6.90 = zero_zero_nat )
% 12.85/6.90 = ( ( A = zero_zero_nat )
% 12.85/6.90 | ( B = zero_zero_nat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_370_mult__eq__0__iff,axiom,
% 12.85/6.90 ! [A: int,B: int] :
% 12.85/6.90 ( ( ( times_times_int @ A @ B )
% 12.85/6.90 = zero_zero_int )
% 12.85/6.90 = ( ( A = zero_zero_int )
% 12.85/6.90 | ( B = zero_zero_int ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_371_mult__cancel__left,axiom,
% 12.85/6.90 ! [C: complex,A: complex,B: complex] :
% 12.85/6.90 ( ( ( times_times_complex @ C @ A )
% 12.85/6.90 = ( times_times_complex @ C @ B ) )
% 12.85/6.90 = ( ( C = zero_zero_complex )
% 12.85/6.90 | ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_372_mult__cancel__left,axiom,
% 12.85/6.90 ! [C: real,A: real,B: real] :
% 12.85/6.90 ( ( ( times_times_real @ C @ A )
% 12.85/6.90 = ( times_times_real @ C @ B ) )
% 12.85/6.90 = ( ( C = zero_zero_real )
% 12.85/6.90 | ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_373_mult__cancel__left,axiom,
% 12.85/6.90 ! [C: rat,A: rat,B: rat] :
% 12.85/6.90 ( ( ( times_times_rat @ C @ A )
% 12.85/6.90 = ( times_times_rat @ C @ B ) )
% 12.85/6.90 = ( ( C = zero_zero_rat )
% 12.85/6.90 | ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_374_mult__cancel__left,axiom,
% 12.85/6.90 ! [C: nat,A: nat,B: nat] :
% 12.85/6.90 ( ( ( times_times_nat @ C @ A )
% 12.85/6.90 = ( times_times_nat @ C @ B ) )
% 12.85/6.90 = ( ( C = zero_zero_nat )
% 12.85/6.90 | ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_375_mult__cancel__left,axiom,
% 12.85/6.90 ! [C: int,A: int,B: int] :
% 12.85/6.90 ( ( ( times_times_int @ C @ A )
% 12.85/6.90 = ( times_times_int @ C @ B ) )
% 12.85/6.90 = ( ( C = zero_zero_int )
% 12.85/6.90 | ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_376_mult__cancel__right,axiom,
% 12.85/6.90 ! [A: complex,C: complex,B: complex] :
% 12.85/6.90 ( ( ( times_times_complex @ A @ C )
% 12.85/6.90 = ( times_times_complex @ B @ C ) )
% 12.85/6.90 = ( ( C = zero_zero_complex )
% 12.85/6.90 | ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_377_mult__cancel__right,axiom,
% 12.85/6.90 ! [A: real,C: real,B: real] :
% 12.85/6.90 ( ( ( times_times_real @ A @ C )
% 12.85/6.90 = ( times_times_real @ B @ C ) )
% 12.85/6.90 = ( ( C = zero_zero_real )
% 12.85/6.90 | ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_378_mult__cancel__right,axiom,
% 12.85/6.90 ! [A: rat,C: rat,B: rat] :
% 12.85/6.90 ( ( ( times_times_rat @ A @ C )
% 12.85/6.90 = ( times_times_rat @ B @ C ) )
% 12.85/6.90 = ( ( C = zero_zero_rat )
% 12.85/6.90 | ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_379_mult__cancel__right,axiom,
% 12.85/6.90 ! [A: nat,C: nat,B: nat] :
% 12.85/6.90 ( ( ( times_times_nat @ A @ C )
% 12.85/6.90 = ( times_times_nat @ B @ C ) )
% 12.85/6.90 = ( ( C = zero_zero_nat )
% 12.85/6.90 | ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_380_mult__cancel__right,axiom,
% 12.85/6.90 ! [A: int,C: int,B: int] :
% 12.85/6.90 ( ( ( times_times_int @ A @ C )
% 12.85/6.90 = ( times_times_int @ B @ C ) )
% 12.85/6.90 = ( ( C = zero_zero_int )
% 12.85/6.90 | ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_381_div__0,axiom,
% 12.85/6.90 ! [A: complex] :
% 12.85/6.90 ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 12.85/6.90 = zero_zero_complex ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_382_div__0,axiom,
% 12.85/6.90 ! [A: real] :
% 12.85/6.90 ( ( divide_divide_real @ zero_zero_real @ A )
% 12.85/6.90 = zero_zero_real ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_383_div__0,axiom,
% 12.85/6.90 ! [A: rat] :
% 12.85/6.90 ( ( divide_divide_rat @ zero_zero_rat @ A )
% 12.85/6.90 = zero_zero_rat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_384_div__0,axiom,
% 12.85/6.90 ! [A: nat] :
% 12.85/6.90 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 12.85/6.90 = zero_zero_nat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_385_div__0,axiom,
% 12.85/6.90 ! [A: int] :
% 12.85/6.90 ( ( divide_divide_int @ zero_zero_int @ A )
% 12.85/6.90 = zero_zero_int ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_386_div__by__0,axiom,
% 12.85/6.90 ! [A: complex] :
% 12.85/6.90 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 12.85/6.90 = zero_zero_complex ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_387_div__by__0,axiom,
% 12.85/6.90 ! [A: real] :
% 12.85/6.90 ( ( divide_divide_real @ A @ zero_zero_real )
% 12.85/6.90 = zero_zero_real ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_388_div__by__0,axiom,
% 12.85/6.90 ! [A: rat] :
% 12.85/6.90 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 12.85/6.90 = zero_zero_rat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_389_div__by__0,axiom,
% 12.85/6.90 ! [A: nat] :
% 12.85/6.90 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 12.85/6.90 = zero_zero_nat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_390_div__by__0,axiom,
% 12.85/6.90 ! [A: int] :
% 12.85/6.90 ( ( divide_divide_int @ A @ zero_zero_int )
% 12.85/6.90 = zero_zero_int ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_391_divide__eq__0__iff,axiom,
% 12.85/6.90 ! [A: complex,B: complex] :
% 12.85/6.90 ( ( ( divide1717551699836669952omplex @ A @ B )
% 12.85/6.90 = zero_zero_complex )
% 12.85/6.90 = ( ( A = zero_zero_complex )
% 12.85/6.90 | ( B = zero_zero_complex ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_392_divide__eq__0__iff,axiom,
% 12.85/6.90 ! [A: real,B: real] :
% 12.85/6.90 ( ( ( divide_divide_real @ A @ B )
% 12.85/6.90 = zero_zero_real )
% 12.85/6.90 = ( ( A = zero_zero_real )
% 12.85/6.90 | ( B = zero_zero_real ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_393_divide__eq__0__iff,axiom,
% 12.85/6.90 ! [A: rat,B: rat] :
% 12.85/6.90 ( ( ( divide_divide_rat @ A @ B )
% 12.85/6.90 = zero_zero_rat )
% 12.85/6.90 = ( ( A = zero_zero_rat )
% 12.85/6.90 | ( B = zero_zero_rat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_394_divide__cancel__left,axiom,
% 12.85/6.90 ! [C: complex,A: complex,B: complex] :
% 12.85/6.90 ( ( ( divide1717551699836669952omplex @ C @ A )
% 12.85/6.90 = ( divide1717551699836669952omplex @ C @ B ) )
% 12.85/6.90 = ( ( C = zero_zero_complex )
% 12.85/6.90 | ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_395_divide__cancel__left,axiom,
% 12.85/6.90 ! [C: real,A: real,B: real] :
% 12.85/6.90 ( ( ( divide_divide_real @ C @ A )
% 12.85/6.90 = ( divide_divide_real @ C @ B ) )
% 12.85/6.90 = ( ( C = zero_zero_real )
% 12.85/6.90 | ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_396_divide__cancel__left,axiom,
% 12.85/6.90 ! [C: rat,A: rat,B: rat] :
% 12.85/6.90 ( ( ( divide_divide_rat @ C @ A )
% 12.85/6.90 = ( divide_divide_rat @ C @ B ) )
% 12.85/6.90 = ( ( C = zero_zero_rat )
% 12.85/6.90 | ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_397_divide__cancel__right,axiom,
% 12.85/6.90 ! [A: complex,C: complex,B: complex] :
% 12.85/6.90 ( ( ( divide1717551699836669952omplex @ A @ C )
% 12.85/6.90 = ( divide1717551699836669952omplex @ B @ C ) )
% 12.85/6.90 = ( ( C = zero_zero_complex )
% 12.85/6.90 | ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_398_divide__cancel__right,axiom,
% 12.85/6.90 ! [A: real,C: real,B: real] :
% 12.85/6.90 ( ( ( divide_divide_real @ A @ C )
% 12.85/6.90 = ( divide_divide_real @ B @ C ) )
% 12.85/6.90 = ( ( C = zero_zero_real )
% 12.85/6.90 | ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_399_divide__cancel__right,axiom,
% 12.85/6.90 ! [A: rat,C: rat,B: rat] :
% 12.85/6.90 ( ( ( divide_divide_rat @ A @ C )
% 12.85/6.90 = ( divide_divide_rat @ B @ C ) )
% 12.85/6.90 = ( ( C = zero_zero_rat )
% 12.85/6.90 | ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_400_bits__div__0,axiom,
% 12.85/6.90 ! [A: nat] :
% 12.85/6.90 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 12.85/6.90 = zero_zero_nat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_401_bits__div__0,axiom,
% 12.85/6.90 ! [A: int] :
% 12.85/6.90 ( ( divide_divide_int @ zero_zero_int @ A )
% 12.85/6.90 = zero_zero_int ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_402_bits__div__by__0,axiom,
% 12.85/6.90 ! [A: nat] :
% 12.85/6.90 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 12.85/6.90 = zero_zero_nat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_403_bits__div__by__0,axiom,
% 12.85/6.90 ! [A: int] :
% 12.85/6.90 ( ( divide_divide_int @ A @ zero_zero_int )
% 12.85/6.90 = zero_zero_int ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_404_division__ring__divide__zero,axiom,
% 12.85/6.90 ! [A: complex] :
% 12.85/6.90 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 12.85/6.90 = zero_zero_complex ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_405_division__ring__divide__zero,axiom,
% 12.85/6.90 ! [A: real] :
% 12.85/6.90 ( ( divide_divide_real @ A @ zero_zero_real )
% 12.85/6.90 = zero_zero_real ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_406_division__ring__divide__zero,axiom,
% 12.85/6.90 ! [A: rat] :
% 12.85/6.90 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 12.85/6.90 = zero_zero_rat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_407_dvd__0__left__iff,axiom,
% 12.85/6.90 ! [A: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 12.85/6.90 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_408_dvd__0__left__iff,axiom,
% 12.85/6.90 ! [A: complex] :
% 12.85/6.90 ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 12.85/6.90 = ( A = zero_zero_complex ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_409_dvd__0__left__iff,axiom,
% 12.85/6.90 ! [A: real] :
% 12.85/6.90 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 12.85/6.90 = ( A = zero_zero_real ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_410_dvd__0__left__iff,axiom,
% 12.85/6.90 ! [A: rat] :
% 12.85/6.90 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 12.85/6.90 = ( A = zero_zero_rat ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_411_dvd__0__left__iff,axiom,
% 12.85/6.90 ! [A: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 12.85/6.90 = ( A = zero_zero_nat ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_412_dvd__0__left__iff,axiom,
% 12.85/6.90 ! [A: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 12.85/6.90 = ( A = zero_zero_int ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_413_dvd__0__right,axiom,
% 12.85/6.90 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_414_dvd__0__right,axiom,
% 12.85/6.90 ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_415_dvd__0__right,axiom,
% 12.85/6.90 ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_416_dvd__0__right,axiom,
% 12.85/6.90 ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_417_dvd__0__right,axiom,
% 12.85/6.90 ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_418_dvd__0__right,axiom,
% 12.85/6.90 ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_419_le0,axiom,
% 12.85/6.90 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_420_bot__nat__0_Oextremum,axiom,
% 12.85/6.90 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_421_mult__cancel2,axiom,
% 12.85/6.90 ! [M: nat,K: nat,N: nat] :
% 12.85/6.90 ( ( ( times_times_nat @ M @ K )
% 12.85/6.90 = ( times_times_nat @ N @ K ) )
% 12.85/6.90 = ( ( M = N )
% 12.85/6.90 | ( K = zero_zero_nat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_422_mult__cancel1,axiom,
% 12.85/6.90 ! [K: nat,M: nat,N: nat] :
% 12.85/6.90 ( ( ( times_times_nat @ K @ M )
% 12.85/6.90 = ( times_times_nat @ K @ N ) )
% 12.85/6.90 = ( ( M = N )
% 12.85/6.90 | ( K = zero_zero_nat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_423_mult__0__right,axiom,
% 12.85/6.90 ! [M: nat] :
% 12.85/6.90 ( ( times_times_nat @ M @ zero_zero_nat )
% 12.85/6.90 = zero_zero_nat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_424_mult__is__0,axiom,
% 12.85/6.90 ! [M: nat,N: nat] :
% 12.85/6.90 ( ( ( times_times_nat @ M @ N )
% 12.85/6.90 = zero_zero_nat )
% 12.85/6.90 = ( ( M = zero_zero_nat )
% 12.85/6.90 | ( N = zero_zero_nat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_425_triangle__0,axiom,
% 12.85/6.90 ( ( nat_triangle @ zero_zero_nat )
% 12.85/6.90 = zero_zero_nat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_426_nonzero__mult__div__cancel__left,axiom,
% 12.85/6.90 ! [A: complex,B: complex] :
% 12.85/6.90 ( ( A != zero_zero_complex )
% 12.85/6.90 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
% 12.85/6.90 = B ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_427_nonzero__mult__div__cancel__left,axiom,
% 12.85/6.90 ! [A: real,B: real] :
% 12.85/6.90 ( ( A != zero_zero_real )
% 12.85/6.90 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
% 12.85/6.90 = B ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_428_nonzero__mult__div__cancel__left,axiom,
% 12.85/6.90 ! [A: rat,B: rat] :
% 12.85/6.90 ( ( A != zero_zero_rat )
% 12.85/6.90 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
% 12.85/6.90 = B ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_429_nonzero__mult__div__cancel__left,axiom,
% 12.85/6.90 ! [A: nat,B: nat] :
% 12.85/6.90 ( ( A != zero_zero_nat )
% 12.85/6.90 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
% 12.85/6.90 = B ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_430_nonzero__mult__div__cancel__left,axiom,
% 12.85/6.90 ! [A: int,B: int] :
% 12.85/6.90 ( ( A != zero_zero_int )
% 12.85/6.90 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
% 12.85/6.90 = B ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_431_nonzero__mult__div__cancel__right,axiom,
% 12.85/6.90 ! [B: complex,A: complex] :
% 12.85/6.90 ( ( B != zero_zero_complex )
% 12.85/6.90 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
% 12.85/6.90 = A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_432_nonzero__mult__div__cancel__right,axiom,
% 12.85/6.90 ! [B: real,A: real] :
% 12.85/6.90 ( ( B != zero_zero_real )
% 12.85/6.90 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
% 12.85/6.90 = A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_433_nonzero__mult__div__cancel__right,axiom,
% 12.85/6.90 ! [B: rat,A: rat] :
% 12.85/6.90 ( ( B != zero_zero_rat )
% 12.85/6.90 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
% 12.85/6.90 = A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_434_nonzero__mult__div__cancel__right,axiom,
% 12.85/6.90 ! [B: nat,A: nat] :
% 12.85/6.90 ( ( B != zero_zero_nat )
% 12.85/6.90 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
% 12.85/6.90 = A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_435_nonzero__mult__div__cancel__right,axiom,
% 12.85/6.90 ! [B: int,A: int] :
% 12.85/6.90 ( ( B != zero_zero_int )
% 12.85/6.90 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
% 12.85/6.90 = A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_436_div__mult__mult1,axiom,
% 12.85/6.90 ! [C: nat,A: nat,B: nat] :
% 12.85/6.90 ( ( C != zero_zero_nat )
% 12.85/6.90 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 12.85/6.90 = ( divide_divide_nat @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_437_div__mult__mult1,axiom,
% 12.85/6.90 ! [C: int,A: int,B: int] :
% 12.85/6.90 ( ( C != zero_zero_int )
% 12.85/6.90 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 12.85/6.90 = ( divide_divide_int @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_438_div__mult__mult2,axiom,
% 12.85/6.90 ! [C: nat,A: nat,B: nat] :
% 12.85/6.90 ( ( C != zero_zero_nat )
% 12.85/6.90 => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 12.85/6.90 = ( divide_divide_nat @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_439_div__mult__mult2,axiom,
% 12.85/6.90 ! [C: int,A: int,B: int] :
% 12.85/6.90 ( ( C != zero_zero_int )
% 12.85/6.90 => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 12.85/6.90 = ( divide_divide_int @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_440_div__mult__mult1__if,axiom,
% 12.85/6.90 ! [C: nat,A: nat,B: nat] :
% 12.85/6.90 ( ( ( C = zero_zero_nat )
% 12.85/6.90 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 12.85/6.90 = zero_zero_nat ) )
% 12.85/6.90 & ( ( C != zero_zero_nat )
% 12.85/6.90 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 12.85/6.90 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_441_div__mult__mult1__if,axiom,
% 12.85/6.90 ! [C: int,A: int,B: int] :
% 12.85/6.90 ( ( ( C = zero_zero_int )
% 12.85/6.90 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 12.85/6.90 = zero_zero_int ) )
% 12.85/6.90 & ( ( C != zero_zero_int )
% 12.85/6.90 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 12.85/6.90 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_442_mult__divide__mult__cancel__left__if,axiom,
% 12.85/6.90 ! [C: complex,A: complex,B: complex] :
% 12.85/6.90 ( ( ( C = zero_zero_complex )
% 12.85/6.90 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 12.85/6.90 = zero_zero_complex ) )
% 12.85/6.90 & ( ( C != zero_zero_complex )
% 12.85/6.90 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 12.85/6.90 = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_443_mult__divide__mult__cancel__left__if,axiom,
% 12.85/6.90 ! [C: real,A: real,B: real] :
% 12.85/6.90 ( ( ( C = zero_zero_real )
% 12.85/6.90 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 12.85/6.90 = zero_zero_real ) )
% 12.85/6.90 & ( ( C != zero_zero_real )
% 12.85/6.90 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 12.85/6.90 = ( divide_divide_real @ A @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_444_mult__divide__mult__cancel__left__if,axiom,
% 12.85/6.90 ! [C: rat,A: rat,B: rat] :
% 12.85/6.90 ( ( ( C = zero_zero_rat )
% 12.85/6.90 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 12.85/6.90 = zero_zero_rat ) )
% 12.85/6.90 & ( ( C != zero_zero_rat )
% 12.85/6.90 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 12.85/6.90 = ( divide_divide_rat @ A @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_445_nonzero__mult__divide__mult__cancel__left,axiom,
% 12.85/6.90 ! [C: complex,A: complex,B: complex] :
% 12.85/6.90 ( ( C != zero_zero_complex )
% 12.85/6.90 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 12.85/6.90 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_446_nonzero__mult__divide__mult__cancel__left,axiom,
% 12.85/6.90 ! [C: real,A: real,B: real] :
% 12.85/6.90 ( ( C != zero_zero_real )
% 12.85/6.90 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 12.85/6.90 = ( divide_divide_real @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_447_nonzero__mult__divide__mult__cancel__left,axiom,
% 12.85/6.90 ! [C: rat,A: rat,B: rat] :
% 12.85/6.90 ( ( C != zero_zero_rat )
% 12.85/6.90 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 12.85/6.90 = ( divide_divide_rat @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_448_nonzero__mult__divide__mult__cancel__left2,axiom,
% 12.85/6.90 ! [C: complex,A: complex,B: complex] :
% 12.85/6.90 ( ( C != zero_zero_complex )
% 12.85/6.90 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
% 12.85/6.90 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_449_nonzero__mult__divide__mult__cancel__left2,axiom,
% 12.85/6.90 ! [C: real,A: real,B: real] :
% 12.85/6.90 ( ( C != zero_zero_real )
% 12.85/6.90 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
% 12.85/6.90 = ( divide_divide_real @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_450_nonzero__mult__divide__mult__cancel__left2,axiom,
% 12.85/6.90 ! [C: rat,A: rat,B: rat] :
% 12.85/6.90 ( ( C != zero_zero_rat )
% 12.85/6.90 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
% 12.85/6.90 = ( divide_divide_rat @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_451_nonzero__mult__divide__mult__cancel__right,axiom,
% 12.85/6.90 ! [C: complex,A: complex,B: complex] :
% 12.85/6.90 ( ( C != zero_zero_complex )
% 12.85/6.90 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 12.85/6.90 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_452_nonzero__mult__divide__mult__cancel__right,axiom,
% 12.85/6.90 ! [C: real,A: real,B: real] :
% 12.85/6.90 ( ( C != zero_zero_real )
% 12.85/6.90 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 12.85/6.90 = ( divide_divide_real @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_453_nonzero__mult__divide__mult__cancel__right,axiom,
% 12.85/6.90 ! [C: rat,A: rat,B: rat] :
% 12.85/6.90 ( ( C != zero_zero_rat )
% 12.85/6.90 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 12.85/6.90 = ( divide_divide_rat @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_454_nonzero__mult__divide__mult__cancel__right2,axiom,
% 12.85/6.90 ! [C: complex,A: complex,B: complex] :
% 12.85/6.90 ( ( C != zero_zero_complex )
% 12.85/6.90 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
% 12.85/6.90 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_455_nonzero__mult__divide__mult__cancel__right2,axiom,
% 12.85/6.90 ! [C: real,A: real,B: real] :
% 12.85/6.90 ( ( C != zero_zero_real )
% 12.85/6.90 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
% 12.85/6.90 = ( divide_divide_real @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_456_nonzero__mult__divide__mult__cancel__right2,axiom,
% 12.85/6.90 ! [C: rat,A: rat,B: rat] :
% 12.85/6.90 ( ( C != zero_zero_rat )
% 12.85/6.90 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
% 12.85/6.90 = ( divide_divide_rat @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_457_dvd__mult__cancel__left,axiom,
% 12.85/6.90 ! [C: code_integer,A: code_integer,B: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 12.85/6.90 = ( ( C = zero_z3403309356797280102nteger )
% 12.85/6.90 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_458_dvd__mult__cancel__left,axiom,
% 12.85/6.90 ! [C: complex,A: complex,B: complex] :
% 12.85/6.90 ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 12.85/6.90 = ( ( C = zero_zero_complex )
% 12.85/6.90 | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_459_dvd__mult__cancel__left,axiom,
% 12.85/6.90 ! [C: real,A: real,B: real] :
% 12.85/6.90 ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 12.85/6.90 = ( ( C = zero_zero_real )
% 12.85/6.90 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_460_dvd__mult__cancel__left,axiom,
% 12.85/6.90 ! [C: rat,A: rat,B: rat] :
% 12.85/6.90 ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 12.85/6.90 = ( ( C = zero_zero_rat )
% 12.85/6.90 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_461_dvd__mult__cancel__left,axiom,
% 12.85/6.90 ! [C: int,A: int,B: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 12.85/6.90 = ( ( C = zero_zero_int )
% 12.85/6.90 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_462_dvd__mult__cancel__right,axiom,
% 12.85/6.90 ! [A: code_integer,C: code_integer,B: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 12.85/6.90 = ( ( C = zero_z3403309356797280102nteger )
% 12.85/6.90 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_463_dvd__mult__cancel__right,axiom,
% 12.85/6.90 ! [A: complex,C: complex,B: complex] :
% 12.85/6.90 ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 12.85/6.90 = ( ( C = zero_zero_complex )
% 12.85/6.90 | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_464_dvd__mult__cancel__right,axiom,
% 12.85/6.90 ! [A: real,C: real,B: real] :
% 12.85/6.90 ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 12.85/6.90 = ( ( C = zero_zero_real )
% 12.85/6.90 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_465_dvd__mult__cancel__right,axiom,
% 12.85/6.90 ! [A: rat,C: rat,B: rat] :
% 12.85/6.90 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 12.85/6.90 = ( ( C = zero_zero_rat )
% 12.85/6.90 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_466_dvd__mult__cancel__right,axiom,
% 12.85/6.90 ! [A: int,C: int,B: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 12.85/6.90 = ( ( C = zero_zero_int )
% 12.85/6.90 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_467_dvd__times__left__cancel__iff,axiom,
% 12.85/6.90 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.90 ( ( A != zero_z3403309356797280102nteger )
% 12.85/6.90 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 12.85/6.90 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_468_dvd__times__left__cancel__iff,axiom,
% 12.85/6.90 ! [A: nat,B: nat,C: nat] :
% 12.85/6.90 ( ( A != zero_zero_nat )
% 12.85/6.90 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
% 12.85/6.90 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_469_dvd__times__left__cancel__iff,axiom,
% 12.85/6.90 ! [A: int,B: int,C: int] :
% 12.85/6.90 ( ( A != zero_zero_int )
% 12.85/6.90 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
% 12.85/6.90 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_470_dvd__times__right__cancel__iff,axiom,
% 12.85/6.90 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.90 ( ( A != zero_z3403309356797280102nteger )
% 12.85/6.90 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 12.85/6.90 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_471_dvd__times__right__cancel__iff,axiom,
% 12.85/6.90 ! [A: nat,B: nat,C: nat] :
% 12.85/6.90 ( ( A != zero_zero_nat )
% 12.85/6.90 => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
% 12.85/6.90 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_472_dvd__times__right__cancel__iff,axiom,
% 12.85/6.90 ! [A: int,B: int,C: int] :
% 12.85/6.90 ( ( A != zero_zero_int )
% 12.85/6.90 => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
% 12.85/6.90 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_473_of__nat__eq__0__iff,axiom,
% 12.85/6.90 ! [M: nat] :
% 12.85/6.90 ( ( ( semiri8010041392384452111omplex @ M )
% 12.85/6.90 = zero_zero_complex )
% 12.85/6.90 = ( M = zero_zero_nat ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_474_of__nat__eq__0__iff,axiom,
% 12.85/6.90 ! [M: nat] :
% 12.85/6.90 ( ( ( semiri5074537144036343181t_real @ M )
% 12.85/6.90 = zero_zero_real )
% 12.85/6.90 = ( M = zero_zero_nat ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_475_of__nat__eq__0__iff,axiom,
% 12.85/6.90 ! [M: nat] :
% 12.85/6.90 ( ( ( semiri681578069525770553at_rat @ M )
% 12.85/6.90 = zero_zero_rat )
% 12.85/6.90 = ( M = zero_zero_nat ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_476_of__nat__eq__0__iff,axiom,
% 12.85/6.90 ! [M: nat] :
% 12.85/6.90 ( ( ( semiri1316708129612266289at_nat @ M )
% 12.85/6.90 = zero_zero_nat )
% 12.85/6.90 = ( M = zero_zero_nat ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_477_of__nat__eq__0__iff,axiom,
% 12.85/6.90 ! [M: nat] :
% 12.85/6.90 ( ( ( semiri1314217659103216013at_int @ M )
% 12.85/6.90 = zero_zero_int )
% 12.85/6.90 = ( M = zero_zero_nat ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_478_of__nat__0__eq__iff,axiom,
% 12.85/6.90 ! [N: nat] :
% 12.85/6.90 ( ( zero_zero_complex
% 12.85/6.90 = ( semiri8010041392384452111omplex @ N ) )
% 12.85/6.90 = ( zero_zero_nat = N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_479_of__nat__0__eq__iff,axiom,
% 12.85/6.90 ! [N: nat] :
% 12.85/6.90 ( ( zero_zero_real
% 12.85/6.90 = ( semiri5074537144036343181t_real @ N ) )
% 12.85/6.90 = ( zero_zero_nat = N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_480_of__nat__0__eq__iff,axiom,
% 12.85/6.90 ! [N: nat] :
% 12.85/6.90 ( ( zero_zero_rat
% 12.85/6.90 = ( semiri681578069525770553at_rat @ N ) )
% 12.85/6.90 = ( zero_zero_nat = N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_481_of__nat__0__eq__iff,axiom,
% 12.85/6.90 ! [N: nat] :
% 12.85/6.90 ( ( zero_zero_nat
% 12.85/6.90 = ( semiri1316708129612266289at_nat @ N ) )
% 12.85/6.90 = ( zero_zero_nat = N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_482_of__nat__0__eq__iff,axiom,
% 12.85/6.90 ! [N: nat] :
% 12.85/6.90 ( ( zero_zero_int
% 12.85/6.90 = ( semiri1314217659103216013at_int @ N ) )
% 12.85/6.90 = ( zero_zero_nat = N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_483_of__nat__0,axiom,
% 12.85/6.90 ( ( semiri8010041392384452111omplex @ zero_zero_nat )
% 12.85/6.90 = zero_zero_complex ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_484_of__nat__0,axiom,
% 12.85/6.90 ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 12.85/6.90 = zero_zero_real ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_485_of__nat__0,axiom,
% 12.85/6.90 ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 12.85/6.90 = zero_zero_rat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_486_of__nat__0,axiom,
% 12.85/6.90 ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 12.85/6.90 = zero_zero_nat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_487_of__nat__0,axiom,
% 12.85/6.90 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 12.85/6.90 = zero_zero_int ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_488_one__eq__mult__iff,axiom,
% 12.85/6.90 ! [M: nat,N: nat] :
% 12.85/6.90 ( ( ( suc @ zero_zero_nat )
% 12.85/6.90 = ( times_times_nat @ M @ N ) )
% 12.85/6.90 = ( ( M
% 12.85/6.90 = ( suc @ zero_zero_nat ) )
% 12.85/6.90 & ( N
% 12.85/6.90 = ( suc @ zero_zero_nat ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_489_mult__eq__1__iff,axiom,
% 12.85/6.90 ! [M: nat,N: nat] :
% 12.85/6.90 ( ( ( times_times_nat @ M @ N )
% 12.85/6.90 = ( suc @ zero_zero_nat ) )
% 12.85/6.90 = ( ( M
% 12.85/6.90 = ( suc @ zero_zero_nat ) )
% 12.85/6.90 & ( N
% 12.85/6.90 = ( suc @ zero_zero_nat ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_490_dvd__1__left,axiom,
% 12.85/6.90 ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_491_dvd__1__iff__1,axiom,
% 12.85/6.90 ! [M: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 12.85/6.90 = ( M
% 12.85/6.90 = ( suc @ zero_zero_nat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_492_div__by__Suc__0,axiom,
% 12.85/6.90 ! [M: nat] :
% 12.85/6.90 ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 12.85/6.90 = M ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_493_nat__mult__dvd__cancel__disj,axiom,
% 12.85/6.90 ! [K: nat,M: nat,N: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.85/6.90 = ( ( K = zero_zero_nat )
% 12.85/6.90 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_494_nat__mult__div__cancel__disj,axiom,
% 12.85/6.90 ! [K: nat,M: nat,N: nat] :
% 12.85/6.90 ( ( ( K = zero_zero_nat )
% 12.85/6.90 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.85/6.90 = zero_zero_nat ) )
% 12.85/6.90 & ( ( K != zero_zero_nat )
% 12.85/6.90 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.85/6.90 = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_495_divide__eq__eq__numeral1_I1_J,axiom,
% 12.85/6.90 ! [B: complex,W: num,A: complex] :
% 12.85/6.90 ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
% 12.85/6.90 = A )
% 12.85/6.90 = ( ( ( ( numera6690914467698888265omplex @ W )
% 12.85/6.90 != zero_zero_complex )
% 12.85/6.90 => ( B
% 12.85/6.90 = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
% 12.85/6.90 & ( ( ( numera6690914467698888265omplex @ W )
% 12.85/6.90 = zero_zero_complex )
% 12.85/6.90 => ( A = zero_zero_complex ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_496_divide__eq__eq__numeral1_I1_J,axiom,
% 12.85/6.90 ! [B: real,W: num,A: real] :
% 12.85/6.90 ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
% 12.85/6.90 = A )
% 12.85/6.90 = ( ( ( ( numeral_numeral_real @ W )
% 12.85/6.90 != zero_zero_real )
% 12.85/6.90 => ( B
% 12.85/6.90 = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
% 12.85/6.90 & ( ( ( numeral_numeral_real @ W )
% 12.85/6.90 = zero_zero_real )
% 12.85/6.90 => ( A = zero_zero_real ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_497_divide__eq__eq__numeral1_I1_J,axiom,
% 12.85/6.90 ! [B: rat,W: num,A: rat] :
% 12.85/6.90 ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
% 12.85/6.90 = A )
% 12.85/6.90 = ( ( ( ( numeral_numeral_rat @ W )
% 12.85/6.90 != zero_zero_rat )
% 12.85/6.90 => ( B
% 12.85/6.90 = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
% 12.85/6.90 & ( ( ( numeral_numeral_rat @ W )
% 12.85/6.90 = zero_zero_rat )
% 12.85/6.90 => ( A = zero_zero_rat ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_498_eq__divide__eq__numeral1_I1_J,axiom,
% 12.85/6.90 ! [A: complex,B: complex,W: num] :
% 12.85/6.90 ( ( A
% 12.85/6.90 = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
% 12.85/6.90 = ( ( ( ( numera6690914467698888265omplex @ W )
% 12.85/6.90 != zero_zero_complex )
% 12.85/6.90 => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
% 12.85/6.90 = B ) )
% 12.85/6.90 & ( ( ( numera6690914467698888265omplex @ W )
% 12.85/6.90 = zero_zero_complex )
% 12.85/6.90 => ( A = zero_zero_complex ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_499_eq__divide__eq__numeral1_I1_J,axiom,
% 12.85/6.90 ! [A: real,B: real,W: num] :
% 12.85/6.90 ( ( A
% 12.85/6.90 = ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 12.85/6.90 = ( ( ( ( numeral_numeral_real @ W )
% 12.85/6.90 != zero_zero_real )
% 12.85/6.90 => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
% 12.85/6.90 = B ) )
% 12.85/6.90 & ( ( ( numeral_numeral_real @ W )
% 12.85/6.90 = zero_zero_real )
% 12.85/6.90 => ( A = zero_zero_real ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_500_eq__divide__eq__numeral1_I1_J,axiom,
% 12.85/6.90 ! [A: rat,B: rat,W: num] :
% 12.85/6.90 ( ( A
% 12.85/6.90 = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 12.85/6.90 = ( ( ( ( numeral_numeral_rat @ W )
% 12.85/6.90 != zero_zero_rat )
% 12.85/6.90 => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
% 12.85/6.90 = B ) )
% 12.85/6.90 & ( ( ( numeral_numeral_rat @ W )
% 12.85/6.90 = zero_zero_rat )
% 12.85/6.90 => ( A = zero_zero_rat ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_501_of__nat__le__0__iff,axiom,
% 12.85/6.90 ! [M: nat] :
% 12.85/6.90 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 12.85/6.90 = ( M = zero_zero_nat ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_502_of__nat__le__0__iff,axiom,
% 12.85/6.90 ! [M: nat] :
% 12.85/6.90 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 12.85/6.90 = ( M = zero_zero_nat ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_503_of__nat__le__0__iff,axiom,
% 12.85/6.90 ! [M: nat] :
% 12.85/6.90 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 12.85/6.90 = ( M = zero_zero_nat ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_504_of__nat__le__0__iff,axiom,
% 12.85/6.90 ! [M: nat] :
% 12.85/6.90 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 12.85/6.90 = ( M = zero_zero_nat ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_505_one__le__mult__iff,axiom,
% 12.85/6.90 ! [M: nat,N: nat] :
% 12.85/6.90 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 12.85/6.90 = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 12.85/6.90 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_506_verit__la__generic,axiom,
% 12.85/6.90 ! [A: int,X: int] :
% 12.85/6.90 ( ( ord_less_eq_int @ A @ X )
% 12.85/6.90 | ( A = X )
% 12.85/6.90 | ( ord_less_eq_int @ X @ A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_507_le__numeral__extra_I3_J,axiom,
% 12.85/6.90 ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 12.85/6.90
% 12.85/6.90 thf(fact_508_le__numeral__extra_I3_J,axiom,
% 12.85/6.90 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 12.85/6.90
% 12.85/6.90 thf(fact_509_le__numeral__extra_I3_J,axiom,
% 12.85/6.90 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 12.85/6.90
% 12.85/6.90 thf(fact_510_le__numeral__extra_I3_J,axiom,
% 12.85/6.90 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 12.85/6.90
% 12.85/6.90 thf(fact_511_zero__neq__numeral,axiom,
% 12.85/6.90 ! [N: num] :
% 12.85/6.90 ( zero_zero_complex
% 12.85/6.90 != ( numera6690914467698888265omplex @ N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_512_zero__neq__numeral,axiom,
% 12.85/6.90 ! [N: num] :
% 12.85/6.90 ( zero_zero_real
% 12.85/6.90 != ( numeral_numeral_real @ N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_513_zero__neq__numeral,axiom,
% 12.85/6.90 ! [N: num] :
% 12.85/6.90 ( zero_zero_rat
% 12.85/6.90 != ( numeral_numeral_rat @ N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_514_zero__neq__numeral,axiom,
% 12.85/6.90 ! [N: num] :
% 12.85/6.90 ( zero_zero_nat
% 12.85/6.90 != ( numeral_numeral_nat @ N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_515_zero__neq__numeral,axiom,
% 12.85/6.90 ! [N: num] :
% 12.85/6.90 ( zero_zero_int
% 12.85/6.90 != ( numeral_numeral_int @ N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_516_mult__not__zero,axiom,
% 12.85/6.90 ! [A: complex,B: complex] :
% 12.85/6.90 ( ( ( times_times_complex @ A @ B )
% 12.85/6.90 != zero_zero_complex )
% 12.85/6.90 => ( ( A != zero_zero_complex )
% 12.85/6.90 & ( B != zero_zero_complex ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_517_mult__not__zero,axiom,
% 12.85/6.90 ! [A: real,B: real] :
% 12.85/6.90 ( ( ( times_times_real @ A @ B )
% 12.85/6.90 != zero_zero_real )
% 12.85/6.90 => ( ( A != zero_zero_real )
% 12.85/6.90 & ( B != zero_zero_real ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_518_mult__not__zero,axiom,
% 12.85/6.90 ! [A: rat,B: rat] :
% 12.85/6.90 ( ( ( times_times_rat @ A @ B )
% 12.85/6.90 != zero_zero_rat )
% 12.85/6.90 => ( ( A != zero_zero_rat )
% 12.85/6.90 & ( B != zero_zero_rat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_519_mult__not__zero,axiom,
% 12.85/6.90 ! [A: nat,B: nat] :
% 12.85/6.90 ( ( ( times_times_nat @ A @ B )
% 12.85/6.90 != zero_zero_nat )
% 12.85/6.90 => ( ( A != zero_zero_nat )
% 12.85/6.90 & ( B != zero_zero_nat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_520_mult__not__zero,axiom,
% 12.85/6.90 ! [A: int,B: int] :
% 12.85/6.90 ( ( ( times_times_int @ A @ B )
% 12.85/6.90 != zero_zero_int )
% 12.85/6.90 => ( ( A != zero_zero_int )
% 12.85/6.90 & ( B != zero_zero_int ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_521_divisors__zero,axiom,
% 12.85/6.90 ! [A: complex,B: complex] :
% 12.85/6.90 ( ( ( times_times_complex @ A @ B )
% 12.85/6.90 = zero_zero_complex )
% 12.85/6.90 => ( ( A = zero_zero_complex )
% 12.85/6.90 | ( B = zero_zero_complex ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_522_divisors__zero,axiom,
% 12.85/6.90 ! [A: real,B: real] :
% 12.85/6.90 ( ( ( times_times_real @ A @ B )
% 12.85/6.90 = zero_zero_real )
% 12.85/6.90 => ( ( A = zero_zero_real )
% 12.85/6.90 | ( B = zero_zero_real ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_523_divisors__zero,axiom,
% 12.85/6.90 ! [A: rat,B: rat] :
% 12.85/6.90 ( ( ( times_times_rat @ A @ B )
% 12.85/6.90 = zero_zero_rat )
% 12.85/6.90 => ( ( A = zero_zero_rat )
% 12.85/6.90 | ( B = zero_zero_rat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_524_divisors__zero,axiom,
% 12.85/6.90 ! [A: nat,B: nat] :
% 12.85/6.90 ( ( ( times_times_nat @ A @ B )
% 12.85/6.90 = zero_zero_nat )
% 12.85/6.90 => ( ( A = zero_zero_nat )
% 12.85/6.90 | ( B = zero_zero_nat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_525_divisors__zero,axiom,
% 12.85/6.90 ! [A: int,B: int] :
% 12.85/6.90 ( ( ( times_times_int @ A @ B )
% 12.85/6.90 = zero_zero_int )
% 12.85/6.90 => ( ( A = zero_zero_int )
% 12.85/6.90 | ( B = zero_zero_int ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_526_no__zero__divisors,axiom,
% 12.85/6.90 ! [A: complex,B: complex] :
% 12.85/6.90 ( ( A != zero_zero_complex )
% 12.85/6.90 => ( ( B != zero_zero_complex )
% 12.85/6.90 => ( ( times_times_complex @ A @ B )
% 12.85/6.90 != zero_zero_complex ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_527_no__zero__divisors,axiom,
% 12.85/6.90 ! [A: real,B: real] :
% 12.85/6.90 ( ( A != zero_zero_real )
% 12.85/6.90 => ( ( B != zero_zero_real )
% 12.85/6.90 => ( ( times_times_real @ A @ B )
% 12.85/6.90 != zero_zero_real ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_528_no__zero__divisors,axiom,
% 12.85/6.90 ! [A: rat,B: rat] :
% 12.85/6.90 ( ( A != zero_zero_rat )
% 12.85/6.90 => ( ( B != zero_zero_rat )
% 12.85/6.90 => ( ( times_times_rat @ A @ B )
% 12.85/6.90 != zero_zero_rat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_529_no__zero__divisors,axiom,
% 12.85/6.90 ! [A: nat,B: nat] :
% 12.85/6.90 ( ( A != zero_zero_nat )
% 12.85/6.90 => ( ( B != zero_zero_nat )
% 12.85/6.90 => ( ( times_times_nat @ A @ B )
% 12.85/6.90 != zero_zero_nat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_530_no__zero__divisors,axiom,
% 12.85/6.90 ! [A: int,B: int] :
% 12.85/6.90 ( ( A != zero_zero_int )
% 12.85/6.90 => ( ( B != zero_zero_int )
% 12.85/6.90 => ( ( times_times_int @ A @ B )
% 12.85/6.90 != zero_zero_int ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_531_mult__left__cancel,axiom,
% 12.85/6.90 ! [C: complex,A: complex,B: complex] :
% 12.85/6.90 ( ( C != zero_zero_complex )
% 12.85/6.90 => ( ( ( times_times_complex @ C @ A )
% 12.85/6.90 = ( times_times_complex @ C @ B ) )
% 12.85/6.90 = ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_532_mult__left__cancel,axiom,
% 12.85/6.90 ! [C: real,A: real,B: real] :
% 12.85/6.90 ( ( C != zero_zero_real )
% 12.85/6.90 => ( ( ( times_times_real @ C @ A )
% 12.85/6.90 = ( times_times_real @ C @ B ) )
% 12.85/6.90 = ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_533_mult__left__cancel,axiom,
% 12.85/6.90 ! [C: rat,A: rat,B: rat] :
% 12.85/6.90 ( ( C != zero_zero_rat )
% 12.85/6.90 => ( ( ( times_times_rat @ C @ A )
% 12.85/6.90 = ( times_times_rat @ C @ B ) )
% 12.85/6.90 = ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_534_mult__left__cancel,axiom,
% 12.85/6.90 ! [C: nat,A: nat,B: nat] :
% 12.85/6.90 ( ( C != zero_zero_nat )
% 12.85/6.90 => ( ( ( times_times_nat @ C @ A )
% 12.85/6.90 = ( times_times_nat @ C @ B ) )
% 12.85/6.90 = ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_535_mult__left__cancel,axiom,
% 12.85/6.90 ! [C: int,A: int,B: int] :
% 12.85/6.90 ( ( C != zero_zero_int )
% 12.85/6.90 => ( ( ( times_times_int @ C @ A )
% 12.85/6.90 = ( times_times_int @ C @ B ) )
% 12.85/6.90 = ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_536_mult__right__cancel,axiom,
% 12.85/6.90 ! [C: complex,A: complex,B: complex] :
% 12.85/6.90 ( ( C != zero_zero_complex )
% 12.85/6.90 => ( ( ( times_times_complex @ A @ C )
% 12.85/6.90 = ( times_times_complex @ B @ C ) )
% 12.85/6.90 = ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_537_mult__right__cancel,axiom,
% 12.85/6.90 ! [C: real,A: real,B: real] :
% 12.85/6.90 ( ( C != zero_zero_real )
% 12.85/6.90 => ( ( ( times_times_real @ A @ C )
% 12.85/6.90 = ( times_times_real @ B @ C ) )
% 12.85/6.90 = ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_538_mult__right__cancel,axiom,
% 12.85/6.90 ! [C: rat,A: rat,B: rat] :
% 12.85/6.90 ( ( C != zero_zero_rat )
% 12.85/6.90 => ( ( ( times_times_rat @ A @ C )
% 12.85/6.90 = ( times_times_rat @ B @ C ) )
% 12.85/6.90 = ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_539_mult__right__cancel,axiom,
% 12.85/6.90 ! [C: nat,A: nat,B: nat] :
% 12.85/6.90 ( ( C != zero_zero_nat )
% 12.85/6.90 => ( ( ( times_times_nat @ A @ C )
% 12.85/6.90 = ( times_times_nat @ B @ C ) )
% 12.85/6.90 = ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_540_mult__right__cancel,axiom,
% 12.85/6.90 ! [C: int,A: int,B: int] :
% 12.85/6.90 ( ( C != zero_zero_int )
% 12.85/6.90 => ( ( ( times_times_int @ A @ C )
% 12.85/6.90 = ( times_times_int @ B @ C ) )
% 12.85/6.90 = ( A = B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_541_dvd__0__left,axiom,
% 12.85/6.90 ! [A: code_integer] :
% 12.85/6.90 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 12.85/6.90 => ( A = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_542_dvd__0__left,axiom,
% 12.85/6.90 ! [A: complex] :
% 12.85/6.90 ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 12.85/6.90 => ( A = zero_zero_complex ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_543_dvd__0__left,axiom,
% 12.85/6.90 ! [A: real] :
% 12.85/6.90 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 12.85/6.90 => ( A = zero_zero_real ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_544_dvd__0__left,axiom,
% 12.85/6.90 ! [A: rat] :
% 12.85/6.90 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 12.85/6.90 => ( A = zero_zero_rat ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_545_dvd__0__left,axiom,
% 12.85/6.90 ! [A: nat] :
% 12.85/6.90 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 12.85/6.90 => ( A = zero_zero_nat ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_546_dvd__0__left,axiom,
% 12.85/6.90 ! [A: int] :
% 12.85/6.90 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 12.85/6.90 => ( A = zero_zero_int ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_547_dvd__field__iff,axiom,
% 12.85/6.90 ( dvd_dvd_complex
% 12.85/6.90 = ( ^ [A3: complex,B2: complex] :
% 12.85/6.90 ( ( A3 = zero_zero_complex )
% 12.85/6.90 => ( B2 = zero_zero_complex ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_548_dvd__field__iff,axiom,
% 12.85/6.90 ( dvd_dvd_real
% 12.85/6.90 = ( ^ [A3: real,B2: real] :
% 12.85/6.90 ( ( A3 = zero_zero_real )
% 12.85/6.90 => ( B2 = zero_zero_real ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_549_dvd__field__iff,axiom,
% 12.85/6.90 ( dvd_dvd_rat
% 12.85/6.90 = ( ^ [A3: rat,B2: rat] :
% 12.85/6.90 ( ( A3 = zero_zero_rat )
% 12.85/6.90 => ( B2 = zero_zero_rat ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_550_list__decode_Ocases,axiom,
% 12.85/6.90 ! [X: nat] :
% 12.85/6.90 ( ( X != zero_zero_nat )
% 12.85/6.90 => ~ ! [N2: nat] :
% 12.85/6.90 ( X
% 12.85/6.90 != ( suc @ N2 ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_551_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
% 12.85/6.90 ! [X: nat] :
% 12.85/6.90 ( ( X != zero_zero_nat )
% 12.85/6.90 => ( ( X
% 12.85/6.90 != ( suc @ zero_zero_nat ) )
% 12.85/6.90 => ~ ! [Va: nat] :
% 12.85/6.90 ( X
% 12.85/6.90 != ( suc @ ( suc @ Va ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_552_nat_Odistinct_I1_J,axiom,
% 12.85/6.90 ! [X2: nat] :
% 12.85/6.90 ( zero_zero_nat
% 12.85/6.90 != ( suc @ X2 ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_553_old_Onat_Odistinct_I2_J,axiom,
% 12.85/6.90 ! [Nat2: nat] :
% 12.85/6.90 ( ( suc @ Nat2 )
% 12.85/6.90 != zero_zero_nat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_554_old_Onat_Odistinct_I1_J,axiom,
% 12.85/6.90 ! [Nat2: nat] :
% 12.85/6.90 ( zero_zero_nat
% 12.85/6.90 != ( suc @ Nat2 ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_555_nat_OdiscI,axiom,
% 12.85/6.90 ! [Nat: nat,X2: nat] :
% 12.85/6.90 ( ( Nat
% 12.85/6.90 = ( suc @ X2 ) )
% 12.85/6.90 => ( Nat != zero_zero_nat ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_556_old_Onat_Oexhaust,axiom,
% 12.85/6.90 ! [Y4: nat] :
% 12.85/6.90 ( ( Y4 != zero_zero_nat )
% 12.85/6.90 => ~ ! [Nat3: nat] :
% 12.85/6.90 ( Y4
% 12.85/6.90 != ( suc @ Nat3 ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_557_nat__induct,axiom,
% 12.85/6.90 ! [P: nat > $o,N: nat] :
% 12.85/6.90 ( ( P @ zero_zero_nat )
% 12.85/6.90 => ( ! [N2: nat] :
% 12.85/6.90 ( ( P @ N2 )
% 12.85/6.90 => ( P @ ( suc @ N2 ) ) )
% 12.85/6.90 => ( P @ N ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_558_diff__induct,axiom,
% 12.85/6.90 ! [P: nat > nat > $o,M: nat,N: nat] :
% 12.85/6.90 ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
% 12.85/6.90 => ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
% 12.85/6.90 => ( ! [X4: nat,Y: nat] :
% 12.85/6.90 ( ( P @ X4 @ Y )
% 12.85/6.90 => ( P @ ( suc @ X4 ) @ ( suc @ Y ) ) )
% 12.85/6.90 => ( P @ M @ N ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_559_zero__induct,axiom,
% 12.85/6.90 ! [P: nat > $o,K: nat] :
% 12.85/6.90 ( ( P @ K )
% 12.85/6.90 => ( ! [N2: nat] :
% 12.85/6.90 ( ( P @ ( suc @ N2 ) )
% 12.85/6.90 => ( P @ N2 ) )
% 12.85/6.90 => ( P @ zero_zero_nat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_560_Suc__neq__Zero,axiom,
% 12.85/6.90 ! [M: nat] :
% 12.85/6.90 ( ( suc @ M )
% 12.85/6.90 != zero_zero_nat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_561_Zero__neq__Suc,axiom,
% 12.85/6.90 ! [M: nat] :
% 12.85/6.90 ( zero_zero_nat
% 12.85/6.90 != ( suc @ M ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_562_Zero__not__Suc,axiom,
% 12.85/6.90 ! [M: nat] :
% 12.85/6.90 ( zero_zero_nat
% 12.85/6.90 != ( suc @ M ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_563_not0__implies__Suc,axiom,
% 12.85/6.90 ! [N: nat] :
% 12.85/6.90 ( ( N != zero_zero_nat )
% 12.85/6.90 => ? [M4: nat] :
% 12.85/6.90 ( N
% 12.85/6.90 = ( suc @ M4 ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_564_le__0__eq,axiom,
% 12.85/6.90 ! [N: nat] :
% 12.85/6.90 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 12.85/6.90 = ( N = zero_zero_nat ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_565_bot__nat__0_Oextremum__uniqueI,axiom,
% 12.85/6.90 ! [A: nat] :
% 12.85/6.90 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 12.85/6.90 => ( A = zero_zero_nat ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_566_bot__nat__0_Oextremum__unique,axiom,
% 12.85/6.90 ! [A: nat] :
% 12.85/6.90 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 12.85/6.90 = ( A = zero_zero_nat ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_567_less__eq__nat_Osimps_I1_J,axiom,
% 12.85/6.90 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_568_nat__mult__eq__cancel__disj,axiom,
% 12.85/6.90 ! [K: nat,M: nat,N: nat] :
% 12.85/6.90 ( ( ( times_times_nat @ K @ M )
% 12.85/6.90 = ( times_times_nat @ K @ N ) )
% 12.85/6.90 = ( ( K = zero_zero_nat )
% 12.85/6.90 | ( M = N ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_569_mult__0,axiom,
% 12.85/6.90 ! [N: nat] :
% 12.85/6.90 ( ( times_times_nat @ zero_zero_nat @ N )
% 12.85/6.90 = zero_zero_nat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_570_zero__le__numeral,axiom,
% 12.85/6.90 ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_571_zero__le__numeral,axiom,
% 12.85/6.90 ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_572_zero__le__numeral,axiom,
% 12.85/6.90 ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_573_zero__le__numeral,axiom,
% 12.85/6.90 ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_574_not__numeral__le__zero,axiom,
% 12.85/6.90 ! [N: num] :
% 12.85/6.90 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_575_not__numeral__le__zero,axiom,
% 12.85/6.90 ! [N: num] :
% 12.85/6.90 ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_576_not__numeral__le__zero,axiom,
% 12.85/6.90 ! [N: num] :
% 12.85/6.90 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_577_not__numeral__le__zero,axiom,
% 12.85/6.90 ! [N: num] :
% 12.85/6.90 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_578_mult__mono,axiom,
% 12.85/6.90 ! [A: rat,B: rat,C: rat,D: rat] :
% 12.85/6.90 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_rat @ C @ D )
% 12.85/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 12.85/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.85/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_579_mult__mono,axiom,
% 12.85/6.90 ! [A: nat,B: nat,C: nat,D: nat] :
% 12.85/6.90 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_nat @ C @ D )
% 12.85/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 12.85/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.85/6.90 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_580_mult__mono,axiom,
% 12.85/6.90 ! [A: int,B: int,C: int,D: int] :
% 12.85/6.90 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_int @ C @ D )
% 12.85/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 12.85/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.90 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_581_mult__mono,axiom,
% 12.85/6.90 ! [A: real,B: real,C: real,D: real] :
% 12.85/6.90 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_real @ C @ D )
% 12.85/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 12.85/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.90 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_582_mult__mono_H,axiom,
% 12.85/6.90 ! [A: rat,B: rat,C: rat,D: rat] :
% 12.85/6.90 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_rat @ C @ D )
% 12.85/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.85/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_583_mult__mono_H,axiom,
% 12.85/6.90 ! [A: nat,B: nat,C: nat,D: nat] :
% 12.85/6.90 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_nat @ C @ D )
% 12.85/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.85/6.90 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_584_mult__mono_H,axiom,
% 12.85/6.90 ! [A: int,B: int,C: int,D: int] :
% 12.85/6.90 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_int @ C @ D )
% 12.85/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.90 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_585_mult__mono_H,axiom,
% 12.85/6.90 ! [A: real,B: real,C: real,D: real] :
% 12.85/6.90 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_real @ C @ D )
% 12.85/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.90 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_586_zero__le__square,axiom,
% 12.85/6.90 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_587_zero__le__square,axiom,
% 12.85/6.90 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_588_zero__le__square,axiom,
% 12.85/6.90 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_589_split__mult__pos__le,axiom,
% 12.85/6.90 ! [A: rat,B: rat] :
% 12.85/6.90 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.90 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 12.85/6.90 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.85/6.90 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 12.85/6.90 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_590_split__mult__pos__le,axiom,
% 12.85/6.90 ! [A: int,B: int] :
% 12.85/6.90 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.90 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 12.85/6.90 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.85/6.90 & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 12.85/6.90 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_591_split__mult__pos__le,axiom,
% 12.85/6.90 ! [A: real,B: real] :
% 12.85/6.90 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.90 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 12.85/6.90 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.85/6.90 & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 12.85/6.90 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_592_mult__left__mono__neg,axiom,
% 12.85/6.90 ! [B: rat,A: rat,C: rat] :
% 12.85/6.90 ( ( ord_less_eq_rat @ B @ A )
% 12.85/6.90 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.85/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_593_mult__left__mono__neg,axiom,
% 12.85/6.90 ! [B: int,A: int,C: int] :
% 12.85/6.90 ( ( ord_less_eq_int @ B @ A )
% 12.85/6.90 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 12.85/6.90 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_594_mult__left__mono__neg,axiom,
% 12.85/6.90 ! [B: real,A: real,C: real] :
% 12.85/6.90 ( ( ord_less_eq_real @ B @ A )
% 12.85/6.90 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.85/6.90 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_595_mult__nonpos__nonpos,axiom,
% 12.85/6.90 ! [A: rat,B: rat] :
% 12.85/6.90 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.85/6.90 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 12.85/6.90 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_596_mult__nonpos__nonpos,axiom,
% 12.85/6.90 ! [A: int,B: int] :
% 12.85/6.90 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.85/6.90 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 12.85/6.90 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_597_mult__nonpos__nonpos,axiom,
% 12.85/6.90 ! [A: real,B: real] :
% 12.85/6.90 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.85/6.90 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 12.85/6.90 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_598_mult__left__mono,axiom,
% 12.85/6.90 ! [A: rat,B: rat,C: rat] :
% 12.85/6.90 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.85/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_599_mult__left__mono,axiom,
% 12.85/6.90 ! [A: nat,B: nat,C: nat] :
% 12.85/6.90 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.85/6.90 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_600_mult__left__mono,axiom,
% 12.85/6.90 ! [A: int,B: int,C: int] :
% 12.85/6.90 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.90 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_601_mult__left__mono,axiom,
% 12.85/6.90 ! [A: real,B: real,C: real] :
% 12.85/6.90 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.90 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_602_mult__right__mono__neg,axiom,
% 12.85/6.90 ! [B: rat,A: rat,C: rat] :
% 12.85/6.90 ( ( ord_less_eq_rat @ B @ A )
% 12.85/6.90 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.85/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_603_mult__right__mono__neg,axiom,
% 12.85/6.90 ! [B: int,A: int,C: int] :
% 12.85/6.90 ( ( ord_less_eq_int @ B @ A )
% 12.85/6.90 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 12.85/6.90 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_604_mult__right__mono__neg,axiom,
% 12.85/6.90 ! [B: real,A: real,C: real] :
% 12.85/6.90 ( ( ord_less_eq_real @ B @ A )
% 12.85/6.90 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.85/6.90 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_605_mult__right__mono,axiom,
% 12.85/6.90 ! [A: rat,B: rat,C: rat] :
% 12.85/6.90 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.85/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_606_mult__right__mono,axiom,
% 12.85/6.90 ! [A: nat,B: nat,C: nat] :
% 12.85/6.90 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.85/6.90 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_607_mult__right__mono,axiom,
% 12.85/6.90 ! [A: int,B: int,C: int] :
% 12.85/6.90 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.90 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_608_mult__right__mono,axiom,
% 12.85/6.90 ! [A: real,B: real,C: real] :
% 12.85/6.90 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.90 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_609_mult__le__0__iff,axiom,
% 12.85/6.90 ! [A: rat,B: rat] :
% 12.85/6.90 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 12.85/6.90 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.90 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 12.85/6.90 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.85/6.90 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_610_mult__le__0__iff,axiom,
% 12.85/6.90 ! [A: int,B: int] :
% 12.85/6.90 ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 12.85/6.90 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.90 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 12.85/6.90 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.85/6.90 & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_611_mult__le__0__iff,axiom,
% 12.85/6.90 ! [A: real,B: real] :
% 12.85/6.90 ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 12.85/6.90 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.90 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 12.85/6.90 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.85/6.90 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_612_split__mult__neg__le,axiom,
% 12.85/6.90 ! [A: rat,B: rat] :
% 12.85/6.90 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.90 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 12.85/6.90 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.85/6.90 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
% 12.85/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_613_split__mult__neg__le,axiom,
% 12.85/6.90 ! [A: nat,B: nat] :
% 12.85/6.90 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.90 & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 12.85/6.90 | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 12.85/6.90 & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 12.85/6.90 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_614_split__mult__neg__le,axiom,
% 12.85/6.90 ! [A: int,B: int] :
% 12.85/6.90 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.90 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 12.85/6.90 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.85/6.90 & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
% 12.85/6.90 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_615_split__mult__neg__le,axiom,
% 12.85/6.90 ! [A: real,B: real] :
% 12.85/6.90 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.90 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 12.85/6.90 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.85/6.90 & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
% 12.85/6.90 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_616_mult__nonneg__nonneg,axiom,
% 12.85/6.90 ! [A: rat,B: rat] :
% 12.85/6.90 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 12.85/6.90 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_617_mult__nonneg__nonneg,axiom,
% 12.85/6.90 ! [A: nat,B: nat] :
% 12.85/6.90 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 12.85/6.90 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_618_mult__nonneg__nonneg,axiom,
% 12.85/6.90 ! [A: int,B: int] :
% 12.85/6.90 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 12.85/6.90 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_619_mult__nonneg__nonneg,axiom,
% 12.85/6.90 ! [A: real,B: real] :
% 12.85/6.90 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 12.85/6.90 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_620_mult__nonneg__nonpos,axiom,
% 12.85/6.90 ! [A: rat,B: rat] :
% 12.85/6.90 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.90 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 12.85/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_621_mult__nonneg__nonpos,axiom,
% 12.85/6.90 ! [A: nat,B: nat] :
% 12.85/6.90 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.90 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 12.85/6.90 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_622_mult__nonneg__nonpos,axiom,
% 12.85/6.90 ! [A: int,B: int] :
% 12.85/6.90 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.90 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 12.85/6.90 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_623_mult__nonneg__nonpos,axiom,
% 12.85/6.90 ! [A: real,B: real] :
% 12.85/6.90 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.90 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 12.85/6.90 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_624_mult__nonpos__nonneg,axiom,
% 12.85/6.90 ! [A: rat,B: rat] :
% 12.85/6.90 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.85/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 12.85/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_625_mult__nonpos__nonneg,axiom,
% 12.85/6.90 ! [A: nat,B: nat] :
% 12.85/6.90 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 12.85/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 12.85/6.90 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_626_mult__nonpos__nonneg,axiom,
% 12.85/6.90 ! [A: int,B: int] :
% 12.85/6.90 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.85/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 12.85/6.90 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_627_mult__nonpos__nonneg,axiom,
% 12.85/6.90 ! [A: real,B: real] :
% 12.85/6.90 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.85/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 12.85/6.90 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_628_mult__nonneg__nonpos2,axiom,
% 12.85/6.90 ! [A: rat,B: rat] :
% 12.85/6.90 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.90 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 12.85/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_629_mult__nonneg__nonpos2,axiom,
% 12.85/6.90 ! [A: nat,B: nat] :
% 12.85/6.90 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.90 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 12.85/6.90 => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_630_mult__nonneg__nonpos2,axiom,
% 12.85/6.90 ! [A: int,B: int] :
% 12.85/6.90 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.90 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 12.85/6.90 => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_631_mult__nonneg__nonpos2,axiom,
% 12.85/6.90 ! [A: real,B: real] :
% 12.85/6.90 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.90 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 12.85/6.90 => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_632_zero__le__mult__iff,axiom,
% 12.85/6.90 ! [A: rat,B: rat] :
% 12.85/6.90 ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 12.85/6.90 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.90 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 12.85/6.90 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.85/6.90 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_633_zero__le__mult__iff,axiom,
% 12.85/6.90 ! [A: int,B: int] :
% 12.85/6.90 ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 12.85/6.90 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.90 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 12.85/6.90 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.85/6.90 & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_634_zero__le__mult__iff,axiom,
% 12.85/6.90 ! [A: real,B: real] :
% 12.85/6.90 ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 12.85/6.90 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.90 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 12.85/6.90 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.85/6.90 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_635_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 12.85/6.90 ! [A: rat,B: rat,C: rat] :
% 12.85/6.90 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.85/6.90 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_636_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 12.85/6.90 ! [A: nat,B: nat,C: nat] :
% 12.85/6.90 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.85/6.90 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_637_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 12.85/6.90 ! [A: int,B: int,C: int] :
% 12.85/6.90 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.90 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_638_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 12.85/6.90 ! [A: real,B: real,C: real] :
% 12.85/6.90 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.90 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_639_divide__le__0__iff,axiom,
% 12.85/6.90 ! [A: rat,B: rat] :
% 12.85/6.90 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 12.85/6.90 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.90 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 12.85/6.90 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.85/6.90 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_640_divide__le__0__iff,axiom,
% 12.85/6.90 ! [A: real,B: real] :
% 12.85/6.90 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 12.85/6.90 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.90 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 12.85/6.90 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.85/6.90 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_641_divide__right__mono,axiom,
% 12.85/6.90 ! [A: rat,B: rat,C: rat] :
% 12.85/6.90 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.85/6.90 => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_642_divide__right__mono,axiom,
% 12.85/6.90 ! [A: real,B: real,C: real] :
% 12.85/6.90 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.90 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.90 => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_643_zero__le__divide__iff,axiom,
% 12.85/6.90 ! [A: rat,B: rat] :
% 12.85/6.90 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 12.85/6.90 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.90 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 12.85/6.90 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.85/6.90 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 12.85/6.90
% 12.85/6.90 thf(fact_644_zero__le__divide__iff,axiom,
% 12.85/6.90 ! [A: real,B: real] :
% 12.85/6.90 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 12.85/6.91 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.91 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 12.85/6.91 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.85/6.91 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_645_divide__nonneg__nonneg,axiom,
% 12.85/6.91 ! [X: rat,Y4: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 12.85/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 12.85/6.91 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_646_divide__nonneg__nonneg,axiom,
% 12.85/6.91 ! [X: real,Y4: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/6.91 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_647_divide__nonneg__nonpos,axiom,
% 12.85/6.91 ! [X: rat,Y4: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 12.85/6.91 => ( ( ord_less_eq_rat @ Y4 @ zero_zero_rat )
% 12.85/6.91 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y4 ) @ zero_zero_rat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_648_divide__nonneg__nonpos,axiom,
% 12.85/6.91 ! [X: real,Y4: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/6.91 => ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
% 12.85/6.91 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_649_divide__nonpos__nonneg,axiom,
% 12.85/6.91 ! [X: rat,Y4: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 12.85/6.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 12.85/6.91 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y4 ) @ zero_zero_rat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_650_divide__nonpos__nonneg,axiom,
% 12.85/6.91 ! [X: real,Y4: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 12.85/6.91 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/6.91 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_651_divide__nonpos__nonpos,axiom,
% 12.85/6.91 ! [X: rat,Y4: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 12.85/6.91 => ( ( ord_less_eq_rat @ Y4 @ zero_zero_rat )
% 12.85/6.91 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_652_divide__nonpos__nonpos,axiom,
% 12.85/6.91 ! [X: real,Y4: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 12.85/6.91 => ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
% 12.85/6.91 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_653_divide__right__mono__neg,axiom,
% 12.85/6.91 ! [A: rat,B: rat,C: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.85/6.91 => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_654_divide__right__mono__neg,axiom,
% 12.85/6.91 ! [A: real,B: real,C: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.85/6.91 => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_655_frac__eq__eq,axiom,
% 12.85/6.91 ! [Y4: complex,Z: complex,X: complex,W: complex] :
% 12.85/6.91 ( ( Y4 != zero_zero_complex )
% 12.85/6.91 => ( ( Z != zero_zero_complex )
% 12.85/6.91 => ( ( ( divide1717551699836669952omplex @ X @ Y4 )
% 12.85/6.91 = ( divide1717551699836669952omplex @ W @ Z ) )
% 12.85/6.91 = ( ( times_times_complex @ X @ Z )
% 12.85/6.91 = ( times_times_complex @ W @ Y4 ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_656_frac__eq__eq,axiom,
% 12.85/6.91 ! [Y4: real,Z: real,X: real,W: real] :
% 12.85/6.91 ( ( Y4 != zero_zero_real )
% 12.85/6.91 => ( ( Z != zero_zero_real )
% 12.85/6.91 => ( ( ( divide_divide_real @ X @ Y4 )
% 12.85/6.91 = ( divide_divide_real @ W @ Z ) )
% 12.85/6.91 = ( ( times_times_real @ X @ Z )
% 12.85/6.91 = ( times_times_real @ W @ Y4 ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_657_frac__eq__eq,axiom,
% 12.85/6.91 ! [Y4: rat,Z: rat,X: rat,W: rat] :
% 12.85/6.91 ( ( Y4 != zero_zero_rat )
% 12.85/6.91 => ( ( Z != zero_zero_rat )
% 12.85/6.91 => ( ( ( divide_divide_rat @ X @ Y4 )
% 12.85/6.91 = ( divide_divide_rat @ W @ Z ) )
% 12.85/6.91 = ( ( times_times_rat @ X @ Z )
% 12.85/6.91 = ( times_times_rat @ W @ Y4 ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_658_divide__eq__eq,axiom,
% 12.85/6.91 ! [B: complex,C: complex,A: complex] :
% 12.85/6.91 ( ( ( divide1717551699836669952omplex @ B @ C )
% 12.85/6.91 = A )
% 12.85/6.91 = ( ( ( C != zero_zero_complex )
% 12.85/6.91 => ( B
% 12.85/6.91 = ( times_times_complex @ A @ C ) ) )
% 12.85/6.91 & ( ( C = zero_zero_complex )
% 12.85/6.91 => ( A = zero_zero_complex ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_659_divide__eq__eq,axiom,
% 12.85/6.91 ! [B: real,C: real,A: real] :
% 12.85/6.91 ( ( ( divide_divide_real @ B @ C )
% 12.85/6.91 = A )
% 12.85/6.91 = ( ( ( C != zero_zero_real )
% 12.85/6.91 => ( B
% 12.85/6.91 = ( times_times_real @ A @ C ) ) )
% 12.85/6.91 & ( ( C = zero_zero_real )
% 12.85/6.91 => ( A = zero_zero_real ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_660_divide__eq__eq,axiom,
% 12.85/6.91 ! [B: rat,C: rat,A: rat] :
% 12.85/6.91 ( ( ( divide_divide_rat @ B @ C )
% 12.85/6.91 = A )
% 12.85/6.91 = ( ( ( C != zero_zero_rat )
% 12.85/6.91 => ( B
% 12.85/6.91 = ( times_times_rat @ A @ C ) ) )
% 12.85/6.91 & ( ( C = zero_zero_rat )
% 12.85/6.91 => ( A = zero_zero_rat ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_661_eq__divide__eq,axiom,
% 12.85/6.91 ! [A: complex,B: complex,C: complex] :
% 12.85/6.91 ( ( A
% 12.85/6.91 = ( divide1717551699836669952omplex @ B @ C ) )
% 12.85/6.91 = ( ( ( C != zero_zero_complex )
% 12.85/6.91 => ( ( times_times_complex @ A @ C )
% 12.85/6.91 = B ) )
% 12.85/6.91 & ( ( C = zero_zero_complex )
% 12.85/6.91 => ( A = zero_zero_complex ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_662_eq__divide__eq,axiom,
% 12.85/6.91 ! [A: real,B: real,C: real] :
% 12.85/6.91 ( ( A
% 12.85/6.91 = ( divide_divide_real @ B @ C ) )
% 12.85/6.91 = ( ( ( C != zero_zero_real )
% 12.85/6.91 => ( ( times_times_real @ A @ C )
% 12.85/6.91 = B ) )
% 12.85/6.91 & ( ( C = zero_zero_real )
% 12.85/6.91 => ( A = zero_zero_real ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_663_eq__divide__eq,axiom,
% 12.85/6.91 ! [A: rat,B: rat,C: rat] :
% 12.85/6.91 ( ( A
% 12.85/6.91 = ( divide_divide_rat @ B @ C ) )
% 12.85/6.91 = ( ( ( C != zero_zero_rat )
% 12.85/6.91 => ( ( times_times_rat @ A @ C )
% 12.85/6.91 = B ) )
% 12.85/6.91 & ( ( C = zero_zero_rat )
% 12.85/6.91 => ( A = zero_zero_rat ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_664_divide__eq__imp,axiom,
% 12.85/6.91 ! [C: complex,B: complex,A: complex] :
% 12.85/6.91 ( ( C != zero_zero_complex )
% 12.85/6.91 => ( ( B
% 12.85/6.91 = ( times_times_complex @ A @ C ) )
% 12.85/6.91 => ( ( divide1717551699836669952omplex @ B @ C )
% 12.85/6.91 = A ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_665_divide__eq__imp,axiom,
% 12.85/6.91 ! [C: real,B: real,A: real] :
% 12.85/6.91 ( ( C != zero_zero_real )
% 12.85/6.91 => ( ( B
% 12.85/6.91 = ( times_times_real @ A @ C ) )
% 12.85/6.91 => ( ( divide_divide_real @ B @ C )
% 12.85/6.91 = A ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_666_divide__eq__imp,axiom,
% 12.85/6.91 ! [C: rat,B: rat,A: rat] :
% 12.85/6.91 ( ( C != zero_zero_rat )
% 12.85/6.91 => ( ( B
% 12.85/6.91 = ( times_times_rat @ A @ C ) )
% 12.85/6.91 => ( ( divide_divide_rat @ B @ C )
% 12.85/6.91 = A ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_667_eq__divide__imp,axiom,
% 12.85/6.91 ! [C: complex,A: complex,B: complex] :
% 12.85/6.91 ( ( C != zero_zero_complex )
% 12.85/6.91 => ( ( ( times_times_complex @ A @ C )
% 12.85/6.91 = B )
% 12.85/6.91 => ( A
% 12.85/6.91 = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_668_eq__divide__imp,axiom,
% 12.85/6.91 ! [C: real,A: real,B: real] :
% 12.85/6.91 ( ( C != zero_zero_real )
% 12.85/6.91 => ( ( ( times_times_real @ A @ C )
% 12.85/6.91 = B )
% 12.85/6.91 => ( A
% 12.85/6.91 = ( divide_divide_real @ B @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_669_eq__divide__imp,axiom,
% 12.85/6.91 ! [C: rat,A: rat,B: rat] :
% 12.85/6.91 ( ( C != zero_zero_rat )
% 12.85/6.91 => ( ( ( times_times_rat @ A @ C )
% 12.85/6.91 = B )
% 12.85/6.91 => ( A
% 12.85/6.91 = ( divide_divide_rat @ B @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_670_nonzero__divide__eq__eq,axiom,
% 12.85/6.91 ! [C: complex,B: complex,A: complex] :
% 12.85/6.91 ( ( C != zero_zero_complex )
% 12.85/6.91 => ( ( ( divide1717551699836669952omplex @ B @ C )
% 12.85/6.91 = A )
% 12.85/6.91 = ( B
% 12.85/6.91 = ( times_times_complex @ A @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_671_nonzero__divide__eq__eq,axiom,
% 12.85/6.91 ! [C: real,B: real,A: real] :
% 12.85/6.91 ( ( C != zero_zero_real )
% 12.85/6.91 => ( ( ( divide_divide_real @ B @ C )
% 12.85/6.91 = A )
% 12.85/6.91 = ( B
% 12.85/6.91 = ( times_times_real @ A @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_672_nonzero__divide__eq__eq,axiom,
% 12.85/6.91 ! [C: rat,B: rat,A: rat] :
% 12.85/6.91 ( ( C != zero_zero_rat )
% 12.85/6.91 => ( ( ( divide_divide_rat @ B @ C )
% 12.85/6.91 = A )
% 12.85/6.91 = ( B
% 12.85/6.91 = ( times_times_rat @ A @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_673_nonzero__eq__divide__eq,axiom,
% 12.85/6.91 ! [C: complex,A: complex,B: complex] :
% 12.85/6.91 ( ( C != zero_zero_complex )
% 12.85/6.91 => ( ( A
% 12.85/6.91 = ( divide1717551699836669952omplex @ B @ C ) )
% 12.85/6.91 = ( ( times_times_complex @ A @ C )
% 12.85/6.91 = B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_674_nonzero__eq__divide__eq,axiom,
% 12.85/6.91 ! [C: real,A: real,B: real] :
% 12.85/6.91 ( ( C != zero_zero_real )
% 12.85/6.91 => ( ( A
% 12.85/6.91 = ( divide_divide_real @ B @ C ) )
% 12.85/6.91 = ( ( times_times_real @ A @ C )
% 12.85/6.91 = B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_675_nonzero__eq__divide__eq,axiom,
% 12.85/6.91 ! [C: rat,A: rat,B: rat] :
% 12.85/6.91 ( ( C != zero_zero_rat )
% 12.85/6.91 => ( ( A
% 12.85/6.91 = ( divide_divide_rat @ B @ C ) )
% 12.85/6.91 = ( ( times_times_rat @ A @ C )
% 12.85/6.91 = B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_676_of__nat__0__le__iff,axiom,
% 12.85/6.91 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_677_of__nat__0__le__iff,axiom,
% 12.85/6.91 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_678_of__nat__0__le__iff,axiom,
% 12.85/6.91 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_679_of__nat__0__le__iff,axiom,
% 12.85/6.91 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_680_dvd__div__eq__0__iff,axiom,
% 12.85/6.91 ! [B: code_integer,A: code_integer] :
% 12.85/6.91 ( ( dvd_dvd_Code_integer @ B @ A )
% 12.85/6.91 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 12.85/6.91 = zero_z3403309356797280102nteger )
% 12.85/6.91 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_681_dvd__div__eq__0__iff,axiom,
% 12.85/6.91 ! [B: complex,A: complex] :
% 12.85/6.91 ( ( dvd_dvd_complex @ B @ A )
% 12.85/6.91 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 12.85/6.91 = zero_zero_complex )
% 12.85/6.91 = ( A = zero_zero_complex ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_682_dvd__div__eq__0__iff,axiom,
% 12.85/6.91 ! [B: real,A: real] :
% 12.85/6.91 ( ( dvd_dvd_real @ B @ A )
% 12.85/6.91 => ( ( ( divide_divide_real @ A @ B )
% 12.85/6.91 = zero_zero_real )
% 12.85/6.91 = ( A = zero_zero_real ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_683_dvd__div__eq__0__iff,axiom,
% 12.85/6.91 ! [B: rat,A: rat] :
% 12.85/6.91 ( ( dvd_dvd_rat @ B @ A )
% 12.85/6.91 => ( ( ( divide_divide_rat @ A @ B )
% 12.85/6.91 = zero_zero_rat )
% 12.85/6.91 = ( A = zero_zero_rat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_684_dvd__div__eq__0__iff,axiom,
% 12.85/6.91 ! [B: nat,A: nat] :
% 12.85/6.91 ( ( dvd_dvd_nat @ B @ A )
% 12.85/6.91 => ( ( ( divide_divide_nat @ A @ B )
% 12.85/6.91 = zero_zero_nat )
% 12.85/6.91 = ( A = zero_zero_nat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_685_dvd__div__eq__0__iff,axiom,
% 12.85/6.91 ! [B: int,A: int] :
% 12.85/6.91 ( ( dvd_dvd_int @ B @ A )
% 12.85/6.91 => ( ( ( divide_divide_int @ A @ B )
% 12.85/6.91 = zero_zero_int )
% 12.85/6.91 = ( A = zero_zero_int ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_686_of__nat__neq__0,axiom,
% 12.85/6.91 ! [N: nat] :
% 12.85/6.91 ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
% 12.85/6.91 != zero_zero_complex ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_687_of__nat__neq__0,axiom,
% 12.85/6.91 ! [N: nat] :
% 12.85/6.91 ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
% 12.85/6.91 != zero_zero_real ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_688_of__nat__neq__0,axiom,
% 12.85/6.91 ! [N: nat] :
% 12.85/6.91 ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
% 12.85/6.91 != zero_zero_rat ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_689_of__nat__neq__0,axiom,
% 12.85/6.91 ! [N: nat] :
% 12.85/6.91 ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
% 12.85/6.91 != zero_zero_nat ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_690_of__nat__neq__0,axiom,
% 12.85/6.91 ! [N: nat] :
% 12.85/6.91 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 12.85/6.91 != zero_zero_int ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_691_verit__comp__simplify1_I2_J,axiom,
% 12.85/6.91 ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_692_verit__comp__simplify1_I2_J,axiom,
% 12.85/6.91 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_693_verit__comp__simplify1_I2_J,axiom,
% 12.85/6.91 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_694_verit__comp__simplify1_I2_J,axiom,
% 12.85/6.91 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_695_verit__comp__simplify1_I2_J,axiom,
% 12.85/6.91 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_696_verit__comp__simplify1_I2_J,axiom,
% 12.85/6.91 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_697_nle__le,axiom,
% 12.85/6.91 ! [A: rat,B: rat] :
% 12.85/6.91 ( ( ~ ( ord_less_eq_rat @ A @ B ) )
% 12.85/6.91 = ( ( ord_less_eq_rat @ B @ A )
% 12.85/6.91 & ( B != A ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_698_nle__le,axiom,
% 12.85/6.91 ! [A: num,B: num] :
% 12.85/6.91 ( ( ~ ( ord_less_eq_num @ A @ B ) )
% 12.85/6.91 = ( ( ord_less_eq_num @ B @ A )
% 12.85/6.91 & ( B != A ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_699_nle__le,axiom,
% 12.85/6.91 ! [A: nat,B: nat] :
% 12.85/6.91 ( ( ~ ( ord_less_eq_nat @ A @ B ) )
% 12.85/6.91 = ( ( ord_less_eq_nat @ B @ A )
% 12.85/6.91 & ( B != A ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_700_nle__le,axiom,
% 12.85/6.91 ! [A: int,B: int] :
% 12.85/6.91 ( ( ~ ( ord_less_eq_int @ A @ B ) )
% 12.85/6.91 = ( ( ord_less_eq_int @ B @ A )
% 12.85/6.91 & ( B != A ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_701_nle__le,axiom,
% 12.85/6.91 ! [A: real,B: real] :
% 12.85/6.91 ( ( ~ ( ord_less_eq_real @ A @ B ) )
% 12.85/6.91 = ( ( ord_less_eq_real @ B @ A )
% 12.85/6.91 & ( B != A ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_702_le__cases3,axiom,
% 12.85/6.91 ! [X: rat,Y4: rat,Z: rat] :
% 12.85/6.91 ( ( ( ord_less_eq_rat @ X @ Y4 )
% 12.85/6.91 => ~ ( ord_less_eq_rat @ Y4 @ Z ) )
% 12.85/6.91 => ( ( ( ord_less_eq_rat @ Y4 @ X )
% 12.85/6.91 => ~ ( ord_less_eq_rat @ X @ Z ) )
% 12.85/6.91 => ( ( ( ord_less_eq_rat @ X @ Z )
% 12.85/6.91 => ~ ( ord_less_eq_rat @ Z @ Y4 ) )
% 12.85/6.91 => ( ( ( ord_less_eq_rat @ Z @ Y4 )
% 12.85/6.91 => ~ ( ord_less_eq_rat @ Y4 @ X ) )
% 12.85/6.91 => ( ( ( ord_less_eq_rat @ Y4 @ Z )
% 12.85/6.91 => ~ ( ord_less_eq_rat @ Z @ X ) )
% 12.85/6.91 => ~ ( ( ord_less_eq_rat @ Z @ X )
% 12.85/6.91 => ~ ( ord_less_eq_rat @ X @ Y4 ) ) ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_703_le__cases3,axiom,
% 12.85/6.91 ! [X: num,Y4: num,Z: num] :
% 12.85/6.91 ( ( ( ord_less_eq_num @ X @ Y4 )
% 12.85/6.91 => ~ ( ord_less_eq_num @ Y4 @ Z ) )
% 12.85/6.91 => ( ( ( ord_less_eq_num @ Y4 @ X )
% 12.85/6.91 => ~ ( ord_less_eq_num @ X @ Z ) )
% 12.85/6.91 => ( ( ( ord_less_eq_num @ X @ Z )
% 12.85/6.91 => ~ ( ord_less_eq_num @ Z @ Y4 ) )
% 12.85/6.91 => ( ( ( ord_less_eq_num @ Z @ Y4 )
% 12.85/6.91 => ~ ( ord_less_eq_num @ Y4 @ X ) )
% 12.85/6.91 => ( ( ( ord_less_eq_num @ Y4 @ Z )
% 12.85/6.91 => ~ ( ord_less_eq_num @ Z @ X ) )
% 12.85/6.91 => ~ ( ( ord_less_eq_num @ Z @ X )
% 12.85/6.91 => ~ ( ord_less_eq_num @ X @ Y4 ) ) ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_704_le__cases3,axiom,
% 12.85/6.91 ! [X: nat,Y4: nat,Z: nat] :
% 12.85/6.91 ( ( ( ord_less_eq_nat @ X @ Y4 )
% 12.85/6.91 => ~ ( ord_less_eq_nat @ Y4 @ Z ) )
% 12.85/6.91 => ( ( ( ord_less_eq_nat @ Y4 @ X )
% 12.85/6.91 => ~ ( ord_less_eq_nat @ X @ Z ) )
% 12.85/6.91 => ( ( ( ord_less_eq_nat @ X @ Z )
% 12.85/6.91 => ~ ( ord_less_eq_nat @ Z @ Y4 ) )
% 12.85/6.91 => ( ( ( ord_less_eq_nat @ Z @ Y4 )
% 12.85/6.91 => ~ ( ord_less_eq_nat @ Y4 @ X ) )
% 12.85/6.91 => ( ( ( ord_less_eq_nat @ Y4 @ Z )
% 12.85/6.91 => ~ ( ord_less_eq_nat @ Z @ X ) )
% 12.85/6.91 => ~ ( ( ord_less_eq_nat @ Z @ X )
% 12.85/6.91 => ~ ( ord_less_eq_nat @ X @ Y4 ) ) ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_705_le__cases3,axiom,
% 12.85/6.91 ! [X: int,Y4: int,Z: int] :
% 12.85/6.91 ( ( ( ord_less_eq_int @ X @ Y4 )
% 12.85/6.91 => ~ ( ord_less_eq_int @ Y4 @ Z ) )
% 12.85/6.91 => ( ( ( ord_less_eq_int @ Y4 @ X )
% 12.85/6.91 => ~ ( ord_less_eq_int @ X @ Z ) )
% 12.85/6.91 => ( ( ( ord_less_eq_int @ X @ Z )
% 12.85/6.91 => ~ ( ord_less_eq_int @ Z @ Y4 ) )
% 12.85/6.91 => ( ( ( ord_less_eq_int @ Z @ Y4 )
% 12.85/6.91 => ~ ( ord_less_eq_int @ Y4 @ X ) )
% 12.85/6.91 => ( ( ( ord_less_eq_int @ Y4 @ Z )
% 12.85/6.91 => ~ ( ord_less_eq_int @ Z @ X ) )
% 12.85/6.91 => ~ ( ( ord_less_eq_int @ Z @ X )
% 12.85/6.91 => ~ ( ord_less_eq_int @ X @ Y4 ) ) ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_706_le__cases3,axiom,
% 12.85/6.91 ! [X: real,Y4: real,Z: real] :
% 12.85/6.91 ( ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/6.91 => ~ ( ord_less_eq_real @ Y4 @ Z ) )
% 12.85/6.91 => ( ( ( ord_less_eq_real @ Y4 @ X )
% 12.85/6.91 => ~ ( ord_less_eq_real @ X @ Z ) )
% 12.85/6.91 => ( ( ( ord_less_eq_real @ X @ Z )
% 12.85/6.91 => ~ ( ord_less_eq_real @ Z @ Y4 ) )
% 12.85/6.91 => ( ( ( ord_less_eq_real @ Z @ Y4 )
% 12.85/6.91 => ~ ( ord_less_eq_real @ Y4 @ X ) )
% 12.85/6.91 => ( ( ( ord_less_eq_real @ Y4 @ Z )
% 12.85/6.91 => ~ ( ord_less_eq_real @ Z @ X ) )
% 12.85/6.91 => ~ ( ( ord_less_eq_real @ Z @ X )
% 12.85/6.91 => ~ ( ord_less_eq_real @ X @ Y4 ) ) ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_707_order__class_Oorder__eq__iff,axiom,
% 12.85/6.91 ( ( ^ [Y5: set_int,Z4: set_int] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [X3: set_int,Y6: set_int] :
% 12.85/6.91 ( ( ord_less_eq_set_int @ X3 @ Y6 )
% 12.85/6.91 & ( ord_less_eq_set_int @ Y6 @ X3 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_708_order__class_Oorder__eq__iff,axiom,
% 12.85/6.91 ( ( ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [X3: rat,Y6: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X3 @ Y6 )
% 12.85/6.91 & ( ord_less_eq_rat @ Y6 @ X3 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_709_order__class_Oorder__eq__iff,axiom,
% 12.85/6.91 ( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [X3: num,Y6: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X3 @ Y6 )
% 12.85/6.91 & ( ord_less_eq_num @ Y6 @ X3 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_710_order__class_Oorder__eq__iff,axiom,
% 12.85/6.91 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [X3: nat,Y6: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ X3 @ Y6 )
% 12.85/6.91 & ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_711_order__class_Oorder__eq__iff,axiom,
% 12.85/6.91 ( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [X3: int,Y6: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ X3 @ Y6 )
% 12.85/6.91 & ( ord_less_eq_int @ Y6 @ X3 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_712_order__class_Oorder__eq__iff,axiom,
% 12.85/6.91 ( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [X3: real,Y6: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ X3 @ Y6 )
% 12.85/6.91 & ( ord_less_eq_real @ Y6 @ X3 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_713_ord__eq__le__trans,axiom,
% 12.85/6.91 ! [A: set_int,B: set_int,C: set_int] :
% 12.85/6.91 ( ( A = B )
% 12.85/6.91 => ( ( ord_less_eq_set_int @ B @ C )
% 12.85/6.91 => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_714_ord__eq__le__trans,axiom,
% 12.85/6.91 ! [A: rat,B: rat,C: rat] :
% 12.85/6.91 ( ( A = B )
% 12.85/6.91 => ( ( ord_less_eq_rat @ B @ C )
% 12.85/6.91 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_715_ord__eq__le__trans,axiom,
% 12.85/6.91 ! [A: num,B: num,C: num] :
% 12.85/6.91 ( ( A = B )
% 12.85/6.91 => ( ( ord_less_eq_num @ B @ C )
% 12.85/6.91 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_716_ord__eq__le__trans,axiom,
% 12.85/6.91 ! [A: nat,B: nat,C: nat] :
% 12.85/6.91 ( ( A = B )
% 12.85/6.91 => ( ( ord_less_eq_nat @ B @ C )
% 12.85/6.91 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_717_ord__eq__le__trans,axiom,
% 12.85/6.91 ! [A: int,B: int,C: int] :
% 12.85/6.91 ( ( A = B )
% 12.85/6.91 => ( ( ord_less_eq_int @ B @ C )
% 12.85/6.91 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_718_ord__eq__le__trans,axiom,
% 12.85/6.91 ! [A: real,B: real,C: real] :
% 12.85/6.91 ( ( A = B )
% 12.85/6.91 => ( ( ord_less_eq_real @ B @ C )
% 12.85/6.91 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_719_ord__le__eq__trans,axiom,
% 12.85/6.91 ! [A: set_int,B: set_int,C: set_int] :
% 12.85/6.91 ( ( ord_less_eq_set_int @ A @ B )
% 12.85/6.91 => ( ( B = C )
% 12.85/6.91 => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_720_ord__le__eq__trans,axiom,
% 12.85/6.91 ! [A: rat,B: rat,C: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.91 => ( ( B = C )
% 12.85/6.91 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_721_ord__le__eq__trans,axiom,
% 12.85/6.91 ! [A: num,B: num,C: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.91 => ( ( B = C )
% 12.85/6.91 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_722_ord__le__eq__trans,axiom,
% 12.85/6.91 ! [A: nat,B: nat,C: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.91 => ( ( B = C )
% 12.85/6.91 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_723_ord__le__eq__trans,axiom,
% 12.85/6.91 ! [A: int,B: int,C: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.91 => ( ( B = C )
% 12.85/6.91 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_724_ord__le__eq__trans,axiom,
% 12.85/6.91 ! [A: real,B: real,C: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.91 => ( ( B = C )
% 12.85/6.91 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_725_order__antisym,axiom,
% 12.85/6.91 ! [X: set_int,Y4: set_int] :
% 12.85/6.91 ( ( ord_less_eq_set_int @ X @ Y4 )
% 12.85/6.91 => ( ( ord_less_eq_set_int @ Y4 @ X )
% 12.85/6.91 => ( X = Y4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_726_order__antisym,axiom,
% 12.85/6.91 ! [X: rat,Y4: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X @ Y4 )
% 12.85/6.91 => ( ( ord_less_eq_rat @ Y4 @ X )
% 12.85/6.91 => ( X = Y4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_727_order__antisym,axiom,
% 12.85/6.91 ! [X: num,Y4: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X @ Y4 )
% 12.85/6.91 => ( ( ord_less_eq_num @ Y4 @ X )
% 12.85/6.91 => ( X = Y4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_728_order__antisym,axiom,
% 12.85/6.91 ! [X: nat,Y4: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ X @ Y4 )
% 12.85/6.91 => ( ( ord_less_eq_nat @ Y4 @ X )
% 12.85/6.91 => ( X = Y4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_729_order__antisym,axiom,
% 12.85/6.91 ! [X: int,Y4: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ X @ Y4 )
% 12.85/6.91 => ( ( ord_less_eq_int @ Y4 @ X )
% 12.85/6.91 => ( X = Y4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_730_order__antisym,axiom,
% 12.85/6.91 ! [X: real,Y4: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/6.91 => ( ( ord_less_eq_real @ Y4 @ X )
% 12.85/6.91 => ( X = Y4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_731_order_Otrans,axiom,
% 12.85/6.91 ! [A: set_int,B: set_int,C: set_int] :
% 12.85/6.91 ( ( ord_less_eq_set_int @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_set_int @ B @ C )
% 12.85/6.91 => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_732_order_Otrans,axiom,
% 12.85/6.91 ! [A: rat,B: rat,C: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_rat @ B @ C )
% 12.85/6.91 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_733_order_Otrans,axiom,
% 12.85/6.91 ! [A: num,B: num,C: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_num @ B @ C )
% 12.85/6.91 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_734_order_Otrans,axiom,
% 12.85/6.91 ! [A: nat,B: nat,C: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_nat @ B @ C )
% 12.85/6.91 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_735_order_Otrans,axiom,
% 12.85/6.91 ! [A: int,B: int,C: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_int @ B @ C )
% 12.85/6.91 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_736_order_Otrans,axiom,
% 12.85/6.91 ! [A: real,B: real,C: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_real @ B @ C )
% 12.85/6.91 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_737_order__trans,axiom,
% 12.85/6.91 ! [X: set_int,Y4: set_int,Z: set_int] :
% 12.85/6.91 ( ( ord_less_eq_set_int @ X @ Y4 )
% 12.85/6.91 => ( ( ord_less_eq_set_int @ Y4 @ Z )
% 12.85/6.91 => ( ord_less_eq_set_int @ X @ Z ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_738_order__trans,axiom,
% 12.85/6.91 ! [X: rat,Y4: rat,Z: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X @ Y4 )
% 12.85/6.91 => ( ( ord_less_eq_rat @ Y4 @ Z )
% 12.85/6.91 => ( ord_less_eq_rat @ X @ Z ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_739_order__trans,axiom,
% 12.85/6.91 ! [X: num,Y4: num,Z: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X @ Y4 )
% 12.85/6.91 => ( ( ord_less_eq_num @ Y4 @ Z )
% 12.85/6.91 => ( ord_less_eq_num @ X @ Z ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_740_order__trans,axiom,
% 12.85/6.91 ! [X: nat,Y4: nat,Z: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ X @ Y4 )
% 12.85/6.91 => ( ( ord_less_eq_nat @ Y4 @ Z )
% 12.85/6.91 => ( ord_less_eq_nat @ X @ Z ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_741_order__trans,axiom,
% 12.85/6.91 ! [X: int,Y4: int,Z: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ X @ Y4 )
% 12.85/6.91 => ( ( ord_less_eq_int @ Y4 @ Z )
% 12.85/6.91 => ( ord_less_eq_int @ X @ Z ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_742_order__trans,axiom,
% 12.85/6.91 ! [X: real,Y4: real,Z: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/6.91 => ( ( ord_less_eq_real @ Y4 @ Z )
% 12.85/6.91 => ( ord_less_eq_real @ X @ Z ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_743_linorder__wlog,axiom,
% 12.85/6.91 ! [P: rat > rat > $o,A: rat,B: rat] :
% 12.85/6.91 ( ! [A4: rat,B3: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A4 @ B3 )
% 12.85/6.91 => ( P @ A4 @ B3 ) )
% 12.85/6.91 => ( ! [A4: rat,B3: rat] :
% 12.85/6.91 ( ( P @ B3 @ A4 )
% 12.85/6.91 => ( P @ A4 @ B3 ) )
% 12.85/6.91 => ( P @ A @ B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_744_linorder__wlog,axiom,
% 12.85/6.91 ! [P: num > num > $o,A: num,B: num] :
% 12.85/6.91 ( ! [A4: num,B3: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ A4 @ B3 )
% 12.85/6.91 => ( P @ A4 @ B3 ) )
% 12.85/6.91 => ( ! [A4: num,B3: num] :
% 12.85/6.91 ( ( P @ B3 @ A4 )
% 12.85/6.91 => ( P @ A4 @ B3 ) )
% 12.85/6.91 => ( P @ A @ B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_745_linorder__wlog,axiom,
% 12.85/6.91 ! [P: nat > nat > $o,A: nat,B: nat] :
% 12.85/6.91 ( ! [A4: nat,B3: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ A4 @ B3 )
% 12.85/6.91 => ( P @ A4 @ B3 ) )
% 12.85/6.91 => ( ! [A4: nat,B3: nat] :
% 12.85/6.91 ( ( P @ B3 @ A4 )
% 12.85/6.91 => ( P @ A4 @ B3 ) )
% 12.85/6.91 => ( P @ A @ B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_746_linorder__wlog,axiom,
% 12.85/6.91 ! [P: int > int > $o,A: int,B: int] :
% 12.85/6.91 ( ! [A4: int,B3: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ A4 @ B3 )
% 12.85/6.91 => ( P @ A4 @ B3 ) )
% 12.85/6.91 => ( ! [A4: int,B3: int] :
% 12.85/6.91 ( ( P @ B3 @ A4 )
% 12.85/6.91 => ( P @ A4 @ B3 ) )
% 12.85/6.91 => ( P @ A @ B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_747_linorder__wlog,axiom,
% 12.85/6.91 ! [P: real > real > $o,A: real,B: real] :
% 12.85/6.91 ( ! [A4: real,B3: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ A4 @ B3 )
% 12.85/6.91 => ( P @ A4 @ B3 ) )
% 12.85/6.91 => ( ! [A4: real,B3: real] :
% 12.85/6.91 ( ( P @ B3 @ A4 )
% 12.85/6.91 => ( P @ A4 @ B3 ) )
% 12.85/6.91 => ( P @ A @ B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_748_dual__order_Oeq__iff,axiom,
% 12.85/6.91 ( ( ^ [Y5: set_int,Z4: set_int] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [A3: set_int,B2: set_int] :
% 12.85/6.91 ( ( ord_less_eq_set_int @ B2 @ A3 )
% 12.85/6.91 & ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_749_dual__order_Oeq__iff,axiom,
% 12.85/6.91 ( ( ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [A3: rat,B2: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ B2 @ A3 )
% 12.85/6.91 & ( ord_less_eq_rat @ A3 @ B2 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_750_dual__order_Oeq__iff,axiom,
% 12.85/6.91 ( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [A3: num,B2: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ B2 @ A3 )
% 12.85/6.91 & ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_751_dual__order_Oeq__iff,axiom,
% 12.85/6.91 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [A3: nat,B2: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ B2 @ A3 )
% 12.85/6.91 & ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_752_dual__order_Oeq__iff,axiom,
% 12.85/6.91 ( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [A3: int,B2: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ B2 @ A3 )
% 12.85/6.91 & ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_753_dual__order_Oeq__iff,axiom,
% 12.85/6.91 ( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [A3: real,B2: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ B2 @ A3 )
% 12.85/6.91 & ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_754_dual__order_Oantisym,axiom,
% 12.85/6.91 ! [B: set_int,A: set_int] :
% 12.85/6.91 ( ( ord_less_eq_set_int @ B @ A )
% 12.85/6.91 => ( ( ord_less_eq_set_int @ A @ B )
% 12.85/6.91 => ( A = B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_755_dual__order_Oantisym,axiom,
% 12.85/6.91 ! [B: rat,A: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ B @ A )
% 12.85/6.91 => ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.91 => ( A = B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_756_dual__order_Oantisym,axiom,
% 12.85/6.91 ! [B: num,A: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ B @ A )
% 12.85/6.91 => ( ( ord_less_eq_num @ A @ B )
% 12.85/6.91 => ( A = B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_757_dual__order_Oantisym,axiom,
% 12.85/6.91 ! [B: nat,A: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ B @ A )
% 12.85/6.91 => ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.91 => ( A = B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_758_dual__order_Oantisym,axiom,
% 12.85/6.91 ! [B: int,A: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ B @ A )
% 12.85/6.91 => ( ( ord_less_eq_int @ A @ B )
% 12.85/6.91 => ( A = B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_759_dual__order_Oantisym,axiom,
% 12.85/6.91 ! [B: real,A: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ B @ A )
% 12.85/6.91 => ( ( ord_less_eq_real @ A @ B )
% 12.85/6.91 => ( A = B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_760_dual__order_Otrans,axiom,
% 12.85/6.91 ! [B: set_int,A: set_int,C: set_int] :
% 12.85/6.91 ( ( ord_less_eq_set_int @ B @ A )
% 12.85/6.91 => ( ( ord_less_eq_set_int @ C @ B )
% 12.85/6.91 => ( ord_less_eq_set_int @ C @ A ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_761_dual__order_Otrans,axiom,
% 12.85/6.91 ! [B: rat,A: rat,C: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ B @ A )
% 12.85/6.91 => ( ( ord_less_eq_rat @ C @ B )
% 12.85/6.91 => ( ord_less_eq_rat @ C @ A ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_762_dual__order_Otrans,axiom,
% 12.85/6.91 ! [B: num,A: num,C: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ B @ A )
% 12.85/6.91 => ( ( ord_less_eq_num @ C @ B )
% 12.85/6.91 => ( ord_less_eq_num @ C @ A ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_763_dual__order_Otrans,axiom,
% 12.85/6.91 ! [B: nat,A: nat,C: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ B @ A )
% 12.85/6.91 => ( ( ord_less_eq_nat @ C @ B )
% 12.85/6.91 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_764_dual__order_Otrans,axiom,
% 12.85/6.91 ! [B: int,A: int,C: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ B @ A )
% 12.85/6.91 => ( ( ord_less_eq_int @ C @ B )
% 12.85/6.91 => ( ord_less_eq_int @ C @ A ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_765_dual__order_Otrans,axiom,
% 12.85/6.91 ! [B: real,A: real,C: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ B @ A )
% 12.85/6.91 => ( ( ord_less_eq_real @ C @ B )
% 12.85/6.91 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_766_antisym,axiom,
% 12.85/6.91 ! [A: set_int,B: set_int] :
% 12.85/6.91 ( ( ord_less_eq_set_int @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_set_int @ B @ A )
% 12.85/6.91 => ( A = B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_767_antisym,axiom,
% 12.85/6.91 ! [A: rat,B: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_rat @ B @ A )
% 12.85/6.91 => ( A = B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_768_antisym,axiom,
% 12.85/6.91 ! [A: num,B: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_num @ B @ A )
% 12.85/6.91 => ( A = B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_769_antisym,axiom,
% 12.85/6.91 ! [A: nat,B: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_nat @ B @ A )
% 12.85/6.91 => ( A = B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_770_antisym,axiom,
% 12.85/6.91 ! [A: int,B: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_int @ B @ A )
% 12.85/6.91 => ( A = B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_771_antisym,axiom,
% 12.85/6.91 ! [A: real,B: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_real @ B @ A )
% 12.85/6.91 => ( A = B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_772_Orderings_Oorder__eq__iff,axiom,
% 12.85/6.91 ( ( ^ [Y5: set_int,Z4: set_int] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [A3: set_int,B2: set_int] :
% 12.85/6.91 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 12.85/6.91 & ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_773_Orderings_Oorder__eq__iff,axiom,
% 12.85/6.91 ( ( ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [A3: rat,B2: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A3 @ B2 )
% 12.85/6.91 & ( ord_less_eq_rat @ B2 @ A3 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_774_Orderings_Oorder__eq__iff,axiom,
% 12.85/6.91 ( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [A3: num,B2: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ A3 @ B2 )
% 12.85/6.91 & ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_775_Orderings_Oorder__eq__iff,axiom,
% 12.85/6.91 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [A3: nat,B2: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ A3 @ B2 )
% 12.85/6.91 & ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_776_Orderings_Oorder__eq__iff,axiom,
% 12.85/6.91 ( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [A3: int,B2: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ A3 @ B2 )
% 12.85/6.91 & ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_777_Orderings_Oorder__eq__iff,axiom,
% 12.85/6.91 ( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [A3: real,B2: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ A3 @ B2 )
% 12.85/6.91 & ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_778_order__subst1,axiom,
% 12.85/6.91 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_rat @ B @ C )
% 12.85/6.91 => ( ! [X4: rat,Y: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_779_order__subst1,axiom,
% 12.85/6.91 ! [A: rat,F: num > rat,B: num,C: num] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_num @ B @ C )
% 12.85/6.91 => ( ! [X4: num,Y: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_780_order__subst1,axiom,
% 12.85/6.91 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_nat @ B @ C )
% 12.85/6.91 => ( ! [X4: nat,Y: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_781_order__subst1,axiom,
% 12.85/6.91 ! [A: rat,F: int > rat,B: int,C: int] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_int @ B @ C )
% 12.85/6.91 => ( ! [X4: int,Y: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_782_order__subst1,axiom,
% 12.85/6.91 ! [A: rat,F: real > rat,B: real,C: real] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_real @ B @ C )
% 12.85/6.91 => ( ! [X4: real,Y: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_783_order__subst1,axiom,
% 12.85/6.91 ! [A: num,F: rat > num,B: rat,C: rat] :
% 12.85/6.91 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_rat @ B @ C )
% 12.85/6.91 => ( ! [X4: rat,Y: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_784_order__subst1,axiom,
% 12.85/6.91 ! [A: num,F: num > num,B: num,C: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_num @ B @ C )
% 12.85/6.91 => ( ! [X4: num,Y: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_785_order__subst1,axiom,
% 12.85/6.91 ! [A: num,F: nat > num,B: nat,C: nat] :
% 12.85/6.91 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_nat @ B @ C )
% 12.85/6.91 => ( ! [X4: nat,Y: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_786_order__subst1,axiom,
% 12.85/6.91 ! [A: num,F: int > num,B: int,C: int] :
% 12.85/6.91 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_int @ B @ C )
% 12.85/6.91 => ( ! [X4: int,Y: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_787_order__subst1,axiom,
% 12.85/6.91 ! [A: num,F: real > num,B: real,C: real] :
% 12.85/6.91 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_real @ B @ C )
% 12.85/6.91 => ( ! [X4: real,Y: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_788_order__subst2,axiom,
% 12.85/6.91 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 12.85/6.91 => ( ! [X4: rat,Y: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_789_order__subst2,axiom,
% 12.85/6.91 ! [A: rat,B: rat,F: rat > num,C: num] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 12.85/6.91 => ( ! [X4: rat,Y: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_790_order__subst2,axiom,
% 12.85/6.91 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 12.85/6.91 => ( ! [X4: rat,Y: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_791_order__subst2,axiom,
% 12.85/6.91 ! [A: rat,B: rat,F: rat > int,C: int] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 12.85/6.91 => ( ! [X4: rat,Y: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_792_order__subst2,axiom,
% 12.85/6.91 ! [A: rat,B: rat,F: rat > real,C: real] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 12.85/6.91 => ( ! [X4: rat,Y: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_793_order__subst2,axiom,
% 12.85/6.91 ! [A: num,B: num,F: num > rat,C: rat] :
% 12.85/6.91 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 12.85/6.91 => ( ! [X4: num,Y: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_794_order__subst2,axiom,
% 12.85/6.91 ! [A: num,B: num,F: num > num,C: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 12.85/6.91 => ( ! [X4: num,Y: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_795_order__subst2,axiom,
% 12.85/6.91 ! [A: num,B: num,F: num > nat,C: nat] :
% 12.85/6.91 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 12.85/6.91 => ( ! [X4: num,Y: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_796_order__subst2,axiom,
% 12.85/6.91 ! [A: num,B: num,F: num > int,C: int] :
% 12.85/6.91 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 12.85/6.91 => ( ! [X4: num,Y: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_797_order__subst2,axiom,
% 12.85/6.91 ! [A: num,B: num,F: num > real,C: real] :
% 12.85/6.91 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 12.85/6.91 => ( ! [X4: num,Y: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_798_order__eq__refl,axiom,
% 12.85/6.91 ! [X: set_int,Y4: set_int] :
% 12.85/6.91 ( ( X = Y4 )
% 12.85/6.91 => ( ord_less_eq_set_int @ X @ Y4 ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_799_order__eq__refl,axiom,
% 12.85/6.91 ! [X: rat,Y4: rat] :
% 12.85/6.91 ( ( X = Y4 )
% 12.85/6.91 => ( ord_less_eq_rat @ X @ Y4 ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_800_order__eq__refl,axiom,
% 12.85/6.91 ! [X: num,Y4: num] :
% 12.85/6.91 ( ( X = Y4 )
% 12.85/6.91 => ( ord_less_eq_num @ X @ Y4 ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_801_order__eq__refl,axiom,
% 12.85/6.91 ! [X: nat,Y4: nat] :
% 12.85/6.91 ( ( X = Y4 )
% 12.85/6.91 => ( ord_less_eq_nat @ X @ Y4 ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_802_order__eq__refl,axiom,
% 12.85/6.91 ! [X: int,Y4: int] :
% 12.85/6.91 ( ( X = Y4 )
% 12.85/6.91 => ( ord_less_eq_int @ X @ Y4 ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_803_order__eq__refl,axiom,
% 12.85/6.91 ! [X: real,Y4: real] :
% 12.85/6.91 ( ( X = Y4 )
% 12.85/6.91 => ( ord_less_eq_real @ X @ Y4 ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_804_verit__la__disequality,axiom,
% 12.85/6.91 ! [A: rat,B: rat] :
% 12.85/6.91 ( ( A = B )
% 12.85/6.91 | ~ ( ord_less_eq_rat @ A @ B )
% 12.85/6.91 | ~ ( ord_less_eq_rat @ B @ A ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_805_verit__la__disequality,axiom,
% 12.85/6.91 ! [A: num,B: num] :
% 12.85/6.91 ( ( A = B )
% 12.85/6.91 | ~ ( ord_less_eq_num @ A @ B )
% 12.85/6.91 | ~ ( ord_less_eq_num @ B @ A ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_806_verit__la__disequality,axiom,
% 12.85/6.91 ! [A: nat,B: nat] :
% 12.85/6.91 ( ( A = B )
% 12.85/6.91 | ~ ( ord_less_eq_nat @ A @ B )
% 12.85/6.91 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_807_verit__la__disequality,axiom,
% 12.85/6.91 ! [A: int,B: int] :
% 12.85/6.91 ( ( A = B )
% 12.85/6.91 | ~ ( ord_less_eq_int @ A @ B )
% 12.85/6.91 | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_808_verit__la__disequality,axiom,
% 12.85/6.91 ! [A: real,B: real] :
% 12.85/6.91 ( ( A = B )
% 12.85/6.91 | ~ ( ord_less_eq_real @ A @ B )
% 12.85/6.91 | ~ ( ord_less_eq_real @ B @ A ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_809_linorder__linear,axiom,
% 12.85/6.91 ! [X: rat,Y4: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X @ Y4 )
% 12.85/6.91 | ( ord_less_eq_rat @ Y4 @ X ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_810_linorder__linear,axiom,
% 12.85/6.91 ! [X: num,Y4: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X @ Y4 )
% 12.85/6.91 | ( ord_less_eq_num @ Y4 @ X ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_811_linorder__linear,axiom,
% 12.85/6.91 ! [X: nat,Y4: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ X @ Y4 )
% 12.85/6.91 | ( ord_less_eq_nat @ Y4 @ X ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_812_linorder__linear,axiom,
% 12.85/6.91 ! [X: int,Y4: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ X @ Y4 )
% 12.85/6.91 | ( ord_less_eq_int @ Y4 @ X ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_813_linorder__linear,axiom,
% 12.85/6.91 ! [X: real,Y4: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/6.91 | ( ord_less_eq_real @ Y4 @ X ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_814_ord__eq__le__subst,axiom,
% 12.85/6.91 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 12.85/6.91 ( ( A
% 12.85/6.91 = ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_rat @ B @ C )
% 12.85/6.91 => ( ! [X4: rat,Y: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_815_ord__eq__le__subst,axiom,
% 12.85/6.91 ! [A: num,F: rat > num,B: rat,C: rat] :
% 12.85/6.91 ( ( A
% 12.85/6.91 = ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_rat @ B @ C )
% 12.85/6.91 => ( ! [X4: rat,Y: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_816_ord__eq__le__subst,axiom,
% 12.85/6.91 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 12.85/6.91 ( ( A
% 12.85/6.91 = ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_rat @ B @ C )
% 12.85/6.91 => ( ! [X4: rat,Y: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_817_ord__eq__le__subst,axiom,
% 12.85/6.91 ! [A: int,F: rat > int,B: rat,C: rat] :
% 12.85/6.91 ( ( A
% 12.85/6.91 = ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_rat @ B @ C )
% 12.85/6.91 => ( ! [X4: rat,Y: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_818_ord__eq__le__subst,axiom,
% 12.85/6.91 ! [A: real,F: rat > real,B: rat,C: rat] :
% 12.85/6.91 ( ( A
% 12.85/6.91 = ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_rat @ B @ C )
% 12.85/6.91 => ( ! [X4: rat,Y: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_819_ord__eq__le__subst,axiom,
% 12.85/6.91 ! [A: rat,F: num > rat,B: num,C: num] :
% 12.85/6.91 ( ( A
% 12.85/6.91 = ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_num @ B @ C )
% 12.85/6.91 => ( ! [X4: num,Y: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_820_ord__eq__le__subst,axiom,
% 12.85/6.91 ! [A: num,F: num > num,B: num,C: num] :
% 12.85/6.91 ( ( A
% 12.85/6.91 = ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_num @ B @ C )
% 12.85/6.91 => ( ! [X4: num,Y: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_821_ord__eq__le__subst,axiom,
% 12.85/6.91 ! [A: nat,F: num > nat,B: num,C: num] :
% 12.85/6.91 ( ( A
% 12.85/6.91 = ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_num @ B @ C )
% 12.85/6.91 => ( ! [X4: num,Y: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_822_ord__eq__le__subst,axiom,
% 12.85/6.91 ! [A: int,F: num > int,B: num,C: num] :
% 12.85/6.91 ( ( A
% 12.85/6.91 = ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_num @ B @ C )
% 12.85/6.91 => ( ! [X4: num,Y: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_823_ord__eq__le__subst,axiom,
% 12.85/6.91 ! [A: real,F: num > real,B: num,C: num] :
% 12.85/6.91 ( ( A
% 12.85/6.91 = ( F @ B ) )
% 12.85/6.91 => ( ( ord_less_eq_num @ B @ C )
% 12.85/6.91 => ( ! [X4: num,Y: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_824_ord__le__eq__subst,axiom,
% 12.85/6.91 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.91 => ( ( ( F @ B )
% 12.85/6.91 = C )
% 12.85/6.91 => ( ! [X4: rat,Y: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_825_ord__le__eq__subst,axiom,
% 12.85/6.91 ! [A: rat,B: rat,F: rat > num,C: num] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.91 => ( ( ( F @ B )
% 12.85/6.91 = C )
% 12.85/6.91 => ( ! [X4: rat,Y: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_826_ord__le__eq__subst,axiom,
% 12.85/6.91 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.91 => ( ( ( F @ B )
% 12.85/6.91 = C )
% 12.85/6.91 => ( ! [X4: rat,Y: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_827_ord__le__eq__subst,axiom,
% 12.85/6.91 ! [A: rat,B: rat,F: rat > int,C: int] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.91 => ( ( ( F @ B )
% 12.85/6.91 = C )
% 12.85/6.91 => ( ! [X4: rat,Y: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_828_ord__le__eq__subst,axiom,
% 12.85/6.91 ! [A: rat,B: rat,F: rat > real,C: real] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.91 => ( ( ( F @ B )
% 12.85/6.91 = C )
% 12.85/6.91 => ( ! [X4: rat,Y: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_829_ord__le__eq__subst,axiom,
% 12.85/6.91 ! [A: num,B: num,F: num > rat,C: rat] :
% 12.85/6.91 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.91 => ( ( ( F @ B )
% 12.85/6.91 = C )
% 12.85/6.91 => ( ! [X4: num,Y: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_830_ord__le__eq__subst,axiom,
% 12.85/6.91 ! [A: num,B: num,F: num > num,C: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.91 => ( ( ( F @ B )
% 12.85/6.91 = C )
% 12.85/6.91 => ( ! [X4: num,Y: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_831_ord__le__eq__subst,axiom,
% 12.85/6.91 ! [A: num,B: num,F: num > nat,C: nat] :
% 12.85/6.91 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.91 => ( ( ( F @ B )
% 12.85/6.91 = C )
% 12.85/6.91 => ( ! [X4: num,Y: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_832_ord__le__eq__subst,axiom,
% 12.85/6.91 ! [A: num,B: num,F: num > int,C: int] :
% 12.85/6.91 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.91 => ( ( ( F @ B )
% 12.85/6.91 = C )
% 12.85/6.91 => ( ! [X4: num,Y: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_833_ord__le__eq__subst,axiom,
% 12.85/6.91 ! [A: num,B: num,F: num > real,C: real] :
% 12.85/6.91 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.91 => ( ( ( F @ B )
% 12.85/6.91 = C )
% 12.85/6.91 => ( ! [X4: num,Y: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.91 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.91 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_834_linorder__le__cases,axiom,
% 12.85/6.91 ! [X: rat,Y4: rat] :
% 12.85/6.91 ( ~ ( ord_less_eq_rat @ X @ Y4 )
% 12.85/6.91 => ( ord_less_eq_rat @ Y4 @ X ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_835_linorder__le__cases,axiom,
% 12.85/6.91 ! [X: num,Y4: num] :
% 12.85/6.91 ( ~ ( ord_less_eq_num @ X @ Y4 )
% 12.85/6.91 => ( ord_less_eq_num @ Y4 @ X ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_836_linorder__le__cases,axiom,
% 12.85/6.91 ! [X: nat,Y4: nat] :
% 12.85/6.91 ( ~ ( ord_less_eq_nat @ X @ Y4 )
% 12.85/6.91 => ( ord_less_eq_nat @ Y4 @ X ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_837_linorder__le__cases,axiom,
% 12.85/6.91 ! [X: int,Y4: int] :
% 12.85/6.91 ( ~ ( ord_less_eq_int @ X @ Y4 )
% 12.85/6.91 => ( ord_less_eq_int @ Y4 @ X ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_838_linorder__le__cases,axiom,
% 12.85/6.91 ! [X: real,Y4: real] :
% 12.85/6.91 ( ~ ( ord_less_eq_real @ X @ Y4 )
% 12.85/6.91 => ( ord_less_eq_real @ Y4 @ X ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_839_order__antisym__conv,axiom,
% 12.85/6.91 ! [Y4: set_int,X: set_int] :
% 12.85/6.91 ( ( ord_less_eq_set_int @ Y4 @ X )
% 12.85/6.91 => ( ( ord_less_eq_set_int @ X @ Y4 )
% 12.85/6.91 = ( X = Y4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_840_order__antisym__conv,axiom,
% 12.85/6.91 ! [Y4: rat,X: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ Y4 @ X )
% 12.85/6.91 => ( ( ord_less_eq_rat @ X @ Y4 )
% 12.85/6.91 = ( X = Y4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_841_order__antisym__conv,axiom,
% 12.85/6.91 ! [Y4: num,X: num] :
% 12.85/6.91 ( ( ord_less_eq_num @ Y4 @ X )
% 12.85/6.91 => ( ( ord_less_eq_num @ X @ Y4 )
% 12.85/6.91 = ( X = Y4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_842_order__antisym__conv,axiom,
% 12.85/6.91 ! [Y4: nat,X: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ Y4 @ X )
% 12.85/6.91 => ( ( ord_less_eq_nat @ X @ Y4 )
% 12.85/6.91 = ( X = Y4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_843_order__antisym__conv,axiom,
% 12.85/6.91 ! [Y4: int,X: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ Y4 @ X )
% 12.85/6.91 => ( ( ord_less_eq_int @ X @ Y4 )
% 12.85/6.91 = ( X = Y4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_844_order__antisym__conv,axiom,
% 12.85/6.91 ! [Y4: real,X: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ Y4 @ X )
% 12.85/6.91 => ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/6.91 = ( X = Y4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_845_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 12.85/6.91 ! [C: nat,A: nat,B: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.85/6.91 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 12.85/6.91 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_846_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 12.85/6.91 ! [C: int,A: int,B: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.91 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 12.85/6.91 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_847_divide__eq__eq__numeral_I1_J,axiom,
% 12.85/6.91 ! [B: complex,C: complex,W: num] :
% 12.85/6.91 ( ( ( divide1717551699836669952omplex @ B @ C )
% 12.85/6.91 = ( numera6690914467698888265omplex @ W ) )
% 12.85/6.91 = ( ( ( C != zero_zero_complex )
% 12.85/6.91 => ( B
% 12.85/6.91 = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
% 12.85/6.91 & ( ( C = zero_zero_complex )
% 12.85/6.91 => ( ( numera6690914467698888265omplex @ W )
% 12.85/6.91 = zero_zero_complex ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_848_divide__eq__eq__numeral_I1_J,axiom,
% 12.85/6.91 ! [B: real,C: real,W: num] :
% 12.85/6.91 ( ( ( divide_divide_real @ B @ C )
% 12.85/6.91 = ( numeral_numeral_real @ W ) )
% 12.85/6.91 = ( ( ( C != zero_zero_real )
% 12.85/6.91 => ( B
% 12.85/6.91 = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 12.85/6.91 & ( ( C = zero_zero_real )
% 12.85/6.91 => ( ( numeral_numeral_real @ W )
% 12.85/6.91 = zero_zero_real ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_849_divide__eq__eq__numeral_I1_J,axiom,
% 12.85/6.91 ! [B: rat,C: rat,W: num] :
% 12.85/6.91 ( ( ( divide_divide_rat @ B @ C )
% 12.85/6.91 = ( numeral_numeral_rat @ W ) )
% 12.85/6.91 = ( ( ( C != zero_zero_rat )
% 12.85/6.91 => ( B
% 12.85/6.91 = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 12.85/6.91 & ( ( C = zero_zero_rat )
% 12.85/6.91 => ( ( numeral_numeral_rat @ W )
% 12.85/6.91 = zero_zero_rat ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_850_eq__divide__eq__numeral_I1_J,axiom,
% 12.85/6.91 ! [W: num,B: complex,C: complex] :
% 12.85/6.91 ( ( ( numera6690914467698888265omplex @ W )
% 12.85/6.91 = ( divide1717551699836669952omplex @ B @ C ) )
% 12.85/6.91 = ( ( ( C != zero_zero_complex )
% 12.85/6.91 => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
% 12.85/6.91 = B ) )
% 12.85/6.91 & ( ( C = zero_zero_complex )
% 12.85/6.91 => ( ( numera6690914467698888265omplex @ W )
% 12.85/6.91 = zero_zero_complex ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_851_eq__divide__eq__numeral_I1_J,axiom,
% 12.85/6.91 ! [W: num,B: real,C: real] :
% 12.85/6.91 ( ( ( numeral_numeral_real @ W )
% 12.85/6.91 = ( divide_divide_real @ B @ C ) )
% 12.85/6.91 = ( ( ( C != zero_zero_real )
% 12.85/6.91 => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
% 12.85/6.91 = B ) )
% 12.85/6.91 & ( ( C = zero_zero_real )
% 12.85/6.91 => ( ( numeral_numeral_real @ W )
% 12.85/6.91 = zero_zero_real ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_852_eq__divide__eq__numeral_I1_J,axiom,
% 12.85/6.91 ! [W: num,B: rat,C: rat] :
% 12.85/6.91 ( ( ( numeral_numeral_rat @ W )
% 12.85/6.91 = ( divide_divide_rat @ B @ C ) )
% 12.85/6.91 = ( ( ( C != zero_zero_rat )
% 12.85/6.91 => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
% 12.85/6.91 = B ) )
% 12.85/6.91 & ( ( C = zero_zero_rat )
% 12.85/6.91 => ( ( numeral_numeral_rat @ W )
% 12.85/6.91 = zero_zero_rat ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_853_dvd__div__eq__mult,axiom,
% 12.85/6.91 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.91 ( ( A != zero_z3403309356797280102nteger )
% 12.85/6.91 => ( ( dvd_dvd_Code_integer @ A @ B )
% 12.85/6.91 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 12.85/6.91 = C )
% 12.85/6.91 = ( B
% 12.85/6.91 = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_854_dvd__div__eq__mult,axiom,
% 12.85/6.91 ! [A: nat,B: nat,C: nat] :
% 12.85/6.91 ( ( A != zero_zero_nat )
% 12.85/6.91 => ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.91 => ( ( ( divide_divide_nat @ B @ A )
% 12.85/6.91 = C )
% 12.85/6.91 = ( B
% 12.85/6.91 = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_855_dvd__div__eq__mult,axiom,
% 12.85/6.91 ! [A: int,B: int,C: int] :
% 12.85/6.91 ( ( A != zero_zero_int )
% 12.85/6.91 => ( ( dvd_dvd_int @ A @ B )
% 12.85/6.91 => ( ( ( divide_divide_int @ B @ A )
% 12.85/6.91 = C )
% 12.85/6.91 = ( B
% 12.85/6.91 = ( times_times_int @ C @ A ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_856_div__dvd__iff__mult,axiom,
% 12.85/6.91 ! [B: code_integer,A: code_integer,C: code_integer] :
% 12.85/6.91 ( ( B != zero_z3403309356797280102nteger )
% 12.85/6.91 => ( ( dvd_dvd_Code_integer @ B @ A )
% 12.85/6.91 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 12.85/6.91 = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_857_div__dvd__iff__mult,axiom,
% 12.85/6.91 ! [B: nat,A: nat,C: nat] :
% 12.85/6.91 ( ( B != zero_zero_nat )
% 12.85/6.91 => ( ( dvd_dvd_nat @ B @ A )
% 12.85/6.91 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 12.85/6.91 = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_858_div__dvd__iff__mult,axiom,
% 12.85/6.91 ! [B: int,A: int,C: int] :
% 12.85/6.91 ( ( B != zero_zero_int )
% 12.85/6.91 => ( ( dvd_dvd_int @ B @ A )
% 12.85/6.91 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 12.85/6.91 = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_859_dvd__div__iff__mult,axiom,
% 12.85/6.91 ! [C: code_integer,B: code_integer,A: code_integer] :
% 12.85/6.91 ( ( C != zero_z3403309356797280102nteger )
% 12.85/6.91 => ( ( dvd_dvd_Code_integer @ C @ B )
% 12.85/6.91 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 12.85/6.91 = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_860_dvd__div__iff__mult,axiom,
% 12.85/6.91 ! [C: nat,B: nat,A: nat] :
% 12.85/6.91 ( ( C != zero_zero_nat )
% 12.85/6.91 => ( ( dvd_dvd_nat @ C @ B )
% 12.85/6.91 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 12.85/6.91 = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_861_dvd__div__iff__mult,axiom,
% 12.85/6.91 ! [C: int,B: int,A: int] :
% 12.85/6.91 ( ( C != zero_zero_int )
% 12.85/6.91 => ( ( dvd_dvd_int @ C @ B )
% 12.85/6.91 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
% 12.85/6.91 = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_862_dvd__div__div__eq__mult,axiom,
% 12.85/6.91 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 12.85/6.91 ( ( A != zero_z3403309356797280102nteger )
% 12.85/6.91 => ( ( C != zero_z3403309356797280102nteger )
% 12.85/6.91 => ( ( dvd_dvd_Code_integer @ A @ B )
% 12.85/6.91 => ( ( dvd_dvd_Code_integer @ C @ D )
% 12.85/6.91 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 12.85/6.91 = ( divide6298287555418463151nteger @ D @ C ) )
% 12.85/6.91 = ( ( times_3573771949741848930nteger @ B @ C )
% 12.85/6.91 = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_863_dvd__div__div__eq__mult,axiom,
% 12.85/6.91 ! [A: nat,C: nat,B: nat,D: nat] :
% 12.85/6.91 ( ( A != zero_zero_nat )
% 12.85/6.91 => ( ( C != zero_zero_nat )
% 12.85/6.91 => ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.91 => ( ( dvd_dvd_nat @ C @ D )
% 12.85/6.91 => ( ( ( divide_divide_nat @ B @ A )
% 12.85/6.91 = ( divide_divide_nat @ D @ C ) )
% 12.85/6.91 = ( ( times_times_nat @ B @ C )
% 12.85/6.91 = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_864_dvd__div__div__eq__mult,axiom,
% 12.85/6.91 ! [A: int,C: int,B: int,D: int] :
% 12.85/6.91 ( ( A != zero_zero_int )
% 12.85/6.91 => ( ( C != zero_zero_int )
% 12.85/6.91 => ( ( dvd_dvd_int @ A @ B )
% 12.85/6.91 => ( ( dvd_dvd_int @ C @ D )
% 12.85/6.91 => ( ( ( divide_divide_int @ B @ A )
% 12.85/6.91 = ( divide_divide_int @ D @ C ) )
% 12.85/6.91 = ( ( times_times_int @ B @ C )
% 12.85/6.91 = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_865_numeral__1__eq__Suc__0,axiom,
% 12.85/6.91 ( ( numeral_numeral_nat @ one )
% 12.85/6.91 = ( suc @ zero_zero_nat ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_866_nat__int__comparison_I1_J,axiom,
% 12.85/6.91 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [A3: nat,B2: nat] :
% 12.85/6.91 ( ( semiri1314217659103216013at_int @ A3 )
% 12.85/6.91 = ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_867_int__if,axiom,
% 12.85/6.91 ! [P: $o,A: nat,B: nat] :
% 12.85/6.91 ( ( P
% 12.85/6.91 => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
% 12.85/6.91 = ( semiri1314217659103216013at_int @ A ) ) )
% 12.85/6.91 & ( ~ P
% 12.85/6.91 => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
% 12.85/6.91 = ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_868_even__zero,axiom,
% 12.85/6.91 dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 12.85/6.91
% 12.85/6.91 thf(fact_869_even__zero,axiom,
% 12.85/6.91 dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 12.85/6.91
% 12.85/6.91 thf(fact_870_even__zero,axiom,
% 12.85/6.91 dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 12.85/6.91
% 12.85/6.91 thf(fact_871_numeral__2__eq__2,axiom,
% 12.85/6.91 ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 12.85/6.91 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_872_numeral__3__eq__3,axiom,
% 12.85/6.91 ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 12.85/6.91 = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_873_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
% 12.85/6.91 ( ( vEBT_V8646137997579335489_i_l_d @ zero_zero_nat )
% 12.85/6.91 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_874_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
% 12.85/6.91 ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ zero_zero_nat ) )
% 12.85/6.91 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_875_verit__eq__simplify_I10_J,axiom,
% 12.85/6.91 ! [X2: num] :
% 12.85/6.91 ( one
% 12.85/6.91 != ( bit0 @ X2 ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_876_le__zero__eq,axiom,
% 12.85/6.91 ! [N: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 12.85/6.91 = ( N = zero_zero_nat ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_877_unset__bit__0,axiom,
% 12.85/6.91 ! [A: int] :
% 12.85/6.91 ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
% 12.85/6.91 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_878_unset__bit__0,axiom,
% 12.85/6.91 ! [A: nat] :
% 12.85/6.91 ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
% 12.85/6.91 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_879_unset__bit__0,axiom,
% 12.85/6.91 ! [A: code_integer] :
% 12.85/6.91 ( ( bit_se8260200283734997820nteger @ zero_zero_nat @ A )
% 12.85/6.91 = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_880_even__set__bit__iff,axiom,
% 12.85/6.91 ! [M: nat,A: int] :
% 12.85/6.91 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 12.85/6.91 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.91 & ( M != zero_zero_nat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_881_even__set__bit__iff,axiom,
% 12.85/6.91 ! [M: nat,A: nat] :
% 12.85/6.91 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 12.85/6.91 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.91 & ( M != zero_zero_nat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_882_even__set__bit__iff,axiom,
% 12.85/6.91 ! [M: nat,A: code_integer] :
% 12.85/6.91 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 12.85/6.91 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.91 & ( M != zero_zero_nat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_883_even__flip__bit__iff,axiom,
% 12.85/6.91 ! [M: nat,A: int] :
% 12.85/6.91 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 12.85/6.91 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.91 != ( M = zero_zero_nat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_884_even__flip__bit__iff,axiom,
% 12.85/6.91 ! [M: nat,A: nat] :
% 12.85/6.91 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 12.85/6.91 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.91 != ( M = zero_zero_nat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_885_even__flip__bit__iff,axiom,
% 12.85/6.91 ! [M: nat,A: code_integer] :
% 12.85/6.91 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 12.85/6.91 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.91 != ( M = zero_zero_nat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_886_even__unset__bit__iff,axiom,
% 12.85/6.91 ! [M: nat,A: int] :
% 12.85/6.91 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 12.85/6.91 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.91 | ( M = zero_zero_nat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_887_even__unset__bit__iff,axiom,
% 12.85/6.91 ! [M: nat,A: nat] :
% 12.85/6.91 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 12.85/6.91 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.91 | ( M = zero_zero_nat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_888_even__unset__bit__iff,axiom,
% 12.85/6.91 ! [M: nat,A: code_integer] :
% 12.85/6.91 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 12.85/6.91 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.91 | ( M = zero_zero_nat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_889_subset__antisym,axiom,
% 12.85/6.91 ! [A2: set_int,B4: set_int] :
% 12.85/6.91 ( ( ord_less_eq_set_int @ A2 @ B4 )
% 12.85/6.91 => ( ( ord_less_eq_set_int @ B4 @ A2 )
% 12.85/6.91 => ( A2 = B4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_890_subsetI,axiom,
% 12.85/6.91 ! [A2: set_complex,B4: set_complex] :
% 12.85/6.91 ( ! [X4: complex] :
% 12.85/6.91 ( ( member_complex @ X4 @ A2 )
% 12.85/6.91 => ( member_complex @ X4 @ B4 ) )
% 12.85/6.91 => ( ord_le211207098394363844omplex @ A2 @ B4 ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_891_subsetI,axiom,
% 12.85/6.91 ! [A2: set_real,B4: set_real] :
% 12.85/6.91 ( ! [X4: real] :
% 12.85/6.91 ( ( member_real @ X4 @ A2 )
% 12.85/6.91 => ( member_real @ X4 @ B4 ) )
% 12.85/6.91 => ( ord_less_eq_set_real @ A2 @ B4 ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_892_subsetI,axiom,
% 12.85/6.91 ! [A2: set_set_nat,B4: set_set_nat] :
% 12.85/6.91 ( ! [X4: set_nat] :
% 12.85/6.91 ( ( member_set_nat @ X4 @ A2 )
% 12.85/6.91 => ( member_set_nat @ X4 @ B4 ) )
% 12.85/6.91 => ( ord_le6893508408891458716et_nat @ A2 @ B4 ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_893_subsetI,axiom,
% 12.85/6.91 ! [A2: set_nat,B4: set_nat] :
% 12.85/6.91 ( ! [X4: nat] :
% 12.85/6.91 ( ( member_nat @ X4 @ A2 )
% 12.85/6.91 => ( member_nat @ X4 @ B4 ) )
% 12.85/6.91 => ( ord_less_eq_set_nat @ A2 @ B4 ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_894_subsetI,axiom,
% 12.85/6.91 ! [A2: set_int,B4: set_int] :
% 12.85/6.91 ( ! [X4: int] :
% 12.85/6.91 ( ( member_int @ X4 @ A2 )
% 12.85/6.91 => ( member_int @ X4 @ B4 ) )
% 12.85/6.91 => ( ord_less_eq_set_int @ A2 @ B4 ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_895_odd__Suc__minus__one,axiom,
% 12.85/6.91 ! [N: nat] :
% 12.85/6.91 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.91 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 12.85/6.91 = N ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_896_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
% 12.85/6.91 ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ zero_zero_nat ) )
% 12.85/6.91 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_897_unset__bit__nonnegative__int__iff,axiom,
% 12.85/6.91 ! [N: nat,K: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
% 12.85/6.91 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_898_set__bit__nonnegative__int__iff,axiom,
% 12.85/6.91 ! [N: nat,K: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
% 12.85/6.91 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_899_flip__bit__nonnegative__int__iff,axiom,
% 12.85/6.91 ! [N: nat,K: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
% 12.85/6.91 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_900_diff__self,axiom,
% 12.85/6.91 ! [A: complex] :
% 12.85/6.91 ( ( minus_minus_complex @ A @ A )
% 12.85/6.91 = zero_zero_complex ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_901_diff__self,axiom,
% 12.85/6.91 ! [A: real] :
% 12.85/6.91 ( ( minus_minus_real @ A @ A )
% 12.85/6.91 = zero_zero_real ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_902_diff__self,axiom,
% 12.85/6.91 ! [A: rat] :
% 12.85/6.91 ( ( minus_minus_rat @ A @ A )
% 12.85/6.91 = zero_zero_rat ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_903_diff__self,axiom,
% 12.85/6.91 ! [A: int] :
% 12.85/6.91 ( ( minus_minus_int @ A @ A )
% 12.85/6.91 = zero_zero_int ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_904_diff__0__right,axiom,
% 12.85/6.91 ! [A: complex] :
% 12.85/6.91 ( ( minus_minus_complex @ A @ zero_zero_complex )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_905_diff__0__right,axiom,
% 12.85/6.91 ! [A: real] :
% 12.85/6.91 ( ( minus_minus_real @ A @ zero_zero_real )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_906_diff__0__right,axiom,
% 12.85/6.91 ! [A: rat] :
% 12.85/6.91 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_907_diff__0__right,axiom,
% 12.85/6.91 ! [A: int] :
% 12.85/6.91 ( ( minus_minus_int @ A @ zero_zero_int )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_908_zero__diff,axiom,
% 12.85/6.91 ! [A: nat] :
% 12.85/6.91 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 12.85/6.91 = zero_zero_nat ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_909_diff__zero,axiom,
% 12.85/6.91 ! [A: complex] :
% 12.85/6.91 ( ( minus_minus_complex @ A @ zero_zero_complex )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_910_diff__zero,axiom,
% 12.85/6.91 ! [A: real] :
% 12.85/6.91 ( ( minus_minus_real @ A @ zero_zero_real )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_911_diff__zero,axiom,
% 12.85/6.91 ! [A: rat] :
% 12.85/6.91 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_912_diff__zero,axiom,
% 12.85/6.91 ! [A: nat] :
% 12.85/6.91 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_913_diff__zero,axiom,
% 12.85/6.91 ! [A: int] :
% 12.85/6.91 ( ( minus_minus_int @ A @ zero_zero_int )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_914_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 12.85/6.91 ! [A: complex] :
% 12.85/6.91 ( ( minus_minus_complex @ A @ A )
% 12.85/6.91 = zero_zero_complex ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_915_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 12.85/6.91 ! [A: real] :
% 12.85/6.91 ( ( minus_minus_real @ A @ A )
% 12.85/6.91 = zero_zero_real ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_916_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 12.85/6.91 ! [A: rat] :
% 12.85/6.91 ( ( minus_minus_rat @ A @ A )
% 12.85/6.91 = zero_zero_rat ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_917_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 12.85/6.91 ! [A: nat] :
% 12.85/6.91 ( ( minus_minus_nat @ A @ A )
% 12.85/6.91 = zero_zero_nat ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_918_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 12.85/6.91 ! [A: int] :
% 12.85/6.91 ( ( minus_minus_int @ A @ A )
% 12.85/6.91 = zero_zero_int ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_919_Suc__diff__diff,axiom,
% 12.85/6.91 ! [M: nat,N: nat,K: nat] :
% 12.85/6.91 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 12.85/6.91 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_920_diff__Suc__Suc,axiom,
% 12.85/6.91 ! [M: nat,N: nat] :
% 12.85/6.91 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 12.85/6.91 = ( minus_minus_nat @ M @ N ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_921_diff__self__eq__0,axiom,
% 12.85/6.91 ! [M: nat] :
% 12.85/6.91 ( ( minus_minus_nat @ M @ M )
% 12.85/6.91 = zero_zero_nat ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_922_diff__0__eq__0,axiom,
% 12.85/6.91 ! [N: nat] :
% 12.85/6.91 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 12.85/6.91 = zero_zero_nat ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_923_diff__diff__cancel,axiom,
% 12.85/6.91 ! [I: nat,N: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ I @ N )
% 12.85/6.91 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 12.85/6.91 = I ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_924_diff__ge__0__iff__ge,axiom,
% 12.85/6.91 ! [A: rat,B: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 12.85/6.91 = ( ord_less_eq_rat @ B @ A ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_925_diff__ge__0__iff__ge,axiom,
% 12.85/6.91 ! [A: int,B: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 12.85/6.91 = ( ord_less_eq_int @ B @ A ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_926_diff__ge__0__iff__ge,axiom,
% 12.85/6.91 ! [A: real,B: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 12.85/6.91 = ( ord_less_eq_real @ B @ A ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_927_right__diff__distrib__numeral,axiom,
% 12.85/6.91 ! [V: num,B: complex,C: complex] :
% 12.85/6.91 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B @ C ) )
% 12.85/6.91 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_928_right__diff__distrib__numeral,axiom,
% 12.85/6.91 ! [V: num,B: real,C: real] :
% 12.85/6.91 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
% 12.85/6.91 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_929_right__diff__distrib__numeral,axiom,
% 12.85/6.91 ! [V: num,B: rat,C: rat] :
% 12.85/6.91 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
% 12.85/6.91 = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_930_right__diff__distrib__numeral,axiom,
% 12.85/6.91 ! [V: num,B: int,C: int] :
% 12.85/6.91 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
% 12.85/6.91 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_931_left__diff__distrib__numeral,axiom,
% 12.85/6.91 ! [A: complex,B: complex,V: num] :
% 12.85/6.91 ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 12.85/6.91 = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_932_left__diff__distrib__numeral,axiom,
% 12.85/6.91 ! [A: real,B: real,V: num] :
% 12.85/6.91 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 12.85/6.91 = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_933_left__diff__distrib__numeral,axiom,
% 12.85/6.91 ! [A: rat,B: rat,V: num] :
% 12.85/6.91 ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 12.85/6.91 = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_934_left__diff__distrib__numeral,axiom,
% 12.85/6.91 ! [A: int,B: int,V: num] :
% 12.85/6.91 ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 12.85/6.91 = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_935_div__diff,axiom,
% 12.85/6.91 ! [C: code_integer,A: code_integer,B: code_integer] :
% 12.85/6.91 ( ( dvd_dvd_Code_integer @ C @ A )
% 12.85/6.91 => ( ( dvd_dvd_Code_integer @ C @ B )
% 12.85/6.91 => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 12.85/6.91 = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_936_div__diff,axiom,
% 12.85/6.91 ! [C: int,A: int,B: int] :
% 12.85/6.91 ( ( dvd_dvd_int @ C @ A )
% 12.85/6.91 => ( ( dvd_dvd_int @ C @ B )
% 12.85/6.91 => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
% 12.85/6.91 = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_937_diff__is__0__eq,axiom,
% 12.85/6.91 ! [M: nat,N: nat] :
% 12.85/6.91 ( ( ( minus_minus_nat @ M @ N )
% 12.85/6.91 = zero_zero_nat )
% 12.85/6.91 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_938_diff__is__0__eq_H,axiom,
% 12.85/6.91 ! [M: nat,N: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.91 => ( ( minus_minus_nat @ M @ N )
% 12.85/6.91 = zero_zero_nat ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_939_half__nonnegative__int__iff,axiom,
% 12.85/6.91 ! [K: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 12.85/6.91 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_940_diff__eq__diff__eq,axiom,
% 12.85/6.91 ! [A: real,B: real,C: real,D: real] :
% 12.85/6.91 ( ( ( minus_minus_real @ A @ B )
% 12.85/6.91 = ( minus_minus_real @ C @ D ) )
% 12.85/6.91 => ( ( A = B )
% 12.85/6.91 = ( C = D ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_941_diff__eq__diff__eq,axiom,
% 12.85/6.91 ! [A: rat,B: rat,C: rat,D: rat] :
% 12.85/6.91 ( ( ( minus_minus_rat @ A @ B )
% 12.85/6.91 = ( minus_minus_rat @ C @ D ) )
% 12.85/6.91 => ( ( A = B )
% 12.85/6.91 = ( C = D ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_942_diff__eq__diff__eq,axiom,
% 12.85/6.91 ! [A: int,B: int,C: int,D: int] :
% 12.85/6.91 ( ( ( minus_minus_int @ A @ B )
% 12.85/6.91 = ( minus_minus_int @ C @ D ) )
% 12.85/6.91 => ( ( A = B )
% 12.85/6.91 = ( C = D ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_943_diff__right__commute,axiom,
% 12.85/6.91 ! [A: real,C: real,B: real] :
% 12.85/6.91 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 12.85/6.91 = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_944_diff__right__commute,axiom,
% 12.85/6.91 ! [A: rat,C: rat,B: rat] :
% 12.85/6.91 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
% 12.85/6.91 = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_945_diff__right__commute,axiom,
% 12.85/6.91 ! [A: nat,C: nat,B: nat] :
% 12.85/6.91 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 12.85/6.91 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_946_diff__right__commute,axiom,
% 12.85/6.91 ! [A: int,C: int,B: int] :
% 12.85/6.91 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 12.85/6.91 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_947_diff__commute,axiom,
% 12.85/6.91 ! [I: nat,J: nat,K: nat] :
% 12.85/6.91 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 12.85/6.91 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_948_diff__eq__diff__less__eq,axiom,
% 12.85/6.91 ! [A: rat,B: rat,C: rat,D: rat] :
% 12.85/6.91 ( ( ( minus_minus_rat @ A @ B )
% 12.85/6.91 = ( minus_minus_rat @ C @ D ) )
% 12.85/6.91 => ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.91 = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_949_diff__eq__diff__less__eq,axiom,
% 12.85/6.91 ! [A: int,B: int,C: int,D: int] :
% 12.85/6.91 ( ( ( minus_minus_int @ A @ B )
% 12.85/6.91 = ( minus_minus_int @ C @ D ) )
% 12.85/6.91 => ( ( ord_less_eq_int @ A @ B )
% 12.85/6.91 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_950_diff__eq__diff__less__eq,axiom,
% 12.85/6.91 ! [A: real,B: real,C: real,D: real] :
% 12.85/6.91 ( ( ( minus_minus_real @ A @ B )
% 12.85/6.91 = ( minus_minus_real @ C @ D ) )
% 12.85/6.91 => ( ( ord_less_eq_real @ A @ B )
% 12.85/6.91 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_951_diff__right__mono,axiom,
% 12.85/6.91 ! [A: rat,B: rat,C: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.91 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_952_diff__right__mono,axiom,
% 12.85/6.91 ! [A: int,B: int,C: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.91 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_953_diff__right__mono,axiom,
% 12.85/6.91 ! [A: real,B: real,C: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.91 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_954_diff__left__mono,axiom,
% 12.85/6.91 ! [B: rat,A: rat,C: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ B @ A )
% 12.85/6.91 => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_955_diff__left__mono,axiom,
% 12.85/6.91 ! [B: int,A: int,C: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ B @ A )
% 12.85/6.91 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_956_diff__left__mono,axiom,
% 12.85/6.91 ! [B: real,A: real,C: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ B @ A )
% 12.85/6.91 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_957_diff__mono,axiom,
% 12.85/6.91 ! [A: rat,B: rat,D: rat,C: rat] :
% 12.85/6.91 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_rat @ D @ C )
% 12.85/6.91 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_958_diff__mono,axiom,
% 12.85/6.91 ! [A: int,B: int,D: int,C: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_int @ D @ C )
% 12.85/6.91 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_959_diff__mono,axiom,
% 12.85/6.91 ! [A: real,B: real,D: real,C: real] :
% 12.85/6.91 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.91 => ( ( ord_less_eq_real @ D @ C )
% 12.85/6.91 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_960_eq__iff__diff__eq__0,axiom,
% 12.85/6.91 ( ( ^ [Y5: complex,Z4: complex] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [A3: complex,B2: complex] :
% 12.85/6.91 ( ( minus_minus_complex @ A3 @ B2 )
% 12.85/6.91 = zero_zero_complex ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_961_eq__iff__diff__eq__0,axiom,
% 12.85/6.91 ( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [A3: real,B2: real] :
% 12.85/6.91 ( ( minus_minus_real @ A3 @ B2 )
% 12.85/6.91 = zero_zero_real ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_962_eq__iff__diff__eq__0,axiom,
% 12.85/6.91 ( ( ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [A3: rat,B2: rat] :
% 12.85/6.91 ( ( minus_minus_rat @ A3 @ B2 )
% 12.85/6.91 = zero_zero_rat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_963_eq__iff__diff__eq__0,axiom,
% 12.85/6.91 ( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [A3: int,B2: int] :
% 12.85/6.91 ( ( minus_minus_int @ A3 @ B2 )
% 12.85/6.91 = zero_zero_int ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_964_of__nat__diff,axiom,
% 12.85/6.91 ! [N: nat,M: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.91 => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N ) )
% 12.85/6.91 = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_965_of__nat__diff,axiom,
% 12.85/6.91 ! [N: nat,M: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.91 => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
% 12.85/6.91 = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_966_of__nat__diff,axiom,
% 12.85/6.91 ! [N: nat,M: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.91 => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
% 12.85/6.91 = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_967_of__nat__diff,axiom,
% 12.85/6.91 ! [N: nat,M: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.91 => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
% 12.85/6.91 = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_968_of__nat__diff,axiom,
% 12.85/6.91 ! [N: nat,M: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.91 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
% 12.85/6.91 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_969_right__diff__distrib_H,axiom,
% 12.85/6.91 ! [A: real,B: real,C: real] :
% 12.85/6.91 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 12.85/6.91 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_970_right__diff__distrib_H,axiom,
% 12.85/6.91 ! [A: rat,B: rat,C: rat] :
% 12.85/6.91 ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 12.85/6.91 = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_971_right__diff__distrib_H,axiom,
% 12.85/6.91 ! [A: nat,B: nat,C: nat] :
% 12.85/6.91 ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
% 12.85/6.91 = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_972_right__diff__distrib_H,axiom,
% 12.85/6.91 ! [A: int,B: int,C: int] :
% 12.85/6.91 ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 12.85/6.91 = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_973_left__diff__distrib_H,axiom,
% 12.85/6.91 ! [B: real,C: real,A: real] :
% 12.85/6.91 ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
% 12.85/6.91 = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_974_left__diff__distrib_H,axiom,
% 12.85/6.91 ! [B: rat,C: rat,A: rat] :
% 12.85/6.91 ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
% 12.85/6.91 = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_975_left__diff__distrib_H,axiom,
% 12.85/6.91 ! [B: nat,C: nat,A: nat] :
% 12.85/6.91 ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
% 12.85/6.91 = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_976_left__diff__distrib_H,axiom,
% 12.85/6.91 ! [B: int,C: int,A: int] :
% 12.85/6.91 ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
% 12.85/6.91 = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_977_right__diff__distrib,axiom,
% 12.85/6.91 ! [A: real,B: real,C: real] :
% 12.85/6.91 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 12.85/6.91 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_978_right__diff__distrib,axiom,
% 12.85/6.91 ! [A: rat,B: rat,C: rat] :
% 12.85/6.91 ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 12.85/6.91 = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_979_right__diff__distrib,axiom,
% 12.85/6.91 ! [A: int,B: int,C: int] :
% 12.85/6.91 ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 12.85/6.91 = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_980_left__diff__distrib,axiom,
% 12.85/6.91 ! [A: real,B: real,C: real] :
% 12.85/6.91 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
% 12.85/6.91 = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_981_left__diff__distrib,axiom,
% 12.85/6.91 ! [A: rat,B: rat,C: rat] :
% 12.85/6.91 ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 12.85/6.91 = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_982_left__diff__distrib,axiom,
% 12.85/6.91 ! [A: int,B: int,C: int] :
% 12.85/6.91 ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
% 12.85/6.91 = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_983_diff__divide__distrib,axiom,
% 12.85/6.91 ! [A: complex,B: complex,C: complex] :
% 12.85/6.91 ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
% 12.85/6.91 = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_984_diff__divide__distrib,axiom,
% 12.85/6.91 ! [A: real,B: real,C: real] :
% 12.85/6.91 ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
% 12.85/6.91 = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_985_diff__divide__distrib,axiom,
% 12.85/6.91 ! [A: rat,B: rat,C: rat] :
% 12.85/6.91 ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 12.85/6.91 = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_986_dvd__diff__commute,axiom,
% 12.85/6.91 ! [A: code_integer,C: code_integer,B: code_integer] :
% 12.85/6.91 ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
% 12.85/6.91 = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_987_dvd__diff__commute,axiom,
% 12.85/6.91 ! [A: int,C: int,B: int] :
% 12.85/6.91 ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
% 12.85/6.91 = ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_988_dvd__diff,axiom,
% 12.85/6.91 ! [X: code_integer,Y4: code_integer,Z: code_integer] :
% 12.85/6.91 ( ( dvd_dvd_Code_integer @ X @ Y4 )
% 12.85/6.91 => ( ( dvd_dvd_Code_integer @ X @ Z )
% 12.85/6.91 => ( dvd_dvd_Code_integer @ X @ ( minus_8373710615458151222nteger @ Y4 @ Z ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_989_dvd__diff,axiom,
% 12.85/6.91 ! [X: real,Y4: real,Z: real] :
% 12.85/6.91 ( ( dvd_dvd_real @ X @ Y4 )
% 12.85/6.91 => ( ( dvd_dvd_real @ X @ Z )
% 12.85/6.91 => ( dvd_dvd_real @ X @ ( minus_minus_real @ Y4 @ Z ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_990_dvd__diff,axiom,
% 12.85/6.91 ! [X: rat,Y4: rat,Z: rat] :
% 12.85/6.91 ( ( dvd_dvd_rat @ X @ Y4 )
% 12.85/6.91 => ( ( dvd_dvd_rat @ X @ Z )
% 12.85/6.91 => ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y4 @ Z ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_991_dvd__diff,axiom,
% 12.85/6.91 ! [X: int,Y4: int,Z: int] :
% 12.85/6.91 ( ( dvd_dvd_int @ X @ Y4 )
% 12.85/6.91 => ( ( dvd_dvd_int @ X @ Z )
% 12.85/6.91 => ( dvd_dvd_int @ X @ ( minus_minus_int @ Y4 @ Z ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_992_zero__induct__lemma,axiom,
% 12.85/6.91 ! [P: nat > $o,K: nat,I: nat] :
% 12.85/6.91 ( ( P @ K )
% 12.85/6.91 => ( ! [N2: nat] :
% 12.85/6.91 ( ( P @ ( suc @ N2 ) )
% 12.85/6.91 => ( P @ N2 ) )
% 12.85/6.91 => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_993_diffs0__imp__equal,axiom,
% 12.85/6.91 ! [M: nat,N: nat] :
% 12.85/6.91 ( ( ( minus_minus_nat @ M @ N )
% 12.85/6.91 = zero_zero_nat )
% 12.85/6.91 => ( ( ( minus_minus_nat @ N @ M )
% 12.85/6.91 = zero_zero_nat )
% 12.85/6.91 => ( M = N ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_994_minus__nat_Odiff__0,axiom,
% 12.85/6.91 ! [M: nat] :
% 12.85/6.91 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 12.85/6.91 = M ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_995_eq__diff__iff,axiom,
% 12.85/6.91 ! [K: nat,M: nat,N: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ K @ M )
% 12.85/6.91 => ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.91 => ( ( ( minus_minus_nat @ M @ K )
% 12.85/6.91 = ( minus_minus_nat @ N @ K ) )
% 12.85/6.91 = ( M = N ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_996_le__diff__iff,axiom,
% 12.85/6.91 ! [K: nat,M: nat,N: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ K @ M )
% 12.85/6.91 => ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.91 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 12.85/6.91 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_997_Nat_Odiff__diff__eq,axiom,
% 12.85/6.91 ! [K: nat,M: nat,N: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ K @ M )
% 12.85/6.91 => ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.91 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 12.85/6.91 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_998_diff__le__mono,axiom,
% 12.85/6.91 ! [M: nat,N: nat,L: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.91 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_999_diff__le__self,axiom,
% 12.85/6.91 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1000_le__diff__iff_H,axiom,
% 12.85/6.91 ! [A: nat,C: nat,B: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ A @ C )
% 12.85/6.91 => ( ( ord_less_eq_nat @ B @ C )
% 12.85/6.91 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 12.85/6.91 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1001_diff__le__mono2,axiom,
% 12.85/6.91 ! [M: nat,N: nat,L: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.91 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1002_diff__mult__distrib,axiom,
% 12.85/6.91 ! [M: nat,N: nat,K: nat] :
% 12.85/6.91 ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 12.85/6.91 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1003_diff__mult__distrib2,axiom,
% 12.85/6.91 ! [K: nat,M: nat,N: nat] :
% 12.85/6.91 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 12.85/6.91 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1004_dvd__diff__nat,axiom,
% 12.85/6.91 ! [K: nat,M: nat,N: nat] :
% 12.85/6.91 ( ( dvd_dvd_nat @ K @ M )
% 12.85/6.91 => ( ( dvd_dvd_nat @ K @ N )
% 12.85/6.91 => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1005_le__iff__diff__le__0,axiom,
% 12.85/6.91 ( ord_less_eq_rat
% 12.85/6.91 = ( ^ [A3: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B2 ) @ zero_zero_rat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1006_le__iff__diff__le__0,axiom,
% 12.85/6.91 ( ord_less_eq_int
% 12.85/6.91 = ( ^ [A3: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1007_le__iff__diff__le__0,axiom,
% 12.85/6.91 ( ord_less_eq_real
% 12.85/6.91 = ( ^ [A3: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1008_less__eq__int__code_I1_J,axiom,
% 12.85/6.91 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1009_times__int__code_I1_J,axiom,
% 12.85/6.91 ! [K: int] :
% 12.85/6.91 ( ( times_times_int @ K @ zero_zero_int )
% 12.85/6.91 = zero_zero_int ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1010_times__int__code_I2_J,axiom,
% 12.85/6.91 ! [L: int] :
% 12.85/6.91 ( ( times_times_int @ zero_zero_int @ L )
% 12.85/6.91 = zero_zero_int ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1011_ile0__eq,axiom,
% 12.85/6.91 ! [N: extended_enat] :
% 12.85/6.91 ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
% 12.85/6.91 = ( N = zero_z5237406670263579293d_enat ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1012_i0__lb,axiom,
% 12.85/6.91 ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1013_unset__bit__less__eq,axiom,
% 12.85/6.91 ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1014_set__bit__greater__eq,axiom,
% 12.85/6.91 ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1015_Suc__diff__le,axiom,
% 12.85/6.91 ! [N: nat,M: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.91 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 12.85/6.91 = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1016_dvd__diffD,axiom,
% 12.85/6.91 ! [K: nat,M: nat,N: nat] :
% 12.85/6.91 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 12.85/6.91 => ( ( dvd_dvd_nat @ K @ N )
% 12.85/6.91 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.91 => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1017_dvd__diffD1,axiom,
% 12.85/6.91 ! [K: nat,M: nat,N: nat] :
% 12.85/6.91 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 12.85/6.91 => ( ( dvd_dvd_nat @ K @ M )
% 12.85/6.91 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.91 => ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1018_less__eq__dvd__minus,axiom,
% 12.85/6.91 ! [M: nat,N: nat] :
% 12.85/6.91 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.91 => ( ( dvd_dvd_nat @ M @ N )
% 12.85/6.91 = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1019_int__ops_I1_J,axiom,
% 12.85/6.91 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 12.85/6.91 = zero_zero_int ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1020_nonneg__int__cases,axiom,
% 12.85/6.91 ! [K: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.85/6.91 => ~ ! [N2: nat] :
% 12.85/6.91 ( K
% 12.85/6.91 != ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1021_zero__le__imp__eq__int,axiom,
% 12.85/6.91 ! [K: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.85/6.91 => ? [N2: nat] :
% 12.85/6.91 ( K
% 12.85/6.91 = ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1022_zdvd__antisym__nonneg,axiom,
% 12.85/6.91 ! [M: int,N: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 12.85/6.91 => ( ( ord_less_eq_int @ zero_zero_int @ N )
% 12.85/6.91 => ( ( dvd_dvd_int @ M @ N )
% 12.85/6.91 => ( ( dvd_dvd_int @ N @ M )
% 12.85/6.91 => ( M = N ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1023_zdvd__mult__cancel,axiom,
% 12.85/6.91 ! [K: int,M: int,N: int] :
% 12.85/6.91 ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
% 12.85/6.91 => ( ( K != zero_zero_int )
% 12.85/6.91 => ( dvd_dvd_int @ M @ N ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1024_divide__diff__eq__iff,axiom,
% 12.85/6.91 ! [Z: complex,X: complex,Y4: complex] :
% 12.85/6.91 ( ( Z != zero_zero_complex )
% 12.85/6.91 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y4 )
% 12.85/6.91 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y4 @ Z ) ) @ Z ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1025_divide__diff__eq__iff,axiom,
% 12.85/6.91 ! [Z: real,X: real,Y4: real] :
% 12.85/6.91 ( ( Z != zero_zero_real )
% 12.85/6.91 => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y4 )
% 12.85/6.91 = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y4 @ Z ) ) @ Z ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1026_divide__diff__eq__iff,axiom,
% 12.85/6.91 ! [Z: rat,X: rat,Y4: rat] :
% 12.85/6.91 ( ( Z != zero_zero_rat )
% 12.85/6.91 => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y4 )
% 12.85/6.91 = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y4 @ Z ) ) @ Z ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1027_diff__divide__eq__iff,axiom,
% 12.85/6.91 ! [Z: complex,X: complex,Y4: complex] :
% 12.85/6.91 ( ( Z != zero_zero_complex )
% 12.85/6.91 => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y4 @ Z ) )
% 12.85/6.91 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ Y4 ) @ Z ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1028_diff__divide__eq__iff,axiom,
% 12.85/6.91 ! [Z: real,X: real,Y4: real] :
% 12.85/6.91 ( ( Z != zero_zero_real )
% 12.85/6.91 => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y4 @ Z ) )
% 12.85/6.91 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y4 ) @ Z ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1029_diff__divide__eq__iff,axiom,
% 12.85/6.91 ! [Z: rat,X: rat,Y4: rat] :
% 12.85/6.91 ( ( Z != zero_zero_rat )
% 12.85/6.91 => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y4 @ Z ) )
% 12.85/6.91 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y4 ) @ Z ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1030_diff__frac__eq,axiom,
% 12.85/6.91 ! [Y4: complex,Z: complex,X: complex,W: complex] :
% 12.85/6.91 ( ( Y4 != zero_zero_complex )
% 12.85/6.91 => ( ( Z != zero_zero_complex )
% 12.85/6.91 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y4 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 12.85/6.91 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y4 ) ) @ ( times_times_complex @ Y4 @ Z ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1031_diff__frac__eq,axiom,
% 12.85/6.91 ! [Y4: real,Z: real,X: real,W: real] :
% 12.85/6.91 ( ( Y4 != zero_zero_real )
% 12.85/6.91 => ( ( Z != zero_zero_real )
% 12.85/6.91 => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y4 ) @ ( divide_divide_real @ W @ Z ) )
% 12.85/6.91 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y4 ) ) @ ( times_times_real @ Y4 @ Z ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1032_diff__frac__eq,axiom,
% 12.85/6.91 ! [Y4: rat,Z: rat,X: rat,W: rat] :
% 12.85/6.91 ( ( Y4 != zero_zero_rat )
% 12.85/6.91 => ( ( Z != zero_zero_rat )
% 12.85/6.91 => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y4 ) @ ( divide_divide_rat @ W @ Z ) )
% 12.85/6.91 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y4 ) ) @ ( times_times_rat @ Y4 @ Z ) ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1033_add__divide__eq__if__simps_I4_J,axiom,
% 12.85/6.91 ! [Z: complex,A: complex,B: complex] :
% 12.85/6.91 ( ( ( Z = zero_zero_complex )
% 12.85/6.91 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 12.85/6.91 = A ) )
% 12.85/6.91 & ( ( Z != zero_zero_complex )
% 12.85/6.91 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 12.85/6.91 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1034_add__divide__eq__if__simps_I4_J,axiom,
% 12.85/6.91 ! [Z: real,A: real,B: real] :
% 12.85/6.91 ( ( ( Z = zero_zero_real )
% 12.85/6.91 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 12.85/6.91 = A ) )
% 12.85/6.91 & ( ( Z != zero_zero_real )
% 12.85/6.91 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 12.85/6.91 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1035_add__divide__eq__if__simps_I4_J,axiom,
% 12.85/6.91 ! [Z: rat,A: rat,B: rat] :
% 12.85/6.91 ( ( ( Z = zero_zero_rat )
% 12.85/6.91 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 12.85/6.91 = A ) )
% 12.85/6.91 & ( ( Z != zero_zero_rat )
% 12.85/6.91 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 12.85/6.91 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1036_zero__reorient,axiom,
% 12.85/6.91 ! [X: complex] :
% 12.85/6.91 ( ( zero_zero_complex = X )
% 12.85/6.91 = ( X = zero_zero_complex ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1037_zero__reorient,axiom,
% 12.85/6.91 ! [X: real] :
% 12.85/6.91 ( ( zero_zero_real = X )
% 12.85/6.91 = ( X = zero_zero_real ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1038_zero__reorient,axiom,
% 12.85/6.91 ! [X: rat] :
% 12.85/6.91 ( ( zero_zero_rat = X )
% 12.85/6.91 = ( X = zero_zero_rat ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1039_zero__reorient,axiom,
% 12.85/6.91 ! [X: nat] :
% 12.85/6.91 ( ( zero_zero_nat = X )
% 12.85/6.91 = ( X = zero_zero_nat ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1040_zero__reorient,axiom,
% 12.85/6.91 ! [X: int] :
% 12.85/6.91 ( ( zero_zero_int = X )
% 12.85/6.91 = ( X = zero_zero_int ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1041_mult_Oleft__commute,axiom,
% 12.85/6.91 ! [B: real,A: real,C: real] :
% 12.85/6.91 ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 12.85/6.91 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1042_mult_Oleft__commute,axiom,
% 12.85/6.91 ! [B: rat,A: rat,C: rat] :
% 12.85/6.91 ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
% 12.85/6.91 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1043_mult_Oleft__commute,axiom,
% 12.85/6.91 ! [B: nat,A: nat,C: nat] :
% 12.85/6.91 ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 12.85/6.91 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1044_mult_Oleft__commute,axiom,
% 12.85/6.91 ! [B: int,A: int,C: int] :
% 12.85/6.91 ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
% 12.85/6.91 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1045_mult_Ocommute,axiom,
% 12.85/6.91 ( times_times_real
% 12.85/6.91 = ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1046_mult_Ocommute,axiom,
% 12.85/6.91 ( times_times_rat
% 12.85/6.91 = ( ^ [A3: rat,B2: rat] : ( times_times_rat @ B2 @ A3 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1047_mult_Ocommute,axiom,
% 12.85/6.91 ( times_times_nat
% 12.85/6.91 = ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1048_mult_Ocommute,axiom,
% 12.85/6.91 ( times_times_int
% 12.85/6.91 = ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1049_mult_Oassoc,axiom,
% 12.85/6.91 ! [A: real,B: real,C: real] :
% 12.85/6.91 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 12.85/6.91 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1050_mult_Oassoc,axiom,
% 12.85/6.91 ! [A: rat,B: rat,C: rat] :
% 12.85/6.91 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 12.85/6.91 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1051_mult_Oassoc,axiom,
% 12.85/6.91 ! [A: nat,B: nat,C: nat] :
% 12.85/6.91 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 12.85/6.91 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1052_mult_Oassoc,axiom,
% 12.85/6.91 ! [A: int,B: int,C: int] :
% 12.85/6.91 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 12.85/6.91 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1053_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 12.85/6.91 ! [A: real,B: real,C: real] :
% 12.85/6.91 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 12.85/6.91 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1054_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 12.85/6.91 ! [A: rat,B: rat,C: rat] :
% 12.85/6.91 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 12.85/6.91 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1055_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 12.85/6.91 ! [A: nat,B: nat,C: nat] :
% 12.85/6.91 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 12.85/6.91 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1056_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 12.85/6.91 ! [A: int,B: int,C: int] :
% 12.85/6.91 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 12.85/6.91 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1057_in__mono,axiom,
% 12.85/6.91 ! [A2: set_complex,B4: set_complex,X: complex] :
% 12.85/6.91 ( ( ord_le211207098394363844omplex @ A2 @ B4 )
% 12.85/6.91 => ( ( member_complex @ X @ A2 )
% 12.85/6.91 => ( member_complex @ X @ B4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1058_in__mono,axiom,
% 12.85/6.91 ! [A2: set_real,B4: set_real,X: real] :
% 12.85/6.91 ( ( ord_less_eq_set_real @ A2 @ B4 )
% 12.85/6.91 => ( ( member_real @ X @ A2 )
% 12.85/6.91 => ( member_real @ X @ B4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1059_in__mono,axiom,
% 12.85/6.91 ! [A2: set_set_nat,B4: set_set_nat,X: set_nat] :
% 12.85/6.91 ( ( ord_le6893508408891458716et_nat @ A2 @ B4 )
% 12.85/6.91 => ( ( member_set_nat @ X @ A2 )
% 12.85/6.91 => ( member_set_nat @ X @ B4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1060_in__mono,axiom,
% 12.85/6.91 ! [A2: set_nat,B4: set_nat,X: nat] :
% 12.85/6.91 ( ( ord_less_eq_set_nat @ A2 @ B4 )
% 12.85/6.91 => ( ( member_nat @ X @ A2 )
% 12.85/6.91 => ( member_nat @ X @ B4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1061_in__mono,axiom,
% 12.85/6.91 ! [A2: set_int,B4: set_int,X: int] :
% 12.85/6.91 ( ( ord_less_eq_set_int @ A2 @ B4 )
% 12.85/6.91 => ( ( member_int @ X @ A2 )
% 12.85/6.91 => ( member_int @ X @ B4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1062_subsetD,axiom,
% 12.85/6.91 ! [A2: set_complex,B4: set_complex,C: complex] :
% 12.85/6.91 ( ( ord_le211207098394363844omplex @ A2 @ B4 )
% 12.85/6.91 => ( ( member_complex @ C @ A2 )
% 12.85/6.91 => ( member_complex @ C @ B4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1063_subsetD,axiom,
% 12.85/6.91 ! [A2: set_real,B4: set_real,C: real] :
% 12.85/6.91 ( ( ord_less_eq_set_real @ A2 @ B4 )
% 12.85/6.91 => ( ( member_real @ C @ A2 )
% 12.85/6.91 => ( member_real @ C @ B4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1064_subsetD,axiom,
% 12.85/6.91 ! [A2: set_set_nat,B4: set_set_nat,C: set_nat] :
% 12.85/6.91 ( ( ord_le6893508408891458716et_nat @ A2 @ B4 )
% 12.85/6.91 => ( ( member_set_nat @ C @ A2 )
% 12.85/6.91 => ( member_set_nat @ C @ B4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1065_subsetD,axiom,
% 12.85/6.91 ! [A2: set_nat,B4: set_nat,C: nat] :
% 12.85/6.91 ( ( ord_less_eq_set_nat @ A2 @ B4 )
% 12.85/6.91 => ( ( member_nat @ C @ A2 )
% 12.85/6.91 => ( member_nat @ C @ B4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1066_subsetD,axiom,
% 12.85/6.91 ! [A2: set_int,B4: set_int,C: int] :
% 12.85/6.91 ( ( ord_less_eq_set_int @ A2 @ B4 )
% 12.85/6.91 => ( ( member_int @ C @ A2 )
% 12.85/6.91 => ( member_int @ C @ B4 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1067_equalityE,axiom,
% 12.85/6.91 ! [A2: set_int,B4: set_int] :
% 12.85/6.91 ( ( A2 = B4 )
% 12.85/6.91 => ~ ( ( ord_less_eq_set_int @ A2 @ B4 )
% 12.85/6.91 => ~ ( ord_less_eq_set_int @ B4 @ A2 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1068_subset__eq,axiom,
% 12.85/6.91 ( ord_le211207098394363844omplex
% 12.85/6.91 = ( ^ [A5: set_complex,B5: set_complex] :
% 12.85/6.91 ! [X3: complex] :
% 12.85/6.91 ( ( member_complex @ X3 @ A5 )
% 12.85/6.91 => ( member_complex @ X3 @ B5 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1069_subset__eq,axiom,
% 12.85/6.91 ( ord_less_eq_set_real
% 12.85/6.91 = ( ^ [A5: set_real,B5: set_real] :
% 12.85/6.91 ! [X3: real] :
% 12.85/6.91 ( ( member_real @ X3 @ A5 )
% 12.85/6.91 => ( member_real @ X3 @ B5 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1070_subset__eq,axiom,
% 12.85/6.91 ( ord_le6893508408891458716et_nat
% 12.85/6.91 = ( ^ [A5: set_set_nat,B5: set_set_nat] :
% 12.85/6.91 ! [X3: set_nat] :
% 12.85/6.91 ( ( member_set_nat @ X3 @ A5 )
% 12.85/6.91 => ( member_set_nat @ X3 @ B5 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1071_subset__eq,axiom,
% 12.85/6.91 ( ord_less_eq_set_nat
% 12.85/6.91 = ( ^ [A5: set_nat,B5: set_nat] :
% 12.85/6.91 ! [X3: nat] :
% 12.85/6.91 ( ( member_nat @ X3 @ A5 )
% 12.85/6.91 => ( member_nat @ X3 @ B5 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1072_subset__eq,axiom,
% 12.85/6.91 ( ord_less_eq_set_int
% 12.85/6.91 = ( ^ [A5: set_int,B5: set_int] :
% 12.85/6.91 ! [X3: int] :
% 12.85/6.91 ( ( member_int @ X3 @ A5 )
% 12.85/6.91 => ( member_int @ X3 @ B5 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1073_equalityD1,axiom,
% 12.85/6.91 ! [A2: set_int,B4: set_int] :
% 12.85/6.91 ( ( A2 = B4 )
% 12.85/6.91 => ( ord_less_eq_set_int @ A2 @ B4 ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1074_equalityD2,axiom,
% 12.85/6.91 ! [A2: set_int,B4: set_int] :
% 12.85/6.91 ( ( A2 = B4 )
% 12.85/6.91 => ( ord_less_eq_set_int @ B4 @ A2 ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1075_subset__iff,axiom,
% 12.85/6.91 ( ord_le211207098394363844omplex
% 12.85/6.91 = ( ^ [A5: set_complex,B5: set_complex] :
% 12.85/6.91 ! [T2: complex] :
% 12.85/6.91 ( ( member_complex @ T2 @ A5 )
% 12.85/6.91 => ( member_complex @ T2 @ B5 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1076_subset__iff,axiom,
% 12.85/6.91 ( ord_less_eq_set_real
% 12.85/6.91 = ( ^ [A5: set_real,B5: set_real] :
% 12.85/6.91 ! [T2: real] :
% 12.85/6.91 ( ( member_real @ T2 @ A5 )
% 12.85/6.91 => ( member_real @ T2 @ B5 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1077_subset__iff,axiom,
% 12.85/6.91 ( ord_le6893508408891458716et_nat
% 12.85/6.91 = ( ^ [A5: set_set_nat,B5: set_set_nat] :
% 12.85/6.91 ! [T2: set_nat] :
% 12.85/6.91 ( ( member_set_nat @ T2 @ A5 )
% 12.85/6.91 => ( member_set_nat @ T2 @ B5 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1078_subset__iff,axiom,
% 12.85/6.91 ( ord_less_eq_set_nat
% 12.85/6.91 = ( ^ [A5: set_nat,B5: set_nat] :
% 12.85/6.91 ! [T2: nat] :
% 12.85/6.91 ( ( member_nat @ T2 @ A5 )
% 12.85/6.91 => ( member_nat @ T2 @ B5 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1079_subset__iff,axiom,
% 12.85/6.91 ( ord_less_eq_set_int
% 12.85/6.91 = ( ^ [A5: set_int,B5: set_int] :
% 12.85/6.91 ! [T2: int] :
% 12.85/6.91 ( ( member_int @ T2 @ A5 )
% 12.85/6.91 => ( member_int @ T2 @ B5 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1080_subset__refl,axiom,
% 12.85/6.91 ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1081_Collect__mono,axiom,
% 12.85/6.91 ! [P: real > $o,Q: real > $o] :
% 12.85/6.91 ( ! [X4: real] :
% 12.85/6.91 ( ( P @ X4 )
% 12.85/6.91 => ( Q @ X4 ) )
% 12.85/6.91 => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1082_Collect__mono,axiom,
% 12.85/6.91 ! [P: list_nat > $o,Q: list_nat > $o] :
% 12.85/6.91 ( ! [X4: list_nat] :
% 12.85/6.91 ( ( P @ X4 )
% 12.85/6.91 => ( Q @ X4 ) )
% 12.85/6.91 => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1083_Collect__mono,axiom,
% 12.85/6.91 ! [P: set_nat > $o,Q: set_nat > $o] :
% 12.85/6.91 ( ! [X4: set_nat] :
% 12.85/6.91 ( ( P @ X4 )
% 12.85/6.91 => ( Q @ X4 ) )
% 12.85/6.91 => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1084_Collect__mono,axiom,
% 12.85/6.91 ! [P: nat > $o,Q: nat > $o] :
% 12.85/6.91 ( ! [X4: nat] :
% 12.85/6.91 ( ( P @ X4 )
% 12.85/6.91 => ( Q @ X4 ) )
% 12.85/6.91 => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1085_Collect__mono,axiom,
% 12.85/6.91 ! [P: int > $o,Q: int > $o] :
% 12.85/6.91 ( ! [X4: int] :
% 12.85/6.91 ( ( P @ X4 )
% 12.85/6.91 => ( Q @ X4 ) )
% 12.85/6.91 => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1086_subset__trans,axiom,
% 12.85/6.91 ! [A2: set_int,B4: set_int,C2: set_int] :
% 12.85/6.91 ( ( ord_less_eq_set_int @ A2 @ B4 )
% 12.85/6.91 => ( ( ord_less_eq_set_int @ B4 @ C2 )
% 12.85/6.91 => ( ord_less_eq_set_int @ A2 @ C2 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1087_set__eq__subset,axiom,
% 12.85/6.91 ( ( ^ [Y5: set_int,Z4: set_int] : ( Y5 = Z4 ) )
% 12.85/6.91 = ( ^ [A5: set_int,B5: set_int] :
% 12.85/6.91 ( ( ord_less_eq_set_int @ A5 @ B5 )
% 12.85/6.91 & ( ord_less_eq_set_int @ B5 @ A5 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1088_Collect__mono__iff,axiom,
% 12.85/6.91 ! [P: real > $o,Q: real > $o] :
% 12.85/6.91 ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
% 12.85/6.91 = ( ! [X3: real] :
% 12.85/6.91 ( ( P @ X3 )
% 12.85/6.91 => ( Q @ X3 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1089_Collect__mono__iff,axiom,
% 12.85/6.91 ! [P: list_nat > $o,Q: list_nat > $o] :
% 12.85/6.91 ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
% 12.85/6.91 = ( ! [X3: list_nat] :
% 12.85/6.91 ( ( P @ X3 )
% 12.85/6.91 => ( Q @ X3 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1090_Collect__mono__iff,axiom,
% 12.85/6.91 ! [P: set_nat > $o,Q: set_nat > $o] :
% 12.85/6.91 ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
% 12.85/6.91 = ( ! [X3: set_nat] :
% 12.85/6.91 ( ( P @ X3 )
% 12.85/6.91 => ( Q @ X3 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1091_Collect__mono__iff,axiom,
% 12.85/6.91 ! [P: nat > $o,Q: nat > $o] :
% 12.85/6.91 ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 12.85/6.91 = ( ! [X3: nat] :
% 12.85/6.91 ( ( P @ X3 )
% 12.85/6.91 => ( Q @ X3 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1092_Collect__mono__iff,axiom,
% 12.85/6.91 ! [P: int > $o,Q: int > $o] :
% 12.85/6.91 ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
% 12.85/6.91 = ( ! [X3: int] :
% 12.85/6.91 ( ( P @ X3 )
% 12.85/6.91 => ( Q @ X3 ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1093_zdiv__zmult2__eq,axiom,
% 12.85/6.91 ! [C: int,A: int,B: int] :
% 12.85/6.91 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.91 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 12.85/6.91 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1094_frac__le__eq,axiom,
% 12.85/6.91 ! [Y4: rat,Z: rat,X: rat,W: rat] :
% 12.85/6.91 ( ( Y4 != zero_zero_rat )
% 12.85/6.91 => ( ( Z != zero_zero_rat )
% 12.85/6.91 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y4 ) @ ( divide_divide_rat @ W @ Z ) )
% 12.85/6.91 = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y4 ) ) @ ( times_times_rat @ Y4 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1095_frac__le__eq,axiom,
% 12.85/6.91 ! [Y4: real,Z: real,X: real,W: real] :
% 12.85/6.91 ( ( Y4 != zero_zero_real )
% 12.85/6.91 => ( ( Z != zero_zero_real )
% 12.85/6.91 => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ ( divide_divide_real @ W @ Z ) )
% 12.85/6.91 = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y4 ) ) @ ( times_times_real @ Y4 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1096_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
% 12.85/6.91 ( ( vEBT_V8346862874174094_d_u_p @ zero_zero_nat )
% 12.85/6.91 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1097_zero__le,axiom,
% 12.85/6.91 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1098_bezout1__nat,axiom,
% 12.85/6.91 ! [A: nat,B: nat] :
% 12.85/6.91 ? [D2: nat,X4: nat,Y: nat] :
% 12.85/6.91 ( ( dvd_dvd_nat @ D2 @ A )
% 12.85/6.91 & ( dvd_dvd_nat @ D2 @ B )
% 12.85/6.91 & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y ) )
% 12.85/6.91 = D2 )
% 12.85/6.91 | ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y ) )
% 12.85/6.91 = D2 ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1099_odd__two__times__div__two__nat,axiom,
% 12.85/6.91 ! [N: nat] :
% 12.85/6.91 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.91 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.91 = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1100_buildup__build__time,axiom,
% 12.85/6.91 ! [N: nat] : ( ord_less_nat @ ( vEBT_V8346862874174094_d_u_p @ N ) @ ( vEBT_V8646137997579335489_i_l_d @ N ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1101_zdvd__mono,axiom,
% 12.85/6.91 ! [K: int,M: int,T: int] :
% 12.85/6.91 ( ( K != zero_zero_int )
% 12.85/6.91 => ( ( dvd_dvd_int @ M @ T )
% 12.85/6.91 = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1102_zero__le__power__eq__numeral,axiom,
% 12.85/6.91 ! [A: rat,W: num] :
% 12.85/6.91 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 12.85/6.91 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.91 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.91 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1103_zero__le__power__eq__numeral,axiom,
% 12.85/6.91 ! [A: int,W: num] :
% 12.85/6.91 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 12.85/6.91 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.91 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.91 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1104_zero__le__power__eq__numeral,axiom,
% 12.85/6.91 ! [A: real,W: num] :
% 12.85/6.91 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 12.85/6.91 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.91 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.91 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1105_gcd__nat_Oextremum,axiom,
% 12.85/6.91 ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1106_gcd__nat_Oextremum__strict,axiom,
% 12.85/6.91 ! [A: nat] :
% 12.85/6.91 ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 12.85/6.91 & ( zero_zero_nat != A ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1107_gcd__nat_Oextremum__unique,axiom,
% 12.85/6.91 ! [A: nat] :
% 12.85/6.91 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 12.85/6.91 = ( A = zero_zero_nat ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1108_gcd__nat_Onot__eq__extremum,axiom,
% 12.85/6.91 ! [A: nat] :
% 12.85/6.91 ( ( A != zero_zero_nat )
% 12.85/6.91 = ( ( dvd_dvd_nat @ A @ zero_zero_nat )
% 12.85/6.91 & ( A != zero_zero_nat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1109_gcd__nat_Oextremum__uniqueI,axiom,
% 12.85/6.91 ! [A: nat] :
% 12.85/6.91 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 12.85/6.91 => ( A = zero_zero_nat ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1110_idiff__0,axiom,
% 12.85/6.91 ! [N: extended_enat] :
% 12.85/6.91 ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
% 12.85/6.91 = zero_z5237406670263579293d_enat ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1111_idiff__0__right,axiom,
% 12.85/6.91 ! [N: extended_enat] :
% 12.85/6.91 ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
% 12.85/6.91 = N ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1112_not__gr__zero,axiom,
% 12.85/6.91 ! [N: nat] :
% 12.85/6.91 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 12.85/6.91 = ( N = zero_zero_nat ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1113_numeral__less__iff,axiom,
% 12.85/6.91 ! [M: num,N: num] :
% 12.85/6.91 ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 12.85/6.91 = ( ord_less_num @ M @ N ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1114_numeral__less__iff,axiom,
% 12.85/6.91 ! [M: num,N: num] :
% 12.85/6.91 ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 12.85/6.91 = ( ord_less_num @ M @ N ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1115_numeral__less__iff,axiom,
% 12.85/6.91 ! [M: num,N: num] :
% 12.85/6.91 ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 12.85/6.91 = ( ord_less_num @ M @ N ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1116_numeral__less__iff,axiom,
% 12.85/6.91 ! [M: num,N: num] :
% 12.85/6.91 ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 12.85/6.91 = ( ord_less_num @ M @ N ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1117_mult_Oright__neutral,axiom,
% 12.85/6.91 ! [A: complex] :
% 12.85/6.91 ( ( times_times_complex @ A @ one_one_complex )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1118_mult_Oright__neutral,axiom,
% 12.85/6.91 ! [A: real] :
% 12.85/6.91 ( ( times_times_real @ A @ one_one_real )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1119_mult_Oright__neutral,axiom,
% 12.85/6.91 ! [A: rat] :
% 12.85/6.91 ( ( times_times_rat @ A @ one_one_rat )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1120_mult_Oright__neutral,axiom,
% 12.85/6.91 ! [A: nat] :
% 12.85/6.91 ( ( times_times_nat @ A @ one_one_nat )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1121_mult_Oright__neutral,axiom,
% 12.85/6.91 ! [A: int] :
% 12.85/6.91 ( ( times_times_int @ A @ one_one_int )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1122_mult__1,axiom,
% 12.85/6.91 ! [A: complex] :
% 12.85/6.91 ( ( times_times_complex @ one_one_complex @ A )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1123_mult__1,axiom,
% 12.85/6.91 ! [A: real] :
% 12.85/6.91 ( ( times_times_real @ one_one_real @ A )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1124_mult__1,axiom,
% 12.85/6.91 ! [A: rat] :
% 12.85/6.91 ( ( times_times_rat @ one_one_rat @ A )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1125_mult__1,axiom,
% 12.85/6.91 ! [A: nat] :
% 12.85/6.91 ( ( times_times_nat @ one_one_nat @ A )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1126_mult__1,axiom,
% 12.85/6.91 ! [A: int] :
% 12.85/6.91 ( ( times_times_int @ one_one_int @ A )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1127_div__by__1,axiom,
% 12.85/6.91 ! [A: complex] :
% 12.85/6.91 ( ( divide1717551699836669952omplex @ A @ one_one_complex )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1128_div__by__1,axiom,
% 12.85/6.91 ! [A: real] :
% 12.85/6.91 ( ( divide_divide_real @ A @ one_one_real )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1129_div__by__1,axiom,
% 12.85/6.91 ! [A: rat] :
% 12.85/6.91 ( ( divide_divide_rat @ A @ one_one_rat )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1130_div__by__1,axiom,
% 12.85/6.91 ! [A: nat] :
% 12.85/6.91 ( ( divide_divide_nat @ A @ one_one_nat )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1131_div__by__1,axiom,
% 12.85/6.91 ! [A: int] :
% 12.85/6.91 ( ( divide_divide_int @ A @ one_one_int )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1132_bits__div__by__1,axiom,
% 12.85/6.91 ! [A: nat] :
% 12.85/6.91 ( ( divide_divide_nat @ A @ one_one_nat )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1133_bits__div__by__1,axiom,
% 12.85/6.91 ! [A: int] :
% 12.85/6.91 ( ( divide_divide_int @ A @ one_one_int )
% 12.85/6.91 = A ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1134_lessI,axiom,
% 12.85/6.91 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1135_Suc__mono,axiom,
% 12.85/6.91 ! [M: nat,N: nat] :
% 12.85/6.91 ( ( ord_less_nat @ M @ N )
% 12.85/6.91 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1136_Suc__less__eq,axiom,
% 12.85/6.91 ! [M: nat,N: nat] :
% 12.85/6.91 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 12.85/6.91 = ( ord_less_nat @ M @ N ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1137_bot__nat__0_Onot__eq__extremum,axiom,
% 12.85/6.91 ! [A: nat] :
% 12.85/6.91 ( ( A != zero_zero_nat )
% 12.85/6.91 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1138_neq0__conv,axiom,
% 12.85/6.91 ! [N: nat] :
% 12.85/6.91 ( ( N != zero_zero_nat )
% 12.85/6.91 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1139_less__nat__zero__code,axiom,
% 12.85/6.91 ! [N: nat] :
% 12.85/6.91 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1140_nat__mult__eq__1__iff,axiom,
% 12.85/6.91 ! [M: nat,N: nat] :
% 12.85/6.91 ( ( ( times_times_nat @ M @ N )
% 12.85/6.91 = one_one_nat )
% 12.85/6.91 = ( ( M = one_one_nat )
% 12.85/6.91 & ( N = one_one_nat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1141_nat__1__eq__mult__iff,axiom,
% 12.85/6.91 ! [M: nat,N: nat] :
% 12.85/6.91 ( ( one_one_nat
% 12.85/6.91 = ( times_times_nat @ M @ N ) )
% 12.85/6.91 = ( ( M = one_one_nat )
% 12.85/6.91 & ( N = one_one_nat ) ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1142_nat__dvd__1__iff__1,axiom,
% 12.85/6.91 ! [M: nat] :
% 12.85/6.91 ( ( dvd_dvd_nat @ M @ one_one_nat )
% 12.85/6.91 = ( M = one_one_nat ) ) ).
% 12.85/6.91
% 12.85/6.91 thf(fact_1143_diff__gt__0__iff__gt,axiom,
% 12.85/6.91 ! [A: real,B: real] :
% 12.85/6.91 ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 12.85/6.91 = ( ord_less_real @ B @ A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1144_diff__gt__0__iff__gt,axiom,
% 12.85/6.92 ! [A: rat,B: rat] :
% 12.85/6.92 ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 12.85/6.92 = ( ord_less_rat @ B @ A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1145_diff__gt__0__iff__gt,axiom,
% 12.85/6.92 ! [A: int,B: int] :
% 12.85/6.92 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 12.85/6.92 = ( ord_less_int @ B @ A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1146_mult__cancel__left1,axiom,
% 12.85/6.92 ! [C: complex,B: complex] :
% 12.85/6.92 ( ( C
% 12.85/6.92 = ( times_times_complex @ C @ B ) )
% 12.85/6.92 = ( ( C = zero_zero_complex )
% 12.85/6.92 | ( B = one_one_complex ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1147_mult__cancel__left1,axiom,
% 12.85/6.92 ! [C: real,B: real] :
% 12.85/6.92 ( ( C
% 12.85/6.92 = ( times_times_real @ C @ B ) )
% 12.85/6.92 = ( ( C = zero_zero_real )
% 12.85/6.92 | ( B = one_one_real ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1148_mult__cancel__left1,axiom,
% 12.85/6.92 ! [C: rat,B: rat] :
% 12.85/6.92 ( ( C
% 12.85/6.92 = ( times_times_rat @ C @ B ) )
% 12.85/6.92 = ( ( C = zero_zero_rat )
% 12.85/6.92 | ( B = one_one_rat ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1149_mult__cancel__left1,axiom,
% 12.85/6.92 ! [C: int,B: int] :
% 12.85/6.92 ( ( C
% 12.85/6.92 = ( times_times_int @ C @ B ) )
% 12.85/6.92 = ( ( C = zero_zero_int )
% 12.85/6.92 | ( B = one_one_int ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1150_mult__cancel__left2,axiom,
% 12.85/6.92 ! [C: complex,A: complex] :
% 12.85/6.92 ( ( ( times_times_complex @ C @ A )
% 12.85/6.92 = C )
% 12.85/6.92 = ( ( C = zero_zero_complex )
% 12.85/6.92 | ( A = one_one_complex ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1151_mult__cancel__left2,axiom,
% 12.85/6.92 ! [C: real,A: real] :
% 12.85/6.92 ( ( ( times_times_real @ C @ A )
% 12.85/6.92 = C )
% 12.85/6.92 = ( ( C = zero_zero_real )
% 12.85/6.92 | ( A = one_one_real ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1152_mult__cancel__left2,axiom,
% 12.85/6.92 ! [C: rat,A: rat] :
% 12.85/6.92 ( ( ( times_times_rat @ C @ A )
% 12.85/6.92 = C )
% 12.85/6.92 = ( ( C = zero_zero_rat )
% 12.85/6.92 | ( A = one_one_rat ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1153_mult__cancel__left2,axiom,
% 12.85/6.92 ! [C: int,A: int] :
% 12.85/6.92 ( ( ( times_times_int @ C @ A )
% 12.85/6.92 = C )
% 12.85/6.92 = ( ( C = zero_zero_int )
% 12.85/6.92 | ( A = one_one_int ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1154_mult__cancel__right1,axiom,
% 12.85/6.92 ! [C: complex,B: complex] :
% 12.85/6.92 ( ( C
% 12.85/6.92 = ( times_times_complex @ B @ C ) )
% 12.85/6.92 = ( ( C = zero_zero_complex )
% 12.85/6.92 | ( B = one_one_complex ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1155_mult__cancel__right1,axiom,
% 12.85/6.92 ! [C: real,B: real] :
% 12.85/6.92 ( ( C
% 12.85/6.92 = ( times_times_real @ B @ C ) )
% 12.85/6.92 = ( ( C = zero_zero_real )
% 12.85/6.92 | ( B = one_one_real ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1156_mult__cancel__right1,axiom,
% 12.85/6.92 ! [C: rat,B: rat] :
% 12.85/6.92 ( ( C
% 12.85/6.92 = ( times_times_rat @ B @ C ) )
% 12.85/6.92 = ( ( C = zero_zero_rat )
% 12.85/6.92 | ( B = one_one_rat ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1157_mult__cancel__right1,axiom,
% 12.85/6.92 ! [C: int,B: int] :
% 12.85/6.92 ( ( C
% 12.85/6.92 = ( times_times_int @ B @ C ) )
% 12.85/6.92 = ( ( C = zero_zero_int )
% 12.85/6.92 | ( B = one_one_int ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1158_mult__cancel__right2,axiom,
% 12.85/6.92 ! [A: complex,C: complex] :
% 12.85/6.92 ( ( ( times_times_complex @ A @ C )
% 12.85/6.92 = C )
% 12.85/6.92 = ( ( C = zero_zero_complex )
% 12.85/6.92 | ( A = one_one_complex ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1159_mult__cancel__right2,axiom,
% 12.85/6.92 ! [A: real,C: real] :
% 12.85/6.92 ( ( ( times_times_real @ A @ C )
% 12.85/6.92 = C )
% 12.85/6.92 = ( ( C = zero_zero_real )
% 12.85/6.92 | ( A = one_one_real ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1160_mult__cancel__right2,axiom,
% 12.85/6.92 ! [A: rat,C: rat] :
% 12.85/6.92 ( ( ( times_times_rat @ A @ C )
% 12.85/6.92 = C )
% 12.85/6.92 = ( ( C = zero_zero_rat )
% 12.85/6.92 | ( A = one_one_rat ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1161_mult__cancel__right2,axiom,
% 12.85/6.92 ! [A: int,C: int] :
% 12.85/6.92 ( ( ( times_times_int @ A @ C )
% 12.85/6.92 = C )
% 12.85/6.92 = ( ( C = zero_zero_int )
% 12.85/6.92 | ( A = one_one_int ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1162_diff__numeral__special_I9_J,axiom,
% 12.85/6.92 ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 12.85/6.92 = zero_zero_complex ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1163_diff__numeral__special_I9_J,axiom,
% 12.85/6.92 ( ( minus_minus_real @ one_one_real @ one_one_real )
% 12.85/6.92 = zero_zero_real ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1164_diff__numeral__special_I9_J,axiom,
% 12.85/6.92 ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 12.85/6.92 = zero_zero_rat ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1165_diff__numeral__special_I9_J,axiom,
% 12.85/6.92 ( ( minus_minus_int @ one_one_int @ one_one_int )
% 12.85/6.92 = zero_zero_int ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1166_div__self,axiom,
% 12.85/6.92 ! [A: complex] :
% 12.85/6.92 ( ( A != zero_zero_complex )
% 12.85/6.92 => ( ( divide1717551699836669952omplex @ A @ A )
% 12.85/6.92 = one_one_complex ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1167_div__self,axiom,
% 12.85/6.92 ! [A: real] :
% 12.85/6.92 ( ( A != zero_zero_real )
% 12.85/6.92 => ( ( divide_divide_real @ A @ A )
% 12.85/6.92 = one_one_real ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1168_div__self,axiom,
% 12.85/6.92 ! [A: rat] :
% 12.85/6.92 ( ( A != zero_zero_rat )
% 12.85/6.92 => ( ( divide_divide_rat @ A @ A )
% 12.85/6.92 = one_one_rat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1169_div__self,axiom,
% 12.85/6.92 ! [A: nat] :
% 12.85/6.92 ( ( A != zero_zero_nat )
% 12.85/6.92 => ( ( divide_divide_nat @ A @ A )
% 12.85/6.92 = one_one_nat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1170_div__self,axiom,
% 12.85/6.92 ! [A: int] :
% 12.85/6.92 ( ( A != zero_zero_int )
% 12.85/6.92 => ( ( divide_divide_int @ A @ A )
% 12.85/6.92 = one_one_int ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1171_divide__eq__1__iff,axiom,
% 12.85/6.92 ! [A: complex,B: complex] :
% 12.85/6.92 ( ( ( divide1717551699836669952omplex @ A @ B )
% 12.85/6.92 = one_one_complex )
% 12.85/6.92 = ( ( B != zero_zero_complex )
% 12.85/6.92 & ( A = B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1172_divide__eq__1__iff,axiom,
% 12.85/6.92 ! [A: real,B: real] :
% 12.85/6.92 ( ( ( divide_divide_real @ A @ B )
% 12.85/6.92 = one_one_real )
% 12.85/6.92 = ( ( B != zero_zero_real )
% 12.85/6.92 & ( A = B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1173_divide__eq__1__iff,axiom,
% 12.85/6.92 ! [A: rat,B: rat] :
% 12.85/6.92 ( ( ( divide_divide_rat @ A @ B )
% 12.85/6.92 = one_one_rat )
% 12.85/6.92 = ( ( B != zero_zero_rat )
% 12.85/6.92 & ( A = B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1174_one__eq__divide__iff,axiom,
% 12.85/6.92 ! [A: complex,B: complex] :
% 12.85/6.92 ( ( one_one_complex
% 12.85/6.92 = ( divide1717551699836669952omplex @ A @ B ) )
% 12.85/6.92 = ( ( B != zero_zero_complex )
% 12.85/6.92 & ( A = B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1175_one__eq__divide__iff,axiom,
% 12.85/6.92 ! [A: real,B: real] :
% 12.85/6.92 ( ( one_one_real
% 12.85/6.92 = ( divide_divide_real @ A @ B ) )
% 12.85/6.92 = ( ( B != zero_zero_real )
% 12.85/6.92 & ( A = B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1176_one__eq__divide__iff,axiom,
% 12.85/6.92 ! [A: rat,B: rat] :
% 12.85/6.92 ( ( one_one_rat
% 12.85/6.92 = ( divide_divide_rat @ A @ B ) )
% 12.85/6.92 = ( ( B != zero_zero_rat )
% 12.85/6.92 & ( A = B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1177_divide__self,axiom,
% 12.85/6.92 ! [A: complex] :
% 12.85/6.92 ( ( A != zero_zero_complex )
% 12.85/6.92 => ( ( divide1717551699836669952omplex @ A @ A )
% 12.85/6.92 = one_one_complex ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1178_divide__self,axiom,
% 12.85/6.92 ! [A: real] :
% 12.85/6.92 ( ( A != zero_zero_real )
% 12.85/6.92 => ( ( divide_divide_real @ A @ A )
% 12.85/6.92 = one_one_real ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1179_divide__self,axiom,
% 12.85/6.92 ! [A: rat] :
% 12.85/6.92 ( ( A != zero_zero_rat )
% 12.85/6.92 => ( ( divide_divide_rat @ A @ A )
% 12.85/6.92 = one_one_rat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1180_divide__self__if,axiom,
% 12.85/6.92 ! [A: complex] :
% 12.85/6.92 ( ( ( A = zero_zero_complex )
% 12.85/6.92 => ( ( divide1717551699836669952omplex @ A @ A )
% 12.85/6.92 = zero_zero_complex ) )
% 12.85/6.92 & ( ( A != zero_zero_complex )
% 12.85/6.92 => ( ( divide1717551699836669952omplex @ A @ A )
% 12.85/6.92 = one_one_complex ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1181_divide__self__if,axiom,
% 12.85/6.92 ! [A: real] :
% 12.85/6.92 ( ( ( A = zero_zero_real )
% 12.85/6.92 => ( ( divide_divide_real @ A @ A )
% 12.85/6.92 = zero_zero_real ) )
% 12.85/6.92 & ( ( A != zero_zero_real )
% 12.85/6.92 => ( ( divide_divide_real @ A @ A )
% 12.85/6.92 = one_one_real ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1182_divide__self__if,axiom,
% 12.85/6.92 ! [A: rat] :
% 12.85/6.92 ( ( ( A = zero_zero_rat )
% 12.85/6.92 => ( ( divide_divide_rat @ A @ A )
% 12.85/6.92 = zero_zero_rat ) )
% 12.85/6.92 & ( ( A != zero_zero_rat )
% 12.85/6.92 => ( ( divide_divide_rat @ A @ A )
% 12.85/6.92 = one_one_rat ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1183_divide__eq__eq__1,axiom,
% 12.85/6.92 ! [B: real,A: real] :
% 12.85/6.92 ( ( ( divide_divide_real @ B @ A )
% 12.85/6.92 = one_one_real )
% 12.85/6.92 = ( ( A != zero_zero_real )
% 12.85/6.92 & ( A = B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1184_divide__eq__eq__1,axiom,
% 12.85/6.92 ! [B: rat,A: rat] :
% 12.85/6.92 ( ( ( divide_divide_rat @ B @ A )
% 12.85/6.92 = one_one_rat )
% 12.85/6.92 = ( ( A != zero_zero_rat )
% 12.85/6.92 & ( A = B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1185_eq__divide__eq__1,axiom,
% 12.85/6.92 ! [B: real,A: real] :
% 12.85/6.92 ( ( one_one_real
% 12.85/6.92 = ( divide_divide_real @ B @ A ) )
% 12.85/6.92 = ( ( A != zero_zero_real )
% 12.85/6.92 & ( A = B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1186_eq__divide__eq__1,axiom,
% 12.85/6.92 ! [B: rat,A: rat] :
% 12.85/6.92 ( ( one_one_rat
% 12.85/6.92 = ( divide_divide_rat @ B @ A ) )
% 12.85/6.92 = ( ( A != zero_zero_rat )
% 12.85/6.92 & ( A = B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1187_one__divide__eq__0__iff,axiom,
% 12.85/6.92 ! [A: real] :
% 12.85/6.92 ( ( ( divide_divide_real @ one_one_real @ A )
% 12.85/6.92 = zero_zero_real )
% 12.85/6.92 = ( A = zero_zero_real ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1188_one__divide__eq__0__iff,axiom,
% 12.85/6.92 ! [A: rat] :
% 12.85/6.92 ( ( ( divide_divide_rat @ one_one_rat @ A )
% 12.85/6.92 = zero_zero_rat )
% 12.85/6.92 = ( A = zero_zero_rat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1189_zero__eq__1__divide__iff,axiom,
% 12.85/6.92 ! [A: real] :
% 12.85/6.92 ( ( zero_zero_real
% 12.85/6.92 = ( divide_divide_real @ one_one_real @ A ) )
% 12.85/6.92 = ( A = zero_zero_real ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1190_zero__eq__1__divide__iff,axiom,
% 12.85/6.92 ! [A: rat] :
% 12.85/6.92 ( ( zero_zero_rat
% 12.85/6.92 = ( divide_divide_rat @ one_one_rat @ A ) )
% 12.85/6.92 = ( A = zero_zero_rat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1191_numeral__eq__one__iff,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ( ( ( numera6690914467698888265omplex @ N )
% 12.85/6.92 = one_one_complex )
% 12.85/6.92 = ( N = one ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1192_numeral__eq__one__iff,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ( ( ( numeral_numeral_real @ N )
% 12.85/6.92 = one_one_real )
% 12.85/6.92 = ( N = one ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1193_numeral__eq__one__iff,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ( ( ( numeral_numeral_rat @ N )
% 12.85/6.92 = one_one_rat )
% 12.85/6.92 = ( N = one ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1194_numeral__eq__one__iff,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ( ( ( numeral_numeral_nat @ N )
% 12.85/6.92 = one_one_nat )
% 12.85/6.92 = ( N = one ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1195_numeral__eq__one__iff,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ( ( ( numeral_numeral_int @ N )
% 12.85/6.92 = one_one_int )
% 12.85/6.92 = ( N = one ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1196_one__eq__numeral__iff,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ( ( one_one_complex
% 12.85/6.92 = ( numera6690914467698888265omplex @ N ) )
% 12.85/6.92 = ( one = N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1197_one__eq__numeral__iff,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ( ( one_one_real
% 12.85/6.92 = ( numeral_numeral_real @ N ) )
% 12.85/6.92 = ( one = N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1198_one__eq__numeral__iff,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ( ( one_one_rat
% 12.85/6.92 = ( numeral_numeral_rat @ N ) )
% 12.85/6.92 = ( one = N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1199_one__eq__numeral__iff,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ( ( one_one_nat
% 12.85/6.92 = ( numeral_numeral_nat @ N ) )
% 12.85/6.92 = ( one = N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1200_one__eq__numeral__iff,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ( ( one_one_int
% 12.85/6.92 = ( numeral_numeral_int @ N ) )
% 12.85/6.92 = ( one = N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1201_unit__prod,axiom,
% 12.85/6.92 ! [A: code_integer,B: code_integer] :
% 12.85/6.92 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.85/6.92 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 12.85/6.92 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1202_unit__prod,axiom,
% 12.85/6.92 ! [A: nat,B: nat] :
% 12.85/6.92 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.85/6.92 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 12.85/6.92 => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1203_unit__prod,axiom,
% 12.85/6.92 ! [A: int,B: int] :
% 12.85/6.92 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.85/6.92 => ( ( dvd_dvd_int @ B @ one_one_int )
% 12.85/6.92 => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1204_unit__div,axiom,
% 12.85/6.92 ! [A: code_integer,B: code_integer] :
% 12.85/6.92 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.85/6.92 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 12.85/6.92 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1205_unit__div,axiom,
% 12.85/6.92 ! [A: nat,B: nat] :
% 12.85/6.92 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.85/6.92 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 12.85/6.92 => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1206_unit__div,axiom,
% 12.85/6.92 ! [A: int,B: int] :
% 12.85/6.92 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.85/6.92 => ( ( dvd_dvd_int @ B @ one_one_int )
% 12.85/6.92 => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1207_unit__div__1__unit,axiom,
% 12.85/6.92 ! [A: code_integer] :
% 12.85/6.92 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.85/6.92 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1208_unit__div__1__unit,axiom,
% 12.85/6.92 ! [A: nat] :
% 12.85/6.92 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.85/6.92 => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1209_unit__div__1__unit,axiom,
% 12.85/6.92 ! [A: int] :
% 12.85/6.92 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.85/6.92 => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1210_unit__div__1__div__1,axiom,
% 12.85/6.92 ! [A: code_integer] :
% 12.85/6.92 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.85/6.92 => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 12.85/6.92 = A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1211_unit__div__1__div__1,axiom,
% 12.85/6.92 ! [A: nat] :
% 12.85/6.92 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.85/6.92 => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
% 12.85/6.92 = A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1212_unit__div__1__div__1,axiom,
% 12.85/6.92 ! [A: int] :
% 12.85/6.92 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.85/6.92 => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
% 12.85/6.92 = A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1213_of__nat__less__iff,axiom,
% 12.85/6.92 ! [M: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 12.85/6.92 = ( ord_less_nat @ M @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1214_of__nat__less__iff,axiom,
% 12.85/6.92 ! [M: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 12.85/6.92 = ( ord_less_nat @ M @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1215_of__nat__less__iff,axiom,
% 12.85/6.92 ! [M: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 12.85/6.92 = ( ord_less_nat @ M @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1216_of__nat__less__iff,axiom,
% 12.85/6.92 ! [M: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 12.85/6.92 = ( ord_less_nat @ M @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1217_zero__less__Suc,axiom,
% 12.85/6.92 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1218_less__Suc0,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 12.85/6.92 = ( N = zero_zero_nat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1219_zero__less__diff,axiom,
% 12.85/6.92 ! [N: nat,M: nat] :
% 12.85/6.92 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 12.85/6.92 = ( ord_less_nat @ M @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1220_nat__mult__less__cancel__disj,axiom,
% 12.85/6.92 ! [K: nat,M: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.85/6.92 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/6.92 & ( ord_less_nat @ M @ N ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1221_nat__0__less__mult__iff,axiom,
% 12.85/6.92 ! [M: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 12.85/6.92 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/6.92 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1222_mult__less__cancel2,axiom,
% 12.85/6.92 ! [M: nat,K: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 12.85/6.92 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/6.92 & ( ord_less_nat @ M @ N ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1223_of__nat__eq__1__iff,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( ( semiri8010041392384452111omplex @ N )
% 12.85/6.92 = one_one_complex )
% 12.85/6.92 = ( N = one_one_nat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1224_of__nat__eq__1__iff,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( ( semiri5074537144036343181t_real @ N )
% 12.85/6.92 = one_one_real )
% 12.85/6.92 = ( N = one_one_nat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1225_of__nat__eq__1__iff,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( ( semiri681578069525770553at_rat @ N )
% 12.85/6.92 = one_one_rat )
% 12.85/6.92 = ( N = one_one_nat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1226_of__nat__eq__1__iff,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( ( semiri1316708129612266289at_nat @ N )
% 12.85/6.92 = one_one_nat )
% 12.85/6.92 = ( N = one_one_nat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1227_of__nat__eq__1__iff,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( ( semiri1314217659103216013at_int @ N )
% 12.85/6.92 = one_one_int )
% 12.85/6.92 = ( N = one_one_nat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1228_of__nat__1__eq__iff,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( one_one_complex
% 12.85/6.92 = ( semiri8010041392384452111omplex @ N ) )
% 12.85/6.92 = ( N = one_one_nat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1229_of__nat__1__eq__iff,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( one_one_real
% 12.85/6.92 = ( semiri5074537144036343181t_real @ N ) )
% 12.85/6.92 = ( N = one_one_nat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1230_of__nat__1__eq__iff,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( one_one_rat
% 12.85/6.92 = ( semiri681578069525770553at_rat @ N ) )
% 12.85/6.92 = ( N = one_one_nat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1231_of__nat__1__eq__iff,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( one_one_nat
% 12.85/6.92 = ( semiri1316708129612266289at_nat @ N ) )
% 12.85/6.92 = ( N = one_one_nat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1232_of__nat__1__eq__iff,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( one_one_int
% 12.85/6.92 = ( semiri1314217659103216013at_int @ N ) )
% 12.85/6.92 = ( N = one_one_nat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1233_of__nat__1,axiom,
% 12.85/6.92 ( ( semiri8010041392384452111omplex @ one_one_nat )
% 12.85/6.92 = one_one_complex ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1234_of__nat__1,axiom,
% 12.85/6.92 ( ( semiri5074537144036343181t_real @ one_one_nat )
% 12.85/6.92 = one_one_real ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1235_of__nat__1,axiom,
% 12.85/6.92 ( ( semiri681578069525770553at_rat @ one_one_nat )
% 12.85/6.92 = one_one_rat ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1236_of__nat__1,axiom,
% 12.85/6.92 ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 12.85/6.92 = one_one_nat ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1237_of__nat__1,axiom,
% 12.85/6.92 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 12.85/6.92 = one_one_int ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1238_less__one,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( ord_less_nat @ N @ one_one_nat )
% 12.85/6.92 = ( N = zero_zero_nat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1239_div__less,axiom,
% 12.85/6.92 ! [M: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_nat @ M @ N )
% 12.85/6.92 => ( ( divide_divide_nat @ M @ N )
% 12.85/6.92 = zero_zero_nat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1240_diff__Suc__1,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 12.85/6.92 = N ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1241_divide__le__0__1__iff,axiom,
% 12.85/6.92 ! [A: rat] :
% 12.85/6.92 ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 12.85/6.92 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1242_divide__le__0__1__iff,axiom,
% 12.85/6.92 ! [A: real] :
% 12.85/6.92 ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 12.85/6.92 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1243_zero__le__divide__1__iff,axiom,
% 12.85/6.92 ! [A: rat] :
% 12.85/6.92 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 12.85/6.92 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1244_zero__le__divide__1__iff,axiom,
% 12.85/6.92 ! [A: real] :
% 12.85/6.92 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 12.85/6.92 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1245_zero__less__divide__1__iff,axiom,
% 12.85/6.92 ! [A: real] :
% 12.85/6.92 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 12.85/6.92 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1246_zero__less__divide__1__iff,axiom,
% 12.85/6.92 ! [A: rat] :
% 12.85/6.92 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 12.85/6.92 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1247_less__divide__eq__1__pos,axiom,
% 12.85/6.92 ! [A: real,B: real] :
% 12.85/6.92 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.92 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 12.85/6.92 = ( ord_less_real @ A @ B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1248_less__divide__eq__1__pos,axiom,
% 12.85/6.92 ! [A: rat,B: rat] :
% 12.85/6.92 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.92 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 12.85/6.92 = ( ord_less_rat @ A @ B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1249_less__divide__eq__1__neg,axiom,
% 12.85/6.92 ! [A: real,B: real] :
% 12.85/6.92 ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/6.92 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 12.85/6.92 = ( ord_less_real @ B @ A ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1250_less__divide__eq__1__neg,axiom,
% 12.85/6.92 ! [A: rat,B: rat] :
% 12.85/6.92 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/6.92 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 12.85/6.92 = ( ord_less_rat @ B @ A ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1251_divide__less__eq__1__pos,axiom,
% 12.85/6.92 ! [A: real,B: real] :
% 12.85/6.92 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.92 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 12.85/6.92 = ( ord_less_real @ B @ A ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1252_divide__less__eq__1__pos,axiom,
% 12.85/6.92 ! [A: rat,B: rat] :
% 12.85/6.92 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.92 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 12.85/6.92 = ( ord_less_rat @ B @ A ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1253_divide__less__eq__1__neg,axiom,
% 12.85/6.92 ! [A: real,B: real] :
% 12.85/6.92 ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/6.92 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 12.85/6.92 = ( ord_less_real @ A @ B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1254_divide__less__eq__1__neg,axiom,
% 12.85/6.92 ! [A: rat,B: rat] :
% 12.85/6.92 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/6.92 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 12.85/6.92 = ( ord_less_rat @ A @ B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1255_divide__less__0__1__iff,axiom,
% 12.85/6.92 ! [A: real] :
% 12.85/6.92 ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 12.85/6.92 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1256_divide__less__0__1__iff,axiom,
% 12.85/6.92 ! [A: rat] :
% 12.85/6.92 ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 12.85/6.92 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1257_one__less__numeral__iff,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 12.85/6.92 = ( ord_less_num @ one @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1258_one__less__numeral__iff,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 12.85/6.92 = ( ord_less_num @ one @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1259_one__less__numeral__iff,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 12.85/6.92 = ( ord_less_num @ one @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1260_one__less__numeral__iff,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 12.85/6.92 = ( ord_less_num @ one @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1261_divide__less__eq__numeral1_I1_J,axiom,
% 12.85/6.92 ! [B: real,W: num,A: real] :
% 12.85/6.92 ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 12.85/6.92 = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1262_divide__less__eq__numeral1_I1_J,axiom,
% 12.85/6.92 ! [B: rat,W: num,A: rat] :
% 12.85/6.92 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 12.85/6.92 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1263_less__divide__eq__numeral1_I1_J,axiom,
% 12.85/6.92 ! [A: real,B: real,W: num] :
% 12.85/6.92 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 12.85/6.92 = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1264_less__divide__eq__numeral1_I1_J,axiom,
% 12.85/6.92 ! [A: rat,B: rat,W: num] :
% 12.85/6.92 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 12.85/6.92 = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1265_nonzero__divide__mult__cancel__left,axiom,
% 12.85/6.92 ! [A: complex,B: complex] :
% 12.85/6.92 ( ( A != zero_zero_complex )
% 12.85/6.92 => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
% 12.85/6.92 = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1266_nonzero__divide__mult__cancel__left,axiom,
% 12.85/6.92 ! [A: real,B: real] :
% 12.85/6.92 ( ( A != zero_zero_real )
% 12.85/6.92 => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
% 12.85/6.92 = ( divide_divide_real @ one_one_real @ B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1267_nonzero__divide__mult__cancel__left,axiom,
% 12.85/6.92 ! [A: rat,B: rat] :
% 12.85/6.92 ( ( A != zero_zero_rat )
% 12.85/6.92 => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
% 12.85/6.92 = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1268_nonzero__divide__mult__cancel__right,axiom,
% 12.85/6.92 ! [B: complex,A: complex] :
% 12.85/6.92 ( ( B != zero_zero_complex )
% 12.85/6.92 => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
% 12.85/6.92 = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1269_nonzero__divide__mult__cancel__right,axiom,
% 12.85/6.92 ! [B: real,A: real] :
% 12.85/6.92 ( ( B != zero_zero_real )
% 12.85/6.92 => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
% 12.85/6.92 = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1270_nonzero__divide__mult__cancel__right,axiom,
% 12.85/6.92 ! [B: rat,A: rat] :
% 12.85/6.92 ( ( B != zero_zero_rat )
% 12.85/6.92 => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
% 12.85/6.92 = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1271_unit__mult__div__div,axiom,
% 12.85/6.92 ! [A: code_integer,B: code_integer] :
% 12.85/6.92 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.85/6.92 => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 12.85/6.92 = ( divide6298287555418463151nteger @ B @ A ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1272_unit__mult__div__div,axiom,
% 12.85/6.92 ! [A: nat,B: nat] :
% 12.85/6.92 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.85/6.92 => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
% 12.85/6.92 = ( divide_divide_nat @ B @ A ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1273_unit__mult__div__div,axiom,
% 12.85/6.92 ! [A: int,B: int] :
% 12.85/6.92 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.85/6.92 => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
% 12.85/6.92 = ( divide_divide_int @ B @ A ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1274_unit__div__mult__self,axiom,
% 12.85/6.92 ! [A: code_integer,B: code_integer] :
% 12.85/6.92 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.85/6.92 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 12.85/6.92 = B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1275_unit__div__mult__self,axiom,
% 12.85/6.92 ! [A: nat,B: nat] :
% 12.85/6.92 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.85/6.92 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 12.85/6.92 = B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1276_unit__div__mult__self,axiom,
% 12.85/6.92 ! [A: int,B: int] :
% 12.85/6.92 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.85/6.92 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 12.85/6.92 = B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1277_pow__divides__pow__iff,axiom,
% 12.85/6.92 ! [N: nat,A: int,B: int] :
% 12.85/6.92 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.92 => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 12.85/6.92 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1278_pow__divides__pow__iff,axiom,
% 12.85/6.92 ! [N: nat,A: nat,B: nat] :
% 12.85/6.92 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.92 => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 12.85/6.92 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1279_Suc__pred,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.92 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 12.85/6.92 = N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1280_nat__mult__le__cancel__disj,axiom,
% 12.85/6.92 ! [K: nat,M: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.85/6.92 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/6.92 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1281_mult__le__cancel2,axiom,
% 12.85/6.92 ! [M: nat,K: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 12.85/6.92 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/6.92 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1282_div__mult__self1__is__m,axiom,
% 12.85/6.92 ! [N: nat,M: nat] :
% 12.85/6.92 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.92 => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 12.85/6.92 = M ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1283_div__mult__self__is__m,axiom,
% 12.85/6.92 ! [N: nat,M: nat] :
% 12.85/6.92 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.92 => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 12.85/6.92 = M ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1284_divide__le__eq__1__neg,axiom,
% 12.85/6.92 ! [A: rat,B: rat] :
% 12.85/6.92 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/6.92 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 12.85/6.92 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1285_divide__le__eq__1__neg,axiom,
% 12.85/6.92 ! [A: real,B: real] :
% 12.85/6.92 ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/6.92 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 12.85/6.92 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1286_divide__le__eq__1__pos,axiom,
% 12.85/6.92 ! [A: rat,B: rat] :
% 12.85/6.92 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.92 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 12.85/6.92 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1287_divide__le__eq__1__pos,axiom,
% 12.85/6.92 ! [A: real,B: real] :
% 12.85/6.92 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.92 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 12.85/6.92 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1288_le__divide__eq__1__neg,axiom,
% 12.85/6.92 ! [A: rat,B: rat] :
% 12.85/6.92 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/6.92 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 12.85/6.92 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1289_le__divide__eq__1__neg,axiom,
% 12.85/6.92 ! [A: real,B: real] :
% 12.85/6.92 ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/6.92 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 12.85/6.92 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1290_le__divide__eq__1__pos,axiom,
% 12.85/6.92 ! [A: rat,B: rat] :
% 12.85/6.92 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.92 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 12.85/6.92 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1291_le__divide__eq__1__pos,axiom,
% 12.85/6.92 ! [A: real,B: real] :
% 12.85/6.92 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.92 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 12.85/6.92 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1292_of__nat__0__less__iff,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
% 12.85/6.92 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1293_of__nat__0__less__iff,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
% 12.85/6.92 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1294_of__nat__0__less__iff,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
% 12.85/6.92 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1295_of__nat__0__less__iff,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
% 12.85/6.92 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1296_Suc__1,axiom,
% 12.85/6.92 ( ( suc @ one_one_nat )
% 12.85/6.92 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1297_Suc__diff__1,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.92 => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 12.85/6.92 = N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1298_numeral__le__one__iff,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 12.85/6.92 = ( ord_less_eq_num @ N @ one ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1299_numeral__le__one__iff,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 12.85/6.92 = ( ord_less_eq_num @ N @ one ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1300_numeral__le__one__iff,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 12.85/6.92 = ( ord_less_eq_num @ N @ one ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1301_numeral__le__one__iff,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 12.85/6.92 = ( ord_less_eq_num @ N @ one ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1302_one__div__two__eq__zero,axiom,
% 12.85/6.92 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.92 = zero_zero_nat ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1303_one__div__two__eq__zero,axiom,
% 12.85/6.92 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.92 = zero_zero_int ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1304_bits__1__div__2,axiom,
% 12.85/6.92 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.92 = zero_zero_nat ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1305_bits__1__div__2,axiom,
% 12.85/6.92 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.92 = zero_zero_int ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1306_even__power,axiom,
% 12.85/6.92 ! [A: code_integer,N: nat] :
% 12.85/6.92 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 12.85/6.92 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.92 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1307_even__power,axiom,
% 12.85/6.92 ! [A: nat,N: nat] :
% 12.85/6.92 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
% 12.85/6.92 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.92 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1308_even__power,axiom,
% 12.85/6.92 ! [A: int,N: nat] :
% 12.85/6.92 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
% 12.85/6.92 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.92 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1309_power__less__zero__eq,axiom,
% 12.85/6.92 ! [A: real,N: nat] :
% 12.85/6.92 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 12.85/6.92 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.92 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1310_power__less__zero__eq,axiom,
% 12.85/6.92 ! [A: rat,N: nat] :
% 12.85/6.92 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 12.85/6.92 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.92 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1311_power__less__zero__eq,axiom,
% 12.85/6.92 ! [A: int,N: nat] :
% 12.85/6.92 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 12.85/6.92 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.92 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1312_power__less__zero__eq__numeral,axiom,
% 12.85/6.92 ! [A: real,W: num] :
% 12.85/6.92 ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 12.85/6.92 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.92 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1313_power__less__zero__eq__numeral,axiom,
% 12.85/6.92 ! [A: rat,W: num] :
% 12.85/6.92 ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 12.85/6.92 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.92 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1314_power__less__zero__eq__numeral,axiom,
% 12.85/6.92 ! [A: int,W: num] :
% 12.85/6.92 ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 12.85/6.92 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.92 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1315_semiring__parity__class_Oeven__mask__iff,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
% 12.85/6.92 = ( N = zero_zero_nat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1316_semiring__parity__class_Oeven__mask__iff,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
% 12.85/6.92 = ( N = zero_zero_nat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1317_semiring__parity__class_Oeven__mask__iff,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 12.85/6.92 = ( N = zero_zero_nat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1318_zero__less__power__eq__numeral,axiom,
% 12.85/6.92 ! [A: real,W: num] :
% 12.85/6.92 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 12.85/6.92 = ( ( ( numeral_numeral_nat @ W )
% 12.85/6.92 = zero_zero_nat )
% 12.85/6.92 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.92 & ( A != zero_zero_real ) )
% 12.85/6.92 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.92 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1319_zero__less__power__eq__numeral,axiom,
% 12.85/6.92 ! [A: rat,W: num] :
% 12.85/6.92 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 12.85/6.92 = ( ( ( numeral_numeral_nat @ W )
% 12.85/6.92 = zero_zero_nat )
% 12.85/6.92 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.92 & ( A != zero_zero_rat ) )
% 12.85/6.92 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.92 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1320_zero__less__power__eq__numeral,axiom,
% 12.85/6.92 ! [A: int,W: num] :
% 12.85/6.92 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 12.85/6.92 = ( ( ( numeral_numeral_nat @ W )
% 12.85/6.92 = zero_zero_nat )
% 12.85/6.92 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.92 & ( A != zero_zero_int ) )
% 12.85/6.92 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.92 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1321_power__le__zero__eq__numeral,axiom,
% 12.85/6.92 ! [A: rat,W: num] :
% 12.85/6.92 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 12.85/6.92 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 12.85/6.92 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.92 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 12.85/6.92 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.92 & ( A = zero_zero_rat ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1322_power__le__zero__eq__numeral,axiom,
% 12.85/6.92 ! [A: int,W: num] :
% 12.85/6.92 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 12.85/6.92 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 12.85/6.92 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.92 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 12.85/6.92 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.92 & ( A = zero_zero_int ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1323_power__le__zero__eq__numeral,axiom,
% 12.85/6.92 ! [A: real,W: num] :
% 12.85/6.92 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 12.85/6.92 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 12.85/6.92 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.92 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 12.85/6.92 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.92 & ( A = zero_zero_real ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1324_minus__int__code_I1_J,axiom,
% 12.85/6.92 ! [K: int] :
% 12.85/6.92 ( ( minus_minus_int @ K @ zero_zero_int )
% 12.85/6.92 = K ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1325_imult__is__0,axiom,
% 12.85/6.92 ! [M: extended_enat,N: extended_enat] :
% 12.85/6.92 ( ( ( times_7803423173614009249d_enat @ M @ N )
% 12.85/6.92 = zero_z5237406670263579293d_enat )
% 12.85/6.92 = ( ( M = zero_z5237406670263579293d_enat )
% 12.85/6.92 | ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1326_one__reorient,axiom,
% 12.85/6.92 ! [X: complex] :
% 12.85/6.92 ( ( one_one_complex = X )
% 12.85/6.92 = ( X = one_one_complex ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1327_one__reorient,axiom,
% 12.85/6.92 ! [X: real] :
% 12.85/6.92 ( ( one_one_real = X )
% 12.85/6.92 = ( X = one_one_real ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1328_one__reorient,axiom,
% 12.85/6.92 ! [X: rat] :
% 12.85/6.92 ( ( one_one_rat = X )
% 12.85/6.92 = ( X = one_one_rat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1329_one__reorient,axiom,
% 12.85/6.92 ! [X: nat] :
% 12.85/6.92 ( ( one_one_nat = X )
% 12.85/6.92 = ( X = one_one_nat ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1330_one__reorient,axiom,
% 12.85/6.92 ! [X: int] :
% 12.85/6.92 ( ( one_one_int = X )
% 12.85/6.92 = ( X = one_one_int ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1331_linordered__field__no__lb,axiom,
% 12.85/6.92 ! [X5: real] :
% 12.85/6.92 ? [Y: real] : ( ord_less_real @ Y @ X5 ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1332_linordered__field__no__lb,axiom,
% 12.85/6.92 ! [X5: rat] :
% 12.85/6.92 ? [Y: rat] : ( ord_less_rat @ Y @ X5 ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1333_linordered__field__no__ub,axiom,
% 12.85/6.92 ! [X5: real] :
% 12.85/6.92 ? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1334_linordered__field__no__ub,axiom,
% 12.85/6.92 ! [X5: rat] :
% 12.85/6.92 ? [X_1: rat] : ( ord_less_rat @ X5 @ X_1 ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1335_less__numeral__extra_I4_J,axiom,
% 12.85/6.92 ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1336_less__numeral__extra_I4_J,axiom,
% 12.85/6.92 ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1337_less__numeral__extra_I4_J,axiom,
% 12.85/6.92 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1338_less__numeral__extra_I4_J,axiom,
% 12.85/6.92 ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1339_nat__neq__iff,axiom,
% 12.85/6.92 ! [M: nat,N: nat] :
% 12.85/6.92 ( ( M != N )
% 12.85/6.92 = ( ( ord_less_nat @ M @ N )
% 12.85/6.92 | ( ord_less_nat @ N @ M ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1340_less__not__refl,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ~ ( ord_less_nat @ N @ N ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1341_less__not__refl2,axiom,
% 12.85/6.92 ! [N: nat,M: nat] :
% 12.85/6.92 ( ( ord_less_nat @ N @ M )
% 12.85/6.92 => ( M != N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1342_less__not__refl3,axiom,
% 12.85/6.92 ! [S2: nat,T: nat] :
% 12.85/6.92 ( ( ord_less_nat @ S2 @ T )
% 12.85/6.92 => ( S2 != T ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1343_less__irrefl__nat,axiom,
% 12.85/6.92 ! [N: nat] :
% 12.85/6.92 ~ ( ord_less_nat @ N @ N ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1344_nat__less__induct,axiom,
% 12.85/6.92 ! [P: nat > $o,N: nat] :
% 12.85/6.92 ( ! [N2: nat] :
% 12.85/6.92 ( ! [M2: nat] :
% 12.85/6.92 ( ( ord_less_nat @ M2 @ N2 )
% 12.85/6.92 => ( P @ M2 ) )
% 12.85/6.92 => ( P @ N2 ) )
% 12.85/6.92 => ( P @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1345_infinite__descent,axiom,
% 12.85/6.92 ! [P: nat > $o,N: nat] :
% 12.85/6.92 ( ! [N2: nat] :
% 12.85/6.92 ( ~ ( P @ N2 )
% 12.85/6.92 => ? [M2: nat] :
% 12.85/6.92 ( ( ord_less_nat @ M2 @ N2 )
% 12.85/6.92 & ~ ( P @ M2 ) ) )
% 12.85/6.92 => ( P @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1346_linorder__neqE__nat,axiom,
% 12.85/6.92 ! [X: nat,Y4: nat] :
% 12.85/6.92 ( ( X != Y4 )
% 12.85/6.92 => ( ~ ( ord_less_nat @ X @ Y4 )
% 12.85/6.92 => ( ord_less_nat @ Y4 @ X ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1347_linorder__neqE__linordered__idom,axiom,
% 12.85/6.92 ! [X: real,Y4: real] :
% 12.85/6.92 ( ( X != Y4 )
% 12.85/6.92 => ( ~ ( ord_less_real @ X @ Y4 )
% 12.85/6.92 => ( ord_less_real @ Y4 @ X ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1348_linorder__neqE__linordered__idom,axiom,
% 12.85/6.92 ! [X: rat,Y4: rat] :
% 12.85/6.92 ( ( X != Y4 )
% 12.85/6.92 => ( ~ ( ord_less_rat @ X @ Y4 )
% 12.85/6.92 => ( ord_less_rat @ Y4 @ X ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1349_linorder__neqE__linordered__idom,axiom,
% 12.85/6.92 ! [X: int,Y4: int] :
% 12.85/6.92 ( ( X != Y4 )
% 12.85/6.92 => ( ~ ( ord_less_int @ X @ Y4 )
% 12.85/6.92 => ( ord_less_int @ Y4 @ X ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1350_verit__comp__simplify1_I1_J,axiom,
% 12.85/6.92 ! [A: real] :
% 12.85/6.92 ~ ( ord_less_real @ A @ A ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1351_verit__comp__simplify1_I1_J,axiom,
% 12.85/6.92 ! [A: rat] :
% 12.85/6.92 ~ ( ord_less_rat @ A @ A ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1352_verit__comp__simplify1_I1_J,axiom,
% 12.85/6.92 ! [A: num] :
% 12.85/6.92 ~ ( ord_less_num @ A @ A ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1353_verit__comp__simplify1_I1_J,axiom,
% 12.85/6.92 ! [A: nat] :
% 12.85/6.92 ~ ( ord_less_nat @ A @ A ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1354_verit__comp__simplify1_I1_J,axiom,
% 12.85/6.92 ! [A: int] :
% 12.85/6.92 ~ ( ord_less_int @ A @ A ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1355_lt__ex,axiom,
% 12.85/6.92 ! [X: real] :
% 12.85/6.92 ? [Y: real] : ( ord_less_real @ Y @ X ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1356_lt__ex,axiom,
% 12.85/6.92 ! [X: rat] :
% 12.85/6.92 ? [Y: rat] : ( ord_less_rat @ Y @ X ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1357_lt__ex,axiom,
% 12.85/6.92 ! [X: int] :
% 12.85/6.92 ? [Y: int] : ( ord_less_int @ Y @ X ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1358_gt__ex,axiom,
% 12.85/6.92 ! [X: real] :
% 12.85/6.92 ? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1359_gt__ex,axiom,
% 12.85/6.92 ! [X: rat] :
% 12.85/6.92 ? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1360_gt__ex,axiom,
% 12.85/6.92 ! [X: nat] :
% 12.85/6.92 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1361_gt__ex,axiom,
% 12.85/6.92 ! [X: int] :
% 12.85/6.92 ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1362_dense,axiom,
% 12.85/6.92 ! [X: real,Y4: real] :
% 12.85/6.92 ( ( ord_less_real @ X @ Y4 )
% 12.85/6.92 => ? [Z2: real] :
% 12.85/6.92 ( ( ord_less_real @ X @ Z2 )
% 12.85/6.92 & ( ord_less_real @ Z2 @ Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1363_dense,axiom,
% 12.85/6.92 ! [X: rat,Y4: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X @ Y4 )
% 12.85/6.92 => ? [Z2: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X @ Z2 )
% 12.85/6.92 & ( ord_less_rat @ Z2 @ Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1364_less__imp__neq,axiom,
% 12.85/6.92 ! [X: real,Y4: real] :
% 12.85/6.92 ( ( ord_less_real @ X @ Y4 )
% 12.85/6.92 => ( X != Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1365_less__imp__neq,axiom,
% 12.85/6.92 ! [X: rat,Y4: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X @ Y4 )
% 12.85/6.92 => ( X != Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1366_less__imp__neq,axiom,
% 12.85/6.92 ! [X: num,Y4: num] :
% 12.85/6.92 ( ( ord_less_num @ X @ Y4 )
% 12.85/6.92 => ( X != Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1367_less__imp__neq,axiom,
% 12.85/6.92 ! [X: nat,Y4: nat] :
% 12.85/6.92 ( ( ord_less_nat @ X @ Y4 )
% 12.85/6.92 => ( X != Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1368_less__imp__neq,axiom,
% 12.85/6.92 ! [X: int,Y4: int] :
% 12.85/6.92 ( ( ord_less_int @ X @ Y4 )
% 12.85/6.92 => ( X != Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1369_order_Oasym,axiom,
% 12.85/6.92 ! [A: real,B: real] :
% 12.85/6.92 ( ( ord_less_real @ A @ B )
% 12.85/6.92 => ~ ( ord_less_real @ B @ A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1370_order_Oasym,axiom,
% 12.85/6.92 ! [A: rat,B: rat] :
% 12.85/6.92 ( ( ord_less_rat @ A @ B )
% 12.85/6.92 => ~ ( ord_less_rat @ B @ A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1371_order_Oasym,axiom,
% 12.85/6.92 ! [A: num,B: num] :
% 12.85/6.92 ( ( ord_less_num @ A @ B )
% 12.85/6.92 => ~ ( ord_less_num @ B @ A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1372_order_Oasym,axiom,
% 12.85/6.92 ! [A: nat,B: nat] :
% 12.85/6.92 ( ( ord_less_nat @ A @ B )
% 12.85/6.92 => ~ ( ord_less_nat @ B @ A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1373_order_Oasym,axiom,
% 12.85/6.92 ! [A: int,B: int] :
% 12.85/6.92 ( ( ord_less_int @ A @ B )
% 12.85/6.92 => ~ ( ord_less_int @ B @ A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1374_ord__eq__less__trans,axiom,
% 12.85/6.92 ! [A: real,B: real,C: real] :
% 12.85/6.92 ( ( A = B )
% 12.85/6.92 => ( ( ord_less_real @ B @ C )
% 12.85/6.92 => ( ord_less_real @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1375_ord__eq__less__trans,axiom,
% 12.85/6.92 ! [A: rat,B: rat,C: rat] :
% 12.85/6.92 ( ( A = B )
% 12.85/6.92 => ( ( ord_less_rat @ B @ C )
% 12.85/6.92 => ( ord_less_rat @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1376_ord__eq__less__trans,axiom,
% 12.85/6.92 ! [A: num,B: num,C: num] :
% 12.85/6.92 ( ( A = B )
% 12.85/6.92 => ( ( ord_less_num @ B @ C )
% 12.85/6.92 => ( ord_less_num @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1377_ord__eq__less__trans,axiom,
% 12.85/6.92 ! [A: nat,B: nat,C: nat] :
% 12.85/6.92 ( ( A = B )
% 12.85/6.92 => ( ( ord_less_nat @ B @ C )
% 12.85/6.92 => ( ord_less_nat @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1378_ord__eq__less__trans,axiom,
% 12.85/6.92 ! [A: int,B: int,C: int] :
% 12.85/6.92 ( ( A = B )
% 12.85/6.92 => ( ( ord_less_int @ B @ C )
% 12.85/6.92 => ( ord_less_int @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1379_ord__less__eq__trans,axiom,
% 12.85/6.92 ! [A: real,B: real,C: real] :
% 12.85/6.92 ( ( ord_less_real @ A @ B )
% 12.85/6.92 => ( ( B = C )
% 12.85/6.92 => ( ord_less_real @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1380_ord__less__eq__trans,axiom,
% 12.85/6.92 ! [A: rat,B: rat,C: rat] :
% 12.85/6.92 ( ( ord_less_rat @ A @ B )
% 12.85/6.92 => ( ( B = C )
% 12.85/6.92 => ( ord_less_rat @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1381_ord__less__eq__trans,axiom,
% 12.85/6.92 ! [A: num,B: num,C: num] :
% 12.85/6.92 ( ( ord_less_num @ A @ B )
% 12.85/6.92 => ( ( B = C )
% 12.85/6.92 => ( ord_less_num @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1382_ord__less__eq__trans,axiom,
% 12.85/6.92 ! [A: nat,B: nat,C: nat] :
% 12.85/6.92 ( ( ord_less_nat @ A @ B )
% 12.85/6.92 => ( ( B = C )
% 12.85/6.92 => ( ord_less_nat @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1383_ord__less__eq__trans,axiom,
% 12.85/6.92 ! [A: int,B: int,C: int] :
% 12.85/6.92 ( ( ord_less_int @ A @ B )
% 12.85/6.92 => ( ( B = C )
% 12.85/6.92 => ( ord_less_int @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1384_less__induct,axiom,
% 12.85/6.92 ! [P: nat > $o,A: nat] :
% 12.85/6.92 ( ! [X4: nat] :
% 12.85/6.92 ( ! [Y3: nat] :
% 12.85/6.92 ( ( ord_less_nat @ Y3 @ X4 )
% 12.85/6.92 => ( P @ Y3 ) )
% 12.85/6.92 => ( P @ X4 ) )
% 12.85/6.92 => ( P @ A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1385_antisym__conv3,axiom,
% 12.85/6.92 ! [Y4: real,X: real] :
% 12.85/6.92 ( ~ ( ord_less_real @ Y4 @ X )
% 12.85/6.92 => ( ( ~ ( ord_less_real @ X @ Y4 ) )
% 12.85/6.92 = ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1386_antisym__conv3,axiom,
% 12.85/6.92 ! [Y4: rat,X: rat] :
% 12.85/6.92 ( ~ ( ord_less_rat @ Y4 @ X )
% 12.85/6.92 => ( ( ~ ( ord_less_rat @ X @ Y4 ) )
% 12.85/6.92 = ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1387_antisym__conv3,axiom,
% 12.85/6.92 ! [Y4: num,X: num] :
% 12.85/6.92 ( ~ ( ord_less_num @ Y4 @ X )
% 12.85/6.92 => ( ( ~ ( ord_less_num @ X @ Y4 ) )
% 12.85/6.92 = ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1388_antisym__conv3,axiom,
% 12.85/6.92 ! [Y4: nat,X: nat] :
% 12.85/6.92 ( ~ ( ord_less_nat @ Y4 @ X )
% 12.85/6.92 => ( ( ~ ( ord_less_nat @ X @ Y4 ) )
% 12.85/6.92 = ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1389_antisym__conv3,axiom,
% 12.85/6.92 ! [Y4: int,X: int] :
% 12.85/6.92 ( ~ ( ord_less_int @ Y4 @ X )
% 12.85/6.92 => ( ( ~ ( ord_less_int @ X @ Y4 ) )
% 12.85/6.92 = ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1390_linorder__cases,axiom,
% 12.85/6.92 ! [X: real,Y4: real] :
% 12.85/6.92 ( ~ ( ord_less_real @ X @ Y4 )
% 12.85/6.92 => ( ( X != Y4 )
% 12.85/6.92 => ( ord_less_real @ Y4 @ X ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1391_linorder__cases,axiom,
% 12.85/6.92 ! [X: rat,Y4: rat] :
% 12.85/6.92 ( ~ ( ord_less_rat @ X @ Y4 )
% 12.85/6.92 => ( ( X != Y4 )
% 12.85/6.92 => ( ord_less_rat @ Y4 @ X ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1392_linorder__cases,axiom,
% 12.85/6.92 ! [X: num,Y4: num] :
% 12.85/6.92 ( ~ ( ord_less_num @ X @ Y4 )
% 12.85/6.92 => ( ( X != Y4 )
% 12.85/6.92 => ( ord_less_num @ Y4 @ X ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1393_linorder__cases,axiom,
% 12.85/6.92 ! [X: nat,Y4: nat] :
% 12.85/6.92 ( ~ ( ord_less_nat @ X @ Y4 )
% 12.85/6.92 => ( ( X != Y4 )
% 12.85/6.92 => ( ord_less_nat @ Y4 @ X ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1394_linorder__cases,axiom,
% 12.85/6.92 ! [X: int,Y4: int] :
% 12.85/6.92 ( ~ ( ord_less_int @ X @ Y4 )
% 12.85/6.92 => ( ( X != Y4 )
% 12.85/6.92 => ( ord_less_int @ Y4 @ X ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1395_dual__order_Oasym,axiom,
% 12.85/6.92 ! [B: real,A: real] :
% 12.85/6.92 ( ( ord_less_real @ B @ A )
% 12.85/6.92 => ~ ( ord_less_real @ A @ B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1396_dual__order_Oasym,axiom,
% 12.85/6.92 ! [B: rat,A: rat] :
% 12.85/6.92 ( ( ord_less_rat @ B @ A )
% 12.85/6.92 => ~ ( ord_less_rat @ A @ B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1397_dual__order_Oasym,axiom,
% 12.85/6.92 ! [B: num,A: num] :
% 12.85/6.92 ( ( ord_less_num @ B @ A )
% 12.85/6.92 => ~ ( ord_less_num @ A @ B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1398_dual__order_Oasym,axiom,
% 12.85/6.92 ! [B: nat,A: nat] :
% 12.85/6.92 ( ( ord_less_nat @ B @ A )
% 12.85/6.92 => ~ ( ord_less_nat @ A @ B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1399_dual__order_Oasym,axiom,
% 12.85/6.92 ! [B: int,A: int] :
% 12.85/6.92 ( ( ord_less_int @ B @ A )
% 12.85/6.92 => ~ ( ord_less_int @ A @ B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1400_dual__order_Oirrefl,axiom,
% 12.85/6.92 ! [A: real] :
% 12.85/6.92 ~ ( ord_less_real @ A @ A ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1401_dual__order_Oirrefl,axiom,
% 12.85/6.92 ! [A: rat] :
% 12.85/6.92 ~ ( ord_less_rat @ A @ A ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1402_dual__order_Oirrefl,axiom,
% 12.85/6.92 ! [A: num] :
% 12.85/6.92 ~ ( ord_less_num @ A @ A ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1403_dual__order_Oirrefl,axiom,
% 12.85/6.92 ! [A: nat] :
% 12.85/6.92 ~ ( ord_less_nat @ A @ A ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1404_dual__order_Oirrefl,axiom,
% 12.85/6.92 ! [A: int] :
% 12.85/6.92 ~ ( ord_less_int @ A @ A ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1405_exists__least__iff,axiom,
% 12.85/6.92 ( ( ^ [P2: nat > $o] :
% 12.85/6.92 ? [X6: nat] : ( P2 @ X6 ) )
% 12.85/6.92 = ( ^ [P3: nat > $o] :
% 12.85/6.92 ? [N4: nat] :
% 12.85/6.92 ( ( P3 @ N4 )
% 12.85/6.92 & ! [M5: nat] :
% 12.85/6.92 ( ( ord_less_nat @ M5 @ N4 )
% 12.85/6.92 => ~ ( P3 @ M5 ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1406_linorder__less__wlog,axiom,
% 12.85/6.92 ! [P: real > real > $o,A: real,B: real] :
% 12.85/6.92 ( ! [A4: real,B3: real] :
% 12.85/6.92 ( ( ord_less_real @ A4 @ B3 )
% 12.85/6.92 => ( P @ A4 @ B3 ) )
% 12.85/6.92 => ( ! [A4: real] : ( P @ A4 @ A4 )
% 12.85/6.92 => ( ! [A4: real,B3: real] :
% 12.85/6.92 ( ( P @ B3 @ A4 )
% 12.85/6.92 => ( P @ A4 @ B3 ) )
% 12.85/6.92 => ( P @ A @ B ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1407_linorder__less__wlog,axiom,
% 12.85/6.92 ! [P: rat > rat > $o,A: rat,B: rat] :
% 12.85/6.92 ( ! [A4: rat,B3: rat] :
% 12.85/6.92 ( ( ord_less_rat @ A4 @ B3 )
% 12.85/6.92 => ( P @ A4 @ B3 ) )
% 12.85/6.92 => ( ! [A4: rat] : ( P @ A4 @ A4 )
% 12.85/6.92 => ( ! [A4: rat,B3: rat] :
% 12.85/6.92 ( ( P @ B3 @ A4 )
% 12.85/6.92 => ( P @ A4 @ B3 ) )
% 12.85/6.92 => ( P @ A @ B ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1408_linorder__less__wlog,axiom,
% 12.85/6.92 ! [P: num > num > $o,A: num,B: num] :
% 12.85/6.92 ( ! [A4: num,B3: num] :
% 12.85/6.92 ( ( ord_less_num @ A4 @ B3 )
% 12.85/6.92 => ( P @ A4 @ B3 ) )
% 12.85/6.92 => ( ! [A4: num] : ( P @ A4 @ A4 )
% 12.85/6.92 => ( ! [A4: num,B3: num] :
% 12.85/6.92 ( ( P @ B3 @ A4 )
% 12.85/6.92 => ( P @ A4 @ B3 ) )
% 12.85/6.92 => ( P @ A @ B ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1409_linorder__less__wlog,axiom,
% 12.85/6.92 ! [P: nat > nat > $o,A: nat,B: nat] :
% 12.85/6.92 ( ! [A4: nat,B3: nat] :
% 12.85/6.92 ( ( ord_less_nat @ A4 @ B3 )
% 12.85/6.92 => ( P @ A4 @ B3 ) )
% 12.85/6.92 => ( ! [A4: nat] : ( P @ A4 @ A4 )
% 12.85/6.92 => ( ! [A4: nat,B3: nat] :
% 12.85/6.92 ( ( P @ B3 @ A4 )
% 12.85/6.92 => ( P @ A4 @ B3 ) )
% 12.85/6.92 => ( P @ A @ B ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1410_linorder__less__wlog,axiom,
% 12.85/6.92 ! [P: int > int > $o,A: int,B: int] :
% 12.85/6.92 ( ! [A4: int,B3: int] :
% 12.85/6.92 ( ( ord_less_int @ A4 @ B3 )
% 12.85/6.92 => ( P @ A4 @ B3 ) )
% 12.85/6.92 => ( ! [A4: int] : ( P @ A4 @ A4 )
% 12.85/6.92 => ( ! [A4: int,B3: int] :
% 12.85/6.92 ( ( P @ B3 @ A4 )
% 12.85/6.92 => ( P @ A4 @ B3 ) )
% 12.85/6.92 => ( P @ A @ B ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1411_order_Ostrict__trans,axiom,
% 12.85/6.92 ! [A: real,B: real,C: real] :
% 12.85/6.92 ( ( ord_less_real @ A @ B )
% 12.85/6.92 => ( ( ord_less_real @ B @ C )
% 12.85/6.92 => ( ord_less_real @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1412_order_Ostrict__trans,axiom,
% 12.85/6.92 ! [A: rat,B: rat,C: rat] :
% 12.85/6.92 ( ( ord_less_rat @ A @ B )
% 12.85/6.92 => ( ( ord_less_rat @ B @ C )
% 12.85/6.92 => ( ord_less_rat @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1413_order_Ostrict__trans,axiom,
% 12.85/6.92 ! [A: num,B: num,C: num] :
% 12.85/6.92 ( ( ord_less_num @ A @ B )
% 12.85/6.92 => ( ( ord_less_num @ B @ C )
% 12.85/6.92 => ( ord_less_num @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1414_order_Ostrict__trans,axiom,
% 12.85/6.92 ! [A: nat,B: nat,C: nat] :
% 12.85/6.92 ( ( ord_less_nat @ A @ B )
% 12.85/6.92 => ( ( ord_less_nat @ B @ C )
% 12.85/6.92 => ( ord_less_nat @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1415_order_Ostrict__trans,axiom,
% 12.85/6.92 ! [A: int,B: int,C: int] :
% 12.85/6.92 ( ( ord_less_int @ A @ B )
% 12.85/6.92 => ( ( ord_less_int @ B @ C )
% 12.85/6.92 => ( ord_less_int @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1416_not__less__iff__gr__or__eq,axiom,
% 12.85/6.92 ! [X: real,Y4: real] :
% 12.85/6.92 ( ( ~ ( ord_less_real @ X @ Y4 ) )
% 12.85/6.92 = ( ( ord_less_real @ Y4 @ X )
% 12.85/6.92 | ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1417_not__less__iff__gr__or__eq,axiom,
% 12.85/6.92 ! [X: rat,Y4: rat] :
% 12.85/6.92 ( ( ~ ( ord_less_rat @ X @ Y4 ) )
% 12.85/6.92 = ( ( ord_less_rat @ Y4 @ X )
% 12.85/6.92 | ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1418_not__less__iff__gr__or__eq,axiom,
% 12.85/6.92 ! [X: num,Y4: num] :
% 12.85/6.92 ( ( ~ ( ord_less_num @ X @ Y4 ) )
% 12.85/6.92 = ( ( ord_less_num @ Y4 @ X )
% 12.85/6.92 | ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1419_not__less__iff__gr__or__eq,axiom,
% 12.85/6.92 ! [X: nat,Y4: nat] :
% 12.85/6.92 ( ( ~ ( ord_less_nat @ X @ Y4 ) )
% 12.85/6.92 = ( ( ord_less_nat @ Y4 @ X )
% 12.85/6.92 | ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1420_not__less__iff__gr__or__eq,axiom,
% 12.85/6.92 ! [X: int,Y4: int] :
% 12.85/6.92 ( ( ~ ( ord_less_int @ X @ Y4 ) )
% 12.85/6.92 = ( ( ord_less_int @ Y4 @ X )
% 12.85/6.92 | ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1421_dual__order_Ostrict__trans,axiom,
% 12.85/6.92 ! [B: real,A: real,C: real] :
% 12.85/6.92 ( ( ord_less_real @ B @ A )
% 12.85/6.92 => ( ( ord_less_real @ C @ B )
% 12.85/6.92 => ( ord_less_real @ C @ A ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1422_dual__order_Ostrict__trans,axiom,
% 12.85/6.92 ! [B: rat,A: rat,C: rat] :
% 12.85/6.92 ( ( ord_less_rat @ B @ A )
% 12.85/6.92 => ( ( ord_less_rat @ C @ B )
% 12.85/6.92 => ( ord_less_rat @ C @ A ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1423_dual__order_Ostrict__trans,axiom,
% 12.85/6.92 ! [B: num,A: num,C: num] :
% 12.85/6.92 ( ( ord_less_num @ B @ A )
% 12.85/6.92 => ( ( ord_less_num @ C @ B )
% 12.85/6.92 => ( ord_less_num @ C @ A ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1424_dual__order_Ostrict__trans,axiom,
% 12.85/6.92 ! [B: nat,A: nat,C: nat] :
% 12.85/6.92 ( ( ord_less_nat @ B @ A )
% 12.85/6.92 => ( ( ord_less_nat @ C @ B )
% 12.85/6.92 => ( ord_less_nat @ C @ A ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1425_dual__order_Ostrict__trans,axiom,
% 12.85/6.92 ! [B: int,A: int,C: int] :
% 12.85/6.92 ( ( ord_less_int @ B @ A )
% 12.85/6.92 => ( ( ord_less_int @ C @ B )
% 12.85/6.92 => ( ord_less_int @ C @ A ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1426_order_Ostrict__implies__not__eq,axiom,
% 12.85/6.92 ! [A: real,B: real] :
% 12.85/6.92 ( ( ord_less_real @ A @ B )
% 12.85/6.92 => ( A != B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1427_order_Ostrict__implies__not__eq,axiom,
% 12.85/6.92 ! [A: rat,B: rat] :
% 12.85/6.92 ( ( ord_less_rat @ A @ B )
% 12.85/6.92 => ( A != B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1428_order_Ostrict__implies__not__eq,axiom,
% 12.85/6.92 ! [A: num,B: num] :
% 12.85/6.92 ( ( ord_less_num @ A @ B )
% 12.85/6.92 => ( A != B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1429_order_Ostrict__implies__not__eq,axiom,
% 12.85/6.92 ! [A: nat,B: nat] :
% 12.85/6.92 ( ( ord_less_nat @ A @ B )
% 12.85/6.92 => ( A != B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1430_order_Ostrict__implies__not__eq,axiom,
% 12.85/6.92 ! [A: int,B: int] :
% 12.85/6.92 ( ( ord_less_int @ A @ B )
% 12.85/6.92 => ( A != B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1431_dual__order_Ostrict__implies__not__eq,axiom,
% 12.85/6.92 ! [B: real,A: real] :
% 12.85/6.92 ( ( ord_less_real @ B @ A )
% 12.85/6.92 => ( A != B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1432_dual__order_Ostrict__implies__not__eq,axiom,
% 12.85/6.92 ! [B: rat,A: rat] :
% 12.85/6.92 ( ( ord_less_rat @ B @ A )
% 12.85/6.92 => ( A != B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1433_dual__order_Ostrict__implies__not__eq,axiom,
% 12.85/6.92 ! [B: num,A: num] :
% 12.85/6.92 ( ( ord_less_num @ B @ A )
% 12.85/6.92 => ( A != B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1434_dual__order_Ostrict__implies__not__eq,axiom,
% 12.85/6.92 ! [B: nat,A: nat] :
% 12.85/6.92 ( ( ord_less_nat @ B @ A )
% 12.85/6.92 => ( A != B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1435_dual__order_Ostrict__implies__not__eq,axiom,
% 12.85/6.92 ! [B: int,A: int] :
% 12.85/6.92 ( ( ord_less_int @ B @ A )
% 12.85/6.92 => ( A != B ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1436_linorder__neqE,axiom,
% 12.85/6.92 ! [X: real,Y4: real] :
% 12.85/6.92 ( ( X != Y4 )
% 12.85/6.92 => ( ~ ( ord_less_real @ X @ Y4 )
% 12.85/6.92 => ( ord_less_real @ Y4 @ X ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1437_linorder__neqE,axiom,
% 12.85/6.92 ! [X: rat,Y4: rat] :
% 12.85/6.92 ( ( X != Y4 )
% 12.85/6.92 => ( ~ ( ord_less_rat @ X @ Y4 )
% 12.85/6.92 => ( ord_less_rat @ Y4 @ X ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1438_linorder__neqE,axiom,
% 12.85/6.92 ! [X: num,Y4: num] :
% 12.85/6.92 ( ( X != Y4 )
% 12.85/6.92 => ( ~ ( ord_less_num @ X @ Y4 )
% 12.85/6.92 => ( ord_less_num @ Y4 @ X ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1439_linorder__neqE,axiom,
% 12.85/6.92 ! [X: nat,Y4: nat] :
% 12.85/6.92 ( ( X != Y4 )
% 12.85/6.92 => ( ~ ( ord_less_nat @ X @ Y4 )
% 12.85/6.92 => ( ord_less_nat @ Y4 @ X ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1440_linorder__neqE,axiom,
% 12.85/6.92 ! [X: int,Y4: int] :
% 12.85/6.92 ( ( X != Y4 )
% 12.85/6.92 => ( ~ ( ord_less_int @ X @ Y4 )
% 12.85/6.92 => ( ord_less_int @ Y4 @ X ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1441_order__less__asym,axiom,
% 12.85/6.92 ! [X: real,Y4: real] :
% 12.85/6.92 ( ( ord_less_real @ X @ Y4 )
% 12.85/6.92 => ~ ( ord_less_real @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1442_order__less__asym,axiom,
% 12.85/6.92 ! [X: rat,Y4: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X @ Y4 )
% 12.85/6.92 => ~ ( ord_less_rat @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1443_order__less__asym,axiom,
% 12.85/6.92 ! [X: num,Y4: num] :
% 12.85/6.92 ( ( ord_less_num @ X @ Y4 )
% 12.85/6.92 => ~ ( ord_less_num @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1444_order__less__asym,axiom,
% 12.85/6.92 ! [X: nat,Y4: nat] :
% 12.85/6.92 ( ( ord_less_nat @ X @ Y4 )
% 12.85/6.92 => ~ ( ord_less_nat @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1445_order__less__asym,axiom,
% 12.85/6.92 ! [X: int,Y4: int] :
% 12.85/6.92 ( ( ord_less_int @ X @ Y4 )
% 12.85/6.92 => ~ ( ord_less_int @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1446_linorder__neq__iff,axiom,
% 12.85/6.92 ! [X: real,Y4: real] :
% 12.85/6.92 ( ( X != Y4 )
% 12.85/6.92 = ( ( ord_less_real @ X @ Y4 )
% 12.85/6.92 | ( ord_less_real @ Y4 @ X ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1447_linorder__neq__iff,axiom,
% 12.85/6.92 ! [X: rat,Y4: rat] :
% 12.85/6.92 ( ( X != Y4 )
% 12.85/6.92 = ( ( ord_less_rat @ X @ Y4 )
% 12.85/6.92 | ( ord_less_rat @ Y4 @ X ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1448_linorder__neq__iff,axiom,
% 12.85/6.92 ! [X: num,Y4: num] :
% 12.85/6.92 ( ( X != Y4 )
% 12.85/6.92 = ( ( ord_less_num @ X @ Y4 )
% 12.85/6.92 | ( ord_less_num @ Y4 @ X ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1449_linorder__neq__iff,axiom,
% 12.85/6.92 ! [X: nat,Y4: nat] :
% 12.85/6.92 ( ( X != Y4 )
% 12.85/6.92 = ( ( ord_less_nat @ X @ Y4 )
% 12.85/6.92 | ( ord_less_nat @ Y4 @ X ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1450_linorder__neq__iff,axiom,
% 12.85/6.92 ! [X: int,Y4: int] :
% 12.85/6.92 ( ( X != Y4 )
% 12.85/6.92 = ( ( ord_less_int @ X @ Y4 )
% 12.85/6.92 | ( ord_less_int @ Y4 @ X ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1451_order__less__asym_H,axiom,
% 12.85/6.92 ! [A: real,B: real] :
% 12.85/6.92 ( ( ord_less_real @ A @ B )
% 12.85/6.92 => ~ ( ord_less_real @ B @ A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1452_order__less__asym_H,axiom,
% 12.85/6.92 ! [A: rat,B: rat] :
% 12.85/6.92 ( ( ord_less_rat @ A @ B )
% 12.85/6.92 => ~ ( ord_less_rat @ B @ A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1453_order__less__asym_H,axiom,
% 12.85/6.92 ! [A: num,B: num] :
% 12.85/6.92 ( ( ord_less_num @ A @ B )
% 12.85/6.92 => ~ ( ord_less_num @ B @ A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1454_order__less__asym_H,axiom,
% 12.85/6.92 ! [A: nat,B: nat] :
% 12.85/6.92 ( ( ord_less_nat @ A @ B )
% 12.85/6.92 => ~ ( ord_less_nat @ B @ A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1455_order__less__asym_H,axiom,
% 12.85/6.92 ! [A: int,B: int] :
% 12.85/6.92 ( ( ord_less_int @ A @ B )
% 12.85/6.92 => ~ ( ord_less_int @ B @ A ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1456_order__less__trans,axiom,
% 12.85/6.92 ! [X: real,Y4: real,Z: real] :
% 12.85/6.92 ( ( ord_less_real @ X @ Y4 )
% 12.85/6.92 => ( ( ord_less_real @ Y4 @ Z )
% 12.85/6.92 => ( ord_less_real @ X @ Z ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1457_order__less__trans,axiom,
% 12.85/6.92 ! [X: rat,Y4: rat,Z: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X @ Y4 )
% 12.85/6.92 => ( ( ord_less_rat @ Y4 @ Z )
% 12.85/6.92 => ( ord_less_rat @ X @ Z ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1458_order__less__trans,axiom,
% 12.85/6.92 ! [X: num,Y4: num,Z: num] :
% 12.85/6.92 ( ( ord_less_num @ X @ Y4 )
% 12.85/6.92 => ( ( ord_less_num @ Y4 @ Z )
% 12.85/6.92 => ( ord_less_num @ X @ Z ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1459_order__less__trans,axiom,
% 12.85/6.92 ! [X: nat,Y4: nat,Z: nat] :
% 12.85/6.92 ( ( ord_less_nat @ X @ Y4 )
% 12.85/6.92 => ( ( ord_less_nat @ Y4 @ Z )
% 12.85/6.92 => ( ord_less_nat @ X @ Z ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1460_order__less__trans,axiom,
% 12.85/6.92 ! [X: int,Y4: int,Z: int] :
% 12.85/6.92 ( ( ord_less_int @ X @ Y4 )
% 12.85/6.92 => ( ( ord_less_int @ Y4 @ Z )
% 12.85/6.92 => ( ord_less_int @ X @ Z ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1461_ord__eq__less__subst,axiom,
% 12.85/6.92 ! [A: real,F: real > real,B: real,C: real] :
% 12.85/6.92 ( ( A
% 12.85/6.92 = ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_real @ B @ C )
% 12.85/6.92 => ( ! [X4: real,Y: real] :
% 12.85/6.92 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.92 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1462_ord__eq__less__subst,axiom,
% 12.85/6.92 ! [A: rat,F: real > rat,B: real,C: real] :
% 12.85/6.92 ( ( A
% 12.85/6.92 = ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_real @ B @ C )
% 12.85/6.92 => ( ! [X4: real,Y: real] :
% 12.85/6.92 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.92 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1463_ord__eq__less__subst,axiom,
% 12.85/6.92 ! [A: num,F: real > num,B: real,C: real] :
% 12.85/6.92 ( ( A
% 12.85/6.92 = ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_real @ B @ C )
% 12.85/6.92 => ( ! [X4: real,Y: real] :
% 12.85/6.92 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.92 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1464_ord__eq__less__subst,axiom,
% 12.85/6.92 ! [A: nat,F: real > nat,B: real,C: real] :
% 12.85/6.92 ( ( A
% 12.85/6.92 = ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_real @ B @ C )
% 12.85/6.92 => ( ! [X4: real,Y: real] :
% 12.85/6.92 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.92 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1465_ord__eq__less__subst,axiom,
% 12.85/6.92 ! [A: int,F: real > int,B: real,C: real] :
% 12.85/6.92 ( ( A
% 12.85/6.92 = ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_real @ B @ C )
% 12.85/6.92 => ( ! [X4: real,Y: real] :
% 12.85/6.92 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.92 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1466_ord__eq__less__subst,axiom,
% 12.85/6.92 ! [A: real,F: rat > real,B: rat,C: rat] :
% 12.85/6.92 ( ( A
% 12.85/6.92 = ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_rat @ B @ C )
% 12.85/6.92 => ( ! [X4: rat,Y: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.92 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1467_ord__eq__less__subst,axiom,
% 12.85/6.92 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 12.85/6.92 ( ( A
% 12.85/6.92 = ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_rat @ B @ C )
% 12.85/6.92 => ( ! [X4: rat,Y: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.92 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1468_ord__eq__less__subst,axiom,
% 12.85/6.92 ! [A: num,F: rat > num,B: rat,C: rat] :
% 12.85/6.92 ( ( A
% 12.85/6.92 = ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_rat @ B @ C )
% 12.85/6.92 => ( ! [X4: rat,Y: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.92 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1469_ord__eq__less__subst,axiom,
% 12.85/6.92 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 12.85/6.92 ( ( A
% 12.85/6.92 = ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_rat @ B @ C )
% 12.85/6.92 => ( ! [X4: rat,Y: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.92 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1470_ord__eq__less__subst,axiom,
% 12.85/6.92 ! [A: int,F: rat > int,B: rat,C: rat] :
% 12.85/6.92 ( ( A
% 12.85/6.92 = ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_rat @ B @ C )
% 12.85/6.92 => ( ! [X4: rat,Y: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.92 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1471_ord__less__eq__subst,axiom,
% 12.85/6.92 ! [A: real,B: real,F: real > real,C: real] :
% 12.85/6.92 ( ( ord_less_real @ A @ B )
% 12.85/6.92 => ( ( ( F @ B )
% 12.85/6.92 = C )
% 12.85/6.92 => ( ! [X4: real,Y: real] :
% 12.85/6.92 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.92 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1472_ord__less__eq__subst,axiom,
% 12.85/6.92 ! [A: real,B: real,F: real > rat,C: rat] :
% 12.85/6.92 ( ( ord_less_real @ A @ B )
% 12.85/6.92 => ( ( ( F @ B )
% 12.85/6.92 = C )
% 12.85/6.92 => ( ! [X4: real,Y: real] :
% 12.85/6.92 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.92 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1473_ord__less__eq__subst,axiom,
% 12.85/6.92 ! [A: real,B: real,F: real > num,C: num] :
% 12.85/6.92 ( ( ord_less_real @ A @ B )
% 12.85/6.92 => ( ( ( F @ B )
% 12.85/6.92 = C )
% 12.85/6.92 => ( ! [X4: real,Y: real] :
% 12.85/6.92 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.92 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1474_ord__less__eq__subst,axiom,
% 12.85/6.92 ! [A: real,B: real,F: real > nat,C: nat] :
% 12.85/6.92 ( ( ord_less_real @ A @ B )
% 12.85/6.92 => ( ( ( F @ B )
% 12.85/6.92 = C )
% 12.85/6.92 => ( ! [X4: real,Y: real] :
% 12.85/6.92 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.92 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1475_ord__less__eq__subst,axiom,
% 12.85/6.92 ! [A: real,B: real,F: real > int,C: int] :
% 12.85/6.92 ( ( ord_less_real @ A @ B )
% 12.85/6.92 => ( ( ( F @ B )
% 12.85/6.92 = C )
% 12.85/6.92 => ( ! [X4: real,Y: real] :
% 12.85/6.92 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.92 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1476_ord__less__eq__subst,axiom,
% 12.85/6.92 ! [A: rat,B: rat,F: rat > real,C: real] :
% 12.85/6.92 ( ( ord_less_rat @ A @ B )
% 12.85/6.92 => ( ( ( F @ B )
% 12.85/6.92 = C )
% 12.85/6.92 => ( ! [X4: rat,Y: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.92 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1477_ord__less__eq__subst,axiom,
% 12.85/6.92 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 12.85/6.92 ( ( ord_less_rat @ A @ B )
% 12.85/6.92 => ( ( ( F @ B )
% 12.85/6.92 = C )
% 12.85/6.92 => ( ! [X4: rat,Y: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.92 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1478_ord__less__eq__subst,axiom,
% 12.85/6.92 ! [A: rat,B: rat,F: rat > num,C: num] :
% 12.85/6.92 ( ( ord_less_rat @ A @ B )
% 12.85/6.92 => ( ( ( F @ B )
% 12.85/6.92 = C )
% 12.85/6.92 => ( ! [X4: rat,Y: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.92 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1479_ord__less__eq__subst,axiom,
% 12.85/6.92 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 12.85/6.92 ( ( ord_less_rat @ A @ B )
% 12.85/6.92 => ( ( ( F @ B )
% 12.85/6.92 = C )
% 12.85/6.92 => ( ! [X4: rat,Y: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.92 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1480_ord__less__eq__subst,axiom,
% 12.85/6.92 ! [A: rat,B: rat,F: rat > int,C: int] :
% 12.85/6.92 ( ( ord_less_rat @ A @ B )
% 12.85/6.92 => ( ( ( F @ B )
% 12.85/6.92 = C )
% 12.85/6.92 => ( ! [X4: rat,Y: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.92 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1481_order__less__irrefl,axiom,
% 12.85/6.92 ! [X: real] :
% 12.85/6.92 ~ ( ord_less_real @ X @ X ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1482_order__less__irrefl,axiom,
% 12.85/6.92 ! [X: rat] :
% 12.85/6.92 ~ ( ord_less_rat @ X @ X ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1483_order__less__irrefl,axiom,
% 12.85/6.92 ! [X: num] :
% 12.85/6.92 ~ ( ord_less_num @ X @ X ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1484_order__less__irrefl,axiom,
% 12.85/6.92 ! [X: nat] :
% 12.85/6.92 ~ ( ord_less_nat @ X @ X ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1485_order__less__irrefl,axiom,
% 12.85/6.92 ! [X: int] :
% 12.85/6.92 ~ ( ord_less_int @ X @ X ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1486_order__less__subst1,axiom,
% 12.85/6.92 ! [A: real,F: real > real,B: real,C: real] :
% 12.85/6.92 ( ( ord_less_real @ A @ ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_real @ B @ C )
% 12.85/6.92 => ( ! [X4: real,Y: real] :
% 12.85/6.92 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.92 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1487_order__less__subst1,axiom,
% 12.85/6.92 ! [A: real,F: rat > real,B: rat,C: rat] :
% 12.85/6.92 ( ( ord_less_real @ A @ ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_rat @ B @ C )
% 12.85/6.92 => ( ! [X4: rat,Y: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.92 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1488_order__less__subst1,axiom,
% 12.85/6.92 ! [A: real,F: num > real,B: num,C: num] :
% 12.85/6.92 ( ( ord_less_real @ A @ ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_num @ B @ C )
% 12.85/6.92 => ( ! [X4: num,Y: num] :
% 12.85/6.92 ( ( ord_less_num @ X4 @ Y )
% 12.85/6.92 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1489_order__less__subst1,axiom,
% 12.85/6.92 ! [A: real,F: nat > real,B: nat,C: nat] :
% 12.85/6.92 ( ( ord_less_real @ A @ ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_nat @ B @ C )
% 12.85/6.92 => ( ! [X4: nat,Y: nat] :
% 12.85/6.92 ( ( ord_less_nat @ X4 @ Y )
% 12.85/6.92 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1490_order__less__subst1,axiom,
% 12.85/6.92 ! [A: real,F: int > real,B: int,C: int] :
% 12.85/6.92 ( ( ord_less_real @ A @ ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_int @ B @ C )
% 12.85/6.92 => ( ! [X4: int,Y: int] :
% 12.85/6.92 ( ( ord_less_int @ X4 @ Y )
% 12.85/6.92 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1491_order__less__subst1,axiom,
% 12.85/6.92 ! [A: rat,F: real > rat,B: real,C: real] :
% 12.85/6.92 ( ( ord_less_rat @ A @ ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_real @ B @ C )
% 12.85/6.92 => ( ! [X4: real,Y: real] :
% 12.85/6.92 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.92 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1492_order__less__subst1,axiom,
% 12.85/6.92 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 12.85/6.92 ( ( ord_less_rat @ A @ ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_rat @ B @ C )
% 12.85/6.92 => ( ! [X4: rat,Y: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.92 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1493_order__less__subst1,axiom,
% 12.85/6.92 ! [A: rat,F: num > rat,B: num,C: num] :
% 12.85/6.92 ( ( ord_less_rat @ A @ ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_num @ B @ C )
% 12.85/6.92 => ( ! [X4: num,Y: num] :
% 12.85/6.92 ( ( ord_less_num @ X4 @ Y )
% 12.85/6.92 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1494_order__less__subst1,axiom,
% 12.85/6.92 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 12.85/6.92 ( ( ord_less_rat @ A @ ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_nat @ B @ C )
% 12.85/6.92 => ( ! [X4: nat,Y: nat] :
% 12.85/6.92 ( ( ord_less_nat @ X4 @ Y )
% 12.85/6.92 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1495_order__less__subst1,axiom,
% 12.85/6.92 ! [A: rat,F: int > rat,B: int,C: int] :
% 12.85/6.92 ( ( ord_less_rat @ A @ ( F @ B ) )
% 12.85/6.92 => ( ( ord_less_int @ B @ C )
% 12.85/6.92 => ( ! [X4: int,Y: int] :
% 12.85/6.92 ( ( ord_less_int @ X4 @ Y )
% 12.85/6.92 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1496_order__less__subst2,axiom,
% 12.85/6.92 ! [A: real,B: real,F: real > real,C: real] :
% 12.85/6.92 ( ( ord_less_real @ A @ B )
% 12.85/6.92 => ( ( ord_less_real @ ( F @ B ) @ C )
% 12.85/6.92 => ( ! [X4: real,Y: real] :
% 12.85/6.92 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.92 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1497_order__less__subst2,axiom,
% 12.85/6.92 ! [A: real,B: real,F: real > rat,C: rat] :
% 12.85/6.92 ( ( ord_less_real @ A @ B )
% 12.85/6.92 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 12.85/6.92 => ( ! [X4: real,Y: real] :
% 12.85/6.92 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.92 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1498_order__less__subst2,axiom,
% 12.85/6.92 ! [A: real,B: real,F: real > num,C: num] :
% 12.85/6.92 ( ( ord_less_real @ A @ B )
% 12.85/6.92 => ( ( ord_less_num @ ( F @ B ) @ C )
% 12.85/6.92 => ( ! [X4: real,Y: real] :
% 12.85/6.92 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.92 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1499_order__less__subst2,axiom,
% 12.85/6.92 ! [A: real,B: real,F: real > nat,C: nat] :
% 12.85/6.92 ( ( ord_less_real @ A @ B )
% 12.85/6.92 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 12.85/6.92 => ( ! [X4: real,Y: real] :
% 12.85/6.92 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.92 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1500_order__less__subst2,axiom,
% 12.85/6.92 ! [A: real,B: real,F: real > int,C: int] :
% 12.85/6.92 ( ( ord_less_real @ A @ B )
% 12.85/6.92 => ( ( ord_less_int @ ( F @ B ) @ C )
% 12.85/6.92 => ( ! [X4: real,Y: real] :
% 12.85/6.92 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.92 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1501_order__less__subst2,axiom,
% 12.85/6.92 ! [A: rat,B: rat,F: rat > real,C: real] :
% 12.85/6.92 ( ( ord_less_rat @ A @ B )
% 12.85/6.92 => ( ( ord_less_real @ ( F @ B ) @ C )
% 12.85/6.92 => ( ! [X4: rat,Y: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.92 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1502_order__less__subst2,axiom,
% 12.85/6.92 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 12.85/6.92 ( ( ord_less_rat @ A @ B )
% 12.85/6.92 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 12.85/6.92 => ( ! [X4: rat,Y: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.92 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1503_order__less__subst2,axiom,
% 12.85/6.92 ! [A: rat,B: rat,F: rat > num,C: num] :
% 12.85/6.92 ( ( ord_less_rat @ A @ B )
% 12.85/6.92 => ( ( ord_less_num @ ( F @ B ) @ C )
% 12.85/6.92 => ( ! [X4: rat,Y: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.92 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1504_order__less__subst2,axiom,
% 12.85/6.92 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 12.85/6.92 ( ( ord_less_rat @ A @ B )
% 12.85/6.92 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 12.85/6.92 => ( ! [X4: rat,Y: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.92 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1505_order__less__subst2,axiom,
% 12.85/6.92 ! [A: rat,B: rat,F: rat > int,C: int] :
% 12.85/6.92 ( ( ord_less_rat @ A @ B )
% 12.85/6.92 => ( ( ord_less_int @ ( F @ B ) @ C )
% 12.85/6.92 => ( ! [X4: rat,Y: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.92 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.92 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1506_order__less__not__sym,axiom,
% 12.85/6.92 ! [X: real,Y4: real] :
% 12.85/6.92 ( ( ord_less_real @ X @ Y4 )
% 12.85/6.92 => ~ ( ord_less_real @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1507_order__less__not__sym,axiom,
% 12.85/6.92 ! [X: rat,Y4: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X @ Y4 )
% 12.85/6.92 => ~ ( ord_less_rat @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1508_order__less__not__sym,axiom,
% 12.85/6.92 ! [X: num,Y4: num] :
% 12.85/6.92 ( ( ord_less_num @ X @ Y4 )
% 12.85/6.92 => ~ ( ord_less_num @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1509_order__less__not__sym,axiom,
% 12.85/6.92 ! [X: nat,Y4: nat] :
% 12.85/6.92 ( ( ord_less_nat @ X @ Y4 )
% 12.85/6.92 => ~ ( ord_less_nat @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1510_order__less__not__sym,axiom,
% 12.85/6.92 ! [X: int,Y4: int] :
% 12.85/6.92 ( ( ord_less_int @ X @ Y4 )
% 12.85/6.92 => ~ ( ord_less_int @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1511_order__less__imp__triv,axiom,
% 12.85/6.92 ! [X: real,Y4: real,P: $o] :
% 12.85/6.92 ( ( ord_less_real @ X @ Y4 )
% 12.85/6.92 => ( ( ord_less_real @ Y4 @ X )
% 12.85/6.92 => P ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1512_order__less__imp__triv,axiom,
% 12.85/6.92 ! [X: rat,Y4: rat,P: $o] :
% 12.85/6.92 ( ( ord_less_rat @ X @ Y4 )
% 12.85/6.92 => ( ( ord_less_rat @ Y4 @ X )
% 12.85/6.92 => P ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1513_order__less__imp__triv,axiom,
% 12.85/6.92 ! [X: num,Y4: num,P: $o] :
% 12.85/6.92 ( ( ord_less_num @ X @ Y4 )
% 12.85/6.92 => ( ( ord_less_num @ Y4 @ X )
% 12.85/6.92 => P ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1514_order__less__imp__triv,axiom,
% 12.85/6.92 ! [X: nat,Y4: nat,P: $o] :
% 12.85/6.92 ( ( ord_less_nat @ X @ Y4 )
% 12.85/6.92 => ( ( ord_less_nat @ Y4 @ X )
% 12.85/6.92 => P ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1515_order__less__imp__triv,axiom,
% 12.85/6.92 ! [X: int,Y4: int,P: $o] :
% 12.85/6.92 ( ( ord_less_int @ X @ Y4 )
% 12.85/6.92 => ( ( ord_less_int @ Y4 @ X )
% 12.85/6.92 => P ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1516_linorder__less__linear,axiom,
% 12.85/6.92 ! [X: real,Y4: real] :
% 12.85/6.92 ( ( ord_less_real @ X @ Y4 )
% 12.85/6.92 | ( X = Y4 )
% 12.85/6.92 | ( ord_less_real @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1517_linorder__less__linear,axiom,
% 12.85/6.92 ! [X: rat,Y4: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X @ Y4 )
% 12.85/6.92 | ( X = Y4 )
% 12.85/6.92 | ( ord_less_rat @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1518_linorder__less__linear,axiom,
% 12.85/6.92 ! [X: num,Y4: num] :
% 12.85/6.92 ( ( ord_less_num @ X @ Y4 )
% 12.85/6.92 | ( X = Y4 )
% 12.85/6.92 | ( ord_less_num @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1519_linorder__less__linear,axiom,
% 12.85/6.92 ! [X: nat,Y4: nat] :
% 12.85/6.92 ( ( ord_less_nat @ X @ Y4 )
% 12.85/6.92 | ( X = Y4 )
% 12.85/6.92 | ( ord_less_nat @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1520_linorder__less__linear,axiom,
% 12.85/6.92 ! [X: int,Y4: int] :
% 12.85/6.92 ( ( ord_less_int @ X @ Y4 )
% 12.85/6.92 | ( X = Y4 )
% 12.85/6.92 | ( ord_less_int @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1521_order__less__imp__not__eq,axiom,
% 12.85/6.92 ! [X: real,Y4: real] :
% 12.85/6.92 ( ( ord_less_real @ X @ Y4 )
% 12.85/6.92 => ( X != Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1522_order__less__imp__not__eq,axiom,
% 12.85/6.92 ! [X: rat,Y4: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X @ Y4 )
% 12.85/6.92 => ( X != Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1523_order__less__imp__not__eq,axiom,
% 12.85/6.92 ! [X: num,Y4: num] :
% 12.85/6.92 ( ( ord_less_num @ X @ Y4 )
% 12.85/6.92 => ( X != Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1524_order__less__imp__not__eq,axiom,
% 12.85/6.92 ! [X: nat,Y4: nat] :
% 12.85/6.92 ( ( ord_less_nat @ X @ Y4 )
% 12.85/6.92 => ( X != Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1525_order__less__imp__not__eq,axiom,
% 12.85/6.92 ! [X: int,Y4: int] :
% 12.85/6.92 ( ( ord_less_int @ X @ Y4 )
% 12.85/6.92 => ( X != Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1526_order__less__imp__not__eq2,axiom,
% 12.85/6.92 ! [X: real,Y4: real] :
% 12.85/6.92 ( ( ord_less_real @ X @ Y4 )
% 12.85/6.92 => ( Y4 != X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1527_order__less__imp__not__eq2,axiom,
% 12.85/6.92 ! [X: rat,Y4: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X @ Y4 )
% 12.85/6.92 => ( Y4 != X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1528_order__less__imp__not__eq2,axiom,
% 12.85/6.92 ! [X: num,Y4: num] :
% 12.85/6.92 ( ( ord_less_num @ X @ Y4 )
% 12.85/6.92 => ( Y4 != X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1529_order__less__imp__not__eq2,axiom,
% 12.85/6.92 ! [X: nat,Y4: nat] :
% 12.85/6.92 ( ( ord_less_nat @ X @ Y4 )
% 12.85/6.92 => ( Y4 != X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1530_order__less__imp__not__eq2,axiom,
% 12.85/6.92 ! [X: int,Y4: int] :
% 12.85/6.92 ( ( ord_less_int @ X @ Y4 )
% 12.85/6.92 => ( Y4 != X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1531_order__less__imp__not__less,axiom,
% 12.85/6.92 ! [X: real,Y4: real] :
% 12.85/6.92 ( ( ord_less_real @ X @ Y4 )
% 12.85/6.92 => ~ ( ord_less_real @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1532_order__less__imp__not__less,axiom,
% 12.85/6.92 ! [X: rat,Y4: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X @ Y4 )
% 12.85/6.92 => ~ ( ord_less_rat @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1533_order__less__imp__not__less,axiom,
% 12.85/6.92 ! [X: num,Y4: num] :
% 12.85/6.92 ( ( ord_less_num @ X @ Y4 )
% 12.85/6.92 => ~ ( ord_less_num @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1534_order__less__imp__not__less,axiom,
% 12.85/6.92 ! [X: nat,Y4: nat] :
% 12.85/6.92 ( ( ord_less_nat @ X @ Y4 )
% 12.85/6.92 => ~ ( ord_less_nat @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1535_order__less__imp__not__less,axiom,
% 12.85/6.92 ! [X: int,Y4: int] :
% 12.85/6.92 ( ( ord_less_int @ X @ Y4 )
% 12.85/6.92 => ~ ( ord_less_int @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1536_less__numeral__extra_I1_J,axiom,
% 12.85/6.92 ord_less_real @ zero_zero_real @ one_one_real ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1537_less__numeral__extra_I1_J,axiom,
% 12.85/6.92 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1538_less__numeral__extra_I1_J,axiom,
% 12.85/6.92 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1539_less__numeral__extra_I1_J,axiom,
% 12.85/6.92 ord_less_int @ zero_zero_int @ one_one_int ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1540_zero__less__one,axiom,
% 12.85/6.92 ord_less_real @ zero_zero_real @ one_one_real ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1541_zero__less__one,axiom,
% 12.85/6.92 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1542_zero__less__one,axiom,
% 12.85/6.92 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1543_zero__less__one,axiom,
% 12.85/6.92 ord_less_int @ zero_zero_int @ one_one_int ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1544_not__one__less__zero,axiom,
% 12.85/6.92 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1545_not__one__less__zero,axiom,
% 12.85/6.92 ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1546_not__one__less__zero,axiom,
% 12.85/6.92 ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1547_not__one__less__zero,axiom,
% 12.85/6.92 ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1548_not__numeral__less__one,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1549_not__numeral__less__one,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1550_not__numeral__less__one,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1551_not__numeral__less__one,axiom,
% 12.85/6.92 ! [N: num] :
% 12.85/6.92 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1552_less__1__mult,axiom,
% 12.85/6.92 ! [M: real,N: real] :
% 12.85/6.92 ( ( ord_less_real @ one_one_real @ M )
% 12.85/6.92 => ( ( ord_less_real @ one_one_real @ N )
% 12.85/6.92 => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1553_less__1__mult,axiom,
% 12.85/6.92 ! [M: rat,N: rat] :
% 12.85/6.92 ( ( ord_less_rat @ one_one_rat @ M )
% 12.85/6.92 => ( ( ord_less_rat @ one_one_rat @ N )
% 12.85/6.92 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1554_less__1__mult,axiom,
% 12.85/6.92 ! [M: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_nat @ one_one_nat @ M )
% 12.85/6.92 => ( ( ord_less_nat @ one_one_nat @ N )
% 12.85/6.92 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1555_less__1__mult,axiom,
% 12.85/6.92 ! [M: int,N: int] :
% 12.85/6.92 ( ( ord_less_int @ one_one_int @ M )
% 12.85/6.92 => ( ( ord_less_int @ one_one_int @ N )
% 12.85/6.92 => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1556_lift__Suc__mono__less__iff,axiom,
% 12.85/6.92 ! [F: nat > real,N: nat,M: nat] :
% 12.85/6.92 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 12.85/6.92 => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 12.85/6.92 = ( ord_less_nat @ N @ M ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1557_lift__Suc__mono__less__iff,axiom,
% 12.85/6.92 ! [F: nat > rat,N: nat,M: nat] :
% 12.85/6.92 ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 12.85/6.92 => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
% 12.85/6.92 = ( ord_less_nat @ N @ M ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1558_lift__Suc__mono__less__iff,axiom,
% 12.85/6.92 ! [F: nat > num,N: nat,M: nat] :
% 12.85/6.92 ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 12.85/6.92 => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
% 12.85/6.92 = ( ord_less_nat @ N @ M ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1559_lift__Suc__mono__less__iff,axiom,
% 12.85/6.92 ! [F: nat > nat,N: nat,M: nat] :
% 12.85/6.92 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 12.85/6.92 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 12.85/6.92 = ( ord_less_nat @ N @ M ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1560_lift__Suc__mono__less__iff,axiom,
% 12.85/6.92 ! [F: nat > int,N: nat,M: nat] :
% 12.85/6.92 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 12.85/6.92 => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 12.85/6.92 = ( ord_less_nat @ N @ M ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1561_lift__Suc__mono__less,axiom,
% 12.85/6.92 ! [F: nat > real,N: nat,N3: nat] :
% 12.85/6.92 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 12.85/6.92 => ( ( ord_less_nat @ N @ N3 )
% 12.85/6.92 => ( ord_less_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1562_lift__Suc__mono__less,axiom,
% 12.85/6.92 ! [F: nat > rat,N: nat,N3: nat] :
% 12.85/6.92 ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 12.85/6.92 => ( ( ord_less_nat @ N @ N3 )
% 12.85/6.92 => ( ord_less_rat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1563_lift__Suc__mono__less,axiom,
% 12.85/6.92 ! [F: nat > num,N: nat,N3: nat] :
% 12.85/6.92 ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 12.85/6.92 => ( ( ord_less_nat @ N @ N3 )
% 12.85/6.92 => ( ord_less_num @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1564_lift__Suc__mono__less,axiom,
% 12.85/6.92 ! [F: nat > nat,N: nat,N3: nat] :
% 12.85/6.92 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 12.85/6.92 => ( ( ord_less_nat @ N @ N3 )
% 12.85/6.92 => ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1565_lift__Suc__mono__less,axiom,
% 12.85/6.92 ! [F: nat > int,N: nat,N3: nat] :
% 12.85/6.92 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 12.85/6.92 => ( ( ord_less_nat @ N @ N3 )
% 12.85/6.92 => ( ord_less_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1566_of__nat__less__imp__less,axiom,
% 12.85/6.92 ! [M: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 12.85/6.92 => ( ord_less_nat @ M @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1567_of__nat__less__imp__less,axiom,
% 12.85/6.92 ! [M: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 12.85/6.92 => ( ord_less_nat @ M @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1568_of__nat__less__imp__less,axiom,
% 12.85/6.92 ! [M: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 12.85/6.92 => ( ord_less_nat @ M @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1569_of__nat__less__imp__less,axiom,
% 12.85/6.92 ! [M: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 12.85/6.92 => ( ord_less_nat @ M @ N ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1570_less__imp__of__nat__less,axiom,
% 12.85/6.92 ! [M: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_nat @ M @ N )
% 12.85/6.92 => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1571_less__imp__of__nat__less,axiom,
% 12.85/6.92 ! [M: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_nat @ M @ N )
% 12.85/6.92 => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1572_less__imp__of__nat__less,axiom,
% 12.85/6.92 ! [M: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_nat @ M @ N )
% 12.85/6.92 => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1573_less__imp__of__nat__less,axiom,
% 12.85/6.92 ! [M: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_nat @ M @ N )
% 12.85/6.92 => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1574_minf_I8_J,axiom,
% 12.85/6.92 ! [T: rat] :
% 12.85/6.92 ? [Z2: rat] :
% 12.85/6.92 ! [X5: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X5 @ Z2 )
% 12.85/6.92 => ~ ( ord_less_eq_rat @ T @ X5 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1575_minf_I8_J,axiom,
% 12.85/6.92 ! [T: num] :
% 12.85/6.92 ? [Z2: num] :
% 12.85/6.92 ! [X5: num] :
% 12.85/6.92 ( ( ord_less_num @ X5 @ Z2 )
% 12.85/6.92 => ~ ( ord_less_eq_num @ T @ X5 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1576_minf_I8_J,axiom,
% 12.85/6.92 ! [T: nat] :
% 12.85/6.92 ? [Z2: nat] :
% 12.85/6.92 ! [X5: nat] :
% 12.85/6.92 ( ( ord_less_nat @ X5 @ Z2 )
% 12.85/6.92 => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1577_minf_I8_J,axiom,
% 12.85/6.92 ! [T: int] :
% 12.85/6.92 ? [Z2: int] :
% 12.85/6.92 ! [X5: int] :
% 12.85/6.92 ( ( ord_less_int @ X5 @ Z2 )
% 12.85/6.92 => ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1578_minf_I8_J,axiom,
% 12.85/6.92 ! [T: real] :
% 12.85/6.92 ? [Z2: real] :
% 12.85/6.92 ! [X5: real] :
% 12.85/6.92 ( ( ord_less_real @ X5 @ Z2 )
% 12.85/6.92 => ~ ( ord_less_eq_real @ T @ X5 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1579_minf_I6_J,axiom,
% 12.85/6.92 ! [T: rat] :
% 12.85/6.92 ? [Z2: rat] :
% 12.85/6.92 ! [X5: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X5 @ Z2 )
% 12.85/6.92 => ( ord_less_eq_rat @ X5 @ T ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1580_minf_I6_J,axiom,
% 12.85/6.92 ! [T: num] :
% 12.85/6.92 ? [Z2: num] :
% 12.85/6.92 ! [X5: num] :
% 12.85/6.92 ( ( ord_less_num @ X5 @ Z2 )
% 12.85/6.92 => ( ord_less_eq_num @ X5 @ T ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1581_minf_I6_J,axiom,
% 12.85/6.92 ! [T: nat] :
% 12.85/6.92 ? [Z2: nat] :
% 12.85/6.92 ! [X5: nat] :
% 12.85/6.92 ( ( ord_less_nat @ X5 @ Z2 )
% 12.85/6.92 => ( ord_less_eq_nat @ X5 @ T ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1582_minf_I6_J,axiom,
% 12.85/6.92 ! [T: int] :
% 12.85/6.92 ? [Z2: int] :
% 12.85/6.92 ! [X5: int] :
% 12.85/6.92 ( ( ord_less_int @ X5 @ Z2 )
% 12.85/6.92 => ( ord_less_eq_int @ X5 @ T ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1583_minf_I6_J,axiom,
% 12.85/6.92 ! [T: real] :
% 12.85/6.92 ? [Z2: real] :
% 12.85/6.92 ! [X5: real] :
% 12.85/6.92 ( ( ord_less_real @ X5 @ Z2 )
% 12.85/6.92 => ( ord_less_eq_real @ X5 @ T ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1584_pinf_I8_J,axiom,
% 12.85/6.92 ! [T: rat] :
% 12.85/6.92 ? [Z2: rat] :
% 12.85/6.92 ! [X5: rat] :
% 12.85/6.92 ( ( ord_less_rat @ Z2 @ X5 )
% 12.85/6.92 => ( ord_less_eq_rat @ T @ X5 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1585_pinf_I8_J,axiom,
% 12.85/6.92 ! [T: num] :
% 12.85/6.92 ? [Z2: num] :
% 12.85/6.92 ! [X5: num] :
% 12.85/6.92 ( ( ord_less_num @ Z2 @ X5 )
% 12.85/6.92 => ( ord_less_eq_num @ T @ X5 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1586_pinf_I8_J,axiom,
% 12.85/6.92 ! [T: nat] :
% 12.85/6.92 ? [Z2: nat] :
% 12.85/6.92 ! [X5: nat] :
% 12.85/6.92 ( ( ord_less_nat @ Z2 @ X5 )
% 12.85/6.92 => ( ord_less_eq_nat @ T @ X5 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1587_pinf_I8_J,axiom,
% 12.85/6.92 ! [T: int] :
% 12.85/6.92 ? [Z2: int] :
% 12.85/6.92 ! [X5: int] :
% 12.85/6.92 ( ( ord_less_int @ Z2 @ X5 )
% 12.85/6.92 => ( ord_less_eq_int @ T @ X5 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1588_pinf_I8_J,axiom,
% 12.85/6.92 ! [T: real] :
% 12.85/6.92 ? [Z2: real] :
% 12.85/6.92 ! [X5: real] :
% 12.85/6.92 ( ( ord_less_real @ Z2 @ X5 )
% 12.85/6.92 => ( ord_less_eq_real @ T @ X5 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1589_pinf_I6_J,axiom,
% 12.85/6.92 ! [T: rat] :
% 12.85/6.92 ? [Z2: rat] :
% 12.85/6.92 ! [X5: rat] :
% 12.85/6.92 ( ( ord_less_rat @ Z2 @ X5 )
% 12.85/6.92 => ~ ( ord_less_eq_rat @ X5 @ T ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1590_pinf_I6_J,axiom,
% 12.85/6.92 ! [T: num] :
% 12.85/6.92 ? [Z2: num] :
% 12.85/6.92 ! [X5: num] :
% 12.85/6.92 ( ( ord_less_num @ Z2 @ X5 )
% 12.85/6.92 => ~ ( ord_less_eq_num @ X5 @ T ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1591_pinf_I6_J,axiom,
% 12.85/6.92 ! [T: nat] :
% 12.85/6.92 ? [Z2: nat] :
% 12.85/6.92 ! [X5: nat] :
% 12.85/6.92 ( ( ord_less_nat @ Z2 @ X5 )
% 12.85/6.92 => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1592_pinf_I6_J,axiom,
% 12.85/6.92 ! [T: int] :
% 12.85/6.92 ? [Z2: int] :
% 12.85/6.92 ! [X5: int] :
% 12.85/6.92 ( ( ord_less_int @ Z2 @ X5 )
% 12.85/6.92 => ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1593_pinf_I6_J,axiom,
% 12.85/6.92 ! [T: real] :
% 12.85/6.92 ? [Z2: real] :
% 12.85/6.92 ! [X5: real] :
% 12.85/6.92 ( ( ord_less_real @ Z2 @ X5 )
% 12.85/6.92 => ~ ( ord_less_eq_real @ X5 @ T ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1594_less__divide__eq__1,axiom,
% 12.85/6.92 ! [B: real,A: real] :
% 12.85/6.92 ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 12.85/6.92 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.92 & ( ord_less_real @ A @ B ) )
% 12.85/6.92 | ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/6.92 & ( ord_less_real @ B @ A ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1595_less__divide__eq__1,axiom,
% 12.85/6.92 ! [B: rat,A: rat] :
% 12.85/6.92 ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 12.85/6.92 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.92 & ( ord_less_rat @ A @ B ) )
% 12.85/6.92 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/6.92 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1596_divide__less__eq__1,axiom,
% 12.85/6.92 ! [B: real,A: real] :
% 12.85/6.92 ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 12.85/6.92 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.92 & ( ord_less_real @ B @ A ) )
% 12.85/6.92 | ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/6.92 & ( ord_less_real @ A @ B ) )
% 12.85/6.92 | ( A = zero_zero_real ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1597_divide__less__eq__1,axiom,
% 12.85/6.92 ! [B: rat,A: rat] :
% 12.85/6.92 ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 12.85/6.92 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.92 & ( ord_less_rat @ B @ A ) )
% 12.85/6.92 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/6.92 & ( ord_less_rat @ A @ B ) )
% 12.85/6.92 | ( A = zero_zero_rat ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1598_nat__induct__non__zero,axiom,
% 12.85/6.92 ! [N: nat,P: nat > $o] :
% 12.85/6.92 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.92 => ( ( P @ one_one_nat )
% 12.85/6.92 => ( ! [N2: nat] :
% 12.85/6.92 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 12.85/6.92 => ( ( P @ N2 )
% 12.85/6.92 => ( P @ ( suc @ N2 ) ) ) )
% 12.85/6.92 => ( P @ N ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1599_div__eq__dividend__iff,axiom,
% 12.85/6.92 ! [M: nat,N: nat] :
% 12.85/6.92 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/6.92 => ( ( ( divide_divide_nat @ M @ N )
% 12.85/6.92 = M )
% 12.85/6.92 = ( N = one_one_nat ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1600_div__less__dividend,axiom,
% 12.85/6.92 ! [N: nat,M: nat] :
% 12.85/6.92 ( ( ord_less_nat @ one_one_nat @ N )
% 12.85/6.92 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/6.92 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1601_verit__comp__simplify1_I3_J,axiom,
% 12.85/6.92 ! [B6: rat,A6: rat] :
% 12.85/6.92 ( ( ~ ( ord_less_eq_rat @ B6 @ A6 ) )
% 12.85/6.92 = ( ord_less_rat @ A6 @ B6 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1602_verit__comp__simplify1_I3_J,axiom,
% 12.85/6.92 ! [B6: num,A6: num] :
% 12.85/6.92 ( ( ~ ( ord_less_eq_num @ B6 @ A6 ) )
% 12.85/6.92 = ( ord_less_num @ A6 @ B6 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1603_verit__comp__simplify1_I3_J,axiom,
% 12.85/6.92 ! [B6: nat,A6: nat] :
% 12.85/6.92 ( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
% 12.85/6.92 = ( ord_less_nat @ A6 @ B6 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1604_verit__comp__simplify1_I3_J,axiom,
% 12.85/6.92 ! [B6: int,A6: int] :
% 12.85/6.92 ( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
% 12.85/6.92 = ( ord_less_int @ A6 @ B6 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1605_verit__comp__simplify1_I3_J,axiom,
% 12.85/6.92 ! [B6: real,A6: real] :
% 12.85/6.92 ( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
% 12.85/6.92 = ( ord_less_real @ A6 @ B6 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1606_leD,axiom,
% 12.85/6.92 ! [Y4: set_int,X: set_int] :
% 12.85/6.92 ( ( ord_less_eq_set_int @ Y4 @ X )
% 12.85/6.92 => ~ ( ord_less_set_int @ X @ Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1607_leD,axiom,
% 12.85/6.92 ! [Y4: rat,X: rat] :
% 12.85/6.92 ( ( ord_less_eq_rat @ Y4 @ X )
% 12.85/6.92 => ~ ( ord_less_rat @ X @ Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1608_leD,axiom,
% 12.85/6.92 ! [Y4: num,X: num] :
% 12.85/6.92 ( ( ord_less_eq_num @ Y4 @ X )
% 12.85/6.92 => ~ ( ord_less_num @ X @ Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1609_leD,axiom,
% 12.85/6.92 ! [Y4: nat,X: nat] :
% 12.85/6.92 ( ( ord_less_eq_nat @ Y4 @ X )
% 12.85/6.92 => ~ ( ord_less_nat @ X @ Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1610_leD,axiom,
% 12.85/6.92 ! [Y4: int,X: int] :
% 12.85/6.92 ( ( ord_less_eq_int @ Y4 @ X )
% 12.85/6.92 => ~ ( ord_less_int @ X @ Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1611_leD,axiom,
% 12.85/6.92 ! [Y4: real,X: real] :
% 12.85/6.92 ( ( ord_less_eq_real @ Y4 @ X )
% 12.85/6.92 => ~ ( ord_less_real @ X @ Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1612_leI,axiom,
% 12.85/6.92 ! [X: rat,Y4: rat] :
% 12.85/6.92 ( ~ ( ord_less_rat @ X @ Y4 )
% 12.85/6.92 => ( ord_less_eq_rat @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1613_leI,axiom,
% 12.85/6.92 ! [X: num,Y4: num] :
% 12.85/6.92 ( ~ ( ord_less_num @ X @ Y4 )
% 12.85/6.92 => ( ord_less_eq_num @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1614_leI,axiom,
% 12.85/6.92 ! [X: nat,Y4: nat] :
% 12.85/6.92 ( ~ ( ord_less_nat @ X @ Y4 )
% 12.85/6.92 => ( ord_less_eq_nat @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1615_leI,axiom,
% 12.85/6.92 ! [X: int,Y4: int] :
% 12.85/6.92 ( ~ ( ord_less_int @ X @ Y4 )
% 12.85/6.92 => ( ord_less_eq_int @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1616_leI,axiom,
% 12.85/6.92 ! [X: real,Y4: real] :
% 12.85/6.92 ( ~ ( ord_less_real @ X @ Y4 )
% 12.85/6.92 => ( ord_less_eq_real @ Y4 @ X ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1617_nless__le,axiom,
% 12.85/6.92 ! [A: set_int,B: set_int] :
% 12.85/6.92 ( ( ~ ( ord_less_set_int @ A @ B ) )
% 12.85/6.92 = ( ~ ( ord_less_eq_set_int @ A @ B )
% 12.85/6.92 | ( A = B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1618_nless__le,axiom,
% 12.85/6.92 ! [A: rat,B: rat] :
% 12.85/6.92 ( ( ~ ( ord_less_rat @ A @ B ) )
% 12.85/6.92 = ( ~ ( ord_less_eq_rat @ A @ B )
% 12.85/6.92 | ( A = B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1619_nless__le,axiom,
% 12.85/6.92 ! [A: num,B: num] :
% 12.85/6.92 ( ( ~ ( ord_less_num @ A @ B ) )
% 12.85/6.92 = ( ~ ( ord_less_eq_num @ A @ B )
% 12.85/6.92 | ( A = B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1620_nless__le,axiom,
% 12.85/6.92 ! [A: nat,B: nat] :
% 12.85/6.92 ( ( ~ ( ord_less_nat @ A @ B ) )
% 12.85/6.92 = ( ~ ( ord_less_eq_nat @ A @ B )
% 12.85/6.92 | ( A = B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1621_nless__le,axiom,
% 12.85/6.92 ! [A: int,B: int] :
% 12.85/6.92 ( ( ~ ( ord_less_int @ A @ B ) )
% 12.85/6.92 = ( ~ ( ord_less_eq_int @ A @ B )
% 12.85/6.92 | ( A = B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1622_nless__le,axiom,
% 12.85/6.92 ! [A: real,B: real] :
% 12.85/6.92 ( ( ~ ( ord_less_real @ A @ B ) )
% 12.85/6.92 = ( ~ ( ord_less_eq_real @ A @ B )
% 12.85/6.92 | ( A = B ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1623_antisym__conv1,axiom,
% 12.85/6.92 ! [X: set_int,Y4: set_int] :
% 12.85/6.92 ( ~ ( ord_less_set_int @ X @ Y4 )
% 12.85/6.92 => ( ( ord_less_eq_set_int @ X @ Y4 )
% 12.85/6.92 = ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1624_antisym__conv1,axiom,
% 12.85/6.92 ! [X: rat,Y4: rat] :
% 12.85/6.92 ( ~ ( ord_less_rat @ X @ Y4 )
% 12.85/6.92 => ( ( ord_less_eq_rat @ X @ Y4 )
% 12.85/6.92 = ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1625_antisym__conv1,axiom,
% 12.85/6.92 ! [X: num,Y4: num] :
% 12.85/6.92 ( ~ ( ord_less_num @ X @ Y4 )
% 12.85/6.92 => ( ( ord_less_eq_num @ X @ Y4 )
% 12.85/6.92 = ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1626_antisym__conv1,axiom,
% 12.85/6.92 ! [X: nat,Y4: nat] :
% 12.85/6.92 ( ~ ( ord_less_nat @ X @ Y4 )
% 12.85/6.92 => ( ( ord_less_eq_nat @ X @ Y4 )
% 12.85/6.92 = ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1627_antisym__conv1,axiom,
% 12.85/6.92 ! [X: int,Y4: int] :
% 12.85/6.92 ( ~ ( ord_less_int @ X @ Y4 )
% 12.85/6.92 => ( ( ord_less_eq_int @ X @ Y4 )
% 12.85/6.92 = ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1628_antisym__conv1,axiom,
% 12.85/6.92 ! [X: real,Y4: real] :
% 12.85/6.92 ( ~ ( ord_less_real @ X @ Y4 )
% 12.85/6.92 => ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/6.92 = ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1629_antisym__conv2,axiom,
% 12.85/6.92 ! [X: set_int,Y4: set_int] :
% 12.85/6.92 ( ( ord_less_eq_set_int @ X @ Y4 )
% 12.85/6.92 => ( ( ~ ( ord_less_set_int @ X @ Y4 ) )
% 12.85/6.92 = ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1630_antisym__conv2,axiom,
% 12.85/6.92 ! [X: rat,Y4: rat] :
% 12.85/6.92 ( ( ord_less_eq_rat @ X @ Y4 )
% 12.85/6.92 => ( ( ~ ( ord_less_rat @ X @ Y4 ) )
% 12.85/6.92 = ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1631_antisym__conv2,axiom,
% 12.85/6.92 ! [X: num,Y4: num] :
% 12.85/6.92 ( ( ord_less_eq_num @ X @ Y4 )
% 12.85/6.92 => ( ( ~ ( ord_less_num @ X @ Y4 ) )
% 12.85/6.92 = ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1632_antisym__conv2,axiom,
% 12.85/6.92 ! [X: nat,Y4: nat] :
% 12.85/6.92 ( ( ord_less_eq_nat @ X @ Y4 )
% 12.85/6.92 => ( ( ~ ( ord_less_nat @ X @ Y4 ) )
% 12.85/6.92 = ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1633_antisym__conv2,axiom,
% 12.85/6.92 ! [X: int,Y4: int] :
% 12.85/6.92 ( ( ord_less_eq_int @ X @ Y4 )
% 12.85/6.92 => ( ( ~ ( ord_less_int @ X @ Y4 ) )
% 12.85/6.92 = ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1634_antisym__conv2,axiom,
% 12.85/6.92 ! [X: real,Y4: real] :
% 12.85/6.92 ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/6.92 => ( ( ~ ( ord_less_real @ X @ Y4 ) )
% 12.85/6.92 = ( X = Y4 ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1635_dense__ge,axiom,
% 12.85/6.92 ! [Z: rat,Y4: rat] :
% 12.85/6.92 ( ! [X4: rat] :
% 12.85/6.92 ( ( ord_less_rat @ Z @ X4 )
% 12.85/6.92 => ( ord_less_eq_rat @ Y4 @ X4 ) )
% 12.85/6.92 => ( ord_less_eq_rat @ Y4 @ Z ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1636_dense__ge,axiom,
% 12.85/6.92 ! [Z: real,Y4: real] :
% 12.85/6.92 ( ! [X4: real] :
% 12.85/6.92 ( ( ord_less_real @ Z @ X4 )
% 12.85/6.92 => ( ord_less_eq_real @ Y4 @ X4 ) )
% 12.85/6.92 => ( ord_less_eq_real @ Y4 @ Z ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1637_dense__le,axiom,
% 12.85/6.92 ! [Y4: rat,Z: rat] :
% 12.85/6.92 ( ! [X4: rat] :
% 12.85/6.92 ( ( ord_less_rat @ X4 @ Y4 )
% 12.85/6.92 => ( ord_less_eq_rat @ X4 @ Z ) )
% 12.85/6.92 => ( ord_less_eq_rat @ Y4 @ Z ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1638_dense__le,axiom,
% 12.85/6.92 ! [Y4: real,Z: real] :
% 12.85/6.92 ( ! [X4: real] :
% 12.85/6.92 ( ( ord_less_real @ X4 @ Y4 )
% 12.85/6.92 => ( ord_less_eq_real @ X4 @ Z ) )
% 12.85/6.92 => ( ord_less_eq_real @ Y4 @ Z ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1639_less__le__not__le,axiom,
% 12.85/6.92 ( ord_less_set_int
% 12.85/6.92 = ( ^ [X3: set_int,Y6: set_int] :
% 12.85/6.92 ( ( ord_less_eq_set_int @ X3 @ Y6 )
% 12.85/6.92 & ~ ( ord_less_eq_set_int @ Y6 @ X3 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1640_less__le__not__le,axiom,
% 12.85/6.92 ( ord_less_rat
% 12.85/6.92 = ( ^ [X3: rat,Y6: rat] :
% 12.85/6.92 ( ( ord_less_eq_rat @ X3 @ Y6 )
% 12.85/6.92 & ~ ( ord_less_eq_rat @ Y6 @ X3 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1641_less__le__not__le,axiom,
% 12.85/6.92 ( ord_less_num
% 12.85/6.92 = ( ^ [X3: num,Y6: num] :
% 12.85/6.92 ( ( ord_less_eq_num @ X3 @ Y6 )
% 12.85/6.92 & ~ ( ord_less_eq_num @ Y6 @ X3 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1642_less__le__not__le,axiom,
% 12.85/6.92 ( ord_less_nat
% 12.85/6.92 = ( ^ [X3: nat,Y6: nat] :
% 12.85/6.92 ( ( ord_less_eq_nat @ X3 @ Y6 )
% 12.85/6.92 & ~ ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1643_less__le__not__le,axiom,
% 12.85/6.92 ( ord_less_int
% 12.85/6.92 = ( ^ [X3: int,Y6: int] :
% 12.85/6.92 ( ( ord_less_eq_int @ X3 @ Y6 )
% 12.85/6.92 & ~ ( ord_less_eq_int @ Y6 @ X3 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1644_less__le__not__le,axiom,
% 12.85/6.92 ( ord_less_real
% 12.85/6.92 = ( ^ [X3: real,Y6: real] :
% 12.85/6.92 ( ( ord_less_eq_real @ X3 @ Y6 )
% 12.85/6.92 & ~ ( ord_less_eq_real @ Y6 @ X3 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1645_not__le__imp__less,axiom,
% 12.85/6.92 ! [Y4: rat,X: rat] :
% 12.85/6.92 ( ~ ( ord_less_eq_rat @ Y4 @ X )
% 12.85/6.92 => ( ord_less_rat @ X @ Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1646_not__le__imp__less,axiom,
% 12.85/6.92 ! [Y4: num,X: num] :
% 12.85/6.92 ( ~ ( ord_less_eq_num @ Y4 @ X )
% 12.85/6.92 => ( ord_less_num @ X @ Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1647_not__le__imp__less,axiom,
% 12.85/6.92 ! [Y4: nat,X: nat] :
% 12.85/6.92 ( ~ ( ord_less_eq_nat @ Y4 @ X )
% 12.85/6.92 => ( ord_less_nat @ X @ Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1648_not__le__imp__less,axiom,
% 12.85/6.92 ! [Y4: int,X: int] :
% 12.85/6.92 ( ~ ( ord_less_eq_int @ Y4 @ X )
% 12.85/6.92 => ( ord_less_int @ X @ Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1649_not__le__imp__less,axiom,
% 12.85/6.92 ! [Y4: real,X: real] :
% 12.85/6.92 ( ~ ( ord_less_eq_real @ Y4 @ X )
% 12.85/6.92 => ( ord_less_real @ X @ Y4 ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1650_order_Oorder__iff__strict,axiom,
% 12.85/6.92 ( ord_less_eq_set_int
% 12.85/6.92 = ( ^ [A3: set_int,B2: set_int] :
% 12.85/6.92 ( ( ord_less_set_int @ A3 @ B2 )
% 12.85/6.92 | ( A3 = B2 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1651_order_Oorder__iff__strict,axiom,
% 12.85/6.92 ( ord_less_eq_rat
% 12.85/6.92 = ( ^ [A3: rat,B2: rat] :
% 12.85/6.92 ( ( ord_less_rat @ A3 @ B2 )
% 12.85/6.92 | ( A3 = B2 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1652_order_Oorder__iff__strict,axiom,
% 12.85/6.92 ( ord_less_eq_num
% 12.85/6.92 = ( ^ [A3: num,B2: num] :
% 12.85/6.92 ( ( ord_less_num @ A3 @ B2 )
% 12.85/6.92 | ( A3 = B2 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1653_order_Oorder__iff__strict,axiom,
% 12.85/6.92 ( ord_less_eq_nat
% 12.85/6.92 = ( ^ [A3: nat,B2: nat] :
% 12.85/6.92 ( ( ord_less_nat @ A3 @ B2 )
% 12.85/6.92 | ( A3 = B2 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1654_order_Oorder__iff__strict,axiom,
% 12.85/6.92 ( ord_less_eq_int
% 12.85/6.92 = ( ^ [A3: int,B2: int] :
% 12.85/6.92 ( ( ord_less_int @ A3 @ B2 )
% 12.85/6.92 | ( A3 = B2 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1655_order_Oorder__iff__strict,axiom,
% 12.85/6.92 ( ord_less_eq_real
% 12.85/6.92 = ( ^ [A3: real,B2: real] :
% 12.85/6.92 ( ( ord_less_real @ A3 @ B2 )
% 12.85/6.92 | ( A3 = B2 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1656_order_Ostrict__iff__order,axiom,
% 12.85/6.92 ( ord_less_set_int
% 12.85/6.92 = ( ^ [A3: set_int,B2: set_int] :
% 12.85/6.92 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 12.85/6.92 & ( A3 != B2 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1657_order_Ostrict__iff__order,axiom,
% 12.85/6.92 ( ord_less_rat
% 12.85/6.92 = ( ^ [A3: rat,B2: rat] :
% 12.85/6.92 ( ( ord_less_eq_rat @ A3 @ B2 )
% 12.85/6.92 & ( A3 != B2 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1658_order_Ostrict__iff__order,axiom,
% 12.85/6.92 ( ord_less_num
% 12.85/6.92 = ( ^ [A3: num,B2: num] :
% 12.85/6.92 ( ( ord_less_eq_num @ A3 @ B2 )
% 12.85/6.92 & ( A3 != B2 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1659_order_Ostrict__iff__order,axiom,
% 12.85/6.92 ( ord_less_nat
% 12.85/6.92 = ( ^ [A3: nat,B2: nat] :
% 12.85/6.92 ( ( ord_less_eq_nat @ A3 @ B2 )
% 12.85/6.92 & ( A3 != B2 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1660_order_Ostrict__iff__order,axiom,
% 12.85/6.92 ( ord_less_int
% 12.85/6.92 = ( ^ [A3: int,B2: int] :
% 12.85/6.92 ( ( ord_less_eq_int @ A3 @ B2 )
% 12.85/6.92 & ( A3 != B2 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1661_order_Ostrict__iff__order,axiom,
% 12.85/6.92 ( ord_less_real
% 12.85/6.92 = ( ^ [A3: real,B2: real] :
% 12.85/6.92 ( ( ord_less_eq_real @ A3 @ B2 )
% 12.85/6.92 & ( A3 != B2 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1662_order_Ostrict__trans1,axiom,
% 12.85/6.92 ! [A: set_int,B: set_int,C: set_int] :
% 12.85/6.92 ( ( ord_less_eq_set_int @ A @ B )
% 12.85/6.92 => ( ( ord_less_set_int @ B @ C )
% 12.85/6.92 => ( ord_less_set_int @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1663_order_Ostrict__trans1,axiom,
% 12.85/6.92 ! [A: rat,B: rat,C: rat] :
% 12.85/6.92 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.92 => ( ( ord_less_rat @ B @ C )
% 12.85/6.92 => ( ord_less_rat @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1664_order_Ostrict__trans1,axiom,
% 12.85/6.92 ! [A: num,B: num,C: num] :
% 12.85/6.92 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.92 => ( ( ord_less_num @ B @ C )
% 12.85/6.92 => ( ord_less_num @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1665_order_Ostrict__trans1,axiom,
% 12.85/6.92 ! [A: nat,B: nat,C: nat] :
% 12.85/6.92 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.92 => ( ( ord_less_nat @ B @ C )
% 12.85/6.92 => ( ord_less_nat @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1666_order_Ostrict__trans1,axiom,
% 12.85/6.92 ! [A: int,B: int,C: int] :
% 12.85/6.92 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.92 => ( ( ord_less_int @ B @ C )
% 12.85/6.92 => ( ord_less_int @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1667_order_Ostrict__trans1,axiom,
% 12.85/6.92 ! [A: real,B: real,C: real] :
% 12.85/6.92 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.92 => ( ( ord_less_real @ B @ C )
% 12.85/6.92 => ( ord_less_real @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1668_order_Ostrict__trans2,axiom,
% 12.85/6.92 ! [A: set_int,B: set_int,C: set_int] :
% 12.85/6.92 ( ( ord_less_set_int @ A @ B )
% 12.85/6.92 => ( ( ord_less_eq_set_int @ B @ C )
% 12.85/6.92 => ( ord_less_set_int @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1669_order_Ostrict__trans2,axiom,
% 12.85/6.92 ! [A: rat,B: rat,C: rat] :
% 12.85/6.92 ( ( ord_less_rat @ A @ B )
% 12.85/6.92 => ( ( ord_less_eq_rat @ B @ C )
% 12.85/6.92 => ( ord_less_rat @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1670_order_Ostrict__trans2,axiom,
% 12.85/6.92 ! [A: num,B: num,C: num] :
% 12.85/6.92 ( ( ord_less_num @ A @ B )
% 12.85/6.92 => ( ( ord_less_eq_num @ B @ C )
% 12.85/6.92 => ( ord_less_num @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1671_order_Ostrict__trans2,axiom,
% 12.85/6.92 ! [A: nat,B: nat,C: nat] :
% 12.85/6.92 ( ( ord_less_nat @ A @ B )
% 12.85/6.92 => ( ( ord_less_eq_nat @ B @ C )
% 12.85/6.92 => ( ord_less_nat @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1672_order_Ostrict__trans2,axiom,
% 12.85/6.92 ! [A: int,B: int,C: int] :
% 12.85/6.92 ( ( ord_less_int @ A @ B )
% 12.85/6.92 => ( ( ord_less_eq_int @ B @ C )
% 12.85/6.92 => ( ord_less_int @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1673_order_Ostrict__trans2,axiom,
% 12.85/6.92 ! [A: real,B: real,C: real] :
% 12.85/6.92 ( ( ord_less_real @ A @ B )
% 12.85/6.92 => ( ( ord_less_eq_real @ B @ C )
% 12.85/6.92 => ( ord_less_real @ A @ C ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1674_order_Ostrict__iff__not,axiom,
% 12.85/6.92 ( ord_less_set_int
% 12.85/6.92 = ( ^ [A3: set_int,B2: set_int] :
% 12.85/6.92 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 12.85/6.92 & ~ ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1675_order_Ostrict__iff__not,axiom,
% 12.85/6.92 ( ord_less_rat
% 12.85/6.92 = ( ^ [A3: rat,B2: rat] :
% 12.85/6.92 ( ( ord_less_eq_rat @ A3 @ B2 )
% 12.85/6.92 & ~ ( ord_less_eq_rat @ B2 @ A3 ) ) ) ) ).
% 12.85/6.92
% 12.85/6.92 thf(fact_1676_order_Ostrict__iff__not,axiom,
% 12.85/6.93 ( ord_less_num
% 12.85/6.93 = ( ^ [A3: num,B2: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ A3 @ B2 )
% 12.85/6.93 & ~ ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1677_order_Ostrict__iff__not,axiom,
% 12.85/6.93 ( ord_less_nat
% 12.85/6.93 = ( ^ [A3: nat,B2: nat] :
% 12.85/6.93 ( ( ord_less_eq_nat @ A3 @ B2 )
% 12.85/6.93 & ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1678_order_Ostrict__iff__not,axiom,
% 12.85/6.93 ( ord_less_int
% 12.85/6.93 = ( ^ [A3: int,B2: int] :
% 12.85/6.93 ( ( ord_less_eq_int @ A3 @ B2 )
% 12.85/6.93 & ~ ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1679_order_Ostrict__iff__not,axiom,
% 12.85/6.93 ( ord_less_real
% 12.85/6.93 = ( ^ [A3: real,B2: real] :
% 12.85/6.93 ( ( ord_less_eq_real @ A3 @ B2 )
% 12.85/6.93 & ~ ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1680_dense__ge__bounded,axiom,
% 12.85/6.93 ! [Z: rat,X: rat,Y4: rat] :
% 12.85/6.93 ( ( ord_less_rat @ Z @ X )
% 12.85/6.93 => ( ! [W2: rat] :
% 12.85/6.93 ( ( ord_less_rat @ Z @ W2 )
% 12.85/6.93 => ( ( ord_less_rat @ W2 @ X )
% 12.85/6.93 => ( ord_less_eq_rat @ Y4 @ W2 ) ) )
% 12.85/6.93 => ( ord_less_eq_rat @ Y4 @ Z ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1681_dense__ge__bounded,axiom,
% 12.85/6.93 ! [Z: real,X: real,Y4: real] :
% 12.85/6.93 ( ( ord_less_real @ Z @ X )
% 12.85/6.93 => ( ! [W2: real] :
% 12.85/6.93 ( ( ord_less_real @ Z @ W2 )
% 12.85/6.93 => ( ( ord_less_real @ W2 @ X )
% 12.85/6.93 => ( ord_less_eq_real @ Y4 @ W2 ) ) )
% 12.85/6.93 => ( ord_less_eq_real @ Y4 @ Z ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1682_dense__le__bounded,axiom,
% 12.85/6.93 ! [X: rat,Y4: rat,Z: rat] :
% 12.85/6.93 ( ( ord_less_rat @ X @ Y4 )
% 12.85/6.93 => ( ! [W2: rat] :
% 12.85/6.93 ( ( ord_less_rat @ X @ W2 )
% 12.85/6.93 => ( ( ord_less_rat @ W2 @ Y4 )
% 12.85/6.93 => ( ord_less_eq_rat @ W2 @ Z ) ) )
% 12.85/6.93 => ( ord_less_eq_rat @ Y4 @ Z ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1683_dense__le__bounded,axiom,
% 12.85/6.93 ! [X: real,Y4: real,Z: real] :
% 12.85/6.93 ( ( ord_less_real @ X @ Y4 )
% 12.85/6.93 => ( ! [W2: real] :
% 12.85/6.93 ( ( ord_less_real @ X @ W2 )
% 12.85/6.93 => ( ( ord_less_real @ W2 @ Y4 )
% 12.85/6.93 => ( ord_less_eq_real @ W2 @ Z ) ) )
% 12.85/6.93 => ( ord_less_eq_real @ Y4 @ Z ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1684_dual__order_Oorder__iff__strict,axiom,
% 12.85/6.93 ( ord_less_eq_set_int
% 12.85/6.93 = ( ^ [B2: set_int,A3: set_int] :
% 12.85/6.93 ( ( ord_less_set_int @ B2 @ A3 )
% 12.85/6.93 | ( A3 = B2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1685_dual__order_Oorder__iff__strict,axiom,
% 12.85/6.93 ( ord_less_eq_rat
% 12.85/6.93 = ( ^ [B2: rat,A3: rat] :
% 12.85/6.93 ( ( ord_less_rat @ B2 @ A3 )
% 12.85/6.93 | ( A3 = B2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1686_dual__order_Oorder__iff__strict,axiom,
% 12.85/6.93 ( ord_less_eq_num
% 12.85/6.93 = ( ^ [B2: num,A3: num] :
% 12.85/6.93 ( ( ord_less_num @ B2 @ A3 )
% 12.85/6.93 | ( A3 = B2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1687_dual__order_Oorder__iff__strict,axiom,
% 12.85/6.93 ( ord_less_eq_nat
% 12.85/6.93 = ( ^ [B2: nat,A3: nat] :
% 12.85/6.93 ( ( ord_less_nat @ B2 @ A3 )
% 12.85/6.93 | ( A3 = B2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1688_dual__order_Oorder__iff__strict,axiom,
% 12.85/6.93 ( ord_less_eq_int
% 12.85/6.93 = ( ^ [B2: int,A3: int] :
% 12.85/6.93 ( ( ord_less_int @ B2 @ A3 )
% 12.85/6.93 | ( A3 = B2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1689_dual__order_Oorder__iff__strict,axiom,
% 12.85/6.93 ( ord_less_eq_real
% 12.85/6.93 = ( ^ [B2: real,A3: real] :
% 12.85/6.93 ( ( ord_less_real @ B2 @ A3 )
% 12.85/6.93 | ( A3 = B2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1690_dual__order_Ostrict__iff__order,axiom,
% 12.85/6.93 ( ord_less_set_int
% 12.85/6.93 = ( ^ [B2: set_int,A3: set_int] :
% 12.85/6.93 ( ( ord_less_eq_set_int @ B2 @ A3 )
% 12.85/6.93 & ( A3 != B2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1691_dual__order_Ostrict__iff__order,axiom,
% 12.85/6.93 ( ord_less_rat
% 12.85/6.93 = ( ^ [B2: rat,A3: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ B2 @ A3 )
% 12.85/6.93 & ( A3 != B2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1692_dual__order_Ostrict__iff__order,axiom,
% 12.85/6.93 ( ord_less_num
% 12.85/6.93 = ( ^ [B2: num,A3: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ B2 @ A3 )
% 12.85/6.93 & ( A3 != B2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1693_dual__order_Ostrict__iff__order,axiom,
% 12.85/6.93 ( ord_less_nat
% 12.85/6.93 = ( ^ [B2: nat,A3: nat] :
% 12.85/6.93 ( ( ord_less_eq_nat @ B2 @ A3 )
% 12.85/6.93 & ( A3 != B2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1694_dual__order_Ostrict__iff__order,axiom,
% 12.85/6.93 ( ord_less_int
% 12.85/6.93 = ( ^ [B2: int,A3: int] :
% 12.85/6.93 ( ( ord_less_eq_int @ B2 @ A3 )
% 12.85/6.93 & ( A3 != B2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1695_dual__order_Ostrict__iff__order,axiom,
% 12.85/6.93 ( ord_less_real
% 12.85/6.93 = ( ^ [B2: real,A3: real] :
% 12.85/6.93 ( ( ord_less_eq_real @ B2 @ A3 )
% 12.85/6.93 & ( A3 != B2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1696_dual__order_Ostrict__trans1,axiom,
% 12.85/6.93 ! [B: set_int,A: set_int,C: set_int] :
% 12.85/6.93 ( ( ord_less_eq_set_int @ B @ A )
% 12.85/6.93 => ( ( ord_less_set_int @ C @ B )
% 12.85/6.93 => ( ord_less_set_int @ C @ A ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1697_dual__order_Ostrict__trans1,axiom,
% 12.85/6.93 ! [B: rat,A: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ B @ A )
% 12.85/6.93 => ( ( ord_less_rat @ C @ B )
% 12.85/6.93 => ( ord_less_rat @ C @ A ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1698_dual__order_Ostrict__trans1,axiom,
% 12.85/6.93 ! [B: num,A: num,C: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ B @ A )
% 12.85/6.93 => ( ( ord_less_num @ C @ B )
% 12.85/6.93 => ( ord_less_num @ C @ A ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1699_dual__order_Ostrict__trans1,axiom,
% 12.85/6.93 ! [B: nat,A: nat,C: nat] :
% 12.85/6.93 ( ( ord_less_eq_nat @ B @ A )
% 12.85/6.93 => ( ( ord_less_nat @ C @ B )
% 12.85/6.93 => ( ord_less_nat @ C @ A ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1700_dual__order_Ostrict__trans1,axiom,
% 12.85/6.93 ! [B: int,A: int,C: int] :
% 12.85/6.93 ( ( ord_less_eq_int @ B @ A )
% 12.85/6.93 => ( ( ord_less_int @ C @ B )
% 12.85/6.93 => ( ord_less_int @ C @ A ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1701_dual__order_Ostrict__trans1,axiom,
% 12.85/6.93 ! [B: real,A: real,C: real] :
% 12.85/6.93 ( ( ord_less_eq_real @ B @ A )
% 12.85/6.93 => ( ( ord_less_real @ C @ B )
% 12.85/6.93 => ( ord_less_real @ C @ A ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1702_dual__order_Ostrict__trans2,axiom,
% 12.85/6.93 ! [B: set_int,A: set_int,C: set_int] :
% 12.85/6.93 ( ( ord_less_set_int @ B @ A )
% 12.85/6.93 => ( ( ord_less_eq_set_int @ C @ B )
% 12.85/6.93 => ( ord_less_set_int @ C @ A ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1703_dual__order_Ostrict__trans2,axiom,
% 12.85/6.93 ! [B: rat,A: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_rat @ B @ A )
% 12.85/6.93 => ( ( ord_less_eq_rat @ C @ B )
% 12.85/6.93 => ( ord_less_rat @ C @ A ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1704_dual__order_Ostrict__trans2,axiom,
% 12.85/6.93 ! [B: num,A: num,C: num] :
% 12.85/6.93 ( ( ord_less_num @ B @ A )
% 12.85/6.93 => ( ( ord_less_eq_num @ C @ B )
% 12.85/6.93 => ( ord_less_num @ C @ A ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1705_dual__order_Ostrict__trans2,axiom,
% 12.85/6.93 ! [B: nat,A: nat,C: nat] :
% 12.85/6.93 ( ( ord_less_nat @ B @ A )
% 12.85/6.93 => ( ( ord_less_eq_nat @ C @ B )
% 12.85/6.93 => ( ord_less_nat @ C @ A ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1706_dual__order_Ostrict__trans2,axiom,
% 12.85/6.93 ! [B: int,A: int,C: int] :
% 12.85/6.93 ( ( ord_less_int @ B @ A )
% 12.85/6.93 => ( ( ord_less_eq_int @ C @ B )
% 12.85/6.93 => ( ord_less_int @ C @ A ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1707_dual__order_Ostrict__trans2,axiom,
% 12.85/6.93 ! [B: real,A: real,C: real] :
% 12.85/6.93 ( ( ord_less_real @ B @ A )
% 12.85/6.93 => ( ( ord_less_eq_real @ C @ B )
% 12.85/6.93 => ( ord_less_real @ C @ A ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1708_dual__order_Ostrict__iff__not,axiom,
% 12.85/6.93 ( ord_less_set_int
% 12.85/6.93 = ( ^ [B2: set_int,A3: set_int] :
% 12.85/6.93 ( ( ord_less_eq_set_int @ B2 @ A3 )
% 12.85/6.93 & ~ ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1709_dual__order_Ostrict__iff__not,axiom,
% 12.85/6.93 ( ord_less_rat
% 12.85/6.93 = ( ^ [B2: rat,A3: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ B2 @ A3 )
% 12.85/6.93 & ~ ( ord_less_eq_rat @ A3 @ B2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1710_dual__order_Ostrict__iff__not,axiom,
% 12.85/6.93 ( ord_less_num
% 12.85/6.93 = ( ^ [B2: num,A3: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ B2 @ A3 )
% 12.85/6.93 & ~ ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1711_dual__order_Ostrict__iff__not,axiom,
% 12.85/6.93 ( ord_less_nat
% 12.85/6.93 = ( ^ [B2: nat,A3: nat] :
% 12.85/6.93 ( ( ord_less_eq_nat @ B2 @ A3 )
% 12.85/6.93 & ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1712_dual__order_Ostrict__iff__not,axiom,
% 12.85/6.93 ( ord_less_int
% 12.85/6.93 = ( ^ [B2: int,A3: int] :
% 12.85/6.93 ( ( ord_less_eq_int @ B2 @ A3 )
% 12.85/6.93 & ~ ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1713_dual__order_Ostrict__iff__not,axiom,
% 12.85/6.93 ( ord_less_real
% 12.85/6.93 = ( ^ [B2: real,A3: real] :
% 12.85/6.93 ( ( ord_less_eq_real @ B2 @ A3 )
% 12.85/6.93 & ~ ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1714_order_Ostrict__implies__order,axiom,
% 12.85/6.93 ! [A: set_int,B: set_int] :
% 12.85/6.93 ( ( ord_less_set_int @ A @ B )
% 12.85/6.93 => ( ord_less_eq_set_int @ A @ B ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1715_order_Ostrict__implies__order,axiom,
% 12.85/6.93 ! [A: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ A @ B )
% 12.85/6.93 => ( ord_less_eq_rat @ A @ B ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1716_order_Ostrict__implies__order,axiom,
% 12.85/6.93 ! [A: num,B: num] :
% 12.85/6.93 ( ( ord_less_num @ A @ B )
% 12.85/6.93 => ( ord_less_eq_num @ A @ B ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1717_order_Ostrict__implies__order,axiom,
% 12.85/6.93 ! [A: nat,B: nat] :
% 12.85/6.93 ( ( ord_less_nat @ A @ B )
% 12.85/6.93 => ( ord_less_eq_nat @ A @ B ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1718_order_Ostrict__implies__order,axiom,
% 12.85/6.93 ! [A: int,B: int] :
% 12.85/6.93 ( ( ord_less_int @ A @ B )
% 12.85/6.93 => ( ord_less_eq_int @ A @ B ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1719_order_Ostrict__implies__order,axiom,
% 12.85/6.93 ! [A: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ A @ B )
% 12.85/6.93 => ( ord_less_eq_real @ A @ B ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1720_dual__order_Ostrict__implies__order,axiom,
% 12.85/6.93 ! [B: set_int,A: set_int] :
% 12.85/6.93 ( ( ord_less_set_int @ B @ A )
% 12.85/6.93 => ( ord_less_eq_set_int @ B @ A ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1721_dual__order_Ostrict__implies__order,axiom,
% 12.85/6.93 ! [B: rat,A: rat] :
% 12.85/6.93 ( ( ord_less_rat @ B @ A )
% 12.85/6.93 => ( ord_less_eq_rat @ B @ A ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1722_dual__order_Ostrict__implies__order,axiom,
% 12.85/6.93 ! [B: num,A: num] :
% 12.85/6.93 ( ( ord_less_num @ B @ A )
% 12.85/6.93 => ( ord_less_eq_num @ B @ A ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1723_dual__order_Ostrict__implies__order,axiom,
% 12.85/6.93 ! [B: nat,A: nat] :
% 12.85/6.93 ( ( ord_less_nat @ B @ A )
% 12.85/6.93 => ( ord_less_eq_nat @ B @ A ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1724_dual__order_Ostrict__implies__order,axiom,
% 12.85/6.93 ! [B: int,A: int] :
% 12.85/6.93 ( ( ord_less_int @ B @ A )
% 12.85/6.93 => ( ord_less_eq_int @ B @ A ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1725_dual__order_Ostrict__implies__order,axiom,
% 12.85/6.93 ! [B: real,A: real] :
% 12.85/6.93 ( ( ord_less_real @ B @ A )
% 12.85/6.93 => ( ord_less_eq_real @ B @ A ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1726_order__le__less,axiom,
% 12.85/6.93 ( ord_less_eq_set_int
% 12.85/6.93 = ( ^ [X3: set_int,Y6: set_int] :
% 12.85/6.93 ( ( ord_less_set_int @ X3 @ Y6 )
% 12.85/6.93 | ( X3 = Y6 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1727_order__le__less,axiom,
% 12.85/6.93 ( ord_less_eq_rat
% 12.85/6.93 = ( ^ [X3: rat,Y6: rat] :
% 12.85/6.93 ( ( ord_less_rat @ X3 @ Y6 )
% 12.85/6.93 | ( X3 = Y6 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1728_order__le__less,axiom,
% 12.85/6.93 ( ord_less_eq_num
% 12.85/6.93 = ( ^ [X3: num,Y6: num] :
% 12.85/6.93 ( ( ord_less_num @ X3 @ Y6 )
% 12.85/6.93 | ( X3 = Y6 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1729_order__le__less,axiom,
% 12.85/6.93 ( ord_less_eq_nat
% 12.85/6.93 = ( ^ [X3: nat,Y6: nat] :
% 12.85/6.93 ( ( ord_less_nat @ X3 @ Y6 )
% 12.85/6.93 | ( X3 = Y6 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1730_order__le__less,axiom,
% 12.85/6.93 ( ord_less_eq_int
% 12.85/6.93 = ( ^ [X3: int,Y6: int] :
% 12.85/6.93 ( ( ord_less_int @ X3 @ Y6 )
% 12.85/6.93 | ( X3 = Y6 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1731_order__le__less,axiom,
% 12.85/6.93 ( ord_less_eq_real
% 12.85/6.93 = ( ^ [X3: real,Y6: real] :
% 12.85/6.93 ( ( ord_less_real @ X3 @ Y6 )
% 12.85/6.93 | ( X3 = Y6 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1732_order__less__le,axiom,
% 12.85/6.93 ( ord_less_set_int
% 12.85/6.93 = ( ^ [X3: set_int,Y6: set_int] :
% 12.85/6.93 ( ( ord_less_eq_set_int @ X3 @ Y6 )
% 12.85/6.93 & ( X3 != Y6 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1733_order__less__le,axiom,
% 12.85/6.93 ( ord_less_rat
% 12.85/6.93 = ( ^ [X3: rat,Y6: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ X3 @ Y6 )
% 12.85/6.93 & ( X3 != Y6 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1734_order__less__le,axiom,
% 12.85/6.93 ( ord_less_num
% 12.85/6.93 = ( ^ [X3: num,Y6: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ X3 @ Y6 )
% 12.85/6.93 & ( X3 != Y6 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1735_order__less__le,axiom,
% 12.85/6.93 ( ord_less_nat
% 12.85/6.93 = ( ^ [X3: nat,Y6: nat] :
% 12.85/6.93 ( ( ord_less_eq_nat @ X3 @ Y6 )
% 12.85/6.93 & ( X3 != Y6 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1736_order__less__le,axiom,
% 12.85/6.93 ( ord_less_int
% 12.85/6.93 = ( ^ [X3: int,Y6: int] :
% 12.85/6.93 ( ( ord_less_eq_int @ X3 @ Y6 )
% 12.85/6.93 & ( X3 != Y6 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1737_order__less__le,axiom,
% 12.85/6.93 ( ord_less_real
% 12.85/6.93 = ( ^ [X3: real,Y6: real] :
% 12.85/6.93 ( ( ord_less_eq_real @ X3 @ Y6 )
% 12.85/6.93 & ( X3 != Y6 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1738_linorder__not__le,axiom,
% 12.85/6.93 ! [X: rat,Y4: rat] :
% 12.85/6.93 ( ( ~ ( ord_less_eq_rat @ X @ Y4 ) )
% 12.85/6.93 = ( ord_less_rat @ Y4 @ X ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1739_linorder__not__le,axiom,
% 12.85/6.93 ! [X: num,Y4: num] :
% 12.85/6.93 ( ( ~ ( ord_less_eq_num @ X @ Y4 ) )
% 12.85/6.93 = ( ord_less_num @ Y4 @ X ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1740_linorder__not__le,axiom,
% 12.85/6.93 ! [X: nat,Y4: nat] :
% 12.85/6.93 ( ( ~ ( ord_less_eq_nat @ X @ Y4 ) )
% 12.85/6.93 = ( ord_less_nat @ Y4 @ X ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1741_linorder__not__le,axiom,
% 12.85/6.93 ! [X: int,Y4: int] :
% 12.85/6.93 ( ( ~ ( ord_less_eq_int @ X @ Y4 ) )
% 12.85/6.93 = ( ord_less_int @ Y4 @ X ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1742_linorder__not__le,axiom,
% 12.85/6.93 ! [X: real,Y4: real] :
% 12.85/6.93 ( ( ~ ( ord_less_eq_real @ X @ Y4 ) )
% 12.85/6.93 = ( ord_less_real @ Y4 @ X ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1743_linorder__not__less,axiom,
% 12.85/6.93 ! [X: rat,Y4: rat] :
% 12.85/6.93 ( ( ~ ( ord_less_rat @ X @ Y4 ) )
% 12.85/6.93 = ( ord_less_eq_rat @ Y4 @ X ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1744_linorder__not__less,axiom,
% 12.85/6.93 ! [X: num,Y4: num] :
% 12.85/6.93 ( ( ~ ( ord_less_num @ X @ Y4 ) )
% 12.85/6.93 = ( ord_less_eq_num @ Y4 @ X ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1745_linorder__not__less,axiom,
% 12.85/6.93 ! [X: nat,Y4: nat] :
% 12.85/6.93 ( ( ~ ( ord_less_nat @ X @ Y4 ) )
% 12.85/6.93 = ( ord_less_eq_nat @ Y4 @ X ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1746_linorder__not__less,axiom,
% 12.85/6.93 ! [X: int,Y4: int] :
% 12.85/6.93 ( ( ~ ( ord_less_int @ X @ Y4 ) )
% 12.85/6.93 = ( ord_less_eq_int @ Y4 @ X ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1747_linorder__not__less,axiom,
% 12.85/6.93 ! [X: real,Y4: real] :
% 12.85/6.93 ( ( ~ ( ord_less_real @ X @ Y4 ) )
% 12.85/6.93 = ( ord_less_eq_real @ Y4 @ X ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1748_order__less__imp__le,axiom,
% 12.85/6.93 ! [X: set_int,Y4: set_int] :
% 12.85/6.93 ( ( ord_less_set_int @ X @ Y4 )
% 12.85/6.93 => ( ord_less_eq_set_int @ X @ Y4 ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1749_order__less__imp__le,axiom,
% 12.85/6.93 ! [X: rat,Y4: rat] :
% 12.85/6.93 ( ( ord_less_rat @ X @ Y4 )
% 12.85/6.93 => ( ord_less_eq_rat @ X @ Y4 ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1750_order__less__imp__le,axiom,
% 12.85/6.93 ! [X: num,Y4: num] :
% 12.85/6.93 ( ( ord_less_num @ X @ Y4 )
% 12.85/6.93 => ( ord_less_eq_num @ X @ Y4 ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1751_order__less__imp__le,axiom,
% 12.85/6.93 ! [X: nat,Y4: nat] :
% 12.85/6.93 ( ( ord_less_nat @ X @ Y4 )
% 12.85/6.93 => ( ord_less_eq_nat @ X @ Y4 ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1752_order__less__imp__le,axiom,
% 12.85/6.93 ! [X: int,Y4: int] :
% 12.85/6.93 ( ( ord_less_int @ X @ Y4 )
% 12.85/6.93 => ( ord_less_eq_int @ X @ Y4 ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1753_order__less__imp__le,axiom,
% 12.85/6.93 ! [X: real,Y4: real] :
% 12.85/6.93 ( ( ord_less_real @ X @ Y4 )
% 12.85/6.93 => ( ord_less_eq_real @ X @ Y4 ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1754_order__le__neq__trans,axiom,
% 12.85/6.93 ! [A: set_int,B: set_int] :
% 12.85/6.93 ( ( ord_less_eq_set_int @ A @ B )
% 12.85/6.93 => ( ( A != B )
% 12.85/6.93 => ( ord_less_set_int @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1755_order__le__neq__trans,axiom,
% 12.85/6.93 ! [A: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.93 => ( ( A != B )
% 12.85/6.93 => ( ord_less_rat @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1756_order__le__neq__trans,axiom,
% 12.85/6.93 ! [A: num,B: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.93 => ( ( A != B )
% 12.85/6.93 => ( ord_less_num @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1757_order__le__neq__trans,axiom,
% 12.85/6.93 ! [A: nat,B: nat] :
% 12.85/6.93 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.93 => ( ( A != B )
% 12.85/6.93 => ( ord_less_nat @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1758_order__le__neq__trans,axiom,
% 12.85/6.93 ! [A: int,B: int] :
% 12.85/6.93 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.93 => ( ( A != B )
% 12.85/6.93 => ( ord_less_int @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1759_order__le__neq__trans,axiom,
% 12.85/6.93 ! [A: real,B: real] :
% 12.85/6.93 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.93 => ( ( A != B )
% 12.85/6.93 => ( ord_less_real @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1760_order__neq__le__trans,axiom,
% 12.85/6.93 ! [A: set_int,B: set_int] :
% 12.85/6.93 ( ( A != B )
% 12.85/6.93 => ( ( ord_less_eq_set_int @ A @ B )
% 12.85/6.93 => ( ord_less_set_int @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1761_order__neq__le__trans,axiom,
% 12.85/6.93 ! [A: rat,B: rat] :
% 12.85/6.93 ( ( A != B )
% 12.85/6.93 => ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.93 => ( ord_less_rat @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1762_order__neq__le__trans,axiom,
% 12.85/6.93 ! [A: num,B: num] :
% 12.85/6.93 ( ( A != B )
% 12.85/6.93 => ( ( ord_less_eq_num @ A @ B )
% 12.85/6.93 => ( ord_less_num @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1763_order__neq__le__trans,axiom,
% 12.85/6.93 ! [A: nat,B: nat] :
% 12.85/6.93 ( ( A != B )
% 12.85/6.93 => ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.93 => ( ord_less_nat @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1764_order__neq__le__trans,axiom,
% 12.85/6.93 ! [A: int,B: int] :
% 12.85/6.93 ( ( A != B )
% 12.85/6.93 => ( ( ord_less_eq_int @ A @ B )
% 12.85/6.93 => ( ord_less_int @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1765_order__neq__le__trans,axiom,
% 12.85/6.93 ! [A: real,B: real] :
% 12.85/6.93 ( ( A != B )
% 12.85/6.93 => ( ( ord_less_eq_real @ A @ B )
% 12.85/6.93 => ( ord_less_real @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1766_order__le__less__trans,axiom,
% 12.85/6.93 ! [X: set_int,Y4: set_int,Z: set_int] :
% 12.85/6.93 ( ( ord_less_eq_set_int @ X @ Y4 )
% 12.85/6.93 => ( ( ord_less_set_int @ Y4 @ Z )
% 12.85/6.93 => ( ord_less_set_int @ X @ Z ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1767_order__le__less__trans,axiom,
% 12.85/6.93 ! [X: rat,Y4: rat,Z: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ X @ Y4 )
% 12.85/6.93 => ( ( ord_less_rat @ Y4 @ Z )
% 12.85/6.93 => ( ord_less_rat @ X @ Z ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1768_order__le__less__trans,axiom,
% 12.85/6.93 ! [X: num,Y4: num,Z: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ X @ Y4 )
% 12.85/6.93 => ( ( ord_less_num @ Y4 @ Z )
% 12.85/6.93 => ( ord_less_num @ X @ Z ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1769_order__le__less__trans,axiom,
% 12.85/6.93 ! [X: nat,Y4: nat,Z: nat] :
% 12.85/6.93 ( ( ord_less_eq_nat @ X @ Y4 )
% 12.85/6.93 => ( ( ord_less_nat @ Y4 @ Z )
% 12.85/6.93 => ( ord_less_nat @ X @ Z ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1770_order__le__less__trans,axiom,
% 12.85/6.93 ! [X: int,Y4: int,Z: int] :
% 12.85/6.93 ( ( ord_less_eq_int @ X @ Y4 )
% 12.85/6.93 => ( ( ord_less_int @ Y4 @ Z )
% 12.85/6.93 => ( ord_less_int @ X @ Z ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1771_order__le__less__trans,axiom,
% 12.85/6.93 ! [X: real,Y4: real,Z: real] :
% 12.85/6.93 ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/6.93 => ( ( ord_less_real @ Y4 @ Z )
% 12.85/6.93 => ( ord_less_real @ X @ Z ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1772_order__less__le__trans,axiom,
% 12.85/6.93 ! [X: set_int,Y4: set_int,Z: set_int] :
% 12.85/6.93 ( ( ord_less_set_int @ X @ Y4 )
% 12.85/6.93 => ( ( ord_less_eq_set_int @ Y4 @ Z )
% 12.85/6.93 => ( ord_less_set_int @ X @ Z ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1773_order__less__le__trans,axiom,
% 12.85/6.93 ! [X: rat,Y4: rat,Z: rat] :
% 12.85/6.93 ( ( ord_less_rat @ X @ Y4 )
% 12.85/6.93 => ( ( ord_less_eq_rat @ Y4 @ Z )
% 12.85/6.93 => ( ord_less_rat @ X @ Z ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1774_order__less__le__trans,axiom,
% 12.85/6.93 ! [X: num,Y4: num,Z: num] :
% 12.85/6.93 ( ( ord_less_num @ X @ Y4 )
% 12.85/6.93 => ( ( ord_less_eq_num @ Y4 @ Z )
% 12.85/6.93 => ( ord_less_num @ X @ Z ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1775_order__less__le__trans,axiom,
% 12.85/6.93 ! [X: nat,Y4: nat,Z: nat] :
% 12.85/6.93 ( ( ord_less_nat @ X @ Y4 )
% 12.85/6.93 => ( ( ord_less_eq_nat @ Y4 @ Z )
% 12.85/6.93 => ( ord_less_nat @ X @ Z ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1776_order__less__le__trans,axiom,
% 12.85/6.93 ! [X: int,Y4: int,Z: int] :
% 12.85/6.93 ( ( ord_less_int @ X @ Y4 )
% 12.85/6.93 => ( ( ord_less_eq_int @ Y4 @ Z )
% 12.85/6.93 => ( ord_less_int @ X @ Z ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1777_order__less__le__trans,axiom,
% 12.85/6.93 ! [X: real,Y4: real,Z: real] :
% 12.85/6.93 ( ( ord_less_real @ X @ Y4 )
% 12.85/6.93 => ( ( ord_less_eq_real @ Y4 @ Z )
% 12.85/6.93 => ( ord_less_real @ X @ Z ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1778_order__le__less__subst1,axiom,
% 12.85/6.93 ! [A: rat,F: real > rat,B: real,C: real] :
% 12.85/6.93 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_real @ B @ C )
% 12.85/6.93 => ( ! [X4: real,Y: real] :
% 12.85/6.93 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.93 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1779_order__le__less__subst1,axiom,
% 12.85/6.93 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_rat @ B @ C )
% 12.85/6.93 => ( ! [X4: rat,Y: rat] :
% 12.85/6.93 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.93 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1780_order__le__less__subst1,axiom,
% 12.85/6.93 ! [A: rat,F: num > rat,B: num,C: num] :
% 12.85/6.93 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_num @ B @ C )
% 12.85/6.93 => ( ! [X4: num,Y: num] :
% 12.85/6.93 ( ( ord_less_num @ X4 @ Y )
% 12.85/6.93 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1781_order__le__less__subst1,axiom,
% 12.85/6.93 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_nat @ B @ C )
% 12.85/6.93 => ( ! [X4: nat,Y: nat] :
% 12.85/6.93 ( ( ord_less_nat @ X4 @ Y )
% 12.85/6.93 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1782_order__le__less__subst1,axiom,
% 12.85/6.93 ! [A: rat,F: int > rat,B: int,C: int] :
% 12.85/6.93 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_int @ B @ C )
% 12.85/6.93 => ( ! [X4: int,Y: int] :
% 12.85/6.93 ( ( ord_less_int @ X4 @ Y )
% 12.85/6.93 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1783_order__le__less__subst1,axiom,
% 12.85/6.93 ! [A: num,F: real > num,B: real,C: real] :
% 12.85/6.93 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_real @ B @ C )
% 12.85/6.93 => ( ! [X4: real,Y: real] :
% 12.85/6.93 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.93 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1784_order__le__less__subst1,axiom,
% 12.85/6.93 ! [A: num,F: rat > num,B: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_rat @ B @ C )
% 12.85/6.93 => ( ! [X4: rat,Y: rat] :
% 12.85/6.93 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.93 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1785_order__le__less__subst1,axiom,
% 12.85/6.93 ! [A: num,F: num > num,B: num,C: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_num @ B @ C )
% 12.85/6.93 => ( ! [X4: num,Y: num] :
% 12.85/6.93 ( ( ord_less_num @ X4 @ Y )
% 12.85/6.93 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1786_order__le__less__subst1,axiom,
% 12.85/6.93 ! [A: num,F: nat > num,B: nat,C: nat] :
% 12.85/6.93 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_nat @ B @ C )
% 12.85/6.93 => ( ! [X4: nat,Y: nat] :
% 12.85/6.93 ( ( ord_less_nat @ X4 @ Y )
% 12.85/6.93 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1787_order__le__less__subst1,axiom,
% 12.85/6.93 ! [A: num,F: int > num,B: int,C: int] :
% 12.85/6.93 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_int @ B @ C )
% 12.85/6.93 => ( ! [X4: int,Y: int] :
% 12.85/6.93 ( ( ord_less_int @ X4 @ Y )
% 12.85/6.93 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1788_order__le__less__subst2,axiom,
% 12.85/6.93 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.93 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: rat,Y: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1789_order__le__less__subst2,axiom,
% 12.85/6.93 ! [A: rat,B: rat,F: rat > num,C: num] :
% 12.85/6.93 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.93 => ( ( ord_less_num @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: rat,Y: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1790_order__le__less__subst2,axiom,
% 12.85/6.93 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.93 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: rat,Y: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1791_order__le__less__subst2,axiom,
% 12.85/6.93 ! [A: rat,B: rat,F: rat > int,C: int] :
% 12.85/6.93 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.93 => ( ( ord_less_int @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: rat,Y: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1792_order__le__less__subst2,axiom,
% 12.85/6.93 ! [A: rat,B: rat,F: rat > real,C: real] :
% 12.85/6.93 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.93 => ( ( ord_less_real @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: rat,Y: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1793_order__le__less__subst2,axiom,
% 12.85/6.93 ! [A: num,B: num,F: num > rat,C: rat] :
% 12.85/6.93 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.93 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: num,Y: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1794_order__le__less__subst2,axiom,
% 12.85/6.93 ! [A: num,B: num,F: num > num,C: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.93 => ( ( ord_less_num @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: num,Y: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1795_order__le__less__subst2,axiom,
% 12.85/6.93 ! [A: num,B: num,F: num > nat,C: nat] :
% 12.85/6.93 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.93 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: num,Y: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1796_order__le__less__subst2,axiom,
% 12.85/6.93 ! [A: num,B: num,F: num > int,C: int] :
% 12.85/6.93 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.93 => ( ( ord_less_int @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: num,Y: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1797_order__le__less__subst2,axiom,
% 12.85/6.93 ! [A: num,B: num,F: num > real,C: real] :
% 12.85/6.93 ( ( ord_less_eq_num @ A @ B )
% 12.85/6.93 => ( ( ord_less_real @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: num,Y: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1798_order__less__le__subst1,axiom,
% 12.85/6.93 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_rat @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_eq_rat @ B @ C )
% 12.85/6.93 => ( ! [X4: rat,Y: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1799_order__less__le__subst1,axiom,
% 12.85/6.93 ! [A: num,F: rat > num,B: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_num @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_eq_rat @ B @ C )
% 12.85/6.93 => ( ! [X4: rat,Y: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1800_order__less__le__subst1,axiom,
% 12.85/6.93 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_nat @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_eq_rat @ B @ C )
% 12.85/6.93 => ( ! [X4: rat,Y: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1801_order__less__le__subst1,axiom,
% 12.85/6.93 ! [A: int,F: rat > int,B: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_int @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_eq_rat @ B @ C )
% 12.85/6.93 => ( ! [X4: rat,Y: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1802_order__less__le__subst1,axiom,
% 12.85/6.93 ! [A: real,F: rat > real,B: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_real @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_eq_rat @ B @ C )
% 12.85/6.93 => ( ! [X4: rat,Y: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1803_order__less__le__subst1,axiom,
% 12.85/6.93 ! [A: rat,F: num > rat,B: num,C: num] :
% 12.85/6.93 ( ( ord_less_rat @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_eq_num @ B @ C )
% 12.85/6.93 => ( ! [X4: num,Y: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1804_order__less__le__subst1,axiom,
% 12.85/6.93 ! [A: num,F: num > num,B: num,C: num] :
% 12.85/6.93 ( ( ord_less_num @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_eq_num @ B @ C )
% 12.85/6.93 => ( ! [X4: num,Y: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1805_order__less__le__subst1,axiom,
% 12.85/6.93 ! [A: nat,F: num > nat,B: num,C: num] :
% 12.85/6.93 ( ( ord_less_nat @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_eq_num @ B @ C )
% 12.85/6.93 => ( ! [X4: num,Y: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1806_order__less__le__subst1,axiom,
% 12.85/6.93 ! [A: int,F: num > int,B: num,C: num] :
% 12.85/6.93 ( ( ord_less_int @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_eq_num @ B @ C )
% 12.85/6.93 => ( ! [X4: num,Y: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1807_order__less__le__subst1,axiom,
% 12.85/6.93 ! [A: real,F: num > real,B: num,C: num] :
% 12.85/6.93 ( ( ord_less_real @ A @ ( F @ B ) )
% 12.85/6.93 => ( ( ord_less_eq_num @ B @ C )
% 12.85/6.93 => ( ! [X4: num,Y: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ X4 @ Y )
% 12.85/6.93 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1808_order__less__le__subst2,axiom,
% 12.85/6.93 ! [A: real,B: real,F: real > rat,C: rat] :
% 12.85/6.93 ( ( ord_less_real @ A @ B )
% 12.85/6.93 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: real,Y: real] :
% 12.85/6.93 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.93 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1809_order__less__le__subst2,axiom,
% 12.85/6.93 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 12.85/6.93 ( ( ord_less_rat @ A @ B )
% 12.85/6.93 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: rat,Y: rat] :
% 12.85/6.93 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.93 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1810_order__less__le__subst2,axiom,
% 12.85/6.93 ! [A: num,B: num,F: num > rat,C: rat] :
% 12.85/6.93 ( ( ord_less_num @ A @ B )
% 12.85/6.93 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: num,Y: num] :
% 12.85/6.93 ( ( ord_less_num @ X4 @ Y )
% 12.85/6.93 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1811_order__less__le__subst2,axiom,
% 12.85/6.93 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 12.85/6.93 ( ( ord_less_nat @ A @ B )
% 12.85/6.93 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: nat,Y: nat] :
% 12.85/6.93 ( ( ord_less_nat @ X4 @ Y )
% 12.85/6.93 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1812_order__less__le__subst2,axiom,
% 12.85/6.93 ! [A: int,B: int,F: int > rat,C: rat] :
% 12.85/6.93 ( ( ord_less_int @ A @ B )
% 12.85/6.93 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: int,Y: int] :
% 12.85/6.93 ( ( ord_less_int @ X4 @ Y )
% 12.85/6.93 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1813_order__less__le__subst2,axiom,
% 12.85/6.93 ! [A: real,B: real,F: real > num,C: num] :
% 12.85/6.93 ( ( ord_less_real @ A @ B )
% 12.85/6.93 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: real,Y: real] :
% 12.85/6.93 ( ( ord_less_real @ X4 @ Y )
% 12.85/6.93 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1814_order__less__le__subst2,axiom,
% 12.85/6.93 ! [A: rat,B: rat,F: rat > num,C: num] :
% 12.85/6.93 ( ( ord_less_rat @ A @ B )
% 12.85/6.93 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: rat,Y: rat] :
% 12.85/6.93 ( ( ord_less_rat @ X4 @ Y )
% 12.85/6.93 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1815_order__less__le__subst2,axiom,
% 12.85/6.93 ! [A: num,B: num,F: num > num,C: num] :
% 12.85/6.93 ( ( ord_less_num @ A @ B )
% 12.85/6.93 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: num,Y: num] :
% 12.85/6.93 ( ( ord_less_num @ X4 @ Y )
% 12.85/6.93 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1816_order__less__le__subst2,axiom,
% 12.85/6.93 ! [A: nat,B: nat,F: nat > num,C: num] :
% 12.85/6.93 ( ( ord_less_nat @ A @ B )
% 12.85/6.93 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: nat,Y: nat] :
% 12.85/6.93 ( ( ord_less_nat @ X4 @ Y )
% 12.85/6.93 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1817_order__less__le__subst2,axiom,
% 12.85/6.93 ! [A: int,B: int,F: int > num,C: num] :
% 12.85/6.93 ( ( ord_less_int @ A @ B )
% 12.85/6.93 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 12.85/6.93 => ( ! [X4: int,Y: int] :
% 12.85/6.93 ( ( ord_less_int @ X4 @ Y )
% 12.85/6.93 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y ) ) )
% 12.85/6.93 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1818_linorder__le__less__linear,axiom,
% 12.85/6.93 ! [X: rat,Y4: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ X @ Y4 )
% 12.85/6.93 | ( ord_less_rat @ Y4 @ X ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1819_linorder__le__less__linear,axiom,
% 12.85/6.93 ! [X: num,Y4: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ X @ Y4 )
% 12.85/6.93 | ( ord_less_num @ Y4 @ X ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1820_linorder__le__less__linear,axiom,
% 12.85/6.93 ! [X: nat,Y4: nat] :
% 12.85/6.93 ( ( ord_less_eq_nat @ X @ Y4 )
% 12.85/6.93 | ( ord_less_nat @ Y4 @ X ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1821_linorder__le__less__linear,axiom,
% 12.85/6.93 ! [X: int,Y4: int] :
% 12.85/6.93 ( ( ord_less_eq_int @ X @ Y4 )
% 12.85/6.93 | ( ord_less_int @ Y4 @ X ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1822_linorder__le__less__linear,axiom,
% 12.85/6.93 ! [X: real,Y4: real] :
% 12.85/6.93 ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/6.93 | ( ord_less_real @ Y4 @ X ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1823_order__le__imp__less__or__eq,axiom,
% 12.85/6.93 ! [X: set_int,Y4: set_int] :
% 12.85/6.93 ( ( ord_less_eq_set_int @ X @ Y4 )
% 12.85/6.93 => ( ( ord_less_set_int @ X @ Y4 )
% 12.85/6.93 | ( X = Y4 ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1824_order__le__imp__less__or__eq,axiom,
% 12.85/6.93 ! [X: rat,Y4: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ X @ Y4 )
% 12.85/6.93 => ( ( ord_less_rat @ X @ Y4 )
% 12.85/6.93 | ( X = Y4 ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1825_order__le__imp__less__or__eq,axiom,
% 12.85/6.93 ! [X: num,Y4: num] :
% 12.85/6.93 ( ( ord_less_eq_num @ X @ Y4 )
% 12.85/6.93 => ( ( ord_less_num @ X @ Y4 )
% 12.85/6.93 | ( X = Y4 ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1826_order__le__imp__less__or__eq,axiom,
% 12.85/6.93 ! [X: nat,Y4: nat] :
% 12.85/6.93 ( ( ord_less_eq_nat @ X @ Y4 )
% 12.85/6.93 => ( ( ord_less_nat @ X @ Y4 )
% 12.85/6.93 | ( X = Y4 ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1827_order__le__imp__less__or__eq,axiom,
% 12.85/6.93 ! [X: int,Y4: int] :
% 12.85/6.93 ( ( ord_less_eq_int @ X @ Y4 )
% 12.85/6.93 => ( ( ord_less_int @ X @ Y4 )
% 12.85/6.93 | ( X = Y4 ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1828_order__le__imp__less__or__eq,axiom,
% 12.85/6.93 ! [X: real,Y4: real] :
% 12.85/6.93 ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/6.93 => ( ( ord_less_real @ X @ Y4 )
% 12.85/6.93 | ( X = Y4 ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1829_less__numeral__extra_I3_J,axiom,
% 12.85/6.93 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1830_less__numeral__extra_I3_J,axiom,
% 12.85/6.93 ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1831_less__numeral__extra_I3_J,axiom,
% 12.85/6.93 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1832_less__numeral__extra_I3_J,axiom,
% 12.85/6.93 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1833_field__lbound__gt__zero,axiom,
% 12.85/6.93 ! [D1: real,D22: real] :
% 12.85/6.93 ( ( ord_less_real @ zero_zero_real @ D1 )
% 12.85/6.93 => ( ( ord_less_real @ zero_zero_real @ D22 )
% 12.85/6.93 => ? [E: real] :
% 12.85/6.93 ( ( ord_less_real @ zero_zero_real @ E )
% 12.85/6.93 & ( ord_less_real @ E @ D1 )
% 12.85/6.93 & ( ord_less_real @ E @ D22 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1834_field__lbound__gt__zero,axiom,
% 12.85/6.93 ! [D1: rat,D22: rat] :
% 12.85/6.93 ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 12.85/6.93 => ( ( ord_less_rat @ zero_zero_rat @ D22 )
% 12.85/6.93 => ? [E: rat] :
% 12.85/6.93 ( ( ord_less_rat @ zero_zero_rat @ E )
% 12.85/6.93 & ( ord_less_rat @ E @ D1 )
% 12.85/6.93 & ( ord_less_rat @ E @ D22 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1835_zero__less__iff__neq__zero,axiom,
% 12.85/6.93 ! [N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.93 = ( N != zero_zero_nat ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1836_gr__implies__not__zero,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ M @ N )
% 12.85/6.93 => ( N != zero_zero_nat ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1837_not__less__zero,axiom,
% 12.85/6.93 ! [N: nat] :
% 12.85/6.93 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1838_gr__zeroI,axiom,
% 12.85/6.93 ! [N: nat] :
% 12.85/6.93 ( ( N != zero_zero_nat )
% 12.85/6.93 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1839_zdiff__int__split,axiom,
% 12.85/6.93 ! [P: int > $o,X: nat,Y4: nat] :
% 12.85/6.93 ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y4 ) ) )
% 12.85/6.93 = ( ( ( ord_less_eq_nat @ Y4 @ X )
% 12.85/6.93 => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y4 ) ) ) )
% 12.85/6.93 & ( ( ord_less_nat @ X @ Y4 )
% 12.85/6.93 => ( P @ zero_zero_int ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1840_diff__strict__right__mono,axiom,
% 12.85/6.93 ! [A: real,B: real,C: real] :
% 12.85/6.93 ( ( ord_less_real @ A @ B )
% 12.85/6.93 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1841_diff__strict__right__mono,axiom,
% 12.85/6.93 ! [A: rat,B: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_rat @ A @ B )
% 12.85/6.93 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1842_diff__strict__right__mono,axiom,
% 12.85/6.93 ! [A: int,B: int,C: int] :
% 12.85/6.93 ( ( ord_less_int @ A @ B )
% 12.85/6.93 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1843_diff__strict__left__mono,axiom,
% 12.85/6.93 ! [B: real,A: real,C: real] :
% 12.85/6.93 ( ( ord_less_real @ B @ A )
% 12.85/6.93 => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1844_diff__strict__left__mono,axiom,
% 12.85/6.93 ! [B: rat,A: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_rat @ B @ A )
% 12.85/6.93 => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1845_diff__strict__left__mono,axiom,
% 12.85/6.93 ! [B: int,A: int,C: int] :
% 12.85/6.93 ( ( ord_less_int @ B @ A )
% 12.85/6.93 => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1846_diff__eq__diff__less,axiom,
% 12.85/6.93 ! [A: real,B: real,C: real,D: real] :
% 12.85/6.93 ( ( ( minus_minus_real @ A @ B )
% 12.85/6.93 = ( minus_minus_real @ C @ D ) )
% 12.85/6.93 => ( ( ord_less_real @ A @ B )
% 12.85/6.93 = ( ord_less_real @ C @ D ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1847_diff__eq__diff__less,axiom,
% 12.85/6.93 ! [A: rat,B: rat,C: rat,D: rat] :
% 12.85/6.93 ( ( ( minus_minus_rat @ A @ B )
% 12.85/6.93 = ( minus_minus_rat @ C @ D ) )
% 12.85/6.93 => ( ( ord_less_rat @ A @ B )
% 12.85/6.93 = ( ord_less_rat @ C @ D ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1848_diff__eq__diff__less,axiom,
% 12.85/6.93 ! [A: int,B: int,C: int,D: int] :
% 12.85/6.93 ( ( ( minus_minus_int @ A @ B )
% 12.85/6.93 = ( minus_minus_int @ C @ D ) )
% 12.85/6.93 => ( ( ord_less_int @ A @ B )
% 12.85/6.93 = ( ord_less_int @ C @ D ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1849_diff__strict__mono,axiom,
% 12.85/6.93 ! [A: real,B: real,D: real,C: real] :
% 12.85/6.93 ( ( ord_less_real @ A @ B )
% 12.85/6.93 => ( ( ord_less_real @ D @ C )
% 12.85/6.93 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1850_diff__strict__mono,axiom,
% 12.85/6.93 ! [A: rat,B: rat,D: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_rat @ A @ B )
% 12.85/6.93 => ( ( ord_less_rat @ D @ C )
% 12.85/6.93 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1851_diff__strict__mono,axiom,
% 12.85/6.93 ! [A: int,B: int,D: int,C: int] :
% 12.85/6.93 ( ( ord_less_int @ A @ B )
% 12.85/6.93 => ( ( ord_less_int @ D @ C )
% 12.85/6.93 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1852_double__diff,axiom,
% 12.85/6.93 ! [A2: set_nat,B4: set_nat,C2: set_nat] :
% 12.85/6.93 ( ( ord_less_eq_set_nat @ A2 @ B4 )
% 12.85/6.93 => ( ( ord_less_eq_set_nat @ B4 @ C2 )
% 12.85/6.93 => ( ( minus_minus_set_nat @ B4 @ ( minus_minus_set_nat @ C2 @ A2 ) )
% 12.85/6.93 = A2 ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1853_double__diff,axiom,
% 12.85/6.93 ! [A2: set_int,B4: set_int,C2: set_int] :
% 12.85/6.93 ( ( ord_less_eq_set_int @ A2 @ B4 )
% 12.85/6.93 => ( ( ord_less_eq_set_int @ B4 @ C2 )
% 12.85/6.93 => ( ( minus_minus_set_int @ B4 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 12.85/6.93 = A2 ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1854_Diff__subset,axiom,
% 12.85/6.93 ! [A2: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B4 ) @ A2 ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1855_Diff__subset,axiom,
% 12.85/6.93 ! [A2: set_int,B4: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B4 ) @ A2 ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1856_Diff__mono,axiom,
% 12.85/6.93 ! [A2: set_nat,C2: set_nat,D3: set_nat,B4: set_nat] :
% 12.85/6.93 ( ( ord_less_eq_set_nat @ A2 @ C2 )
% 12.85/6.93 => ( ( ord_less_eq_set_nat @ D3 @ B4 )
% 12.85/6.93 => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B4 ) @ ( minus_minus_set_nat @ C2 @ D3 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1857_Diff__mono,axiom,
% 12.85/6.93 ! [A2: set_int,C2: set_int,D3: set_int,B4: set_int] :
% 12.85/6.93 ( ( ord_less_eq_set_int @ A2 @ C2 )
% 12.85/6.93 => ( ( ord_less_eq_set_int @ D3 @ B4 )
% 12.85/6.93 => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B4 ) @ ( minus_minus_set_int @ C2 @ D3 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1858_le__numeral__extra_I4_J,axiom,
% 12.85/6.93 ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1859_le__numeral__extra_I4_J,axiom,
% 12.85/6.93 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1860_le__numeral__extra_I4_J,axiom,
% 12.85/6.93 ord_less_eq_int @ one_one_int @ one_one_int ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1861_le__numeral__extra_I4_J,axiom,
% 12.85/6.93 ord_less_eq_real @ one_one_real @ one_one_real ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1862_Nat_OlessE,axiom,
% 12.85/6.93 ! [I: nat,K: nat] :
% 12.85/6.93 ( ( ord_less_nat @ I @ K )
% 12.85/6.93 => ( ( K
% 12.85/6.93 != ( suc @ I ) )
% 12.85/6.93 => ~ ! [J2: nat] :
% 12.85/6.93 ( ( ord_less_nat @ I @ J2 )
% 12.85/6.93 => ( K
% 12.85/6.93 != ( suc @ J2 ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1863_Suc__lessD,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 12.85/6.93 => ( ord_less_nat @ M @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1864_Suc__lessE,axiom,
% 12.85/6.93 ! [I: nat,K: nat] :
% 12.85/6.93 ( ( ord_less_nat @ ( suc @ I ) @ K )
% 12.85/6.93 => ~ ! [J2: nat] :
% 12.85/6.93 ( ( ord_less_nat @ I @ J2 )
% 12.85/6.93 => ( K
% 12.85/6.93 != ( suc @ J2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1865_Suc__lessI,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ M @ N )
% 12.85/6.93 => ( ( ( suc @ M )
% 12.85/6.93 != N )
% 12.85/6.93 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1866_less__SucE,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 12.85/6.93 => ( ~ ( ord_less_nat @ M @ N )
% 12.85/6.93 => ( M = N ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1867_less__SucI,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ M @ N )
% 12.85/6.93 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1868_Ex__less__Suc,axiom,
% 12.85/6.93 ! [N: nat,P: nat > $o] :
% 12.85/6.93 ( ( ? [I2: nat] :
% 12.85/6.93 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 12.85/6.93 & ( P @ I2 ) ) )
% 12.85/6.93 = ( ( P @ N )
% 12.85/6.93 | ? [I2: nat] :
% 12.85/6.93 ( ( ord_less_nat @ I2 @ N )
% 12.85/6.93 & ( P @ I2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1869_less__Suc__eq,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 12.85/6.93 = ( ( ord_less_nat @ M @ N )
% 12.85/6.93 | ( M = N ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1870_not__less__eq,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ~ ( ord_less_nat @ M @ N ) )
% 12.85/6.93 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1871_All__less__Suc,axiom,
% 12.85/6.93 ! [N: nat,P: nat > $o] :
% 12.85/6.93 ( ( ! [I2: nat] :
% 12.85/6.93 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 12.85/6.93 => ( P @ I2 ) ) )
% 12.85/6.93 = ( ( P @ N )
% 12.85/6.93 & ! [I2: nat] :
% 12.85/6.93 ( ( ord_less_nat @ I2 @ N )
% 12.85/6.93 => ( P @ I2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1872_Suc__less__eq2,axiom,
% 12.85/6.93 ! [N: nat,M: nat] :
% 12.85/6.93 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 12.85/6.93 = ( ? [M6: nat] :
% 12.85/6.93 ( ( M
% 12.85/6.93 = ( suc @ M6 ) )
% 12.85/6.93 & ( ord_less_nat @ N @ M6 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1873_less__antisym,axiom,
% 12.85/6.93 ! [N: nat,M: nat] :
% 12.85/6.93 ( ~ ( ord_less_nat @ N @ M )
% 12.85/6.93 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 12.85/6.93 => ( M = N ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1874_Suc__less__SucD,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 12.85/6.93 => ( ord_less_nat @ M @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1875_less__trans__Suc,axiom,
% 12.85/6.93 ! [I: nat,J: nat,K: nat] :
% 12.85/6.93 ( ( ord_less_nat @ I @ J )
% 12.85/6.93 => ( ( ord_less_nat @ J @ K )
% 12.85/6.93 => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1876_less__Suc__induct,axiom,
% 12.85/6.93 ! [I: nat,J: nat,P: nat > nat > $o] :
% 12.85/6.93 ( ( ord_less_nat @ I @ J )
% 12.85/6.93 => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
% 12.85/6.93 => ( ! [I3: nat,J2: nat,K2: nat] :
% 12.85/6.93 ( ( ord_less_nat @ I3 @ J2 )
% 12.85/6.93 => ( ( ord_less_nat @ J2 @ K2 )
% 12.85/6.93 => ( ( P @ I3 @ J2 )
% 12.85/6.93 => ( ( P @ J2 @ K2 )
% 12.85/6.93 => ( P @ I3 @ K2 ) ) ) ) )
% 12.85/6.93 => ( P @ I @ J ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1877_strict__inc__induct,axiom,
% 12.85/6.93 ! [I: nat,J: nat,P: nat > $o] :
% 12.85/6.93 ( ( ord_less_nat @ I @ J )
% 12.85/6.93 => ( ! [I3: nat] :
% 12.85/6.93 ( ( J
% 12.85/6.93 = ( suc @ I3 ) )
% 12.85/6.93 => ( P @ I3 ) )
% 12.85/6.93 => ( ! [I3: nat] :
% 12.85/6.93 ( ( ord_less_nat @ I3 @ J )
% 12.85/6.93 => ( ( P @ ( suc @ I3 ) )
% 12.85/6.93 => ( P @ I3 ) ) )
% 12.85/6.93 => ( P @ I ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1878_not__less__less__Suc__eq,axiom,
% 12.85/6.93 ! [N: nat,M: nat] :
% 12.85/6.93 ( ~ ( ord_less_nat @ N @ M )
% 12.85/6.93 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 12.85/6.93 = ( N = M ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1879_zero__neq__one,axiom,
% 12.85/6.93 zero_zero_complex != one_one_complex ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1880_zero__neq__one,axiom,
% 12.85/6.93 zero_zero_real != one_one_real ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1881_zero__neq__one,axiom,
% 12.85/6.93 zero_zero_rat != one_one_rat ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1882_zero__neq__one,axiom,
% 12.85/6.93 zero_zero_nat != one_one_nat ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1883_zero__neq__one,axiom,
% 12.85/6.93 zero_zero_int != one_one_int ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1884_bot__nat__0_Oextremum__strict,axiom,
% 12.85/6.93 ! [A: nat] :
% 12.85/6.93 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1885_gr0I,axiom,
% 12.85/6.93 ! [N: nat] :
% 12.85/6.93 ( ( N != zero_zero_nat )
% 12.85/6.93 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1886_not__gr0,axiom,
% 12.85/6.93 ! [N: nat] :
% 12.85/6.93 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 12.85/6.93 = ( N = zero_zero_nat ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1887_not__less0,axiom,
% 12.85/6.93 ! [N: nat] :
% 12.85/6.93 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1888_less__zeroE,axiom,
% 12.85/6.93 ! [N: nat] :
% 12.85/6.93 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1889_gr__implies__not0,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ M @ N )
% 12.85/6.93 => ( N != zero_zero_nat ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1890_infinite__descent0,axiom,
% 12.85/6.93 ! [P: nat > $o,N: nat] :
% 12.85/6.93 ( ( P @ zero_zero_nat )
% 12.85/6.93 => ( ! [N2: nat] :
% 12.85/6.93 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 12.85/6.93 => ( ~ ( P @ N2 )
% 12.85/6.93 => ? [M2: nat] :
% 12.85/6.93 ( ( ord_less_nat @ M2 @ N2 )
% 12.85/6.93 & ~ ( P @ M2 ) ) ) )
% 12.85/6.93 => ( P @ N ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1891_comm__monoid__mult__class_Omult__1,axiom,
% 12.85/6.93 ! [A: complex] :
% 12.85/6.93 ( ( times_times_complex @ one_one_complex @ A )
% 12.85/6.93 = A ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1892_comm__monoid__mult__class_Omult__1,axiom,
% 12.85/6.93 ! [A: real] :
% 12.85/6.93 ( ( times_times_real @ one_one_real @ A )
% 12.85/6.93 = A ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1893_comm__monoid__mult__class_Omult__1,axiom,
% 12.85/6.93 ! [A: rat] :
% 12.85/6.93 ( ( times_times_rat @ one_one_rat @ A )
% 12.85/6.93 = A ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1894_comm__monoid__mult__class_Omult__1,axiom,
% 12.85/6.93 ! [A: nat] :
% 12.85/6.93 ( ( times_times_nat @ one_one_nat @ A )
% 12.85/6.93 = A ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1895_comm__monoid__mult__class_Omult__1,axiom,
% 12.85/6.93 ! [A: int] :
% 12.85/6.93 ( ( times_times_int @ one_one_int @ A )
% 12.85/6.93 = A ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1896_mult_Ocomm__neutral,axiom,
% 12.85/6.93 ! [A: complex] :
% 12.85/6.93 ( ( times_times_complex @ A @ one_one_complex )
% 12.85/6.93 = A ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1897_mult_Ocomm__neutral,axiom,
% 12.85/6.93 ! [A: real] :
% 12.85/6.93 ( ( times_times_real @ A @ one_one_real )
% 12.85/6.93 = A ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1898_mult_Ocomm__neutral,axiom,
% 12.85/6.93 ! [A: rat] :
% 12.85/6.93 ( ( times_times_rat @ A @ one_one_rat )
% 12.85/6.93 = A ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1899_mult_Ocomm__neutral,axiom,
% 12.85/6.93 ! [A: nat] :
% 12.85/6.93 ( ( times_times_nat @ A @ one_one_nat )
% 12.85/6.93 = A ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1900_mult_Ocomm__neutral,axiom,
% 12.85/6.93 ! [A: int] :
% 12.85/6.93 ( ( times_times_int @ A @ one_one_int )
% 12.85/6.93 = A ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1901_int__distrib_I4_J,axiom,
% 12.85/6.93 ! [W: int,Z1: int,Z22: int] :
% 12.85/6.93 ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
% 12.85/6.93 = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1902_int__distrib_I3_J,axiom,
% 12.85/6.93 ! [Z1: int,Z22: int,W: int] :
% 12.85/6.93 ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
% 12.85/6.93 = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1903_less__mono__imp__le__mono,axiom,
% 12.85/6.93 ! [F: nat > nat,I: nat,J: nat] :
% 12.85/6.93 ( ! [I3: nat,J2: nat] :
% 12.85/6.93 ( ( ord_less_nat @ I3 @ J2 )
% 12.85/6.93 => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
% 12.85/6.93 => ( ( ord_less_eq_nat @ I @ J )
% 12.85/6.93 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1904_le__neq__implies__less,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.93 => ( ( M != N )
% 12.85/6.93 => ( ord_less_nat @ M @ N ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1905_less__or__eq__imp__le,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ( ord_less_nat @ M @ N )
% 12.85/6.93 | ( M = N ) )
% 12.85/6.93 => ( ord_less_eq_nat @ M @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1906_le__eq__less__or__eq,axiom,
% 12.85/6.93 ( ord_less_eq_nat
% 12.85/6.93 = ( ^ [M5: nat,N4: nat] :
% 12.85/6.93 ( ( ord_less_nat @ M5 @ N4 )
% 12.85/6.93 | ( M5 = N4 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1907_less__imp__le__nat,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ M @ N )
% 12.85/6.93 => ( ord_less_eq_nat @ M @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1908_nat__less__le,axiom,
% 12.85/6.93 ( ord_less_nat
% 12.85/6.93 = ( ^ [M5: nat,N4: nat] :
% 12.85/6.93 ( ( ord_less_eq_nat @ M5 @ N4 )
% 12.85/6.93 & ( M5 != N4 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1909_int__diff__cases,axiom,
% 12.85/6.93 ! [Z: int] :
% 12.85/6.93 ~ ! [M4: nat,N2: nat] :
% 12.85/6.93 ( Z
% 12.85/6.93 != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1910_diff__less__mono2,axiom,
% 12.85/6.93 ! [M: nat,N: nat,L: nat] :
% 12.85/6.93 ( ( ord_less_nat @ M @ N )
% 12.85/6.93 => ( ( ord_less_nat @ M @ L )
% 12.85/6.93 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1911_less__imp__diff__less,axiom,
% 12.85/6.93 ! [J: nat,K: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ J @ K )
% 12.85/6.93 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1912_dvd__unit__imp__unit,axiom,
% 12.85/6.93 ! [A: code_integer,B: code_integer] :
% 12.85/6.93 ( ( dvd_dvd_Code_integer @ A @ B )
% 12.85/6.93 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 12.85/6.93 => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1913_dvd__unit__imp__unit,axiom,
% 12.85/6.93 ! [A: nat,B: nat] :
% 12.85/6.93 ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.93 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 12.85/6.93 => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1914_dvd__unit__imp__unit,axiom,
% 12.85/6.93 ! [A: int,B: int] :
% 12.85/6.93 ( ( dvd_dvd_int @ A @ B )
% 12.85/6.93 => ( ( dvd_dvd_int @ B @ one_one_int )
% 12.85/6.93 => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1915_unit__imp__dvd,axiom,
% 12.85/6.93 ! [B: code_integer,A: code_integer] :
% 12.85/6.93 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 12.85/6.93 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1916_unit__imp__dvd,axiom,
% 12.85/6.93 ! [B: nat,A: nat] :
% 12.85/6.93 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 12.85/6.93 => ( dvd_dvd_nat @ B @ A ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1917_unit__imp__dvd,axiom,
% 12.85/6.93 ! [B: int,A: int] :
% 12.85/6.93 ( ( dvd_dvd_int @ B @ one_one_int )
% 12.85/6.93 => ( dvd_dvd_int @ B @ A ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1918_one__dvd,axiom,
% 12.85/6.93 ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1919_one__dvd,axiom,
% 12.85/6.93 ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1920_one__dvd,axiom,
% 12.85/6.93 ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1921_one__dvd,axiom,
% 12.85/6.93 ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1922_one__dvd,axiom,
% 12.85/6.93 ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1923_one__dvd,axiom,
% 12.85/6.93 ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1924_zdvd__zdiffD,axiom,
% 12.85/6.93 ! [K: int,M: int,N: int] :
% 12.85/6.93 ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
% 12.85/6.93 => ( ( dvd_dvd_int @ K @ N )
% 12.85/6.93 => ( dvd_dvd_int @ K @ M ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1925_nat__mult__1__right,axiom,
% 12.85/6.93 ! [N: nat] :
% 12.85/6.93 ( ( times_times_nat @ N @ one_one_nat )
% 12.85/6.93 = N ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1926_nat__mult__1,axiom,
% 12.85/6.93 ! [N: nat] :
% 12.85/6.93 ( ( times_times_nat @ one_one_nat @ N )
% 12.85/6.93 = N ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1927_dvd__pos__nat,axiom,
% 12.85/6.93 ! [N: nat,M: nat] :
% 12.85/6.93 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.93 => ( ( dvd_dvd_nat @ M @ N )
% 12.85/6.93 => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1928_mult__le__cancel__left1,axiom,
% 12.85/6.93 ! [C: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
% 12.85/6.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 12.85/6.93 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1929_mult__le__cancel__left1,axiom,
% 12.85/6.93 ! [C: int,B: int] :
% 12.85/6.93 ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
% 12.85/6.93 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 12.85/6.93 => ( ord_less_eq_int @ one_one_int @ B ) )
% 12.85/6.93 & ( ( ord_less_int @ C @ zero_zero_int )
% 12.85/6.93 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1930_mult__le__cancel__left1,axiom,
% 12.85/6.93 ! [C: real,B: real] :
% 12.85/6.93 ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
% 12.85/6.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ord_less_eq_real @ one_one_real @ B ) )
% 12.85/6.93 & ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.93 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1931_mult__le__cancel__left2,axiom,
% 12.85/6.93 ! [C: rat,A: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 12.85/6.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 12.85/6.93 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1932_mult__le__cancel__left2,axiom,
% 12.85/6.93 ! [C: int,A: int] :
% 12.85/6.93 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 12.85/6.93 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 12.85/6.93 => ( ord_less_eq_int @ A @ one_one_int ) )
% 12.85/6.93 & ( ( ord_less_int @ C @ zero_zero_int )
% 12.85/6.93 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1933_mult__le__cancel__left2,axiom,
% 12.85/6.93 ! [C: real,A: real] :
% 12.85/6.93 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 12.85/6.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ord_less_eq_real @ A @ one_one_real ) )
% 12.85/6.93 & ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.93 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1934_mult__le__cancel__right1,axiom,
% 12.85/6.93 ! [C: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
% 12.85/6.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 12.85/6.93 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1935_mult__le__cancel__right1,axiom,
% 12.85/6.93 ! [C: int,B: int] :
% 12.85/6.93 ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
% 12.85/6.93 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 12.85/6.93 => ( ord_less_eq_int @ one_one_int @ B ) )
% 12.85/6.93 & ( ( ord_less_int @ C @ zero_zero_int )
% 12.85/6.93 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1936_mult__le__cancel__right1,axiom,
% 12.85/6.93 ! [C: real,B: real] :
% 12.85/6.93 ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
% 12.85/6.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ord_less_eq_real @ one_one_real @ B ) )
% 12.85/6.93 & ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.93 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1937_mult__le__cancel__right2,axiom,
% 12.85/6.93 ! [A: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 12.85/6.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 12.85/6.93 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1938_mult__le__cancel__right2,axiom,
% 12.85/6.93 ! [A: int,C: int] :
% 12.85/6.93 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 12.85/6.93 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 12.85/6.93 => ( ord_less_eq_int @ A @ one_one_int ) )
% 12.85/6.93 & ( ( ord_less_int @ C @ zero_zero_int )
% 12.85/6.93 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1939_mult__le__cancel__right2,axiom,
% 12.85/6.93 ! [A: real,C: real] :
% 12.85/6.93 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 12.85/6.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ord_less_eq_real @ A @ one_one_real ) )
% 12.85/6.93 & ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.93 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1940_mult__less__cancel__left1,axiom,
% 12.85/6.93 ! [C: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
% 12.85/6.93 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ord_less_rat @ one_one_rat @ B ) )
% 12.85/6.93 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1941_mult__less__cancel__left1,axiom,
% 12.85/6.93 ! [C: int,B: int] :
% 12.85/6.93 ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
% 12.85/6.93 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.93 => ( ord_less_int @ one_one_int @ B ) )
% 12.85/6.93 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 12.85/6.93 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1942_mult__less__cancel__left1,axiom,
% 12.85/6.93 ! [C: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
% 12.85/6.93 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ord_less_real @ one_one_real @ B ) )
% 12.85/6.93 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.85/6.93 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1943_mult__less__cancel__left2,axiom,
% 12.85/6.93 ! [C: rat,A: rat] :
% 12.85/6.93 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 12.85/6.93 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ord_less_rat @ A @ one_one_rat ) )
% 12.85/6.93 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1944_mult__less__cancel__left2,axiom,
% 12.85/6.93 ! [C: int,A: int] :
% 12.85/6.93 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 12.85/6.93 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.93 => ( ord_less_int @ A @ one_one_int ) )
% 12.85/6.93 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 12.85/6.93 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1945_mult__less__cancel__left2,axiom,
% 12.85/6.93 ! [C: real,A: real] :
% 12.85/6.93 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 12.85/6.93 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ord_less_real @ A @ one_one_real ) )
% 12.85/6.93 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.85/6.93 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1946_mult__less__cancel__right1,axiom,
% 12.85/6.93 ! [C: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
% 12.85/6.93 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ord_less_rat @ one_one_rat @ B ) )
% 12.85/6.93 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1947_mult__less__cancel__right1,axiom,
% 12.85/6.93 ! [C: int,B: int] :
% 12.85/6.93 ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
% 12.85/6.93 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.93 => ( ord_less_int @ one_one_int @ B ) )
% 12.85/6.93 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 12.85/6.93 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1948_mult__less__cancel__right1,axiom,
% 12.85/6.93 ! [C: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
% 12.85/6.93 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ord_less_real @ one_one_real @ B ) )
% 12.85/6.93 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.85/6.93 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1949_mult__less__cancel__right2,axiom,
% 12.85/6.93 ! [A: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 12.85/6.93 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ord_less_rat @ A @ one_one_rat ) )
% 12.85/6.93 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1950_mult__less__cancel__right2,axiom,
% 12.85/6.93 ! [A: int,C: int] :
% 12.85/6.93 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 12.85/6.93 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.93 => ( ord_less_int @ A @ one_one_int ) )
% 12.85/6.93 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 12.85/6.93 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1951_mult__less__cancel__right2,axiom,
% 12.85/6.93 ! [A: real,C: real] :
% 12.85/6.93 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 12.85/6.93 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ord_less_real @ A @ one_one_real ) )
% 12.85/6.93 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.85/6.93 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1952_field__le__mult__one__interval,axiom,
% 12.85/6.93 ! [X: rat,Y4: rat] :
% 12.85/6.93 ( ! [Z2: rat] :
% 12.85/6.93 ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 12.85/6.93 => ( ( ord_less_rat @ Z2 @ one_one_rat )
% 12.85/6.93 => ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X ) @ Y4 ) ) )
% 12.85/6.93 => ( ord_less_eq_rat @ X @ Y4 ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1953_field__le__mult__one__interval,axiom,
% 12.85/6.93 ! [X: real,Y4: real] :
% 12.85/6.93 ( ! [Z2: real] :
% 12.85/6.93 ( ( ord_less_real @ zero_zero_real @ Z2 )
% 12.85/6.93 => ( ( ord_less_real @ Z2 @ one_one_real )
% 12.85/6.93 => ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ Y4 ) ) )
% 12.85/6.93 => ( ord_less_eq_real @ X @ Y4 ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1954_divide__le__eq__1,axiom,
% 12.85/6.93 ! [B: rat,A: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 12.85/6.93 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.93 & ( ord_less_eq_rat @ B @ A ) )
% 12.85/6.93 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/6.93 & ( ord_less_eq_rat @ A @ B ) )
% 12.85/6.93 | ( A = zero_zero_rat ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1955_divide__le__eq__1,axiom,
% 12.85/6.93 ! [B: real,A: real] :
% 12.85/6.93 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 12.85/6.93 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.93 & ( ord_less_eq_real @ B @ A ) )
% 12.85/6.93 | ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/6.93 & ( ord_less_eq_real @ A @ B ) )
% 12.85/6.93 | ( A = zero_zero_real ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1956_le__divide__eq__1,axiom,
% 12.85/6.93 ! [B: rat,A: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 12.85/6.93 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.93 & ( ord_less_eq_rat @ A @ B ) )
% 12.85/6.93 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/6.93 & ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1957_le__divide__eq__1,axiom,
% 12.85/6.93 ! [B: real,A: real] :
% 12.85/6.93 ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 12.85/6.93 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.93 & ( ord_less_eq_real @ A @ B ) )
% 12.85/6.93 | ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/6.93 & ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1958_Suc__pred_H,axiom,
% 12.85/6.93 ! [N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.93 => ( N
% 12.85/6.93 = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1959_Suc__diff__eq__diff__pred,axiom,
% 12.85/6.93 ! [N: nat,M: nat] :
% 12.85/6.93 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.93 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 12.85/6.93 = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1960_dvd__mult__cancel2,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/6.93 => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
% 12.85/6.93 = ( N = one_one_nat ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1961_dvd__mult__cancel1,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/6.93 => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
% 12.85/6.93 = ( N = one_one_nat ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1962_of__nat__less__two__power,axiom,
% 12.85/6.93 ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1963_of__nat__less__two__power,axiom,
% 12.85/6.93 ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1964_of__nat__less__two__power,axiom,
% 12.85/6.93 ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1965_power__diff__power__eq,axiom,
% 12.85/6.93 ! [A: nat,N: nat,M: nat] :
% 12.85/6.93 ( ( A != zero_zero_nat )
% 12.85/6.93 => ( ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.93 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 12.85/6.93 = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 12.85/6.93 & ( ~ ( ord_less_eq_nat @ N @ M )
% 12.85/6.93 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 12.85/6.93 = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1966_power__diff__power__eq,axiom,
% 12.85/6.93 ! [A: int,N: nat,M: nat] :
% 12.85/6.93 ( ( A != zero_zero_int )
% 12.85/6.93 => ( ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.93 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 12.85/6.93 = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 12.85/6.93 & ( ~ ( ord_less_eq_nat @ N @ M )
% 12.85/6.93 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 12.85/6.93 = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1967_not__numeral__less__zero,axiom,
% 12.85/6.93 ! [N: num] :
% 12.85/6.93 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1968_not__numeral__less__zero,axiom,
% 12.85/6.93 ! [N: num] :
% 12.85/6.93 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1969_not__numeral__less__zero,axiom,
% 12.85/6.93 ! [N: num] :
% 12.85/6.93 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1970_not__numeral__less__zero,axiom,
% 12.85/6.93 ! [N: num] :
% 12.85/6.93 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1971_zero__less__numeral,axiom,
% 12.85/6.93 ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1972_zero__less__numeral,axiom,
% 12.85/6.93 ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1973_zero__less__numeral,axiom,
% 12.85/6.93 ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1974_zero__less__numeral,axiom,
% 12.85/6.93 ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1975_mult__neg__neg,axiom,
% 12.85/6.93 ! [A: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/6.93 => ( ( ord_less_real @ B @ zero_zero_real )
% 12.85/6.93 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1976_mult__neg__neg,axiom,
% 12.85/6.93 ! [A: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/6.93 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1977_mult__neg__neg,axiom,
% 12.85/6.93 ! [A: int,B: int] :
% 12.85/6.93 ( ( ord_less_int @ A @ zero_zero_int )
% 12.85/6.93 => ( ( ord_less_int @ B @ zero_zero_int )
% 12.85/6.93 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1978_not__square__less__zero,axiom,
% 12.85/6.93 ! [A: real] :
% 12.85/6.93 ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1979_not__square__less__zero,axiom,
% 12.85/6.93 ! [A: rat] :
% 12.85/6.93 ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1980_not__square__less__zero,axiom,
% 12.85/6.93 ! [A: int] :
% 12.85/6.93 ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1981_mult__less__0__iff,axiom,
% 12.85/6.93 ! [A: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 12.85/6.93 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.93 & ( ord_less_real @ B @ zero_zero_real ) )
% 12.85/6.93 | ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/6.93 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1982_mult__less__0__iff,axiom,
% 12.85/6.93 ! [A: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 12.85/6.93 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.93 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 12.85/6.93 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/6.93 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1983_mult__less__0__iff,axiom,
% 12.85/6.93 ! [A: int,B: int] :
% 12.85/6.93 ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 12.85/6.93 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 12.85/6.93 & ( ord_less_int @ B @ zero_zero_int ) )
% 12.85/6.93 | ( ( ord_less_int @ A @ zero_zero_int )
% 12.85/6.93 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1984_mult__neg__pos,axiom,
% 12.85/6.93 ! [A: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/6.93 => ( ( ord_less_real @ zero_zero_real @ B )
% 12.85/6.93 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1985_mult__neg__pos,axiom,
% 12.85/6.93 ! [A: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/6.93 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 12.85/6.93 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1986_mult__neg__pos,axiom,
% 12.85/6.93 ! [A: nat,B: nat] :
% 12.85/6.93 ( ( ord_less_nat @ A @ zero_zero_nat )
% 12.85/6.93 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 12.85/6.93 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1987_mult__neg__pos,axiom,
% 12.85/6.93 ! [A: int,B: int] :
% 12.85/6.93 ( ( ord_less_int @ A @ zero_zero_int )
% 12.85/6.93 => ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.93 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1988_mult__pos__neg,axiom,
% 12.85/6.93 ! [A: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.93 => ( ( ord_less_real @ B @ zero_zero_real )
% 12.85/6.93 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1989_mult__pos__neg,axiom,
% 12.85/6.93 ! [A: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.93 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1990_mult__pos__neg,axiom,
% 12.85/6.93 ! [A: nat,B: nat] :
% 12.85/6.93 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.85/6.93 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 12.85/6.93 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1991_mult__pos__neg,axiom,
% 12.85/6.93 ! [A: int,B: int] :
% 12.85/6.93 ( ( ord_less_int @ zero_zero_int @ A )
% 12.85/6.93 => ( ( ord_less_int @ B @ zero_zero_int )
% 12.85/6.93 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1992_mult__pos__pos,axiom,
% 12.85/6.93 ! [A: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.93 => ( ( ord_less_real @ zero_zero_real @ B )
% 12.85/6.93 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1993_mult__pos__pos,axiom,
% 12.85/6.93 ! [A: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.93 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 12.85/6.93 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1994_mult__pos__pos,axiom,
% 12.85/6.93 ! [A: nat,B: nat] :
% 12.85/6.93 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.85/6.93 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 12.85/6.93 => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1995_mult__pos__pos,axiom,
% 12.85/6.93 ! [A: int,B: int] :
% 12.85/6.93 ( ( ord_less_int @ zero_zero_int @ A )
% 12.85/6.93 => ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.93 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1996_mult__pos__neg2,axiom,
% 12.85/6.93 ! [A: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.93 => ( ( ord_less_real @ B @ zero_zero_real )
% 12.85/6.93 => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1997_mult__pos__neg2,axiom,
% 12.85/6.93 ! [A: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.93 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1998_mult__pos__neg2,axiom,
% 12.85/6.93 ! [A: nat,B: nat] :
% 12.85/6.93 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.85/6.93 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 12.85/6.93 => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_1999_mult__pos__neg2,axiom,
% 12.85/6.93 ! [A: int,B: int] :
% 12.85/6.93 ( ( ord_less_int @ zero_zero_int @ A )
% 12.85/6.93 => ( ( ord_less_int @ B @ zero_zero_int )
% 12.85/6.93 => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2000_zero__less__mult__iff,axiom,
% 12.85/6.93 ! [A: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 12.85/6.93 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.93 & ( ord_less_real @ zero_zero_real @ B ) )
% 12.85/6.93 | ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/6.93 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2001_zero__less__mult__iff,axiom,
% 12.85/6.93 ! [A: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 12.85/6.93 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.93 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 12.85/6.93 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/6.93 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2002_zero__less__mult__iff,axiom,
% 12.85/6.93 ! [A: int,B: int] :
% 12.85/6.93 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 12.85/6.93 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 12.85/6.93 & ( ord_less_int @ zero_zero_int @ B ) )
% 12.85/6.93 | ( ( ord_less_int @ A @ zero_zero_int )
% 12.85/6.93 & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2003_zero__less__mult__pos,axiom,
% 12.85/6.93 ! [A: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 12.85/6.93 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.93 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2004_zero__less__mult__pos,axiom,
% 12.85/6.93 ! [A: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 12.85/6.93 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.93 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2005_zero__less__mult__pos,axiom,
% 12.85/6.93 ! [A: nat,B: nat] :
% 12.85/6.93 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
% 12.85/6.93 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.85/6.93 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2006_zero__less__mult__pos,axiom,
% 12.85/6.93 ! [A: int,B: int] :
% 12.85/6.93 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 12.85/6.93 => ( ( ord_less_int @ zero_zero_int @ A )
% 12.85/6.93 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2007_zero__less__mult__pos2,axiom,
% 12.85/6.93 ! [B: real,A: real] :
% 12.85/6.93 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
% 12.85/6.93 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.93 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2008_zero__less__mult__pos2,axiom,
% 12.85/6.93 ! [B: rat,A: rat] :
% 12.85/6.93 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
% 12.85/6.93 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.93 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2009_zero__less__mult__pos2,axiom,
% 12.85/6.93 ! [B: nat,A: nat] :
% 12.85/6.93 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
% 12.85/6.93 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.85/6.93 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2010_zero__less__mult__pos2,axiom,
% 12.85/6.93 ! [B: int,A: int] :
% 12.85/6.93 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
% 12.85/6.93 => ( ( ord_less_int @ zero_zero_int @ A )
% 12.85/6.93 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2011_mult__less__cancel__left__neg,axiom,
% 12.85/6.93 ! [C: real,A: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.93 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 12.85/6.93 = ( ord_less_real @ B @ A ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2012_mult__less__cancel__left__neg,axiom,
% 12.85/6.93 ! [C: rat,A: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.93 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 12.85/6.93 = ( ord_less_rat @ B @ A ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2013_mult__less__cancel__left__neg,axiom,
% 12.85/6.93 ! [C: int,A: int,B: int] :
% 12.85/6.93 ( ( ord_less_int @ C @ zero_zero_int )
% 12.85/6.93 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 12.85/6.93 = ( ord_less_int @ B @ A ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2014_mult__less__cancel__left__pos,axiom,
% 12.85/6.93 ! [C: real,A: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 12.85/6.93 = ( ord_less_real @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2015_mult__less__cancel__left__pos,axiom,
% 12.85/6.93 ! [C: rat,A: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 12.85/6.93 = ( ord_less_rat @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2016_mult__less__cancel__left__pos,axiom,
% 12.85/6.93 ! [C: int,A: int,B: int] :
% 12.85/6.93 ( ( ord_less_int @ zero_zero_int @ C )
% 12.85/6.93 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 12.85/6.93 = ( ord_less_int @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2017_mult__strict__left__mono__neg,axiom,
% 12.85/6.93 ! [B: real,A: real,C: real] :
% 12.85/6.93 ( ( ord_less_real @ B @ A )
% 12.85/6.93 => ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.93 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2018_mult__strict__left__mono__neg,axiom,
% 12.85/6.93 ! [B: rat,A: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_rat @ B @ A )
% 12.85/6.93 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2019_mult__strict__left__mono__neg,axiom,
% 12.85/6.93 ! [B: int,A: int,C: int] :
% 12.85/6.93 ( ( ord_less_int @ B @ A )
% 12.85/6.93 => ( ( ord_less_int @ C @ zero_zero_int )
% 12.85/6.93 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2020_mult__strict__left__mono,axiom,
% 12.85/6.93 ! [A: real,B: real,C: real] :
% 12.85/6.93 ( ( ord_less_real @ A @ B )
% 12.85/6.93 => ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2021_mult__strict__left__mono,axiom,
% 12.85/6.93 ! [A: rat,B: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_rat @ A @ B )
% 12.85/6.93 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2022_mult__strict__left__mono,axiom,
% 12.85/6.93 ! [A: nat,B: nat,C: nat] :
% 12.85/6.93 ( ( ord_less_nat @ A @ B )
% 12.85/6.93 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 12.85/6.93 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2023_mult__strict__left__mono,axiom,
% 12.85/6.93 ! [A: int,B: int,C: int] :
% 12.85/6.93 ( ( ord_less_int @ A @ B )
% 12.85/6.93 => ( ( ord_less_int @ zero_zero_int @ C )
% 12.85/6.93 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2024_mult__less__cancel__left__disj,axiom,
% 12.85/6.93 ! [C: real,A: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 12.85/6.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.93 & ( ord_less_real @ A @ B ) )
% 12.85/6.93 | ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.93 & ( ord_less_real @ B @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2025_mult__less__cancel__left__disj,axiom,
% 12.85/6.93 ! [C: rat,A: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 12.85/6.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.93 & ( ord_less_rat @ A @ B ) )
% 12.85/6.93 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.93 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2026_mult__less__cancel__left__disj,axiom,
% 12.85/6.93 ! [C: int,A: int,B: int] :
% 12.85/6.93 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 12.85/6.93 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 12.85/6.93 & ( ord_less_int @ A @ B ) )
% 12.85/6.93 | ( ( ord_less_int @ C @ zero_zero_int )
% 12.85/6.93 & ( ord_less_int @ B @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2027_mult__strict__right__mono__neg,axiom,
% 12.85/6.93 ! [B: real,A: real,C: real] :
% 12.85/6.93 ( ( ord_less_real @ B @ A )
% 12.85/6.93 => ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.93 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2028_mult__strict__right__mono__neg,axiom,
% 12.85/6.93 ! [B: rat,A: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_rat @ B @ A )
% 12.85/6.93 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2029_mult__strict__right__mono__neg,axiom,
% 12.85/6.93 ! [B: int,A: int,C: int] :
% 12.85/6.93 ( ( ord_less_int @ B @ A )
% 12.85/6.93 => ( ( ord_less_int @ C @ zero_zero_int )
% 12.85/6.93 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2030_mult__strict__right__mono,axiom,
% 12.85/6.93 ! [A: real,B: real,C: real] :
% 12.85/6.93 ( ( ord_less_real @ A @ B )
% 12.85/6.93 => ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2031_mult__strict__right__mono,axiom,
% 12.85/6.93 ! [A: rat,B: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_rat @ A @ B )
% 12.85/6.93 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2032_mult__strict__right__mono,axiom,
% 12.85/6.93 ! [A: nat,B: nat,C: nat] :
% 12.85/6.93 ( ( ord_less_nat @ A @ B )
% 12.85/6.93 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 12.85/6.93 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2033_mult__strict__right__mono,axiom,
% 12.85/6.93 ! [A: int,B: int,C: int] :
% 12.85/6.93 ( ( ord_less_int @ A @ B )
% 12.85/6.93 => ( ( ord_less_int @ zero_zero_int @ C )
% 12.85/6.93 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2034_mult__less__cancel__right__disj,axiom,
% 12.85/6.93 ! [A: real,C: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 12.85/6.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.93 & ( ord_less_real @ A @ B ) )
% 12.85/6.93 | ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.93 & ( ord_less_real @ B @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2035_mult__less__cancel__right__disj,axiom,
% 12.85/6.93 ! [A: rat,C: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 12.85/6.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.93 & ( ord_less_rat @ A @ B ) )
% 12.85/6.93 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.93 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2036_mult__less__cancel__right__disj,axiom,
% 12.85/6.93 ! [A: int,C: int,B: int] :
% 12.85/6.93 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 12.85/6.93 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 12.85/6.93 & ( ord_less_int @ A @ B ) )
% 12.85/6.93 | ( ( ord_less_int @ C @ zero_zero_int )
% 12.85/6.93 & ( ord_less_int @ B @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2037_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 12.85/6.93 ! [A: real,B: real,C: real] :
% 12.85/6.93 ( ( ord_less_real @ A @ B )
% 12.85/6.93 => ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2038_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 12.85/6.93 ! [A: rat,B: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_rat @ A @ B )
% 12.85/6.93 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2039_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 12.85/6.93 ! [A: nat,B: nat,C: nat] :
% 12.85/6.93 ( ( ord_less_nat @ A @ B )
% 12.85/6.93 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 12.85/6.93 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2040_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 12.85/6.93 ! [A: int,B: int,C: int] :
% 12.85/6.93 ( ( ord_less_int @ A @ B )
% 12.85/6.93 => ( ( ord_less_int @ zero_zero_int @ C )
% 12.85/6.93 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2041_less__iff__diff__less__0,axiom,
% 12.85/6.93 ( ord_less_real
% 12.85/6.93 = ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2042_less__iff__diff__less__0,axiom,
% 12.85/6.93 ( ord_less_rat
% 12.85/6.93 = ( ^ [A3: rat,B2: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B2 ) @ zero_zero_rat ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2043_less__iff__diff__less__0,axiom,
% 12.85/6.93 ( ord_less_int
% 12.85/6.93 = ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2044_divide__neg__neg,axiom,
% 12.85/6.93 ! [X: real,Y4: real] :
% 12.85/6.93 ( ( ord_less_real @ X @ zero_zero_real )
% 12.85/6.93 => ( ( ord_less_real @ Y4 @ zero_zero_real )
% 12.85/6.93 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2045_divide__neg__neg,axiom,
% 12.85/6.93 ! [X: rat,Y4: rat] :
% 12.85/6.93 ( ( ord_less_rat @ X @ zero_zero_rat )
% 12.85/6.93 => ( ( ord_less_rat @ Y4 @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2046_divide__neg__pos,axiom,
% 12.85/6.93 ! [X: real,Y4: real] :
% 12.85/6.93 ( ( ord_less_real @ X @ zero_zero_real )
% 12.85/6.93 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/6.93 => ( ord_less_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2047_divide__neg__pos,axiom,
% 12.85/6.93 ! [X: rat,Y4: rat] :
% 12.85/6.93 ( ( ord_less_rat @ X @ zero_zero_rat )
% 12.85/6.93 => ( ( ord_less_rat @ zero_zero_rat @ Y4 )
% 12.85/6.93 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y4 ) @ zero_zero_rat ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2048_divide__pos__neg,axiom,
% 12.85/6.93 ! [X: real,Y4: real] :
% 12.85/6.93 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.93 => ( ( ord_less_real @ Y4 @ zero_zero_real )
% 12.85/6.93 => ( ord_less_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2049_divide__pos__neg,axiom,
% 12.85/6.93 ! [X: rat,Y4: rat] :
% 12.85/6.93 ( ( ord_less_rat @ zero_zero_rat @ X )
% 12.85/6.93 => ( ( ord_less_rat @ Y4 @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y4 ) @ zero_zero_rat ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2050_divide__pos__pos,axiom,
% 12.85/6.93 ! [X: real,Y4: real] :
% 12.85/6.93 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.93 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/6.93 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2051_divide__pos__pos,axiom,
% 12.85/6.93 ! [X: rat,Y4: rat] :
% 12.85/6.93 ( ( ord_less_rat @ zero_zero_rat @ X )
% 12.85/6.93 => ( ( ord_less_rat @ zero_zero_rat @ Y4 )
% 12.85/6.93 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2052_divide__less__0__iff,axiom,
% 12.85/6.93 ! [A: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 12.85/6.93 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.93 & ( ord_less_real @ B @ zero_zero_real ) )
% 12.85/6.93 | ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/6.93 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2053_divide__less__0__iff,axiom,
% 12.85/6.93 ! [A: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 12.85/6.93 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.93 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 12.85/6.93 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/6.93 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2054_divide__less__cancel,axiom,
% 12.85/6.93 ! [A: real,C: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 12.85/6.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ord_less_real @ A @ B ) )
% 12.85/6.93 & ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.93 => ( ord_less_real @ B @ A ) )
% 12.85/6.93 & ( C != zero_zero_real ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2055_divide__less__cancel,axiom,
% 12.85/6.93 ! [A: rat,C: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 12.85/6.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ord_less_rat @ A @ B ) )
% 12.85/6.93 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_rat @ B @ A ) )
% 12.85/6.93 & ( C != zero_zero_rat ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2056_zero__less__divide__iff,axiom,
% 12.85/6.93 ! [A: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 12.85/6.93 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.93 & ( ord_less_real @ zero_zero_real @ B ) )
% 12.85/6.93 | ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/6.93 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2057_zero__less__divide__iff,axiom,
% 12.85/6.93 ! [A: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 12.85/6.93 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.93 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 12.85/6.93 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/6.93 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2058_divide__strict__right__mono,axiom,
% 12.85/6.93 ! [A: real,B: real,C: real] :
% 12.85/6.93 ( ( ord_less_real @ A @ B )
% 12.85/6.93 => ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2059_divide__strict__right__mono,axiom,
% 12.85/6.93 ! [A: rat,B: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_rat @ A @ B )
% 12.85/6.93 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2060_divide__strict__right__mono__neg,axiom,
% 12.85/6.93 ! [B: real,A: real,C: real] :
% 12.85/6.93 ( ( ord_less_real @ B @ A )
% 12.85/6.93 => ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.93 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2061_divide__strict__right__mono__neg,axiom,
% 12.85/6.93 ! [B: rat,A: rat,C: rat] :
% 12.85/6.93 ( ( ord_less_rat @ B @ A )
% 12.85/6.93 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2062_of__nat__less__0__iff,axiom,
% 12.85/6.93 ! [M: nat] :
% 12.85/6.93 ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2063_of__nat__less__0__iff,axiom,
% 12.85/6.93 ! [M: nat] :
% 12.85/6.93 ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2064_of__nat__less__0__iff,axiom,
% 12.85/6.93 ! [M: nat] :
% 12.85/6.93 ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2065_of__nat__less__0__iff,axiom,
% 12.85/6.93 ! [M: nat] :
% 12.85/6.93 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2066_zero__less__one__class_Ozero__le__one,axiom,
% 12.85/6.93 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2067_zero__less__one__class_Ozero__le__one,axiom,
% 12.85/6.93 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2068_zero__less__one__class_Ozero__le__one,axiom,
% 12.85/6.93 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2069_zero__less__one__class_Ozero__le__one,axiom,
% 12.85/6.93 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2070_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 12.85/6.93 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2071_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 12.85/6.93 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2072_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 12.85/6.93 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2073_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 12.85/6.93 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2074_not__one__le__zero,axiom,
% 12.85/6.93 ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2075_not__one__le__zero,axiom,
% 12.85/6.93 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2076_not__one__le__zero,axiom,
% 12.85/6.93 ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2077_not__one__le__zero,axiom,
% 12.85/6.93 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2078_one__le__numeral,axiom,
% 12.85/6.93 ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2079_one__le__numeral,axiom,
% 12.85/6.93 ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2080_one__le__numeral,axiom,
% 12.85/6.93 ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2081_one__le__numeral,axiom,
% 12.85/6.93 ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2082_less__Suc__eq__0__disj,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 12.85/6.93 = ( ( M = zero_zero_nat )
% 12.85/6.93 | ? [J3: nat] :
% 12.85/6.93 ( ( M
% 12.85/6.93 = ( suc @ J3 ) )
% 12.85/6.93 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2083_gr0__implies__Suc,axiom,
% 12.85/6.93 ! [N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.93 => ? [M4: nat] :
% 12.85/6.93 ( N
% 12.85/6.93 = ( suc @ M4 ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2084_All__less__Suc2,axiom,
% 12.85/6.93 ! [N: nat,P: nat > $o] :
% 12.85/6.93 ( ( ! [I2: nat] :
% 12.85/6.93 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 12.85/6.93 => ( P @ I2 ) ) )
% 12.85/6.93 = ( ( P @ zero_zero_nat )
% 12.85/6.93 & ! [I2: nat] :
% 12.85/6.93 ( ( ord_less_nat @ I2 @ N )
% 12.85/6.93 => ( P @ ( suc @ I2 ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2085_gr0__conv__Suc,axiom,
% 12.85/6.93 ! [N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.93 = ( ? [M5: nat] :
% 12.85/6.93 ( N
% 12.85/6.93 = ( suc @ M5 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2086_Ex__less__Suc2,axiom,
% 12.85/6.93 ! [N: nat,P: nat > $o] :
% 12.85/6.93 ( ( ? [I2: nat] :
% 12.85/6.93 ( ( ord_less_nat @ I2 @ ( suc @ N ) )
% 12.85/6.93 & ( P @ I2 ) ) )
% 12.85/6.93 = ( ( P @ zero_zero_nat )
% 12.85/6.93 | ? [I2: nat] :
% 12.85/6.93 ( ( ord_less_nat @ I2 @ N )
% 12.85/6.93 & ( P @ ( suc @ I2 ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2087_le__imp__less__Suc,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.93 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2088_less__eq__Suc__le,axiom,
% 12.85/6.93 ( ord_less_nat
% 12.85/6.93 = ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2089_less__Suc__eq__le,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 12.85/6.93 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2090_le__less__Suc__eq,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.93 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 12.85/6.93 = ( N = M ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2091_Suc__le__lessD,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 12.85/6.93 => ( ord_less_nat @ M @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2092_inc__induct,axiom,
% 12.85/6.93 ! [I: nat,J: nat,P: nat > $o] :
% 12.85/6.93 ( ( ord_less_eq_nat @ I @ J )
% 12.85/6.93 => ( ( P @ J )
% 12.85/6.93 => ( ! [N2: nat] :
% 12.85/6.93 ( ( ord_less_eq_nat @ I @ N2 )
% 12.85/6.93 => ( ( ord_less_nat @ N2 @ J )
% 12.85/6.93 => ( ( P @ ( suc @ N2 ) )
% 12.85/6.93 => ( P @ N2 ) ) ) )
% 12.85/6.93 => ( P @ I ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2093_dec__induct,axiom,
% 12.85/6.93 ! [I: nat,J: nat,P: nat > $o] :
% 12.85/6.93 ( ( ord_less_eq_nat @ I @ J )
% 12.85/6.93 => ( ( P @ I )
% 12.85/6.93 => ( ! [N2: nat] :
% 12.85/6.93 ( ( ord_less_eq_nat @ I @ N2 )
% 12.85/6.93 => ( ( ord_less_nat @ N2 @ J )
% 12.85/6.93 => ( ( P @ N2 )
% 12.85/6.93 => ( P @ ( suc @ N2 ) ) ) ) )
% 12.85/6.93 => ( P @ J ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2094_Suc__le__eq,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 12.85/6.93 = ( ord_less_nat @ M @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2095_Suc__leI,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ M @ N )
% 12.85/6.93 => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2096_right__inverse__eq,axiom,
% 12.85/6.93 ! [B: complex,A: complex] :
% 12.85/6.93 ( ( B != zero_zero_complex )
% 12.85/6.93 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 12.85/6.93 = one_one_complex )
% 12.85/6.93 = ( A = B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2097_right__inverse__eq,axiom,
% 12.85/6.93 ! [B: real,A: real] :
% 12.85/6.93 ( ( B != zero_zero_real )
% 12.85/6.93 => ( ( ( divide_divide_real @ A @ B )
% 12.85/6.93 = one_one_real )
% 12.85/6.93 = ( A = B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2098_right__inverse__eq,axiom,
% 12.85/6.93 ! [B: rat,A: rat] :
% 12.85/6.93 ( ( B != zero_zero_rat )
% 12.85/6.93 => ( ( ( divide_divide_rat @ A @ B )
% 12.85/6.93 = one_one_rat )
% 12.85/6.93 = ( A = B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2099_numeral__One,axiom,
% 12.85/6.93 ( ( numera6690914467698888265omplex @ one )
% 12.85/6.93 = one_one_complex ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2100_numeral__One,axiom,
% 12.85/6.93 ( ( numeral_numeral_real @ one )
% 12.85/6.93 = one_one_real ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2101_numeral__One,axiom,
% 12.85/6.93 ( ( numeral_numeral_rat @ one )
% 12.85/6.93 = one_one_rat ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2102_numeral__One,axiom,
% 12.85/6.93 ( ( numeral_numeral_nat @ one )
% 12.85/6.93 = one_one_nat ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2103_numeral__One,axiom,
% 12.85/6.93 ( ( numeral_numeral_int @ one )
% 12.85/6.93 = one_one_int ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2104_ex__least__nat__le,axiom,
% 12.85/6.93 ! [P: nat > $o,N: nat] :
% 12.85/6.93 ( ( P @ N )
% 12.85/6.93 => ( ~ ( P @ zero_zero_nat )
% 12.85/6.93 => ? [K2: nat] :
% 12.85/6.93 ( ( ord_less_eq_nat @ K2 @ N )
% 12.85/6.93 & ! [I4: nat] :
% 12.85/6.93 ( ( ord_less_nat @ I4 @ K2 )
% 12.85/6.93 => ~ ( P @ I4 ) )
% 12.85/6.93 & ( P @ K2 ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2105_not__is__unit__0,axiom,
% 12.85/6.93 ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2106_not__is__unit__0,axiom,
% 12.85/6.93 ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2107_not__is__unit__0,axiom,
% 12.85/6.93 ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2108_diff__less__Suc,axiom,
% 12.85/6.93 ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2109_Suc__diff__Suc,axiom,
% 12.85/6.93 ! [N: nat,M: nat] :
% 12.85/6.93 ( ( ord_less_nat @ N @ M )
% 12.85/6.93 => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 12.85/6.93 = ( minus_minus_nat @ M @ N ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2110_Suc__mult__less__cancel1,axiom,
% 12.85/6.93 ! [K: nat,M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 12.85/6.93 = ( ord_less_nat @ M @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2111_diff__less,axiom,
% 12.85/6.93 ! [N: nat,M: nat] :
% 12.85/6.93 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.93 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/6.93 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2112_nat__mult__less__cancel1,axiom,
% 12.85/6.93 ! [K: nat,M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/6.93 => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.85/6.93 = ( ord_less_nat @ M @ N ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2113_nat__mult__eq__cancel1,axiom,
% 12.85/6.93 ! [K: nat,M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/6.93 => ( ( ( times_times_nat @ K @ M )
% 12.85/6.93 = ( times_times_nat @ K @ N ) )
% 12.85/6.93 = ( M = N ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2114_mult__less__mono2,axiom,
% 12.85/6.93 ! [I: nat,J: nat,K: nat] :
% 12.85/6.93 ( ( ord_less_nat @ I @ J )
% 12.85/6.93 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/6.93 => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2115_mult__less__mono1,axiom,
% 12.85/6.93 ! [I: nat,J: nat,K: nat] :
% 12.85/6.93 ( ( ord_less_nat @ I @ J )
% 12.85/6.93 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/6.93 => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2116_is__unit__mult__iff,axiom,
% 12.85/6.93 ! [A: code_integer,B: code_integer] :
% 12.85/6.93 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
% 12.85/6.93 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.85/6.93 & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2117_is__unit__mult__iff,axiom,
% 12.85/6.93 ! [A: nat,B: nat] :
% 12.85/6.93 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
% 12.85/6.93 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.85/6.93 & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2118_is__unit__mult__iff,axiom,
% 12.85/6.93 ! [A: int,B: int] :
% 12.85/6.93 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
% 12.85/6.93 = ( ( dvd_dvd_int @ A @ one_one_int )
% 12.85/6.93 & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2119_dvd__mult__unit__iff,axiom,
% 12.85/6.93 ! [B: code_integer,A: code_integer,C: code_integer] :
% 12.85/6.93 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 12.85/6.93 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
% 12.85/6.93 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2120_dvd__mult__unit__iff,axiom,
% 12.85/6.93 ! [B: nat,A: nat,C: nat] :
% 12.85/6.93 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 12.85/6.93 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
% 12.85/6.93 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2121_dvd__mult__unit__iff,axiom,
% 12.85/6.93 ! [B: int,A: int,C: int] :
% 12.85/6.93 ( ( dvd_dvd_int @ B @ one_one_int )
% 12.85/6.93 => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
% 12.85/6.93 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2122_mult__unit__dvd__iff,axiom,
% 12.85/6.93 ! [B: code_integer,A: code_integer,C: code_integer] :
% 12.85/6.93 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 12.85/6.93 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 12.85/6.93 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2123_mult__unit__dvd__iff,axiom,
% 12.85/6.93 ! [B: nat,A: nat,C: nat] :
% 12.85/6.93 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 12.85/6.93 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 12.85/6.93 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2124_mult__unit__dvd__iff,axiom,
% 12.85/6.93 ! [B: int,A: int,C: int] :
% 12.85/6.93 ( ( dvd_dvd_int @ B @ one_one_int )
% 12.85/6.93 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 12.85/6.93 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2125_dvd__mult__unit__iff_H,axiom,
% 12.85/6.93 ! [B: code_integer,A: code_integer,C: code_integer] :
% 12.85/6.93 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 12.85/6.93 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 12.85/6.93 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2126_dvd__mult__unit__iff_H,axiom,
% 12.85/6.93 ! [B: nat,A: nat,C: nat] :
% 12.85/6.93 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 12.85/6.93 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 12.85/6.93 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2127_dvd__mult__unit__iff_H,axiom,
% 12.85/6.93 ! [B: int,A: int,C: int] :
% 12.85/6.93 ( ( dvd_dvd_int @ B @ one_one_int )
% 12.85/6.93 => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 12.85/6.93 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2128_mult__unit__dvd__iff_H,axiom,
% 12.85/6.93 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.93 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.85/6.93 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 12.85/6.93 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2129_mult__unit__dvd__iff_H,axiom,
% 12.85/6.93 ! [A: nat,B: nat,C: nat] :
% 12.85/6.93 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.85/6.93 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 12.85/6.93 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2130_mult__unit__dvd__iff_H,axiom,
% 12.85/6.93 ! [A: int,B: int,C: int] :
% 12.85/6.93 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.85/6.93 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 12.85/6.93 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2131_unit__mult__left__cancel,axiom,
% 12.85/6.93 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.93 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.85/6.93 => ( ( ( times_3573771949741848930nteger @ A @ B )
% 12.85/6.93 = ( times_3573771949741848930nteger @ A @ C ) )
% 12.85/6.93 = ( B = C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2132_unit__mult__left__cancel,axiom,
% 12.85/6.93 ! [A: nat,B: nat,C: nat] :
% 12.85/6.93 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.85/6.93 => ( ( ( times_times_nat @ A @ B )
% 12.85/6.93 = ( times_times_nat @ A @ C ) )
% 12.85/6.93 = ( B = C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2133_unit__mult__left__cancel,axiom,
% 12.85/6.93 ! [A: int,B: int,C: int] :
% 12.85/6.93 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.85/6.93 => ( ( ( times_times_int @ A @ B )
% 12.85/6.93 = ( times_times_int @ A @ C ) )
% 12.85/6.93 = ( B = C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2134_unit__mult__right__cancel,axiom,
% 12.85/6.93 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.93 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.85/6.93 => ( ( ( times_3573771949741848930nteger @ B @ A )
% 12.85/6.93 = ( times_3573771949741848930nteger @ C @ A ) )
% 12.85/6.93 = ( B = C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2135_unit__mult__right__cancel,axiom,
% 12.85/6.93 ! [A: nat,B: nat,C: nat] :
% 12.85/6.93 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.85/6.93 => ( ( ( times_times_nat @ B @ A )
% 12.85/6.93 = ( times_times_nat @ C @ A ) )
% 12.85/6.93 = ( B = C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2136_unit__mult__right__cancel,axiom,
% 12.85/6.93 ! [A: int,B: int,C: int] :
% 12.85/6.93 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.85/6.93 => ( ( ( times_times_int @ B @ A )
% 12.85/6.93 = ( times_times_int @ C @ A ) )
% 12.85/6.93 = ( B = C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2137_diff__less__mono,axiom,
% 12.85/6.93 ! [A: nat,B: nat,C: nat] :
% 12.85/6.93 ( ( ord_less_nat @ A @ B )
% 12.85/6.93 => ( ( ord_less_eq_nat @ C @ A )
% 12.85/6.93 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2138_less__diff__iff,axiom,
% 12.85/6.93 ! [K: nat,M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_eq_nat @ K @ M )
% 12.85/6.93 => ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.93 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 12.85/6.93 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2139_nat__dvd__not__less,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/6.93 => ( ( ord_less_nat @ M @ N )
% 12.85/6.93 => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2140_unit__div__cancel,axiom,
% 12.85/6.93 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.93 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.85/6.93 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 12.85/6.93 = ( divide6298287555418463151nteger @ C @ A ) )
% 12.85/6.93 = ( B = C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2141_unit__div__cancel,axiom,
% 12.85/6.93 ! [A: nat,B: nat,C: nat] :
% 12.85/6.93 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.85/6.93 => ( ( ( divide_divide_nat @ B @ A )
% 12.85/6.93 = ( divide_divide_nat @ C @ A ) )
% 12.85/6.93 = ( B = C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2142_unit__div__cancel,axiom,
% 12.85/6.93 ! [A: int,B: int,C: int] :
% 12.85/6.93 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.85/6.93 => ( ( ( divide_divide_int @ B @ A )
% 12.85/6.93 = ( divide_divide_int @ C @ A ) )
% 12.85/6.93 = ( B = C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2143_div__unit__dvd__iff,axiom,
% 12.85/6.93 ! [B: code_integer,A: code_integer,C: code_integer] :
% 12.85/6.93 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 12.85/6.93 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 12.85/6.93 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2144_div__unit__dvd__iff,axiom,
% 12.85/6.93 ! [B: nat,A: nat,C: nat] :
% 12.85/6.93 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 12.85/6.93 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 12.85/6.93 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2145_div__unit__dvd__iff,axiom,
% 12.85/6.93 ! [B: int,A: int,C: int] :
% 12.85/6.93 ( ( dvd_dvd_int @ B @ one_one_int )
% 12.85/6.93 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 12.85/6.93 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2146_dvd__div__unit__iff,axiom,
% 12.85/6.93 ! [B: code_integer,A: code_integer,C: code_integer] :
% 12.85/6.93 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 12.85/6.93 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
% 12.85/6.93 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2147_dvd__div__unit__iff,axiom,
% 12.85/6.93 ! [B: nat,A: nat,C: nat] :
% 12.85/6.93 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 12.85/6.93 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
% 12.85/6.93 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2148_dvd__div__unit__iff,axiom,
% 12.85/6.93 ! [B: int,A: int,C: int] :
% 12.85/6.93 ( ( dvd_dvd_int @ B @ one_one_int )
% 12.85/6.93 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
% 12.85/6.93 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2149_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( ( divide_divide_nat @ M @ N )
% 12.85/6.93 = zero_zero_nat )
% 12.85/6.93 = ( ( ord_less_nat @ M @ N )
% 12.85/6.93 | ( N = zero_zero_nat ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2150_numerals_I1_J,axiom,
% 12.85/6.93 ( ( numeral_numeral_nat @ one )
% 12.85/6.93 = one_one_nat ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2151_One__nat__def,axiom,
% 12.85/6.93 ( one_one_nat
% 12.85/6.93 = ( suc @ zero_zero_nat ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2152_dvd__minus__self,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
% 12.85/6.93 = ( ( ord_less_nat @ N @ M )
% 12.85/6.93 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2153_less__mult__imp__div__less,axiom,
% 12.85/6.93 ! [M: nat,I: nat,N: nat] :
% 12.85/6.93 ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
% 12.85/6.93 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2154_diff__Suc__eq__diff__pred,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 12.85/6.93 = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2155_mult__eq__self__implies__10,axiom,
% 12.85/6.93 ! [M: nat,N: nat] :
% 12.85/6.93 ( ( M
% 12.85/6.93 = ( times_times_nat @ M @ N ) )
% 12.85/6.93 => ( ( N = one_one_nat )
% 12.85/6.93 | ( M = zero_zero_nat ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2156_zero__less__power__eq,axiom,
% 12.85/6.93 ! [A: real,N: nat] :
% 12.85/6.93 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 12.85/6.93 = ( ( N = zero_zero_nat )
% 12.85/6.93 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.93 & ( A != zero_zero_real ) )
% 12.85/6.93 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.93 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2157_zero__less__power__eq,axiom,
% 12.85/6.93 ! [A: rat,N: nat] :
% 12.85/6.93 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 12.85/6.93 = ( ( N = zero_zero_nat )
% 12.85/6.93 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.93 & ( A != zero_zero_rat ) )
% 12.85/6.93 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.93 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2158_zero__less__power__eq,axiom,
% 12.85/6.93 ! [A: int,N: nat] :
% 12.85/6.93 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 12.85/6.93 = ( ( N = zero_zero_nat )
% 12.85/6.93 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.93 & ( A != zero_zero_int ) )
% 12.85/6.93 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.93 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2159_mult__le__cancel__left,axiom,
% 12.85/6.93 ! [C: rat,A: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 12.85/6.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ord_less_eq_rat @ A @ B ) )
% 12.85/6.93 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2160_mult__le__cancel__left,axiom,
% 12.85/6.93 ! [C: int,A: int,B: int] :
% 12.85/6.93 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 12.85/6.93 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 12.85/6.93 => ( ord_less_eq_int @ A @ B ) )
% 12.85/6.93 & ( ( ord_less_int @ C @ zero_zero_int )
% 12.85/6.93 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2161_mult__le__cancel__left,axiom,
% 12.85/6.93 ! [C: real,A: real,B: real] :
% 12.85/6.93 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 12.85/6.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ord_less_eq_real @ A @ B ) )
% 12.85/6.93 & ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.93 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2162_mult__le__cancel__right,axiom,
% 12.85/6.93 ! [A: rat,C: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 12.85/6.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ord_less_eq_rat @ A @ B ) )
% 12.85/6.93 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2163_mult__le__cancel__right,axiom,
% 12.85/6.93 ! [A: int,C: int,B: int] :
% 12.85/6.93 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 12.85/6.93 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 12.85/6.93 => ( ord_less_eq_int @ A @ B ) )
% 12.85/6.93 & ( ( ord_less_int @ C @ zero_zero_int )
% 12.85/6.93 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2164_mult__le__cancel__right,axiom,
% 12.85/6.93 ! [A: real,C: real,B: real] :
% 12.85/6.93 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 12.85/6.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ord_less_eq_real @ A @ B ) )
% 12.85/6.93 & ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.93 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2165_mult__left__less__imp__less,axiom,
% 12.85/6.93 ! [C: rat,A: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 12.85/6.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ord_less_rat @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2166_mult__left__less__imp__less,axiom,
% 12.85/6.93 ! [C: nat,A: nat,B: nat] :
% 12.85/6.93 ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 12.85/6.93 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.85/6.93 => ( ord_less_nat @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2167_mult__left__less__imp__less,axiom,
% 12.85/6.93 ! [C: int,A: int,B: int] :
% 12.85/6.93 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 12.85/6.93 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.93 => ( ord_less_int @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2168_mult__left__less__imp__less,axiom,
% 12.85/6.93 ! [C: real,A: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 12.85/6.93 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ord_less_real @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2169_mult__strict__mono,axiom,
% 12.85/6.93 ! [A: rat,B: rat,C: rat,D: rat] :
% 12.85/6.93 ( ( ord_less_rat @ A @ B )
% 12.85/6.93 => ( ( ord_less_rat @ C @ D )
% 12.85/6.93 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 12.85/6.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2170_mult__strict__mono,axiom,
% 12.85/6.93 ! [A: nat,B: nat,C: nat,D: nat] :
% 12.85/6.93 ( ( ord_less_nat @ A @ B )
% 12.85/6.93 => ( ( ord_less_nat @ C @ D )
% 12.85/6.93 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 12.85/6.93 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.85/6.93 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2171_mult__strict__mono,axiom,
% 12.85/6.93 ! [A: int,B: int,C: int,D: int] :
% 12.85/6.93 ( ( ord_less_int @ A @ B )
% 12.85/6.93 => ( ( ord_less_int @ C @ D )
% 12.85/6.93 => ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.93 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.93 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2172_mult__strict__mono,axiom,
% 12.85/6.93 ! [A: real,B: real,C: real,D: real] :
% 12.85/6.93 ( ( ord_less_real @ A @ B )
% 12.85/6.93 => ( ( ord_less_real @ C @ D )
% 12.85/6.93 => ( ( ord_less_real @ zero_zero_real @ B )
% 12.85/6.93 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2173_mult__less__cancel__left,axiom,
% 12.85/6.93 ! [C: rat,A: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 12.85/6.93 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ord_less_rat @ A @ B ) )
% 12.85/6.93 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.85/6.93 => ( ord_less_rat @ B @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2174_mult__less__cancel__left,axiom,
% 12.85/6.93 ! [C: int,A: int,B: int] :
% 12.85/6.93 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 12.85/6.93 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.93 => ( ord_less_int @ A @ B ) )
% 12.85/6.93 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 12.85/6.93 => ( ord_less_int @ B @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2175_mult__less__cancel__left,axiom,
% 12.85/6.93 ! [C: real,A: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 12.85/6.93 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ord_less_real @ A @ B ) )
% 12.85/6.93 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.85/6.93 => ( ord_less_real @ B @ A ) ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2176_mult__right__less__imp__less,axiom,
% 12.85/6.93 ! [A: rat,C: rat,B: rat] :
% 12.85/6.93 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 12.85/6.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.85/6.93 => ( ord_less_rat @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2177_mult__right__less__imp__less,axiom,
% 12.85/6.93 ! [A: nat,C: nat,B: nat] :
% 12.85/6.93 ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 12.85/6.93 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.85/6.93 => ( ord_less_nat @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2178_mult__right__less__imp__less,axiom,
% 12.85/6.93 ! [A: int,C: int,B: int] :
% 12.85/6.93 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 12.85/6.93 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.93 => ( ord_less_int @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2179_mult__right__less__imp__less,axiom,
% 12.85/6.93 ! [A: real,C: real,B: real] :
% 12.85/6.93 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 12.85/6.93 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.93 => ( ord_less_real @ A @ B ) ) ) ).
% 12.85/6.93
% 12.85/6.93 thf(fact_2180_mult__strict__mono_H,axiom,
% 12.85/6.93 ! [A: rat,B: rat,C: rat,D: rat] :
% 12.85/6.93 ( ( ord_less_rat @ A @ B )
% 12.85/6.93 => ( ( ord_less_rat @ C @ D )
% 12.85/6.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2181_mult__strict__mono_H,axiom,
% 12.85/6.94 ! [A: nat,B: nat,C: nat,D: nat] :
% 12.85/6.94 ( ( ord_less_nat @ A @ B )
% 12.85/6.94 => ( ( ord_less_nat @ C @ D )
% 12.85/6.94 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.94 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.85/6.94 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2182_mult__strict__mono_H,axiom,
% 12.85/6.94 ! [A: int,B: int,C: int,D: int] :
% 12.85/6.94 ( ( ord_less_int @ A @ B )
% 12.85/6.94 => ( ( ord_less_int @ C @ D )
% 12.85/6.94 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.94 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.94 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2183_mult__strict__mono_H,axiom,
% 12.85/6.94 ! [A: real,B: real,C: real,D: real] :
% 12.85/6.94 ( ( ord_less_real @ A @ B )
% 12.85/6.94 => ( ( ord_less_real @ C @ D )
% 12.85/6.94 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.94 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2184_mult__less__cancel__right,axiom,
% 12.85/6.94 ! [A: rat,C: rat,B: rat] :
% 12.85/6.94 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 12.85/6.94 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ord_less_rat @ A @ B ) )
% 12.85/6.94 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_rat @ B @ A ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2185_mult__less__cancel__right,axiom,
% 12.85/6.94 ! [A: int,C: int,B: int] :
% 12.85/6.94 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 12.85/6.94 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.94 => ( ord_less_int @ A @ B ) )
% 12.85/6.94 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 12.85/6.94 => ( ord_less_int @ B @ A ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2186_mult__less__cancel__right,axiom,
% 12.85/6.94 ! [A: real,C: real,B: real] :
% 12.85/6.94 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 12.85/6.94 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ord_less_real @ A @ B ) )
% 12.85/6.94 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ord_less_real @ B @ A ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2187_mult__le__cancel__left__neg,axiom,
% 12.85/6.94 ! [C: rat,A: rat,B: rat] :
% 12.85/6.94 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 12.85/6.94 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2188_mult__le__cancel__left__neg,axiom,
% 12.85/6.94 ! [C: int,A: int,B: int] :
% 12.85/6.94 ( ( ord_less_int @ C @ zero_zero_int )
% 12.85/6.94 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 12.85/6.94 = ( ord_less_eq_int @ B @ A ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2189_mult__le__cancel__left__neg,axiom,
% 12.85/6.94 ! [C: real,A: real,B: real] :
% 12.85/6.94 ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 12.85/6.94 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2190_mult__le__cancel__left__pos,axiom,
% 12.85/6.94 ! [C: rat,A: rat,B: rat] :
% 12.85/6.94 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 12.85/6.94 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2191_mult__le__cancel__left__pos,axiom,
% 12.85/6.94 ! [C: int,A: int,B: int] :
% 12.85/6.94 ( ( ord_less_int @ zero_zero_int @ C )
% 12.85/6.94 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 12.85/6.94 = ( ord_less_eq_int @ A @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2192_mult__le__cancel__left__pos,axiom,
% 12.85/6.94 ! [C: real,A: real,B: real] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 12.85/6.94 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2193_mult__left__le__imp__le,axiom,
% 12.85/6.94 ! [C: rat,A: rat,B: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 12.85/6.94 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2194_mult__left__le__imp__le,axiom,
% 12.85/6.94 ! [C: nat,A: nat,B: nat] :
% 12.85/6.94 ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 12.85/6.94 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 12.85/6.94 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2195_mult__left__le__imp__le,axiom,
% 12.85/6.94 ! [C: int,A: int,B: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 12.85/6.94 => ( ( ord_less_int @ zero_zero_int @ C )
% 12.85/6.94 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2196_mult__left__le__imp__le,axiom,
% 12.85/6.94 ! [C: real,A: real,B: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 12.85/6.94 => ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2197_mult__right__le__imp__le,axiom,
% 12.85/6.94 ! [A: rat,C: rat,B: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 12.85/6.94 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2198_mult__right__le__imp__le,axiom,
% 12.85/6.94 ! [A: nat,C: nat,B: nat] :
% 12.85/6.94 ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 12.85/6.94 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 12.85/6.94 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2199_mult__right__le__imp__le,axiom,
% 12.85/6.94 ! [A: int,C: int,B: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 12.85/6.94 => ( ( ord_less_int @ zero_zero_int @ C )
% 12.85/6.94 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2200_mult__right__le__imp__le,axiom,
% 12.85/6.94 ! [A: real,C: real,B: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 12.85/6.94 => ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2201_mult__le__less__imp__less,axiom,
% 12.85/6.94 ! [A: rat,B: rat,C: rat,D: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.94 => ( ( ord_less_rat @ C @ D )
% 12.85/6.94 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.94 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2202_mult__le__less__imp__less,axiom,
% 12.85/6.94 ! [A: nat,B: nat,C: nat,D: nat] :
% 12.85/6.94 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.94 => ( ( ord_less_nat @ C @ D )
% 12.85/6.94 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.85/6.94 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.85/6.94 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2203_mult__le__less__imp__less,axiom,
% 12.85/6.94 ! [A: int,B: int,C: int,D: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.94 => ( ( ord_less_int @ C @ D )
% 12.85/6.94 => ( ( ord_less_int @ zero_zero_int @ A )
% 12.85/6.94 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.94 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2204_mult__le__less__imp__less,axiom,
% 12.85/6.94 ! [A: real,B: real,C: real,D: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.94 => ( ( ord_less_real @ C @ D )
% 12.85/6.94 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.94 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2205_mult__less__le__imp__less,axiom,
% 12.85/6.94 ! [A: rat,B: rat,C: rat,D: rat] :
% 12.85/6.94 ( ( ord_less_rat @ A @ B )
% 12.85/6.94 => ( ( ord_less_eq_rat @ C @ D )
% 12.85/6.94 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.94 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2206_mult__less__le__imp__less,axiom,
% 12.85/6.94 ! [A: nat,B: nat,C: nat,D: nat] :
% 12.85/6.94 ( ( ord_less_nat @ A @ B )
% 12.85/6.94 => ( ( ord_less_eq_nat @ C @ D )
% 12.85/6.94 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.94 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 12.85/6.94 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2207_mult__less__le__imp__less,axiom,
% 12.85/6.94 ! [A: int,B: int,C: int,D: int] :
% 12.85/6.94 ( ( ord_less_int @ A @ B )
% 12.85/6.94 => ( ( ord_less_eq_int @ C @ D )
% 12.85/6.94 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.94 => ( ( ord_less_int @ zero_zero_int @ C )
% 12.85/6.94 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2208_mult__less__le__imp__less,axiom,
% 12.85/6.94 ! [A: real,B: real,C: real,D: real] :
% 12.85/6.94 ( ( ord_less_real @ A @ B )
% 12.85/6.94 => ( ( ord_less_eq_real @ C @ D )
% 12.85/6.94 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.94 => ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2209_frac__le,axiom,
% 12.85/6.94 ! [Y4: rat,X: rat,W: rat,Z: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 12.85/6.94 => ( ( ord_less_eq_rat @ X @ Y4 )
% 12.85/6.94 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 12.85/6.94 => ( ( ord_less_eq_rat @ W @ Z )
% 12.85/6.94 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y4 @ W ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2210_frac__le,axiom,
% 12.85/6.94 ! [Y4: real,X: real,W: real,Z: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/6.94 => ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/6.94 => ( ( ord_less_real @ zero_zero_real @ W )
% 12.85/6.94 => ( ( ord_less_eq_real @ W @ Z )
% 12.85/6.94 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y4 @ W ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2211_frac__less,axiom,
% 12.85/6.94 ! [X: rat,Y4: rat,W: rat,Z: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 12.85/6.94 => ( ( ord_less_rat @ X @ Y4 )
% 12.85/6.94 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 12.85/6.94 => ( ( ord_less_eq_rat @ W @ Z )
% 12.85/6.94 => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y4 @ W ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2212_frac__less,axiom,
% 12.85/6.94 ! [X: real,Y4: real,W: real,Z: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/6.94 => ( ( ord_less_real @ X @ Y4 )
% 12.85/6.94 => ( ( ord_less_real @ zero_zero_real @ W )
% 12.85/6.94 => ( ( ord_less_eq_real @ W @ Z )
% 12.85/6.94 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y4 @ W ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2213_frac__less2,axiom,
% 12.85/6.94 ! [X: rat,Y4: rat,W: rat,Z: rat] :
% 12.85/6.94 ( ( ord_less_rat @ zero_zero_rat @ X )
% 12.85/6.94 => ( ( ord_less_eq_rat @ X @ Y4 )
% 12.85/6.94 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 12.85/6.94 => ( ( ord_less_rat @ W @ Z )
% 12.85/6.94 => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y4 @ W ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2214_frac__less2,axiom,
% 12.85/6.94 ! [X: real,Y4: real,W: real,Z: real] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.94 => ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/6.94 => ( ( ord_less_real @ zero_zero_real @ W )
% 12.85/6.94 => ( ( ord_less_real @ W @ Z )
% 12.85/6.94 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y4 @ W ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2215_divide__le__cancel,axiom,
% 12.85/6.94 ! [A: rat,C: rat,B: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 12.85/6.94 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ord_less_eq_rat @ A @ B ) )
% 12.85/6.94 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2216_divide__le__cancel,axiom,
% 12.85/6.94 ! [A: real,C: real,B: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 12.85/6.94 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ord_less_eq_real @ A @ B ) )
% 12.85/6.94 & ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2217_divide__nonneg__neg,axiom,
% 12.85/6.94 ! [X: rat,Y4: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 12.85/6.94 => ( ( ord_less_rat @ Y4 @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y4 ) @ zero_zero_rat ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2218_divide__nonneg__neg,axiom,
% 12.85/6.94 ! [X: real,Y4: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/6.94 => ( ( ord_less_real @ Y4 @ zero_zero_real )
% 12.85/6.94 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2219_divide__nonneg__pos,axiom,
% 12.85/6.94 ! [X: rat,Y4: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 12.85/6.94 => ( ( ord_less_rat @ zero_zero_rat @ Y4 )
% 12.85/6.94 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2220_divide__nonneg__pos,axiom,
% 12.85/6.94 ! [X: real,Y4: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/6.94 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/6.94 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2221_divide__nonpos__neg,axiom,
% 12.85/6.94 ! [X: rat,Y4: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 12.85/6.94 => ( ( ord_less_rat @ Y4 @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2222_divide__nonpos__neg,axiom,
% 12.85/6.94 ! [X: real,Y4: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 12.85/6.94 => ( ( ord_less_real @ Y4 @ zero_zero_real )
% 12.85/6.94 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2223_divide__nonpos__pos,axiom,
% 12.85/6.94 ! [X: rat,Y4: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 12.85/6.94 => ( ( ord_less_rat @ zero_zero_rat @ Y4 )
% 12.85/6.94 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y4 ) @ zero_zero_rat ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2224_divide__nonpos__pos,axiom,
% 12.85/6.94 ! [X: real,Y4: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 12.85/6.94 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/6.94 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2225_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 12.85/6.94 ! [A: nat,B: nat] :
% 12.85/6.94 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.94 => ( ( ord_less_nat @ A @ B )
% 12.85/6.94 => ( ( divide_divide_nat @ A @ B )
% 12.85/6.94 = zero_zero_nat ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2226_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 12.85/6.94 ! [A: int,B: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.94 => ( ( ord_less_int @ A @ B )
% 12.85/6.94 => ( ( divide_divide_int @ A @ B )
% 12.85/6.94 = zero_zero_int ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2227_div__positive,axiom,
% 12.85/6.94 ! [B: nat,A: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ B )
% 12.85/6.94 => ( ( ord_less_eq_nat @ B @ A )
% 12.85/6.94 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2228_div__positive,axiom,
% 12.85/6.94 ! [B: int,A: int] :
% 12.85/6.94 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.94 => ( ( ord_less_eq_int @ B @ A )
% 12.85/6.94 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2229_divide__less__eq,axiom,
% 12.85/6.94 ! [B: real,C: real,A: real] :
% 12.85/6.94 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 12.85/6.94 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 12.85/6.94 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 12.85/6.94 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2230_divide__less__eq,axiom,
% 12.85/6.94 ! [B: rat,C: rat,A: rat] :
% 12.85/6.94 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 12.85/6.94 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 12.85/6.94 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 12.85/6.94 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2231_less__divide__eq,axiom,
% 12.85/6.94 ! [A: real,B: real,C: real] :
% 12.85/6.94 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 12.85/6.94 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 12.85/6.94 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 12.85/6.94 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2232_less__divide__eq,axiom,
% 12.85/6.94 ! [A: rat,B: rat,C: rat] :
% 12.85/6.94 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 12.85/6.94 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 12.85/6.94 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 12.85/6.94 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2233_neg__divide__less__eq,axiom,
% 12.85/6.94 ! [C: real,B: real,A: real] :
% 12.85/6.94 ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 12.85/6.94 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2234_neg__divide__less__eq,axiom,
% 12.85/6.94 ! [C: rat,B: rat,A: rat] :
% 12.85/6.94 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 12.85/6.94 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2235_neg__less__divide__eq,axiom,
% 12.85/6.94 ! [C: real,A: real,B: real] :
% 12.85/6.94 ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 12.85/6.94 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2236_neg__less__divide__eq,axiom,
% 12.85/6.94 ! [C: rat,A: rat,B: rat] :
% 12.85/6.94 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 12.85/6.94 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2237_pos__divide__less__eq,axiom,
% 12.85/6.94 ! [C: real,B: real,A: real] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 12.85/6.94 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2238_pos__divide__less__eq,axiom,
% 12.85/6.94 ! [C: rat,B: rat,A: rat] :
% 12.85/6.94 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 12.85/6.94 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2239_pos__less__divide__eq,axiom,
% 12.85/6.94 ! [C: real,A: real,B: real] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 12.85/6.94 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2240_pos__less__divide__eq,axiom,
% 12.85/6.94 ! [C: rat,A: rat,B: rat] :
% 12.85/6.94 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 12.85/6.94 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2241_mult__imp__div__pos__less,axiom,
% 12.85/6.94 ! [Y4: real,X: real,Z: real] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/6.94 => ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y4 ) )
% 12.85/6.94 => ( ord_less_real @ ( divide_divide_real @ X @ Y4 ) @ Z ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2242_mult__imp__div__pos__less,axiom,
% 12.85/6.94 ! [Y4: rat,X: rat,Z: rat] :
% 12.85/6.94 ( ( ord_less_rat @ zero_zero_rat @ Y4 )
% 12.85/6.94 => ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y4 ) )
% 12.85/6.94 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y4 ) @ Z ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2243_mult__imp__less__div__pos,axiom,
% 12.85/6.94 ! [Y4: real,Z: real,X: real] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/6.94 => ( ( ord_less_real @ ( times_times_real @ Z @ Y4 ) @ X )
% 12.85/6.94 => ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2244_mult__imp__less__div__pos,axiom,
% 12.85/6.94 ! [Y4: rat,Z: rat,X: rat] :
% 12.85/6.94 ( ( ord_less_rat @ zero_zero_rat @ Y4 )
% 12.85/6.94 => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y4 ) @ X )
% 12.85/6.94 => ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2245_divide__strict__left__mono,axiom,
% 12.85/6.94 ! [B: real,A: real,C: real] :
% 12.85/6.94 ( ( ord_less_real @ B @ A )
% 12.85/6.94 => ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 12.85/6.94 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2246_divide__strict__left__mono,axiom,
% 12.85/6.94 ! [B: rat,A: rat,C: rat] :
% 12.85/6.94 ( ( ord_less_rat @ B @ A )
% 12.85/6.94 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 12.85/6.94 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2247_divide__strict__left__mono__neg,axiom,
% 12.85/6.94 ! [A: real,B: real,C: real] :
% 12.85/6.94 ( ( ord_less_real @ A @ B )
% 12.85/6.94 => ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 12.85/6.94 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2248_divide__strict__left__mono__neg,axiom,
% 12.85/6.94 ! [A: rat,B: rat,C: rat] :
% 12.85/6.94 ( ( ord_less_rat @ A @ B )
% 12.85/6.94 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 12.85/6.94 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2249_mult__left__le__one__le,axiom,
% 12.85/6.94 ! [X: rat,Y4: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 12.85/6.94 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 12.85/6.94 => ( ( ord_less_eq_rat @ Y4 @ one_one_rat )
% 12.85/6.94 => ( ord_less_eq_rat @ ( times_times_rat @ Y4 @ X ) @ X ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2250_mult__left__le__one__le,axiom,
% 12.85/6.94 ! [X: int,Y4: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/6.94 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/6.94 => ( ( ord_less_eq_int @ Y4 @ one_one_int )
% 12.85/6.94 => ( ord_less_eq_int @ ( times_times_int @ Y4 @ X ) @ X ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2251_mult__left__le__one__le,axiom,
% 12.85/6.94 ! [X: real,Y4: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/6.94 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/6.94 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/6.94 => ( ord_less_eq_real @ ( times_times_real @ Y4 @ X ) @ X ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2252_mult__right__le__one__le,axiom,
% 12.85/6.94 ! [X: rat,Y4: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 12.85/6.94 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 12.85/6.94 => ( ( ord_less_eq_rat @ Y4 @ one_one_rat )
% 12.85/6.94 => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y4 ) @ X ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2253_mult__right__le__one__le,axiom,
% 12.85/6.94 ! [X: int,Y4: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/6.94 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/6.94 => ( ( ord_less_eq_int @ Y4 @ one_one_int )
% 12.85/6.94 => ( ord_less_eq_int @ ( times_times_int @ X @ Y4 ) @ X ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2254_mult__right__le__one__le,axiom,
% 12.85/6.94 ! [X: real,Y4: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/6.94 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/6.94 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/6.94 => ( ord_less_eq_real @ ( times_times_real @ X @ Y4 ) @ X ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2255_mult__le__one,axiom,
% 12.85/6.94 ! [A: rat,B: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ A @ one_one_rat )
% 12.85/6.94 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 12.85/6.94 => ( ( ord_less_eq_rat @ B @ one_one_rat )
% 12.85/6.94 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2256_mult__le__one,axiom,
% 12.85/6.94 ! [A: nat,B: nat] :
% 12.85/6.94 ( ( ord_less_eq_nat @ A @ one_one_nat )
% 12.85/6.94 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 12.85/6.94 => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 12.85/6.94 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2257_mult__le__one,axiom,
% 12.85/6.94 ! [A: int,B: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ A @ one_one_int )
% 12.85/6.94 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 12.85/6.94 => ( ( ord_less_eq_int @ B @ one_one_int )
% 12.85/6.94 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2258_mult__le__one,axiom,
% 12.85/6.94 ! [A: real,B: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ A @ one_one_real )
% 12.85/6.94 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 12.85/6.94 => ( ( ord_less_eq_real @ B @ one_one_real )
% 12.85/6.94 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2259_mult__left__le,axiom,
% 12.85/6.94 ! [C: rat,A: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ C @ one_one_rat )
% 12.85/6.94 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.94 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2260_mult__left__le,axiom,
% 12.85/6.94 ! [C: nat,A: nat] :
% 12.85/6.94 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 12.85/6.94 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.94 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2261_mult__left__le,axiom,
% 12.85/6.94 ! [C: int,A: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ C @ one_one_int )
% 12.85/6.94 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.94 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2262_mult__left__le,axiom,
% 12.85/6.94 ! [C: real,A: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ C @ one_one_real )
% 12.85/6.94 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.94 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2263_ex__least__nat__less,axiom,
% 12.85/6.94 ! [P: nat > $o,N: nat] :
% 12.85/6.94 ( ( P @ N )
% 12.85/6.94 => ( ~ ( P @ zero_zero_nat )
% 12.85/6.94 => ? [K2: nat] :
% 12.85/6.94 ( ( ord_less_nat @ K2 @ N )
% 12.85/6.94 & ! [I4: nat] :
% 12.85/6.94 ( ( ord_less_eq_nat @ I4 @ K2 )
% 12.85/6.94 => ~ ( P @ I4 ) )
% 12.85/6.94 & ( P @ ( suc @ K2 ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2264_unit__dvdE,axiom,
% 12.85/6.94 ! [A: code_integer,B: code_integer] :
% 12.85/6.94 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.85/6.94 => ~ ( ( A != zero_z3403309356797280102nteger )
% 12.85/6.94 => ! [C3: code_integer] :
% 12.85/6.94 ( B
% 12.85/6.94 != ( times_3573771949741848930nteger @ A @ C3 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2265_unit__dvdE,axiom,
% 12.85/6.94 ! [A: nat,B: nat] :
% 12.85/6.94 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.85/6.94 => ~ ( ( A != zero_zero_nat )
% 12.85/6.94 => ! [C3: nat] :
% 12.85/6.94 ( B
% 12.85/6.94 != ( times_times_nat @ A @ C3 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2266_unit__dvdE,axiom,
% 12.85/6.94 ! [A: int,B: int] :
% 12.85/6.94 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.85/6.94 => ~ ( ( A != zero_zero_int )
% 12.85/6.94 => ! [C3: int] :
% 12.85/6.94 ( B
% 12.85/6.94 != ( times_times_int @ A @ C3 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2267_diff__Suc__less,axiom,
% 12.85/6.94 ! [N: nat,I: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.94 => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2268_n__less__n__mult__m,axiom,
% 12.85/6.94 ! [N: nat,M: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.94 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 12.85/6.94 => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2269_n__less__m__mult__n,axiom,
% 12.85/6.94 ! [N: nat,M: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.94 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 12.85/6.94 => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2270_one__less__mult,axiom,
% 12.85/6.94 ! [N: nat,M: nat] :
% 12.85/6.94 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.85/6.94 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 12.85/6.94 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2271_unit__div__eq__0__iff,axiom,
% 12.85/6.94 ! [B: code_integer,A: code_integer] :
% 12.85/6.94 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 12.85/6.94 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 12.85/6.94 = zero_z3403309356797280102nteger )
% 12.85/6.94 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2272_unit__div__eq__0__iff,axiom,
% 12.85/6.94 ! [B: nat,A: nat] :
% 12.85/6.94 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 12.85/6.94 => ( ( ( divide_divide_nat @ A @ B )
% 12.85/6.94 = zero_zero_nat )
% 12.85/6.94 = ( A = zero_zero_nat ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2273_unit__div__eq__0__iff,axiom,
% 12.85/6.94 ! [B: int,A: int] :
% 12.85/6.94 ( ( dvd_dvd_int @ B @ one_one_int )
% 12.85/6.94 => ( ( ( divide_divide_int @ A @ B )
% 12.85/6.94 = zero_zero_int )
% 12.85/6.94 = ( A = zero_zero_int ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2274_nat__mult__le__cancel1,axiom,
% 12.85/6.94 ! [K: nat,M: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/6.94 => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.85/6.94 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2275_unit__eq__div1,axiom,
% 12.85/6.94 ! [B: code_integer,A: code_integer,C: code_integer] :
% 12.85/6.94 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 12.85/6.94 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 12.85/6.94 = C )
% 12.85/6.94 = ( A
% 12.85/6.94 = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2276_unit__eq__div1,axiom,
% 12.85/6.94 ! [B: nat,A: nat,C: nat] :
% 12.85/6.94 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 12.85/6.94 => ( ( ( divide_divide_nat @ A @ B )
% 12.85/6.94 = C )
% 12.85/6.94 = ( A
% 12.85/6.94 = ( times_times_nat @ C @ B ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2277_unit__eq__div1,axiom,
% 12.85/6.94 ! [B: int,A: int,C: int] :
% 12.85/6.94 ( ( dvd_dvd_int @ B @ one_one_int )
% 12.85/6.94 => ( ( ( divide_divide_int @ A @ B )
% 12.85/6.94 = C )
% 12.85/6.94 = ( A
% 12.85/6.94 = ( times_times_int @ C @ B ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2278_unit__eq__div2,axiom,
% 12.85/6.94 ! [B: code_integer,A: code_integer,C: code_integer] :
% 12.85/6.94 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 12.85/6.94 => ( ( A
% 12.85/6.94 = ( divide6298287555418463151nteger @ C @ B ) )
% 12.85/6.94 = ( ( times_3573771949741848930nteger @ A @ B )
% 12.85/6.94 = C ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2279_unit__eq__div2,axiom,
% 12.85/6.94 ! [B: nat,A: nat,C: nat] :
% 12.85/6.94 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 12.85/6.94 => ( ( A
% 12.85/6.94 = ( divide_divide_nat @ C @ B ) )
% 12.85/6.94 = ( ( times_times_nat @ A @ B )
% 12.85/6.94 = C ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2280_unit__eq__div2,axiom,
% 12.85/6.94 ! [B: int,A: int,C: int] :
% 12.85/6.94 ( ( dvd_dvd_int @ B @ one_one_int )
% 12.85/6.94 => ( ( A
% 12.85/6.94 = ( divide_divide_int @ C @ B ) )
% 12.85/6.94 = ( ( times_times_int @ A @ B )
% 12.85/6.94 = C ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2281_div__mult__unit2,axiom,
% 12.85/6.94 ! [C: code_integer,B: code_integer,A: code_integer] :
% 12.85/6.94 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 12.85/6.94 => ( ( dvd_dvd_Code_integer @ B @ A )
% 12.85/6.94 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 12.85/6.94 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2282_div__mult__unit2,axiom,
% 12.85/6.94 ! [C: nat,B: nat,A: nat] :
% 12.85/6.94 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 12.85/6.94 => ( ( dvd_dvd_nat @ B @ A )
% 12.85/6.94 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 12.85/6.94 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2283_div__mult__unit2,axiom,
% 12.85/6.94 ! [C: int,B: int,A: int] :
% 12.85/6.94 ( ( dvd_dvd_int @ C @ one_one_int )
% 12.85/6.94 => ( ( dvd_dvd_int @ B @ A )
% 12.85/6.94 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 12.85/6.94 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2284_unit__div__commute,axiom,
% 12.85/6.94 ! [B: code_integer,A: code_integer,C: code_integer] :
% 12.85/6.94 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 12.85/6.94 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 12.85/6.94 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2285_unit__div__commute,axiom,
% 12.85/6.94 ! [B: nat,A: nat,C: nat] :
% 12.85/6.94 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 12.85/6.94 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 12.85/6.94 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2286_unit__div__commute,axiom,
% 12.85/6.94 ! [B: int,A: int,C: int] :
% 12.85/6.94 ( ( dvd_dvd_int @ B @ one_one_int )
% 12.85/6.94 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
% 12.85/6.94 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2287_unit__div__mult__swap,axiom,
% 12.85/6.94 ! [C: code_integer,A: code_integer,B: code_integer] :
% 12.85/6.94 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 12.85/6.94 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 12.85/6.94 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2288_unit__div__mult__swap,axiom,
% 12.85/6.94 ! [C: nat,A: nat,B: nat] :
% 12.85/6.94 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 12.85/6.94 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 12.85/6.94 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2289_unit__div__mult__swap,axiom,
% 12.85/6.94 ! [C: int,A: int,B: int] :
% 12.85/6.94 ( ( dvd_dvd_int @ C @ one_one_int )
% 12.85/6.94 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 12.85/6.94 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2290_is__unit__div__mult2__eq,axiom,
% 12.85/6.94 ! [B: code_integer,C: code_integer,A: code_integer] :
% 12.85/6.94 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 12.85/6.94 => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 12.85/6.94 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 12.85/6.94 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2291_is__unit__div__mult2__eq,axiom,
% 12.85/6.94 ! [B: nat,C: nat,A: nat] :
% 12.85/6.94 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 12.85/6.94 => ( ( dvd_dvd_nat @ C @ one_one_nat )
% 12.85/6.94 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 12.85/6.94 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2292_is__unit__div__mult2__eq,axiom,
% 12.85/6.94 ! [B: int,C: int,A: int] :
% 12.85/6.94 ( ( dvd_dvd_int @ B @ one_one_int )
% 12.85/6.94 => ( ( dvd_dvd_int @ C @ one_one_int )
% 12.85/6.94 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 12.85/6.94 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2293_dvd__imp__le,axiom,
% 12.85/6.94 ! [K: nat,N: nat] :
% 12.85/6.94 ( ( dvd_dvd_nat @ K @ N )
% 12.85/6.94 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.94 => ( ord_less_eq_nat @ K @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2294_div__greater__zero__iff,axiom,
% 12.85/6.94 ! [M: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 12.85/6.94 = ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.94 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2295_div__le__mono2,axiom,
% 12.85/6.94 ! [M: nat,N: nat,K: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/6.94 => ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.94 => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2296_nat__mult__dvd__cancel1,axiom,
% 12.85/6.94 ! [K: nat,M: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/6.94 => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.85/6.94 = ( dvd_dvd_nat @ M @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2297_dvd__mult__cancel,axiom,
% 12.85/6.94 ! [K: nat,M: nat,N: nat] :
% 12.85/6.94 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.85/6.94 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/6.94 => ( dvd_dvd_nat @ M @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2298_div__less__iff__less__mult,axiom,
% 12.85/6.94 ! [Q2: nat,M: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 12.85/6.94 => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
% 12.85/6.94 = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2299_nat__mult__div__cancel1,axiom,
% 12.85/6.94 ! [K: nat,M: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/6.94 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 12.85/6.94 = ( divide_divide_nat @ M @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2300_power__le__zero__eq,axiom,
% 12.85/6.94 ! [A: rat,N: nat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 12.85/6.94 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.94 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 12.85/6.94 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 & ( A = zero_zero_rat ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2301_power__le__zero__eq,axiom,
% 12.85/6.94 ! [A: int,N: nat] :
% 12.85/6.94 ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 12.85/6.94 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.94 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 12.85/6.94 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 & ( A = zero_zero_int ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2302_power__le__zero__eq,axiom,
% 12.85/6.94 ! [A: real,N: nat] :
% 12.85/6.94 ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 12.85/6.94 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.94 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 12.85/6.94 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 & ( A = zero_zero_real ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2303_even__mask__div__iff_H,axiom,
% 12.85/6.94 ! [M: nat,N: nat] :
% 12.85/6.94 ( ( 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 ) ) )
% 12.85/6.94 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2304_even__mask__div__iff_H,axiom,
% 12.85/6.94 ! [M: nat,N: nat] :
% 12.85/6.94 ( ( 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 ) ) )
% 12.85/6.94 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2305_even__mask__div__iff_H,axiom,
% 12.85/6.94 ! [M: nat,N: nat] :
% 12.85/6.94 ( ( 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 ) ) )
% 12.85/6.94 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2306_conj__le__cong,axiom,
% 12.85/6.94 ! [X: int,X7: int,P: $o,P4: $o] :
% 12.85/6.94 ( ( X = X7 )
% 12.85/6.94 => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 12.85/6.94 => ( P = P4 ) )
% 12.85/6.94 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/6.94 & P )
% 12.85/6.94 = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 12.85/6.94 & P4 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2307_imp__le__cong,axiom,
% 12.85/6.94 ! [X: int,X7: int,P: $o,P4: $o] :
% 12.85/6.94 ( ( X = X7 )
% 12.85/6.94 => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 12.85/6.94 => ( P = P4 ) )
% 12.85/6.94 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/6.94 => P )
% 12.85/6.94 = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 12.85/6.94 => P4 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2308_gcd__nat_Onot__eq__order__implies__strict,axiom,
% 12.85/6.94 ! [A: nat,B: nat] :
% 12.85/6.94 ( ( A != B )
% 12.85/6.94 => ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.94 => ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.94 & ( A != B ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2309_gcd__nat_Ostrict__implies__not__eq,axiom,
% 12.85/6.94 ! [A: nat,B: nat] :
% 12.85/6.94 ( ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.94 & ( A != B ) )
% 12.85/6.94 => ( A != B ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2310_gcd__nat_Ostrict__implies__order,axiom,
% 12.85/6.94 ! [A: nat,B: nat] :
% 12.85/6.94 ( ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.94 & ( A != B ) )
% 12.85/6.94 => ( dvd_dvd_nat @ A @ B ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2311_gcd__nat_Ostrict__iff__order,axiom,
% 12.85/6.94 ! [A: nat,B: nat] :
% 12.85/6.94 ( ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.94 & ( A != B ) )
% 12.85/6.94 = ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.94 & ( A != B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2312_gcd__nat_Oorder__iff__strict,axiom,
% 12.85/6.94 ( dvd_dvd_nat
% 12.85/6.94 = ( ^ [A3: nat,B2: nat] :
% 12.85/6.94 ( ( ( dvd_dvd_nat @ A3 @ B2 )
% 12.85/6.94 & ( A3 != B2 ) )
% 12.85/6.94 | ( A3 = B2 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2313_gcd__nat_Ostrict__iff__not,axiom,
% 12.85/6.94 ! [A: nat,B: nat] :
% 12.85/6.94 ( ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.94 & ( A != B ) )
% 12.85/6.94 = ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.94 & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2314_gcd__nat_Ostrict__trans2,axiom,
% 12.85/6.94 ! [A: nat,B: nat,C: nat] :
% 12.85/6.94 ( ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.94 & ( A != B ) )
% 12.85/6.94 => ( ( dvd_dvd_nat @ B @ C )
% 12.85/6.94 => ( ( dvd_dvd_nat @ A @ C )
% 12.85/6.94 & ( A != C ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2315_gcd__nat_Ostrict__trans1,axiom,
% 12.85/6.94 ! [A: nat,B: nat,C: nat] :
% 12.85/6.94 ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.94 => ( ( ( dvd_dvd_nat @ B @ C )
% 12.85/6.94 & ( B != C ) )
% 12.85/6.94 => ( ( dvd_dvd_nat @ A @ C )
% 12.85/6.94 & ( A != C ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2316_gcd__nat_Ostrict__trans,axiom,
% 12.85/6.94 ! [A: nat,B: nat,C: nat] :
% 12.85/6.94 ( ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.94 & ( A != B ) )
% 12.85/6.94 => ( ( ( dvd_dvd_nat @ B @ C )
% 12.85/6.94 & ( B != C ) )
% 12.85/6.94 => ( ( dvd_dvd_nat @ A @ C )
% 12.85/6.94 & ( A != C ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2317_gcd__nat_Oantisym,axiom,
% 12.85/6.94 ! [A: nat,B: nat] :
% 12.85/6.94 ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.94 => ( ( dvd_dvd_nat @ B @ A )
% 12.85/6.94 => ( A = B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2318_gcd__nat_Oirrefl,axiom,
% 12.85/6.94 ! [A: nat] :
% 12.85/6.94 ~ ( ( dvd_dvd_nat @ A @ A )
% 12.85/6.94 & ( A != A ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2319_gcd__nat_Oeq__iff,axiom,
% 12.85/6.94 ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 12.85/6.94 = ( ^ [A3: nat,B2: nat] :
% 12.85/6.94 ( ( dvd_dvd_nat @ A3 @ B2 )
% 12.85/6.94 & ( dvd_dvd_nat @ B2 @ A3 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2320_gcd__nat_Otrans,axiom,
% 12.85/6.94 ! [A: nat,B: nat,C: nat] :
% 12.85/6.94 ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.94 => ( ( dvd_dvd_nat @ B @ C )
% 12.85/6.94 => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2321_gcd__nat_Orefl,axiom,
% 12.85/6.94 ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2322_gcd__nat_Oasym,axiom,
% 12.85/6.94 ! [A: nat,B: nat] :
% 12.85/6.94 ( ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.94 & ( A != B ) )
% 12.85/6.94 => ~ ( ( dvd_dvd_nat @ B @ A )
% 12.85/6.94 & ( B != A ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2323_even__mask__div__iff,axiom,
% 12.85/6.94 ! [M: nat,N: nat] :
% 12.85/6.94 ( ( 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 ) ) )
% 12.85/6.94 = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 = zero_z3403309356797280102nteger )
% 12.85/6.94 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2324_even__mask__div__iff,axiom,
% 12.85/6.94 ! [M: nat,N: nat] :
% 12.85/6.94 ( ( 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 ) ) )
% 12.85/6.94 = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 = zero_zero_nat )
% 12.85/6.94 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2325_even__mask__div__iff,axiom,
% 12.85/6.94 ! [M: nat,N: nat] :
% 12.85/6.94 ( ( 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 ) ) )
% 12.85/6.94 = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 = zero_zero_int )
% 12.85/6.94 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2326_divide__le__eq,axiom,
% 12.85/6.94 ! [B: rat,C: rat,A: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 12.85/6.94 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 12.85/6.94 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 12.85/6.94 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2327_divide__le__eq,axiom,
% 12.85/6.94 ! [B: real,C: real,A: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 12.85/6.94 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 12.85/6.94 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 12.85/6.94 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2328_le__divide__eq,axiom,
% 12.85/6.94 ! [A: rat,B: rat,C: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 12.85/6.94 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 12.85/6.94 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 12.85/6.94 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2329_le__divide__eq,axiom,
% 12.85/6.94 ! [A: real,B: real,C: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 12.85/6.94 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 12.85/6.94 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 12.85/6.94 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2330_divide__left__mono,axiom,
% 12.85/6.94 ! [B: rat,A: rat,C: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ B @ A )
% 12.85/6.94 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 12.85/6.94 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2331_divide__left__mono,axiom,
% 12.85/6.94 ! [B: real,A: real,C: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ B @ A )
% 12.85/6.94 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 12.85/6.94 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2332_neg__divide__le__eq,axiom,
% 12.85/6.94 ! [C: rat,B: rat,A: rat] :
% 12.85/6.94 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 12.85/6.94 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2333_neg__divide__le__eq,axiom,
% 12.85/6.94 ! [C: real,B: real,A: real] :
% 12.85/6.94 ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 12.85/6.94 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2334_neg__le__divide__eq,axiom,
% 12.85/6.94 ! [C: rat,A: rat,B: rat] :
% 12.85/6.94 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 12.85/6.94 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2335_neg__le__divide__eq,axiom,
% 12.85/6.94 ! [C: real,A: real,B: real] :
% 12.85/6.94 ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 12.85/6.94 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2336_pos__divide__le__eq,axiom,
% 12.85/6.94 ! [C: rat,B: rat,A: rat] :
% 12.85/6.94 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 12.85/6.94 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2337_pos__divide__le__eq,axiom,
% 12.85/6.94 ! [C: real,B: real,A: real] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 12.85/6.94 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2338_pos__le__divide__eq,axiom,
% 12.85/6.94 ! [C: rat,A: rat,B: rat] :
% 12.85/6.94 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 12.85/6.94 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2339_pos__le__divide__eq,axiom,
% 12.85/6.94 ! [C: real,A: real,B: real] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 12.85/6.94 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2340_mult__imp__div__pos__le,axiom,
% 12.85/6.94 ! [Y4: rat,X: rat,Z: rat] :
% 12.85/6.94 ( ( ord_less_rat @ zero_zero_rat @ Y4 )
% 12.85/6.94 => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y4 ) )
% 12.85/6.94 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y4 ) @ Z ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2341_mult__imp__div__pos__le,axiom,
% 12.85/6.94 ! [Y4: real,X: real,Z: real] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/6.94 => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y4 ) )
% 12.85/6.94 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ Z ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2342_mult__imp__le__div__pos,axiom,
% 12.85/6.94 ! [Y4: rat,Z: rat,X: rat] :
% 12.85/6.94 ( ( ord_less_rat @ zero_zero_rat @ Y4 )
% 12.85/6.94 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y4 ) @ X )
% 12.85/6.94 => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2343_mult__imp__le__div__pos,axiom,
% 12.85/6.94 ! [Y4: real,Z: real,X: real] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/6.94 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y4 ) @ X )
% 12.85/6.94 => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2344_divide__left__mono__neg,axiom,
% 12.85/6.94 ! [A: rat,B: rat,C: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.94 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 12.85/6.94 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2345_divide__left__mono__neg,axiom,
% 12.85/6.94 ! [A: real,B: real,C: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.94 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 12.85/6.94 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2346_divide__less__eq__numeral_I1_J,axiom,
% 12.85/6.94 ! [B: real,C: real,W: num] :
% 12.85/6.94 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 12.85/6.94 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 12.85/6.94 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 12.85/6.94 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2347_divide__less__eq__numeral_I1_J,axiom,
% 12.85/6.94 ! [B: rat,C: rat,W: num] :
% 12.85/6.94 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 12.85/6.94 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 12.85/6.94 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 12.85/6.94 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2348_less__divide__eq__numeral_I1_J,axiom,
% 12.85/6.94 ! [W: num,B: real,C: real] :
% 12.85/6.94 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 12.85/6.94 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 12.85/6.94 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 12.85/6.94 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2349_less__divide__eq__numeral_I1_J,axiom,
% 12.85/6.94 ! [W: num,B: rat,C: rat] :
% 12.85/6.94 ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 12.85/6.94 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 12.85/6.94 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 12.85/6.94 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2350_frac__less__eq,axiom,
% 12.85/6.94 ! [Y4: real,Z: real,X: real,W: real] :
% 12.85/6.94 ( ( Y4 != zero_zero_real )
% 12.85/6.94 => ( ( Z != zero_zero_real )
% 12.85/6.94 => ( ( ord_less_real @ ( divide_divide_real @ X @ Y4 ) @ ( divide_divide_real @ W @ Z ) )
% 12.85/6.94 = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y4 ) ) @ ( times_times_real @ Y4 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2351_frac__less__eq,axiom,
% 12.85/6.94 ! [Y4: rat,Z: rat,X: rat,W: rat] :
% 12.85/6.94 ( ( Y4 != zero_zero_rat )
% 12.85/6.94 => ( ( Z != zero_zero_rat )
% 12.85/6.94 => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y4 ) @ ( divide_divide_rat @ W @ Z ) )
% 12.85/6.94 = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y4 ) ) @ ( times_times_rat @ Y4 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2352_not__exp__less__eq__0__int,axiom,
% 12.85/6.94 ! [N: nat] :
% 12.85/6.94 ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2353_pos2,axiom,
% 12.85/6.94 ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2354_is__unitE,axiom,
% 12.85/6.94 ! [A: code_integer,C: code_integer] :
% 12.85/6.94 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.85/6.94 => ~ ( ( A != zero_z3403309356797280102nteger )
% 12.85/6.94 => ! [B3: code_integer] :
% 12.85/6.94 ( ( B3 != zero_z3403309356797280102nteger )
% 12.85/6.94 => ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
% 12.85/6.94 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 12.85/6.94 = B3 )
% 12.85/6.94 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B3 )
% 12.85/6.94 = A )
% 12.85/6.94 => ( ( ( times_3573771949741848930nteger @ A @ B3 )
% 12.85/6.94 = one_one_Code_integer )
% 12.85/6.94 => ( ( divide6298287555418463151nteger @ C @ A )
% 12.85/6.94 != ( times_3573771949741848930nteger @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2355_is__unitE,axiom,
% 12.85/6.94 ! [A: nat,C: nat] :
% 12.85/6.94 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.85/6.94 => ~ ( ( A != zero_zero_nat )
% 12.85/6.94 => ! [B3: nat] :
% 12.85/6.94 ( ( B3 != zero_zero_nat )
% 12.85/6.94 => ( ( dvd_dvd_nat @ B3 @ one_one_nat )
% 12.85/6.94 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 12.85/6.94 = B3 )
% 12.85/6.94 => ( ( ( divide_divide_nat @ one_one_nat @ B3 )
% 12.85/6.94 = A )
% 12.85/6.94 => ( ( ( times_times_nat @ A @ B3 )
% 12.85/6.94 = one_one_nat )
% 12.85/6.94 => ( ( divide_divide_nat @ C @ A )
% 12.85/6.94 != ( times_times_nat @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2356_is__unitE,axiom,
% 12.85/6.94 ! [A: int,C: int] :
% 12.85/6.94 ( ( dvd_dvd_int @ A @ one_one_int )
% 12.85/6.94 => ~ ( ( A != zero_zero_int )
% 12.85/6.94 => ! [B3: int] :
% 12.85/6.94 ( ( B3 != zero_zero_int )
% 12.85/6.94 => ( ( dvd_dvd_int @ B3 @ one_one_int )
% 12.85/6.94 => ( ( ( divide_divide_int @ one_one_int @ A )
% 12.85/6.94 = B3 )
% 12.85/6.94 => ( ( ( divide_divide_int @ one_one_int @ B3 )
% 12.85/6.94 = A )
% 12.85/6.94 => ( ( ( times_times_int @ A @ B3 )
% 12.85/6.94 = one_one_int )
% 12.85/6.94 => ( ( divide_divide_int @ C @ A )
% 12.85/6.94 != ( times_times_int @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2357_is__unit__div__mult__cancel__left,axiom,
% 12.85/6.94 ! [A: code_integer,B: code_integer] :
% 12.85/6.94 ( ( A != zero_z3403309356797280102nteger )
% 12.85/6.94 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 12.85/6.94 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
% 12.85/6.94 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2358_is__unit__div__mult__cancel__left,axiom,
% 12.85/6.94 ! [A: nat,B: nat] :
% 12.85/6.94 ( ( A != zero_zero_nat )
% 12.85/6.94 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 12.85/6.94 => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
% 12.85/6.94 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2359_is__unit__div__mult__cancel__left,axiom,
% 12.85/6.94 ! [A: int,B: int] :
% 12.85/6.94 ( ( A != zero_zero_int )
% 12.85/6.94 => ( ( dvd_dvd_int @ B @ one_one_int )
% 12.85/6.94 => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
% 12.85/6.94 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2360_is__unit__div__mult__cancel__right,axiom,
% 12.85/6.94 ! [A: code_integer,B: code_integer] :
% 12.85/6.94 ( ( A != zero_z3403309356797280102nteger )
% 12.85/6.94 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 12.85/6.94 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
% 12.85/6.94 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2361_is__unit__div__mult__cancel__right,axiom,
% 12.85/6.94 ! [A: nat,B: nat] :
% 12.85/6.94 ( ( A != zero_zero_nat )
% 12.85/6.94 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 12.85/6.94 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
% 12.85/6.94 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2362_is__unit__div__mult__cancel__right,axiom,
% 12.85/6.94 ! [A: int,B: int] :
% 12.85/6.94 ( ( A != zero_zero_int )
% 12.85/6.94 => ( ( dvd_dvd_int @ B @ one_one_int )
% 12.85/6.94 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
% 12.85/6.94 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2363_odd__one,axiom,
% 12.85/6.94 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2364_odd__one,axiom,
% 12.85/6.94 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2365_odd__one,axiom,
% 12.85/6.94 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2366_div__if,axiom,
% 12.85/6.94 ( divide_divide_nat
% 12.85/6.94 = ( ^ [M5: nat,N4: nat] :
% 12.85/6.94 ( if_nat
% 12.85/6.94 @ ( ( ord_less_nat @ M5 @ N4 )
% 12.85/6.94 | ( N4 = zero_zero_nat ) )
% 12.85/6.94 @ zero_zero_nat
% 12.85/6.94 @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M5 @ N4 ) @ N4 ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2367_div__geq,axiom,
% 12.85/6.94 ! [N: nat,M: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.94 => ( ~ ( ord_less_nat @ M @ N )
% 12.85/6.94 => ( ( divide_divide_nat @ M @ N )
% 12.85/6.94 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2368_div__nat__eqI,axiom,
% 12.85/6.94 ! [N: nat,Q2: nat,M: nat] :
% 12.85/6.94 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
% 12.85/6.94 => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
% 12.85/6.94 => ( ( divide_divide_nat @ M @ N )
% 12.85/6.94 = Q2 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2369_less__eq__div__iff__mult__less__eq,axiom,
% 12.85/6.94 ! [Q2: nat,M: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 12.85/6.94 => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
% 12.85/6.94 = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2370_even__mult__exp__div__exp__iff,axiom,
% 12.85/6.94 ! [A: code_integer,M: nat,N: nat] :
% 12.85/6.94 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 12.85/6.94 = ( ( ord_less_nat @ N @ M )
% 12.85/6.94 | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 = zero_z3403309356797280102nteger )
% 12.85/6.94 | ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.94 & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2371_even__mult__exp__div__exp__iff,axiom,
% 12.85/6.94 ! [A: nat,M: nat,N: nat] :
% 12.85/6.94 ( ( 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 ) ) )
% 12.85/6.94 = ( ( ord_less_nat @ N @ M )
% 12.85/6.94 | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 = zero_zero_nat )
% 12.85/6.94 | ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.94 & ( 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 ) ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2372_even__mult__exp__div__exp__iff,axiom,
% 12.85/6.94 ! [A: int,M: nat,N: nat] :
% 12.85/6.94 ( ( 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 ) ) )
% 12.85/6.94 = ( ( ord_less_nat @ N @ M )
% 12.85/6.94 | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 = zero_zero_int )
% 12.85/6.94 | ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.94 & ( 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 ) ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2373_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 12.85/6.94 ! [N: nat,M: nat] :
% 12.85/6.94 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 != zero_zero_nat )
% 12.85/6.94 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 12.85/6.94 != zero_zero_nat ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2374_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 12.85/6.94 ! [N: nat,M: nat] :
% 12.85/6.94 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 != zero_zero_int )
% 12.85/6.94 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 12.85/6.94 != zero_zero_int ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2375_power__mono__odd,axiom,
% 12.85/6.94 ! [N: nat,A: rat,B: rat] :
% 12.85/6.94 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 => ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.94 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2376_power__mono__odd,axiom,
% 12.85/6.94 ! [N: nat,A: int,B: int] :
% 12.85/6.94 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 => ( ( ord_less_eq_int @ A @ B )
% 12.85/6.94 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2377_power__mono__odd,axiom,
% 12.85/6.94 ! [N: nat,A: real,B: real] :
% 12.85/6.94 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 => ( ( ord_less_eq_real @ A @ B )
% 12.85/6.94 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2378_divide__le__eq__numeral_I1_J,axiom,
% 12.85/6.94 ! [B: rat,C: rat,W: num] :
% 12.85/6.94 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 12.85/6.94 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 12.85/6.94 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 12.85/6.94 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2379_divide__le__eq__numeral_I1_J,axiom,
% 12.85/6.94 ! [B: real,C: real,W: num] :
% 12.85/6.94 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 12.85/6.94 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 12.85/6.94 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 12.85/6.94 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2380_le__divide__eq__numeral_I1_J,axiom,
% 12.85/6.94 ! [W: num,B: rat,C: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 12.85/6.94 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 12.85/6.94 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.94 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 12.85/6.94 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.94 => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2381_le__divide__eq__numeral_I1_J,axiom,
% 12.85/6.94 ! [W: num,B: real,C: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 12.85/6.94 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 12.85/6.94 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 12.85/6.94 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.94 => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2382_half__gt__zero,axiom,
% 12.85/6.94 ! [A: real] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.94 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2383_half__gt__zero,axiom,
% 12.85/6.94 ! [A: rat] :
% 12.85/6.94 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.94 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2384_half__gt__zero__iff,axiom,
% 12.85/6.94 ! [A: real] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/6.94 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2385_half__gt__zero__iff,axiom,
% 12.85/6.94 ! [A: rat] :
% 12.85/6.94 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 12.85/6.94 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2386_four__x__squared,axiom,
% 12.85/6.94 ! [X: real] :
% 12.85/6.94 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.94 = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2387_less__2__cases,axiom,
% 12.85/6.94 ! [N: nat] :
% 12.85/6.94 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.94 => ( ( N = zero_zero_nat )
% 12.85/6.94 | ( N
% 12.85/6.94 = ( suc @ zero_zero_nat ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2388_less__2__cases__iff,axiom,
% 12.85/6.94 ! [N: nat] :
% 12.85/6.94 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.94 = ( ( N = zero_zero_nat )
% 12.85/6.94 | ( N
% 12.85/6.94 = ( suc @ zero_zero_nat ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2389_real__of__nat__div2,axiom,
% 12.85/6.94 ! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2390_odd__pos,axiom,
% 12.85/6.94 ! [N: nat] :
% 12.85/6.94 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2391_le__div__geq,axiom,
% 12.85/6.94 ! [N: nat,M: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.94 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.94 => ( ( divide_divide_nat @ M @ N )
% 12.85/6.94 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2392_split__div_H,axiom,
% 12.85/6.94 ! [P: nat > $o,M: nat,N: nat] :
% 12.85/6.94 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 12.85/6.94 = ( ( ( N = zero_zero_nat )
% 12.85/6.94 & ( P @ zero_zero_nat ) )
% 12.85/6.94 | ? [Q3: nat] :
% 12.85/6.94 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
% 12.85/6.94 & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
% 12.85/6.94 & ( P @ Q3 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2393_inverse__of__nat__le,axiom,
% 12.85/6.94 ! [N: nat,M: nat] :
% 12.85/6.94 ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.94 => ( ( N != zero_zero_nat )
% 12.85/6.94 => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2394_inverse__of__nat__le,axiom,
% 12.85/6.94 ! [N: nat,M: nat] :
% 12.85/6.94 ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.94 => ( ( N != zero_zero_nat )
% 12.85/6.94 => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2395_real__archimedian__rdiv__eq__0,axiom,
% 12.85/6.94 ! [X: real,C: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/6.94 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.94 => ( ! [M4: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ M4 )
% 12.85/6.94 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
% 12.85/6.94 => ( X = zero_zero_real ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2396_zero__le__power__eq,axiom,
% 12.85/6.94 ! [A: rat,N: nat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 12.85/6.94 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2397_zero__le__power__eq,axiom,
% 12.85/6.94 ! [A: int,N: nat] :
% 12.85/6.94 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 12.85/6.94 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2398_zero__le__power__eq,axiom,
% 12.85/6.94 ! [A: real,N: nat] :
% 12.85/6.94 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 12.85/6.94 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2399_zero__le__odd__power,axiom,
% 12.85/6.94 ! [N: nat,A: rat] :
% 12.85/6.94 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 12.85/6.94 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2400_zero__le__odd__power,axiom,
% 12.85/6.94 ! [N: nat,A: int] :
% 12.85/6.94 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 12.85/6.94 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2401_zero__le__odd__power,axiom,
% 12.85/6.94 ! [N: nat,A: real] :
% 12.85/6.94 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 12.85/6.94 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2402_zero__le__even__power,axiom,
% 12.85/6.94 ! [N: nat,A: rat] :
% 12.85/6.94 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2403_zero__le__even__power,axiom,
% 12.85/6.94 ! [N: nat,A: int] :
% 12.85/6.94 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2404_zero__le__even__power,axiom,
% 12.85/6.94 ! [N: nat,A: real] :
% 12.85/6.94 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.94 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2405_nat__bit__induct,axiom,
% 12.85/6.94 ! [P: nat > $o,N: nat] :
% 12.85/6.94 ( ( P @ zero_zero_nat )
% 12.85/6.94 => ( ! [N2: nat] :
% 12.85/6.94 ( ( P @ N2 )
% 12.85/6.94 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 12.85/6.94 => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 12.85/6.94 => ( ! [N2: nat] :
% 12.85/6.94 ( ( P @ N2 )
% 12.85/6.94 => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 12.85/6.94 => ( P @ N ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2406_div__2__gt__zero,axiom,
% 12.85/6.94 ! [N: nat] :
% 12.85/6.94 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.85/6.94 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2407_Suc__n__div__2__gt__zero,axiom,
% 12.85/6.94 ! [N: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.94 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2408_inf__period_I2_J,axiom,
% 12.85/6.94 ! [P: real > $o,D3: real,Q: real > $o] :
% 12.85/6.94 ( ! [X4: real,K2: real] :
% 12.85/6.94 ( ( P @ X4 )
% 12.85/6.94 = ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D3 ) ) ) )
% 12.85/6.94 => ( ! [X4: real,K2: real] :
% 12.85/6.94 ( ( Q @ X4 )
% 12.85/6.94 = ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D3 ) ) ) )
% 12.85/6.94 => ! [X5: real,K4: real] :
% 12.85/6.94 ( ( ( P @ X5 )
% 12.85/6.94 | ( Q @ X5 ) )
% 12.85/6.94 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) )
% 12.85/6.94 | ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2409_inf__period_I2_J,axiom,
% 12.85/6.94 ! [P: rat > $o,D3: rat,Q: rat > $o] :
% 12.85/6.94 ( ! [X4: rat,K2: rat] :
% 12.85/6.94 ( ( P @ X4 )
% 12.85/6.94 = ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D3 ) ) ) )
% 12.85/6.94 => ( ! [X4: rat,K2: rat] :
% 12.85/6.94 ( ( Q @ X4 )
% 12.85/6.94 = ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D3 ) ) ) )
% 12.85/6.94 => ! [X5: rat,K4: rat] :
% 12.85/6.94 ( ( ( P @ X5 )
% 12.85/6.94 | ( Q @ X5 ) )
% 12.85/6.94 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) )
% 12.85/6.94 | ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2410_inf__period_I2_J,axiom,
% 12.85/6.94 ! [P: int > $o,D3: int,Q: int > $o] :
% 12.85/6.94 ( ! [X4: int,K2: int] :
% 12.85/6.94 ( ( P @ X4 )
% 12.85/6.94 = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 12.85/6.94 => ( ! [X4: int,K2: int] :
% 12.85/6.94 ( ( Q @ X4 )
% 12.85/6.94 = ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 12.85/6.94 => ! [X5: int,K4: int] :
% 12.85/6.94 ( ( ( P @ X5 )
% 12.85/6.94 | ( Q @ X5 ) )
% 12.85/6.94 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
% 12.85/6.94 | ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2411_inf__period_I1_J,axiom,
% 12.85/6.94 ! [P: real > $o,D3: real,Q: real > $o] :
% 12.85/6.94 ( ! [X4: real,K2: real] :
% 12.85/6.94 ( ( P @ X4 )
% 12.85/6.94 = ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D3 ) ) ) )
% 12.85/6.94 => ( ! [X4: real,K2: real] :
% 12.85/6.94 ( ( Q @ X4 )
% 12.85/6.94 = ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D3 ) ) ) )
% 12.85/6.94 => ! [X5: real,K4: real] :
% 12.85/6.94 ( ( ( P @ X5 )
% 12.85/6.94 & ( Q @ X5 ) )
% 12.85/6.94 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) )
% 12.85/6.94 & ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2412_inf__period_I1_J,axiom,
% 12.85/6.94 ! [P: rat > $o,D3: rat,Q: rat > $o] :
% 12.85/6.94 ( ! [X4: rat,K2: rat] :
% 12.85/6.94 ( ( P @ X4 )
% 12.85/6.94 = ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D3 ) ) ) )
% 12.85/6.94 => ( ! [X4: rat,K2: rat] :
% 12.85/6.94 ( ( Q @ X4 )
% 12.85/6.94 = ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D3 ) ) ) )
% 12.85/6.94 => ! [X5: rat,K4: rat] :
% 12.85/6.94 ( ( ( P @ X5 )
% 12.85/6.94 & ( Q @ X5 ) )
% 12.85/6.94 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) )
% 12.85/6.94 & ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2413_inf__period_I1_J,axiom,
% 12.85/6.94 ! [P: int > $o,D3: int,Q: int > $o] :
% 12.85/6.94 ( ! [X4: int,K2: int] :
% 12.85/6.94 ( ( P @ X4 )
% 12.85/6.94 = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 12.85/6.94 => ( ! [X4: int,K2: int] :
% 12.85/6.94 ( ( Q @ X4 )
% 12.85/6.94 = ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 12.85/6.94 => ! [X5: int,K4: int] :
% 12.85/6.94 ( ( ( P @ X5 )
% 12.85/6.94 & ( Q @ X5 ) )
% 12.85/6.94 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
% 12.85/6.94 & ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2414_dvd__productE,axiom,
% 12.85/6.94 ! [P5: nat,A: nat,B: nat] :
% 12.85/6.94 ( ( dvd_dvd_nat @ P5 @ ( times_times_nat @ A @ B ) )
% 12.85/6.94 => ~ ! [X4: nat,Y: nat] :
% 12.85/6.94 ( ( P5
% 12.85/6.94 = ( times_times_nat @ X4 @ Y ) )
% 12.85/6.94 => ( ( dvd_dvd_nat @ X4 @ A )
% 12.85/6.94 => ~ ( dvd_dvd_nat @ Y @ B ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2415_dvd__productE,axiom,
% 12.85/6.94 ! [P5: int,A: int,B: int] :
% 12.85/6.94 ( ( dvd_dvd_int @ P5 @ ( times_times_int @ A @ B ) )
% 12.85/6.94 => ~ ! [X4: int,Y: int] :
% 12.85/6.94 ( ( P5
% 12.85/6.94 = ( times_times_int @ X4 @ Y ) )
% 12.85/6.94 => ( ( dvd_dvd_int @ X4 @ A )
% 12.85/6.94 => ~ ( dvd_dvd_int @ Y @ B ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2416_division__decomp,axiom,
% 12.85/6.94 ! [A: nat,B: nat,C: nat] :
% 12.85/6.94 ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 12.85/6.94 => ? [B7: nat,C4: nat] :
% 12.85/6.94 ( ( A
% 12.85/6.94 = ( times_times_nat @ B7 @ C4 ) )
% 12.85/6.94 & ( dvd_dvd_nat @ B7 @ B )
% 12.85/6.94 & ( dvd_dvd_nat @ C4 @ C ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2417_division__decomp,axiom,
% 12.85/6.94 ! [A: int,B: int,C: int] :
% 12.85/6.94 ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 12.85/6.94 => ? [B7: int,C4: int] :
% 12.85/6.94 ( ( A
% 12.85/6.94 = ( times_times_int @ B7 @ C4 ) )
% 12.85/6.94 & ( dvd_dvd_int @ B7 @ B )
% 12.85/6.94 & ( dvd_dvd_int @ C4 @ C ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2418_space__space_H,axiom,
% 12.85/6.94 ! [T: vEBT_VEBT] : ( ord_less_nat @ ( vEBT_VEBT_space @ T ) @ ( vEBT_VEBT_space2 @ T ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2419_numeral__power__le__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [I: num,N: nat,X: nat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 12.85/6.94 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2420_numeral__power__le__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [I: num,N: nat,X: nat] :
% 12.85/6.94 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 12.85/6.94 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2421_numeral__power__le__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [I: num,N: nat,X: nat] :
% 12.85/6.94 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 12.85/6.94 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2422_numeral__power__le__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [I: num,N: nat,X: nat] :
% 12.85/6.94 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 12.85/6.94 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2423_of__nat__le__numeral__power__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,I: num,N: nat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 12.85/6.94 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2424_of__nat__le__numeral__power__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,I: num,N: nat] :
% 12.85/6.94 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 12.85/6.94 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2425_of__nat__le__numeral__power__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,I: num,N: nat] :
% 12.85/6.94 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 12.85/6.94 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2426_of__nat__le__numeral__power__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,I: num,N: nat] :
% 12.85/6.94 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 12.85/6.94 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2427_numeral__power__less__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [I: num,N: nat,X: nat] :
% 12.85/6.94 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 12.85/6.94 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2428_numeral__power__less__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [I: num,N: nat,X: nat] :
% 12.85/6.94 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 12.85/6.94 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2429_numeral__power__less__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [I: num,N: nat,X: nat] :
% 12.85/6.94 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 12.85/6.94 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2430_numeral__power__less__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [I: num,N: nat,X: nat] :
% 12.85/6.94 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 12.85/6.94 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2431_of__nat__less__numeral__power__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,I: num,N: nat] :
% 12.85/6.94 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 12.85/6.94 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2432_of__nat__less__numeral__power__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,I: num,N: nat] :
% 12.85/6.94 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 12.85/6.94 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2433_of__nat__less__numeral__power__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,I: num,N: nat] :
% 12.85/6.94 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 12.85/6.94 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2434_of__nat__less__numeral__power__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,I: num,N: nat] :
% 12.85/6.94 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 12.85/6.94 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2435_of__nat__zero__less__power__iff,axiom,
% 12.85/6.94 ! [X: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
% 12.85/6.94 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 12.85/6.94 | ( N = zero_zero_nat ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2436_of__nat__zero__less__power__iff,axiom,
% 12.85/6.94 ! [X: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
% 12.85/6.94 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 12.85/6.94 | ( N = zero_zero_nat ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2437_of__nat__zero__less__power__iff,axiom,
% 12.85/6.94 ! [X: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
% 12.85/6.94 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 12.85/6.94 | ( N = zero_zero_nat ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2438_of__nat__zero__less__power__iff,axiom,
% 12.85/6.94 ! [X: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
% 12.85/6.94 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 12.85/6.94 | ( N = zero_zero_nat ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2439_zero__less__power2,axiom,
% 12.85/6.94 ! [A: real] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.94 = ( A != zero_zero_real ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2440_zero__less__power2,axiom,
% 12.85/6.94 ! [A: rat] :
% 12.85/6.94 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.94 = ( A != zero_zero_rat ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2441_zero__less__power2,axiom,
% 12.85/6.94 ! [A: int] :
% 12.85/6.94 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.94 = ( A != zero_zero_int ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2442_power2__less__eq__zero__iff,axiom,
% 12.85/6.94 ! [A: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 12.85/6.94 = ( A = zero_zero_rat ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2443_power2__less__eq__zero__iff,axiom,
% 12.85/6.94 ! [A: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 12.85/6.94 = ( A = zero_zero_int ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2444_power2__less__eq__zero__iff,axiom,
% 12.85/6.94 ! [A: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 12.85/6.94 = ( A = zero_zero_real ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2445_power2__eq__iff__nonneg,axiom,
% 12.85/6.94 ! [X: rat,Y4: rat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 12.85/6.94 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 12.85/6.94 => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.94 = ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.94 = ( X = Y4 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2446_power2__eq__iff__nonneg,axiom,
% 12.85/6.94 ! [X: nat,Y4: nat] :
% 12.85/6.94 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 12.85/6.94 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
% 12.85/6.94 => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.94 = ( power_power_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.94 = ( X = Y4 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2447_power2__eq__iff__nonneg,axiom,
% 12.85/6.94 ! [X: int,Y4: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/6.94 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/6.94 => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.94 = ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.94 = ( X = Y4 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2448_power2__eq__iff__nonneg,axiom,
% 12.85/6.94 ! [X: real,Y4: real] :
% 12.85/6.94 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/6.94 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/6.94 => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.94 = ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.94 = ( X = Y4 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2449_power__decreasing__iff,axiom,
% 12.85/6.94 ! [B: rat,M: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_rat @ zero_zero_rat @ B )
% 12.85/6.94 => ( ( ord_less_rat @ B @ one_one_rat )
% 12.85/6.94 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 12.85/6.94 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2450_power__decreasing__iff,axiom,
% 12.85/6.94 ! [B: nat,M: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ B )
% 12.85/6.94 => ( ( ord_less_nat @ B @ one_one_nat )
% 12.85/6.94 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 12.85/6.94 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2451_power__decreasing__iff,axiom,
% 12.85/6.94 ! [B: int,M: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.94 => ( ( ord_less_int @ B @ one_one_int )
% 12.85/6.94 => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 12.85/6.94 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2452_power__decreasing__iff,axiom,
% 12.85/6.94 ! [B: real,M: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ B )
% 12.85/6.94 => ( ( ord_less_real @ B @ one_one_real )
% 12.85/6.94 => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 12.85/6.94 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2453_power__mono__iff,axiom,
% 12.85/6.94 ! [A: rat,B: rat,N: nat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.94 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 12.85/6.94 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.94 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 12.85/6.94 = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2454_power__mono__iff,axiom,
% 12.85/6.94 ! [A: nat,B: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.94 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 12.85/6.94 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.94 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 12.85/6.94 = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2455_power__mono__iff,axiom,
% 12.85/6.94 ! [A: int,B: int,N: nat] :
% 12.85/6.94 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.94 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 12.85/6.94 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.94 => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 12.85/6.94 = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2456_power__mono__iff,axiom,
% 12.85/6.94 ! [A: real,B: real,N: nat] :
% 12.85/6.94 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.94 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 12.85/6.94 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.94 => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 12.85/6.94 = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2457_psubsetI,axiom,
% 12.85/6.94 ! [A2: set_int,B4: set_int] :
% 12.85/6.94 ( ( ord_less_eq_set_int @ A2 @ B4 )
% 12.85/6.94 => ( ( A2 != B4 )
% 12.85/6.94 => ( ord_less_set_int @ A2 @ B4 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2458_DiffI,axiom,
% 12.85/6.94 ! [C: complex,A2: set_complex,B4: set_complex] :
% 12.85/6.94 ( ( member_complex @ C @ A2 )
% 12.85/6.94 => ( ~ ( member_complex @ C @ B4 )
% 12.85/6.94 => ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B4 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2459_DiffI,axiom,
% 12.85/6.94 ! [C: real,A2: set_real,B4: set_real] :
% 12.85/6.94 ( ( member_real @ C @ A2 )
% 12.85/6.94 => ( ~ ( member_real @ C @ B4 )
% 12.85/6.94 => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2460_DiffI,axiom,
% 12.85/6.94 ! [C: set_nat,A2: set_set_nat,B4: set_set_nat] :
% 12.85/6.94 ( ( member_set_nat @ C @ A2 )
% 12.85/6.94 => ( ~ ( member_set_nat @ C @ B4 )
% 12.85/6.94 => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B4 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2461_DiffI,axiom,
% 12.85/6.94 ! [C: int,A2: set_int,B4: set_int] :
% 12.85/6.94 ( ( member_int @ C @ A2 )
% 12.85/6.94 => ( ~ ( member_int @ C @ B4 )
% 12.85/6.94 => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B4 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2462_DiffI,axiom,
% 12.85/6.94 ! [C: nat,A2: set_nat,B4: set_nat] :
% 12.85/6.94 ( ( member_nat @ C @ A2 )
% 12.85/6.94 => ( ~ ( member_nat @ C @ B4 )
% 12.85/6.94 => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B4 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2463_Diff__iff,axiom,
% 12.85/6.94 ! [C: complex,A2: set_complex,B4: set_complex] :
% 12.85/6.94 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B4 ) )
% 12.85/6.94 = ( ( member_complex @ C @ A2 )
% 12.85/6.94 & ~ ( member_complex @ C @ B4 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2464_Diff__iff,axiom,
% 12.85/6.94 ! [C: real,A2: set_real,B4: set_real] :
% 12.85/6.94 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) )
% 12.85/6.94 = ( ( member_real @ C @ A2 )
% 12.85/6.94 & ~ ( member_real @ C @ B4 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2465_Diff__iff,axiom,
% 12.85/6.94 ! [C: set_nat,A2: set_set_nat,B4: set_set_nat] :
% 12.85/6.94 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B4 ) )
% 12.85/6.94 = ( ( member_set_nat @ C @ A2 )
% 12.85/6.94 & ~ ( member_set_nat @ C @ B4 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2466_Diff__iff,axiom,
% 12.85/6.94 ! [C: int,A2: set_int,B4: set_int] :
% 12.85/6.94 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B4 ) )
% 12.85/6.94 = ( ( member_int @ C @ A2 )
% 12.85/6.94 & ~ ( member_int @ C @ B4 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2467_Diff__iff,axiom,
% 12.85/6.94 ! [C: nat,A2: set_nat,B4: set_nat] :
% 12.85/6.94 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B4 ) )
% 12.85/6.94 = ( ( member_nat @ C @ A2 )
% 12.85/6.94 & ~ ( member_nat @ C @ B4 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2468_Diff__idemp,axiom,
% 12.85/6.94 ! [A2: set_nat,B4: set_nat] :
% 12.85/6.94 ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B4 ) @ B4 )
% 12.85/6.94 = ( minus_minus_set_nat @ A2 @ B4 ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2469_i0__less,axiom,
% 12.85/6.94 ! [N: extended_enat] :
% 12.85/6.94 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 12.85/6.94 = ( N != zero_z5237406670263579293d_enat ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2470_two__realpow__ge__two,axiom,
% 12.85/6.94 ! [N: nat] :
% 12.85/6.94 ( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
% 12.85/6.94 => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2471_not__real__square__gt__zero,axiom,
% 12.85/6.94 ! [X: real] :
% 12.85/6.94 ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
% 12.85/6.94 = ( X = zero_zero_real ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2472_zle__diff1__eq,axiom,
% 12.85/6.94 ! [W: int,Z: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
% 12.85/6.94 = ( ord_less_int @ W @ Z ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2473_semiring__norm_I78_J,axiom,
% 12.85/6.94 ! [M: num,N: num] :
% 12.85/6.94 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.85/6.94 = ( ord_less_num @ M @ N ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2474_semiring__norm_I75_J,axiom,
% 12.85/6.94 ! [M: num] :
% 12.85/6.94 ~ ( ord_less_num @ M @ one ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2475_semiring__norm_I80_J,axiom,
% 12.85/6.94 ! [M: num,N: num] :
% 12.85/6.94 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.85/6.94 = ( ord_less_num @ M @ N ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2476_unset__bit__negative__int__iff,axiom,
% 12.85/6.94 ! [N: nat,K: int] :
% 12.85/6.94 ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
% 12.85/6.94 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2477_set__bit__negative__int__iff,axiom,
% 12.85/6.94 ! [N: nat,K: int] :
% 12.85/6.94 ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
% 12.85/6.94 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2478_flip__bit__negative__int__iff,axiom,
% 12.85/6.94 ! [N: nat,K: int] :
% 12.85/6.94 ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
% 12.85/6.94 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2479_power__0__Suc,axiom,
% 12.85/6.94 ! [N: nat] :
% 12.85/6.94 ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
% 12.85/6.94 = zero_zero_rat ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2480_power__0__Suc,axiom,
% 12.85/6.94 ! [N: nat] :
% 12.85/6.94 ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 12.85/6.94 = zero_zero_int ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2481_power__0__Suc,axiom,
% 12.85/6.94 ! [N: nat] :
% 12.85/6.94 ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 12.85/6.94 = zero_zero_real ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2482_power__0__Suc,axiom,
% 12.85/6.94 ! [N: nat] :
% 12.85/6.94 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 12.85/6.94 = zero_zero_nat ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2483_power__0__Suc,axiom,
% 12.85/6.94 ! [N: nat] :
% 12.85/6.94 ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
% 12.85/6.94 = zero_zero_complex ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2484_power__zero__numeral,axiom,
% 12.85/6.94 ! [K: num] :
% 12.85/6.94 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 12.85/6.94 = zero_zero_rat ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2485_power__zero__numeral,axiom,
% 12.85/6.94 ! [K: num] :
% 12.85/6.94 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 12.85/6.94 = zero_zero_int ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2486_power__zero__numeral,axiom,
% 12.85/6.94 ! [K: num] :
% 12.85/6.94 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 12.85/6.94 = zero_zero_real ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2487_power__zero__numeral,axiom,
% 12.85/6.94 ! [K: num] :
% 12.85/6.94 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 12.85/6.94 = zero_zero_nat ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2488_power__zero__numeral,axiom,
% 12.85/6.94 ! [K: num] :
% 12.85/6.94 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 12.85/6.94 = zero_zero_complex ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2489_power__Suc0__right,axiom,
% 12.85/6.94 ! [A: int] :
% 12.85/6.94 ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 12.85/6.94 = A ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2490_power__Suc0__right,axiom,
% 12.85/6.94 ! [A: real] :
% 12.85/6.94 ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 12.85/6.94 = A ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2491_power__Suc0__right,axiom,
% 12.85/6.94 ! [A: nat] :
% 12.85/6.94 ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 12.85/6.94 = A ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2492_power__Suc0__right,axiom,
% 12.85/6.94 ! [A: complex] :
% 12.85/6.94 ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 12.85/6.94 = A ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2493_power__Suc__0,axiom,
% 12.85/6.94 ! [N: nat] :
% 12.85/6.94 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.85/6.94 = ( suc @ zero_zero_nat ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2494_nat__power__eq__Suc__0__iff,axiom,
% 12.85/6.94 ! [X: nat,M: nat] :
% 12.85/6.94 ( ( ( power_power_nat @ X @ M )
% 12.85/6.94 = ( suc @ zero_zero_nat ) )
% 12.85/6.94 = ( ( M = zero_zero_nat )
% 12.85/6.94 | ( X
% 12.85/6.94 = ( suc @ zero_zero_nat ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2495_nat__zero__less__power__iff,axiom,
% 12.85/6.94 ! [X: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
% 12.85/6.94 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 12.85/6.94 | ( N = zero_zero_nat ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2496_of__nat__power,axiom,
% 12.85/6.94 ! [M: nat,N: nat] :
% 12.85/6.94 ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
% 12.85/6.94 = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2497_of__nat__power,axiom,
% 12.85/6.94 ! [M: nat,N: nat] :
% 12.85/6.94 ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
% 12.85/6.94 = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2498_of__nat__power,axiom,
% 12.85/6.94 ! [M: nat,N: nat] :
% 12.85/6.94 ( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N ) )
% 12.85/6.94 = ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2499_of__nat__power,axiom,
% 12.85/6.94 ! [M: nat,N: nat] :
% 12.85/6.94 ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
% 12.85/6.94 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2500_of__nat__power,axiom,
% 12.85/6.94 ! [M: nat,N: nat] :
% 12.85/6.94 ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
% 12.85/6.94 = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2501_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 12.85/6.94 ! [B: nat,W: nat,X: nat] :
% 12.85/6.94 ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
% 12.85/6.94 = ( semiri8010041392384452111omplex @ X ) )
% 12.85/6.94 = ( ( power_power_nat @ B @ W )
% 12.85/6.94 = X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2502_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 12.85/6.94 ! [B: nat,W: nat,X: nat] :
% 12.85/6.94 ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
% 12.85/6.94 = ( semiri5074537144036343181t_real @ X ) )
% 12.85/6.94 = ( ( power_power_nat @ B @ W )
% 12.85/6.94 = X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2503_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 12.85/6.94 ! [B: nat,W: nat,X: nat] :
% 12.85/6.94 ( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W )
% 12.85/6.94 = ( semiri681578069525770553at_rat @ X ) )
% 12.85/6.94 = ( ( power_power_nat @ B @ W )
% 12.85/6.94 = X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2504_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 12.85/6.94 ! [B: nat,W: nat,X: nat] :
% 12.85/6.94 ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
% 12.85/6.94 = ( semiri1316708129612266289at_nat @ X ) )
% 12.85/6.94 = ( ( power_power_nat @ B @ W )
% 12.85/6.94 = X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2505_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 12.85/6.94 ! [B: nat,W: nat,X: nat] :
% 12.85/6.94 ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
% 12.85/6.94 = ( semiri1314217659103216013at_int @ X ) )
% 12.85/6.94 = ( ( power_power_nat @ B @ W )
% 12.85/6.94 = X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2506_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,B: nat,W: nat] :
% 12.85/6.94 ( ( ( semiri8010041392384452111omplex @ X )
% 12.85/6.94 = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
% 12.85/6.94 = ( X
% 12.85/6.94 = ( power_power_nat @ B @ W ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2507_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,B: nat,W: nat] :
% 12.85/6.94 ( ( ( semiri5074537144036343181t_real @ X )
% 12.85/6.94 = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 12.85/6.94 = ( X
% 12.85/6.94 = ( power_power_nat @ B @ W ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2508_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,B: nat,W: nat] :
% 12.85/6.94 ( ( ( semiri681578069525770553at_rat @ X )
% 12.85/6.94 = ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 12.85/6.94 = ( X
% 12.85/6.94 = ( power_power_nat @ B @ W ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2509_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,B: nat,W: nat] :
% 12.85/6.94 ( ( ( semiri1316708129612266289at_nat @ X )
% 12.85/6.94 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 12.85/6.94 = ( X
% 12.85/6.94 = ( power_power_nat @ B @ W ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2510_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,B: nat,W: nat] :
% 12.85/6.94 ( ( ( semiri1314217659103216013at_int @ X )
% 12.85/6.94 = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 12.85/6.94 = ( X
% 12.85/6.94 = ( power_power_nat @ B @ W ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2511_div__neg__neg__trivial,axiom,
% 12.85/6.94 ! [K: int,L: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 12.85/6.94 => ( ( ord_less_int @ L @ K )
% 12.85/6.94 => ( ( divide_divide_int @ K @ L )
% 12.85/6.94 = zero_zero_int ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2512_div__pos__pos__trivial,axiom,
% 12.85/6.94 ! [K: int,L: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.85/6.94 => ( ( ord_less_int @ K @ L )
% 12.85/6.94 => ( ( divide_divide_int @ K @ L )
% 12.85/6.94 = zero_zero_int ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2513_power__mult__numeral,axiom,
% 12.85/6.94 ! [A: int,M: num,N: num] :
% 12.85/6.94 ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 12.85/6.94 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2514_power__mult__numeral,axiom,
% 12.85/6.94 ! [A: real,M: num,N: num] :
% 12.85/6.94 ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 12.85/6.94 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2515_power__mult__numeral,axiom,
% 12.85/6.94 ! [A: nat,M: num,N: num] :
% 12.85/6.94 ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 12.85/6.94 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2516_power__mult__numeral,axiom,
% 12.85/6.94 ! [A: complex,M: num,N: num] :
% 12.85/6.94 ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 12.85/6.94 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2517_semiring__norm_I76_J,axiom,
% 12.85/6.94 ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2518_semiring__norm_I81_J,axiom,
% 12.85/6.94 ! [M: num,N: num] :
% 12.85/6.94 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.85/6.94 = ( ord_less_num @ M @ N ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2519_semiring__norm_I77_J,axiom,
% 12.85/6.94 ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2520_enat__ord__number_I2_J,axiom,
% 12.85/6.94 ! [M: num,N: num] :
% 12.85/6.94 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 12.85/6.94 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2521_power__eq__0__iff,axiom,
% 12.85/6.94 ! [A: rat,N: nat] :
% 12.85/6.94 ( ( ( power_power_rat @ A @ N )
% 12.85/6.94 = zero_zero_rat )
% 12.85/6.94 = ( ( A = zero_zero_rat )
% 12.85/6.94 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2522_power__eq__0__iff,axiom,
% 12.85/6.94 ! [A: int,N: nat] :
% 12.85/6.94 ( ( ( power_power_int @ A @ N )
% 12.85/6.94 = zero_zero_int )
% 12.85/6.94 = ( ( A = zero_zero_int )
% 12.85/6.94 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2523_power__eq__0__iff,axiom,
% 12.85/6.94 ! [A: real,N: nat] :
% 12.85/6.94 ( ( ( power_power_real @ A @ N )
% 12.85/6.94 = zero_zero_real )
% 12.85/6.94 = ( ( A = zero_zero_real )
% 12.85/6.94 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2524_power__eq__0__iff,axiom,
% 12.85/6.94 ! [A: nat,N: nat] :
% 12.85/6.94 ( ( ( power_power_nat @ A @ N )
% 12.85/6.94 = zero_zero_nat )
% 12.85/6.94 = ( ( A = zero_zero_nat )
% 12.85/6.94 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2525_power__eq__0__iff,axiom,
% 12.85/6.94 ! [A: complex,N: nat] :
% 12.85/6.94 ( ( ( power_power_complex @ A @ N )
% 12.85/6.94 = zero_zero_complex )
% 12.85/6.94 = ( ( A = zero_zero_complex )
% 12.85/6.94 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2526_real__of__nat__less__numeral__iff,axiom,
% 12.85/6.94 ! [N: nat,W: num] :
% 12.85/6.94 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
% 12.85/6.94 = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2527_numeral__less__real__of__nat__iff,axiom,
% 12.85/6.94 ! [W: num,N: nat] :
% 12.85/6.94 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
% 12.85/6.94 = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2528_semiring__norm_I74_J,axiom,
% 12.85/6.94 ! [M: num,N: num] :
% 12.85/6.94 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.85/6.94 = ( ord_less_num @ M @ N ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2529_semiring__norm_I79_J,axiom,
% 12.85/6.94 ! [M: num,N: num] :
% 12.85/6.94 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.85/6.94 = ( ord_less_eq_num @ M @ N ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2530_power__strict__decreasing__iff,axiom,
% 12.85/6.94 ! [B: real,M: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ B )
% 12.85/6.94 => ( ( ord_less_real @ B @ one_one_real )
% 12.85/6.94 => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 12.85/6.94 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2531_power__strict__decreasing__iff,axiom,
% 12.85/6.94 ! [B: rat,M: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_rat @ zero_zero_rat @ B )
% 12.85/6.94 => ( ( ord_less_rat @ B @ one_one_rat )
% 12.85/6.94 => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 12.85/6.94 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2532_power__strict__decreasing__iff,axiom,
% 12.85/6.94 ! [B: nat,M: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ B )
% 12.85/6.94 => ( ( ord_less_nat @ B @ one_one_nat )
% 12.85/6.94 => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 12.85/6.94 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2533_power__strict__decreasing__iff,axiom,
% 12.85/6.94 ! [B: int,M: nat,N: nat] :
% 12.85/6.94 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.94 => ( ( ord_less_int @ B @ one_one_int )
% 12.85/6.94 => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 12.85/6.94 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2534_power__increasing__iff,axiom,
% 12.85/6.94 ! [B: rat,X: nat,Y4: nat] :
% 12.85/6.94 ( ( ord_less_rat @ one_one_rat @ B )
% 12.85/6.94 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y4 ) )
% 12.85/6.94 = ( ord_less_eq_nat @ X @ Y4 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2535_power__increasing__iff,axiom,
% 12.85/6.94 ! [B: nat,X: nat,Y4: nat] :
% 12.85/6.94 ( ( ord_less_nat @ one_one_nat @ B )
% 12.85/6.94 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y4 ) )
% 12.85/6.94 = ( ord_less_eq_nat @ X @ Y4 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2536_power__increasing__iff,axiom,
% 12.85/6.94 ! [B: int,X: nat,Y4: nat] :
% 12.85/6.94 ( ( ord_less_int @ one_one_int @ B )
% 12.85/6.94 => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y4 ) )
% 12.85/6.94 = ( ord_less_eq_nat @ X @ Y4 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2537_power__increasing__iff,axiom,
% 12.85/6.94 ! [B: real,X: nat,Y4: nat] :
% 12.85/6.94 ( ( ord_less_real @ one_one_real @ B )
% 12.85/6.94 => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y4 ) )
% 12.85/6.94 = ( ord_less_eq_nat @ X @ Y4 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2538_zero__eq__power2,axiom,
% 12.85/6.94 ! [A: rat] :
% 12.85/6.94 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.94 = zero_zero_rat )
% 12.85/6.94 = ( A = zero_zero_rat ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2539_zero__eq__power2,axiom,
% 12.85/6.94 ! [A: int] :
% 12.85/6.94 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.94 = zero_zero_int )
% 12.85/6.94 = ( A = zero_zero_int ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2540_zero__eq__power2,axiom,
% 12.85/6.94 ! [A: real] :
% 12.85/6.94 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.94 = zero_zero_real )
% 12.85/6.94 = ( A = zero_zero_real ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2541_zero__eq__power2,axiom,
% 12.85/6.94 ! [A: nat] :
% 12.85/6.94 ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.94 = zero_zero_nat )
% 12.85/6.94 = ( A = zero_zero_nat ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2542_zero__eq__power2,axiom,
% 12.85/6.94 ! [A: complex] :
% 12.85/6.94 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.94 = zero_zero_complex )
% 12.85/6.94 = ( A = zero_zero_complex ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2543_of__nat__less__of__nat__power__cancel__iff,axiom,
% 12.85/6.94 ! [B: nat,W: nat,X: nat] :
% 12.85/6.94 ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 12.85/6.94 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2544_of__nat__less__of__nat__power__cancel__iff,axiom,
% 12.85/6.94 ! [B: nat,W: nat,X: nat] :
% 12.85/6.94 ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 12.85/6.94 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2545_of__nat__less__of__nat__power__cancel__iff,axiom,
% 12.85/6.94 ! [B: nat,W: nat,X: nat] :
% 12.85/6.94 ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 12.85/6.94 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2546_of__nat__less__of__nat__power__cancel__iff,axiom,
% 12.85/6.94 ! [B: nat,W: nat,X: nat] :
% 12.85/6.94 ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 12.85/6.94 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2547_of__nat__power__less__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,B: nat,W: nat] :
% 12.85/6.94 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 12.85/6.94 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2548_of__nat__power__less__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,B: nat,W: nat] :
% 12.85/6.94 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 12.85/6.94 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2549_of__nat__power__less__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,B: nat,W: nat] :
% 12.85/6.94 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 12.85/6.94 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2550_of__nat__power__less__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,B: nat,W: nat] :
% 12.85/6.94 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 12.85/6.94 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2551_of__nat__power__le__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,B: nat,W: nat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 12.85/6.94 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2552_of__nat__power__le__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,B: nat,W: nat] :
% 12.85/6.94 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 12.85/6.94 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2553_of__nat__power__le__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,B: nat,W: nat] :
% 12.85/6.94 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 12.85/6.94 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2554_of__nat__power__le__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [X: nat,B: nat,W: nat] :
% 12.85/6.94 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 12.85/6.94 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2555_of__nat__le__of__nat__power__cancel__iff,axiom,
% 12.85/6.94 ! [B: nat,W: nat,X: nat] :
% 12.85/6.94 ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 12.85/6.94 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2556_of__nat__le__of__nat__power__cancel__iff,axiom,
% 12.85/6.94 ! [B: nat,W: nat,X: nat] :
% 12.85/6.94 ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 12.85/6.94 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2557_of__nat__le__of__nat__power__cancel__iff,axiom,
% 12.85/6.94 ! [B: nat,W: nat,X: nat] :
% 12.85/6.94 ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 12.85/6.94 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2558_of__nat__le__of__nat__power__cancel__iff,axiom,
% 12.85/6.94 ! [B: nat,W: nat,X: nat] :
% 12.85/6.94 ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 12.85/6.94 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2559_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 12.85/6.94 ! [Y4: nat,X: num,N: nat] :
% 12.85/6.94 ( ( ( semiri8010041392384452111omplex @ Y4 )
% 12.85/6.94 = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
% 12.85/6.94 = ( Y4
% 12.85/6.94 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2560_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 12.85/6.94 ! [Y4: nat,X: num,N: nat] :
% 12.85/6.94 ( ( ( semiri5074537144036343181t_real @ Y4 )
% 12.85/6.94 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 12.85/6.94 = ( Y4
% 12.85/6.94 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2561_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 12.85/6.94 ! [Y4: nat,X: num,N: nat] :
% 12.85/6.94 ( ( ( semiri681578069525770553at_rat @ Y4 )
% 12.85/6.94 = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 12.85/6.94 = ( Y4
% 12.85/6.94 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2562_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 12.85/6.94 ! [Y4: nat,X: num,N: nat] :
% 12.85/6.94 ( ( ( semiri1316708129612266289at_nat @ Y4 )
% 12.85/6.94 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 12.85/6.94 = ( Y4
% 12.85/6.94 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2563_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 12.85/6.94 ! [Y4: nat,X: num,N: nat] :
% 12.85/6.94 ( ( ( semiri1314217659103216013at_int @ Y4 )
% 12.85/6.94 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 12.85/6.94 = ( Y4
% 12.85/6.94 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2564_numeral__power__eq__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [X: num,N: nat,Y4: nat] :
% 12.85/6.94 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
% 12.85/6.94 = ( semiri8010041392384452111omplex @ Y4 ) )
% 12.85/6.94 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 12.85/6.94 = Y4 ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2565_numeral__power__eq__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [X: num,N: nat,Y4: nat] :
% 12.85/6.94 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 12.85/6.94 = ( semiri5074537144036343181t_real @ Y4 ) )
% 12.85/6.94 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 12.85/6.94 = Y4 ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2566_numeral__power__eq__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [X: num,N: nat,Y4: nat] :
% 12.85/6.94 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
% 12.85/6.94 = ( semiri681578069525770553at_rat @ Y4 ) )
% 12.85/6.94 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 12.85/6.94 = Y4 ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2567_numeral__power__eq__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [X: num,N: nat,Y4: nat] :
% 12.85/6.94 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 12.85/6.94 = ( semiri1316708129612266289at_nat @ Y4 ) )
% 12.85/6.94 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 12.85/6.94 = Y4 ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2568_numeral__power__eq__of__nat__cancel__iff,axiom,
% 12.85/6.94 ! [X: num,N: nat,Y4: nat] :
% 12.85/6.94 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 12.85/6.94 = ( semiri1314217659103216013at_int @ Y4 ) )
% 12.85/6.94 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 12.85/6.94 = Y4 ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2569_half__negative__int__iff,axiom,
% 12.85/6.94 ! [K: int] :
% 12.85/6.94 ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 12.85/6.94 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2570_DiffE,axiom,
% 12.85/6.94 ! [C: complex,A2: set_complex,B4: set_complex] :
% 12.85/6.94 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B4 ) )
% 12.85/6.94 => ~ ( ( member_complex @ C @ A2 )
% 12.85/6.94 => ( member_complex @ C @ B4 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2571_DiffE,axiom,
% 12.85/6.94 ! [C: real,A2: set_real,B4: set_real] :
% 12.85/6.94 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) )
% 12.85/6.94 => ~ ( ( member_real @ C @ A2 )
% 12.85/6.94 => ( member_real @ C @ B4 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2572_DiffE,axiom,
% 12.85/6.94 ! [C: set_nat,A2: set_set_nat,B4: set_set_nat] :
% 12.85/6.94 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B4 ) )
% 12.85/6.94 => ~ ( ( member_set_nat @ C @ A2 )
% 12.85/6.94 => ( member_set_nat @ C @ B4 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2573_DiffE,axiom,
% 12.85/6.94 ! [C: int,A2: set_int,B4: set_int] :
% 12.85/6.94 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B4 ) )
% 12.85/6.94 => ~ ( ( member_int @ C @ A2 )
% 12.85/6.94 => ( member_int @ C @ B4 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2574_DiffE,axiom,
% 12.85/6.94 ! [C: nat,A2: set_nat,B4: set_nat] :
% 12.85/6.94 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B4 ) )
% 12.85/6.94 => ~ ( ( member_nat @ C @ A2 )
% 12.85/6.94 => ( member_nat @ C @ B4 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2575_DiffD1,axiom,
% 12.85/6.94 ! [C: complex,A2: set_complex,B4: set_complex] :
% 12.85/6.94 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B4 ) )
% 12.85/6.94 => ( member_complex @ C @ A2 ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2576_DiffD1,axiom,
% 12.85/6.94 ! [C: real,A2: set_real,B4: set_real] :
% 12.85/6.94 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) )
% 12.85/6.94 => ( member_real @ C @ A2 ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2577_DiffD1,axiom,
% 12.85/6.94 ! [C: set_nat,A2: set_set_nat,B4: set_set_nat] :
% 12.85/6.94 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B4 ) )
% 12.85/6.94 => ( member_set_nat @ C @ A2 ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2578_DiffD1,axiom,
% 12.85/6.94 ! [C: int,A2: set_int,B4: set_int] :
% 12.85/6.94 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B4 ) )
% 12.85/6.94 => ( member_int @ C @ A2 ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2579_DiffD1,axiom,
% 12.85/6.94 ! [C: nat,A2: set_nat,B4: set_nat] :
% 12.85/6.94 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B4 ) )
% 12.85/6.94 => ( member_nat @ C @ A2 ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2580_DiffD2,axiom,
% 12.85/6.94 ! [C: complex,A2: set_complex,B4: set_complex] :
% 12.85/6.94 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B4 ) )
% 12.85/6.94 => ~ ( member_complex @ C @ B4 ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2581_DiffD2,axiom,
% 12.85/6.94 ! [C: real,A2: set_real,B4: set_real] :
% 12.85/6.94 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) )
% 12.85/6.94 => ~ ( member_real @ C @ B4 ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2582_DiffD2,axiom,
% 12.85/6.94 ! [C: set_nat,A2: set_set_nat,B4: set_set_nat] :
% 12.85/6.94 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B4 ) )
% 12.85/6.94 => ~ ( member_set_nat @ C @ B4 ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2583_DiffD2,axiom,
% 12.85/6.94 ! [C: int,A2: set_int,B4: set_int] :
% 12.85/6.94 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B4 ) )
% 12.85/6.94 => ~ ( member_int @ C @ B4 ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2584_DiffD2,axiom,
% 12.85/6.94 ! [C: nat,A2: set_nat,B4: set_nat] :
% 12.85/6.94 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B4 ) )
% 12.85/6.94 => ~ ( member_nat @ C @ B4 ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2585_psubset__imp__ex__mem,axiom,
% 12.85/6.94 ! [A2: set_complex,B4: set_complex] :
% 12.85/6.94 ( ( ord_less_set_complex @ A2 @ B4 )
% 12.85/6.94 => ? [B3: complex] : ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B4 @ A2 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2586_psubset__imp__ex__mem,axiom,
% 12.85/6.94 ! [A2: set_real,B4: set_real] :
% 12.85/6.94 ( ( ord_less_set_real @ A2 @ B4 )
% 12.85/6.94 => ? [B3: real] : ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A2 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2587_psubset__imp__ex__mem,axiom,
% 12.85/6.94 ! [A2: set_set_nat,B4: set_set_nat] :
% 12.85/6.94 ( ( ord_less_set_set_nat @ A2 @ B4 )
% 12.85/6.94 => ? [B3: set_nat] : ( member_set_nat @ B3 @ ( minus_2163939370556025621et_nat @ B4 @ A2 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2588_psubset__imp__ex__mem,axiom,
% 12.85/6.94 ! [A2: set_int,B4: set_int] :
% 12.85/6.94 ( ( ord_less_set_int @ A2 @ B4 )
% 12.85/6.94 => ? [B3: int] : ( member_int @ B3 @ ( minus_minus_set_int @ B4 @ A2 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2589_psubset__imp__ex__mem,axiom,
% 12.85/6.94 ! [A2: set_nat,B4: set_nat] :
% 12.85/6.94 ( ( ord_less_set_nat @ A2 @ B4 )
% 12.85/6.94 => ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B4 @ A2 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2590_int__less__induct,axiom,
% 12.85/6.94 ! [I: int,K: int,P: int > $o] :
% 12.85/6.94 ( ( ord_less_int @ I @ K )
% 12.85/6.94 => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 12.85/6.94 => ( ! [I3: int] :
% 12.85/6.94 ( ( ord_less_int @ I3 @ K )
% 12.85/6.94 => ( ( P @ I3 )
% 12.85/6.94 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 12.85/6.94 => ( P @ I ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2591_real__arch__pow,axiom,
% 12.85/6.94 ! [X: real,Y4: real] :
% 12.85/6.94 ( ( ord_less_real @ one_one_real @ X )
% 12.85/6.94 => ? [N2: nat] : ( ord_less_real @ Y4 @ ( power_power_real @ X @ N2 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2592_real__arch__pow__inv,axiom,
% 12.85/6.94 ! [Y4: real,X: real] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/6.94 => ( ( ord_less_real @ X @ one_one_real )
% 12.85/6.94 => ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X @ N2 ) @ Y4 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2593_int__one__le__iff__zero__less,axiom,
% 12.85/6.94 ! [Z: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ one_one_int @ Z )
% 12.85/6.94 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2594_pos__zmult__eq__1__iff,axiom,
% 12.85/6.94 ! [M: int,N: int] :
% 12.85/6.94 ( ( ord_less_int @ zero_zero_int @ M )
% 12.85/6.94 => ( ( ( times_times_int @ M @ N )
% 12.85/6.94 = one_one_int )
% 12.85/6.94 = ( ( M = one_one_int )
% 12.85/6.94 & ( N = one_one_int ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2595_int__div__less__self,axiom,
% 12.85/6.94 ! [X: int,K: int] :
% 12.85/6.94 ( ( ord_less_int @ zero_zero_int @ X )
% 12.85/6.94 => ( ( ord_less_int @ one_one_int @ K )
% 12.85/6.94 => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2596_psubsetE,axiom,
% 12.85/6.94 ! [A2: set_int,B4: set_int] :
% 12.85/6.94 ( ( ord_less_set_int @ A2 @ B4 )
% 12.85/6.94 => ~ ( ( ord_less_eq_set_int @ A2 @ B4 )
% 12.85/6.94 => ( ord_less_eq_set_int @ B4 @ A2 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2597_psubset__eq,axiom,
% 12.85/6.94 ( ord_less_set_int
% 12.85/6.94 = ( ^ [A5: set_int,B5: set_int] :
% 12.85/6.94 ( ( ord_less_eq_set_int @ A5 @ B5 )
% 12.85/6.94 & ( A5 != B5 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2598_psubset__imp__subset,axiom,
% 12.85/6.94 ! [A2: set_int,B4: set_int] :
% 12.85/6.94 ( ( ord_less_set_int @ A2 @ B4 )
% 12.85/6.94 => ( ord_less_eq_set_int @ A2 @ B4 ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2599_psubset__subset__trans,axiom,
% 12.85/6.94 ! [A2: set_int,B4: set_int,C2: set_int] :
% 12.85/6.94 ( ( ord_less_set_int @ A2 @ B4 )
% 12.85/6.94 => ( ( ord_less_eq_set_int @ B4 @ C2 )
% 12.85/6.94 => ( ord_less_set_int @ A2 @ C2 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2600_subset__not__subset__eq,axiom,
% 12.85/6.94 ( ord_less_set_int
% 12.85/6.94 = ( ^ [A5: set_int,B5: set_int] :
% 12.85/6.94 ( ( ord_less_eq_set_int @ A5 @ B5 )
% 12.85/6.94 & ~ ( ord_less_eq_set_int @ B5 @ A5 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2601_subset__psubset__trans,axiom,
% 12.85/6.94 ! [A2: set_int,B4: set_int,C2: set_int] :
% 12.85/6.94 ( ( ord_less_eq_set_int @ A2 @ B4 )
% 12.85/6.94 => ( ( ord_less_set_int @ B4 @ C2 )
% 12.85/6.94 => ( ord_less_set_int @ A2 @ C2 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2602_subset__iff__psubset__eq,axiom,
% 12.85/6.94 ( ord_less_eq_set_int
% 12.85/6.94 = ( ^ [A5: set_int,B5: set_int] :
% 12.85/6.94 ( ( ord_less_set_int @ A5 @ B5 )
% 12.85/6.94 | ( A5 = B5 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2603_less__int__code_I1_J,axiom,
% 12.85/6.94 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2604_less__eq__real__def,axiom,
% 12.85/6.94 ( ord_less_eq_real
% 12.85/6.94 = ( ^ [X3: real,Y6: real] :
% 12.85/6.94 ( ( ord_less_real @ X3 @ Y6 )
% 12.85/6.94 | ( X3 = Y6 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2605_int__le__induct,axiom,
% 12.85/6.94 ! [I: int,K: int,P: int > $o] :
% 12.85/6.94 ( ( ord_less_eq_int @ I @ K )
% 12.85/6.94 => ( ( P @ K )
% 12.85/6.94 => ( ! [I3: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ I3 @ K )
% 12.85/6.94 => ( ( P @ I3 )
% 12.85/6.94 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 12.85/6.94 => ( P @ I ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2606_enat__0__less__mult__iff,axiom,
% 12.85/6.94 ! [M: extended_enat,N: extended_enat] :
% 12.85/6.94 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
% 12.85/6.94 = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 12.85/6.94 & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2607_not__iless0,axiom,
% 12.85/6.94 ! [N: extended_enat] :
% 12.85/6.94 ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2608_zero__one__enat__neq_I1_J,axiom,
% 12.85/6.94 zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2609_realpow__pos__nth2,axiom,
% 12.85/6.94 ! [A: real,N: nat] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.94 => ? [R3: real] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ R3 )
% 12.85/6.94 & ( ( power_power_real @ R3 @ ( suc @ N ) )
% 12.85/6.94 = A ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2610_nat__int__comparison_I2_J,axiom,
% 12.85/6.94 ( ord_less_nat
% 12.85/6.94 = ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2611_zmult__zless__mono2,axiom,
% 12.85/6.94 ! [I: int,J: int,K: int] :
% 12.85/6.94 ( ( ord_less_int @ I @ J )
% 12.85/6.94 => ( ( ord_less_int @ zero_zero_int @ K )
% 12.85/6.94 => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2612_plusinfinity,axiom,
% 12.85/6.94 ! [D: int,P4: int > $o,P: int > $o] :
% 12.85/6.94 ( ( ord_less_int @ zero_zero_int @ D )
% 12.85/6.94 => ( ! [X4: int,K2: int] :
% 12.85/6.94 ( ( P4 @ X4 )
% 12.85/6.94 = ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 12.85/6.94 => ( ? [Z3: int] :
% 12.85/6.94 ! [X4: int] :
% 12.85/6.94 ( ( ord_less_int @ Z3 @ X4 )
% 12.85/6.94 => ( ( P @ X4 )
% 12.85/6.94 = ( P4 @ X4 ) ) )
% 12.85/6.94 => ( ? [X_12: int] : ( P4 @ X_12 )
% 12.85/6.94 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2613_minusinfinity,axiom,
% 12.85/6.94 ! [D: int,P1: int > $o,P: int > $o] :
% 12.85/6.94 ( ( ord_less_int @ zero_zero_int @ D )
% 12.85/6.94 => ( ! [X4: int,K2: int] :
% 12.85/6.94 ( ( P1 @ X4 )
% 12.85/6.94 = ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 12.85/6.94 => ( ? [Z3: int] :
% 12.85/6.94 ! [X4: int] :
% 12.85/6.94 ( ( ord_less_int @ X4 @ Z3 )
% 12.85/6.94 => ( ( P @ X4 )
% 12.85/6.94 = ( P1 @ X4 ) ) )
% 12.85/6.94 => ( ? [X_12: int] : ( P1 @ X_12 )
% 12.85/6.94 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2614_pos__imp__zdiv__neg__iff,axiom,
% 12.85/6.94 ! [B: int,A: int] :
% 12.85/6.94 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.94 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 12.85/6.94 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2615_neg__imp__zdiv__neg__iff,axiom,
% 12.85/6.94 ! [B: int,A: int] :
% 12.85/6.94 ( ( ord_less_int @ B @ zero_zero_int )
% 12.85/6.94 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 12.85/6.94 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2616_div__neg__pos__less0,axiom,
% 12.85/6.94 ! [A: int,B: int] :
% 12.85/6.94 ( ( ord_less_int @ A @ zero_zero_int )
% 12.85/6.94 => ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.94 => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2617_zdvd__not__zless,axiom,
% 12.85/6.94 ! [M: int,N: int] :
% 12.85/6.94 ( ( ord_less_int @ zero_zero_int @ M )
% 12.85/6.94 => ( ( ord_less_int @ M @ N )
% 12.85/6.94 => ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2618_int__ops_I2_J,axiom,
% 12.85/6.94 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 12.85/6.94 = one_one_int ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2619_realpow__pos__nth__unique,axiom,
% 12.85/6.94 ! [N: nat,A: real] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.94 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.94 => ? [X4: real] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.85/6.94 & ( ( power_power_real @ X4 @ N )
% 12.85/6.94 = A )
% 12.85/6.94 & ! [Y3: real] :
% 12.85/6.94 ( ( ( ord_less_real @ zero_zero_real @ Y3 )
% 12.85/6.94 & ( ( power_power_real @ Y3 @ N )
% 12.85/6.94 = A ) )
% 12.85/6.94 => ( Y3 = X4 ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2620_realpow__pos__nth,axiom,
% 12.85/6.94 ! [N: nat,A: real] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.94 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.94 => ? [R3: real] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ R3 )
% 12.85/6.94 & ( ( power_power_real @ R3 @ N )
% 12.85/6.94 = A ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2621_int__ops_I6_J,axiom,
% 12.85/6.94 ! [A: nat,B: nat] :
% 12.85/6.94 ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 12.85/6.94 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 12.85/6.94 = zero_zero_int ) )
% 12.85/6.94 & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 12.85/6.94 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 12.85/6.94 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2622_decr__mult__lemma,axiom,
% 12.85/6.94 ! [D: int,P: int > $o,K: int] :
% 12.85/6.94 ( ( ord_less_int @ zero_zero_int @ D )
% 12.85/6.94 => ( ! [X4: int] :
% 12.85/6.94 ( ( P @ X4 )
% 12.85/6.94 => ( P @ ( minus_minus_int @ X4 @ D ) ) )
% 12.85/6.94 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.85/6.94 => ! [X5: int] :
% 12.85/6.94 ( ( P @ X5 )
% 12.85/6.94 => ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2623_reals__Archimedean3,axiom,
% 12.85/6.94 ! [X: real] :
% 12.85/6.94 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.94 => ! [Y3: real] :
% 12.85/6.94 ? [N2: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2624_zdiv__mono1,axiom,
% 12.85/6.94 ! [A: int,A6: int,B: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ A @ A6 )
% 12.85/6.94 => ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.94 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A6 @ B ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2625_zdiv__mono2,axiom,
% 12.85/6.94 ! [A: int,B6: int,B: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.94 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 12.85/6.94 => ( ( ord_less_eq_int @ B6 @ B )
% 12.85/6.94 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B6 ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2626_zdiv__eq__0__iff,axiom,
% 12.85/6.94 ! [I: int,K: int] :
% 12.85/6.94 ( ( ( divide_divide_int @ I @ K )
% 12.85/6.94 = zero_zero_int )
% 12.85/6.94 = ( ( K = zero_zero_int )
% 12.85/6.94 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 12.85/6.94 & ( ord_less_int @ I @ K ) )
% 12.85/6.94 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 12.85/6.94 & ( ord_less_int @ K @ I ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2627_zdiv__mono1__neg,axiom,
% 12.85/6.94 ! [A: int,A6: int,B: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ A @ A6 )
% 12.85/6.94 => ( ( ord_less_int @ B @ zero_zero_int )
% 12.85/6.94 => ( ord_less_eq_int @ ( divide_divide_int @ A6 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2628_zdiv__mono2__neg,axiom,
% 12.85/6.94 ! [A: int,B6: int,B: int] :
% 12.85/6.94 ( ( ord_less_int @ A @ zero_zero_int )
% 12.85/6.94 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 12.85/6.94 => ( ( ord_less_eq_int @ B6 @ B )
% 12.85/6.94 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B6 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2629_div__int__pos__iff,axiom,
% 12.85/6.94 ! [K: int,L: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
% 12.85/6.94 = ( ( K = zero_zero_int )
% 12.85/6.94 | ( L = zero_zero_int )
% 12.85/6.94 | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.85/6.94 & ( ord_less_eq_int @ zero_zero_int @ L ) )
% 12.85/6.94 | ( ( ord_less_int @ K @ zero_zero_int )
% 12.85/6.94 & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2630_div__positive__int,axiom,
% 12.85/6.94 ! [L: int,K: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ L @ K )
% 12.85/6.94 => ( ( ord_less_int @ zero_zero_int @ L )
% 12.85/6.94 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2631_div__nonneg__neg__le0,axiom,
% 12.85/6.94 ! [A: int,B: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.94 => ( ( ord_less_int @ B @ zero_zero_int )
% 12.85/6.94 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2632_div__nonpos__pos__le0,axiom,
% 12.85/6.94 ! [A: int,B: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.85/6.94 => ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.94 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2633_pos__imp__zdiv__pos__iff,axiom,
% 12.85/6.94 ! [K: int,I: int] :
% 12.85/6.94 ( ( ord_less_int @ zero_zero_int @ K )
% 12.85/6.94 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
% 12.85/6.94 = ( ord_less_eq_int @ K @ I ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2634_neg__imp__zdiv__nonneg__iff,axiom,
% 12.85/6.94 ! [B: int,A: int] :
% 12.85/6.94 ( ( ord_less_int @ B @ zero_zero_int )
% 12.85/6.94 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 12.85/6.94 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2635_pos__imp__zdiv__nonneg__iff,axiom,
% 12.85/6.94 ! [B: int,A: int] :
% 12.85/6.94 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.94 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 12.85/6.94 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2636_nonneg1__imp__zdiv__pos__iff,axiom,
% 12.85/6.94 ! [A: int,B: int] :
% 12.85/6.94 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.94 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 12.85/6.94 = ( ( ord_less_eq_int @ B @ A )
% 12.85/6.94 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2637_zdvd__imp__le,axiom,
% 12.85/6.94 ! [Z: int,N: int] :
% 12.85/6.94 ( ( dvd_dvd_int @ Z @ N )
% 12.85/6.94 => ( ( ord_less_int @ zero_zero_int @ N )
% 12.85/6.94 => ( ord_less_eq_int @ Z @ N ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2638_pos__int__cases,axiom,
% 12.85/6.94 ! [K: int] :
% 12.85/6.94 ( ( ord_less_int @ zero_zero_int @ K )
% 12.85/6.94 => ~ ! [N2: nat] :
% 12.85/6.94 ( ( K
% 12.85/6.94 = ( semiri1314217659103216013at_int @ N2 ) )
% 12.85/6.94 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2639_zero__less__imp__eq__int,axiom,
% 12.85/6.94 ! [K: int] :
% 12.85/6.94 ( ( ord_less_int @ zero_zero_int @ K )
% 12.85/6.94 => ? [N2: nat] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 12.85/6.94 & ( K
% 12.85/6.94 = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2640_zmult__zless__mono2__lemma,axiom,
% 12.85/6.94 ! [I: int,J: int,K: nat] :
% 12.85/6.94 ( ( ord_less_int @ I @ J )
% 12.85/6.94 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/6.94 => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2641_two__realpow__ge__one,axiom,
% 12.85/6.94 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2642_real__of__nat__div3,axiom,
% 12.85/6.94 ! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) @ one_one_real ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2643_int__power__div__base,axiom,
% 12.85/6.94 ! [M: nat,K: int] :
% 12.85/6.94 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/6.94 => ( ( ord_less_int @ zero_zero_int @ K )
% 12.85/6.94 => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 12.85/6.94 = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2644_power__not__zero,axiom,
% 12.85/6.94 ! [A: rat,N: nat] :
% 12.85/6.94 ( ( A != zero_zero_rat )
% 12.85/6.94 => ( ( power_power_rat @ A @ N )
% 12.85/6.94 != zero_zero_rat ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2645_power__not__zero,axiom,
% 12.85/6.94 ! [A: int,N: nat] :
% 12.85/6.94 ( ( A != zero_zero_int )
% 12.85/6.94 => ( ( power_power_int @ A @ N )
% 12.85/6.94 != zero_zero_int ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2646_power__not__zero,axiom,
% 12.85/6.94 ! [A: real,N: nat] :
% 12.85/6.94 ( ( A != zero_zero_real )
% 12.85/6.94 => ( ( power_power_real @ A @ N )
% 12.85/6.94 != zero_zero_real ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2647_power__not__zero,axiom,
% 12.85/6.94 ! [A: nat,N: nat] :
% 12.85/6.94 ( ( A != zero_zero_nat )
% 12.85/6.94 => ( ( power_power_nat @ A @ N )
% 12.85/6.94 != zero_zero_nat ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2648_power__not__zero,axiom,
% 12.85/6.94 ! [A: complex,N: nat] :
% 12.85/6.94 ( ( A != zero_zero_complex )
% 12.85/6.94 => ( ( power_power_complex @ A @ N )
% 12.85/6.94 != zero_zero_complex ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2649_power__commuting__commutes,axiom,
% 12.85/6.94 ! [X: complex,Y4: complex,N: nat] :
% 12.85/6.94 ( ( ( times_times_complex @ X @ Y4 )
% 12.85/6.94 = ( times_times_complex @ Y4 @ X ) )
% 12.85/6.94 => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ Y4 )
% 12.85/6.94 = ( times_times_complex @ Y4 @ ( power_power_complex @ X @ N ) ) ) ) ).
% 12.85/6.94
% 12.85/6.94 thf(fact_2650_power__commuting__commutes,axiom,
% 12.85/6.94 ! [X: real,Y4: real,N: nat] :
% 12.85/6.94 ( ( ( times_times_real @ X @ Y4 )
% 12.85/6.94 = ( times_times_real @ Y4 @ X ) )
% 12.85/6.94 => ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y4 )
% 12.85/6.95 = ( times_times_real @ Y4 @ ( power_power_real @ X @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2651_power__commuting__commutes,axiom,
% 12.85/6.95 ! [X: rat,Y4: rat,N: nat] :
% 12.85/6.95 ( ( ( times_times_rat @ X @ Y4 )
% 12.85/6.95 = ( times_times_rat @ Y4 @ X ) )
% 12.85/6.95 => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ Y4 )
% 12.85/6.95 = ( times_times_rat @ Y4 @ ( power_power_rat @ X @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2652_power__commuting__commutes,axiom,
% 12.85/6.95 ! [X: nat,Y4: nat,N: nat] :
% 12.85/6.95 ( ( ( times_times_nat @ X @ Y4 )
% 12.85/6.95 = ( times_times_nat @ Y4 @ X ) )
% 12.85/6.95 => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y4 )
% 12.85/6.95 = ( times_times_nat @ Y4 @ ( power_power_nat @ X @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2653_power__commuting__commutes,axiom,
% 12.85/6.95 ! [X: int,Y4: int,N: nat] :
% 12.85/6.95 ( ( ( times_times_int @ X @ Y4 )
% 12.85/6.95 = ( times_times_int @ Y4 @ X ) )
% 12.85/6.95 => ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y4 )
% 12.85/6.95 = ( times_times_int @ Y4 @ ( power_power_int @ X @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2654_power__mult__distrib,axiom,
% 12.85/6.95 ! [A: complex,B: complex,N: nat] :
% 12.85/6.95 ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
% 12.85/6.95 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2655_power__mult__distrib,axiom,
% 12.85/6.95 ! [A: real,B: real,N: nat] :
% 12.85/6.95 ( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
% 12.85/6.95 = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2656_power__mult__distrib,axiom,
% 12.85/6.95 ! [A: rat,B: rat,N: nat] :
% 12.85/6.95 ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
% 12.85/6.95 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2657_power__mult__distrib,axiom,
% 12.85/6.95 ! [A: nat,B: nat,N: nat] :
% 12.85/6.95 ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
% 12.85/6.95 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2658_power__mult__distrib,axiom,
% 12.85/6.95 ! [A: int,B: int,N: nat] :
% 12.85/6.95 ( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
% 12.85/6.95 = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2659_power__commutes,axiom,
% 12.85/6.95 ! [A: complex,N: nat] :
% 12.85/6.95 ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
% 12.85/6.95 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2660_power__commutes,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
% 12.85/6.95 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2661_power__commutes,axiom,
% 12.85/6.95 ! [A: rat,N: nat] :
% 12.85/6.95 ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
% 12.85/6.95 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2662_power__commutes,axiom,
% 12.85/6.95 ! [A: nat,N: nat] :
% 12.85/6.95 ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
% 12.85/6.95 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2663_power__commutes,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
% 12.85/6.95 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2664_power__divide,axiom,
% 12.85/6.95 ! [A: complex,B: complex,N: nat] :
% 12.85/6.95 ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N )
% 12.85/6.95 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2665_power__divide,axiom,
% 12.85/6.95 ! [A: real,B: real,N: nat] :
% 12.85/6.95 ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
% 12.85/6.95 = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2666_power__divide,axiom,
% 12.85/6.95 ! [A: rat,B: rat,N: nat] :
% 12.85/6.95 ( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N )
% 12.85/6.95 = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2667_dvd__power__same,axiom,
% 12.85/6.95 ! [X: code_integer,Y4: code_integer,N: nat] :
% 12.85/6.95 ( ( dvd_dvd_Code_integer @ X @ Y4 )
% 12.85/6.95 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y4 @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2668_dvd__power__same,axiom,
% 12.85/6.95 ! [X: int,Y4: int,N: nat] :
% 12.85/6.95 ( ( dvd_dvd_int @ X @ Y4 )
% 12.85/6.95 => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y4 @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2669_dvd__power__same,axiom,
% 12.85/6.95 ! [X: real,Y4: real,N: nat] :
% 12.85/6.95 ( ( dvd_dvd_real @ X @ Y4 )
% 12.85/6.95 => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y4 @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2670_dvd__power__same,axiom,
% 12.85/6.95 ! [X: nat,Y4: nat,N: nat] :
% 12.85/6.95 ( ( dvd_dvd_nat @ X @ Y4 )
% 12.85/6.95 => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y4 @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2671_dvd__power__same,axiom,
% 12.85/6.95 ! [X: complex,Y4: complex,N: nat] :
% 12.85/6.95 ( ( dvd_dvd_complex @ X @ Y4 )
% 12.85/6.95 => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y4 @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2672_nat__power__less__imp__less,axiom,
% 12.85/6.95 ! [I: nat,M: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ I )
% 12.85/6.95 => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 12.85/6.95 => ( ord_less_nat @ M @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2673_power__mult,axiom,
% 12.85/6.95 ! [A: int,M: nat,N: nat] :
% 12.85/6.95 ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
% 12.85/6.95 = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2674_power__mult,axiom,
% 12.85/6.95 ! [A: real,M: nat,N: nat] :
% 12.85/6.95 ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
% 12.85/6.95 = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2675_power__mult,axiom,
% 12.85/6.95 ! [A: nat,M: nat,N: nat] :
% 12.85/6.95 ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
% 12.85/6.95 = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2676_power__mult,axiom,
% 12.85/6.95 ! [A: complex,M: nat,N: nat] :
% 12.85/6.95 ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
% 12.85/6.95 = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2677_power__mono,axiom,
% 12.85/6.95 ! [A: rat,B: rat,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.95 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.95 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2678_power__mono,axiom,
% 12.85/6.95 ! [A: nat,B: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.95 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.95 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2679_power__mono,axiom,
% 12.85/6.95 ! [A: int,B: int,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.95 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.95 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2680_power__mono,axiom,
% 12.85/6.95 ! [A: real,B: real,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.95 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2681_zero__le__power,axiom,
% 12.85/6.95 ! [A: rat,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.95 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2682_zero__le__power,axiom,
% 12.85/6.95 ! [A: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.95 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2683_zero__le__power,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.95 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2684_zero__le__power,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.95 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2685_zero__less__power,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.95 => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2686_zero__less__power,axiom,
% 12.85/6.95 ! [A: rat,N: nat] :
% 12.85/6.95 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.95 => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2687_zero__less__power,axiom,
% 12.85/6.95 ! [A: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.85/6.95 => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2688_zero__less__power,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( ord_less_int @ zero_zero_int @ A )
% 12.85/6.95 => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2689_one__le__power,axiom,
% 12.85/6.95 ! [A: rat,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 12.85/6.95 => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2690_one__le__power,axiom,
% 12.85/6.95 ! [A: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 12.85/6.95 => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2691_one__le__power,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_int @ one_one_int @ A )
% 12.85/6.95 => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2692_one__le__power,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_real @ one_one_real @ A )
% 12.85/6.95 => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2693_left__right__inverse__power,axiom,
% 12.85/6.95 ! [X: complex,Y4: complex,N: nat] :
% 12.85/6.95 ( ( ( times_times_complex @ X @ Y4 )
% 12.85/6.95 = one_one_complex )
% 12.85/6.95 => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y4 @ N ) )
% 12.85/6.95 = one_one_complex ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2694_left__right__inverse__power,axiom,
% 12.85/6.95 ! [X: real,Y4: real,N: nat] :
% 12.85/6.95 ( ( ( times_times_real @ X @ Y4 )
% 12.85/6.95 = one_one_real )
% 12.85/6.95 => ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y4 @ N ) )
% 12.85/6.95 = one_one_real ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2695_left__right__inverse__power,axiom,
% 12.85/6.95 ! [X: rat,Y4: rat,N: nat] :
% 12.85/6.95 ( ( ( times_times_rat @ X @ Y4 )
% 12.85/6.95 = one_one_rat )
% 12.85/6.95 => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y4 @ N ) )
% 12.85/6.95 = one_one_rat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2696_left__right__inverse__power,axiom,
% 12.85/6.95 ! [X: nat,Y4: nat,N: nat] :
% 12.85/6.95 ( ( ( times_times_nat @ X @ Y4 )
% 12.85/6.95 = one_one_nat )
% 12.85/6.95 => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y4 @ N ) )
% 12.85/6.95 = one_one_nat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2697_left__right__inverse__power,axiom,
% 12.85/6.95 ! [X: int,Y4: int,N: nat] :
% 12.85/6.95 ( ( ( times_times_int @ X @ Y4 )
% 12.85/6.95 = one_one_int )
% 12.85/6.95 => ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y4 @ N ) )
% 12.85/6.95 = one_one_int ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2698_power__Suc2,axiom,
% 12.85/6.95 ! [A: complex,N: nat] :
% 12.85/6.95 ( ( power_power_complex @ A @ ( suc @ N ) )
% 12.85/6.95 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2699_power__Suc2,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( power_power_real @ A @ ( suc @ N ) )
% 12.85/6.95 = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2700_power__Suc2,axiom,
% 12.85/6.95 ! [A: rat,N: nat] :
% 12.85/6.95 ( ( power_power_rat @ A @ ( suc @ N ) )
% 12.85/6.95 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2701_power__Suc2,axiom,
% 12.85/6.95 ! [A: nat,N: nat] :
% 12.85/6.95 ( ( power_power_nat @ A @ ( suc @ N ) )
% 12.85/6.95 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2702_power__Suc2,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( power_power_int @ A @ ( suc @ N ) )
% 12.85/6.95 = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2703_power__Suc,axiom,
% 12.85/6.95 ! [A: complex,N: nat] :
% 12.85/6.95 ( ( power_power_complex @ A @ ( suc @ N ) )
% 12.85/6.95 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2704_power__Suc,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( power_power_real @ A @ ( suc @ N ) )
% 12.85/6.95 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2705_power__Suc,axiom,
% 12.85/6.95 ! [A: rat,N: nat] :
% 12.85/6.95 ( ( power_power_rat @ A @ ( suc @ N ) )
% 12.85/6.95 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2706_power__Suc,axiom,
% 12.85/6.95 ! [A: nat,N: nat] :
% 12.85/6.95 ( ( power_power_nat @ A @ ( suc @ N ) )
% 12.85/6.95 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2707_power__Suc,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( power_power_int @ A @ ( suc @ N ) )
% 12.85/6.95 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2708_power__one__over,axiom,
% 12.85/6.95 ! [A: complex,N: nat] :
% 12.85/6.95 ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
% 12.85/6.95 = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2709_power__one__over,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
% 12.85/6.95 = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2710_power__one__over,axiom,
% 12.85/6.95 ! [A: rat,N: nat] :
% 12.85/6.95 ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
% 12.85/6.95 = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2711_power__0,axiom,
% 12.85/6.95 ! [A: rat] :
% 12.85/6.95 ( ( power_power_rat @ A @ zero_zero_nat )
% 12.85/6.95 = one_one_rat ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2712_power__0,axiom,
% 12.85/6.95 ! [A: int] :
% 12.85/6.95 ( ( power_power_int @ A @ zero_zero_nat )
% 12.85/6.95 = one_one_int ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2713_power__0,axiom,
% 12.85/6.95 ! [A: real] :
% 12.85/6.95 ( ( power_power_real @ A @ zero_zero_nat )
% 12.85/6.95 = one_one_real ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2714_power__0,axiom,
% 12.85/6.95 ! [A: nat] :
% 12.85/6.95 ( ( power_power_nat @ A @ zero_zero_nat )
% 12.85/6.95 = one_one_nat ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2715_power__0,axiom,
% 12.85/6.95 ! [A: complex] :
% 12.85/6.95 ( ( power_power_complex @ A @ zero_zero_nat )
% 12.85/6.95 = one_one_complex ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2716_div__power,axiom,
% 12.85/6.95 ! [B: code_integer,A: code_integer,N: nat] :
% 12.85/6.95 ( ( dvd_dvd_Code_integer @ B @ A )
% 12.85/6.95 => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N )
% 12.85/6.95 = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2717_div__power,axiom,
% 12.85/6.95 ! [B: nat,A: nat,N: nat] :
% 12.85/6.95 ( ( dvd_dvd_nat @ B @ A )
% 12.85/6.95 => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N )
% 12.85/6.95 = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2718_div__power,axiom,
% 12.85/6.95 ! [B: int,A: int,N: nat] :
% 12.85/6.95 ( ( dvd_dvd_int @ B @ A )
% 12.85/6.95 => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N )
% 12.85/6.95 = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2719_power__gt__expt,axiom,
% 12.85/6.95 ! [N: nat,K: nat] :
% 12.85/6.95 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.85/6.95 => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2720_nat__one__le__power,axiom,
% 12.85/6.95 ! [I: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 12.85/6.95 => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2721_dvd__power__le,axiom,
% 12.85/6.95 ! [X: code_integer,Y4: code_integer,N: nat,M: nat] :
% 12.85/6.95 ( ( dvd_dvd_Code_integer @ X @ Y4 )
% 12.85/6.95 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.95 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y4 @ M ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2722_dvd__power__le,axiom,
% 12.85/6.95 ! [X: int,Y4: int,N: nat,M: nat] :
% 12.85/6.95 ( ( dvd_dvd_int @ X @ Y4 )
% 12.85/6.95 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.95 => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y4 @ M ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2723_dvd__power__le,axiom,
% 12.85/6.95 ! [X: real,Y4: real,N: nat,M: nat] :
% 12.85/6.95 ( ( dvd_dvd_real @ X @ Y4 )
% 12.85/6.95 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.95 => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y4 @ M ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2724_dvd__power__le,axiom,
% 12.85/6.95 ! [X: nat,Y4: nat,N: nat,M: nat] :
% 12.85/6.95 ( ( dvd_dvd_nat @ X @ Y4 )
% 12.85/6.95 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.95 => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y4 @ M ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2725_dvd__power__le,axiom,
% 12.85/6.95 ! [X: complex,Y4: complex,N: nat,M: nat] :
% 12.85/6.95 ( ( dvd_dvd_complex @ X @ Y4 )
% 12.85/6.95 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.95 => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y4 @ M ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2726_power__le__dvd,axiom,
% 12.85/6.95 ! [A: code_integer,N: nat,B: code_integer,M: nat] :
% 12.85/6.95 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
% 12.85/6.95 => ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.95 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2727_power__le__dvd,axiom,
% 12.85/6.95 ! [A: int,N: nat,B: int,M: nat] :
% 12.85/6.95 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
% 12.85/6.95 => ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.95 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2728_power__le__dvd,axiom,
% 12.85/6.95 ! [A: real,N: nat,B: real,M: nat] :
% 12.85/6.95 ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
% 12.85/6.95 => ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.95 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2729_power__le__dvd,axiom,
% 12.85/6.95 ! [A: nat,N: nat,B: nat,M: nat] :
% 12.85/6.95 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
% 12.85/6.95 => ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.95 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2730_power__le__dvd,axiom,
% 12.85/6.95 ! [A: complex,N: nat,B: complex,M: nat] :
% 12.85/6.95 ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
% 12.85/6.95 => ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.95 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2731_le__imp__power__dvd,axiom,
% 12.85/6.95 ! [M: nat,N: nat,A: code_integer] :
% 12.85/6.95 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.95 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2732_le__imp__power__dvd,axiom,
% 12.85/6.95 ! [M: nat,N: nat,A: int] :
% 12.85/6.95 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.95 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2733_le__imp__power__dvd,axiom,
% 12.85/6.95 ! [M: nat,N: nat,A: real] :
% 12.85/6.95 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.95 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2734_le__imp__power__dvd,axiom,
% 12.85/6.95 ! [M: nat,N: nat,A: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.95 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2735_le__imp__power__dvd,axiom,
% 12.85/6.95 ! [M: nat,N: nat,A: complex] :
% 12.85/6.95 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.95 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2736_power__less__imp__less__base,axiom,
% 12.85/6.95 ! [A: rat,N: nat,B: rat] :
% 12.85/6.95 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 12.85/6.95 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 12.85/6.95 => ( ord_less_rat @ A @ B ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2737_power__less__imp__less__base,axiom,
% 12.85/6.95 ! [A: nat,N: nat,B: nat] :
% 12.85/6.95 ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 12.85/6.95 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 12.85/6.95 => ( ord_less_nat @ A @ B ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2738_power__less__imp__less__base,axiom,
% 12.85/6.95 ! [A: int,N: nat,B: int] :
% 12.85/6.95 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 12.85/6.95 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 12.85/6.95 => ( ord_less_int @ A @ B ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2739_power__less__imp__less__base,axiom,
% 12.85/6.95 ! [A: real,N: nat,B: real] :
% 12.85/6.95 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 12.85/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 12.85/6.95 => ( ord_less_real @ A @ B ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2740_power__le__one,axiom,
% 12.85/6.95 ! [A: rat,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.95 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 12.85/6.95 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2741_power__le__one,axiom,
% 12.85/6.95 ! [A: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.95 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 12.85/6.95 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2742_power__le__one,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.95 => ( ( ord_less_eq_int @ A @ one_one_int )
% 12.85/6.95 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2743_power__le__one,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.95 => ( ( ord_less_eq_real @ A @ one_one_real )
% 12.85/6.95 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2744_power__le__imp__le__base,axiom,
% 12.85/6.95 ! [A: rat,N: nat,B: rat] :
% 12.85/6.95 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
% 12.85/6.95 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 12.85/6.95 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2745_power__le__imp__le__base,axiom,
% 12.85/6.95 ! [A: nat,N: nat,B: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
% 12.85/6.95 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 12.85/6.95 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2746_power__le__imp__le__base,axiom,
% 12.85/6.95 ! [A: int,N: nat,B: int] :
% 12.85/6.95 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
% 12.85/6.95 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 12.85/6.95 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2747_power__le__imp__le__base,axiom,
% 12.85/6.95 ! [A: real,N: nat,B: real] :
% 12.85/6.95 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
% 12.85/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 12.85/6.95 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2748_power__inject__base,axiom,
% 12.85/6.95 ! [A: rat,N: nat,B: rat] :
% 12.85/6.95 ( ( ( power_power_rat @ A @ ( suc @ N ) )
% 12.85/6.95 = ( power_power_rat @ B @ ( suc @ N ) ) )
% 12.85/6.95 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.95 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 12.85/6.95 => ( A = B ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2749_power__inject__base,axiom,
% 12.85/6.95 ! [A: nat,N: nat,B: nat] :
% 12.85/6.95 ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 12.85/6.95 = ( power_power_nat @ B @ ( suc @ N ) ) )
% 12.85/6.95 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.95 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 12.85/6.95 => ( A = B ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2750_power__inject__base,axiom,
% 12.85/6.95 ! [A: int,N: nat,B: int] :
% 12.85/6.95 ( ( ( power_power_int @ A @ ( suc @ N ) )
% 12.85/6.95 = ( power_power_int @ B @ ( suc @ N ) ) )
% 12.85/6.95 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.95 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 12.85/6.95 => ( A = B ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2751_power__inject__base,axiom,
% 12.85/6.95 ! [A: real,N: nat,B: real] :
% 12.85/6.95 ( ( ( power_power_real @ A @ ( suc @ N ) )
% 12.85/6.95 = ( power_power_real @ B @ ( suc @ N ) ) )
% 12.85/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 12.85/6.95 => ( A = B ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2752_power__less__power__Suc,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( ord_less_real @ one_one_real @ A )
% 12.85/6.95 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2753_power__less__power__Suc,axiom,
% 12.85/6.95 ! [A: rat,N: nat] :
% 12.85/6.95 ( ( ord_less_rat @ one_one_rat @ A )
% 12.85/6.95 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2754_power__less__power__Suc,axiom,
% 12.85/6.95 ! [A: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_nat @ one_one_nat @ A )
% 12.85/6.95 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2755_power__less__power__Suc,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( ord_less_int @ one_one_int @ A )
% 12.85/6.95 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2756_power__gt1__lemma,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( ord_less_real @ one_one_real @ A )
% 12.85/6.95 => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2757_power__gt1__lemma,axiom,
% 12.85/6.95 ! [A: rat,N: nat] :
% 12.85/6.95 ( ( ord_less_rat @ one_one_rat @ A )
% 12.85/6.95 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2758_power__gt1__lemma,axiom,
% 12.85/6.95 ! [A: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_nat @ one_one_nat @ A )
% 12.85/6.95 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2759_power__gt1__lemma,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( ord_less_int @ one_one_int @ A )
% 12.85/6.95 => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2760_power__0__left,axiom,
% 12.85/6.95 ! [N: nat] :
% 12.85/6.95 ( ( ( N = zero_zero_nat )
% 12.85/6.95 => ( ( power_power_rat @ zero_zero_rat @ N )
% 12.85/6.95 = one_one_rat ) )
% 12.85/6.95 & ( ( N != zero_zero_nat )
% 12.85/6.95 => ( ( power_power_rat @ zero_zero_rat @ N )
% 12.85/6.95 = zero_zero_rat ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2761_power__0__left,axiom,
% 12.85/6.95 ! [N: nat] :
% 12.85/6.95 ( ( ( N = zero_zero_nat )
% 12.85/6.95 => ( ( power_power_int @ zero_zero_int @ N )
% 12.85/6.95 = one_one_int ) )
% 12.85/6.95 & ( ( N != zero_zero_nat )
% 12.85/6.95 => ( ( power_power_int @ zero_zero_int @ N )
% 12.85/6.95 = zero_zero_int ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2762_power__0__left,axiom,
% 12.85/6.95 ! [N: nat] :
% 12.85/6.95 ( ( ( N = zero_zero_nat )
% 12.85/6.95 => ( ( power_power_real @ zero_zero_real @ N )
% 12.85/6.95 = one_one_real ) )
% 12.85/6.95 & ( ( N != zero_zero_nat )
% 12.85/6.95 => ( ( power_power_real @ zero_zero_real @ N )
% 12.85/6.95 = zero_zero_real ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2763_power__0__left,axiom,
% 12.85/6.95 ! [N: nat] :
% 12.85/6.95 ( ( ( N = zero_zero_nat )
% 12.85/6.95 => ( ( power_power_nat @ zero_zero_nat @ N )
% 12.85/6.95 = one_one_nat ) )
% 12.85/6.95 & ( ( N != zero_zero_nat )
% 12.85/6.95 => ( ( power_power_nat @ zero_zero_nat @ N )
% 12.85/6.95 = zero_zero_nat ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2764_power__0__left,axiom,
% 12.85/6.95 ! [N: nat] :
% 12.85/6.95 ( ( ( N = zero_zero_nat )
% 12.85/6.95 => ( ( power_power_complex @ zero_zero_complex @ N )
% 12.85/6.95 = one_one_complex ) )
% 12.85/6.95 & ( ( N != zero_zero_nat )
% 12.85/6.95 => ( ( power_power_complex @ zero_zero_complex @ N )
% 12.85/6.95 = zero_zero_complex ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2765_power__gt1,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( ord_less_real @ one_one_real @ A )
% 12.85/6.95 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2766_power__gt1,axiom,
% 12.85/6.95 ! [A: rat,N: nat] :
% 12.85/6.95 ( ( ord_less_rat @ one_one_rat @ A )
% 12.85/6.95 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2767_power__gt1,axiom,
% 12.85/6.95 ! [A: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_nat @ one_one_nat @ A )
% 12.85/6.95 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2768_power__gt1,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( ord_less_int @ one_one_int @ A )
% 12.85/6.95 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2769_power__increasing,axiom,
% 12.85/6.95 ! [N: nat,N5: nat,A: rat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ N @ N5 )
% 12.85/6.95 => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 12.85/6.95 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2770_power__increasing,axiom,
% 12.85/6.95 ! [N: nat,N5: nat,A: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ N @ N5 )
% 12.85/6.95 => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 12.85/6.95 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2771_power__increasing,axiom,
% 12.85/6.95 ! [N: nat,N5: nat,A: int] :
% 12.85/6.95 ( ( ord_less_eq_nat @ N @ N5 )
% 12.85/6.95 => ( ( ord_less_eq_int @ one_one_int @ A )
% 12.85/6.95 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2772_power__increasing,axiom,
% 12.85/6.95 ! [N: nat,N5: nat,A: real] :
% 12.85/6.95 ( ( ord_less_eq_nat @ N @ N5 )
% 12.85/6.95 => ( ( ord_less_eq_real @ one_one_real @ A )
% 12.85/6.95 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2773_zero__power,axiom,
% 12.85/6.95 ! [N: nat] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ( power_power_rat @ zero_zero_rat @ N )
% 12.85/6.95 = zero_zero_rat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2774_zero__power,axiom,
% 12.85/6.95 ! [N: nat] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ( power_power_int @ zero_zero_int @ N )
% 12.85/6.95 = zero_zero_int ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2775_zero__power,axiom,
% 12.85/6.95 ! [N: nat] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ( power_power_real @ zero_zero_real @ N )
% 12.85/6.95 = zero_zero_real ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2776_zero__power,axiom,
% 12.85/6.95 ! [N: nat] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ( power_power_nat @ zero_zero_nat @ N )
% 12.85/6.95 = zero_zero_nat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2777_zero__power,axiom,
% 12.85/6.95 ! [N: nat] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ( power_power_complex @ zero_zero_complex @ N )
% 12.85/6.95 = zero_zero_complex ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2778_less__exp,axiom,
% 12.85/6.95 ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2779_power2__nat__le__imp__le,axiom,
% 12.85/6.95 ! [M: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 12.85/6.95 => ( ord_less_eq_nat @ M @ N ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2780_power2__nat__le__eq__le,axiom,
% 12.85/6.95 ! [M: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.95 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2781_self__le__ge2__pow,axiom,
% 12.85/6.95 ! [K: nat,M: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 12.85/6.95 => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2782_is__unit__power__iff,axiom,
% 12.85/6.95 ! [A: code_integer,N: nat] :
% 12.85/6.95 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
% 12.85/6.95 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 12.85/6.95 | ( N = zero_zero_nat ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2783_is__unit__power__iff,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
% 12.85/6.95 = ( ( dvd_dvd_int @ A @ one_one_int )
% 12.85/6.95 | ( N = zero_zero_nat ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2784_is__unit__power__iff,axiom,
% 12.85/6.95 ! [A: nat,N: nat] :
% 12.85/6.95 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
% 12.85/6.95 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 12.85/6.95 | ( N = zero_zero_nat ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2785_power__dvd__imp__le,axiom,
% 12.85/6.95 ! [I: nat,M: nat,N: nat] :
% 12.85/6.95 ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 12.85/6.95 => ( ( ord_less_nat @ one_one_nat @ I )
% 12.85/6.95 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2786_power__Suc__less,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.95 => ( ( ord_less_real @ A @ one_one_real )
% 12.85/6.95 => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2787_power__Suc__less,axiom,
% 12.85/6.95 ! [A: rat,N: nat] :
% 12.85/6.95 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.95 => ( ( ord_less_rat @ A @ one_one_rat )
% 12.85/6.95 => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2788_power__Suc__less,axiom,
% 12.85/6.95 ! [A: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.85/6.95 => ( ( ord_less_nat @ A @ one_one_nat )
% 12.85/6.95 => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2789_power__Suc__less,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( ord_less_int @ zero_zero_int @ A )
% 12.85/6.95 => ( ( ord_less_int @ A @ one_one_int )
% 12.85/6.95 => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2790_power__Suc__le__self,axiom,
% 12.85/6.95 ! [A: rat,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.95 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 12.85/6.95 => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2791_power__Suc__le__self,axiom,
% 12.85/6.95 ! [A: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.95 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 12.85/6.95 => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2792_power__Suc__le__self,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.95 => ( ( ord_less_eq_int @ A @ one_one_int )
% 12.85/6.95 => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2793_power__Suc__le__self,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.95 => ( ( ord_less_eq_real @ A @ one_one_real )
% 12.85/6.95 => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2794_power__Suc__less__one,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.95 => ( ( ord_less_real @ A @ one_one_real )
% 12.85/6.95 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2795_power__Suc__less__one,axiom,
% 12.85/6.95 ! [A: rat,N: nat] :
% 12.85/6.95 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.95 => ( ( ord_less_rat @ A @ one_one_rat )
% 12.85/6.95 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2796_power__Suc__less__one,axiom,
% 12.85/6.95 ! [A: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.85/6.95 => ( ( ord_less_nat @ A @ one_one_nat )
% 12.85/6.95 => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2797_power__Suc__less__one,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( ord_less_int @ zero_zero_int @ A )
% 12.85/6.95 => ( ( ord_less_int @ A @ one_one_int )
% 12.85/6.95 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2798_power__strict__decreasing,axiom,
% 12.85/6.95 ! [N: nat,N5: nat,A: real] :
% 12.85/6.95 ( ( ord_less_nat @ N @ N5 )
% 12.85/6.95 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.95 => ( ( ord_less_real @ A @ one_one_real )
% 12.85/6.95 => ( ord_less_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2799_power__strict__decreasing,axiom,
% 12.85/6.95 ! [N: nat,N5: nat,A: rat] :
% 12.85/6.95 ( ( ord_less_nat @ N @ N5 )
% 12.85/6.95 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.95 => ( ( ord_less_rat @ A @ one_one_rat )
% 12.85/6.95 => ( ord_less_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2800_power__strict__decreasing,axiom,
% 12.85/6.95 ! [N: nat,N5: nat,A: nat] :
% 12.85/6.95 ( ( ord_less_nat @ N @ N5 )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.85/6.95 => ( ( ord_less_nat @ A @ one_one_nat )
% 12.85/6.95 => ( ord_less_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2801_power__strict__decreasing,axiom,
% 12.85/6.95 ! [N: nat,N5: nat,A: int] :
% 12.85/6.95 ( ( ord_less_nat @ N @ N5 )
% 12.85/6.95 => ( ( ord_less_int @ zero_zero_int @ A )
% 12.85/6.95 => ( ( ord_less_int @ A @ one_one_int )
% 12.85/6.95 => ( ord_less_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2802_power__decreasing,axiom,
% 12.85/6.95 ! [N: nat,N5: nat,A: rat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ N @ N5 )
% 12.85/6.95 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.95 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 12.85/6.95 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2803_power__decreasing,axiom,
% 12.85/6.95 ! [N: nat,N5: nat,A: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ N @ N5 )
% 12.85/6.95 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.95 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 12.85/6.95 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2804_power__decreasing,axiom,
% 12.85/6.95 ! [N: nat,N5: nat,A: int] :
% 12.85/6.95 ( ( ord_less_eq_nat @ N @ N5 )
% 12.85/6.95 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.95 => ( ( ord_less_eq_int @ A @ one_one_int )
% 12.85/6.95 => ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2805_power__decreasing,axiom,
% 12.85/6.95 ! [N: nat,N5: nat,A: real] :
% 12.85/6.95 ( ( ord_less_eq_nat @ N @ N5 )
% 12.85/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.95 => ( ( ord_less_eq_real @ A @ one_one_real )
% 12.85/6.95 => ( ord_less_eq_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2806_power__le__imp__le__exp,axiom,
% 12.85/6.95 ! [A: rat,M: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_rat @ one_one_rat @ A )
% 12.85/6.95 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 12.85/6.95 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2807_power__le__imp__le__exp,axiom,
% 12.85/6.95 ! [A: nat,M: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_nat @ one_one_nat @ A )
% 12.85/6.95 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 12.85/6.95 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2808_power__le__imp__le__exp,axiom,
% 12.85/6.95 ! [A: int,M: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_int @ one_one_int @ A )
% 12.85/6.95 => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 12.85/6.95 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2809_power__le__imp__le__exp,axiom,
% 12.85/6.95 ! [A: real,M: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_real @ one_one_real @ A )
% 12.85/6.95 => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 12.85/6.95 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2810_zero__power2,axiom,
% 12.85/6.95 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = zero_zero_rat ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2811_zero__power2,axiom,
% 12.85/6.95 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = zero_zero_int ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2812_zero__power2,axiom,
% 12.85/6.95 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = zero_zero_real ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2813_zero__power2,axiom,
% 12.85/6.95 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = zero_zero_nat ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2814_zero__power2,axiom,
% 12.85/6.95 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = zero_zero_complex ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2815_power__eq__iff__eq__base,axiom,
% 12.85/6.95 ! [N: nat,A: rat,B: rat] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.95 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 12.85/6.95 => ( ( ( power_power_rat @ A @ N )
% 12.85/6.95 = ( power_power_rat @ B @ N ) )
% 12.85/6.95 = ( A = B ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2816_power__eq__iff__eq__base,axiom,
% 12.85/6.95 ! [N: nat,A: nat,B: nat] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.95 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 12.85/6.95 => ( ( ( power_power_nat @ A @ N )
% 12.85/6.95 = ( power_power_nat @ B @ N ) )
% 12.85/6.95 = ( A = B ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2817_power__eq__iff__eq__base,axiom,
% 12.85/6.95 ! [N: nat,A: int,B: int] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.95 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 12.85/6.95 => ( ( ( power_power_int @ A @ N )
% 12.85/6.95 = ( power_power_int @ B @ N ) )
% 12.85/6.95 = ( A = B ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2818_power__eq__iff__eq__base,axiom,
% 12.85/6.95 ! [N: nat,A: real,B: real] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 12.85/6.95 => ( ( ( power_power_real @ A @ N )
% 12.85/6.95 = ( power_power_real @ B @ N ) )
% 12.85/6.95 = ( A = B ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2819_power__eq__imp__eq__base,axiom,
% 12.85/6.95 ! [A: rat,N: nat,B: rat] :
% 12.85/6.95 ( ( ( power_power_rat @ A @ N )
% 12.85/6.95 = ( power_power_rat @ B @ N ) )
% 12.85/6.95 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.95 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( A = B ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2820_power__eq__imp__eq__base,axiom,
% 12.85/6.95 ! [A: nat,N: nat,B: nat] :
% 12.85/6.95 ( ( ( power_power_nat @ A @ N )
% 12.85/6.95 = ( power_power_nat @ B @ N ) )
% 12.85/6.95 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.95 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( A = B ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2821_power__eq__imp__eq__base,axiom,
% 12.85/6.95 ! [A: int,N: nat,B: int] :
% 12.85/6.95 ( ( ( power_power_int @ A @ N )
% 12.85/6.95 = ( power_power_int @ B @ N ) )
% 12.85/6.95 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.95 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( A = B ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2822_power__eq__imp__eq__base,axiom,
% 12.85/6.95 ! [A: real,N: nat,B: real] :
% 12.85/6.95 ( ( ( power_power_real @ A @ N )
% 12.85/6.95 = ( power_power_real @ B @ N ) )
% 12.85/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( A = B ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2823_power2__eq__square,axiom,
% 12.85/6.95 ! [A: complex] :
% 12.85/6.95 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( times_times_complex @ A @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2824_power2__eq__square,axiom,
% 12.85/6.95 ! [A: real] :
% 12.85/6.95 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( times_times_real @ A @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2825_power2__eq__square,axiom,
% 12.85/6.95 ! [A: rat] :
% 12.85/6.95 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( times_times_rat @ A @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2826_power2__eq__square,axiom,
% 12.85/6.95 ! [A: nat] :
% 12.85/6.95 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( times_times_nat @ A @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2827_power2__eq__square,axiom,
% 12.85/6.95 ! [A: int] :
% 12.85/6.95 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( times_times_int @ A @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2828_power4__eq__xxxx,axiom,
% 12.85/6.95 ! [X: complex] :
% 12.85/6.95 ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.85/6.95 = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X @ X ) @ X ) @ X ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2829_power4__eq__xxxx,axiom,
% 12.85/6.95 ! [X: real] :
% 12.85/6.95 ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.85/6.95 = ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2830_power4__eq__xxxx,axiom,
% 12.85/6.95 ! [X: rat] :
% 12.85/6.95 ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.85/6.95 = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2831_power4__eq__xxxx,axiom,
% 12.85/6.95 ! [X: nat] :
% 12.85/6.95 ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.85/6.95 = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2832_power4__eq__xxxx,axiom,
% 12.85/6.95 ! [X: int] :
% 12.85/6.95 ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.85/6.95 = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2833_one__power2,axiom,
% 12.85/6.95 ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = one_one_rat ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2834_one__power2,axiom,
% 12.85/6.95 ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = one_one_int ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2835_one__power2,axiom,
% 12.85/6.95 ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = one_one_real ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2836_one__power2,axiom,
% 12.85/6.95 ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = one_one_nat ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2837_one__power2,axiom,
% 12.85/6.95 ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = one_one_complex ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2838_self__le__power,axiom,
% 12.85/6.95 ! [A: rat,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2839_self__le__power,axiom,
% 12.85/6.95 ! [A: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2840_self__le__power,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_int @ one_one_int @ A )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2841_self__le__power,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_real @ one_one_real @ A )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2842_power2__commute,axiom,
% 12.85/6.95 ! [X: complex,Y4: complex] :
% 12.85/6.95 ( ( power_power_complex @ ( minus_minus_complex @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( power_power_complex @ ( minus_minus_complex @ Y4 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2843_power2__commute,axiom,
% 12.85/6.95 ! [X: real,Y4: real] :
% 12.85/6.95 ( ( power_power_real @ ( minus_minus_real @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( power_power_real @ ( minus_minus_real @ Y4 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2844_power2__commute,axiom,
% 12.85/6.95 ! [X: rat,Y4: rat] :
% 12.85/6.95 ( ( power_power_rat @ ( minus_minus_rat @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( power_power_rat @ ( minus_minus_rat @ Y4 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2845_power2__commute,axiom,
% 12.85/6.95 ! [X: int,Y4: int] :
% 12.85/6.95 ( ( power_power_int @ ( minus_minus_int @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( power_power_int @ ( minus_minus_int @ Y4 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2846_one__less__power,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( ord_less_real @ one_one_real @ A )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2847_one__less__power,axiom,
% 12.85/6.95 ! [A: rat,N: nat] :
% 12.85/6.95 ( ( ord_less_rat @ one_one_rat @ A )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2848_one__less__power,axiom,
% 12.85/6.95 ! [A: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_nat @ one_one_nat @ A )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2849_one__less__power,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( ord_less_int @ one_one_int @ A )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2850_dvd__power__iff,axiom,
% 12.85/6.95 ! [X: code_integer,M: nat,N: nat] :
% 12.85/6.95 ( ( X != zero_z3403309356797280102nteger )
% 12.85/6.95 => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N ) )
% 12.85/6.95 = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
% 12.85/6.95 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2851_dvd__power__iff,axiom,
% 12.85/6.95 ! [X: int,M: nat,N: nat] :
% 12.85/6.95 ( ( X != zero_zero_int )
% 12.85/6.95 => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
% 12.85/6.95 = ( ( dvd_dvd_int @ X @ one_one_int )
% 12.85/6.95 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2852_dvd__power__iff,axiom,
% 12.85/6.95 ! [X: nat,M: nat,N: nat] :
% 12.85/6.95 ( ( X != zero_zero_nat )
% 12.85/6.95 => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
% 12.85/6.95 = ( ( dvd_dvd_nat @ X @ one_one_nat )
% 12.85/6.95 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2853_power3__eq__cube,axiom,
% 12.85/6.95 ! [A: complex] :
% 12.85/6.95 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 12.85/6.95 = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2854_power3__eq__cube,axiom,
% 12.85/6.95 ! [A: real] :
% 12.85/6.95 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 12.85/6.95 = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2855_power3__eq__cube,axiom,
% 12.85/6.95 ! [A: rat] :
% 12.85/6.95 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 12.85/6.95 = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2856_power3__eq__cube,axiom,
% 12.85/6.95 ! [A: nat] :
% 12.85/6.95 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 12.85/6.95 = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2857_power3__eq__cube,axiom,
% 12.85/6.95 ! [A: int] :
% 12.85/6.95 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 12.85/6.95 = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2858_dvd__power,axiom,
% 12.85/6.95 ! [N: nat,X: code_integer] :
% 12.85/6.95 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 | ( X = one_one_Code_integer ) )
% 12.85/6.95 => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2859_dvd__power,axiom,
% 12.85/6.95 ! [N: nat,X: rat] :
% 12.85/6.95 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 | ( X = one_one_rat ) )
% 12.85/6.95 => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2860_dvd__power,axiom,
% 12.85/6.95 ! [N: nat,X: int] :
% 12.85/6.95 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 | ( X = one_one_int ) )
% 12.85/6.95 => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2861_dvd__power,axiom,
% 12.85/6.95 ! [N: nat,X: real] :
% 12.85/6.95 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 | ( X = one_one_real ) )
% 12.85/6.95 => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2862_dvd__power,axiom,
% 12.85/6.95 ! [N: nat,X: nat] :
% 12.85/6.95 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 | ( X = one_one_nat ) )
% 12.85/6.95 => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2863_dvd__power,axiom,
% 12.85/6.95 ! [N: nat,X: complex] :
% 12.85/6.95 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 | ( X = one_one_complex ) )
% 12.85/6.95 => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2864_power__diff,axiom,
% 12.85/6.95 ! [A: complex,N: nat,M: nat] :
% 12.85/6.95 ( ( A != zero_zero_complex )
% 12.85/6.95 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.95 => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
% 12.85/6.95 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2865_power__diff,axiom,
% 12.85/6.95 ! [A: real,N: nat,M: nat] :
% 12.85/6.95 ( ( A != zero_zero_real )
% 12.85/6.95 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.95 => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
% 12.85/6.95 = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2866_power__diff,axiom,
% 12.85/6.95 ! [A: rat,N: nat,M: nat] :
% 12.85/6.95 ( ( A != zero_zero_rat )
% 12.85/6.95 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.95 => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
% 12.85/6.95 = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2867_power__diff,axiom,
% 12.85/6.95 ! [A: nat,N: nat,M: nat] :
% 12.85/6.95 ( ( A != zero_zero_nat )
% 12.85/6.95 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.95 => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
% 12.85/6.95 = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2868_power__diff,axiom,
% 12.85/6.95 ! [A: int,N: nat,M: nat] :
% 12.85/6.95 ( ( A != zero_zero_int )
% 12.85/6.95 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.95 => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
% 12.85/6.95 = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2869_power__even__eq,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.95 = ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2870_power__even__eq,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.95 = ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2871_power__even__eq,axiom,
% 12.85/6.95 ! [A: nat,N: nat] :
% 12.85/6.95 ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.95 = ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2872_power__even__eq,axiom,
% 12.85/6.95 ! [A: complex,N: nat] :
% 12.85/6.95 ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.95 = ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2873_diff__le__diff__pow,axiom,
% 12.85/6.95 ! [K: nat,M: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 12.85/6.95 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2874_dvd__power__iff__le,axiom,
% 12.85/6.95 ! [K: nat,M: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 12.85/6.95 => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
% 12.85/6.95 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2875_power2__le__imp__le,axiom,
% 12.85/6.95 ! [X: rat,Y4: rat] :
% 12.85/6.95 ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.95 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 12.85/6.95 => ( ord_less_eq_rat @ X @ Y4 ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2876_power2__le__imp__le,axiom,
% 12.85/6.95 ! [X: nat,Y4: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.95 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
% 12.85/6.95 => ( ord_less_eq_nat @ X @ Y4 ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2877_power2__le__imp__le,axiom,
% 12.85/6.95 ! [X: int,Y4: int] :
% 12.85/6.95 ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.95 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/6.95 => ( ord_less_eq_int @ X @ Y4 ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2878_power2__le__imp__le,axiom,
% 12.85/6.95 ! [X: real,Y4: real] :
% 12.85/6.95 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/6.95 => ( ord_less_eq_real @ X @ Y4 ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2879_power2__eq__imp__eq,axiom,
% 12.85/6.95 ! [X: rat,Y4: rat] :
% 12.85/6.95 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.95 => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 12.85/6.95 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 12.85/6.95 => ( X = Y4 ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2880_power2__eq__imp__eq,axiom,
% 12.85/6.95 ! [X: nat,Y4: nat] :
% 12.85/6.95 ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( power_power_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.95 => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 12.85/6.95 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
% 12.85/6.95 => ( X = Y4 ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2881_power2__eq__imp__eq,axiom,
% 12.85/6.95 ! [X: int,Y4: int] :
% 12.85/6.95 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.95 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/6.95 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/6.95 => ( X = Y4 ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2882_power2__eq__imp__eq,axiom,
% 12.85/6.95 ! [X: real,Y4: real] :
% 12.85/6.95 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/6.95 => ( X = Y4 ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2883_zero__le__power2,axiom,
% 12.85/6.95 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2884_zero__le__power2,axiom,
% 12.85/6.95 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2885_zero__le__power2,axiom,
% 12.85/6.95 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2886_power2__less__0,axiom,
% 12.85/6.95 ! [A: real] :
% 12.85/6.95 ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2887_power2__less__0,axiom,
% 12.85/6.95 ! [A: rat] :
% 12.85/6.95 ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2888_power2__less__0,axiom,
% 12.85/6.95 ! [A: int] :
% 12.85/6.95 ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2889_power__strict__mono,axiom,
% 12.85/6.95 ! [A: rat,B: rat,N: nat] :
% 12.85/6.95 ( ( ord_less_rat @ A @ B )
% 12.85/6.95 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2890_power__strict__mono,axiom,
% 12.85/6.95 ! [A: nat,B: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_nat @ A @ B )
% 12.85/6.95 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2891_power__strict__mono,axiom,
% 12.85/6.95 ! [A: int,B: int,N: nat] :
% 12.85/6.95 ( ( ord_less_int @ A @ B )
% 12.85/6.95 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2892_power__strict__mono,axiom,
% 12.85/6.95 ! [A: real,B: real,N: nat] :
% 12.85/6.95 ( ( ord_less_real @ A @ B )
% 12.85/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2893_power__eq__if,axiom,
% 12.85/6.95 ( power_power_complex
% 12.85/6.95 = ( ^ [P6: complex,M5: nat] : ( if_complex @ ( M5 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P6 @ ( power_power_complex @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2894_power__eq__if,axiom,
% 12.85/6.95 ( power_power_real
% 12.85/6.95 = ( ^ [P6: real,M5: nat] : ( if_real @ ( M5 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P6 @ ( power_power_real @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2895_power__eq__if,axiom,
% 12.85/6.95 ( power_power_rat
% 12.85/6.95 = ( ^ [P6: rat,M5: nat] : ( if_rat @ ( M5 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P6 @ ( power_power_rat @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2896_power__eq__if,axiom,
% 12.85/6.95 ( power_power_nat
% 12.85/6.95 = ( ^ [P6: nat,M5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2897_power__eq__if,axiom,
% 12.85/6.95 ( power_power_int
% 12.85/6.95 = ( ^ [P6: int,M5: nat] : ( if_int @ ( M5 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P6 @ ( power_power_int @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2898_power__minus__mult,axiom,
% 12.85/6.95 ! [N: nat,A: complex] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 12.85/6.95 = ( power_power_complex @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2899_power__minus__mult,axiom,
% 12.85/6.95 ! [N: nat,A: real] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 12.85/6.95 = ( power_power_real @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2900_power__minus__mult,axiom,
% 12.85/6.95 ! [N: nat,A: rat] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 12.85/6.95 = ( power_power_rat @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2901_power__minus__mult,axiom,
% 12.85/6.95 ! [N: nat,A: nat] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 12.85/6.95 = ( power_power_nat @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2902_power__minus__mult,axiom,
% 12.85/6.95 ! [N: nat,A: int] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 12.85/6.95 = ( power_power_int @ A @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2903_power2__less__imp__less,axiom,
% 12.85/6.95 ! [X: rat,Y4: rat] :
% 12.85/6.95 ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.95 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 12.85/6.95 => ( ord_less_rat @ X @ Y4 ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2904_power2__less__imp__less,axiom,
% 12.85/6.95 ! [X: nat,Y4: nat] :
% 12.85/6.95 ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.95 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
% 12.85/6.95 => ( ord_less_nat @ X @ Y4 ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2905_power2__less__imp__less,axiom,
% 12.85/6.95 ! [X: int,Y4: int] :
% 12.85/6.95 ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.95 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/6.95 => ( ord_less_int @ X @ Y4 ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2906_power2__less__imp__less,axiom,
% 12.85/6.95 ! [X: real,Y4: real] :
% 12.85/6.95 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.95 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/6.95 => ( ord_less_real @ X @ Y4 ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2907_zero__le__even__power_H,axiom,
% 12.85/6.95 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2908_zero__le__even__power_H,axiom,
% 12.85/6.95 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2909_zero__le__even__power_H,axiom,
% 12.85/6.95 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2910_power__odd__eq,axiom,
% 12.85/6.95 ! [A: complex,N: nat] :
% 12.85/6.95 ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.85/6.95 = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2911_power__odd__eq,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.85/6.95 = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2912_power__odd__eq,axiom,
% 12.85/6.95 ! [A: rat,N: nat] :
% 12.85/6.95 ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.85/6.95 = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2913_power__odd__eq,axiom,
% 12.85/6.95 ! [A: nat,N: nat] :
% 12.85/6.95 ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.85/6.95 = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2914_power__odd__eq,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.85/6.95 = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2915_odd__0__le__power__imp__0__le,axiom,
% 12.85/6.95 ! [A: rat,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 12.85/6.95 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2916_odd__0__le__power__imp__0__le,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 12.85/6.95 => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2917_odd__0__le__power__imp__0__le,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 12.85/6.95 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2918_odd__power__less__zero,axiom,
% 12.85/6.95 ! [A: real,N: nat] :
% 12.85/6.95 ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/6.95 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2919_odd__power__less__zero,axiom,
% 12.85/6.95 ! [A: rat,N: nat] :
% 12.85/6.95 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/6.95 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2920_odd__power__less__zero,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( ord_less_int @ A @ zero_zero_int )
% 12.85/6.95 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2921_space__2__pow__bound,axiom,
% 12.85/6.95 ! [T: vEBT_VEBT,N: nat] :
% 12.85/6.95 ( ( vEBT_invar_vebt @ T @ N )
% 12.85/6.95 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_space2 @ T ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( minus_minus_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) @ one_one_real ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2922_cnt__bound,axiom,
% 12.85/6.95 ! [T: vEBT_VEBT,N: nat] :
% 12.85/6.95 ( ( vEBT_invar_vebt @ T @ N )
% 12.85/6.95 => ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ T ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( minus_minus_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2923_space__bound,axiom,
% 12.85/6.95 ! [T: vEBT_VEBT,N: nat,U: nat] :
% 12.85/6.95 ( ( vEBT_invar_vebt @ T @ N )
% 12.85/6.95 => ( ( U
% 12.85/6.95 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.95 => ( ord_less_eq_nat @ ( vEBT_VEBT_space @ T ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ U ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2924_cnt__bound_H,axiom,
% 12.85/6.95 ! [T: vEBT_VEBT,N: nat] :
% 12.85/6.95 ( ( vEBT_invar_vebt @ T @ N )
% 12.85/6.95 => ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ T ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( minus_minus_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) @ one_one_real ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2925_space_H__bound,axiom,
% 12.85/6.95 ! [T: vEBT_VEBT,N: nat,U: nat] :
% 12.85/6.95 ( ( vEBT_invar_vebt @ T @ N )
% 12.85/6.95 => ( ( U
% 12.85/6.95 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.95 => ( ord_less_eq_nat @ ( vEBT_VEBT_space2 @ T ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ U ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2926_nat__approx__posE,axiom,
% 12.85/6.95 ! [E2: real] :
% 12.85/6.95 ( ( ord_less_real @ zero_zero_real @ E2 )
% 12.85/6.95 => ~ ! [N2: nat] :
% 12.85/6.95 ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ E2 ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2927_nat__approx__posE,axiom,
% 12.85/6.95 ! [E2: rat] :
% 12.85/6.95 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 12.85/6.95 => ~ ! [N2: nat] :
% 12.85/6.95 ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ E2 ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2928_high__def,axiom,
% 12.85/6.95 ( vEBT_VEBT_high
% 12.85/6.95 = ( ^ [X3: nat,N4: nat] : ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2929_pow__sum,axiom,
% 12.85/6.95 ! [A: nat,B: nat] :
% 12.85/6.95 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 12.85/6.95 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2930_even__odd__cases,axiom,
% 12.85/6.95 ! [X: nat] :
% 12.85/6.95 ( ! [N2: nat] :
% 12.85/6.95 ( X
% 12.85/6.95 != ( plus_plus_nat @ N2 @ N2 ) )
% 12.85/6.95 => ~ ! [N2: nat] :
% 12.85/6.95 ( X
% 12.85/6.95 != ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2931_deg__not__0,axiom,
% 12.85/6.95 ! [T: vEBT_VEBT,N: nat] :
% 12.85/6.95 ( ( vEBT_invar_vebt @ T @ N )
% 12.85/6.95 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2932_buildup__gives__valid,axiom,
% 12.85/6.95 ! [N: nat] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2933_add__left__cancel,axiom,
% 12.85/6.95 ! [A: real,B: real,C: real] :
% 12.85/6.95 ( ( ( plus_plus_real @ A @ B )
% 12.85/6.95 = ( plus_plus_real @ A @ C ) )
% 12.85/6.95 = ( B = C ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2934_add__left__cancel,axiom,
% 12.85/6.95 ! [A: rat,B: rat,C: rat] :
% 12.85/6.95 ( ( ( plus_plus_rat @ A @ B )
% 12.85/6.95 = ( plus_plus_rat @ A @ C ) )
% 12.85/6.95 = ( B = C ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2935_add__left__cancel,axiom,
% 12.85/6.95 ! [A: nat,B: nat,C: nat] :
% 12.85/6.95 ( ( ( plus_plus_nat @ A @ B )
% 12.85/6.95 = ( plus_plus_nat @ A @ C ) )
% 12.85/6.95 = ( B = C ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2936_add__left__cancel,axiom,
% 12.85/6.95 ! [A: int,B: int,C: int] :
% 12.85/6.95 ( ( ( plus_plus_int @ A @ B )
% 12.85/6.95 = ( plus_plus_int @ A @ C ) )
% 12.85/6.95 = ( B = C ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2937_add__right__cancel,axiom,
% 12.85/6.95 ! [B: real,A: real,C: real] :
% 12.85/6.95 ( ( ( plus_plus_real @ B @ A )
% 12.85/6.95 = ( plus_plus_real @ C @ A ) )
% 12.85/6.95 = ( B = C ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2938_add__right__cancel,axiom,
% 12.85/6.95 ! [B: rat,A: rat,C: rat] :
% 12.85/6.95 ( ( ( plus_plus_rat @ B @ A )
% 12.85/6.95 = ( plus_plus_rat @ C @ A ) )
% 12.85/6.95 = ( B = C ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2939_add__right__cancel,axiom,
% 12.85/6.95 ! [B: nat,A: nat,C: nat] :
% 12.85/6.95 ( ( ( plus_plus_nat @ B @ A )
% 12.85/6.95 = ( plus_plus_nat @ C @ A ) )
% 12.85/6.95 = ( B = C ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2940_add__right__cancel,axiom,
% 12.85/6.95 ! [B: int,A: int,C: int] :
% 12.85/6.95 ( ( ( plus_plus_int @ B @ A )
% 12.85/6.95 = ( plus_plus_int @ C @ A ) )
% 12.85/6.95 = ( B = C ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2941_high__bound__aux,axiom,
% 12.85/6.95 ! [Ma: nat,N: nat,M: nat] :
% 12.85/6.95 ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 12.85/6.95 => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2942_add__le__cancel__left,axiom,
% 12.85/6.95 ! [C: rat,A: rat,B: rat] :
% 12.85/6.95 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 12.85/6.95 = ( ord_less_eq_rat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2943_add__le__cancel__left,axiom,
% 12.85/6.95 ! [C: nat,A: nat,B: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 12.85/6.95 = ( ord_less_eq_nat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2944_add__le__cancel__left,axiom,
% 12.85/6.95 ! [C: int,A: int,B: int] :
% 12.85/6.95 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 12.85/6.95 = ( ord_less_eq_int @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2945_add__le__cancel__left,axiom,
% 12.85/6.95 ! [C: real,A: real,B: real] :
% 12.85/6.95 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 12.85/6.95 = ( ord_less_eq_real @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2946_add__le__cancel__right,axiom,
% 12.85/6.95 ! [A: rat,C: rat,B: rat] :
% 12.85/6.95 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 12.85/6.95 = ( ord_less_eq_rat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2947_add__le__cancel__right,axiom,
% 12.85/6.95 ! [A: nat,C: nat,B: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 12.85/6.95 = ( ord_less_eq_nat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2948_add__le__cancel__right,axiom,
% 12.85/6.95 ! [A: int,C: int,B: int] :
% 12.85/6.95 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 12.85/6.95 = ( ord_less_eq_int @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2949_add__le__cancel__right,axiom,
% 12.85/6.95 ! [A: real,C: real,B: real] :
% 12.85/6.95 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 12.85/6.95 = ( ord_less_eq_real @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2950_double__eq__0__iff,axiom,
% 12.85/6.95 ! [A: real] :
% 12.85/6.95 ( ( ( plus_plus_real @ A @ A )
% 12.85/6.95 = zero_zero_real )
% 12.85/6.95 = ( A = zero_zero_real ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2951_double__eq__0__iff,axiom,
% 12.85/6.95 ! [A: rat] :
% 12.85/6.95 ( ( ( plus_plus_rat @ A @ A )
% 12.85/6.95 = zero_zero_rat )
% 12.85/6.95 = ( A = zero_zero_rat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2952_double__eq__0__iff,axiom,
% 12.85/6.95 ! [A: int] :
% 12.85/6.95 ( ( ( plus_plus_int @ A @ A )
% 12.85/6.95 = zero_zero_int )
% 12.85/6.95 = ( A = zero_zero_int ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2953_add__0,axiom,
% 12.85/6.95 ! [A: complex] :
% 12.85/6.95 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2954_add__0,axiom,
% 12.85/6.95 ! [A: real] :
% 12.85/6.95 ( ( plus_plus_real @ zero_zero_real @ A )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2955_add__0,axiom,
% 12.85/6.95 ! [A: rat] :
% 12.85/6.95 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2956_add__0,axiom,
% 12.85/6.95 ! [A: nat] :
% 12.85/6.95 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2957_add__0,axiom,
% 12.85/6.95 ! [A: int] :
% 12.85/6.95 ( ( plus_plus_int @ zero_zero_int @ A )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2958_zero__eq__add__iff__both__eq__0,axiom,
% 12.85/6.95 ! [X: nat,Y4: nat] :
% 12.85/6.95 ( ( zero_zero_nat
% 12.85/6.95 = ( plus_plus_nat @ X @ Y4 ) )
% 12.85/6.95 = ( ( X = zero_zero_nat )
% 12.85/6.95 & ( Y4 = zero_zero_nat ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2959_add__eq__0__iff__both__eq__0,axiom,
% 12.85/6.95 ! [X: nat,Y4: nat] :
% 12.85/6.95 ( ( ( plus_plus_nat @ X @ Y4 )
% 12.85/6.95 = zero_zero_nat )
% 12.85/6.95 = ( ( X = zero_zero_nat )
% 12.85/6.95 & ( Y4 = zero_zero_nat ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2960_add__cancel__right__right,axiom,
% 12.85/6.95 ! [A: complex,B: complex] :
% 12.85/6.95 ( ( A
% 12.85/6.95 = ( plus_plus_complex @ A @ B ) )
% 12.85/6.95 = ( B = zero_zero_complex ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2961_add__cancel__right__right,axiom,
% 12.85/6.95 ! [A: real,B: real] :
% 12.85/6.95 ( ( A
% 12.85/6.95 = ( plus_plus_real @ A @ B ) )
% 12.85/6.95 = ( B = zero_zero_real ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2962_add__cancel__right__right,axiom,
% 12.85/6.95 ! [A: rat,B: rat] :
% 12.85/6.95 ( ( A
% 12.85/6.95 = ( plus_plus_rat @ A @ B ) )
% 12.85/6.95 = ( B = zero_zero_rat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2963_add__cancel__right__right,axiom,
% 12.85/6.95 ! [A: nat,B: nat] :
% 12.85/6.95 ( ( A
% 12.85/6.95 = ( plus_plus_nat @ A @ B ) )
% 12.85/6.95 = ( B = zero_zero_nat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2964_add__cancel__right__right,axiom,
% 12.85/6.95 ! [A: int,B: int] :
% 12.85/6.95 ( ( A
% 12.85/6.95 = ( plus_plus_int @ A @ B ) )
% 12.85/6.95 = ( B = zero_zero_int ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2965_add__cancel__right__left,axiom,
% 12.85/6.95 ! [A: complex,B: complex] :
% 12.85/6.95 ( ( A
% 12.85/6.95 = ( plus_plus_complex @ B @ A ) )
% 12.85/6.95 = ( B = zero_zero_complex ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2966_add__cancel__right__left,axiom,
% 12.85/6.95 ! [A: real,B: real] :
% 12.85/6.95 ( ( A
% 12.85/6.95 = ( plus_plus_real @ B @ A ) )
% 12.85/6.95 = ( B = zero_zero_real ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2967_add__cancel__right__left,axiom,
% 12.85/6.95 ! [A: rat,B: rat] :
% 12.85/6.95 ( ( A
% 12.85/6.95 = ( plus_plus_rat @ B @ A ) )
% 12.85/6.95 = ( B = zero_zero_rat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2968_add__cancel__right__left,axiom,
% 12.85/6.95 ! [A: nat,B: nat] :
% 12.85/6.95 ( ( A
% 12.85/6.95 = ( plus_plus_nat @ B @ A ) )
% 12.85/6.95 = ( B = zero_zero_nat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2969_add__cancel__right__left,axiom,
% 12.85/6.95 ! [A: int,B: int] :
% 12.85/6.95 ( ( A
% 12.85/6.95 = ( plus_plus_int @ B @ A ) )
% 12.85/6.95 = ( B = zero_zero_int ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2970_add__cancel__left__right,axiom,
% 12.85/6.95 ! [A: complex,B: complex] :
% 12.85/6.95 ( ( ( plus_plus_complex @ A @ B )
% 12.85/6.95 = A )
% 12.85/6.95 = ( B = zero_zero_complex ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2971_add__cancel__left__right,axiom,
% 12.85/6.95 ! [A: real,B: real] :
% 12.85/6.95 ( ( ( plus_plus_real @ A @ B )
% 12.85/6.95 = A )
% 12.85/6.95 = ( B = zero_zero_real ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2972_add__cancel__left__right,axiom,
% 12.85/6.95 ! [A: rat,B: rat] :
% 12.85/6.95 ( ( ( plus_plus_rat @ A @ B )
% 12.85/6.95 = A )
% 12.85/6.95 = ( B = zero_zero_rat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2973_add__cancel__left__right,axiom,
% 12.85/6.95 ! [A: nat,B: nat] :
% 12.85/6.95 ( ( ( plus_plus_nat @ A @ B )
% 12.85/6.95 = A )
% 12.85/6.95 = ( B = zero_zero_nat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2974_add__cancel__left__right,axiom,
% 12.85/6.95 ! [A: int,B: int] :
% 12.85/6.95 ( ( ( plus_plus_int @ A @ B )
% 12.85/6.95 = A )
% 12.85/6.95 = ( B = zero_zero_int ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2975_add__cancel__left__left,axiom,
% 12.85/6.95 ! [B: complex,A: complex] :
% 12.85/6.95 ( ( ( plus_plus_complex @ B @ A )
% 12.85/6.95 = A )
% 12.85/6.95 = ( B = zero_zero_complex ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2976_add__cancel__left__left,axiom,
% 12.85/6.95 ! [B: real,A: real] :
% 12.85/6.95 ( ( ( plus_plus_real @ B @ A )
% 12.85/6.95 = A )
% 12.85/6.95 = ( B = zero_zero_real ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2977_add__cancel__left__left,axiom,
% 12.85/6.95 ! [B: rat,A: rat] :
% 12.85/6.95 ( ( ( plus_plus_rat @ B @ A )
% 12.85/6.95 = A )
% 12.85/6.95 = ( B = zero_zero_rat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2978_add__cancel__left__left,axiom,
% 12.85/6.95 ! [B: nat,A: nat] :
% 12.85/6.95 ( ( ( plus_plus_nat @ B @ A )
% 12.85/6.95 = A )
% 12.85/6.95 = ( B = zero_zero_nat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2979_add__cancel__left__left,axiom,
% 12.85/6.95 ! [B: int,A: int] :
% 12.85/6.95 ( ( ( plus_plus_int @ B @ A )
% 12.85/6.95 = A )
% 12.85/6.95 = ( B = zero_zero_int ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2980_double__zero__sym,axiom,
% 12.85/6.95 ! [A: real] :
% 12.85/6.95 ( ( zero_zero_real
% 12.85/6.95 = ( plus_plus_real @ A @ A ) )
% 12.85/6.95 = ( A = zero_zero_real ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2981_double__zero__sym,axiom,
% 12.85/6.95 ! [A: rat] :
% 12.85/6.95 ( ( zero_zero_rat
% 12.85/6.95 = ( plus_plus_rat @ A @ A ) )
% 12.85/6.95 = ( A = zero_zero_rat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2982_double__zero__sym,axiom,
% 12.85/6.95 ! [A: int] :
% 12.85/6.95 ( ( zero_zero_int
% 12.85/6.95 = ( plus_plus_int @ A @ A ) )
% 12.85/6.95 = ( A = zero_zero_int ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2983_add_Oright__neutral,axiom,
% 12.85/6.95 ! [A: complex] :
% 12.85/6.95 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2984_add_Oright__neutral,axiom,
% 12.85/6.95 ! [A: real] :
% 12.85/6.95 ( ( plus_plus_real @ A @ zero_zero_real )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2985_add_Oright__neutral,axiom,
% 12.85/6.95 ! [A: rat] :
% 12.85/6.95 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2986_add_Oright__neutral,axiom,
% 12.85/6.95 ! [A: nat] :
% 12.85/6.95 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2987_add_Oright__neutral,axiom,
% 12.85/6.95 ! [A: int] :
% 12.85/6.95 ( ( plus_plus_int @ A @ zero_zero_int )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2988_add__less__cancel__left,axiom,
% 12.85/6.95 ! [C: real,A: real,B: real] :
% 12.85/6.95 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 12.85/6.95 = ( ord_less_real @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2989_add__less__cancel__left,axiom,
% 12.85/6.95 ! [C: rat,A: rat,B: rat] :
% 12.85/6.95 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 12.85/6.95 = ( ord_less_rat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2990_add__less__cancel__left,axiom,
% 12.85/6.95 ! [C: nat,A: nat,B: nat] :
% 12.85/6.95 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 12.85/6.95 = ( ord_less_nat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2991_add__less__cancel__left,axiom,
% 12.85/6.95 ! [C: int,A: int,B: int] :
% 12.85/6.95 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 12.85/6.95 = ( ord_less_int @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2992_add__less__cancel__right,axiom,
% 12.85/6.95 ! [A: real,C: real,B: real] :
% 12.85/6.95 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 12.85/6.95 = ( ord_less_real @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2993_add__less__cancel__right,axiom,
% 12.85/6.95 ! [A: rat,C: rat,B: rat] :
% 12.85/6.95 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 12.85/6.95 = ( ord_less_rat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2994_add__less__cancel__right,axiom,
% 12.85/6.95 ! [A: nat,C: nat,B: nat] :
% 12.85/6.95 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 12.85/6.95 = ( ord_less_nat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2995_add__less__cancel__right,axiom,
% 12.85/6.95 ! [A: int,C: int,B: int] :
% 12.85/6.95 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 12.85/6.95 = ( ord_less_int @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2996_numeral__plus__numeral,axiom,
% 12.85/6.95 ! [M: num,N: num] :
% 12.85/6.95 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 12.85/6.95 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2997_numeral__plus__numeral,axiom,
% 12.85/6.95 ! [M: num,N: num] :
% 12.85/6.95 ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 12.85/6.95 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2998_numeral__plus__numeral,axiom,
% 12.85/6.95 ! [M: num,N: num] :
% 12.85/6.95 ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 12.85/6.95 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_2999_numeral__plus__numeral,axiom,
% 12.85/6.95 ! [M: num,N: num] :
% 12.85/6.95 ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 12.85/6.95 = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3000_numeral__plus__numeral,axiom,
% 12.85/6.95 ! [M: num,N: num] :
% 12.85/6.95 ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 12.85/6.95 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3001_add__numeral__left,axiom,
% 12.85/6.95 ! [V: num,W: num,Z: complex] :
% 12.85/6.95 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 12.85/6.95 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3002_add__numeral__left,axiom,
% 12.85/6.95 ! [V: num,W: num,Z: real] :
% 12.85/6.95 ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 12.85/6.95 = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3003_add__numeral__left,axiom,
% 12.85/6.95 ! [V: num,W: num,Z: rat] :
% 12.85/6.95 ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 12.85/6.95 = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3004_add__numeral__left,axiom,
% 12.85/6.95 ! [V: num,W: num,Z: nat] :
% 12.85/6.95 ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 12.85/6.95 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3005_add__numeral__left,axiom,
% 12.85/6.95 ! [V: num,W: num,Z: int] :
% 12.85/6.95 ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 12.85/6.95 = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3006_add__diff__cancel__right_H,axiom,
% 12.85/6.95 ! [A: real,B: real] :
% 12.85/6.95 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3007_add__diff__cancel__right_H,axiom,
% 12.85/6.95 ! [A: rat,B: rat] :
% 12.85/6.95 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3008_add__diff__cancel__right_H,axiom,
% 12.85/6.95 ! [A: nat,B: nat] :
% 12.85/6.95 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3009_add__diff__cancel__right_H,axiom,
% 12.85/6.95 ! [A: int,B: int] :
% 12.85/6.95 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3010_add__diff__cancel__right,axiom,
% 12.85/6.95 ! [A: real,C: real,B: real] :
% 12.85/6.95 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 12.85/6.95 = ( minus_minus_real @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3011_add__diff__cancel__right,axiom,
% 12.85/6.95 ! [A: rat,C: rat,B: rat] :
% 12.85/6.95 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 12.85/6.95 = ( minus_minus_rat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3012_add__diff__cancel__right,axiom,
% 12.85/6.95 ! [A: nat,C: nat,B: nat] :
% 12.85/6.95 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 12.85/6.95 = ( minus_minus_nat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3013_add__diff__cancel__right,axiom,
% 12.85/6.95 ! [A: int,C: int,B: int] :
% 12.85/6.95 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 12.85/6.95 = ( minus_minus_int @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3014_add__diff__cancel__left_H,axiom,
% 12.85/6.95 ! [A: real,B: real] :
% 12.85/6.95 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 12.85/6.95 = B ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3015_add__diff__cancel__left_H,axiom,
% 12.85/6.95 ! [A: rat,B: rat] :
% 12.85/6.95 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
% 12.85/6.95 = B ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3016_add__diff__cancel__left_H,axiom,
% 12.85/6.95 ! [A: nat,B: nat] :
% 12.85/6.95 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 12.85/6.95 = B ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3017_add__diff__cancel__left_H,axiom,
% 12.85/6.95 ! [A: int,B: int] :
% 12.85/6.95 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 12.85/6.95 = B ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3018_add__diff__cancel__left,axiom,
% 12.85/6.95 ! [C: real,A: real,B: real] :
% 12.85/6.95 ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 12.85/6.95 = ( minus_minus_real @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3019_add__diff__cancel__left,axiom,
% 12.85/6.95 ! [C: rat,A: rat,B: rat] :
% 12.85/6.95 ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 12.85/6.95 = ( minus_minus_rat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3020_add__diff__cancel__left,axiom,
% 12.85/6.95 ! [C: nat,A: nat,B: nat] :
% 12.85/6.95 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 12.85/6.95 = ( minus_minus_nat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3021_add__diff__cancel__left,axiom,
% 12.85/6.95 ! [C: int,A: int,B: int] :
% 12.85/6.95 ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 12.85/6.95 = ( minus_minus_int @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3022_diff__add__cancel,axiom,
% 12.85/6.95 ! [A: real,B: real] :
% 12.85/6.95 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3023_diff__add__cancel,axiom,
% 12.85/6.95 ! [A: rat,B: rat] :
% 12.85/6.95 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3024_diff__add__cancel,axiom,
% 12.85/6.95 ! [A: int,B: int] :
% 12.85/6.95 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3025_add__diff__cancel,axiom,
% 12.85/6.95 ! [A: real,B: real] :
% 12.85/6.95 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3026_add__diff__cancel,axiom,
% 12.85/6.95 ! [A: rat,B: rat] :
% 12.85/6.95 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3027_add__diff__cancel,axiom,
% 12.85/6.95 ! [A: int,B: int] :
% 12.85/6.95 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 12.85/6.95 = A ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3028_high__inv,axiom,
% 12.85/6.95 ! [X: nat,N: nat,Y4: nat] :
% 12.85/6.95 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.95 => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
% 12.85/6.95 = Y4 ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3029_dvd__add__triv__right__iff,axiom,
% 12.85/6.95 ! [A: code_integer,B: code_integer] :
% 12.85/6.95 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
% 12.85/6.95 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3030_dvd__add__triv__right__iff,axiom,
% 12.85/6.95 ! [A: real,B: real] :
% 12.85/6.95 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
% 12.85/6.95 = ( dvd_dvd_real @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3031_dvd__add__triv__right__iff,axiom,
% 12.85/6.95 ! [A: rat,B: rat] :
% 12.85/6.95 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 12.85/6.95 = ( dvd_dvd_rat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3032_dvd__add__triv__right__iff,axiom,
% 12.85/6.95 ! [A: nat,B: nat] :
% 12.85/6.95 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 12.85/6.95 = ( dvd_dvd_nat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3033_dvd__add__triv__right__iff,axiom,
% 12.85/6.95 ! [A: int,B: int] :
% 12.85/6.95 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
% 12.85/6.95 = ( dvd_dvd_int @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3034_dvd__add__triv__left__iff,axiom,
% 12.85/6.95 ! [A: code_integer,B: code_integer] :
% 12.85/6.95 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 12.85/6.95 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3035_dvd__add__triv__left__iff,axiom,
% 12.85/6.95 ! [A: real,B: real] :
% 12.85/6.95 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
% 12.85/6.95 = ( dvd_dvd_real @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3036_dvd__add__triv__left__iff,axiom,
% 12.85/6.95 ! [A: rat,B: rat] :
% 12.85/6.95 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 12.85/6.95 = ( dvd_dvd_rat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3037_dvd__add__triv__left__iff,axiom,
% 12.85/6.95 ! [A: nat,B: nat] :
% 12.85/6.95 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 12.85/6.95 = ( dvd_dvd_nat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3038_dvd__add__triv__left__iff,axiom,
% 12.85/6.95 ! [A: int,B: int] :
% 12.85/6.95 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
% 12.85/6.95 = ( dvd_dvd_int @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3039_add__Suc__right,axiom,
% 12.85/6.95 ! [M: nat,N: nat] :
% 12.85/6.95 ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 12.85/6.95 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3040_add__is__0,axiom,
% 12.85/6.95 ! [M: nat,N: nat] :
% 12.85/6.95 ( ( ( plus_plus_nat @ M @ N )
% 12.85/6.95 = zero_zero_nat )
% 12.85/6.95 = ( ( M = zero_zero_nat )
% 12.85/6.95 & ( N = zero_zero_nat ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3041_Nat_Oadd__0__right,axiom,
% 12.85/6.95 ! [M: nat] :
% 12.85/6.95 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 12.85/6.95 = M ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3042_nat__add__left__cancel__less,axiom,
% 12.85/6.95 ! [K: nat,M: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 12.85/6.95 = ( ord_less_nat @ M @ N ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3043_nat__add__left__cancel__le,axiom,
% 12.85/6.95 ! [K: nat,M: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 12.85/6.95 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3044_diff__diff__left,axiom,
% 12.85/6.95 ! [I: nat,J: nat,K: nat] :
% 12.85/6.95 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 12.85/6.95 = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3045_zero__le__double__add__iff__zero__le__single__add,axiom,
% 12.85/6.95 ! [A: rat] :
% 12.85/6.95 ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 12.85/6.95 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3046_zero__le__double__add__iff__zero__le__single__add,axiom,
% 12.85/6.95 ! [A: int] :
% 12.85/6.95 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 12.85/6.95 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3047_zero__le__double__add__iff__zero__le__single__add,axiom,
% 12.85/6.95 ! [A: real] :
% 12.85/6.95 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 12.85/6.95 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3048_double__add__le__zero__iff__single__add__le__zero,axiom,
% 12.85/6.95 ! [A: rat] :
% 12.85/6.95 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 12.85/6.95 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3049_double__add__le__zero__iff__single__add__le__zero,axiom,
% 12.85/6.95 ! [A: int] :
% 12.85/6.95 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 12.85/6.95 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3050_double__add__le__zero__iff__single__add__le__zero,axiom,
% 12.85/6.95 ! [A: real] :
% 12.85/6.95 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 12.85/6.95 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3051_le__add__same__cancel2,axiom,
% 12.85/6.95 ! [A: rat,B: rat] :
% 12.85/6.95 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 12.85/6.95 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3052_le__add__same__cancel2,axiom,
% 12.85/6.95 ! [A: nat,B: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 12.85/6.95 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3053_le__add__same__cancel2,axiom,
% 12.85/6.95 ! [A: int,B: int] :
% 12.85/6.95 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
% 12.85/6.95 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3054_le__add__same__cancel2,axiom,
% 12.85/6.95 ! [A: real,B: real] :
% 12.85/6.95 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 12.85/6.95 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3055_le__add__same__cancel1,axiom,
% 12.85/6.95 ! [A: rat,B: rat] :
% 12.85/6.95 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 12.85/6.95 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3056_le__add__same__cancel1,axiom,
% 12.85/6.95 ! [A: nat,B: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 12.85/6.95 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3057_le__add__same__cancel1,axiom,
% 12.85/6.95 ! [A: int,B: int] :
% 12.85/6.95 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
% 12.85/6.95 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3058_le__add__same__cancel1,axiom,
% 12.85/6.95 ! [A: real,B: real] :
% 12.85/6.95 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 12.85/6.95 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3059_add__le__same__cancel2,axiom,
% 12.85/6.95 ! [A: rat,B: rat] :
% 12.85/6.95 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 12.85/6.95 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3060_add__le__same__cancel2,axiom,
% 12.85/6.95 ! [A: nat,B: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 12.85/6.95 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3061_add__le__same__cancel2,axiom,
% 12.85/6.95 ! [A: int,B: int] :
% 12.85/6.95 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
% 12.85/6.95 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3062_add__le__same__cancel2,axiom,
% 12.85/6.95 ! [A: real,B: real] :
% 12.85/6.95 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 12.85/6.95 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3063_add__le__same__cancel1,axiom,
% 12.85/6.95 ! [B: rat,A: rat] :
% 12.85/6.95 ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 12.85/6.95 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3064_add__le__same__cancel1,axiom,
% 12.85/6.95 ! [B: nat,A: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 12.85/6.95 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3065_add__le__same__cancel1,axiom,
% 12.85/6.95 ! [B: int,A: int] :
% 12.85/6.95 ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
% 12.85/6.95 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3066_add__le__same__cancel1,axiom,
% 12.85/6.95 ! [B: real,A: real] :
% 12.85/6.95 ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 12.85/6.95 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3067_add__less__same__cancel1,axiom,
% 12.85/6.95 ! [B: real,A: real] :
% 12.85/6.95 ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 12.85/6.95 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3068_add__less__same__cancel1,axiom,
% 12.85/6.95 ! [B: rat,A: rat] :
% 12.85/6.95 ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 12.85/6.95 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3069_add__less__same__cancel1,axiom,
% 12.85/6.95 ! [B: nat,A: nat] :
% 12.85/6.95 ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 12.85/6.95 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3070_add__less__same__cancel1,axiom,
% 12.85/6.95 ! [B: int,A: int] :
% 12.85/6.95 ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
% 12.85/6.95 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3071_add__less__same__cancel2,axiom,
% 12.85/6.95 ! [A: real,B: real] :
% 12.85/6.95 ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 12.85/6.95 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3072_add__less__same__cancel2,axiom,
% 12.85/6.95 ! [A: rat,B: rat] :
% 12.85/6.95 ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 12.85/6.95 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3073_add__less__same__cancel2,axiom,
% 12.85/6.95 ! [A: nat,B: nat] :
% 12.85/6.95 ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 12.85/6.95 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3074_add__less__same__cancel2,axiom,
% 12.85/6.95 ! [A: int,B: int] :
% 12.85/6.95 ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
% 12.85/6.95 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3075_less__add__same__cancel1,axiom,
% 12.85/6.95 ! [A: real,B: real] :
% 12.85/6.95 ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 12.85/6.95 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3076_less__add__same__cancel1,axiom,
% 12.85/6.95 ! [A: rat,B: rat] :
% 12.85/6.95 ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 12.85/6.95 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3077_less__add__same__cancel1,axiom,
% 12.85/6.95 ! [A: nat,B: nat] :
% 12.85/6.95 ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 12.85/6.95 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3078_less__add__same__cancel1,axiom,
% 12.85/6.95 ! [A: int,B: int] :
% 12.85/6.95 ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
% 12.85/6.95 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3079_less__add__same__cancel2,axiom,
% 12.85/6.95 ! [A: real,B: real] :
% 12.85/6.95 ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 12.85/6.95 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3080_less__add__same__cancel2,axiom,
% 12.85/6.95 ! [A: rat,B: rat] :
% 12.85/6.95 ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 12.85/6.95 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3081_less__add__same__cancel2,axiom,
% 12.85/6.95 ! [A: nat,B: nat] :
% 12.85/6.95 ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 12.85/6.95 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3082_less__add__same__cancel2,axiom,
% 12.85/6.95 ! [A: int,B: int] :
% 12.85/6.95 ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
% 12.85/6.95 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3083_double__add__less__zero__iff__single__add__less__zero,axiom,
% 12.85/6.95 ! [A: real] :
% 12.85/6.95 ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 12.85/6.95 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3084_double__add__less__zero__iff__single__add__less__zero,axiom,
% 12.85/6.95 ! [A: rat] :
% 12.85/6.95 ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 12.85/6.95 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3085_double__add__less__zero__iff__single__add__less__zero,axiom,
% 12.85/6.95 ! [A: int] :
% 12.85/6.95 ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 12.85/6.95 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3086_zero__less__double__add__iff__zero__less__single__add,axiom,
% 12.85/6.95 ! [A: real] :
% 12.85/6.95 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 12.85/6.95 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3087_zero__less__double__add__iff__zero__less__single__add,axiom,
% 12.85/6.95 ! [A: rat] :
% 12.85/6.95 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 12.85/6.95 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3088_zero__less__double__add__iff__zero__less__single__add,axiom,
% 12.85/6.95 ! [A: int] :
% 12.85/6.95 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 12.85/6.95 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3089_sum__squares__eq__zero__iff,axiom,
% 12.85/6.95 ! [X: real,Y4: real] :
% 12.85/6.95 ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) )
% 12.85/6.95 = zero_zero_real )
% 12.85/6.95 = ( ( X = zero_zero_real )
% 12.85/6.95 & ( Y4 = zero_zero_real ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3090_sum__squares__eq__zero__iff,axiom,
% 12.85/6.95 ! [X: rat,Y4: rat] :
% 12.85/6.95 ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y4 @ Y4 ) )
% 12.85/6.95 = zero_zero_rat )
% 12.85/6.95 = ( ( X = zero_zero_rat )
% 12.85/6.95 & ( Y4 = zero_zero_rat ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3091_sum__squares__eq__zero__iff,axiom,
% 12.85/6.95 ! [X: int,Y4: int] :
% 12.85/6.95 ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y4 @ Y4 ) )
% 12.85/6.95 = zero_zero_int )
% 12.85/6.95 = ( ( X = zero_zero_int )
% 12.85/6.95 & ( Y4 = zero_zero_int ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3092_le__add__diff__inverse,axiom,
% 12.85/6.95 ! [B: rat,A: rat] :
% 12.85/6.95 ( ( ord_less_eq_rat @ B @ A )
% 12.85/6.95 => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 12.85/6.95 = A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3093_le__add__diff__inverse,axiom,
% 12.85/6.95 ! [B: nat,A: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ B @ A )
% 12.85/6.95 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 12.85/6.95 = A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3094_le__add__diff__inverse,axiom,
% 12.85/6.95 ! [B: int,A: int] :
% 12.85/6.95 ( ( ord_less_eq_int @ B @ A )
% 12.85/6.95 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 12.85/6.95 = A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3095_le__add__diff__inverse,axiom,
% 12.85/6.95 ! [B: real,A: real] :
% 12.85/6.95 ( ( ord_less_eq_real @ B @ A )
% 12.85/6.95 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 12.85/6.95 = A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3096_le__add__diff__inverse2,axiom,
% 12.85/6.95 ! [B: rat,A: rat] :
% 12.85/6.95 ( ( ord_less_eq_rat @ B @ A )
% 12.85/6.95 => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 12.85/6.95 = A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3097_le__add__diff__inverse2,axiom,
% 12.85/6.95 ! [B: nat,A: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ B @ A )
% 12.85/6.95 => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
% 12.85/6.95 = A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3098_le__add__diff__inverse2,axiom,
% 12.85/6.95 ! [B: int,A: int] :
% 12.85/6.95 ( ( ord_less_eq_int @ B @ A )
% 12.85/6.95 => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 12.85/6.95 = A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3099_le__add__diff__inverse2,axiom,
% 12.85/6.95 ! [B: real,A: real] :
% 12.85/6.95 ( ( ord_less_eq_real @ B @ A )
% 12.85/6.95 => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 12.85/6.95 = A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3100_diff__add__zero,axiom,
% 12.85/6.95 ! [A: nat,B: nat] :
% 12.85/6.95 ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 12.85/6.95 = zero_zero_nat ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3101_distrib__left__numeral,axiom,
% 12.85/6.95 ! [V: num,B: complex,C: complex] :
% 12.85/6.95 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
% 12.85/6.95 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3102_distrib__left__numeral,axiom,
% 12.85/6.95 ! [V: num,B: real,C: real] :
% 12.85/6.95 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
% 12.85/6.95 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3103_distrib__left__numeral,axiom,
% 12.85/6.95 ! [V: num,B: rat,C: rat] :
% 12.85/6.95 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
% 12.85/6.95 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3104_distrib__left__numeral,axiom,
% 12.85/6.95 ! [V: num,B: nat,C: nat] :
% 12.85/6.95 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
% 12.85/6.95 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3105_distrib__left__numeral,axiom,
% 12.85/6.95 ! [V: num,B: int,C: int] :
% 12.85/6.95 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
% 12.85/6.95 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3106_distrib__right__numeral,axiom,
% 12.85/6.95 ! [A: complex,B: complex,V: num] :
% 12.85/6.95 ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 12.85/6.95 = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3107_distrib__right__numeral,axiom,
% 12.85/6.95 ! [A: real,B: real,V: num] :
% 12.85/6.95 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 12.85/6.95 = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3108_distrib__right__numeral,axiom,
% 12.85/6.95 ! [A: rat,B: rat,V: num] :
% 12.85/6.95 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 12.85/6.95 = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3109_distrib__right__numeral,axiom,
% 12.85/6.95 ! [A: nat,B: nat,V: num] :
% 12.85/6.95 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
% 12.85/6.95 = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3110_distrib__right__numeral,axiom,
% 12.85/6.95 ! [A: int,B: int,V: num] :
% 12.85/6.95 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 12.85/6.95 = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3111_dvd__add__times__triv__left__iff,axiom,
% 12.85/6.95 ! [A: code_integer,C: code_integer,B: code_integer] :
% 12.85/6.95 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
% 12.85/6.95 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3112_dvd__add__times__triv__left__iff,axiom,
% 12.85/6.95 ! [A: real,C: real,B: real] :
% 12.85/6.95 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
% 12.85/6.95 = ( dvd_dvd_real @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3113_dvd__add__times__triv__left__iff,axiom,
% 12.85/6.95 ! [A: rat,C: rat,B: rat] :
% 12.85/6.95 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
% 12.85/6.95 = ( dvd_dvd_rat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3114_dvd__add__times__triv__left__iff,axiom,
% 12.85/6.95 ! [A: nat,C: nat,B: nat] :
% 12.85/6.95 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
% 12.85/6.95 = ( dvd_dvd_nat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3115_dvd__add__times__triv__left__iff,axiom,
% 12.85/6.95 ! [A: int,C: int,B: int] :
% 12.85/6.95 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
% 12.85/6.95 = ( dvd_dvd_int @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3116_dvd__add__times__triv__right__iff,axiom,
% 12.85/6.95 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.95 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
% 12.85/6.95 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3117_dvd__add__times__triv__right__iff,axiom,
% 12.85/6.95 ! [A: real,B: real,C: real] :
% 12.85/6.95 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
% 12.85/6.95 = ( dvd_dvd_real @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3118_dvd__add__times__triv__right__iff,axiom,
% 12.85/6.95 ! [A: rat,B: rat,C: rat] :
% 12.85/6.95 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
% 12.85/6.95 = ( dvd_dvd_rat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3119_dvd__add__times__triv__right__iff,axiom,
% 12.85/6.95 ! [A: nat,B: nat,C: nat] :
% 12.85/6.95 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
% 12.85/6.95 = ( dvd_dvd_nat @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3120_dvd__add__times__triv__right__iff,axiom,
% 12.85/6.95 ! [A: int,B: int,C: int] :
% 12.85/6.95 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
% 12.85/6.95 = ( dvd_dvd_int @ A @ B ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3121_div__add,axiom,
% 12.85/6.95 ! [C: code_integer,A: code_integer,B: code_integer] :
% 12.85/6.95 ( ( dvd_dvd_Code_integer @ C @ A )
% 12.85/6.95 => ( ( dvd_dvd_Code_integer @ C @ B )
% 12.85/6.95 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 12.85/6.95 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3122_div__add,axiom,
% 12.85/6.95 ! [C: nat,A: nat,B: nat] :
% 12.85/6.95 ( ( dvd_dvd_nat @ C @ A )
% 12.85/6.95 => ( ( dvd_dvd_nat @ C @ B )
% 12.85/6.95 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 12.85/6.95 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3123_div__add,axiom,
% 12.85/6.95 ! [C: int,A: int,B: int] :
% 12.85/6.95 ( ( dvd_dvd_int @ C @ A )
% 12.85/6.95 => ( ( dvd_dvd_int @ C @ B )
% 12.85/6.95 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 12.85/6.95 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3124_of__nat__add,axiom,
% 12.85/6.95 ! [M: nat,N: nat] :
% 12.85/6.95 ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N ) )
% 12.85/6.95 = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3125_of__nat__add,axiom,
% 12.85/6.95 ! [M: nat,N: nat] :
% 12.85/6.95 ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
% 12.85/6.95 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3126_of__nat__add,axiom,
% 12.85/6.95 ! [M: nat,N: nat] :
% 12.85/6.95 ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
% 12.85/6.95 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3127_of__nat__add,axiom,
% 12.85/6.95 ! [M: nat,N: nat] :
% 12.85/6.95 ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
% 12.85/6.95 = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3128_of__nat__add,axiom,
% 12.85/6.95 ! [M: nat,N: nat] :
% 12.85/6.95 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
% 12.85/6.95 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3129_add__gr__0,axiom,
% 12.85/6.95 ! [M: nat,N: nat] :
% 12.85/6.95 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 12.85/6.95 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/6.95 | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3130_mult__Suc__right,axiom,
% 12.85/6.95 ! [M: nat,N: nat] :
% 12.85/6.95 ( ( times_times_nat @ M @ ( suc @ N ) )
% 12.85/6.95 = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3131_Nat_Odiff__diff__right,axiom,
% 12.85/6.95 ! [K: nat,J: nat,I: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ K @ J )
% 12.85/6.95 => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 12.85/6.95 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3132_Nat_Oadd__diff__assoc2,axiom,
% 12.85/6.95 ! [K: nat,J: nat,I: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ K @ J )
% 12.85/6.95 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 12.85/6.95 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3133_Nat_Oadd__diff__assoc,axiom,
% 12.85/6.95 ! [K: nat,J: nat,I: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ K @ J )
% 12.85/6.95 => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 12.85/6.95 = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3134_triangle__Suc,axiom,
% 12.85/6.95 ! [N: nat] :
% 12.85/6.95 ( ( nat_triangle @ ( suc @ N ) )
% 12.85/6.95 = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3135_div__mult__self1,axiom,
% 12.85/6.95 ! [B: nat,A: nat,C: nat] :
% 12.85/6.95 ( ( B != zero_zero_nat )
% 12.85/6.95 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 12.85/6.95 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3136_div__mult__self1,axiom,
% 12.85/6.95 ! [B: int,A: int,C: int] :
% 12.85/6.95 ( ( B != zero_zero_int )
% 12.85/6.95 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 12.85/6.95 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3137_div__mult__self2,axiom,
% 12.85/6.95 ! [B: nat,A: nat,C: nat] :
% 12.85/6.95 ( ( B != zero_zero_nat )
% 12.85/6.95 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 12.85/6.95 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3138_div__mult__self2,axiom,
% 12.85/6.95 ! [B: int,A: int,C: int] :
% 12.85/6.95 ( ( B != zero_zero_int )
% 12.85/6.95 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 12.85/6.95 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3139_div__mult__self3,axiom,
% 12.85/6.95 ! [B: nat,C: nat,A: nat] :
% 12.85/6.95 ( ( B != zero_zero_nat )
% 12.85/6.95 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 12.85/6.95 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3140_div__mult__self3,axiom,
% 12.85/6.95 ! [B: int,C: int,A: int] :
% 12.85/6.95 ( ( B != zero_zero_int )
% 12.85/6.95 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 12.85/6.95 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3141_div__mult__self4,axiom,
% 12.85/6.95 ! [B: nat,C: nat,A: nat] :
% 12.85/6.95 ( ( B != zero_zero_nat )
% 12.85/6.95 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 12.85/6.95 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3142_div__mult__self4,axiom,
% 12.85/6.95 ! [B: int,C: int,A: int] :
% 12.85/6.95 ( ( B != zero_zero_int )
% 12.85/6.95 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 12.85/6.95 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3143_numeral__plus__one,axiom,
% 12.85/6.95 ! [N: num] :
% 12.85/6.95 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
% 12.85/6.95 = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3144_numeral__plus__one,axiom,
% 12.85/6.95 ! [N: num] :
% 12.85/6.95 ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 12.85/6.95 = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3145_numeral__plus__one,axiom,
% 12.85/6.95 ! [N: num] :
% 12.85/6.95 ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 12.85/6.95 = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3146_numeral__plus__one,axiom,
% 12.85/6.95 ! [N: num] :
% 12.85/6.95 ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 12.85/6.95 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3147_numeral__plus__one,axiom,
% 12.85/6.95 ! [N: num] :
% 12.85/6.95 ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 12.85/6.95 = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3148_one__plus__numeral,axiom,
% 12.85/6.95 ! [N: num] :
% 12.85/6.95 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
% 12.85/6.95 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3149_one__plus__numeral,axiom,
% 12.85/6.95 ! [N: num] :
% 12.85/6.95 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 12.85/6.95 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3150_one__plus__numeral,axiom,
% 12.85/6.95 ! [N: num] :
% 12.85/6.95 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 12.85/6.95 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3151_one__plus__numeral,axiom,
% 12.85/6.95 ! [N: num] :
% 12.85/6.95 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 12.85/6.95 = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3152_one__plus__numeral,axiom,
% 12.85/6.95 ! [N: num] :
% 12.85/6.95 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 12.85/6.95 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3153_of__nat__Suc,axiom,
% 12.85/6.95 ! [M: nat] :
% 12.85/6.95 ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 12.85/6.95 = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3154_of__nat__Suc,axiom,
% 12.85/6.95 ! [M: nat] :
% 12.85/6.95 ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 12.85/6.95 = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3155_of__nat__Suc,axiom,
% 12.85/6.95 ! [M: nat] :
% 12.85/6.95 ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 12.85/6.95 = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3156_of__nat__Suc,axiom,
% 12.85/6.95 ! [M: nat] :
% 12.85/6.95 ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 12.85/6.95 = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3157_of__nat__Suc,axiom,
% 12.85/6.95 ! [M: nat] :
% 12.85/6.95 ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 12.85/6.95 = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3158_diff__Suc__diff__eq2,axiom,
% 12.85/6.95 ! [K: nat,J: nat,I: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ K @ J )
% 12.85/6.95 => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
% 12.85/6.95 = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3159_diff__Suc__diff__eq1,axiom,
% 12.85/6.95 ! [K: nat,J: nat,I: nat] :
% 12.85/6.95 ( ( ord_less_eq_nat @ K @ J )
% 12.85/6.95 => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 12.85/6.95 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3160_one__add__one,axiom,
% 12.85/6.95 ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
% 12.85/6.95 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3161_one__add__one,axiom,
% 12.85/6.95 ( ( plus_plus_real @ one_one_real @ one_one_real )
% 12.85/6.95 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3162_one__add__one,axiom,
% 12.85/6.95 ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
% 12.85/6.95 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3163_one__add__one,axiom,
% 12.85/6.95 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 12.85/6.95 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3164_one__add__one,axiom,
% 12.85/6.95 ( ( plus_plus_int @ one_one_int @ one_one_int )
% 12.85/6.95 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3165_odd__add,axiom,
% 12.85/6.95 ! [A: code_integer,B: code_integer] :
% 12.85/6.95 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
% 12.85/6.95 = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 12.85/6.95 != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3166_odd__add,axiom,
% 12.85/6.95 ! [A: nat,B: nat] :
% 12.85/6.95 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
% 12.85/6.95 = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 12.85/6.95 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3167_odd__add,axiom,
% 12.85/6.95 ! [A: int,B: int] :
% 12.85/6.95 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
% 12.85/6.95 = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 12.85/6.95 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3168_even__add,axiom,
% 12.85/6.95 ! [A: code_integer,B: code_integer] :
% 12.85/6.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 12.85/6.95 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.95 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3169_even__add,axiom,
% 12.85/6.95 ! [A: nat,B: nat] :
% 12.85/6.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
% 12.85/6.95 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.95 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3170_even__add,axiom,
% 12.85/6.95 ! [A: int,B: int] :
% 12.85/6.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
% 12.85/6.95 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.95 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3171_add__2__eq__Suc,axiom,
% 12.85/6.95 ! [N: nat] :
% 12.85/6.95 ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.95 = ( suc @ ( suc @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3172_add__2__eq__Suc_H,axiom,
% 12.85/6.95 ! [N: nat] :
% 12.85/6.95 ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( suc @ ( suc @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3173_add__self__div__2,axiom,
% 12.85/6.95 ! [M: nat] :
% 12.85/6.95 ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = M ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3174_even__plus__one__iff,axiom,
% 12.85/6.95 ! [A: code_integer] :
% 12.85/6.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
% 12.85/6.95 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3175_even__plus__one__iff,axiom,
% 12.85/6.95 ! [A: nat] :
% 12.85/6.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
% 12.85/6.95 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3176_even__plus__one__iff,axiom,
% 12.85/6.95 ! [A: int] :
% 12.85/6.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
% 12.85/6.95 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3177_sum__power2__eq__zero__iff,axiom,
% 12.85/6.95 ! [X: rat,Y4: rat] :
% 12.85/6.95 ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.95 = zero_zero_rat )
% 12.85/6.95 = ( ( X = zero_zero_rat )
% 12.85/6.95 & ( Y4 = zero_zero_rat ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3178_sum__power2__eq__zero__iff,axiom,
% 12.85/6.95 ! [X: int,Y4: int] :
% 12.85/6.95 ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.95 = zero_zero_int )
% 12.85/6.95 = ( ( X = zero_zero_int )
% 12.85/6.95 & ( Y4 = zero_zero_int ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3179_sum__power2__eq__zero__iff,axiom,
% 12.85/6.95 ! [X: real,Y4: real] :
% 12.85/6.95 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.95 = zero_zero_real )
% 12.85/6.95 = ( ( X = zero_zero_real )
% 12.85/6.95 & ( Y4 = zero_zero_real ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3180_even__diff,axiom,
% 12.85/6.95 ! [A: code_integer,B: code_integer] :
% 12.85/6.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
% 12.85/6.95 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3181_even__diff,axiom,
% 12.85/6.95 ! [A: int,B: int] :
% 12.85/6.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
% 12.85/6.95 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3182_div__Suc__eq__div__add3,axiom,
% 12.85/6.95 ! [M: nat,N: nat] :
% 12.85/6.95 ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 12.85/6.95 = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3183_Suc__div__eq__add3__div__numeral,axiom,
% 12.85/6.95 ! [M: nat,V: num] :
% 12.85/6.95 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 12.85/6.95 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3184_odd__succ__div__two,axiom,
% 12.85/6.95 ! [A: code_integer] :
% 12.85/6.95 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.95 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3185_odd__succ__div__two,axiom,
% 12.85/6.95 ! [A: nat] :
% 12.85/6.95 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.95 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3186_odd__succ__div__two,axiom,
% 12.85/6.95 ! [A: int] :
% 12.85/6.95 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.95 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3187_even__succ__div__two,axiom,
% 12.85/6.95 ! [A: code_integer] :
% 12.85/6.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.95 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3188_even__succ__div__two,axiom,
% 12.85/6.95 ! [A: nat] :
% 12.85/6.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.95 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3189_even__succ__div__two,axiom,
% 12.85/6.95 ! [A: int] :
% 12.85/6.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.95 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3190_even__succ__div__2,axiom,
% 12.85/6.95 ! [A: code_integer] :
% 12.85/6.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.95 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3191_even__succ__div__2,axiom,
% 12.85/6.95 ! [A: nat] :
% 12.85/6.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.95 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3192_even__succ__div__2,axiom,
% 12.85/6.95 ! [A: int] :
% 12.85/6.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.95 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.95 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3193_even__diff__nat,axiom,
% 12.85/6.95 ! [M: nat,N: nat] :
% 12.85/6.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
% 12.85/6.95 = ( ( ord_less_nat @ M @ N )
% 12.85/6.95 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3194_odd__two__times__div__two__succ,axiom,
% 12.85/6.95 ! [A: code_integer] :
% 12.85/6.95 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.95 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
% 12.85/6.95 = A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3195_odd__two__times__div__two__succ,axiom,
% 12.85/6.95 ! [A: nat] :
% 12.85/6.95 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.95 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
% 12.85/6.95 = A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3196_odd__two__times__div__two__succ,axiom,
% 12.85/6.95 ! [A: int] :
% 12.85/6.95 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.95 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
% 12.85/6.95 = A ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3197_set__bit__0,axiom,
% 12.85/6.95 ! [A: int] :
% 12.85/6.95 ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
% 12.85/6.95 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3198_set__bit__0,axiom,
% 12.85/6.95 ! [A: nat] :
% 12.85/6.95 ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
% 12.85/6.95 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3199_set__bit__0,axiom,
% 12.85/6.95 ! [A: code_integer] :
% 12.85/6.95 ( ( bit_se2793503036327961859nteger @ zero_zero_nat @ A )
% 12.85/6.95 = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3200_even__succ__div__exp,axiom,
% 12.85/6.95 ! [A: code_integer,N: nat] :
% 12.85/6.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.95 = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3201_even__succ__div__exp,axiom,
% 12.85/6.95 ! [A: nat,N: nat] :
% 12.85/6.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.95 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3202_even__succ__div__exp,axiom,
% 12.85/6.95 ! [A: int,N: nat] :
% 12.85/6.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.95 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.95 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.95 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3203_psubsetD,axiom,
% 12.85/6.95 ! [A2: set_complex,B4: set_complex,C: complex] :
% 12.85/6.95 ( ( ord_less_set_complex @ A2 @ B4 )
% 12.85/6.95 => ( ( member_complex @ C @ A2 )
% 12.85/6.95 => ( member_complex @ C @ B4 ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3204_psubsetD,axiom,
% 12.85/6.95 ! [A2: set_real,B4: set_real,C: real] :
% 12.85/6.95 ( ( ord_less_set_real @ A2 @ B4 )
% 12.85/6.95 => ( ( member_real @ C @ A2 )
% 12.85/6.95 => ( member_real @ C @ B4 ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3205_psubsetD,axiom,
% 12.85/6.95 ! [A2: set_set_nat,B4: set_set_nat,C: set_nat] :
% 12.85/6.95 ( ( ord_less_set_set_nat @ A2 @ B4 )
% 12.85/6.95 => ( ( member_set_nat @ C @ A2 )
% 12.85/6.95 => ( member_set_nat @ C @ B4 ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3206_psubsetD,axiom,
% 12.85/6.95 ! [A2: set_nat,B4: set_nat,C: nat] :
% 12.85/6.95 ( ( ord_less_set_nat @ A2 @ B4 )
% 12.85/6.95 => ( ( member_nat @ C @ A2 )
% 12.85/6.95 => ( member_nat @ C @ B4 ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3207_psubsetD,axiom,
% 12.85/6.95 ! [A2: set_int,B4: set_int,C: int] :
% 12.85/6.95 ( ( ord_less_set_int @ A2 @ B4 )
% 12.85/6.95 => ( ( member_int @ C @ A2 )
% 12.85/6.95 => ( member_int @ C @ B4 ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3208_enat__less__induct,axiom,
% 12.85/6.95 ! [P: extended_enat > $o,N: extended_enat] :
% 12.85/6.95 ( ! [N2: extended_enat] :
% 12.85/6.95 ( ! [M2: extended_enat] :
% 12.85/6.95 ( ( ord_le72135733267957522d_enat @ M2 @ N2 )
% 12.85/6.95 => ( P @ M2 ) )
% 12.85/6.95 => ( P @ N2 ) )
% 12.85/6.95 => ( P @ N ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3209_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 12.85/6.95 ! [A: real,B: real,C: real] :
% 12.85/6.95 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 12.85/6.95 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3210_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 12.85/6.95 ! [A: rat,B: rat,C: rat] :
% 12.85/6.95 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 12.85/6.95 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 12.85/6.95
% 12.85/6.95 thf(fact_3211_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 12.85/6.95 ! [A: nat,B: nat,C: nat] :
% 12.85/6.95 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3212_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3213_add__mono__thms__linordered__semiring_I4_J,axiom,
% 12.85/6.96 ! [I: real,J: real,K: real,L: real] :
% 12.85/6.96 ( ( ( I = J )
% 12.85/6.96 & ( K = L ) )
% 12.85/6.96 => ( ( plus_plus_real @ I @ K )
% 12.85/6.96 = ( plus_plus_real @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3214_add__mono__thms__linordered__semiring_I4_J,axiom,
% 12.85/6.96 ! [I: rat,J: rat,K: rat,L: rat] :
% 12.85/6.96 ( ( ( I = J )
% 12.85/6.96 & ( K = L ) )
% 12.85/6.96 => ( ( plus_plus_rat @ I @ K )
% 12.85/6.96 = ( plus_plus_rat @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3215_add__mono__thms__linordered__semiring_I4_J,axiom,
% 12.85/6.96 ! [I: nat,J: nat,K: nat,L: nat] :
% 12.85/6.96 ( ( ( I = J )
% 12.85/6.96 & ( K = L ) )
% 12.85/6.96 => ( ( plus_plus_nat @ I @ K )
% 12.85/6.96 = ( plus_plus_nat @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3216_add__mono__thms__linordered__semiring_I4_J,axiom,
% 12.85/6.96 ! [I: int,J: int,K: int,L: int] :
% 12.85/6.96 ( ( ( I = J )
% 12.85/6.96 & ( K = L ) )
% 12.85/6.96 => ( ( plus_plus_int @ I @ K )
% 12.85/6.96 = ( plus_plus_int @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3217_group__cancel_Oadd1,axiom,
% 12.85/6.96 ! [A2: real,K: real,A: real,B: real] :
% 12.85/6.96 ( ( A2
% 12.85/6.96 = ( plus_plus_real @ K @ A ) )
% 12.85/6.96 => ( ( plus_plus_real @ A2 @ B )
% 12.85/6.96 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3218_group__cancel_Oadd1,axiom,
% 12.85/6.96 ! [A2: rat,K: rat,A: rat,B: rat] :
% 12.85/6.96 ( ( A2
% 12.85/6.96 = ( plus_plus_rat @ K @ A ) )
% 12.85/6.96 => ( ( plus_plus_rat @ A2 @ B )
% 12.85/6.96 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3219_group__cancel_Oadd1,axiom,
% 12.85/6.96 ! [A2: nat,K: nat,A: nat,B: nat] :
% 12.85/6.96 ( ( A2
% 12.85/6.96 = ( plus_plus_nat @ K @ A ) )
% 12.85/6.96 => ( ( plus_plus_nat @ A2 @ B )
% 12.85/6.96 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3220_group__cancel_Oadd1,axiom,
% 12.85/6.96 ! [A2: int,K: int,A: int,B: int] :
% 12.85/6.96 ( ( A2
% 12.85/6.96 = ( plus_plus_int @ K @ A ) )
% 12.85/6.96 => ( ( plus_plus_int @ A2 @ B )
% 12.85/6.96 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3221_group__cancel_Oadd2,axiom,
% 12.85/6.96 ! [B4: real,K: real,B: real,A: real] :
% 12.85/6.96 ( ( B4
% 12.85/6.96 = ( plus_plus_real @ K @ B ) )
% 12.85/6.96 => ( ( plus_plus_real @ A @ B4 )
% 12.85/6.96 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3222_group__cancel_Oadd2,axiom,
% 12.85/6.96 ! [B4: rat,K: rat,B: rat,A: rat] :
% 12.85/6.96 ( ( B4
% 12.85/6.96 = ( plus_plus_rat @ K @ B ) )
% 12.85/6.96 => ( ( plus_plus_rat @ A @ B4 )
% 12.85/6.96 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3223_group__cancel_Oadd2,axiom,
% 12.85/6.96 ! [B4: nat,K: nat,B: nat,A: nat] :
% 12.85/6.96 ( ( B4
% 12.85/6.96 = ( plus_plus_nat @ K @ B ) )
% 12.85/6.96 => ( ( plus_plus_nat @ A @ B4 )
% 12.85/6.96 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3224_group__cancel_Oadd2,axiom,
% 12.85/6.96 ! [B4: int,K: int,B: int,A: int] :
% 12.85/6.96 ( ( B4
% 12.85/6.96 = ( plus_plus_int @ K @ B ) )
% 12.85/6.96 => ( ( plus_plus_int @ A @ B4 )
% 12.85/6.96 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3225_add_Oassoc,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3226_add_Oassoc,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3227_add_Oassoc,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3228_add_Oassoc,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3229_add_Oleft__cancel,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( ( plus_plus_real @ A @ B )
% 12.85/6.96 = ( plus_plus_real @ A @ C ) )
% 12.85/6.96 = ( B = C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3230_add_Oleft__cancel,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( ( plus_plus_rat @ A @ B )
% 12.85/6.96 = ( plus_plus_rat @ A @ C ) )
% 12.85/6.96 = ( B = C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3231_add_Oleft__cancel,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( ( plus_plus_int @ A @ B )
% 12.85/6.96 = ( plus_plus_int @ A @ C ) )
% 12.85/6.96 = ( B = C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3232_add_Oright__cancel,axiom,
% 12.85/6.96 ! [B: real,A: real,C: real] :
% 12.85/6.96 ( ( ( plus_plus_real @ B @ A )
% 12.85/6.96 = ( plus_plus_real @ C @ A ) )
% 12.85/6.96 = ( B = C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3233_add_Oright__cancel,axiom,
% 12.85/6.96 ! [B: rat,A: rat,C: rat] :
% 12.85/6.96 ( ( ( plus_plus_rat @ B @ A )
% 12.85/6.96 = ( plus_plus_rat @ C @ A ) )
% 12.85/6.96 = ( B = C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3234_add_Oright__cancel,axiom,
% 12.85/6.96 ! [B: int,A: int,C: int] :
% 12.85/6.96 ( ( ( plus_plus_int @ B @ A )
% 12.85/6.96 = ( plus_plus_int @ C @ A ) )
% 12.85/6.96 = ( B = C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3235_add_Ocommute,axiom,
% 12.85/6.96 ( plus_plus_real
% 12.85/6.96 = ( ^ [A3: real,B2: real] : ( plus_plus_real @ B2 @ A3 ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3236_add_Ocommute,axiom,
% 12.85/6.96 ( plus_plus_rat
% 12.85/6.96 = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ B2 @ A3 ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3237_add_Ocommute,axiom,
% 12.85/6.96 ( plus_plus_nat
% 12.85/6.96 = ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3238_add_Ocommute,axiom,
% 12.85/6.96 ( plus_plus_int
% 12.85/6.96 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3239_add_Oleft__commute,axiom,
% 12.85/6.96 ! [B: real,A: real,C: real] :
% 12.85/6.96 ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 12.85/6.96 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3240_add_Oleft__commute,axiom,
% 12.85/6.96 ! [B: rat,A: rat,C: rat] :
% 12.85/6.96 ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
% 12.85/6.96 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3241_add_Oleft__commute,axiom,
% 12.85/6.96 ! [B: nat,A: nat,C: nat] :
% 12.85/6.96 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 12.85/6.96 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3242_add_Oleft__commute,axiom,
% 12.85/6.96 ! [B: int,A: int,C: int] :
% 12.85/6.96 ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 12.85/6.96 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3243_add__left__imp__eq,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( ( plus_plus_real @ A @ B )
% 12.85/6.96 = ( plus_plus_real @ A @ C ) )
% 12.85/6.96 => ( B = C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3244_add__left__imp__eq,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( ( plus_plus_rat @ A @ B )
% 12.85/6.96 = ( plus_plus_rat @ A @ C ) )
% 12.85/6.96 => ( B = C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3245_add__left__imp__eq,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( ( plus_plus_nat @ A @ B )
% 12.85/6.96 = ( plus_plus_nat @ A @ C ) )
% 12.85/6.96 => ( B = C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3246_add__left__imp__eq,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( ( plus_plus_int @ A @ B )
% 12.85/6.96 = ( plus_plus_int @ A @ C ) )
% 12.85/6.96 => ( B = C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3247_add__right__imp__eq,axiom,
% 12.85/6.96 ! [B: real,A: real,C: real] :
% 12.85/6.96 ( ( ( plus_plus_real @ B @ A )
% 12.85/6.96 = ( plus_plus_real @ C @ A ) )
% 12.85/6.96 => ( B = C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3248_add__right__imp__eq,axiom,
% 12.85/6.96 ! [B: rat,A: rat,C: rat] :
% 12.85/6.96 ( ( ( plus_plus_rat @ B @ A )
% 12.85/6.96 = ( plus_plus_rat @ C @ A ) )
% 12.85/6.96 => ( B = C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3249_add__right__imp__eq,axiom,
% 12.85/6.96 ! [B: nat,A: nat,C: nat] :
% 12.85/6.96 ( ( ( plus_plus_nat @ B @ A )
% 12.85/6.96 = ( plus_plus_nat @ C @ A ) )
% 12.85/6.96 => ( B = C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3250_add__right__imp__eq,axiom,
% 12.85/6.96 ! [B: int,A: int,C: int] :
% 12.85/6.96 ( ( ( plus_plus_int @ B @ A )
% 12.85/6.96 = ( plus_plus_int @ C @ A ) )
% 12.85/6.96 => ( B = C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3251_is__num__normalize_I1_J,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3252_is__num__normalize_I1_J,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3253_is__num__normalize_I1_J,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3254_add__mono__thms__linordered__semiring_I3_J,axiom,
% 12.85/6.96 ! [I: rat,J: rat,K: rat,L: rat] :
% 12.85/6.96 ( ( ( ord_less_eq_rat @ I @ J )
% 12.85/6.96 & ( K = L ) )
% 12.85/6.96 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3255_add__mono__thms__linordered__semiring_I3_J,axiom,
% 12.85/6.96 ! [I: nat,J: nat,K: nat,L: nat] :
% 12.85/6.96 ( ( ( ord_less_eq_nat @ I @ J )
% 12.85/6.96 & ( K = L ) )
% 12.85/6.96 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3256_add__mono__thms__linordered__semiring_I3_J,axiom,
% 12.85/6.96 ! [I: int,J: int,K: int,L: int] :
% 12.85/6.96 ( ( ( ord_less_eq_int @ I @ J )
% 12.85/6.96 & ( K = L ) )
% 12.85/6.96 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3257_add__mono__thms__linordered__semiring_I3_J,axiom,
% 12.85/6.96 ! [I: real,J: real,K: real,L: real] :
% 12.85/6.96 ( ( ( ord_less_eq_real @ I @ J )
% 12.85/6.96 & ( K = L ) )
% 12.85/6.96 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3258_add__mono__thms__linordered__semiring_I2_J,axiom,
% 12.85/6.96 ! [I: rat,J: rat,K: rat,L: rat] :
% 12.85/6.96 ( ( ( I = J )
% 12.85/6.96 & ( ord_less_eq_rat @ K @ L ) )
% 12.85/6.96 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3259_add__mono__thms__linordered__semiring_I2_J,axiom,
% 12.85/6.96 ! [I: nat,J: nat,K: nat,L: nat] :
% 12.85/6.96 ( ( ( I = J )
% 12.85/6.96 & ( ord_less_eq_nat @ K @ L ) )
% 12.85/6.96 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3260_add__mono__thms__linordered__semiring_I2_J,axiom,
% 12.85/6.96 ! [I: int,J: int,K: int,L: int] :
% 12.85/6.96 ( ( ( I = J )
% 12.85/6.96 & ( ord_less_eq_int @ K @ L ) )
% 12.85/6.96 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3261_add__mono__thms__linordered__semiring_I2_J,axiom,
% 12.85/6.96 ! [I: real,J: real,K: real,L: real] :
% 12.85/6.96 ( ( ( I = J )
% 12.85/6.96 & ( ord_less_eq_real @ K @ L ) )
% 12.85/6.96 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3262_add__mono__thms__linordered__semiring_I1_J,axiom,
% 12.85/6.96 ! [I: rat,J: rat,K: rat,L: rat] :
% 12.85/6.96 ( ( ( ord_less_eq_rat @ I @ J )
% 12.85/6.96 & ( ord_less_eq_rat @ K @ L ) )
% 12.85/6.96 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3263_add__mono__thms__linordered__semiring_I1_J,axiom,
% 12.85/6.96 ! [I: nat,J: nat,K: nat,L: nat] :
% 12.85/6.96 ( ( ( ord_less_eq_nat @ I @ J )
% 12.85/6.96 & ( ord_less_eq_nat @ K @ L ) )
% 12.85/6.96 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3264_add__mono__thms__linordered__semiring_I1_J,axiom,
% 12.85/6.96 ! [I: int,J: int,K: int,L: int] :
% 12.85/6.96 ( ( ( ord_less_eq_int @ I @ J )
% 12.85/6.96 & ( ord_less_eq_int @ K @ L ) )
% 12.85/6.96 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3265_add__mono__thms__linordered__semiring_I1_J,axiom,
% 12.85/6.96 ! [I: real,J: real,K: real,L: real] :
% 12.85/6.96 ( ( ( ord_less_eq_real @ I @ J )
% 12.85/6.96 & ( ord_less_eq_real @ K @ L ) )
% 12.85/6.96 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3266_add__mono,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat,D: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.96 => ( ( ord_less_eq_rat @ C @ D )
% 12.85/6.96 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3267_add__mono,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat,D: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.96 => ( ( ord_less_eq_nat @ C @ D )
% 12.85/6.96 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3268_add__mono,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int,D: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.96 => ( ( ord_less_eq_int @ C @ D )
% 12.85/6.96 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3269_add__mono,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real,D: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.96 => ( ( ord_less_eq_real @ C @ D )
% 12.85/6.96 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3270_add__left__mono,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.96 => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3271_add__left__mono,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.96 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3272_add__left__mono,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.96 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3273_add__left__mono,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.96 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3274_less__eqE,axiom,
% 12.85/6.96 ! [A: nat,B: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.96 => ~ ! [C3: nat] :
% 12.85/6.96 ( B
% 12.85/6.96 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3275_add__right__mono,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.96 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3276_add__right__mono,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.96 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3277_add__right__mono,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.96 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3278_add__right__mono,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.96 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3279_le__iff__add,axiom,
% 12.85/6.96 ( ord_less_eq_nat
% 12.85/6.96 = ( ^ [A3: nat,B2: nat] :
% 12.85/6.96 ? [C5: nat] :
% 12.85/6.96 ( B2
% 12.85/6.96 = ( plus_plus_nat @ A3 @ C5 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3280_add__le__imp__le__left,axiom,
% 12.85/6.96 ! [C: rat,A: rat,B: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 12.85/6.96 => ( ord_less_eq_rat @ A @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3281_add__le__imp__le__left,axiom,
% 12.85/6.96 ! [C: nat,A: nat,B: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 12.85/6.96 => ( ord_less_eq_nat @ A @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3282_add__le__imp__le__left,axiom,
% 12.85/6.96 ! [C: int,A: int,B: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 12.85/6.96 => ( ord_less_eq_int @ A @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3283_add__le__imp__le__left,axiom,
% 12.85/6.96 ! [C: real,A: real,B: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 12.85/6.96 => ( ord_less_eq_real @ A @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3284_add__le__imp__le__right,axiom,
% 12.85/6.96 ! [A: rat,C: rat,B: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 12.85/6.96 => ( ord_less_eq_rat @ A @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3285_add__le__imp__le__right,axiom,
% 12.85/6.96 ! [A: nat,C: nat,B: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 12.85/6.96 => ( ord_less_eq_nat @ A @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3286_add__le__imp__le__right,axiom,
% 12.85/6.96 ! [A: int,C: int,B: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 12.85/6.96 => ( ord_less_eq_int @ A @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3287_add__le__imp__le__right,axiom,
% 12.85/6.96 ! [A: real,C: real,B: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 12.85/6.96 => ( ord_less_eq_real @ A @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3288_verit__sum__simplify,axiom,
% 12.85/6.96 ! [A: complex] :
% 12.85/6.96 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 12.85/6.96 = A ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3289_verit__sum__simplify,axiom,
% 12.85/6.96 ! [A: real] :
% 12.85/6.96 ( ( plus_plus_real @ A @ zero_zero_real )
% 12.85/6.96 = A ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3290_verit__sum__simplify,axiom,
% 12.85/6.96 ! [A: rat] :
% 12.85/6.96 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 12.85/6.96 = A ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3291_verit__sum__simplify,axiom,
% 12.85/6.96 ! [A: nat] :
% 12.85/6.96 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 12.85/6.96 = A ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3292_verit__sum__simplify,axiom,
% 12.85/6.96 ! [A: int] :
% 12.85/6.96 ( ( plus_plus_int @ A @ zero_zero_int )
% 12.85/6.96 = A ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3293_add_Ogroup__left__neutral,axiom,
% 12.85/6.96 ! [A: complex] :
% 12.85/6.96 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 12.85/6.96 = A ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3294_add_Ogroup__left__neutral,axiom,
% 12.85/6.96 ! [A: real] :
% 12.85/6.96 ( ( plus_plus_real @ zero_zero_real @ A )
% 12.85/6.96 = A ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3295_add_Ogroup__left__neutral,axiom,
% 12.85/6.96 ! [A: rat] :
% 12.85/6.96 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 12.85/6.96 = A ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3296_add_Ogroup__left__neutral,axiom,
% 12.85/6.96 ! [A: int] :
% 12.85/6.96 ( ( plus_plus_int @ zero_zero_int @ A )
% 12.85/6.96 = A ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3297_add_Ocomm__neutral,axiom,
% 12.85/6.96 ! [A: complex] :
% 12.85/6.96 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 12.85/6.96 = A ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3298_add_Ocomm__neutral,axiom,
% 12.85/6.96 ! [A: real] :
% 12.85/6.96 ( ( plus_plus_real @ A @ zero_zero_real )
% 12.85/6.96 = A ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3299_add_Ocomm__neutral,axiom,
% 12.85/6.96 ! [A: rat] :
% 12.85/6.96 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 12.85/6.96 = A ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3300_add_Ocomm__neutral,axiom,
% 12.85/6.96 ! [A: nat] :
% 12.85/6.96 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 12.85/6.96 = A ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3301_add_Ocomm__neutral,axiom,
% 12.85/6.96 ! [A: int] :
% 12.85/6.96 ( ( plus_plus_int @ A @ zero_zero_int )
% 12.85/6.96 = A ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3302_comm__monoid__add__class_Oadd__0,axiom,
% 12.85/6.96 ! [A: complex] :
% 12.85/6.96 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 12.85/6.96 = A ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3303_comm__monoid__add__class_Oadd__0,axiom,
% 12.85/6.96 ! [A: real] :
% 12.85/6.96 ( ( plus_plus_real @ zero_zero_real @ A )
% 12.85/6.96 = A ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3304_comm__monoid__add__class_Oadd__0,axiom,
% 12.85/6.96 ! [A: rat] :
% 12.85/6.96 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 12.85/6.96 = A ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3305_comm__monoid__add__class_Oadd__0,axiom,
% 12.85/6.96 ! [A: nat] :
% 12.85/6.96 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 12.85/6.96 = A ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3306_comm__monoid__add__class_Oadd__0,axiom,
% 12.85/6.96 ! [A: int] :
% 12.85/6.96 ( ( plus_plus_int @ zero_zero_int @ A )
% 12.85/6.96 = A ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3307_add__mono__thms__linordered__field_I5_J,axiom,
% 12.85/6.96 ! [I: real,J: real,K: real,L: real] :
% 12.85/6.96 ( ( ( ord_less_real @ I @ J )
% 12.85/6.96 & ( ord_less_real @ K @ L ) )
% 12.85/6.96 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3308_add__mono__thms__linordered__field_I5_J,axiom,
% 12.85/6.96 ! [I: rat,J: rat,K: rat,L: rat] :
% 12.85/6.96 ( ( ( ord_less_rat @ I @ J )
% 12.85/6.96 & ( ord_less_rat @ K @ L ) )
% 12.85/6.96 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3309_add__mono__thms__linordered__field_I5_J,axiom,
% 12.85/6.96 ! [I: nat,J: nat,K: nat,L: nat] :
% 12.85/6.96 ( ( ( ord_less_nat @ I @ J )
% 12.85/6.96 & ( ord_less_nat @ K @ L ) )
% 12.85/6.96 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3310_add__mono__thms__linordered__field_I5_J,axiom,
% 12.85/6.96 ! [I: int,J: int,K: int,L: int] :
% 12.85/6.96 ( ( ( ord_less_int @ I @ J )
% 12.85/6.96 & ( ord_less_int @ K @ L ) )
% 12.85/6.96 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3311_add__mono__thms__linordered__field_I2_J,axiom,
% 12.85/6.96 ! [I: real,J: real,K: real,L: real] :
% 12.85/6.96 ( ( ( I = J )
% 12.85/6.96 & ( ord_less_real @ K @ L ) )
% 12.85/6.96 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3312_add__mono__thms__linordered__field_I2_J,axiom,
% 12.85/6.96 ! [I: rat,J: rat,K: rat,L: rat] :
% 12.85/6.96 ( ( ( I = J )
% 12.85/6.96 & ( ord_less_rat @ K @ L ) )
% 12.85/6.96 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3313_add__mono__thms__linordered__field_I2_J,axiom,
% 12.85/6.96 ! [I: nat,J: nat,K: nat,L: nat] :
% 12.85/6.96 ( ( ( I = J )
% 12.85/6.96 & ( ord_less_nat @ K @ L ) )
% 12.85/6.96 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3314_add__mono__thms__linordered__field_I2_J,axiom,
% 12.85/6.96 ! [I: int,J: int,K: int,L: int] :
% 12.85/6.96 ( ( ( I = J )
% 12.85/6.96 & ( ord_less_int @ K @ L ) )
% 12.85/6.96 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3315_add__mono__thms__linordered__field_I1_J,axiom,
% 12.85/6.96 ! [I: real,J: real,K: real,L: real] :
% 12.85/6.96 ( ( ( ord_less_real @ I @ J )
% 12.85/6.96 & ( K = L ) )
% 12.85/6.96 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3316_add__mono__thms__linordered__field_I1_J,axiom,
% 12.85/6.96 ! [I: rat,J: rat,K: rat,L: rat] :
% 12.85/6.96 ( ( ( ord_less_rat @ I @ J )
% 12.85/6.96 & ( K = L ) )
% 12.85/6.96 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3317_add__mono__thms__linordered__field_I1_J,axiom,
% 12.85/6.96 ! [I: nat,J: nat,K: nat,L: nat] :
% 12.85/6.96 ( ( ( ord_less_nat @ I @ J )
% 12.85/6.96 & ( K = L ) )
% 12.85/6.96 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3318_add__mono__thms__linordered__field_I1_J,axiom,
% 12.85/6.96 ! [I: int,J: int,K: int,L: int] :
% 12.85/6.96 ( ( ( ord_less_int @ I @ J )
% 12.85/6.96 & ( K = L ) )
% 12.85/6.96 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3319_add__strict__mono,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real,D: real] :
% 12.85/6.96 ( ( ord_less_real @ A @ B )
% 12.85/6.96 => ( ( ord_less_real @ C @ D )
% 12.85/6.96 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3320_add__strict__mono,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat,D: rat] :
% 12.85/6.96 ( ( ord_less_rat @ A @ B )
% 12.85/6.96 => ( ( ord_less_rat @ C @ D )
% 12.85/6.96 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3321_add__strict__mono,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat,D: nat] :
% 12.85/6.96 ( ( ord_less_nat @ A @ B )
% 12.85/6.96 => ( ( ord_less_nat @ C @ D )
% 12.85/6.96 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3322_add__strict__mono,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int,D: int] :
% 12.85/6.96 ( ( ord_less_int @ A @ B )
% 12.85/6.96 => ( ( ord_less_int @ C @ D )
% 12.85/6.96 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3323_add__strict__left__mono,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( ord_less_real @ A @ B )
% 12.85/6.96 => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3324_add__strict__left__mono,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( ord_less_rat @ A @ B )
% 12.85/6.96 => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3325_add__strict__left__mono,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( ord_less_nat @ A @ B )
% 12.85/6.96 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3326_add__strict__left__mono,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( ord_less_int @ A @ B )
% 12.85/6.96 => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3327_add__strict__right__mono,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( ord_less_real @ A @ B )
% 12.85/6.96 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3328_add__strict__right__mono,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( ord_less_rat @ A @ B )
% 12.85/6.96 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3329_add__strict__right__mono,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( ord_less_nat @ A @ B )
% 12.85/6.96 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3330_add__strict__right__mono,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( ord_less_int @ A @ B )
% 12.85/6.96 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3331_add__less__imp__less__left,axiom,
% 12.85/6.96 ! [C: real,A: real,B: real] :
% 12.85/6.96 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 12.85/6.96 => ( ord_less_real @ A @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3332_add__less__imp__less__left,axiom,
% 12.85/6.96 ! [C: rat,A: rat,B: rat] :
% 12.85/6.96 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 12.85/6.96 => ( ord_less_rat @ A @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3333_add__less__imp__less__left,axiom,
% 12.85/6.96 ! [C: nat,A: nat,B: nat] :
% 12.85/6.96 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 12.85/6.96 => ( ord_less_nat @ A @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3334_add__less__imp__less__left,axiom,
% 12.85/6.96 ! [C: int,A: int,B: int] :
% 12.85/6.96 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 12.85/6.96 => ( ord_less_int @ A @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3335_add__less__imp__less__right,axiom,
% 12.85/6.96 ! [A: real,C: real,B: real] :
% 12.85/6.96 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 12.85/6.96 => ( ord_less_real @ A @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3336_add__less__imp__less__right,axiom,
% 12.85/6.96 ! [A: rat,C: rat,B: rat] :
% 12.85/6.96 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 12.85/6.96 => ( ord_less_rat @ A @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3337_add__less__imp__less__right,axiom,
% 12.85/6.96 ! [A: nat,C: nat,B: nat] :
% 12.85/6.96 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 12.85/6.96 => ( ord_less_nat @ A @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3338_add__less__imp__less__right,axiom,
% 12.85/6.96 ! [A: int,C: int,B: int] :
% 12.85/6.96 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 12.85/6.96 => ( ord_less_int @ A @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3339_ring__class_Oring__distribs_I2_J,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3340_ring__class_Oring__distribs_I2_J,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3341_ring__class_Oring__distribs_I2_J,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3342_ring__class_Oring__distribs_I1_J,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 12.85/6.96 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3343_ring__class_Oring__distribs_I1_J,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 12.85/6.96 = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3344_ring__class_Oring__distribs_I1_J,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 12.85/6.96 = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3345_comm__semiring__class_Odistrib,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3346_comm__semiring__class_Odistrib,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3347_comm__semiring__class_Odistrib,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3348_comm__semiring__class_Odistrib,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3349_distrib__left,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 12.85/6.96 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3350_distrib__left,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 12.85/6.96 = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3351_distrib__left,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 12.85/6.96 = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3352_distrib__left,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 12.85/6.96 = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3353_distrib__right,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3354_distrib__right,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3355_distrib__right,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3356_distrib__right,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3357_combine__common__factor,axiom,
% 12.85/6.96 ! [A: real,E2: real,B: real,C: real] :
% 12.85/6.96 ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C ) )
% 12.85/6.96 = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E2 ) @ C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3358_combine__common__factor,axiom,
% 12.85/6.96 ! [A: rat,E2: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ C ) )
% 12.85/6.96 = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E2 ) @ C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3359_combine__common__factor,axiom,
% 12.85/6.96 ! [A: nat,E2: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
% 12.85/6.96 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3360_combine__common__factor,axiom,
% 12.85/6.96 ! [A: int,E2: int,B: int,C: int] :
% 12.85/6.96 ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
% 12.85/6.96 = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3361_diff__diff__eq,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 12.85/6.96 = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3362_diff__diff__eq,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 12.85/6.96 = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3363_diff__diff__eq,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 12.85/6.96 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3364_diff__diff__eq,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 12.85/6.96 = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3365_add__implies__diff,axiom,
% 12.85/6.96 ! [C: real,B: real,A: real] :
% 12.85/6.96 ( ( ( plus_plus_real @ C @ B )
% 12.85/6.96 = A )
% 12.85/6.96 => ( C
% 12.85/6.96 = ( minus_minus_real @ A @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3366_add__implies__diff,axiom,
% 12.85/6.96 ! [C: rat,B: rat,A: rat] :
% 12.85/6.96 ( ( ( plus_plus_rat @ C @ B )
% 12.85/6.96 = A )
% 12.85/6.96 => ( C
% 12.85/6.96 = ( minus_minus_rat @ A @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3367_add__implies__diff,axiom,
% 12.85/6.96 ! [C: nat,B: nat,A: nat] :
% 12.85/6.96 ( ( ( plus_plus_nat @ C @ B )
% 12.85/6.96 = A )
% 12.85/6.96 => ( C
% 12.85/6.96 = ( minus_minus_nat @ A @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3368_add__implies__diff,axiom,
% 12.85/6.96 ! [C: int,B: int,A: int] :
% 12.85/6.96 ( ( ( plus_plus_int @ C @ B )
% 12.85/6.96 = A )
% 12.85/6.96 => ( C
% 12.85/6.96 = ( minus_minus_int @ A @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3369_diff__add__eq__diff__diff__swap,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 12.85/6.96 = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3370_diff__add__eq__diff__diff__swap,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 12.85/6.96 = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3371_diff__add__eq__diff__diff__swap,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 12.85/6.96 = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3372_diff__add__eq,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 12.85/6.96 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3373_diff__add__eq,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 12.85/6.96 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3374_diff__add__eq,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 12.85/6.96 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3375_diff__diff__eq2,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 12.85/6.96 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3376_diff__diff__eq2,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 12.85/6.96 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3377_diff__diff__eq2,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 12.85/6.96 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3378_add__diff__eq,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 12.85/6.96 = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3379_add__diff__eq,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 12.85/6.96 = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3380_add__diff__eq,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 12.85/6.96 = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3381_eq__diff__eq,axiom,
% 12.85/6.96 ! [A: real,C: real,B: real] :
% 12.85/6.96 ( ( A
% 12.85/6.96 = ( minus_minus_real @ C @ B ) )
% 12.85/6.96 = ( ( plus_plus_real @ A @ B )
% 12.85/6.96 = C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3382_eq__diff__eq,axiom,
% 12.85/6.96 ! [A: rat,C: rat,B: rat] :
% 12.85/6.96 ( ( A
% 12.85/6.96 = ( minus_minus_rat @ C @ B ) )
% 12.85/6.96 = ( ( plus_plus_rat @ A @ B )
% 12.85/6.96 = C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3383_eq__diff__eq,axiom,
% 12.85/6.96 ! [A: int,C: int,B: int] :
% 12.85/6.96 ( ( A
% 12.85/6.96 = ( minus_minus_int @ C @ B ) )
% 12.85/6.96 = ( ( plus_plus_int @ A @ B )
% 12.85/6.96 = C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3384_diff__eq__eq,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( ( minus_minus_real @ A @ B )
% 12.85/6.96 = C )
% 12.85/6.96 = ( A
% 12.85/6.96 = ( plus_plus_real @ C @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3385_diff__eq__eq,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( ( minus_minus_rat @ A @ B )
% 12.85/6.96 = C )
% 12.85/6.96 = ( A
% 12.85/6.96 = ( plus_plus_rat @ C @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3386_diff__eq__eq,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( ( minus_minus_int @ A @ B )
% 12.85/6.96 = C )
% 12.85/6.96 = ( A
% 12.85/6.96 = ( plus_plus_int @ C @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3387_group__cancel_Osub1,axiom,
% 12.85/6.96 ! [A2: real,K: real,A: real,B: real] :
% 12.85/6.96 ( ( A2
% 12.85/6.96 = ( plus_plus_real @ K @ A ) )
% 12.85/6.96 => ( ( minus_minus_real @ A2 @ B )
% 12.85/6.96 = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3388_group__cancel_Osub1,axiom,
% 12.85/6.96 ! [A2: rat,K: rat,A: rat,B: rat] :
% 12.85/6.96 ( ( A2
% 12.85/6.96 = ( plus_plus_rat @ K @ A ) )
% 12.85/6.96 => ( ( minus_minus_rat @ A2 @ B )
% 12.85/6.96 = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3389_group__cancel_Osub1,axiom,
% 12.85/6.96 ! [A2: int,K: int,A: int,B: int] :
% 12.85/6.96 ( ( A2
% 12.85/6.96 = ( plus_plus_int @ K @ A ) )
% 12.85/6.96 => ( ( minus_minus_int @ A2 @ B )
% 12.85/6.96 = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3390_add__diff__add,axiom,
% 12.85/6.96 ! [A: real,C: real,B: real,D: real] :
% 12.85/6.96 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
% 12.85/6.96 = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3391_add__diff__add,axiom,
% 12.85/6.96 ! [A: rat,C: rat,B: rat,D: rat] :
% 12.85/6.96 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
% 12.85/6.96 = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3392_add__diff__add,axiom,
% 12.85/6.96 ! [A: int,C: int,B: int,D: int] :
% 12.85/6.96 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
% 12.85/6.96 = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3393_add__divide__distrib,axiom,
% 12.85/6.96 ! [A: complex,B: complex,C: complex] :
% 12.85/6.96 ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3394_add__divide__distrib,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3395_add__divide__distrib,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3396_dvd__add__right__iff,axiom,
% 12.85/6.96 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.96 ( ( dvd_dvd_Code_integer @ A @ B )
% 12.85/6.96 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 12.85/6.96 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3397_dvd__add__right__iff,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( dvd_dvd_real @ A @ B )
% 12.85/6.96 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 12.85/6.96 = ( dvd_dvd_real @ A @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3398_dvd__add__right__iff,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( dvd_dvd_rat @ A @ B )
% 12.85/6.96 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 12.85/6.96 = ( dvd_dvd_rat @ A @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3399_dvd__add__right__iff,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.96 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 12.85/6.96 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3400_dvd__add__right__iff,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( dvd_dvd_int @ A @ B )
% 12.85/6.96 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 12.85/6.96 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3401_dvd__add__left__iff,axiom,
% 12.85/6.96 ! [A: code_integer,C: code_integer,B: code_integer] :
% 12.85/6.96 ( ( dvd_dvd_Code_integer @ A @ C )
% 12.85/6.96 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 12.85/6.96 = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3402_dvd__add__left__iff,axiom,
% 12.85/6.96 ! [A: real,C: real,B: real] :
% 12.85/6.96 ( ( dvd_dvd_real @ A @ C )
% 12.85/6.96 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 12.85/6.96 = ( dvd_dvd_real @ A @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3403_dvd__add__left__iff,axiom,
% 12.85/6.96 ! [A: rat,C: rat,B: rat] :
% 12.85/6.96 ( ( dvd_dvd_rat @ A @ C )
% 12.85/6.96 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 12.85/6.96 = ( dvd_dvd_rat @ A @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3404_dvd__add__left__iff,axiom,
% 12.85/6.96 ! [A: nat,C: nat,B: nat] :
% 12.85/6.96 ( ( dvd_dvd_nat @ A @ C )
% 12.85/6.96 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 12.85/6.96 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3405_dvd__add__left__iff,axiom,
% 12.85/6.96 ! [A: int,C: int,B: int] :
% 12.85/6.96 ( ( dvd_dvd_int @ A @ C )
% 12.85/6.96 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 12.85/6.96 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3406_dvd__add,axiom,
% 12.85/6.96 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.96 ( ( dvd_dvd_Code_integer @ A @ B )
% 12.85/6.96 => ( ( dvd_dvd_Code_integer @ A @ C )
% 12.85/6.96 => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3407_dvd__add,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( dvd_dvd_real @ A @ B )
% 12.85/6.96 => ( ( dvd_dvd_real @ A @ C )
% 12.85/6.96 => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3408_dvd__add,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( dvd_dvd_rat @ A @ B )
% 12.85/6.96 => ( ( dvd_dvd_rat @ A @ C )
% 12.85/6.96 => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3409_dvd__add,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.96 => ( ( dvd_dvd_nat @ A @ C )
% 12.85/6.96 => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3410_dvd__add,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( dvd_dvd_int @ A @ B )
% 12.85/6.96 => ( ( dvd_dvd_int @ A @ C )
% 12.85/6.96 => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3411_nat__arith_Osuc1,axiom,
% 12.85/6.96 ! [A2: nat,K: nat,A: nat] :
% 12.85/6.96 ( ( A2
% 12.85/6.96 = ( plus_plus_nat @ K @ A ) )
% 12.85/6.96 => ( ( suc @ A2 )
% 12.85/6.96 = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3412_add__Suc,axiom,
% 12.85/6.96 ! [M: nat,N: nat] :
% 12.85/6.96 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 12.85/6.96 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3413_add__Suc__shift,axiom,
% 12.85/6.96 ! [M: nat,N: nat] :
% 12.85/6.96 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 12.85/6.96 = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3414_Euclid__induct,axiom,
% 12.85/6.96 ! [P: nat > nat > $o,A: nat,B: nat] :
% 12.85/6.96 ( ! [A4: nat,B3: nat] :
% 12.85/6.96 ( ( P @ A4 @ B3 )
% 12.85/6.96 = ( P @ B3 @ A4 ) )
% 12.85/6.96 => ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
% 12.85/6.96 => ( ! [A4: nat,B3: nat] :
% 12.85/6.96 ( ( P @ A4 @ B3 )
% 12.85/6.96 => ( P @ A4 @ ( plus_plus_nat @ A4 @ B3 ) ) )
% 12.85/6.96 => ( P @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3415_plus__nat_Oadd__0,axiom,
% 12.85/6.96 ! [N: nat] :
% 12.85/6.96 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 12.85/6.96 = N ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3416_add__eq__self__zero,axiom,
% 12.85/6.96 ! [M: nat,N: nat] :
% 12.85/6.96 ( ( ( plus_plus_nat @ M @ N )
% 12.85/6.96 = M )
% 12.85/6.96 => ( N = zero_zero_nat ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3417_add__lessD1,axiom,
% 12.85/6.96 ! [I: nat,J: nat,K: nat] :
% 12.85/6.96 ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 12.85/6.96 => ( ord_less_nat @ I @ K ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3418_add__less__mono,axiom,
% 12.85/6.96 ! [I: nat,J: nat,K: nat,L: nat] :
% 12.85/6.96 ( ( ord_less_nat @ I @ J )
% 12.85/6.96 => ( ( ord_less_nat @ K @ L )
% 12.85/6.96 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3419_not__add__less1,axiom,
% 12.85/6.96 ! [I: nat,J: nat] :
% 12.85/6.96 ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3420_not__add__less2,axiom,
% 12.85/6.96 ! [J: nat,I: nat] :
% 12.85/6.96 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3421_add__less__mono1,axiom,
% 12.85/6.96 ! [I: nat,J: nat,K: nat] :
% 12.85/6.96 ( ( ord_less_nat @ I @ J )
% 12.85/6.96 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3422_trans__less__add1,axiom,
% 12.85/6.96 ! [I: nat,J: nat,M: nat] :
% 12.85/6.96 ( ( ord_less_nat @ I @ J )
% 12.85/6.96 => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3423_trans__less__add2,axiom,
% 12.85/6.96 ! [I: nat,J: nat,M: nat] :
% 12.85/6.96 ( ( ord_less_nat @ I @ J )
% 12.85/6.96 => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3424_less__add__eq__less,axiom,
% 12.85/6.96 ! [K: nat,L: nat,M: nat,N: nat] :
% 12.85/6.96 ( ( ord_less_nat @ K @ L )
% 12.85/6.96 => ( ( ( plus_plus_nat @ M @ L )
% 12.85/6.96 = ( plus_plus_nat @ K @ N ) )
% 12.85/6.96 => ( ord_less_nat @ M @ N ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3425_nat__le__iff__add,axiom,
% 12.85/6.96 ( ord_less_eq_nat
% 12.85/6.96 = ( ^ [M5: nat,N4: nat] :
% 12.85/6.96 ? [K3: nat] :
% 12.85/6.96 ( N4
% 12.85/6.96 = ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3426_trans__le__add2,axiom,
% 12.85/6.96 ! [I: nat,J: nat,M: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ I @ J )
% 12.85/6.96 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3427_trans__le__add1,axiom,
% 12.85/6.96 ! [I: nat,J: nat,M: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ I @ J )
% 12.85/6.96 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3428_add__le__mono1,axiom,
% 12.85/6.96 ! [I: nat,J: nat,K: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ I @ J )
% 12.85/6.96 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3429_add__le__mono,axiom,
% 12.85/6.96 ! [I: nat,J: nat,K: nat,L: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ I @ J )
% 12.85/6.96 => ( ( ord_less_eq_nat @ K @ L )
% 12.85/6.96 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3430_le__Suc__ex,axiom,
% 12.85/6.96 ! [K: nat,L: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ K @ L )
% 12.85/6.96 => ? [N2: nat] :
% 12.85/6.96 ( L
% 12.85/6.96 = ( plus_plus_nat @ K @ N2 ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3431_add__leD2,axiom,
% 12.85/6.96 ! [M: nat,K: nat,N: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 12.85/6.96 => ( ord_less_eq_nat @ K @ N ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3432_add__leD1,axiom,
% 12.85/6.96 ! [M: nat,K: nat,N: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 12.85/6.96 => ( ord_less_eq_nat @ M @ N ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3433_le__add2,axiom,
% 12.85/6.96 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3434_le__add1,axiom,
% 12.85/6.96 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3435_add__leE,axiom,
% 12.85/6.96 ! [M: nat,K: nat,N: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 12.85/6.96 => ~ ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.96 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3436_Nat_Odiff__cancel,axiom,
% 12.85/6.96 ! [K: nat,M: nat,N: nat] :
% 12.85/6.96 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 12.85/6.96 = ( minus_minus_nat @ M @ N ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3437_diff__cancel2,axiom,
% 12.85/6.96 ! [M: nat,K: nat,N: nat] :
% 12.85/6.96 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 12.85/6.96 = ( minus_minus_nat @ M @ N ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3438_diff__add__inverse,axiom,
% 12.85/6.96 ! [N: nat,M: nat] :
% 12.85/6.96 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 12.85/6.96 = M ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3439_diff__add__inverse2,axiom,
% 12.85/6.96 ! [M: nat,N: nat] :
% 12.85/6.96 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 12.85/6.96 = M ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3440_left__add__mult__distrib,axiom,
% 12.85/6.96 ! [I: nat,U: nat,J: nat,K: nat] :
% 12.85/6.96 ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 12.85/6.96 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3441_add__mult__distrib2,axiom,
% 12.85/6.96 ! [K: nat,M: nat,N: nat] :
% 12.85/6.96 ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 12.85/6.96 = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3442_add__mult__distrib,axiom,
% 12.85/6.96 ! [M: nat,N: nat,K: nat] :
% 12.85/6.96 ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 12.85/6.96 = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3443_add__nonpos__eq__0__iff,axiom,
% 12.85/6.96 ! [X: rat,Y4: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 12.85/6.96 => ( ( ord_less_eq_rat @ Y4 @ zero_zero_rat )
% 12.85/6.96 => ( ( ( plus_plus_rat @ X @ Y4 )
% 12.85/6.96 = zero_zero_rat )
% 12.85/6.96 = ( ( X = zero_zero_rat )
% 12.85/6.96 & ( Y4 = zero_zero_rat ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3444_add__nonpos__eq__0__iff,axiom,
% 12.85/6.96 ! [X: nat,Y4: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ X @ zero_zero_nat )
% 12.85/6.96 => ( ( ord_less_eq_nat @ Y4 @ zero_zero_nat )
% 12.85/6.96 => ( ( ( plus_plus_nat @ X @ Y4 )
% 12.85/6.96 = zero_zero_nat )
% 12.85/6.96 = ( ( X = zero_zero_nat )
% 12.85/6.96 & ( Y4 = zero_zero_nat ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3445_add__nonpos__eq__0__iff,axiom,
% 12.85/6.96 ! [X: int,Y4: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ X @ zero_zero_int )
% 12.85/6.96 => ( ( ord_less_eq_int @ Y4 @ zero_zero_int )
% 12.85/6.96 => ( ( ( plus_plus_int @ X @ Y4 )
% 12.85/6.96 = zero_zero_int )
% 12.85/6.96 = ( ( X = zero_zero_int )
% 12.85/6.96 & ( Y4 = zero_zero_int ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3446_add__nonpos__eq__0__iff,axiom,
% 12.85/6.96 ! [X: real,Y4: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 12.85/6.96 => ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
% 12.85/6.96 => ( ( ( plus_plus_real @ X @ Y4 )
% 12.85/6.96 = zero_zero_real )
% 12.85/6.96 = ( ( X = zero_zero_real )
% 12.85/6.96 & ( Y4 = zero_zero_real ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3447_add__nonneg__eq__0__iff,axiom,
% 12.85/6.96 ! [X: rat,Y4: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 12.85/6.96 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 12.85/6.96 => ( ( ( plus_plus_rat @ X @ Y4 )
% 12.85/6.96 = zero_zero_rat )
% 12.85/6.96 = ( ( X = zero_zero_rat )
% 12.85/6.96 & ( Y4 = zero_zero_rat ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3448_add__nonneg__eq__0__iff,axiom,
% 12.85/6.96 ! [X: nat,Y4: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 12.85/6.96 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
% 12.85/6.96 => ( ( ( plus_plus_nat @ X @ Y4 )
% 12.85/6.96 = zero_zero_nat )
% 12.85/6.96 = ( ( X = zero_zero_nat )
% 12.85/6.96 & ( Y4 = zero_zero_nat ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3449_add__nonneg__eq__0__iff,axiom,
% 12.85/6.96 ! [X: int,Y4: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/6.96 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/6.96 => ( ( ( plus_plus_int @ X @ Y4 )
% 12.85/6.96 = zero_zero_int )
% 12.85/6.96 = ( ( X = zero_zero_int )
% 12.85/6.96 & ( Y4 = zero_zero_int ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3450_add__nonneg__eq__0__iff,axiom,
% 12.85/6.96 ! [X: real,Y4: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/6.96 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/6.96 => ( ( ( plus_plus_real @ X @ Y4 )
% 12.85/6.96 = zero_zero_real )
% 12.85/6.96 = ( ( X = zero_zero_real )
% 12.85/6.96 & ( Y4 = zero_zero_real ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3451_add__nonpos__nonpos,axiom,
% 12.85/6.96 ! [A: rat,B: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.85/6.96 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 12.85/6.96 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3452_add__nonpos__nonpos,axiom,
% 12.85/6.96 ! [A: nat,B: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 12.85/6.96 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 12.85/6.96 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3453_add__nonpos__nonpos,axiom,
% 12.85/6.96 ! [A: int,B: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.85/6.96 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 12.85/6.96 => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3454_add__nonpos__nonpos,axiom,
% 12.85/6.96 ! [A: real,B: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.85/6.96 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 12.85/6.96 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3455_add__nonneg__nonneg,axiom,
% 12.85/6.96 ! [A: rat,B: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.96 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 12.85/6.96 => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3456_add__nonneg__nonneg,axiom,
% 12.85/6.96 ! [A: nat,B: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.96 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 12.85/6.96 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3457_add__nonneg__nonneg,axiom,
% 12.85/6.96 ! [A: int,B: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.96 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 12.85/6.96 => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3458_add__nonneg__nonneg,axiom,
% 12.85/6.96 ! [A: real,B: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.96 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 12.85/6.96 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3459_add__increasing2,axiom,
% 12.85/6.96 ! [C: rat,B: rat,A: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 12.85/6.96 => ( ( ord_less_eq_rat @ B @ A )
% 12.85/6.96 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3460_add__increasing2,axiom,
% 12.85/6.96 ! [C: nat,B: nat,A: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.85/6.96 => ( ( ord_less_eq_nat @ B @ A )
% 12.85/6.96 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3461_add__increasing2,axiom,
% 12.85/6.96 ! [C: int,B: int,A: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.96 => ( ( ord_less_eq_int @ B @ A )
% 12.85/6.96 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3462_add__increasing2,axiom,
% 12.85/6.96 ! [C: real,B: real,A: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/6.96 => ( ( ord_less_eq_real @ B @ A )
% 12.85/6.96 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3463_add__decreasing2,axiom,
% 12.85/6.96 ! [C: rat,A: rat,B: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 12.85/6.96 => ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.96 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3464_add__decreasing2,axiom,
% 12.85/6.96 ! [C: nat,A: nat,B: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 12.85/6.96 => ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.96 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3465_add__decreasing2,axiom,
% 12.85/6.96 ! [C: int,A: int,B: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ C @ zero_zero_int )
% 12.85/6.96 => ( ( ord_less_eq_int @ A @ B )
% 12.85/6.96 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3466_add__decreasing2,axiom,
% 12.85/6.96 ! [C: real,A: real,B: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.85/6.96 => ( ( ord_less_eq_real @ A @ B )
% 12.85/6.96 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3467_add__increasing,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.96 => ( ( ord_less_eq_rat @ B @ C )
% 12.85/6.96 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3468_add__increasing,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.96 => ( ( ord_less_eq_nat @ B @ C )
% 12.85/6.96 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3469_add__increasing,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.96 => ( ( ord_less_eq_int @ B @ C )
% 12.85/6.96 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3470_add__increasing,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.96 => ( ( ord_less_eq_real @ B @ C )
% 12.85/6.96 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3471_add__decreasing,axiom,
% 12.85/6.96 ! [A: rat,C: rat,B: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.85/6.96 => ( ( ord_less_eq_rat @ C @ B )
% 12.85/6.96 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3472_add__decreasing,axiom,
% 12.85/6.96 ! [A: nat,C: nat,B: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 12.85/6.96 => ( ( ord_less_eq_nat @ C @ B )
% 12.85/6.96 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3473_add__decreasing,axiom,
% 12.85/6.96 ! [A: int,C: int,B: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.85/6.96 => ( ( ord_less_eq_int @ C @ B )
% 12.85/6.96 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3474_add__decreasing,axiom,
% 12.85/6.96 ! [A: real,C: real,B: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.85/6.96 => ( ( ord_less_eq_real @ C @ B )
% 12.85/6.96 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3475_add__mono__thms__linordered__field_I4_J,axiom,
% 12.85/6.96 ! [I: rat,J: rat,K: rat,L: rat] :
% 12.85/6.96 ( ( ( ord_less_eq_rat @ I @ J )
% 12.85/6.96 & ( ord_less_rat @ K @ L ) )
% 12.85/6.96 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3476_add__mono__thms__linordered__field_I4_J,axiom,
% 12.85/6.96 ! [I: nat,J: nat,K: nat,L: nat] :
% 12.85/6.96 ( ( ( ord_less_eq_nat @ I @ J )
% 12.85/6.96 & ( ord_less_nat @ K @ L ) )
% 12.85/6.96 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3477_add__mono__thms__linordered__field_I4_J,axiom,
% 12.85/6.96 ! [I: int,J: int,K: int,L: int] :
% 12.85/6.96 ( ( ( ord_less_eq_int @ I @ J )
% 12.85/6.96 & ( ord_less_int @ K @ L ) )
% 12.85/6.96 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3478_add__mono__thms__linordered__field_I4_J,axiom,
% 12.85/6.96 ! [I: real,J: real,K: real,L: real] :
% 12.85/6.96 ( ( ( ord_less_eq_real @ I @ J )
% 12.85/6.96 & ( ord_less_real @ K @ L ) )
% 12.85/6.96 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3479_add__mono__thms__linordered__field_I3_J,axiom,
% 12.85/6.96 ! [I: rat,J: rat,K: rat,L: rat] :
% 12.85/6.96 ( ( ( ord_less_rat @ I @ J )
% 12.85/6.96 & ( ord_less_eq_rat @ K @ L ) )
% 12.85/6.96 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3480_add__mono__thms__linordered__field_I3_J,axiom,
% 12.85/6.96 ! [I: nat,J: nat,K: nat,L: nat] :
% 12.85/6.96 ( ( ( ord_less_nat @ I @ J )
% 12.85/6.96 & ( ord_less_eq_nat @ K @ L ) )
% 12.85/6.96 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3481_add__mono__thms__linordered__field_I3_J,axiom,
% 12.85/6.96 ! [I: int,J: int,K: int,L: int] :
% 12.85/6.96 ( ( ( ord_less_int @ I @ J )
% 12.85/6.96 & ( ord_less_eq_int @ K @ L ) )
% 12.85/6.96 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3482_add__mono__thms__linordered__field_I3_J,axiom,
% 12.85/6.96 ! [I: real,J: real,K: real,L: real] :
% 12.85/6.96 ( ( ( ord_less_real @ I @ J )
% 12.85/6.96 & ( ord_less_eq_real @ K @ L ) )
% 12.85/6.96 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3483_add__le__less__mono,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat,D: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.96 => ( ( ord_less_rat @ C @ D )
% 12.85/6.96 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3484_add__le__less__mono,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat,D: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.96 => ( ( ord_less_nat @ C @ D )
% 12.85/6.96 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3485_add__le__less__mono,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int,D: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.96 => ( ( ord_less_int @ C @ D )
% 12.85/6.96 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3486_add__le__less__mono,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real,D: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.96 => ( ( ord_less_real @ C @ D )
% 12.85/6.96 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3487_add__less__le__mono,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat,D: rat] :
% 12.85/6.96 ( ( ord_less_rat @ A @ B )
% 12.85/6.96 => ( ( ord_less_eq_rat @ C @ D )
% 12.85/6.96 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3488_add__less__le__mono,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat,D: nat] :
% 12.85/6.96 ( ( ord_less_nat @ A @ B )
% 12.85/6.96 => ( ( ord_less_eq_nat @ C @ D )
% 12.85/6.96 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3489_add__less__le__mono,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int,D: int] :
% 12.85/6.96 ( ( ord_less_int @ A @ B )
% 12.85/6.96 => ( ( ord_less_eq_int @ C @ D )
% 12.85/6.96 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3490_add__less__le__mono,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real,D: real] :
% 12.85/6.96 ( ( ord_less_real @ A @ B )
% 12.85/6.96 => ( ( ord_less_eq_real @ C @ D )
% 12.85/6.96 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3491_add__less__zeroD,axiom,
% 12.85/6.96 ! [X: real,Y4: real] :
% 12.85/6.96 ( ( ord_less_real @ ( plus_plus_real @ X @ Y4 ) @ zero_zero_real )
% 12.85/6.96 => ( ( ord_less_real @ X @ zero_zero_real )
% 12.85/6.96 | ( ord_less_real @ Y4 @ zero_zero_real ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3492_add__less__zeroD,axiom,
% 12.85/6.96 ! [X: rat,Y4: rat] :
% 12.85/6.96 ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y4 ) @ zero_zero_rat )
% 12.85/6.96 => ( ( ord_less_rat @ X @ zero_zero_rat )
% 12.85/6.96 | ( ord_less_rat @ Y4 @ zero_zero_rat ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3493_add__less__zeroD,axiom,
% 12.85/6.96 ! [X: int,Y4: int] :
% 12.85/6.96 ( ( ord_less_int @ ( plus_plus_int @ X @ Y4 ) @ zero_zero_int )
% 12.85/6.96 => ( ( ord_less_int @ X @ zero_zero_int )
% 12.85/6.96 | ( ord_less_int @ Y4 @ zero_zero_int ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3494_add__neg__neg,axiom,
% 12.85/6.96 ! [A: real,B: real] :
% 12.85/6.96 ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/6.96 => ( ( ord_less_real @ B @ zero_zero_real )
% 12.85/6.96 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3495_add__neg__neg,axiom,
% 12.85/6.96 ! [A: rat,B: rat] :
% 12.85/6.96 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/6.96 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 12.85/6.96 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3496_add__neg__neg,axiom,
% 12.85/6.96 ! [A: nat,B: nat] :
% 12.85/6.96 ( ( ord_less_nat @ A @ zero_zero_nat )
% 12.85/6.96 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 12.85/6.96 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3497_add__neg__neg,axiom,
% 12.85/6.96 ! [A: int,B: int] :
% 12.85/6.96 ( ( ord_less_int @ A @ zero_zero_int )
% 12.85/6.96 => ( ( ord_less_int @ B @ zero_zero_int )
% 12.85/6.96 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3498_add__pos__pos,axiom,
% 12.85/6.96 ! [A: real,B: real] :
% 12.85/6.96 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.96 => ( ( ord_less_real @ zero_zero_real @ B )
% 12.85/6.96 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3499_add__pos__pos,axiom,
% 12.85/6.96 ! [A: rat,B: rat] :
% 12.85/6.96 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.96 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 12.85/6.96 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3500_add__pos__pos,axiom,
% 12.85/6.96 ! [A: nat,B: nat] :
% 12.85/6.96 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.85/6.96 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 12.85/6.96 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3501_add__pos__pos,axiom,
% 12.85/6.96 ! [A: int,B: int] :
% 12.85/6.96 ( ( ord_less_int @ zero_zero_int @ A )
% 12.85/6.96 => ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.96 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3502_canonically__ordered__monoid__add__class_OlessE,axiom,
% 12.85/6.96 ! [A: nat,B: nat] :
% 12.85/6.96 ( ( ord_less_nat @ A @ B )
% 12.85/6.96 => ~ ! [C3: nat] :
% 12.85/6.96 ( ( B
% 12.85/6.96 = ( plus_plus_nat @ A @ C3 ) )
% 12.85/6.96 => ( C3 = zero_zero_nat ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3503_pos__add__strict,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.96 => ( ( ord_less_real @ B @ C )
% 12.85/6.96 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3504_pos__add__strict,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.96 => ( ( ord_less_rat @ B @ C )
% 12.85/6.96 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3505_pos__add__strict,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.85/6.96 => ( ( ord_less_nat @ B @ C )
% 12.85/6.96 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3506_pos__add__strict,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( ord_less_int @ zero_zero_int @ A )
% 12.85/6.96 => ( ( ord_less_int @ B @ C )
% 12.85/6.96 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3507_add__le__imp__le__diff,axiom,
% 12.85/6.96 ! [I: rat,K: rat,N: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 12.85/6.96 => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3508_add__le__imp__le__diff,axiom,
% 12.85/6.96 ! [I: nat,K: nat,N: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 12.85/6.96 => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3509_add__le__imp__le__diff,axiom,
% 12.85/6.96 ! [I: int,K: int,N: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 12.85/6.96 => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3510_add__le__imp__le__diff,axiom,
% 12.85/6.96 ! [I: real,K: real,N: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 12.85/6.96 => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3511_add__le__add__imp__diff__le,axiom,
% 12.85/6.96 ! [I: rat,K: rat,N: rat,J: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 12.85/6.96 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 12.85/6.96 => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 12.85/6.96 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 12.85/6.96 => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3512_add__le__add__imp__diff__le,axiom,
% 12.85/6.96 ! [I: nat,K: nat,N: nat,J: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 12.85/6.96 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 12.85/6.96 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 12.85/6.96 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 12.85/6.96 => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3513_add__le__add__imp__diff__le,axiom,
% 12.85/6.96 ! [I: int,K: int,N: int,J: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 12.85/6.96 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 12.85/6.96 => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 12.85/6.96 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 12.85/6.96 => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3514_add__le__add__imp__diff__le,axiom,
% 12.85/6.96 ! [I: real,K: real,N: real,J: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 12.85/6.96 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 12.85/6.96 => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 12.85/6.96 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 12.85/6.96 => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3515_diff__le__eq,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 12.85/6.96 = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3516_diff__le__eq,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
% 12.85/6.96 = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3517_diff__le__eq,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 12.85/6.96 = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3518_le__diff__eq,axiom,
% 12.85/6.96 ! [A: rat,C: rat,B: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 12.85/6.96 = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3519_le__diff__eq,axiom,
% 12.85/6.96 ! [A: int,C: int,B: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
% 12.85/6.96 = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3520_le__diff__eq,axiom,
% 12.85/6.96 ! [A: real,C: real,B: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 12.85/6.96 = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3521_diff__add,axiom,
% 12.85/6.96 ! [A: nat,B: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.96 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 12.85/6.96 = B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3522_le__add__diff,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.96 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3523_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.96 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 12.85/6.96 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3524_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.96 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 12.85/6.96 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3525_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.96 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 12.85/6.96 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3526_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.96 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 12.85/6.96 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3527_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.96 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 12.85/6.96 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3528_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.96 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 12.85/6.96 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3529_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 12.85/6.96 ! [A: nat,B: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.96 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 12.85/6.96 = B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3530_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.96 => ( ( ord_less_eq_nat @ A @ B )
% 12.85/6.96 => ( ( ( minus_minus_nat @ B @ A )
% 12.85/6.96 = C )
% 12.85/6.96 = ( B
% 12.85/6.96 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3531_less__add__one,axiom,
% 12.85/6.96 ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3532_less__add__one,axiom,
% 12.85/6.96 ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3533_less__add__one,axiom,
% 12.85/6.96 ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3534_less__add__one,axiom,
% 12.85/6.96 ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3535_add__mono1,axiom,
% 12.85/6.96 ! [A: real,B: real] :
% 12.85/6.96 ( ( ord_less_real @ A @ B )
% 12.85/6.96 => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3536_add__mono1,axiom,
% 12.85/6.96 ! [A: rat,B: rat] :
% 12.85/6.96 ( ( ord_less_rat @ A @ B )
% 12.85/6.96 => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3537_add__mono1,axiom,
% 12.85/6.96 ! [A: nat,B: nat] :
% 12.85/6.96 ( ( ord_less_nat @ A @ B )
% 12.85/6.96 => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3538_add__mono1,axiom,
% 12.85/6.96 ! [A: int,B: int] :
% 12.85/6.96 ( ( ord_less_int @ A @ B )
% 12.85/6.96 => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3539_numeral__Bit0,axiom,
% 12.85/6.96 ! [N: num] :
% 12.85/6.96 ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 12.85/6.96 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3540_numeral__Bit0,axiom,
% 12.85/6.96 ! [N: num] :
% 12.85/6.96 ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 12.85/6.96 = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3541_numeral__Bit0,axiom,
% 12.85/6.96 ! [N: num] :
% 12.85/6.96 ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 12.85/6.96 = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3542_numeral__Bit0,axiom,
% 12.85/6.96 ! [N: num] :
% 12.85/6.96 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 12.85/6.96 = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3543_numeral__Bit0,axiom,
% 12.85/6.96 ! [N: num] :
% 12.85/6.96 ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 12.85/6.96 = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3544_less__diff__eq,axiom,
% 12.85/6.96 ! [A: real,C: real,B: real] :
% 12.85/6.96 ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
% 12.85/6.96 = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3545_less__diff__eq,axiom,
% 12.85/6.96 ! [A: rat,C: rat,B: rat] :
% 12.85/6.96 ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 12.85/6.96 = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3546_less__diff__eq,axiom,
% 12.85/6.96 ! [A: int,C: int,B: int] :
% 12.85/6.96 ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
% 12.85/6.96 = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3547_diff__less__eq,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
% 12.85/6.96 = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3548_diff__less__eq,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 12.85/6.96 = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3549_diff__less__eq,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
% 12.85/6.96 = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3550_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 12.85/6.96 ! [A: real,B: real] :
% 12.85/6.96 ( ~ ( ord_less_real @ A @ B )
% 12.85/6.96 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 12.85/6.96 = A ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3551_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 12.85/6.96 ! [A: rat,B: rat] :
% 12.85/6.96 ( ~ ( ord_less_rat @ A @ B )
% 12.85/6.96 => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 12.85/6.96 = A ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3552_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 12.85/6.96 ! [A: nat,B: nat] :
% 12.85/6.96 ( ~ ( ord_less_nat @ A @ B )
% 12.85/6.96 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 12.85/6.96 = A ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3553_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 12.85/6.96 ! [A: int,B: int] :
% 12.85/6.96 ( ~ ( ord_less_int @ A @ B )
% 12.85/6.96 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 12.85/6.96 = A ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3554_one__plus__numeral__commute,axiom,
% 12.85/6.96 ! [X: num] :
% 12.85/6.96 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
% 12.85/6.96 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3555_one__plus__numeral__commute,axiom,
% 12.85/6.96 ! [X: num] :
% 12.85/6.96 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
% 12.85/6.96 = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3556_one__plus__numeral__commute,axiom,
% 12.85/6.96 ! [X: num] :
% 12.85/6.96 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
% 12.85/6.96 = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3557_one__plus__numeral__commute,axiom,
% 12.85/6.96 ! [X: num] :
% 12.85/6.96 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
% 12.85/6.96 = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3558_one__plus__numeral__commute,axiom,
% 12.85/6.96 ! [X: num] :
% 12.85/6.96 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
% 12.85/6.96 = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3559_eq__add__iff1,axiom,
% 12.85/6.96 ! [A: real,E2: real,C: real,B: real,D: real] :
% 12.85/6.96 ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
% 12.85/6.96 = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 12.85/6.96 = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C )
% 12.85/6.96 = D ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3560_eq__add__iff1,axiom,
% 12.85/6.96 ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 12.85/6.96 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
% 12.85/6.96 = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 12.85/6.96 = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C )
% 12.85/6.96 = D ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3561_eq__add__iff1,axiom,
% 12.85/6.96 ! [A: int,E2: int,C: int,B: int,D: int] :
% 12.85/6.96 ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
% 12.85/6.96 = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 12.85/6.96 = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
% 12.85/6.96 = D ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3562_eq__add__iff2,axiom,
% 12.85/6.96 ! [A: real,E2: real,C: real,B: real,D: real] :
% 12.85/6.96 ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
% 12.85/6.96 = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 12.85/6.96 = ( C
% 12.85/6.96 = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3563_eq__add__iff2,axiom,
% 12.85/6.96 ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 12.85/6.96 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
% 12.85/6.96 = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 12.85/6.96 = ( C
% 12.85/6.96 = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3564_eq__add__iff2,axiom,
% 12.85/6.96 ! [A: int,E2: int,C: int,B: int,D: int] :
% 12.85/6.96 ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
% 12.85/6.96 = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 12.85/6.96 = ( C
% 12.85/6.96 = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3565_square__diff__square__factored,axiom,
% 12.85/6.96 ! [X: real,Y4: real] :
% 12.85/6.96 ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) )
% 12.85/6.96 = ( times_times_real @ ( plus_plus_real @ X @ Y4 ) @ ( minus_minus_real @ X @ Y4 ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3566_square__diff__square__factored,axiom,
% 12.85/6.96 ! [X: rat,Y4: rat] :
% 12.85/6.96 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y4 @ Y4 ) )
% 12.85/6.96 = ( times_times_rat @ ( plus_plus_rat @ X @ Y4 ) @ ( minus_minus_rat @ X @ Y4 ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3567_square__diff__square__factored,axiom,
% 12.85/6.96 ! [X: int,Y4: int] :
% 12.85/6.96 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y4 @ Y4 ) )
% 12.85/6.96 = ( times_times_int @ ( plus_plus_int @ X @ Y4 ) @ ( minus_minus_int @ X @ Y4 ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3568_mult__diff__mult,axiom,
% 12.85/6.96 ! [X: real,Y4: real,A: real,B: real] :
% 12.85/6.96 ( ( minus_minus_real @ ( times_times_real @ X @ Y4 ) @ ( times_times_real @ A @ B ) )
% 12.85/6.96 = ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y4 @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3569_mult__diff__mult,axiom,
% 12.85/6.96 ! [X: rat,Y4: rat,A: rat,B: rat] :
% 12.85/6.96 ( ( minus_minus_rat @ ( times_times_rat @ X @ Y4 ) @ ( times_times_rat @ A @ B ) )
% 12.85/6.96 = ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y4 @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3570_mult__diff__mult,axiom,
% 12.85/6.96 ! [X: int,Y4: int,A: int,B: int] :
% 12.85/6.96 ( ( minus_minus_int @ ( times_times_int @ X @ Y4 ) @ ( times_times_int @ A @ B ) )
% 12.85/6.96 = ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y4 @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3571_minf_I10_J,axiom,
% 12.85/6.96 ! [D: code_integer,S2: code_integer] :
% 12.85/6.96 ? [Z2: code_integer] :
% 12.85/6.96 ! [X5: code_integer] :
% 12.85/6.96 ( ( ord_le6747313008572928689nteger @ X5 @ Z2 )
% 12.85/6.96 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) )
% 12.85/6.96 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3572_minf_I10_J,axiom,
% 12.85/6.96 ! [D: real,S2: real] :
% 12.85/6.96 ? [Z2: real] :
% 12.85/6.96 ! [X5: real] :
% 12.85/6.96 ( ( ord_less_real @ X5 @ Z2 )
% 12.85/6.96 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) )
% 12.85/6.96 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3573_minf_I10_J,axiom,
% 12.85/6.96 ! [D: rat,S2: rat] :
% 12.85/6.96 ? [Z2: rat] :
% 12.85/6.96 ! [X5: rat] :
% 12.85/6.96 ( ( ord_less_rat @ X5 @ Z2 )
% 12.85/6.96 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) )
% 12.85/6.96 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3574_minf_I10_J,axiom,
% 12.85/6.96 ! [D: nat,S2: nat] :
% 12.85/6.96 ? [Z2: nat] :
% 12.85/6.96 ! [X5: nat] :
% 12.85/6.96 ( ( ord_less_nat @ X5 @ Z2 )
% 12.85/6.96 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) )
% 12.85/6.96 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3575_minf_I10_J,axiom,
% 12.85/6.96 ! [D: int,S2: int] :
% 12.85/6.96 ? [Z2: int] :
% 12.85/6.96 ! [X5: int] :
% 12.85/6.96 ( ( ord_less_int @ X5 @ Z2 )
% 12.85/6.96 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) )
% 12.85/6.96 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3576_minf_I9_J,axiom,
% 12.85/6.96 ! [D: code_integer,S2: code_integer] :
% 12.85/6.96 ? [Z2: code_integer] :
% 12.85/6.96 ! [X5: code_integer] :
% 12.85/6.96 ( ( ord_le6747313008572928689nteger @ X5 @ Z2 )
% 12.85/6.96 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) )
% 12.85/6.96 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3577_minf_I9_J,axiom,
% 12.85/6.96 ! [D: real,S2: real] :
% 12.85/6.96 ? [Z2: real] :
% 12.85/6.96 ! [X5: real] :
% 12.85/6.96 ( ( ord_less_real @ X5 @ Z2 )
% 12.85/6.96 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) )
% 12.85/6.96 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3578_minf_I9_J,axiom,
% 12.85/6.96 ! [D: rat,S2: rat] :
% 12.85/6.96 ? [Z2: rat] :
% 12.85/6.96 ! [X5: rat] :
% 12.85/6.96 ( ( ord_less_rat @ X5 @ Z2 )
% 12.85/6.96 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) )
% 12.85/6.96 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3579_minf_I9_J,axiom,
% 12.85/6.96 ! [D: nat,S2: nat] :
% 12.85/6.96 ? [Z2: nat] :
% 12.85/6.96 ! [X5: nat] :
% 12.85/6.96 ( ( ord_less_nat @ X5 @ Z2 )
% 12.85/6.96 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) )
% 12.85/6.96 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3580_minf_I9_J,axiom,
% 12.85/6.96 ! [D: int,S2: int] :
% 12.85/6.96 ? [Z2: int] :
% 12.85/6.96 ! [X5: int] :
% 12.85/6.96 ( ( ord_less_int @ X5 @ Z2 )
% 12.85/6.96 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) )
% 12.85/6.96 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3581_pinf_I10_J,axiom,
% 12.85/6.96 ! [D: code_integer,S2: code_integer] :
% 12.85/6.96 ? [Z2: code_integer] :
% 12.85/6.96 ! [X5: code_integer] :
% 12.85/6.96 ( ( ord_le6747313008572928689nteger @ Z2 @ X5 )
% 12.85/6.96 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) )
% 12.85/6.96 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3582_pinf_I10_J,axiom,
% 12.85/6.96 ! [D: real,S2: real] :
% 12.85/6.96 ? [Z2: real] :
% 12.85/6.96 ! [X5: real] :
% 12.85/6.96 ( ( ord_less_real @ Z2 @ X5 )
% 12.85/6.96 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) )
% 12.85/6.96 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3583_pinf_I10_J,axiom,
% 12.85/6.96 ! [D: rat,S2: rat] :
% 12.85/6.96 ? [Z2: rat] :
% 12.85/6.96 ! [X5: rat] :
% 12.85/6.96 ( ( ord_less_rat @ Z2 @ X5 )
% 12.85/6.96 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) )
% 12.85/6.96 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3584_pinf_I10_J,axiom,
% 12.85/6.96 ! [D: nat,S2: nat] :
% 12.85/6.96 ? [Z2: nat] :
% 12.85/6.96 ! [X5: nat] :
% 12.85/6.96 ( ( ord_less_nat @ Z2 @ X5 )
% 12.85/6.96 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) )
% 12.85/6.96 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3585_pinf_I10_J,axiom,
% 12.85/6.96 ! [D: int,S2: int] :
% 12.85/6.96 ? [Z2: int] :
% 12.85/6.96 ! [X5: int] :
% 12.85/6.96 ( ( ord_less_int @ Z2 @ X5 )
% 12.85/6.96 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) )
% 12.85/6.96 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3586_pinf_I9_J,axiom,
% 12.85/6.96 ! [D: code_integer,S2: code_integer] :
% 12.85/6.96 ? [Z2: code_integer] :
% 12.85/6.96 ! [X5: code_integer] :
% 12.85/6.96 ( ( ord_le6747313008572928689nteger @ Z2 @ X5 )
% 12.85/6.96 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) )
% 12.85/6.96 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3587_pinf_I9_J,axiom,
% 12.85/6.96 ! [D: real,S2: real] :
% 12.85/6.96 ? [Z2: real] :
% 12.85/6.96 ! [X5: real] :
% 12.85/6.96 ( ( ord_less_real @ Z2 @ X5 )
% 12.85/6.96 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) )
% 12.85/6.96 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3588_pinf_I9_J,axiom,
% 12.85/6.96 ! [D: rat,S2: rat] :
% 12.85/6.96 ? [Z2: rat] :
% 12.85/6.96 ! [X5: rat] :
% 12.85/6.96 ( ( ord_less_rat @ Z2 @ X5 )
% 12.85/6.96 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) )
% 12.85/6.96 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3589_pinf_I9_J,axiom,
% 12.85/6.96 ! [D: nat,S2: nat] :
% 12.85/6.96 ? [Z2: nat] :
% 12.85/6.96 ! [X5: nat] :
% 12.85/6.96 ( ( ord_less_nat @ Z2 @ X5 )
% 12.85/6.96 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) )
% 12.85/6.96 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3590_pinf_I9_J,axiom,
% 12.85/6.96 ! [D: int,S2: int] :
% 12.85/6.96 ? [Z2: int] :
% 12.85/6.96 ! [X5: int] :
% 12.85/6.96 ( ( ord_less_int @ Z2 @ X5 )
% 12.85/6.96 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) )
% 12.85/6.96 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3591_div__plus__div__distrib__dvd__left,axiom,
% 12.85/6.96 ! [C: code_integer,A: code_integer,B: code_integer] :
% 12.85/6.96 ( ( dvd_dvd_Code_integer @ C @ A )
% 12.85/6.96 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 12.85/6.96 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3592_div__plus__div__distrib__dvd__left,axiom,
% 12.85/6.96 ! [C: nat,A: nat,B: nat] :
% 12.85/6.96 ( ( dvd_dvd_nat @ C @ A )
% 12.85/6.96 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3593_div__plus__div__distrib__dvd__left,axiom,
% 12.85/6.96 ! [C: int,A: int,B: int] :
% 12.85/6.96 ( ( dvd_dvd_int @ C @ A )
% 12.85/6.96 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3594_div__plus__div__distrib__dvd__right,axiom,
% 12.85/6.96 ! [C: code_integer,B: code_integer,A: code_integer] :
% 12.85/6.96 ( ( dvd_dvd_Code_integer @ C @ B )
% 12.85/6.96 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 12.85/6.96 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3595_div__plus__div__distrib__dvd__right,axiom,
% 12.85/6.96 ! [C: nat,B: nat,A: nat] :
% 12.85/6.96 ( ( dvd_dvd_nat @ C @ B )
% 12.85/6.96 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3596_div__plus__div__distrib__dvd__right,axiom,
% 12.85/6.96 ! [C: int,B: int,A: int] :
% 12.85/6.96 ( ( dvd_dvd_int @ C @ B )
% 12.85/6.96 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 12.85/6.96 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3597_power__add,axiom,
% 12.85/6.96 ! [A: complex,M: nat,N: nat] :
% 12.85/6.96 ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
% 12.85/6.96 = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3598_power__add,axiom,
% 12.85/6.96 ! [A: real,M: nat,N: nat] :
% 12.85/6.96 ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
% 12.85/6.96 = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3599_power__add,axiom,
% 12.85/6.96 ! [A: rat,M: nat,N: nat] :
% 12.85/6.96 ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
% 12.85/6.96 = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3600_power__add,axiom,
% 12.85/6.96 ! [A: nat,M: nat,N: nat] :
% 12.85/6.96 ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
% 12.85/6.96 = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3601_power__add,axiom,
% 12.85/6.96 ! [A: int,M: nat,N: nat] :
% 12.85/6.96 ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
% 12.85/6.96 = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3602_one__is__add,axiom,
% 12.85/6.96 ! [M: nat,N: nat] :
% 12.85/6.96 ( ( ( suc @ zero_zero_nat )
% 12.85/6.96 = ( plus_plus_nat @ M @ N ) )
% 12.85/6.96 = ( ( ( M
% 12.85/6.96 = ( suc @ zero_zero_nat ) )
% 12.85/6.96 & ( N = zero_zero_nat ) )
% 12.85/6.96 | ( ( M = zero_zero_nat )
% 12.85/6.96 & ( N
% 12.85/6.96 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3603_add__is__1,axiom,
% 12.85/6.96 ! [M: nat,N: nat] :
% 12.85/6.96 ( ( ( plus_plus_nat @ M @ N )
% 12.85/6.96 = ( suc @ zero_zero_nat ) )
% 12.85/6.96 = ( ( ( M
% 12.85/6.96 = ( suc @ zero_zero_nat ) )
% 12.85/6.96 & ( N = zero_zero_nat ) )
% 12.85/6.96 | ( ( M = zero_zero_nat )
% 12.85/6.96 & ( N
% 12.85/6.96 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3604_less__imp__Suc__add,axiom,
% 12.85/6.96 ! [M: nat,N: nat] :
% 12.85/6.96 ( ( ord_less_nat @ M @ N )
% 12.85/6.96 => ? [K2: nat] :
% 12.85/6.96 ( N
% 12.85/6.96 = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3605_less__iff__Suc__add,axiom,
% 12.85/6.96 ( ord_less_nat
% 12.85/6.96 = ( ^ [M5: nat,N4: nat] :
% 12.85/6.96 ? [K3: nat] :
% 12.85/6.96 ( N4
% 12.85/6.96 = ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3606_less__add__Suc2,axiom,
% 12.85/6.96 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3607_less__add__Suc1,axiom,
% 12.85/6.96 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3608_less__natE,axiom,
% 12.85/6.96 ! [M: nat,N: nat] :
% 12.85/6.96 ( ( ord_less_nat @ M @ N )
% 12.85/6.96 => ~ ! [Q4: nat] :
% 12.85/6.96 ( N
% 12.85/6.96 != ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3609_less__imp__add__positive,axiom,
% 12.85/6.96 ! [I: nat,J: nat] :
% 12.85/6.96 ( ( ord_less_nat @ I @ J )
% 12.85/6.96 => ? [K2: nat] :
% 12.85/6.96 ( ( ord_less_nat @ zero_zero_nat @ K2 )
% 12.85/6.96 & ( ( plus_plus_nat @ I @ K2 )
% 12.85/6.96 = J ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3610_mono__nat__linear__lb,axiom,
% 12.85/6.96 ! [F: nat > nat,M: nat,K: nat] :
% 12.85/6.96 ( ! [M4: nat,N2: nat] :
% 12.85/6.96 ( ( ord_less_nat @ M4 @ N2 )
% 12.85/6.96 => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
% 12.85/6.96 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3611_mult__Suc,axiom,
% 12.85/6.96 ! [M: nat,N: nat] :
% 12.85/6.96 ( ( times_times_nat @ ( suc @ M ) @ N )
% 12.85/6.96 = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3612_diff__add__0,axiom,
% 12.85/6.96 ! [N: nat,M: nat] :
% 12.85/6.96 ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 12.85/6.96 = zero_zero_nat ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3613_less__diff__conv,axiom,
% 12.85/6.96 ! [I: nat,J: nat,K: nat] :
% 12.85/6.96 ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 12.85/6.96 = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3614_add__diff__inverse__nat,axiom,
% 12.85/6.96 ! [M: nat,N: nat] :
% 12.85/6.96 ( ~ ( ord_less_nat @ M @ N )
% 12.85/6.96 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 12.85/6.96 = M ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3615_Suc__eq__plus1__left,axiom,
% 12.85/6.96 ( suc
% 12.85/6.96 = ( plus_plus_nat @ one_one_nat ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3616_plus__1__eq__Suc,axiom,
% 12.85/6.96 ( ( plus_plus_nat @ one_one_nat )
% 12.85/6.96 = suc ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3617_Suc__eq__plus1,axiom,
% 12.85/6.96 ( suc
% 12.85/6.96 = ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3618_Nat_Ole__imp__diff__is__add,axiom,
% 12.85/6.96 ! [I: nat,J: nat,K: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ I @ J )
% 12.85/6.96 => ( ( ( minus_minus_nat @ J @ I )
% 12.85/6.96 = K )
% 12.85/6.96 = ( J
% 12.85/6.96 = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3619_Nat_Odiff__add__assoc2,axiom,
% 12.85/6.96 ! [K: nat,J: nat,I: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ K @ J )
% 12.85/6.96 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 12.85/6.96 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3620_Nat_Odiff__add__assoc,axiom,
% 12.85/6.96 ! [K: nat,J: nat,I: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ K @ J )
% 12.85/6.96 => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 12.85/6.96 = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3621_Nat_Ole__diff__conv2,axiom,
% 12.85/6.96 ! [K: nat,J: nat,I: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ K @ J )
% 12.85/6.96 => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 12.85/6.96 = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3622_le__diff__conv,axiom,
% 12.85/6.96 ! [J: nat,K: nat,I: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 12.85/6.96 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3623_bezout__lemma__nat,axiom,
% 12.85/6.96 ! [D: nat,A: nat,B: nat,X: nat,Y4: nat] :
% 12.85/6.96 ( ( dvd_dvd_nat @ D @ A )
% 12.85/6.96 => ( ( dvd_dvd_nat @ D @ B )
% 12.85/6.96 => ( ( ( ( times_times_nat @ A @ X )
% 12.85/6.96 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D ) )
% 12.85/6.96 | ( ( times_times_nat @ B @ X )
% 12.85/6.96 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) )
% 12.85/6.96 => ? [X4: nat,Y: nat] :
% 12.85/6.96 ( ( dvd_dvd_nat @ D @ A )
% 12.85/6.96 & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
% 12.85/6.96 & ( ( ( times_times_nat @ A @ X4 )
% 12.85/6.96 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y ) @ D ) )
% 12.85/6.96 | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X4 )
% 12.85/6.96 = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3624_bezout__add__nat,axiom,
% 12.85/6.96 ! [A: nat,B: nat] :
% 12.85/6.96 ? [D2: nat,X4: nat,Y: nat] :
% 12.85/6.96 ( ( dvd_dvd_nat @ D2 @ A )
% 12.85/6.96 & ( dvd_dvd_nat @ D2 @ B )
% 12.85/6.96 & ( ( ( times_times_nat @ A @ X4 )
% 12.85/6.96 = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D2 ) )
% 12.85/6.96 | ( ( times_times_nat @ B @ X4 )
% 12.85/6.96 = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D2 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3625_field__le__epsilon,axiom,
% 12.85/6.96 ! [X: rat,Y4: rat] :
% 12.85/6.96 ( ! [E: rat] :
% 12.85/6.96 ( ( ord_less_rat @ zero_zero_rat @ E )
% 12.85/6.96 => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y4 @ E ) ) )
% 12.85/6.96 => ( ord_less_eq_rat @ X @ Y4 ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3626_field__le__epsilon,axiom,
% 12.85/6.96 ! [X: real,Y4: real] :
% 12.85/6.96 ( ! [E: real] :
% 12.85/6.96 ( ( ord_less_real @ zero_zero_real @ E )
% 12.85/6.96 => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y4 @ E ) ) )
% 12.85/6.96 => ( ord_less_eq_real @ X @ Y4 ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3627_add__strict__increasing2,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.96 => ( ( ord_less_rat @ B @ C )
% 12.85/6.96 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3628_add__strict__increasing2,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.96 => ( ( ord_less_nat @ B @ C )
% 12.85/6.96 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3629_add__strict__increasing2,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.96 => ( ( ord_less_int @ B @ C )
% 12.85/6.96 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3630_add__strict__increasing2,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.96 => ( ( ord_less_real @ B @ C )
% 12.85/6.96 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3631_add__strict__increasing,axiom,
% 12.85/6.96 ! [A: rat,B: rat,C: rat] :
% 12.85/6.96 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.96 => ( ( ord_less_eq_rat @ B @ C )
% 12.85/6.96 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3632_add__strict__increasing,axiom,
% 12.85/6.96 ! [A: nat,B: nat,C: nat] :
% 12.85/6.96 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.85/6.96 => ( ( ord_less_eq_nat @ B @ C )
% 12.85/6.96 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3633_add__strict__increasing,axiom,
% 12.85/6.96 ! [A: int,B: int,C: int] :
% 12.85/6.96 ( ( ord_less_int @ zero_zero_int @ A )
% 12.85/6.96 => ( ( ord_less_eq_int @ B @ C )
% 12.85/6.96 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3634_add__strict__increasing,axiom,
% 12.85/6.96 ! [A: real,B: real,C: real] :
% 12.85/6.96 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.96 => ( ( ord_less_eq_real @ B @ C )
% 12.85/6.96 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3635_add__pos__nonneg,axiom,
% 12.85/6.96 ! [A: rat,B: rat] :
% 12.85/6.96 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.96 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 12.85/6.96 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3636_add__pos__nonneg,axiom,
% 12.85/6.96 ! [A: nat,B: nat] :
% 12.85/6.96 ( ( ord_less_nat @ zero_zero_nat @ A )
% 12.85/6.96 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 12.85/6.96 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3637_add__pos__nonneg,axiom,
% 12.85/6.96 ! [A: int,B: int] :
% 12.85/6.96 ( ( ord_less_int @ zero_zero_int @ A )
% 12.85/6.96 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 12.85/6.96 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3638_add__pos__nonneg,axiom,
% 12.85/6.96 ! [A: real,B: real] :
% 12.85/6.96 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.96 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 12.85/6.96 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3639_add__nonpos__neg,axiom,
% 12.85/6.96 ! [A: rat,B: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.85/6.96 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 12.85/6.96 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3640_add__nonpos__neg,axiom,
% 12.85/6.96 ! [A: nat,B: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 12.85/6.96 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 12.85/6.96 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3641_add__nonpos__neg,axiom,
% 12.85/6.96 ! [A: int,B: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.85/6.96 => ( ( ord_less_int @ B @ zero_zero_int )
% 12.85/6.96 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3642_add__nonpos__neg,axiom,
% 12.85/6.96 ! [A: real,B: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.85/6.96 => ( ( ord_less_real @ B @ zero_zero_real )
% 12.85/6.96 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3643_add__nonneg__pos,axiom,
% 12.85/6.96 ! [A: rat,B: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.96 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 12.85/6.96 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3644_add__nonneg__pos,axiom,
% 12.85/6.96 ! [A: nat,B: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.96 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 12.85/6.96 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3645_add__nonneg__pos,axiom,
% 12.85/6.96 ! [A: int,B: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.96 => ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.96 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3646_add__nonneg__pos,axiom,
% 12.85/6.96 ! [A: real,B: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.96 => ( ( ord_less_real @ zero_zero_real @ B )
% 12.85/6.96 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3647_add__neg__nonpos,axiom,
% 12.85/6.96 ! [A: rat,B: rat] :
% 12.85/6.96 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/6.96 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 12.85/6.96 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3648_add__neg__nonpos,axiom,
% 12.85/6.96 ! [A: nat,B: nat] :
% 12.85/6.96 ( ( ord_less_nat @ A @ zero_zero_nat )
% 12.85/6.96 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 12.85/6.96 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3649_add__neg__nonpos,axiom,
% 12.85/6.96 ! [A: int,B: int] :
% 12.85/6.96 ( ( ord_less_int @ A @ zero_zero_int )
% 12.85/6.96 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 12.85/6.96 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3650_add__neg__nonpos,axiom,
% 12.85/6.96 ! [A: real,B: real] :
% 12.85/6.96 ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/6.96 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 12.85/6.96 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3651_sum__squares__ge__zero,axiom,
% 12.85/6.96 ! [X: rat,Y4: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y4 @ Y4 ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3652_sum__squares__ge__zero,axiom,
% 12.85/6.96 ! [X: int,Y4: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y4 @ Y4 ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3653_sum__squares__ge__zero,axiom,
% 12.85/6.96 ! [X: real,Y4: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3654_sum__squares__le__zero__iff,axiom,
% 12.85/6.96 ! [X: rat,Y4: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y4 @ Y4 ) ) @ zero_zero_rat )
% 12.85/6.96 = ( ( X = zero_zero_rat )
% 12.85/6.96 & ( Y4 = zero_zero_rat ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3655_sum__squares__le__zero__iff,axiom,
% 12.85/6.96 ! [X: int,Y4: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y4 @ Y4 ) ) @ zero_zero_int )
% 12.85/6.96 = ( ( X = zero_zero_int )
% 12.85/6.96 & ( Y4 = zero_zero_int ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3656_sum__squares__le__zero__iff,axiom,
% 12.85/6.96 ! [X: real,Y4: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) ) @ zero_zero_real )
% 12.85/6.96 = ( ( X = zero_zero_real )
% 12.85/6.96 & ( Y4 = zero_zero_real ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3657_not__sum__squares__lt__zero,axiom,
% 12.85/6.96 ! [X: real,Y4: real] :
% 12.85/6.96 ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) ) @ zero_zero_real ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3658_not__sum__squares__lt__zero,axiom,
% 12.85/6.96 ! [X: rat,Y4: rat] :
% 12.85/6.96 ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y4 @ Y4 ) ) @ zero_zero_rat ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3659_not__sum__squares__lt__zero,axiom,
% 12.85/6.96 ! [X: int,Y4: int] :
% 12.85/6.96 ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y4 @ Y4 ) ) @ zero_zero_int ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3660_sum__squares__gt__zero__iff,axiom,
% 12.85/6.96 ! [X: real,Y4: real] :
% 12.85/6.96 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) ) )
% 12.85/6.96 = ( ( X != zero_zero_real )
% 12.85/6.96 | ( Y4 != zero_zero_real ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3661_sum__squares__gt__zero__iff,axiom,
% 12.85/6.96 ! [X: rat,Y4: rat] :
% 12.85/6.96 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y4 @ Y4 ) ) )
% 12.85/6.96 = ( ( X != zero_zero_rat )
% 12.85/6.96 | ( Y4 != zero_zero_rat ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3662_sum__squares__gt__zero__iff,axiom,
% 12.85/6.96 ! [X: int,Y4: int] :
% 12.85/6.96 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y4 @ Y4 ) ) )
% 12.85/6.96 = ( ( X != zero_zero_int )
% 12.85/6.96 | ( Y4 != zero_zero_int ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3663_discrete,axiom,
% 12.85/6.96 ( ord_less_nat
% 12.85/6.96 = ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3664_discrete,axiom,
% 12.85/6.96 ( ord_less_int
% 12.85/6.96 = ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3665_zero__less__two,axiom,
% 12.85/6.96 ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3666_zero__less__two,axiom,
% 12.85/6.96 ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3667_zero__less__two,axiom,
% 12.85/6.96 ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3668_zero__less__two,axiom,
% 12.85/6.96 ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3669_ordered__ring__class_Ole__add__iff1,axiom,
% 12.85/6.96 ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 12.85/6.96 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3670_ordered__ring__class_Ole__add__iff1,axiom,
% 12.85/6.96 ! [A: int,E2: int,C: int,B: int,D: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 12.85/6.96 = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3671_ordered__ring__class_Ole__add__iff1,axiom,
% 12.85/6.96 ! [A: real,E2: real,C: real,B: real,D: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 12.85/6.96 = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3672_ordered__ring__class_Ole__add__iff2,axiom,
% 12.85/6.96 ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 12.85/6.96 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 12.85/6.96 = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3673_ordered__ring__class_Ole__add__iff2,axiom,
% 12.85/6.96 ! [A: int,E2: int,C: int,B: int,D: int] :
% 12.85/6.96 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 12.85/6.96 = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3674_ordered__ring__class_Ole__add__iff2,axiom,
% 12.85/6.96 ! [A: real,E2: real,C: real,B: real,D: real] :
% 12.85/6.96 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 12.85/6.96 = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3675_add__divide__eq__if__simps_I2_J,axiom,
% 12.85/6.96 ! [Z: complex,A: complex,B: complex] :
% 12.85/6.96 ( ( ( Z = zero_zero_complex )
% 12.85/6.96 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 12.85/6.96 = B ) )
% 12.85/6.96 & ( ( Z != zero_zero_complex )
% 12.85/6.96 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 12.85/6.96 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3676_add__divide__eq__if__simps_I2_J,axiom,
% 12.85/6.96 ! [Z: real,A: real,B: real] :
% 12.85/6.96 ( ( ( Z = zero_zero_real )
% 12.85/6.96 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 12.85/6.96 = B ) )
% 12.85/6.96 & ( ( Z != zero_zero_real )
% 12.85/6.96 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 12.85/6.96 = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3677_add__divide__eq__if__simps_I2_J,axiom,
% 12.85/6.96 ! [Z: rat,A: rat,B: rat] :
% 12.85/6.96 ( ( ( Z = zero_zero_rat )
% 12.85/6.96 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 12.85/6.96 = B ) )
% 12.85/6.96 & ( ( Z != zero_zero_rat )
% 12.85/6.96 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 12.85/6.96 = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3678_add__divide__eq__if__simps_I1_J,axiom,
% 12.85/6.96 ! [Z: complex,A: complex,B: complex] :
% 12.85/6.96 ( ( ( Z = zero_zero_complex )
% 12.85/6.96 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 12.85/6.96 = A ) )
% 12.85/6.96 & ( ( Z != zero_zero_complex )
% 12.85/6.96 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 12.85/6.96 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3679_add__divide__eq__if__simps_I1_J,axiom,
% 12.85/6.96 ! [Z: real,A: real,B: real] :
% 12.85/6.96 ( ( ( Z = zero_zero_real )
% 12.85/6.96 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 12.85/6.96 = A ) )
% 12.85/6.96 & ( ( Z != zero_zero_real )
% 12.85/6.96 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 12.85/6.96 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3680_add__divide__eq__if__simps_I1_J,axiom,
% 12.85/6.96 ! [Z: rat,A: rat,B: rat] :
% 12.85/6.96 ( ( ( Z = zero_zero_rat )
% 12.85/6.96 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 12.85/6.96 = A ) )
% 12.85/6.96 & ( ( Z != zero_zero_rat )
% 12.85/6.96 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 12.85/6.96 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3681_add__frac__eq,axiom,
% 12.85/6.96 ! [Y4: complex,Z: complex,X: complex,W: complex] :
% 12.85/6.96 ( ( Y4 != zero_zero_complex )
% 12.85/6.96 => ( ( Z != zero_zero_complex )
% 12.85/6.96 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y4 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 12.85/6.96 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y4 ) ) @ ( times_times_complex @ Y4 @ Z ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3682_add__frac__eq,axiom,
% 12.85/6.96 ! [Y4: real,Z: real,X: real,W: real] :
% 12.85/6.96 ( ( Y4 != zero_zero_real )
% 12.85/6.96 => ( ( Z != zero_zero_real )
% 12.85/6.96 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y4 ) @ ( divide_divide_real @ W @ Z ) )
% 12.85/6.96 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y4 ) ) @ ( times_times_real @ Y4 @ Z ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3683_add__frac__eq,axiom,
% 12.85/6.96 ! [Y4: rat,Z: rat,X: rat,W: rat] :
% 12.85/6.96 ( ( Y4 != zero_zero_rat )
% 12.85/6.96 => ( ( Z != zero_zero_rat )
% 12.85/6.96 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y4 ) @ ( divide_divide_rat @ W @ Z ) )
% 12.85/6.96 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y4 ) ) @ ( times_times_rat @ Y4 @ Z ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3684_add__frac__num,axiom,
% 12.85/6.96 ! [Y4: complex,X: complex,Z: complex] :
% 12.85/6.96 ( ( Y4 != zero_zero_complex )
% 12.85/6.96 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y4 ) @ Z )
% 12.85/6.96 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y4 ) ) @ Y4 ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3685_add__frac__num,axiom,
% 12.85/6.96 ! [Y4: real,X: real,Z: real] :
% 12.85/6.96 ( ( Y4 != zero_zero_real )
% 12.85/6.96 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y4 ) @ Z )
% 12.85/6.96 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y4 ) ) @ Y4 ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3686_add__frac__num,axiom,
% 12.85/6.96 ! [Y4: rat,X: rat,Z: rat] :
% 12.85/6.96 ( ( Y4 != zero_zero_rat )
% 12.85/6.96 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y4 ) @ Z )
% 12.85/6.96 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y4 ) ) @ Y4 ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3687_add__num__frac,axiom,
% 12.85/6.96 ! [Y4: complex,Z: complex,X: complex] :
% 12.85/6.96 ( ( Y4 != zero_zero_complex )
% 12.85/6.96 => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X @ Y4 ) )
% 12.85/6.96 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y4 ) ) @ Y4 ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3688_add__num__frac,axiom,
% 12.85/6.96 ! [Y4: real,Z: real,X: real] :
% 12.85/6.96 ( ( Y4 != zero_zero_real )
% 12.85/6.96 => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y4 ) )
% 12.85/6.96 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y4 ) ) @ Y4 ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3689_add__num__frac,axiom,
% 12.85/6.96 ! [Y4: rat,Z: rat,X: rat] :
% 12.85/6.96 ( ( Y4 != zero_zero_rat )
% 12.85/6.96 => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y4 ) )
% 12.85/6.96 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y4 ) ) @ Y4 ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3690_add__divide__eq__iff,axiom,
% 12.85/6.96 ! [Z: complex,X: complex,Y4: complex] :
% 12.85/6.96 ( ( Z != zero_zero_complex )
% 12.85/6.96 => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y4 @ Z ) )
% 12.85/6.96 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ Y4 ) @ Z ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3691_add__divide__eq__iff,axiom,
% 12.85/6.96 ! [Z: real,X: real,Y4: real] :
% 12.85/6.96 ( ( Z != zero_zero_real )
% 12.85/6.96 => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y4 @ Z ) )
% 12.85/6.96 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y4 ) @ Z ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3692_add__divide__eq__iff,axiom,
% 12.85/6.96 ! [Z: rat,X: rat,Y4: rat] :
% 12.85/6.96 ( ( Z != zero_zero_rat )
% 12.85/6.96 => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y4 @ Z ) )
% 12.85/6.96 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y4 ) @ Z ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3693_divide__add__eq__iff,axiom,
% 12.85/6.96 ! [Z: complex,X: complex,Y4: complex] :
% 12.85/6.96 ( ( Z != zero_zero_complex )
% 12.85/6.96 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y4 )
% 12.85/6.96 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y4 @ Z ) ) @ Z ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3694_divide__add__eq__iff,axiom,
% 12.85/6.96 ! [Z: real,X: real,Y4: real] :
% 12.85/6.96 ( ( Z != zero_zero_real )
% 12.85/6.96 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y4 )
% 12.85/6.96 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y4 @ Z ) ) @ Z ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3695_divide__add__eq__iff,axiom,
% 12.85/6.96 ! [Z: rat,X: rat,Y4: rat] :
% 12.85/6.96 ( ( Z != zero_zero_rat )
% 12.85/6.96 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y4 )
% 12.85/6.96 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y4 @ Z ) ) @ Z ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3696_less__add__iff1,axiom,
% 12.85/6.96 ! [A: real,E2: real,C: real,B: real,D: real] :
% 12.85/6.96 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 12.85/6.96 = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3697_less__add__iff1,axiom,
% 12.85/6.96 ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 12.85/6.96 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 12.85/6.96 = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3698_less__add__iff1,axiom,
% 12.85/6.96 ! [A: int,E2: int,C: int,B: int,D: int] :
% 12.85/6.96 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 12.85/6.96 = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3699_less__add__iff2,axiom,
% 12.85/6.96 ! [A: real,E2: real,C: real,B: real,D: real] :
% 12.85/6.96 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 12.85/6.96 = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3700_less__add__iff2,axiom,
% 12.85/6.96 ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 12.85/6.96 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 12.85/6.96 = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3701_less__add__iff2,axiom,
% 12.85/6.96 ! [A: int,E2: int,C: int,B: int,D: int] :
% 12.85/6.96 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 12.85/6.96 = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3702_div__add__self2,axiom,
% 12.85/6.96 ! [B: nat,A: nat] :
% 12.85/6.96 ( ( B != zero_zero_nat )
% 12.85/6.96 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 12.85/6.96 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3703_div__add__self2,axiom,
% 12.85/6.96 ! [B: int,A: int] :
% 12.85/6.96 ( ( B != zero_zero_int )
% 12.85/6.96 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
% 12.85/6.96 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3704_div__add__self1,axiom,
% 12.85/6.96 ! [B: nat,A: nat] :
% 12.85/6.96 ( ( B != zero_zero_nat )
% 12.85/6.96 => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 12.85/6.96 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3705_div__add__self1,axiom,
% 12.85/6.96 ! [B: int,A: int] :
% 12.85/6.96 ( ( B != zero_zero_int )
% 12.85/6.96 => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
% 12.85/6.96 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3706_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 12.85/6.96 ! [X: nat,N: nat,M: nat] :
% 12.85/6.96 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 12.85/6.96 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.96 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/6.96 => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3707_less__half__sum,axiom,
% 12.85/6.96 ! [A: real,B: real] :
% 12.85/6.96 ( ( ord_less_real @ A @ B )
% 12.85/6.96 => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3708_less__half__sum,axiom,
% 12.85/6.96 ! [A: rat,B: rat] :
% 12.85/6.96 ( ( ord_less_rat @ A @ B )
% 12.85/6.96 => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3709_gt__half__sum,axiom,
% 12.85/6.96 ! [A: real,B: real] :
% 12.85/6.96 ( ( ord_less_real @ A @ B )
% 12.85/6.96 => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3710_gt__half__sum,axiom,
% 12.85/6.96 ! [A: rat,B: rat] :
% 12.85/6.96 ( ( ord_less_rat @ A @ B )
% 12.85/6.96 => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3711_square__diff__one__factored,axiom,
% 12.85/6.96 ! [X: complex] :
% 12.85/6.96 ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
% 12.85/6.96 = ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3712_square__diff__one__factored,axiom,
% 12.85/6.96 ! [X: real] :
% 12.85/6.96 ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
% 12.85/6.96 = ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3713_square__diff__one__factored,axiom,
% 12.85/6.96 ! [X: rat] :
% 12.85/6.96 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
% 12.85/6.96 = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3714_square__diff__one__factored,axiom,
% 12.85/6.96 ! [X: int] :
% 12.85/6.96 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
% 12.85/6.96 = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3715_unity__coeff__ex,axiom,
% 12.85/6.96 ! [P: code_integer > $o,L: code_integer] :
% 12.85/6.96 ( ( ? [X3: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X3 ) ) )
% 12.85/6.96 = ( ? [X3: code_integer] :
% 12.85/6.96 ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X3 @ zero_z3403309356797280102nteger ) )
% 12.85/6.96 & ( P @ X3 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3716_unity__coeff__ex,axiom,
% 12.85/6.96 ! [P: complex > $o,L: complex] :
% 12.85/6.96 ( ( ? [X3: complex] : ( P @ ( times_times_complex @ L @ X3 ) ) )
% 12.85/6.96 = ( ? [X3: complex] :
% 12.85/6.96 ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X3 @ zero_zero_complex ) )
% 12.85/6.96 & ( P @ X3 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3717_unity__coeff__ex,axiom,
% 12.85/6.96 ! [P: real > $o,L: real] :
% 12.85/6.96 ( ( ? [X3: real] : ( P @ ( times_times_real @ L @ X3 ) ) )
% 12.85/6.96 = ( ? [X3: real] :
% 12.85/6.96 ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X3 @ zero_zero_real ) )
% 12.85/6.96 & ( P @ X3 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3718_unity__coeff__ex,axiom,
% 12.85/6.96 ! [P: rat > $o,L: rat] :
% 12.85/6.96 ( ( ? [X3: rat] : ( P @ ( times_times_rat @ L @ X3 ) ) )
% 12.85/6.96 = ( ? [X3: rat] :
% 12.85/6.96 ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X3 @ zero_zero_rat ) )
% 12.85/6.96 & ( P @ X3 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3719_unity__coeff__ex,axiom,
% 12.85/6.96 ! [P: nat > $o,L: nat] :
% 12.85/6.96 ( ( ? [X3: nat] : ( P @ ( times_times_nat @ L @ X3 ) ) )
% 12.85/6.96 = ( ? [X3: nat] :
% 12.85/6.96 ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X3 @ zero_zero_nat ) )
% 12.85/6.96 & ( P @ X3 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3720_unity__coeff__ex,axiom,
% 12.85/6.96 ! [P: int > $o,L: int] :
% 12.85/6.96 ( ( ? [X3: int] : ( P @ ( times_times_int @ L @ X3 ) ) )
% 12.85/6.96 = ( ? [X3: int] :
% 12.85/6.96 ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X3 @ zero_zero_int ) )
% 12.85/6.96 & ( P @ X3 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3721_inf__period_I3_J,axiom,
% 12.85/6.96 ! [D: code_integer,D3: code_integer,T: code_integer] :
% 12.85/6.96 ( ( dvd_dvd_Code_integer @ D @ D3 )
% 12.85/6.96 => ! [X5: code_integer,K4: code_integer] :
% 12.85/6.96 ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) )
% 12.85/6.96 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3722_inf__period_I3_J,axiom,
% 12.85/6.96 ! [D: real,D3: real,T: real] :
% 12.85/6.96 ( ( dvd_dvd_real @ D @ D3 )
% 12.85/6.96 => ! [X5: real,K4: real] :
% 12.85/6.96 ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) )
% 12.85/6.96 = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3723_inf__period_I3_J,axiom,
% 12.85/6.96 ! [D: rat,D3: rat,T: rat] :
% 12.85/6.96 ( ( dvd_dvd_rat @ D @ D3 )
% 12.85/6.96 => ! [X5: rat,K4: rat] :
% 12.85/6.96 ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) )
% 12.85/6.96 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3724_inf__period_I3_J,axiom,
% 12.85/6.96 ! [D: int,D3: int,T: int] :
% 12.85/6.96 ( ( dvd_dvd_int @ D @ D3 )
% 12.85/6.96 => ! [X5: int,K4: int] :
% 12.85/6.96 ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 12.85/6.96 = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3725_inf__period_I4_J,axiom,
% 12.85/6.96 ! [D: code_integer,D3: code_integer,T: code_integer] :
% 12.85/6.96 ( ( dvd_dvd_Code_integer @ D @ D3 )
% 12.85/6.96 => ! [X5: code_integer,K4: code_integer] :
% 12.85/6.96 ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) ) )
% 12.85/6.96 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3726_inf__period_I4_J,axiom,
% 12.85/6.96 ! [D: real,D3: real,T: real] :
% 12.85/6.96 ( ( dvd_dvd_real @ D @ D3 )
% 12.85/6.96 => ! [X5: real,K4: real] :
% 12.85/6.96 ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) ) )
% 12.85/6.96 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3727_inf__period_I4_J,axiom,
% 12.85/6.96 ! [D: rat,D3: rat,T: rat] :
% 12.85/6.96 ( ( dvd_dvd_rat @ D @ D3 )
% 12.85/6.96 => ! [X5: rat,K4: rat] :
% 12.85/6.96 ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) ) )
% 12.85/6.96 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3728_inf__period_I4_J,axiom,
% 12.85/6.96 ! [D: int,D3: int,T: int] :
% 12.85/6.96 ( ( dvd_dvd_int @ D @ D3 )
% 12.85/6.96 => ! [X5: int,K4: int] :
% 12.85/6.96 ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) ) )
% 12.85/6.96 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3729_numeral__Bit1,axiom,
% 12.85/6.96 ! [N: num] :
% 12.85/6.96 ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 12.85/6.96 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3730_numeral__Bit1,axiom,
% 12.85/6.96 ! [N: num] :
% 12.85/6.96 ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 12.85/6.96 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3731_numeral__Bit1,axiom,
% 12.85/6.96 ! [N: num] :
% 12.85/6.96 ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 12.85/6.96 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3732_numeral__Bit1,axiom,
% 12.85/6.96 ! [N: num] :
% 12.85/6.96 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 12.85/6.96 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3733_numeral__Bit1,axiom,
% 12.85/6.96 ! [N: num] :
% 12.85/6.96 ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 12.85/6.96 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3734_nat__diff__split__asm,axiom,
% 12.85/6.96 ! [P: nat > $o,A: nat,B: nat] :
% 12.85/6.96 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 12.85/6.96 = ( ~ ( ( ( ord_less_nat @ A @ B )
% 12.85/6.96 & ~ ( P @ zero_zero_nat ) )
% 12.85/6.96 | ? [D4: nat] :
% 12.85/6.96 ( ( A
% 12.85/6.96 = ( plus_plus_nat @ B @ D4 ) )
% 12.85/6.96 & ~ ( P @ D4 ) ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3735_nat__diff__split,axiom,
% 12.85/6.96 ! [P: nat > $o,A: nat,B: nat] :
% 12.85/6.96 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 12.85/6.96 = ( ( ( ord_less_nat @ A @ B )
% 12.85/6.96 => ( P @ zero_zero_nat ) )
% 12.85/6.96 & ! [D4: nat] :
% 12.85/6.96 ( ( A
% 12.85/6.96 = ( plus_plus_nat @ B @ D4 ) )
% 12.85/6.96 => ( P @ D4 ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3736_less__diff__conv2,axiom,
% 12.85/6.96 ! [K: nat,J: nat,I: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ K @ J )
% 12.85/6.96 => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 12.85/6.96 = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3737_nat__diff__add__eq2,axiom,
% 12.85/6.96 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ I @ J )
% 12.85/6.96 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 12.85/6.96 = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3738_nat__diff__add__eq1,axiom,
% 12.85/6.96 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ J @ I )
% 12.85/6.96 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 12.85/6.96 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3739_nat__le__add__iff2,axiom,
% 12.85/6.96 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ I @ J )
% 12.85/6.96 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 12.85/6.96 = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3740_nat__le__add__iff1,axiom,
% 12.85/6.96 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ J @ I )
% 12.85/6.96 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 12.85/6.96 = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3741_nat__eq__add__iff2,axiom,
% 12.85/6.96 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ I @ J )
% 12.85/6.96 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 12.85/6.96 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 12.85/6.96 = ( M
% 12.85/6.96 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3742_nat__eq__add__iff1,axiom,
% 12.85/6.96 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 12.85/6.96 ( ( ord_less_eq_nat @ J @ I )
% 12.85/6.96 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 12.85/6.96 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 12.85/6.96 = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
% 12.85/6.96 = N ) ) ) ).
% 12.85/6.96
% 12.85/6.96 thf(fact_3743_bezout__add__strong__nat,axiom,
% 12.85/6.96 ! [A: nat,B: nat] :
% 12.85/6.97 ( ( A != zero_zero_nat )
% 12.85/6.97 => ? [D2: nat,X4: nat,Y: nat] :
% 12.85/6.97 ( ( dvd_dvd_nat @ D2 @ A )
% 12.85/6.97 & ( dvd_dvd_nat @ D2 @ B )
% 12.85/6.97 & ( ( times_times_nat @ A @ X4 )
% 12.85/6.97 = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D2 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3744_convex__bound__le,axiom,
% 12.85/6.97 ! [X: rat,A: rat,Y4: rat,U: rat,V: rat] :
% 12.85/6.97 ( ( ord_less_eq_rat @ X @ A )
% 12.85/6.97 => ( ( ord_less_eq_rat @ Y4 @ A )
% 12.85/6.97 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 12.85/6.97 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 12.85/6.97 => ( ( ( plus_plus_rat @ U @ V )
% 12.85/6.97 = one_one_rat )
% 12.85/6.97 => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y4 ) ) @ A ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3745_convex__bound__le,axiom,
% 12.85/6.97 ! [X: int,A: int,Y4: int,U: int,V: int] :
% 12.85/6.97 ( ( ord_less_eq_int @ X @ A )
% 12.85/6.97 => ( ( ord_less_eq_int @ Y4 @ A )
% 12.85/6.97 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 12.85/6.97 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 12.85/6.97 => ( ( ( plus_plus_int @ U @ V )
% 12.85/6.97 = one_one_int )
% 12.85/6.97 => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y4 ) ) @ A ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3746_convex__bound__le,axiom,
% 12.85/6.97 ! [X: real,A: real,Y4: real,U: real,V: real] :
% 12.85/6.97 ( ( ord_less_eq_real @ X @ A )
% 12.85/6.97 => ( ( ord_less_eq_real @ Y4 @ A )
% 12.85/6.97 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 12.85/6.97 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 12.85/6.97 => ( ( ( plus_plus_real @ U @ V )
% 12.85/6.97 = one_one_real )
% 12.85/6.97 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y4 ) ) @ A ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3747_mult__2,axiom,
% 12.85/6.97 ! [Z: complex] :
% 12.85/6.97 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
% 12.85/6.97 = ( plus_plus_complex @ Z @ Z ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3748_mult__2,axiom,
% 12.85/6.97 ! [Z: real] :
% 12.85/6.97 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
% 12.85/6.97 = ( plus_plus_real @ Z @ Z ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3749_mult__2,axiom,
% 12.85/6.97 ! [Z: rat] :
% 12.85/6.97 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
% 12.85/6.97 = ( plus_plus_rat @ Z @ Z ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3750_mult__2,axiom,
% 12.85/6.97 ! [Z: nat] :
% 12.85/6.97 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
% 12.85/6.97 = ( plus_plus_nat @ Z @ Z ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3751_mult__2,axiom,
% 12.85/6.97 ! [Z: int] :
% 12.85/6.97 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
% 12.85/6.97 = ( plus_plus_int @ Z @ Z ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3752_mult__2__right,axiom,
% 12.85/6.97 ! [Z: complex] :
% 12.85/6.97 ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
% 12.85/6.97 = ( plus_plus_complex @ Z @ Z ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3753_mult__2__right,axiom,
% 12.85/6.97 ! [Z: real] :
% 12.85/6.97 ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.85/6.97 = ( plus_plus_real @ Z @ Z ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3754_mult__2__right,axiom,
% 12.85/6.97 ! [Z: rat] :
% 12.85/6.97 ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
% 12.85/6.97 = ( plus_plus_rat @ Z @ Z ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3755_mult__2__right,axiom,
% 12.85/6.97 ! [Z: nat] :
% 12.85/6.97 ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = ( plus_plus_nat @ Z @ Z ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3756_mult__2__right,axiom,
% 12.85/6.97 ! [Z: int] :
% 12.85/6.97 ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.97 = ( plus_plus_int @ Z @ Z ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3757_left__add__twice,axiom,
% 12.85/6.97 ! [A: complex,B: complex] :
% 12.85/6.97 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
% 12.85/6.97 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3758_left__add__twice,axiom,
% 12.85/6.97 ! [A: real,B: real] :
% 12.85/6.97 ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
% 12.85/6.97 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3759_left__add__twice,axiom,
% 12.85/6.97 ! [A: rat,B: rat] :
% 12.85/6.97 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 12.85/6.97 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3760_left__add__twice,axiom,
% 12.85/6.97 ! [A: nat,B: nat] :
% 12.85/6.97 ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 12.85/6.97 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3761_left__add__twice,axiom,
% 12.85/6.97 ! [A: int,B: int] :
% 12.85/6.97 ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
% 12.85/6.97 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3762_field__sum__of__halves,axiom,
% 12.85/6.97 ! [X: real] :
% 12.85/6.97 ( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/6.97 = X ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3763_field__sum__of__halves,axiom,
% 12.85/6.97 ! [X: rat] :
% 12.85/6.97 ( ( plus_plus_rat @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 12.85/6.97 = X ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3764_odd__even__add,axiom,
% 12.85/6.97 ! [A: code_integer,B: code_integer] :
% 12.85/6.97 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.97 => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
% 12.85/6.97 => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3765_odd__even__add,axiom,
% 12.85/6.97 ! [A: nat,B: nat] :
% 12.85/6.97 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.97 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 12.85/6.97 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3766_odd__even__add,axiom,
% 12.85/6.97 ! [A: int,B: int] :
% 12.85/6.97 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.97 => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
% 12.85/6.97 => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3767_Suc3__eq__add__3,axiom,
% 12.85/6.97 ! [N: nat] :
% 12.85/6.97 ( ( suc @ ( suc @ ( suc @ N ) ) )
% 12.85/6.97 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3768_nat__1__add__1,axiom,
% 12.85/6.97 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 12.85/6.97 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3769_add__eq__if,axiom,
% 12.85/6.97 ( plus_plus_nat
% 12.85/6.97 = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3770_nat__less__add__iff2,axiom,
% 12.85/6.97 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 12.85/6.97 ( ( ord_less_eq_nat @ I @ J )
% 12.85/6.97 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 12.85/6.97 = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3771_nat__less__add__iff1,axiom,
% 12.85/6.97 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 12.85/6.97 ( ( ord_less_eq_nat @ J @ I )
% 12.85/6.97 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 12.85/6.97 = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3772_dividend__less__times__div,axiom,
% 12.85/6.97 ! [N: nat,M: nat] :
% 12.85/6.97 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.97 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3773_dividend__less__div__times,axiom,
% 12.85/6.97 ! [N: nat,M: nat] :
% 12.85/6.97 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.97 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3774_split__div,axiom,
% 12.85/6.97 ! [P: nat > $o,M: nat,N: nat] :
% 12.85/6.97 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 12.85/6.97 = ( ( ( N = zero_zero_nat )
% 12.85/6.97 => ( P @ zero_zero_nat ) )
% 12.85/6.97 & ( ( N != zero_zero_nat )
% 12.85/6.97 => ! [I2: nat,J3: nat] :
% 12.85/6.97 ( ( ord_less_nat @ J3 @ N )
% 12.85/6.97 => ( ( M
% 12.85/6.97 = ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J3 ) )
% 12.85/6.97 => ( P @ I2 ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3775_mult__eq__if,axiom,
% 12.85/6.97 ( times_times_nat
% 12.85/6.97 = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3776_dvd__minus__add,axiom,
% 12.85/6.97 ! [Q2: nat,N: nat,R2: nat,M: nat] :
% 12.85/6.97 ( ( ord_less_eq_nat @ Q2 @ N )
% 12.85/6.97 => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R2 @ M ) )
% 12.85/6.97 => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q2 ) )
% 12.85/6.97 = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q2 ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3777_convex__bound__lt,axiom,
% 12.85/6.97 ! [X: rat,A: rat,Y4: rat,U: rat,V: rat] :
% 12.85/6.97 ( ( ord_less_rat @ X @ A )
% 12.85/6.97 => ( ( ord_less_rat @ Y4 @ A )
% 12.85/6.97 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 12.85/6.97 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 12.85/6.97 => ( ( ( plus_plus_rat @ U @ V )
% 12.85/6.97 = one_one_rat )
% 12.85/6.97 => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y4 ) ) @ A ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3778_convex__bound__lt,axiom,
% 12.85/6.97 ! [X: int,A: int,Y4: int,U: int,V: int] :
% 12.85/6.97 ( ( ord_less_int @ X @ A )
% 12.85/6.97 => ( ( ord_less_int @ Y4 @ A )
% 12.85/6.97 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 12.85/6.97 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 12.85/6.97 => ( ( ( plus_plus_int @ U @ V )
% 12.85/6.97 = one_one_int )
% 12.85/6.97 => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y4 ) ) @ A ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3779_convex__bound__lt,axiom,
% 12.85/6.97 ! [X: real,A: real,Y4: real,U: real,V: real] :
% 12.85/6.97 ( ( ord_less_real @ X @ A )
% 12.85/6.97 => ( ( ord_less_real @ Y4 @ A )
% 12.85/6.97 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 12.85/6.97 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 12.85/6.97 => ( ( ( plus_plus_real @ U @ V )
% 12.85/6.97 = one_one_real )
% 12.85/6.97 => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y4 ) ) @ A ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3780_scaling__mono,axiom,
% 12.85/6.97 ! [U: rat,V: rat,R2: rat,S2: rat] :
% 12.85/6.97 ( ( ord_less_eq_rat @ U @ V )
% 12.85/6.97 => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 12.85/6.97 => ( ( ord_less_eq_rat @ R2 @ S2 )
% 12.85/6.97 => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3781_scaling__mono,axiom,
% 12.85/6.97 ! [U: real,V: real,R2: real,S2: real] :
% 12.85/6.97 ( ( ord_less_eq_real @ U @ V )
% 12.85/6.97 => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 12.85/6.97 => ( ( ord_less_eq_real @ R2 @ S2 )
% 12.85/6.97 => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3782_field__less__half__sum,axiom,
% 12.85/6.97 ! [X: real,Y4: real] :
% 12.85/6.97 ( ( ord_less_real @ X @ Y4 )
% 12.85/6.97 => ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3783_field__less__half__sum,axiom,
% 12.85/6.97 ! [X: rat,Y4: rat] :
% 12.85/6.97 ( ( ord_less_rat @ X @ Y4 )
% 12.85/6.97 => ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y4 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3784_exp__add__not__zero__imp__left,axiom,
% 12.85/6.97 ! [M: nat,N: nat] :
% 12.85/6.97 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 12.85/6.97 != zero_zero_nat )
% 12.85/6.97 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 12.85/6.97 != zero_zero_nat ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3785_exp__add__not__zero__imp__left,axiom,
% 12.85/6.97 ! [M: nat,N: nat] :
% 12.85/6.97 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 12.85/6.97 != zero_zero_int )
% 12.85/6.97 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 12.85/6.97 != zero_zero_int ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3786_exp__add__not__zero__imp__right,axiom,
% 12.85/6.97 ! [M: nat,N: nat] :
% 12.85/6.97 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 12.85/6.97 != zero_zero_nat )
% 12.85/6.97 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.97 != zero_zero_nat ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3787_exp__add__not__zero__imp__right,axiom,
% 12.85/6.97 ! [M: nat,N: nat] :
% 12.85/6.97 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 12.85/6.97 != zero_zero_int )
% 12.85/6.97 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 12.85/6.97 != zero_zero_int ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3788_div__exp__eq,axiom,
% 12.85/6.97 ! [A: nat,M: nat,N: nat] :
% 12.85/6.97 ( ( 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 ) )
% 12.85/6.97 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3789_div__exp__eq,axiom,
% 12.85/6.97 ! [A: int,M: nat,N: nat] :
% 12.85/6.97 ( ( 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 ) )
% 12.85/6.97 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3790_nat__induct2,axiom,
% 12.85/6.97 ! [P: nat > $o,N: nat] :
% 12.85/6.97 ( ( P @ zero_zero_nat )
% 12.85/6.97 => ( ( P @ one_one_nat )
% 12.85/6.97 => ( ! [N2: nat] :
% 12.85/6.97 ( ( P @ N2 )
% 12.85/6.97 => ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/6.97 => ( P @ N ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3791_Suc__div__eq__add3__div,axiom,
% 12.85/6.97 ! [M: nat,N: nat] :
% 12.85/6.97 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 12.85/6.97 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3792_sum__power2__ge__zero,axiom,
% 12.85/6.97 ! [X: rat,Y4: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3793_sum__power2__ge__zero,axiom,
% 12.85/6.97 ! [X: int,Y4: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3794_sum__power2__ge__zero,axiom,
% 12.85/6.97 ! [X: real,Y4: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3795_sum__power2__le__zero__iff,axiom,
% 12.85/6.97 ! [X: rat,Y4: rat] :
% 12.85/6.97 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 12.85/6.97 = ( ( X = zero_zero_rat )
% 12.85/6.97 & ( Y4 = zero_zero_rat ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3796_sum__power2__le__zero__iff,axiom,
% 12.85/6.97 ! [X: int,Y4: int] :
% 12.85/6.97 ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 12.85/6.97 = ( ( X = zero_zero_int )
% 12.85/6.97 & ( Y4 = zero_zero_int ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3797_sum__power2__le__zero__iff,axiom,
% 12.85/6.97 ! [X: real,Y4: real] :
% 12.85/6.97 ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 12.85/6.97 = ( ( X = zero_zero_real )
% 12.85/6.97 & ( Y4 = zero_zero_real ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3798_not__sum__power2__lt__zero,axiom,
% 12.85/6.97 ! [X: real,Y4: real] :
% 12.85/6.97 ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3799_not__sum__power2__lt__zero,axiom,
% 12.85/6.97 ! [X: rat,Y4: rat] :
% 12.85/6.97 ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3800_not__sum__power2__lt__zero,axiom,
% 12.85/6.97 ! [X: int,Y4: int] :
% 12.85/6.97 ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3801_sum__power2__gt__zero__iff,axiom,
% 12.85/6.97 ! [X: real,Y4: real] :
% 12.85/6.97 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/6.97 = ( ( X != zero_zero_real )
% 12.85/6.97 | ( Y4 != zero_zero_real ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3802_sum__power2__gt__zero__iff,axiom,
% 12.85/6.97 ! [X: rat,Y4: rat] :
% 12.85/6.97 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/6.97 = ( ( X != zero_zero_rat )
% 12.85/6.97 | ( Y4 != zero_zero_rat ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3803_sum__power2__gt__zero__iff,axiom,
% 12.85/6.97 ! [X: int,Y4: int] :
% 12.85/6.97 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/6.97 = ( ( X != zero_zero_int )
% 12.85/6.97 | ( Y4 != zero_zero_int ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3804_oddE,axiom,
% 12.85/6.97 ! [A: code_integer] :
% 12.85/6.97 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.97 => ~ ! [B3: code_integer] :
% 12.85/6.97 ( A
% 12.85/6.97 != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) @ one_one_Code_integer ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3805_oddE,axiom,
% 12.85/6.97 ! [A: nat] :
% 12.85/6.97 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.97 => ~ ! [B3: nat] :
% 12.85/6.97 ( A
% 12.85/6.97 != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) @ one_one_nat ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3806_oddE,axiom,
% 12.85/6.97 ! [A: int] :
% 12.85/6.97 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.97 => ~ ! [B3: int] :
% 12.85/6.97 ( A
% 12.85/6.97 != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ one_one_int ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3807_power2__sum,axiom,
% 12.85/6.97 ! [X: complex,Y4: complex] :
% 12.85/6.97 ( ( power_power_complex @ ( plus_plus_complex @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3808_power2__sum,axiom,
% 12.85/6.97 ! [X: real,Y4: real] :
% 12.85/6.97 ( ( power_power_real @ ( plus_plus_real @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3809_power2__sum,axiom,
% 12.85/6.97 ! [X: rat,Y4: rat] :
% 12.85/6.97 ( ( power_power_rat @ ( plus_plus_rat @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3810_power2__sum,axiom,
% 12.85/6.97 ! [X: nat,Y4: nat] :
% 12.85/6.97 ( ( power_power_nat @ ( plus_plus_nat @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3811_power2__sum,axiom,
% 12.85/6.97 ! [X: int,Y4: int] :
% 12.85/6.97 ( ( power_power_int @ ( plus_plus_int @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3812_real__arch__simple,axiom,
% 12.85/6.97 ! [X: rat] :
% 12.85/6.97 ? [N2: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3813_real__arch__simple,axiom,
% 12.85/6.97 ! [X: real] :
% 12.85/6.97 ? [N2: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3814_reals__Archimedean2,axiom,
% 12.85/6.97 ! [X: real] :
% 12.85/6.97 ? [N2: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3815_reals__Archimedean2,axiom,
% 12.85/6.97 ! [X: rat] :
% 12.85/6.97 ? [N2: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3816_exists__least__lemma,axiom,
% 12.85/6.97 ! [P: nat > $o] :
% 12.85/6.97 ( ~ ( P @ zero_zero_nat )
% 12.85/6.97 => ( ? [X_12: nat] : ( P @ X_12 )
% 12.85/6.97 => ? [N2: nat] :
% 12.85/6.97 ( ~ ( P @ N2 )
% 12.85/6.97 & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3817_sum__squares__bound,axiom,
% 12.85/6.97 ! [X: rat,Y4: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y4 ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3818_sum__squares__bound,axiom,
% 12.85/6.97 ! [X: real,Y4: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y4 ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3819_power2__diff,axiom,
% 12.85/6.97 ! [X: complex,Y4: complex] :
% 12.85/6.97 ( ( power_power_complex @ ( minus_minus_complex @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3820_power2__diff,axiom,
% 12.85/6.97 ! [X: real,Y4: real] :
% 12.85/6.97 ( ( power_power_real @ ( minus_minus_real @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3821_power2__diff,axiom,
% 12.85/6.97 ! [X: rat,Y4: rat] :
% 12.85/6.97 ( ( power_power_rat @ ( minus_minus_rat @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3822_power2__diff,axiom,
% 12.85/6.97 ! [X: int,Y4: int] :
% 12.85/6.97 ( ( power_power_int @ ( minus_minus_int @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3823_ex__power__ivl1,axiom,
% 12.85/6.97 ! [B: nat,K: nat] :
% 12.85/6.97 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 12.85/6.97 => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 12.85/6.97 => ? [N2: nat] :
% 12.85/6.97 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 12.85/6.97 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3824_ex__power__ivl2,axiom,
% 12.85/6.97 ! [B: nat,K: nat] :
% 12.85/6.97 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 12.85/6.97 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 12.85/6.97 => ? [N2: nat] :
% 12.85/6.97 ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 12.85/6.97 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3825_arith__geo__mean,axiom,
% 12.85/6.97 ! [U: rat,X: rat,Y4: rat] :
% 12.85/6.97 ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = ( times_times_rat @ X @ Y4 ) )
% 12.85/6.97 => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 12.85/6.97 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 12.85/6.97 => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y4 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3826_arith__geo__mean,axiom,
% 12.85/6.97 ! [U: real,X: real,Y4: real] :
% 12.85/6.97 ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = ( times_times_real @ X @ Y4 ) )
% 12.85/6.97 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/6.97 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/6.97 => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3827_ex__less__of__nat__mult,axiom,
% 12.85/6.97 ! [X: real,Y4: real] :
% 12.85/6.97 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.97 => ? [N2: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3828_ex__less__of__nat__mult,axiom,
% 12.85/6.97 ! [X: rat,Y4: rat] :
% 12.85/6.97 ( ( ord_less_rat @ zero_zero_rat @ X )
% 12.85/6.97 => ? [N2: nat] : ( ord_less_rat @ Y4 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ X ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3829_low__inv,axiom,
% 12.85/6.97 ! [X: nat,N: nat,Y4: nat] :
% 12.85/6.97 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.97 => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
% 12.85/6.97 = X ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3830_Bolzano,axiom,
% 12.85/6.97 ! [A: real,B: real,P: real > real > $o] :
% 12.85/6.97 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.97 => ( ! [A4: real,B3: real,C3: real] :
% 12.85/6.97 ( ( P @ A4 @ B3 )
% 12.85/6.97 => ( ( P @ B3 @ C3 )
% 12.85/6.97 => ( ( ord_less_eq_real @ A4 @ B3 )
% 12.85/6.97 => ( ( ord_less_eq_real @ B3 @ C3 )
% 12.85/6.97 => ( P @ A4 @ C3 ) ) ) ) )
% 12.85/6.97 => ( ! [X4: real] :
% 12.85/6.97 ( ( ord_less_eq_real @ A @ X4 )
% 12.85/6.97 => ( ( ord_less_eq_real @ X4 @ B )
% 12.85/6.97 => ? [D5: real] :
% 12.85/6.97 ( ( ord_less_real @ zero_zero_real @ D5 )
% 12.85/6.97 & ! [A4: real,B3: real] :
% 12.85/6.97 ( ( ( ord_less_eq_real @ A4 @ X4 )
% 12.85/6.97 & ( ord_less_eq_real @ X4 @ B3 )
% 12.85/6.97 & ( ord_less_real @ ( minus_minus_real @ B3 @ A4 ) @ D5 ) )
% 12.85/6.97 => ( P @ A4 @ B3 ) ) ) ) )
% 12.85/6.97 => ( P @ A @ B ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3831_mult__le__cancel__iff1,axiom,
% 12.85/6.97 ! [Z: rat,X: rat,Y4: rat] :
% 12.85/6.97 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 12.85/6.97 => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y4 @ Z ) )
% 12.85/6.97 = ( ord_less_eq_rat @ X @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3832_mult__le__cancel__iff1,axiom,
% 12.85/6.97 ! [Z: int,X: int,Y4: int] :
% 12.85/6.97 ( ( ord_less_int @ zero_zero_int @ Z )
% 12.85/6.97 => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y4 @ Z ) )
% 12.85/6.97 = ( ord_less_eq_int @ X @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3833_mult__le__cancel__iff1,axiom,
% 12.85/6.97 ! [Z: real,X: real,Y4: real] :
% 12.85/6.97 ( ( ord_less_real @ zero_zero_real @ Z )
% 12.85/6.97 => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y4 @ Z ) )
% 12.85/6.97 = ( ord_less_eq_real @ X @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3834_mult__le__cancel__iff2,axiom,
% 12.85/6.97 ! [Z: rat,X: rat,Y4: rat] :
% 12.85/6.97 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 12.85/6.97 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y4 ) )
% 12.85/6.97 = ( ord_less_eq_rat @ X @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3835_mult__le__cancel__iff2,axiom,
% 12.85/6.97 ! [Z: int,X: int,Y4: int] :
% 12.85/6.97 ( ( ord_less_int @ zero_zero_int @ Z )
% 12.85/6.97 => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y4 ) )
% 12.85/6.97 = ( ord_less_eq_int @ X @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3836_mult__le__cancel__iff2,axiom,
% 12.85/6.97 ! [Z: real,X: real,Y4: real] :
% 12.85/6.97 ( ( ord_less_real @ zero_zero_real @ Z )
% 12.85/6.97 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y4 ) )
% 12.85/6.97 = ( ord_less_eq_real @ X @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3837_lemma__termdiff3,axiom,
% 12.85/6.97 ! [H: real,Z: real,K5: real,N: nat] :
% 12.85/6.97 ( ( H != zero_zero_real )
% 12.85/6.97 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
% 12.85/6.97 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H ) ) @ K5 )
% 12.85/6.97 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3838_lemma__termdiff3,axiom,
% 12.85/6.97 ! [H: complex,Z: complex,K5: real,N: nat] :
% 12.85/6.97 ( ( H != zero_zero_complex )
% 12.85/6.97 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
% 12.85/6.97 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H ) ) @ K5 )
% 12.85/6.97 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3839_dbl__inc__simps_I3_J,axiom,
% 12.85/6.97 ( ( neg_nu8557863876264182079omplex @ one_one_complex )
% 12.85/6.97 = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3840_dbl__inc__simps_I3_J,axiom,
% 12.85/6.97 ( ( neg_nu8295874005876285629c_real @ one_one_real )
% 12.85/6.97 = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3841_dbl__inc__simps_I3_J,axiom,
% 12.85/6.97 ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
% 12.85/6.97 = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3842_dbl__inc__simps_I3_J,axiom,
% 12.85/6.97 ( ( neg_nu5851722552734809277nc_int @ one_one_int )
% 12.85/6.97 = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3843_dbl__simps_I3_J,axiom,
% 12.85/6.97 ( ( neg_nu7009210354673126013omplex @ one_one_complex )
% 12.85/6.97 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3844_dbl__simps_I3_J,axiom,
% 12.85/6.97 ( ( neg_numeral_dbl_real @ one_one_real )
% 12.85/6.97 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3845_dbl__simps_I3_J,axiom,
% 12.85/6.97 ( ( neg_numeral_dbl_rat @ one_one_rat )
% 12.85/6.97 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3846_dbl__simps_I3_J,axiom,
% 12.85/6.97 ( ( neg_numeral_dbl_int @ one_one_int )
% 12.85/6.97 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3847_semiring__norm_I6_J,axiom,
% 12.85/6.97 ! [M: num,N: num] :
% 12.85/6.97 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.85/6.97 = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3848_dbl__simps_I2_J,axiom,
% 12.85/6.97 ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
% 12.85/6.97 = zero_zero_complex ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3849_dbl__simps_I2_J,axiom,
% 12.85/6.97 ( ( neg_numeral_dbl_real @ zero_zero_real )
% 12.85/6.97 = zero_zero_real ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3850_dbl__simps_I2_J,axiom,
% 12.85/6.97 ( ( neg_numeral_dbl_rat @ zero_zero_rat )
% 12.85/6.97 = zero_zero_rat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3851_dbl__simps_I2_J,axiom,
% 12.85/6.97 ( ( neg_numeral_dbl_int @ zero_zero_int )
% 12.85/6.97 = zero_zero_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3852_semiring__norm_I2_J,axiom,
% 12.85/6.97 ( ( plus_plus_num @ one @ one )
% 12.85/6.97 = ( bit0 @ one ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3853_zle__add1__eq__le,axiom,
% 12.85/6.97 ! [W: int,Z: int] :
% 12.85/6.97 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 12.85/6.97 = ( ord_less_eq_int @ W @ Z ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3854_semiring__norm_I7_J,axiom,
% 12.85/6.97 ! [M: num,N: num] :
% 12.85/6.97 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.85/6.97 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3855_semiring__norm_I9_J,axiom,
% 12.85/6.97 ! [M: num,N: num] :
% 12.85/6.97 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.85/6.97 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3856_dbl__simps_I5_J,axiom,
% 12.85/6.97 ! [K: num] :
% 12.85/6.97 ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
% 12.85/6.97 = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3857_dbl__simps_I5_J,axiom,
% 12.85/6.97 ! [K: num] :
% 12.85/6.97 ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
% 12.85/6.97 = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3858_dbl__simps_I5_J,axiom,
% 12.85/6.97 ! [K: num] :
% 12.85/6.97 ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
% 12.85/6.97 = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3859_dbl__simps_I5_J,axiom,
% 12.85/6.97 ! [K: num] :
% 12.85/6.97 ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
% 12.85/6.97 = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3860_dbl__inc__simps_I2_J,axiom,
% 12.85/6.97 ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
% 12.85/6.97 = one_one_complex ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3861_dbl__inc__simps_I2_J,axiom,
% 12.85/6.97 ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
% 12.85/6.97 = one_one_real ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3862_dbl__inc__simps_I2_J,axiom,
% 12.85/6.97 ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
% 12.85/6.97 = one_one_rat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3863_dbl__inc__simps_I2_J,axiom,
% 12.85/6.97 ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 12.85/6.97 = one_one_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3864_dbl__inc__simps_I5_J,axiom,
% 12.85/6.97 ! [K: num] :
% 12.85/6.97 ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
% 12.85/6.97 = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3865_dbl__inc__simps_I5_J,axiom,
% 12.85/6.97 ! [K: num] :
% 12.85/6.97 ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
% 12.85/6.97 = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3866_dbl__inc__simps_I5_J,axiom,
% 12.85/6.97 ! [K: num] :
% 12.85/6.97 ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
% 12.85/6.97 = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3867_dbl__inc__simps_I5_J,axiom,
% 12.85/6.97 ! [K: num] :
% 12.85/6.97 ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
% 12.85/6.97 = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3868_power__add__numeral2,axiom,
% 12.85/6.97 ! [A: complex,M: num,N: num,B: complex] :
% 12.85/6.97 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 12.85/6.97 = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3869_power__add__numeral2,axiom,
% 12.85/6.97 ! [A: real,M: num,N: num,B: real] :
% 12.85/6.97 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 12.85/6.97 = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3870_power__add__numeral2,axiom,
% 12.85/6.97 ! [A: rat,M: num,N: num,B: rat] :
% 12.85/6.97 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 12.85/6.97 = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3871_power__add__numeral2,axiom,
% 12.85/6.97 ! [A: nat,M: num,N: num,B: nat] :
% 12.85/6.97 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 12.85/6.97 = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3872_power__add__numeral2,axiom,
% 12.85/6.97 ! [A: int,M: num,N: num,B: int] :
% 12.85/6.97 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 12.85/6.97 = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3873_power__add__numeral,axiom,
% 12.85/6.97 ! [A: complex,M: num,N: num] :
% 12.85/6.97 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
% 12.85/6.97 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3874_power__add__numeral,axiom,
% 12.85/6.97 ! [A: real,M: num,N: num] :
% 12.85/6.97 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
% 12.85/6.97 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3875_power__add__numeral,axiom,
% 12.85/6.97 ! [A: rat,M: num,N: num] :
% 12.85/6.97 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
% 12.85/6.97 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3876_power__add__numeral,axiom,
% 12.85/6.97 ! [A: nat,M: num,N: num] :
% 12.85/6.97 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
% 12.85/6.97 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3877_power__add__numeral,axiom,
% 12.85/6.97 ! [A: int,M: num,N: num] :
% 12.85/6.97 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
% 12.85/6.97 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3878_Suc__numeral,axiom,
% 12.85/6.97 ! [N: num] :
% 12.85/6.97 ( ( suc @ ( numeral_numeral_nat @ N ) )
% 12.85/6.97 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3879_semiring__norm_I3_J,axiom,
% 12.85/6.97 ! [N: num] :
% 12.85/6.97 ( ( plus_plus_num @ one @ ( bit0 @ N ) )
% 12.85/6.97 = ( bit1 @ N ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3880_semiring__norm_I4_J,axiom,
% 12.85/6.97 ! [N: num] :
% 12.85/6.97 ( ( plus_plus_num @ one @ ( bit1 @ N ) )
% 12.85/6.97 = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3881_semiring__norm_I5_J,axiom,
% 12.85/6.97 ! [M: num] :
% 12.85/6.97 ( ( plus_plus_num @ ( bit0 @ M ) @ one )
% 12.85/6.97 = ( bit1 @ M ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3882_semiring__norm_I8_J,axiom,
% 12.85/6.97 ! [M: num] :
% 12.85/6.97 ( ( plus_plus_num @ ( bit1 @ M ) @ one )
% 12.85/6.97 = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3883_semiring__norm_I10_J,axiom,
% 12.85/6.97 ! [M: num,N: num] :
% 12.85/6.97 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.85/6.97 = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3884_semiring__norm_I16_J,axiom,
% 12.85/6.97 ! [M: num,N: num] :
% 12.85/6.97 ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.85/6.97 = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3885_add__One__commute,axiom,
% 12.85/6.97 ! [N: num] :
% 12.85/6.97 ( ( plus_plus_num @ one @ N )
% 12.85/6.97 = ( plus_plus_num @ N @ one ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3886_plus__int__code_I1_J,axiom,
% 12.85/6.97 ! [K: int] :
% 12.85/6.97 ( ( plus_plus_int @ K @ zero_zero_int )
% 12.85/6.97 = K ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3887_plus__int__code_I2_J,axiom,
% 12.85/6.97 ! [L: int] :
% 12.85/6.97 ( ( plus_plus_int @ zero_zero_int @ L )
% 12.85/6.97 = L ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3888_int__distrib_I1_J,axiom,
% 12.85/6.97 ! [Z1: int,Z22: int,W: int] :
% 12.85/6.97 ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
% 12.85/6.97 = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3889_int__distrib_I2_J,axiom,
% 12.85/6.97 ! [W: int,Z1: int,Z22: int] :
% 12.85/6.97 ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
% 12.85/6.97 = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3890_iadd__is__0,axiom,
% 12.85/6.97 ! [M: extended_enat,N: extended_enat] :
% 12.85/6.97 ( ( ( plus_p3455044024723400733d_enat @ M @ N )
% 12.85/6.97 = zero_z5237406670263579293d_enat )
% 12.85/6.97 = ( ( M = zero_z5237406670263579293d_enat )
% 12.85/6.97 & ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3891_dbl__def,axiom,
% 12.85/6.97 ( neg_numeral_dbl_real
% 12.85/6.97 = ( ^ [X3: real] : ( plus_plus_real @ X3 @ X3 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3892_dbl__def,axiom,
% 12.85/6.97 ( neg_numeral_dbl_rat
% 12.85/6.97 = ( ^ [X3: rat] : ( plus_plus_rat @ X3 @ X3 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3893_dbl__def,axiom,
% 12.85/6.97 ( neg_numeral_dbl_int
% 12.85/6.97 = ( ^ [X3: int] : ( plus_plus_int @ X3 @ X3 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3894_odd__nonzero,axiom,
% 12.85/6.97 ! [Z: int] :
% 12.85/6.97 ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
% 12.85/6.97 != zero_zero_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3895_int__plus,axiom,
% 12.85/6.97 ! [N: nat,M: nat] :
% 12.85/6.97 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
% 12.85/6.97 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3896_int__ops_I5_J,axiom,
% 12.85/6.97 ! [A: nat,B: nat] :
% 12.85/6.97 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
% 12.85/6.97 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3897_zadd__int__left,axiom,
% 12.85/6.97 ! [M: nat,N: nat,Z: int] :
% 12.85/6.97 ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
% 12.85/6.97 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3898_int__ge__induct,axiom,
% 12.85/6.97 ! [K: int,I: int,P: int > $o] :
% 12.85/6.97 ( ( ord_less_eq_int @ K @ I )
% 12.85/6.97 => ( ( P @ K )
% 12.85/6.97 => ( ! [I3: int] :
% 12.85/6.97 ( ( ord_less_eq_int @ K @ I3 )
% 12.85/6.97 => ( ( P @ I3 )
% 12.85/6.97 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 12.85/6.97 => ( P @ I ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3899_zless__add1__eq,axiom,
% 12.85/6.97 ! [W: int,Z: int] :
% 12.85/6.97 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 12.85/6.97 = ( ( ord_less_int @ W @ Z )
% 12.85/6.97 | ( W = Z ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3900_int__gr__induct,axiom,
% 12.85/6.97 ! [K: int,I: int,P: int > $o] :
% 12.85/6.97 ( ( ord_less_int @ K @ I )
% 12.85/6.97 => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 12.85/6.97 => ( ! [I3: int] :
% 12.85/6.97 ( ( ord_less_int @ K @ I3 )
% 12.85/6.97 => ( ( P @ I3 )
% 12.85/6.97 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 12.85/6.97 => ( P @ I ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3901_zle__iff__zadd,axiom,
% 12.85/6.97 ( ord_less_eq_int
% 12.85/6.97 = ( ^ [W3: int,Z5: int] :
% 12.85/6.97 ? [N4: nat] :
% 12.85/6.97 ( Z5
% 12.85/6.97 = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3902_zdvd__period,axiom,
% 12.85/6.97 ! [A: int,D: int,X: int,T: int,C: int] :
% 12.85/6.97 ( ( dvd_dvd_int @ A @ D )
% 12.85/6.97 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
% 12.85/6.97 = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3903_zdvd__reduce,axiom,
% 12.85/6.97 ! [K: int,N: int,M: int] :
% 12.85/6.97 ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
% 12.85/6.97 = ( dvd_dvd_int @ K @ N ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3904_add__diff__assoc__enat,axiom,
% 12.85/6.97 ! [Z: extended_enat,Y4: extended_enat,X: extended_enat] :
% 12.85/6.97 ( ( ord_le2932123472753598470d_enat @ Z @ Y4 )
% 12.85/6.97 => ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y4 @ Z ) )
% 12.85/6.97 = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y4 ) @ Z ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3905_int__Suc,axiom,
% 12.85/6.97 ! [N: nat] :
% 12.85/6.97 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 12.85/6.97 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3906_int__ops_I4_J,axiom,
% 12.85/6.97 ! [A: nat] :
% 12.85/6.97 ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
% 12.85/6.97 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3907_zless__iff__Suc__zadd,axiom,
% 12.85/6.97 ( ord_less_int
% 12.85/6.97 = ( ^ [W3: int,Z5: int] :
% 12.85/6.97 ? [N4: nat] :
% 12.85/6.97 ( Z5
% 12.85/6.97 = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3908_odd__less__0__iff,axiom,
% 12.85/6.97 ! [Z: int] :
% 12.85/6.97 ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
% 12.85/6.97 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3909_add1__zle__eq,axiom,
% 12.85/6.97 ! [W: int,Z: int] :
% 12.85/6.97 ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
% 12.85/6.97 = ( ord_less_int @ W @ Z ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3910_zless__imp__add1__zle,axiom,
% 12.85/6.97 ! [W: int,Z: int] :
% 12.85/6.97 ( ( ord_less_int @ W @ Z )
% 12.85/6.97 => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3911_int__induct,axiom,
% 12.85/6.97 ! [P: int > $o,K: int,I: int] :
% 12.85/6.97 ( ( P @ K )
% 12.85/6.97 => ( ! [I3: int] :
% 12.85/6.97 ( ( ord_less_eq_int @ K @ I3 )
% 12.85/6.97 => ( ( P @ I3 )
% 12.85/6.97 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 12.85/6.97 => ( ! [I3: int] :
% 12.85/6.97 ( ( ord_less_eq_int @ I3 @ K )
% 12.85/6.97 => ( ( P @ I3 )
% 12.85/6.97 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 12.85/6.97 => ( P @ I ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3912_Suc__nat__number__of__add,axiom,
% 12.85/6.97 ! [V: num,N: nat] :
% 12.85/6.97 ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
% 12.85/6.97 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3913_dbl__inc__def,axiom,
% 12.85/6.97 ( neg_nu8557863876264182079omplex
% 12.85/6.97 = ( ^ [X3: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X3 @ X3 ) @ one_one_complex ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3914_dbl__inc__def,axiom,
% 12.85/6.97 ( neg_nu8295874005876285629c_real
% 12.85/6.97 = ( ^ [X3: real] : ( plus_plus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3915_dbl__inc__def,axiom,
% 12.85/6.97 ( neg_nu5219082963157363817nc_rat
% 12.85/6.97 = ( ^ [X3: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3916_dbl__inc__def,axiom,
% 12.85/6.97 ( neg_nu5851722552734809277nc_int
% 12.85/6.97 = ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3917_nat__less__real__le,axiom,
% 12.85/6.97 ( ord_less_nat
% 12.85/6.97 = ( ^ [N4: nat,M5: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N4 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M5 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3918_nat__le__real__less,axiom,
% 12.85/6.97 ( ord_less_eq_nat
% 12.85/6.97 = ( ^ [N4: nat,M5: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M5 ) @ one_one_real ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3919_le__imp__0__less,axiom,
% 12.85/6.97 ! [Z: int] :
% 12.85/6.97 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.85/6.97 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3920_unique__quotient__lemma__neg,axiom,
% 12.85/6.97 ! [B: int,Q5: int,R4: int,Q2: int,R2: int] :
% 12.85/6.97 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
% 12.85/6.97 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 12.85/6.97 => ( ( ord_less_int @ B @ R2 )
% 12.85/6.97 => ( ( ord_less_int @ B @ R4 )
% 12.85/6.97 => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3921_unique__quotient__lemma,axiom,
% 12.85/6.97 ! [B: int,Q5: int,R4: int,Q2: int,R2: int] :
% 12.85/6.97 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
% 12.85/6.97 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 12.85/6.97 => ( ( ord_less_int @ R4 @ B )
% 12.85/6.97 => ( ( ord_less_int @ R2 @ B )
% 12.85/6.97 => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3922_zdiv__mono2__neg__lemma,axiom,
% 12.85/6.97 ! [B: int,Q2: int,R2: int,B6: int,Q5: int,R4: int] :
% 12.85/6.97 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 )
% 12.85/6.97 = ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) )
% 12.85/6.97 => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) @ zero_zero_int )
% 12.85/6.97 => ( ( ord_less_int @ R2 @ B )
% 12.85/6.97 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 12.85/6.97 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 12.85/6.97 => ( ( ord_less_eq_int @ B6 @ B )
% 12.85/6.97 => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3923_zdiv__mono2__lemma,axiom,
% 12.85/6.97 ! [B: int,Q2: int,R2: int,B6: int,Q5: int,R4: int] :
% 12.85/6.97 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 )
% 12.85/6.97 = ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) )
% 12.85/6.97 => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) )
% 12.85/6.97 => ( ( ord_less_int @ R4 @ B6 )
% 12.85/6.97 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 12.85/6.97 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 12.85/6.97 => ( ( ord_less_eq_int @ B6 @ B )
% 12.85/6.97 => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3924_q__pos__lemma,axiom,
% 12.85/6.97 ! [B6: int,Q5: int,R4: int] :
% 12.85/6.97 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) )
% 12.85/6.97 => ( ( ord_less_int @ R4 @ B6 )
% 12.85/6.97 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 12.85/6.97 => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3925_incr__mult__lemma,axiom,
% 12.85/6.97 ! [D: int,P: int > $o,K: int] :
% 12.85/6.97 ( ( ord_less_int @ zero_zero_int @ D )
% 12.85/6.97 => ( ! [X4: int] :
% 12.85/6.97 ( ( P @ X4 )
% 12.85/6.97 => ( P @ ( plus_plus_int @ X4 @ D ) ) )
% 12.85/6.97 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.85/6.97 => ! [X5: int] :
% 12.85/6.97 ( ( P @ X5 )
% 12.85/6.97 => ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3926_linear__plus__1__le__power,axiom,
% 12.85/6.97 ! [X: real,N: nat] :
% 12.85/6.97 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/6.97 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3927_even__diff__iff,axiom,
% 12.85/6.97 ! [K: int,L: int] :
% 12.85/6.97 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
% 12.85/6.97 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3928_int__div__pos__eq,axiom,
% 12.85/6.97 ! [A: int,B: int,Q2: int,R2: int] :
% 12.85/6.97 ( ( A
% 12.85/6.97 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
% 12.85/6.97 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 12.85/6.97 => ( ( ord_less_int @ R2 @ B )
% 12.85/6.97 => ( ( divide_divide_int @ A @ B )
% 12.85/6.97 = Q2 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3929_int__div__neg__eq,axiom,
% 12.85/6.97 ! [A: int,B: int,Q2: int,R2: int] :
% 12.85/6.97 ( ( A
% 12.85/6.97 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
% 12.85/6.97 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 12.85/6.97 => ( ( ord_less_int @ B @ R2 )
% 12.85/6.97 => ( ( divide_divide_int @ A @ B )
% 12.85/6.97 = Q2 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3930_split__zdiv,axiom,
% 12.85/6.97 ! [P: int > $o,N: int,K: int] :
% 12.85/6.97 ( ( P @ ( divide_divide_int @ N @ K ) )
% 12.85/6.97 = ( ( ( K = zero_zero_int )
% 12.85/6.97 => ( P @ zero_zero_int ) )
% 12.85/6.97 & ( ( ord_less_int @ zero_zero_int @ K )
% 12.85/6.97 => ! [I2: int,J3: int] :
% 12.85/6.97 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 12.85/6.97 & ( ord_less_int @ J3 @ K )
% 12.85/6.97 & ( N
% 12.85/6.97 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 12.85/6.97 => ( P @ I2 ) ) )
% 12.85/6.97 & ( ( ord_less_int @ K @ zero_zero_int )
% 12.85/6.97 => ! [I2: int,J3: int] :
% 12.85/6.97 ( ( ( ord_less_int @ K @ J3 )
% 12.85/6.97 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 12.85/6.97 & ( N
% 12.85/6.97 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 12.85/6.97 => ( P @ I2 ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3931_div__pos__geq,axiom,
% 12.85/6.97 ! [L: int,K: int] :
% 12.85/6.97 ( ( ord_less_int @ zero_zero_int @ L )
% 12.85/6.97 => ( ( ord_less_eq_int @ L @ K )
% 12.85/6.97 => ( ( divide_divide_int @ K @ L )
% 12.85/6.97 = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3932_L2__set__mult__ineq__lemma,axiom,
% 12.85/6.97 ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3933_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 12.85/6.97 ! [X: nat,N: nat,M: nat] :
% 12.85/6.97 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 12.85/6.97 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.97 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/6.97 => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3934_neg__zdiv__mult__2,axiom,
% 12.85/6.97 ! [A: int,B: int] :
% 12.85/6.97 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.85/6.97 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 12.85/6.97 = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3935_pos__zdiv__mult__2,axiom,
% 12.85/6.97 ! [A: int,B: int] :
% 12.85/6.97 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.97 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 12.85/6.97 = ( divide_divide_int @ B @ A ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3936_Bernoulli__inequality__even,axiom,
% 12.85/6.97 ! [N: nat,X: real] :
% 12.85/6.97 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.97 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3937_mult__less__iff1,axiom,
% 12.85/6.97 ! [Z: real,X: real,Y4: real] :
% 12.85/6.97 ( ( ord_less_real @ zero_zero_real @ Z )
% 12.85/6.97 => ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y4 @ Z ) )
% 12.85/6.97 = ( ord_less_real @ X @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3938_mult__less__iff1,axiom,
% 12.85/6.97 ! [Z: rat,X: rat,Y4: rat] :
% 12.85/6.97 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 12.85/6.97 => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y4 @ Z ) )
% 12.85/6.97 = ( ord_less_rat @ X @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3939_mult__less__iff1,axiom,
% 12.85/6.97 ! [Z: int,X: int,Y4: int] :
% 12.85/6.97 ( ( ord_less_int @ zero_zero_int @ Z )
% 12.85/6.97 => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y4 @ Z ) )
% 12.85/6.97 = ( ord_less_int @ X @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3940_real__average__minus__second,axiom,
% 12.85/6.97 ! [B: real,A: real] :
% 12.85/6.97 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 12.85/6.97 = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3941_real__average__minus__first,axiom,
% 12.85/6.97 ! [A: real,B: real] :
% 12.85/6.97 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 12.85/6.97 = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3942_norm__divide__numeral,axiom,
% 12.85/6.97 ! [A: real,W: num] :
% 12.85/6.97 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
% 12.85/6.97 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3943_norm__divide__numeral,axiom,
% 12.85/6.97 ! [A: complex,W: num] :
% 12.85/6.97 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 12.85/6.97 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3944_norm__mult__numeral2,axiom,
% 12.85/6.97 ! [A: real,W: num] :
% 12.85/6.97 ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
% 12.85/6.97 = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3945_norm__mult__numeral2,axiom,
% 12.85/6.97 ! [A: complex,W: num] :
% 12.85/6.97 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 12.85/6.97 = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3946_norm__mult__numeral1,axiom,
% 12.85/6.97 ! [W: num,A: real] :
% 12.85/6.97 ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 12.85/6.97 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3947_norm__mult__numeral1,axiom,
% 12.85/6.97 ! [W: num,A: complex] :
% 12.85/6.97 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 12.85/6.97 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3948_norm__le__zero__iff,axiom,
% 12.85/6.97 ! [X: real] :
% 12.85/6.97 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
% 12.85/6.97 = ( X = zero_zero_real ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3949_norm__le__zero__iff,axiom,
% 12.85/6.97 ! [X: complex] :
% 12.85/6.97 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
% 12.85/6.97 = ( X = zero_zero_complex ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3950_zero__less__norm__iff,axiom,
% 12.85/6.97 ! [X: real] :
% 12.85/6.97 ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
% 12.85/6.97 = ( X != zero_zero_real ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3951_zero__less__norm__iff,axiom,
% 12.85/6.97 ! [X: complex] :
% 12.85/6.97 ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
% 12.85/6.97 = ( X != zero_zero_complex ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3952_norm__of__nat,axiom,
% 12.85/6.97 ! [N: nat] :
% 12.85/6.97 ( ( real_V7735802525324610683m_real @ ( semiri5074537144036343181t_real @ N ) )
% 12.85/6.97 = ( semiri5074537144036343181t_real @ N ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3953_norm__of__nat,axiom,
% 12.85/6.97 ! [N: nat] :
% 12.85/6.97 ( ( real_V1022390504157884413omplex @ ( semiri8010041392384452111omplex @ N ) )
% 12.85/6.97 = ( semiri5074537144036343181t_real @ N ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3954_norm__eq__zero,axiom,
% 12.85/6.97 ! [X: real] :
% 12.85/6.97 ( ( ( real_V7735802525324610683m_real @ X )
% 12.85/6.97 = zero_zero_real )
% 12.85/6.97 = ( X = zero_zero_real ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3955_norm__eq__zero,axiom,
% 12.85/6.97 ! [X: complex] :
% 12.85/6.97 ( ( ( real_V1022390504157884413omplex @ X )
% 12.85/6.97 = zero_zero_real )
% 12.85/6.97 = ( X = zero_zero_complex ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3956_norm__zero,axiom,
% 12.85/6.97 ( ( real_V7735802525324610683m_real @ zero_zero_real )
% 12.85/6.97 = zero_zero_real ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3957_norm__zero,axiom,
% 12.85/6.97 ( ( real_V1022390504157884413omplex @ zero_zero_complex )
% 12.85/6.97 = zero_zero_real ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3958_norm__numeral,axiom,
% 12.85/6.97 ! [W: num] :
% 12.85/6.97 ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
% 12.85/6.97 = ( numeral_numeral_real @ W ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3959_norm__numeral,axiom,
% 12.85/6.97 ! [W: num] :
% 12.85/6.97 ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
% 12.85/6.97 = ( numeral_numeral_real @ W ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3960_norm__minus__commute,axiom,
% 12.85/6.97 ! [A: real,B: real] :
% 12.85/6.97 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) )
% 12.85/6.97 = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3961_norm__minus__commute,axiom,
% 12.85/6.97 ! [A: complex,B: complex] :
% 12.85/6.97 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) )
% 12.85/6.97 = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ A ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3962_norm__triangle__mono,axiom,
% 12.85/6.97 ! [A: real,R2: real,B: real,S2: real] :
% 12.85/6.97 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
% 12.85/6.97 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S2 )
% 12.85/6.97 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3963_norm__triangle__mono,axiom,
% 12.85/6.97 ! [A: complex,R2: real,B: complex,S2: real] :
% 12.85/6.97 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
% 12.85/6.97 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S2 )
% 12.85/6.97 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3964_norm__triangle__ineq,axiom,
% 12.85/6.97 ! [X: real,Y4: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y4 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3965_norm__triangle__ineq,axiom,
% 12.85/6.97 ! [X: complex,Y4: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y4 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3966_norm__triangle__le,axiom,
% 12.85/6.97 ! [X: real,Y4: real,E2: real] :
% 12.85/6.97 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y4 ) ) @ E2 )
% 12.85/6.97 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y4 ) ) @ E2 ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3967_norm__triangle__le,axiom,
% 12.85/6.97 ! [X: complex,Y4: complex,E2: real] :
% 12.85/6.97 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) ) @ E2 )
% 12.85/6.97 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y4 ) ) @ E2 ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3968_norm__add__leD,axiom,
% 12.85/6.97 ! [A: real,B: real,C: real] :
% 12.85/6.97 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
% 12.85/6.97 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3969_norm__add__leD,axiom,
% 12.85/6.97 ! [A: complex,B: complex,C: real] :
% 12.85/6.97 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
% 12.85/6.97 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3970_norm__diff__triangle__less,axiom,
% 12.85/6.97 ! [X: real,Y4: real,E1: real,Z: real,E22: real] :
% 12.85/6.97 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y4 ) ) @ E1 )
% 12.85/6.97 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y4 @ Z ) ) @ E22 )
% 12.85/6.97 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3971_norm__diff__triangle__less,axiom,
% 12.85/6.97 ! [X: complex,Y4: complex,E1: real,Z: complex,E22: real] :
% 12.85/6.97 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y4 ) ) @ E1 )
% 12.85/6.97 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y4 @ Z ) ) @ E22 )
% 12.85/6.97 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3972_norm__triangle__sub,axiom,
% 12.85/6.97 ! [X: real,Y4: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y4 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y4 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3973_norm__triangle__sub,axiom,
% 12.85/6.97 ! [X: complex,Y4: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y4 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y4 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3974_norm__triangle__ineq4,axiom,
% 12.85/6.97 ! [A: real,B: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3975_norm__triangle__ineq4,axiom,
% 12.85/6.97 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3976_norm__diff__triangle__le,axiom,
% 12.85/6.97 ! [X: real,Y4: real,E1: real,Z: real,E22: real] :
% 12.85/6.97 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y4 ) ) @ E1 )
% 12.85/6.97 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y4 @ Z ) ) @ E22 )
% 12.85/6.97 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3977_norm__diff__triangle__le,axiom,
% 12.85/6.97 ! [X: complex,Y4: complex,E1: real,Z: complex,E22: real] :
% 12.85/6.97 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y4 ) ) @ E1 )
% 12.85/6.97 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y4 @ Z ) ) @ E22 )
% 12.85/6.97 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3978_norm__triangle__le__diff,axiom,
% 12.85/6.97 ! [X: real,Y4: real,E2: real] :
% 12.85/6.97 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y4 ) ) @ E2 )
% 12.85/6.97 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y4 ) ) @ E2 ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3979_norm__triangle__le__diff,axiom,
% 12.85/6.97 ! [X: complex,Y4: complex,E2: real] :
% 12.85/6.97 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) ) @ E2 )
% 12.85/6.97 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y4 ) ) @ E2 ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3980_norm__not__less__zero,axiom,
% 12.85/6.97 ! [X: complex] :
% 12.85/6.97 ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3981_norm__ge__zero,axiom,
% 12.85/6.97 ! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3982_norm__mult,axiom,
% 12.85/6.97 ! [X: real,Y4: real] :
% 12.85/6.97 ( ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y4 ) )
% 12.85/6.97 = ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3983_norm__mult,axiom,
% 12.85/6.97 ! [X: complex,Y4: complex] :
% 12.85/6.97 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y4 ) )
% 12.85/6.97 = ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3984_norm__divide,axiom,
% 12.85/6.97 ! [A: real,B: real] :
% 12.85/6.97 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 12.85/6.97 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3985_norm__divide,axiom,
% 12.85/6.97 ! [A: complex,B: complex] :
% 12.85/6.97 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 12.85/6.97 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3986_norm__diff__triangle__ineq,axiom,
% 12.85/6.97 ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3987_norm__diff__triangle__ineq,axiom,
% 12.85/6.97 ! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3988_nonzero__norm__divide,axiom,
% 12.85/6.97 ! [B: real,A: real] :
% 12.85/6.97 ( ( B != zero_zero_real )
% 12.85/6.97 => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 12.85/6.97 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3989_nonzero__norm__divide,axiom,
% 12.85/6.97 ! [B: complex,A: complex] :
% 12.85/6.97 ( ( B != zero_zero_complex )
% 12.85/6.97 => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 12.85/6.97 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3990_power__eq__imp__eq__norm,axiom,
% 12.85/6.97 ! [W: real,N: nat,Z: real] :
% 12.85/6.97 ( ( ( power_power_real @ W @ N )
% 12.85/6.97 = ( power_power_real @ Z @ N ) )
% 12.85/6.97 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.97 => ( ( real_V7735802525324610683m_real @ W )
% 12.85/6.97 = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3991_power__eq__imp__eq__norm,axiom,
% 12.85/6.97 ! [W: complex,N: nat,Z: complex] :
% 12.85/6.97 ( ( ( power_power_complex @ W @ N )
% 12.85/6.97 = ( power_power_complex @ Z @ N ) )
% 12.85/6.97 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.97 => ( ( real_V1022390504157884413omplex @ W )
% 12.85/6.97 = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3992_norm__mult__less,axiom,
% 12.85/6.97 ! [X: real,R2: real,Y4: real,S2: real] :
% 12.85/6.97 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
% 12.85/6.97 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y4 ) @ S2 )
% 12.85/6.97 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y4 ) ) @ ( times_times_real @ R2 @ S2 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3993_norm__mult__less,axiom,
% 12.85/6.97 ! [X: complex,R2: real,Y4: complex,S2: real] :
% 12.85/6.97 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
% 12.85/6.97 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y4 ) @ S2 )
% 12.85/6.97 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y4 ) ) @ ( times_times_real @ R2 @ S2 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3994_norm__mult__ineq,axiom,
% 12.85/6.97 ! [X: real,Y4: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y4 ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3995_norm__mult__ineq,axiom,
% 12.85/6.97 ! [X: complex,Y4: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y4 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3996_norm__power__ineq,axiom,
% 12.85/6.97 ! [X: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3997_norm__power__ineq,axiom,
% 12.85/6.97 ! [X: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3998_norm__diff__ineq,axiom,
% 12.85/6.97 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_3999_norm__diff__ineq,axiom,
% 12.85/6.97 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4000_norm__triangle__ineq2,axiom,
% 12.85/6.97 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4001_norm__triangle__ineq2,axiom,
% 12.85/6.97 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4002_power__eq__1__iff,axiom,
% 12.85/6.97 ! [W: real,N: nat] :
% 12.85/6.97 ( ( ( power_power_real @ W @ N )
% 12.85/6.97 = one_one_real )
% 12.85/6.97 => ( ( ( real_V7735802525324610683m_real @ W )
% 12.85/6.97 = one_one_real )
% 12.85/6.97 | ( N = zero_zero_nat ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4003_power__eq__1__iff,axiom,
% 12.85/6.97 ! [W: complex,N: nat] :
% 12.85/6.97 ( ( ( power_power_complex @ W @ N )
% 12.85/6.97 = one_one_complex )
% 12.85/6.97 => ( ( ( real_V1022390504157884413omplex @ W )
% 12.85/6.97 = one_one_real )
% 12.85/6.97 | ( N = zero_zero_nat ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4004_square__norm__one,axiom,
% 12.85/6.97 ! [X: real] :
% 12.85/6.97 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = one_one_real )
% 12.85/6.97 => ( ( real_V7735802525324610683m_real @ X )
% 12.85/6.97 = one_one_real ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4005_square__norm__one,axiom,
% 12.85/6.97 ! [X: complex] :
% 12.85/6.97 ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = one_one_complex )
% 12.85/6.97 => ( ( real_V1022390504157884413omplex @ X )
% 12.85/6.97 = one_one_real ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4006_norm__power__diff,axiom,
% 12.85/6.97 ! [Z: real,W: real,M: nat] :
% 12.85/6.97 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 12.85/6.97 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
% 12.85/6.97 => ( 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 ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4007_norm__power__diff,axiom,
% 12.85/6.97 ! [Z: complex,W: complex,M: nat] :
% 12.85/6.97 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 12.85/6.97 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
% 12.85/6.97 => ( 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 ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4008_arcosh__1,axiom,
% 12.85/6.97 ( ( arcosh_real @ one_one_real )
% 12.85/6.97 = zero_zero_real ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4009_artanh__0,axiom,
% 12.85/6.97 ( ( artanh_real @ zero_zero_real )
% 12.85/6.97 = zero_zero_real ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4010_arsinh__0,axiom,
% 12.85/6.97 ( ( arsinh_real @ zero_zero_real )
% 12.85/6.97 = zero_zero_real ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4011_low__def,axiom,
% 12.85/6.97 ( vEBT_VEBT_low
% 12.85/6.97 = ( ^ [X3: nat,N4: nat] : ( modulo_modulo_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4012_pochhammer__double,axiom,
% 12.85/6.97 ! [Z: complex,N: nat] :
% 12.85/6.97 ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.97 = ( 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 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4013_pochhammer__double,axiom,
% 12.85/6.97 ! [Z: real,N: nat] :
% 12.85/6.97 ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.97 = ( 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 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4014_pochhammer__double,axiom,
% 12.85/6.97 ! [Z: rat,N: nat] :
% 12.85/6.97 ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.97 = ( 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 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4015_central__binomial__lower__bound,axiom,
% 12.85/6.97 ! [N: nat] :
% 12.85/6.97 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.97 => ( 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 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4016_flip__bit__0,axiom,
% 12.85/6.97 ! [A: int] :
% 12.85/6.97 ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 12.85/6.97 = ( 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 ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4017_flip__bit__0,axiom,
% 12.85/6.97 ! [A: nat] :
% 12.85/6.97 ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 12.85/6.97 = ( 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 ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4018_flip__bit__0,axiom,
% 12.85/6.97 ! [A: code_integer] :
% 12.85/6.97 ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 12.85/6.97 = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4019_mod__mod__trivial,axiom,
% 12.85/6.97 ! [A: nat,B: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 12.85/6.97 = ( modulo_modulo_nat @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4020_mod__mod__trivial,axiom,
% 12.85/6.97 ! [A: int,B: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 12.85/6.97 = ( modulo_modulo_int @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4021_mod__mod__trivial,axiom,
% 12.85/6.97 ! [A: code_integer,B: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 12.85/6.97 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4022_mod__self,axiom,
% 12.85/6.97 ! [A: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ A @ A )
% 12.85/6.97 = zero_zero_nat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4023_mod__self,axiom,
% 12.85/6.97 ! [A: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ A @ A )
% 12.85/6.97 = zero_zero_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4024_mod__self,axiom,
% 12.85/6.97 ! [A: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ A @ A )
% 12.85/6.97 = zero_z3403309356797280102nteger ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4025_mod__by__0,axiom,
% 12.85/6.97 ! [A: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ A @ zero_zero_nat )
% 12.85/6.97 = A ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4026_mod__by__0,axiom,
% 12.85/6.97 ! [A: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ A @ zero_zero_int )
% 12.85/6.97 = A ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4027_mod__by__0,axiom,
% 12.85/6.97 ! [A: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
% 12.85/6.97 = A ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4028_mod__0,axiom,
% 12.85/6.97 ! [A: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 12.85/6.97 = zero_zero_nat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4029_mod__0,axiom,
% 12.85/6.97 ! [A: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 12.85/6.97 = zero_zero_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4030_mod__0,axiom,
% 12.85/6.97 ! [A: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 12.85/6.97 = zero_z3403309356797280102nteger ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4031_bits__mod__0,axiom,
% 12.85/6.97 ! [A: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 12.85/6.97 = zero_zero_nat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4032_bits__mod__0,axiom,
% 12.85/6.97 ! [A: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 12.85/6.97 = zero_zero_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4033_bits__mod__0,axiom,
% 12.85/6.97 ! [A: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 12.85/6.97 = zero_z3403309356797280102nteger ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4034_mod__add__self2,axiom,
% 12.85/6.97 ! [A: nat,B: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 12.85/6.97 = ( modulo_modulo_nat @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4035_mod__add__self2,axiom,
% 12.85/6.97 ! [A: int,B: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
% 12.85/6.97 = ( modulo_modulo_int @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4036_mod__add__self2,axiom,
% 12.85/6.97 ! [A: code_integer,B: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
% 12.85/6.97 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4037_mod__add__self1,axiom,
% 12.85/6.97 ! [B: nat,A: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 12.85/6.97 = ( modulo_modulo_nat @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4038_mod__add__self1,axiom,
% 12.85/6.97 ! [B: int,A: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
% 12.85/6.97 = ( modulo_modulo_int @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4039_mod__add__self1,axiom,
% 12.85/6.97 ! [B: code_integer,A: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
% 12.85/6.97 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4040_minus__mod__self2,axiom,
% 12.85/6.97 ! [A: int,B: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
% 12.85/6.97 = ( modulo_modulo_int @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4041_minus__mod__self2,axiom,
% 12.85/6.97 ! [A: code_integer,B: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
% 12.85/6.97 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4042_of__bool__less__eq__iff,axiom,
% 12.85/6.97 ! [P: $o,Q: $o] :
% 12.85/6.97 ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 12.85/6.97 = ( P
% 12.85/6.97 => Q ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4043_of__bool__less__eq__iff,axiom,
% 12.85/6.97 ! [P: $o,Q: $o] :
% 12.85/6.97 ( ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 12.85/6.97 = ( P
% 12.85/6.97 => Q ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4044_of__bool__less__eq__iff,axiom,
% 12.85/6.97 ! [P: $o,Q: $o] :
% 12.85/6.97 ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 12.85/6.97 = ( P
% 12.85/6.97 => Q ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4045_of__bool__less__eq__iff,axiom,
% 12.85/6.97 ! [P: $o,Q: $o] :
% 12.85/6.97 ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 12.85/6.97 = ( P
% 12.85/6.97 => Q ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4046_of__bool__less__eq__iff,axiom,
% 12.85/6.97 ! [P: $o,Q: $o] :
% 12.85/6.97 ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 12.85/6.97 = ( P
% 12.85/6.97 => Q ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4047_of__bool__eq__0__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ( zero_n1201886186963655149omplex @ P )
% 12.85/6.97 = zero_zero_complex )
% 12.85/6.97 = ~ P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4048_of__bool__eq__0__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ( zero_n3304061248610475627l_real @ P )
% 12.85/6.97 = zero_zero_real )
% 12.85/6.97 = ~ P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4049_of__bool__eq__0__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 12.85/6.97 = zero_zero_rat )
% 12.85/6.97 = ~ P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4050_of__bool__eq__0__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 12.85/6.97 = zero_zero_nat )
% 12.85/6.97 = ~ P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4051_of__bool__eq__0__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ( zero_n2684676970156552555ol_int @ P )
% 12.85/6.97 = zero_zero_int )
% 12.85/6.97 = ~ P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4052_of__bool__eq__0__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ( zero_n356916108424825756nteger @ P )
% 12.85/6.97 = zero_z3403309356797280102nteger )
% 12.85/6.97 = ~ P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4053_of__bool__eq_I1_J,axiom,
% 12.85/6.97 ( ( zero_n1201886186963655149omplex @ $false )
% 12.85/6.97 = zero_zero_complex ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4054_of__bool__eq_I1_J,axiom,
% 12.85/6.97 ( ( zero_n3304061248610475627l_real @ $false )
% 12.85/6.97 = zero_zero_real ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4055_of__bool__eq_I1_J,axiom,
% 12.85/6.97 ( ( zero_n2052037380579107095ol_rat @ $false )
% 12.85/6.97 = zero_zero_rat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4056_of__bool__eq_I1_J,axiom,
% 12.85/6.97 ( ( zero_n2687167440665602831ol_nat @ $false )
% 12.85/6.97 = zero_zero_nat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4057_of__bool__eq_I1_J,axiom,
% 12.85/6.97 ( ( zero_n2684676970156552555ol_int @ $false )
% 12.85/6.97 = zero_zero_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4058_of__bool__eq_I1_J,axiom,
% 12.85/6.97 ( ( zero_n356916108424825756nteger @ $false )
% 12.85/6.97 = zero_z3403309356797280102nteger ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4059_of__bool__less__iff,axiom,
% 12.85/6.97 ! [P: $o,Q: $o] :
% 12.85/6.97 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 12.85/6.97 = ( ~ P
% 12.85/6.97 & Q ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4060_of__bool__less__iff,axiom,
% 12.85/6.97 ! [P: $o,Q: $o] :
% 12.85/6.97 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 12.85/6.97 = ( ~ P
% 12.85/6.97 & Q ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4061_of__bool__less__iff,axiom,
% 12.85/6.97 ! [P: $o,Q: $o] :
% 12.85/6.97 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 12.85/6.97 = ( ~ P
% 12.85/6.97 & Q ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4062_of__bool__less__iff,axiom,
% 12.85/6.97 ! [P: $o,Q: $o] :
% 12.85/6.97 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 12.85/6.97 = ( ~ P
% 12.85/6.97 & Q ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4063_of__bool__less__iff,axiom,
% 12.85/6.97 ! [P: $o,Q: $o] :
% 12.85/6.97 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 12.85/6.97 = ( ~ P
% 12.85/6.97 & Q ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4064_of__bool__eq__1__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ( zero_n1201886186963655149omplex @ P )
% 12.85/6.97 = one_one_complex )
% 12.85/6.97 = P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4065_of__bool__eq__1__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ( zero_n3304061248610475627l_real @ P )
% 12.85/6.97 = one_one_real )
% 12.85/6.97 = P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4066_of__bool__eq__1__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 12.85/6.97 = one_one_rat )
% 12.85/6.97 = P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4067_of__bool__eq__1__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 12.85/6.97 = one_one_nat )
% 12.85/6.97 = P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4068_of__bool__eq__1__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ( zero_n2684676970156552555ol_int @ P )
% 12.85/6.97 = one_one_int )
% 12.85/6.97 = P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4069_of__bool__eq__1__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ( zero_n356916108424825756nteger @ P )
% 12.85/6.97 = one_one_Code_integer )
% 12.85/6.97 = P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4070_of__bool__eq_I2_J,axiom,
% 12.85/6.97 ( ( zero_n1201886186963655149omplex @ $true )
% 12.85/6.97 = one_one_complex ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4071_of__bool__eq_I2_J,axiom,
% 12.85/6.97 ( ( zero_n3304061248610475627l_real @ $true )
% 12.85/6.97 = one_one_real ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4072_of__bool__eq_I2_J,axiom,
% 12.85/6.97 ( ( zero_n2052037380579107095ol_rat @ $true )
% 12.85/6.97 = one_one_rat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4073_of__bool__eq_I2_J,axiom,
% 12.85/6.97 ( ( zero_n2687167440665602831ol_nat @ $true )
% 12.85/6.97 = one_one_nat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4074_of__bool__eq_I2_J,axiom,
% 12.85/6.97 ( ( zero_n2684676970156552555ol_int @ $true )
% 12.85/6.97 = one_one_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4075_of__bool__eq_I2_J,axiom,
% 12.85/6.97 ( ( zero_n356916108424825756nteger @ $true )
% 12.85/6.97 = one_one_Code_integer ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4076_mod__less,axiom,
% 12.85/6.97 ! [M: nat,N: nat] :
% 12.85/6.97 ( ( ord_less_nat @ M @ N )
% 12.85/6.97 => ( ( modulo_modulo_nat @ M @ N )
% 12.85/6.97 = M ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4077_of__nat__of__bool,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
% 12.85/6.97 = ( zero_n1201886186963655149omplex @ P ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4078_of__nat__of__bool,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
% 12.85/6.97 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4079_of__nat__of__bool,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 12.85/6.97 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4080_of__nat__of__bool,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 12.85/6.97 = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4081_of__nat__of__bool,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
% 12.85/6.97 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4082_of__nat__of__bool,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
% 12.85/6.97 = ( zero_n356916108424825756nteger @ P ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4083_mod__mult__self1__is__0,axiom,
% 12.85/6.97 ! [B: nat,A: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
% 12.85/6.97 = zero_zero_nat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4084_mod__mult__self1__is__0,axiom,
% 12.85/6.97 ! [B: int,A: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
% 12.85/6.97 = zero_zero_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4085_mod__mult__self1__is__0,axiom,
% 12.85/6.97 ! [B: code_integer,A: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
% 12.85/6.97 = zero_z3403309356797280102nteger ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4086_mod__mult__self2__is__0,axiom,
% 12.85/6.97 ! [A: nat,B: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
% 12.85/6.97 = zero_zero_nat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4087_mod__mult__self2__is__0,axiom,
% 12.85/6.97 ! [A: int,B: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
% 12.85/6.97 = zero_zero_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4088_mod__mult__self2__is__0,axiom,
% 12.85/6.97 ! [A: code_integer,B: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 12.85/6.97 = zero_z3403309356797280102nteger ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4089_mod__by__1,axiom,
% 12.85/6.97 ! [A: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 12.85/6.97 = zero_zero_nat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4090_mod__by__1,axiom,
% 12.85/6.97 ! [A: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ A @ one_one_int )
% 12.85/6.97 = zero_zero_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4091_mod__by__1,axiom,
% 12.85/6.97 ! [A: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 12.85/6.97 = zero_z3403309356797280102nteger ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4092_bits__mod__by__1,axiom,
% 12.85/6.97 ! [A: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 12.85/6.97 = zero_zero_nat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4093_bits__mod__by__1,axiom,
% 12.85/6.97 ! [A: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ A @ one_one_int )
% 12.85/6.97 = zero_zero_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4094_bits__mod__by__1,axiom,
% 12.85/6.97 ! [A: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 12.85/6.97 = zero_z3403309356797280102nteger ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4095_bits__mod__div__trivial,axiom,
% 12.85/6.97 ! [A: nat,B: nat] :
% 12.85/6.97 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 12.85/6.97 = zero_zero_nat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4096_bits__mod__div__trivial,axiom,
% 12.85/6.97 ! [A: int,B: int] :
% 12.85/6.97 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 12.85/6.97 = zero_zero_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4097_bits__mod__div__trivial,axiom,
% 12.85/6.97 ! [A: code_integer,B: code_integer] :
% 12.85/6.97 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 12.85/6.97 = zero_z3403309356797280102nteger ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4098_mod__div__trivial,axiom,
% 12.85/6.97 ! [A: nat,B: nat] :
% 12.85/6.97 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 12.85/6.97 = zero_zero_nat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4099_mod__div__trivial,axiom,
% 12.85/6.97 ! [A: int,B: int] :
% 12.85/6.97 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 12.85/6.97 = zero_zero_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4100_mod__div__trivial,axiom,
% 12.85/6.97 ! [A: code_integer,B: code_integer] :
% 12.85/6.97 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 12.85/6.97 = zero_z3403309356797280102nteger ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4101_mod__mult__self4,axiom,
% 12.85/6.97 ! [B: nat,C: nat,A: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 12.85/6.97 = ( modulo_modulo_nat @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4102_mod__mult__self4,axiom,
% 12.85/6.97 ! [B: int,C: int,A: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 12.85/6.97 = ( modulo_modulo_int @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4103_mod__mult__self4,axiom,
% 12.85/6.97 ! [B: code_integer,C: code_integer,A: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
% 12.85/6.97 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4104_mod__mult__self3,axiom,
% 12.85/6.97 ! [C: nat,B: nat,A: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 12.85/6.97 = ( modulo_modulo_nat @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4105_mod__mult__self3,axiom,
% 12.85/6.97 ! [C: int,B: int,A: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 12.85/6.97 = ( modulo_modulo_int @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4106_mod__mult__self3,axiom,
% 12.85/6.97 ! [C: code_integer,B: code_integer,A: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
% 12.85/6.97 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4107_mod__mult__self2,axiom,
% 12.85/6.97 ! [A: nat,B: nat,C: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 12.85/6.97 = ( modulo_modulo_nat @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4108_mod__mult__self2,axiom,
% 12.85/6.97 ! [A: int,B: int,C: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 12.85/6.97 = ( modulo_modulo_int @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4109_mod__mult__self2,axiom,
% 12.85/6.97 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
% 12.85/6.97 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4110_mod__mult__self1,axiom,
% 12.85/6.97 ! [A: nat,C: nat,B: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 12.85/6.97 = ( modulo_modulo_nat @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4111_mod__mult__self1,axiom,
% 12.85/6.97 ! [A: int,C: int,B: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 12.85/6.97 = ( modulo_modulo_int @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4112_mod__mult__self1,axiom,
% 12.85/6.97 ! [A: code_integer,C: code_integer,B: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
% 12.85/6.97 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4113_dvd__imp__mod__0,axiom,
% 12.85/6.97 ! [A: nat,B: nat] :
% 12.85/6.97 ( ( dvd_dvd_nat @ A @ B )
% 12.85/6.97 => ( ( modulo_modulo_nat @ B @ A )
% 12.85/6.97 = zero_zero_nat ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4114_dvd__imp__mod__0,axiom,
% 12.85/6.97 ! [A: int,B: int] :
% 12.85/6.97 ( ( dvd_dvd_int @ A @ B )
% 12.85/6.97 => ( ( modulo_modulo_int @ B @ A )
% 12.85/6.97 = zero_zero_int ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4115_dvd__imp__mod__0,axiom,
% 12.85/6.97 ! [A: code_integer,B: code_integer] :
% 12.85/6.97 ( ( dvd_dvd_Code_integer @ A @ B )
% 12.85/6.97 => ( ( modulo364778990260209775nteger @ B @ A )
% 12.85/6.97 = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4116_zero__less__of__bool__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 12.85/6.97 = P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4117_zero__less__of__bool__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 12.85/6.97 = P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4118_zero__less__of__bool__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 12.85/6.97 = P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4119_zero__less__of__bool__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 12.85/6.97 = P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4120_zero__less__of__bool__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
% 12.85/6.97 = P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4121_of__bool__less__one__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 12.85/6.97 = ~ P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4122_of__bool__less__one__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 12.85/6.97 = ~ P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4123_of__bool__less__one__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 12.85/6.97 = ~ P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4124_of__bool__less__one__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 12.85/6.97 = ~ P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4125_of__bool__less__one__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
% 12.85/6.97 = ~ P ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4126_of__bool__not__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( zero_n1201886186963655149omplex @ ~ P )
% 12.85/6.97 = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4127_of__bool__not__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( zero_n3304061248610475627l_real @ ~ P )
% 12.85/6.97 = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4128_of__bool__not__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( zero_n2052037380579107095ol_rat @ ~ P )
% 12.85/6.97 = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4129_of__bool__not__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( zero_n2684676970156552555ol_int @ ~ P )
% 12.85/6.97 = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4130_of__bool__not__iff,axiom,
% 12.85/6.97 ! [P: $o] :
% 12.85/6.97 ( ( zero_n356916108424825756nteger @ ~ P )
% 12.85/6.97 = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4131_Suc__0__mod__eq,axiom,
% 12.85/6.97 ! [N: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.85/6.97 = ( zero_n2687167440665602831ol_nat
% 12.85/6.97 @ ( N
% 12.85/6.97 != ( suc @ zero_zero_nat ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4132_mod__by__Suc__0,axiom,
% 12.85/6.97 ! [M: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 12.85/6.97 = zero_zero_nat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4133_pochhammer__0,axiom,
% 12.85/6.97 ! [A: complex] :
% 12.85/6.97 ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
% 12.85/6.97 = one_one_complex ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4134_pochhammer__0,axiom,
% 12.85/6.97 ! [A: real] :
% 12.85/6.97 ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
% 12.85/6.97 = one_one_real ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4135_pochhammer__0,axiom,
% 12.85/6.97 ! [A: rat] :
% 12.85/6.97 ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
% 12.85/6.97 = one_one_rat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4136_pochhammer__0,axiom,
% 12.85/6.97 ! [A: nat] :
% 12.85/6.97 ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
% 12.85/6.97 = one_one_nat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4137_pochhammer__0,axiom,
% 12.85/6.97 ! [A: int] :
% 12.85/6.97 ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
% 12.85/6.97 = one_one_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4138_Suc__mod__mult__self1,axiom,
% 12.85/6.97 ! [M: nat,K: nat,N: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
% 12.85/6.97 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4139_Suc__mod__mult__self2,axiom,
% 12.85/6.97 ! [M: nat,N: nat,K: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
% 12.85/6.97 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4140_Suc__mod__mult__self3,axiom,
% 12.85/6.97 ! [K: nat,N: nat,M: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
% 12.85/6.97 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4141_Suc__mod__mult__self4,axiom,
% 12.85/6.97 ! [N: nat,K: nat,M: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
% 12.85/6.97 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4142_bits__one__mod__two__eq__one,axiom,
% 12.85/6.97 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = one_one_nat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4143_bits__one__mod__two__eq__one,axiom,
% 12.85/6.97 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.97 = one_one_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4144_bits__one__mod__two__eq__one,axiom,
% 12.85/6.97 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.97 = one_one_Code_integer ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4145_one__mod__two__eq__one,axiom,
% 12.85/6.97 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = one_one_nat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4146_one__mod__two__eq__one,axiom,
% 12.85/6.97 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.97 = one_one_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4147_one__mod__two__eq__one,axiom,
% 12.85/6.97 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.97 = one_one_Code_integer ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4148_even__mod__2__iff,axiom,
% 12.85/6.97 ! [A: nat] :
% 12.85/6.97 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.97 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4149_even__mod__2__iff,axiom,
% 12.85/6.97 ! [A: int] :
% 12.85/6.97 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 12.85/6.97 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4150_even__mod__2__iff,axiom,
% 12.85/6.97 ! [A: code_integer] :
% 12.85/6.97 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 12.85/6.97 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4151_odd__of__bool__self,axiom,
% 12.85/6.97 ! [P5: $o] :
% 12.85/6.97 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P5 ) ) )
% 12.85/6.97 = P5 ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4152_odd__of__bool__self,axiom,
% 12.85/6.97 ! [P5: $o] :
% 12.85/6.97 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P5 ) ) )
% 12.85/6.97 = P5 ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4153_odd__of__bool__self,axiom,
% 12.85/6.97 ! [P5: $o] :
% 12.85/6.97 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P5 ) ) )
% 12.85/6.97 = P5 ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4154_mod2__Suc__Suc,axiom,
% 12.85/6.97 ! [M: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4155_Suc__times__numeral__mod__eq,axiom,
% 12.85/6.97 ! [K: num,N: nat] :
% 12.85/6.97 ( ( ( numeral_numeral_nat @ K )
% 12.85/6.97 != one_one_nat )
% 12.85/6.97 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
% 12.85/6.97 = one_one_nat ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4156_not__mod__2__eq__1__eq__0,axiom,
% 12.85/6.97 ! [A: nat] :
% 12.85/6.97 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 != one_one_nat )
% 12.85/6.97 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = zero_zero_nat ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4157_not__mod__2__eq__1__eq__0,axiom,
% 12.85/6.97 ! [A: int] :
% 12.85/6.97 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.97 != one_one_int )
% 12.85/6.97 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.97 = zero_zero_int ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4158_not__mod__2__eq__1__eq__0,axiom,
% 12.85/6.97 ! [A: code_integer] :
% 12.85/6.97 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.97 != one_one_Code_integer )
% 12.85/6.97 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.97 = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4159_not__mod__2__eq__0__eq__1,axiom,
% 12.85/6.97 ! [A: nat] :
% 12.85/6.97 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 != zero_zero_nat )
% 12.85/6.97 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = one_one_nat ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4160_not__mod__2__eq__0__eq__1,axiom,
% 12.85/6.97 ! [A: int] :
% 12.85/6.97 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.97 != zero_zero_int )
% 12.85/6.97 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.97 = one_one_int ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4161_not__mod__2__eq__0__eq__1,axiom,
% 12.85/6.97 ! [A: code_integer] :
% 12.85/6.97 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.97 != zero_z3403309356797280102nteger )
% 12.85/6.97 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.97 = one_one_Code_integer ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4162_of__bool__half__eq__0,axiom,
% 12.85/6.97 ! [B: $o] :
% 12.85/6.97 ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = zero_zero_nat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4163_of__bool__half__eq__0,axiom,
% 12.85/6.97 ! [B: $o] :
% 12.85/6.97 ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.97 = zero_zero_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4164_of__bool__half__eq__0,axiom,
% 12.85/6.97 ! [B: $o] :
% 12.85/6.97 ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.97 = zero_z3403309356797280102nteger ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4165_not__mod2__eq__Suc__0__eq__0,axiom,
% 12.85/6.97 ! [N: nat] :
% 12.85/6.97 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 != ( suc @ zero_zero_nat ) )
% 12.85/6.97 = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = zero_zero_nat ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4166_add__self__mod__2,axiom,
% 12.85/6.97 ! [M: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = zero_zero_nat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4167_Suc__mod__eq__add3__mod__numeral,axiom,
% 12.85/6.97 ! [M: nat,V: num] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 12.85/6.97 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4168_mod__Suc__eq__mod__add3,axiom,
% 12.85/6.97 ! [M: nat,N: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 12.85/6.97 = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4169_mod2__gr__0,axiom,
% 12.85/6.97 ! [M: nat] :
% 12.85/6.97 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.97 = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.97 = one_one_nat ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4170_bits__1__div__exp,axiom,
% 12.85/6.97 ! [N: nat] :
% 12.85/6.97 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.97 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4171_bits__1__div__exp,axiom,
% 12.85/6.97 ! [N: nat] :
% 12.85/6.97 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.97 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4172_bits__1__div__exp,axiom,
% 12.85/6.97 ! [N: nat] :
% 12.85/6.97 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.97 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4173_one__div__2__pow__eq,axiom,
% 12.85/6.97 ! [N: nat] :
% 12.85/6.97 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.97 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4174_one__div__2__pow__eq,axiom,
% 12.85/6.97 ! [N: nat] :
% 12.85/6.97 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.97 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4175_one__div__2__pow__eq,axiom,
% 12.85/6.97 ! [N: nat] :
% 12.85/6.97 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.97 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4176_one__mod__2__pow__eq,axiom,
% 12.85/6.97 ! [N: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.97 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4177_one__mod__2__pow__eq,axiom,
% 12.85/6.97 ! [N: nat] :
% 12.85/6.97 ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.97 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4178_one__mod__2__pow__eq,axiom,
% 12.85/6.97 ! [N: nat] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.97 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4179_even__succ__mod__exp,axiom,
% 12.85/6.97 ! [A: nat,N: nat] :
% 12.85/6.97 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.97 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.97 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.97 = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4180_even__succ__mod__exp,axiom,
% 12.85/6.97 ! [A: int,N: nat] :
% 12.85/6.97 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.97 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.97 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.97 = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4181_even__succ__mod__exp,axiom,
% 12.85/6.97 ! [A: code_integer,N: nat] :
% 12.85/6.97 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.97 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.97 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.97 = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4182_of__bool__eq__iff,axiom,
% 12.85/6.97 ! [P5: $o,Q2: $o] :
% 12.85/6.97 ( ( ( zero_n2687167440665602831ol_nat @ P5 )
% 12.85/6.97 = ( zero_n2687167440665602831ol_nat @ Q2 ) )
% 12.85/6.97 = ( P5 = Q2 ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4183_of__bool__eq__iff,axiom,
% 12.85/6.97 ! [P5: $o,Q2: $o] :
% 12.85/6.97 ( ( ( zero_n2684676970156552555ol_int @ P5 )
% 12.85/6.97 = ( zero_n2684676970156552555ol_int @ Q2 ) )
% 12.85/6.97 = ( P5 = Q2 ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4184_of__bool__eq__iff,axiom,
% 12.85/6.97 ! [P5: $o,Q2: $o] :
% 12.85/6.97 ( ( ( zero_n356916108424825756nteger @ P5 )
% 12.85/6.97 = ( zero_n356916108424825756nteger @ Q2 ) )
% 12.85/6.97 = ( P5 = Q2 ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4185_of__nat__mod,axiom,
% 12.85/6.97 ! [M: nat,N: nat] :
% 12.85/6.97 ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
% 12.85/6.97 = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4186_of__nat__mod,axiom,
% 12.85/6.97 ! [M: nat,N: nat] :
% 12.85/6.97 ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
% 12.85/6.97 = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4187_of__nat__mod,axiom,
% 12.85/6.97 ! [M: nat,N: nat] :
% 12.85/6.97 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
% 12.85/6.97 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4188_pochhammer__of__nat,axiom,
% 12.85/6.97 ! [X: nat,N: nat] :
% 12.85/6.97 ( ( comm_s2602460028002588243omplex @ ( semiri8010041392384452111omplex @ X ) @ N )
% 12.85/6.97 = ( semiri8010041392384452111omplex @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4189_pochhammer__of__nat,axiom,
% 12.85/6.97 ! [X: nat,N: nat] :
% 12.85/6.97 ( ( comm_s7457072308508201937r_real @ ( semiri5074537144036343181t_real @ X ) @ N )
% 12.85/6.97 = ( semiri5074537144036343181t_real @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4190_pochhammer__of__nat,axiom,
% 12.85/6.97 ! [X: nat,N: nat] :
% 12.85/6.97 ( ( comm_s4028243227959126397er_rat @ ( semiri681578069525770553at_rat @ X ) @ N )
% 12.85/6.97 = ( semiri681578069525770553at_rat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4191_pochhammer__of__nat,axiom,
% 12.85/6.97 ! [X: nat,N: nat] :
% 12.85/6.97 ( ( comm_s4663373288045622133er_nat @ ( semiri1316708129612266289at_nat @ X ) @ N )
% 12.85/6.97 = ( semiri1316708129612266289at_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4192_pochhammer__of__nat,axiom,
% 12.85/6.97 ! [X: nat,N: nat] :
% 12.85/6.97 ( ( comm_s4660882817536571857er_int @ ( semiri1314217659103216013at_int @ X ) @ N )
% 12.85/6.97 = ( semiri1314217659103216013at_int @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4193_mod__mult__right__eq,axiom,
% 12.85/6.97 ! [A: nat,B: nat,C: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 12.85/6.97 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4194_mod__mult__right__eq,axiom,
% 12.85/6.97 ! [A: int,B: int,C: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 12.85/6.97 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4195_mod__mult__right__eq,axiom,
% 12.85/6.97 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4196_mod__mult__left__eq,axiom,
% 12.85/6.97 ! [A: nat,C: nat,B: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 12.85/6.97 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4197_mod__mult__left__eq,axiom,
% 12.85/6.97 ! [A: int,C: int,B: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 12.85/6.97 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4198_mod__mult__left__eq,axiom,
% 12.85/6.97 ! [A: code_integer,C: code_integer,B: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4199_mult__mod__right,axiom,
% 12.85/6.97 ! [C: nat,A: nat,B: nat] :
% 12.85/6.97 ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 12.85/6.97 = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4200_mult__mod__right,axiom,
% 12.85/6.97 ! [C: int,A: int,B: int] :
% 12.85/6.97 ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 12.85/6.97 = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4201_mult__mod__right,axiom,
% 12.85/6.97 ! [C: code_integer,A: code_integer,B: code_integer] :
% 12.85/6.97 ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 12.85/6.97 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4202_mod__mult__mult2,axiom,
% 12.85/6.97 ! [A: nat,C: nat,B: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 12.85/6.97 = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4203_mod__mult__mult2,axiom,
% 12.85/6.97 ! [A: int,C: int,B: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 12.85/6.97 = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4204_mod__mult__mult2,axiom,
% 12.85/6.97 ! [A: code_integer,C: code_integer,B: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 12.85/6.97 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4205_mod__mult__cong,axiom,
% 12.85/6.97 ! [A: nat,C: nat,A6: nat,B: nat,B6: nat] :
% 12.85/6.97 ( ( ( modulo_modulo_nat @ A @ C )
% 12.85/6.97 = ( modulo_modulo_nat @ A6 @ C ) )
% 12.85/6.97 => ( ( ( modulo_modulo_nat @ B @ C )
% 12.85/6.97 = ( modulo_modulo_nat @ B6 @ C ) )
% 12.85/6.97 => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
% 12.85/6.97 = ( modulo_modulo_nat @ ( times_times_nat @ A6 @ B6 ) @ C ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4206_mod__mult__cong,axiom,
% 12.85/6.97 ! [A: int,C: int,A6: int,B: int,B6: int] :
% 12.85/6.97 ( ( ( modulo_modulo_int @ A @ C )
% 12.85/6.97 = ( modulo_modulo_int @ A6 @ C ) )
% 12.85/6.97 => ( ( ( modulo_modulo_int @ B @ C )
% 12.85/6.97 = ( modulo_modulo_int @ B6 @ C ) )
% 12.85/6.97 => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
% 12.85/6.97 = ( modulo_modulo_int @ ( times_times_int @ A6 @ B6 ) @ C ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4207_mod__mult__cong,axiom,
% 12.85/6.97 ! [A: code_integer,C: code_integer,A6: code_integer,B: code_integer,B6: code_integer] :
% 12.85/6.97 ( ( ( modulo364778990260209775nteger @ A @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ A6 @ C ) )
% 12.85/6.97 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ B6 @ C ) )
% 12.85/6.97 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A6 @ B6 ) @ C ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4208_mod__mult__eq,axiom,
% 12.85/6.97 ! [A: nat,C: nat,B: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 12.85/6.97 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4209_mod__mult__eq,axiom,
% 12.85/6.97 ! [A: int,C: int,B: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 12.85/6.97 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4210_mod__mult__eq,axiom,
% 12.85/6.97 ! [A: code_integer,C: code_integer,B: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4211_mod__add__right__eq,axiom,
% 12.85/6.97 ! [A: nat,B: nat,C: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 12.85/6.97 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4212_mod__add__right__eq,axiom,
% 12.85/6.97 ! [A: int,B: int,C: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 12.85/6.97 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4213_mod__add__right__eq,axiom,
% 12.85/6.97 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4214_mod__add__left__eq,axiom,
% 12.85/6.97 ! [A: nat,C: nat,B: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 12.85/6.97 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4215_mod__add__left__eq,axiom,
% 12.85/6.97 ! [A: int,C: int,B: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 12.85/6.97 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4216_mod__add__left__eq,axiom,
% 12.85/6.97 ! [A: code_integer,C: code_integer,B: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4217_mod__add__cong,axiom,
% 12.85/6.97 ! [A: nat,C: nat,A6: nat,B: nat,B6: nat] :
% 12.85/6.97 ( ( ( modulo_modulo_nat @ A @ C )
% 12.85/6.97 = ( modulo_modulo_nat @ A6 @ C ) )
% 12.85/6.97 => ( ( ( modulo_modulo_nat @ B @ C )
% 12.85/6.97 = ( modulo_modulo_nat @ B6 @ C ) )
% 12.85/6.97 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 12.85/6.97 = ( modulo_modulo_nat @ ( plus_plus_nat @ A6 @ B6 ) @ C ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4218_mod__add__cong,axiom,
% 12.85/6.97 ! [A: int,C: int,A6: int,B: int,B6: int] :
% 12.85/6.97 ( ( ( modulo_modulo_int @ A @ C )
% 12.85/6.97 = ( modulo_modulo_int @ A6 @ C ) )
% 12.85/6.97 => ( ( ( modulo_modulo_int @ B @ C )
% 12.85/6.97 = ( modulo_modulo_int @ B6 @ C ) )
% 12.85/6.97 => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
% 12.85/6.97 = ( modulo_modulo_int @ ( plus_plus_int @ A6 @ B6 ) @ C ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4219_mod__add__cong,axiom,
% 12.85/6.97 ! [A: code_integer,C: code_integer,A6: code_integer,B: code_integer,B6: code_integer] :
% 12.85/6.97 ( ( ( modulo364778990260209775nteger @ A @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ A6 @ C ) )
% 12.85/6.97 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ B6 @ C ) )
% 12.85/6.97 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A6 @ B6 ) @ C ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4220_mod__add__eq,axiom,
% 12.85/6.97 ! [A: nat,C: nat,B: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 12.85/6.97 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4221_mod__add__eq,axiom,
% 12.85/6.97 ! [A: int,C: int,B: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 12.85/6.97 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4222_mod__add__eq,axiom,
% 12.85/6.97 ! [A: code_integer,C: code_integer,B: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4223_mod__diff__right__eq,axiom,
% 12.85/6.97 ! [A: int,B: int,C: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 12.85/6.97 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4224_mod__diff__right__eq,axiom,
% 12.85/6.97 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4225_mod__diff__left__eq,axiom,
% 12.85/6.97 ! [A: int,C: int,B: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 12.85/6.97 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4226_mod__diff__left__eq,axiom,
% 12.85/6.97 ! [A: code_integer,C: code_integer,B: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4227_mod__diff__cong,axiom,
% 12.85/6.97 ! [A: int,C: int,A6: int,B: int,B6: int] :
% 12.85/6.97 ( ( ( modulo_modulo_int @ A @ C )
% 12.85/6.97 = ( modulo_modulo_int @ A6 @ C ) )
% 12.85/6.97 => ( ( ( modulo_modulo_int @ B @ C )
% 12.85/6.97 = ( modulo_modulo_int @ B6 @ C ) )
% 12.85/6.97 => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
% 12.85/6.97 = ( modulo_modulo_int @ ( minus_minus_int @ A6 @ B6 ) @ C ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4228_mod__diff__cong,axiom,
% 12.85/6.97 ! [A: code_integer,C: code_integer,A6: code_integer,B: code_integer,B6: code_integer] :
% 12.85/6.97 ( ( ( modulo364778990260209775nteger @ A @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ A6 @ C ) )
% 12.85/6.97 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ B6 @ C ) )
% 12.85/6.97 => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A6 @ B6 ) @ C ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4229_mod__diff__eq,axiom,
% 12.85/6.97 ! [A: int,C: int,B: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 12.85/6.97 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4230_mod__diff__eq,axiom,
% 12.85/6.97 ! [A: code_integer,C: code_integer,B: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4231_of__bool__conj,axiom,
% 12.85/6.97 ! [P: $o,Q: $o] :
% 12.85/6.97 ( ( zero_n3304061248610475627l_real
% 12.85/6.97 @ ( P
% 12.85/6.97 & Q ) )
% 12.85/6.97 = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4232_of__bool__conj,axiom,
% 12.85/6.97 ! [P: $o,Q: $o] :
% 12.85/6.97 ( ( zero_n2052037380579107095ol_rat
% 12.85/6.97 @ ( P
% 12.85/6.97 & Q ) )
% 12.85/6.97 = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4233_of__bool__conj,axiom,
% 12.85/6.97 ! [P: $o,Q: $o] :
% 12.85/6.97 ( ( zero_n2687167440665602831ol_nat
% 12.85/6.97 @ ( P
% 12.85/6.97 & Q ) )
% 12.85/6.97 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4234_of__bool__conj,axiom,
% 12.85/6.97 ! [P: $o,Q: $o] :
% 12.85/6.97 ( ( zero_n2684676970156552555ol_int
% 12.85/6.97 @ ( P
% 12.85/6.97 & Q ) )
% 12.85/6.97 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4235_of__bool__conj,axiom,
% 12.85/6.97 ! [P: $o,Q: $o] :
% 12.85/6.97 ( ( zero_n356916108424825756nteger
% 12.85/6.97 @ ( P
% 12.85/6.97 & Q ) )
% 12.85/6.97 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4236_power__mod,axiom,
% 12.85/6.97 ! [A: nat,B: nat,N: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N ) @ B )
% 12.85/6.97 = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4237_power__mod,axiom,
% 12.85/6.97 ! [A: int,B: int,N: nat] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N ) @ B )
% 12.85/6.97 = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4238_power__mod,axiom,
% 12.85/6.97 ! [A: code_integer,B: code_integer,N: nat] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N ) @ B )
% 12.85/6.97 = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4239_dvd__mod,axiom,
% 12.85/6.97 ! [K: nat,M: nat,N: nat] :
% 12.85/6.97 ( ( dvd_dvd_nat @ K @ M )
% 12.85/6.97 => ( ( dvd_dvd_nat @ K @ N )
% 12.85/6.97 => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4240_dvd__mod,axiom,
% 12.85/6.97 ! [K: int,M: int,N: int] :
% 12.85/6.97 ( ( dvd_dvd_int @ K @ M )
% 12.85/6.97 => ( ( dvd_dvd_int @ K @ N )
% 12.85/6.97 => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4241_dvd__mod,axiom,
% 12.85/6.97 ! [K: code_integer,M: code_integer,N: code_integer] :
% 12.85/6.97 ( ( dvd_dvd_Code_integer @ K @ M )
% 12.85/6.97 => ( ( dvd_dvd_Code_integer @ K @ N )
% 12.85/6.97 => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4242_mod__mod__cancel,axiom,
% 12.85/6.97 ! [C: nat,B: nat,A: nat] :
% 12.85/6.97 ( ( dvd_dvd_nat @ C @ B )
% 12.85/6.97 => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ C )
% 12.85/6.97 = ( modulo_modulo_nat @ A @ C ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4243_mod__mod__cancel,axiom,
% 12.85/6.97 ! [C: int,B: int,A: int] :
% 12.85/6.97 ( ( dvd_dvd_int @ C @ B )
% 12.85/6.97 => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ C )
% 12.85/6.97 = ( modulo_modulo_int @ A @ C ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4244_mod__mod__cancel,axiom,
% 12.85/6.97 ! [C: code_integer,B: code_integer,A: code_integer] :
% 12.85/6.97 ( ( dvd_dvd_Code_integer @ C @ B )
% 12.85/6.97 => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ A @ C ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4245_dvd__mod__iff,axiom,
% 12.85/6.97 ! [C: nat,B: nat,A: nat] :
% 12.85/6.97 ( ( dvd_dvd_nat @ C @ B )
% 12.85/6.97 => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 12.85/6.97 = ( dvd_dvd_nat @ C @ A ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4246_dvd__mod__iff,axiom,
% 12.85/6.97 ! [C: int,B: int,A: int] :
% 12.85/6.97 ( ( dvd_dvd_int @ C @ B )
% 12.85/6.97 => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 12.85/6.97 = ( dvd_dvd_int @ C @ A ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4247_dvd__mod__iff,axiom,
% 12.85/6.97 ! [C: code_integer,B: code_integer,A: code_integer] :
% 12.85/6.97 ( ( dvd_dvd_Code_integer @ C @ B )
% 12.85/6.97 => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 12.85/6.97 = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4248_dvd__mod__imp__dvd,axiom,
% 12.85/6.97 ! [C: nat,A: nat,B: nat] :
% 12.85/6.97 ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 12.85/6.97 => ( ( dvd_dvd_nat @ C @ B )
% 12.85/6.97 => ( dvd_dvd_nat @ C @ A ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4249_dvd__mod__imp__dvd,axiom,
% 12.85/6.97 ! [C: int,A: int,B: int] :
% 12.85/6.97 ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 12.85/6.97 => ( ( dvd_dvd_int @ C @ B )
% 12.85/6.97 => ( dvd_dvd_int @ C @ A ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4250_dvd__mod__imp__dvd,axiom,
% 12.85/6.97 ! [C: code_integer,A: code_integer,B: code_integer] :
% 12.85/6.97 ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 12.85/6.97 => ( ( dvd_dvd_Code_integer @ C @ B )
% 12.85/6.97 => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4251_mod__Suc__Suc__eq,axiom,
% 12.85/6.97 ! [M: nat,N: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
% 12.85/6.97 = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4252_mod__Suc__eq,axiom,
% 12.85/6.97 ! [M: nat,N: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
% 12.85/6.97 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4253_mod__less__eq__dividend,axiom,
% 12.85/6.97 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4254_of__bool__odd__eq__mod__2,axiom,
% 12.85/6.97 ! [A: nat] :
% 12.85/6.97 ( ( zero_n2687167440665602831ol_nat
% 12.85/6.97 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 12.85/6.97 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4255_of__bool__odd__eq__mod__2,axiom,
% 12.85/6.97 ! [A: int] :
% 12.85/6.97 ( ( zero_n2684676970156552555ol_int
% 12.85/6.97 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 12.85/6.97 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4256_of__bool__odd__eq__mod__2,axiom,
% 12.85/6.97 ! [A: code_integer] :
% 12.85/6.97 ( ( zero_n356916108424825756nteger
% 12.85/6.97 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 12.85/6.97 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4257_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 12.85/6.97 ! [A: code_integer,B: code_integer] :
% 12.85/6.97 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 12.85/6.97 => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4258_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 12.85/6.97 ! [A: nat,B: nat] :
% 12.85/6.97 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.97 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4259_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 12.85/6.97 ! [A: int,B: int] :
% 12.85/6.97 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.97 => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4260_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 12.85/6.97 ! [B: nat,A: nat] :
% 12.85/6.97 ( ( ord_less_nat @ zero_zero_nat @ B )
% 12.85/6.97 => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4261_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 12.85/6.97 ! [B: int,A: int] :
% 12.85/6.97 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.97 => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4262_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 12.85/6.97 ! [B: code_integer,A: code_integer] :
% 12.85/6.97 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 12.85/6.97 => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4263_zero__less__eq__of__bool,axiom,
% 12.85/6.97 ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4264_zero__less__eq__of__bool,axiom,
% 12.85/6.97 ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4265_zero__less__eq__of__bool,axiom,
% 12.85/6.97 ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4266_zero__less__eq__of__bool,axiom,
% 12.85/6.97 ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4267_zero__less__eq__of__bool,axiom,
% 12.85/6.97 ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4268_mod__eq__self__iff__div__eq__0,axiom,
% 12.85/6.97 ! [A: nat,B: nat] :
% 12.85/6.97 ( ( ( modulo_modulo_nat @ A @ B )
% 12.85/6.97 = A )
% 12.85/6.97 = ( ( divide_divide_nat @ A @ B )
% 12.85/6.97 = zero_zero_nat ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4269_mod__eq__self__iff__div__eq__0,axiom,
% 12.85/6.97 ! [A: int,B: int] :
% 12.85/6.97 ( ( ( modulo_modulo_int @ A @ B )
% 12.85/6.97 = A )
% 12.85/6.97 = ( ( divide_divide_int @ A @ B )
% 12.85/6.97 = zero_zero_int ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4270_mod__eq__self__iff__div__eq__0,axiom,
% 12.85/6.97 ! [A: code_integer,B: code_integer] :
% 12.85/6.97 ( ( ( modulo364778990260209775nteger @ A @ B )
% 12.85/6.97 = A )
% 12.85/6.97 = ( ( divide6298287555418463151nteger @ A @ B )
% 12.85/6.97 = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4271_cong__exp__iff__simps_I9_J,axiom,
% 12.85/6.97 ! [M: num,Q2: num,N: num] :
% 12.85/6.97 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.85/6.97 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 12.85/6.97 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 12.85/6.97 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4272_cong__exp__iff__simps_I9_J,axiom,
% 12.85/6.97 ! [M: num,Q2: num,N: num] :
% 12.85/6.97 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.85/6.97 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 12.85/6.97 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 12.85/6.97 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4273_cong__exp__iff__simps_I9_J,axiom,
% 12.85/6.97 ! [M: num,Q2: num,N: num] :
% 12.85/6.97 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.85/6.97 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 12.85/6.97 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 12.85/6.97 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4274_cong__exp__iff__simps_I4_J,axiom,
% 12.85/6.97 ! [M: num,N: num] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
% 12.85/6.97 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4275_cong__exp__iff__simps_I4_J,axiom,
% 12.85/6.97 ! [M: num,N: num] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
% 12.85/6.97 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4276_cong__exp__iff__simps_I4_J,axiom,
% 12.85/6.97 ! [M: num,N: num] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
% 12.85/6.97 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4277_mod__eqE,axiom,
% 12.85/6.97 ! [A: int,C: int,B: int] :
% 12.85/6.97 ( ( ( modulo_modulo_int @ A @ C )
% 12.85/6.97 = ( modulo_modulo_int @ B @ C ) )
% 12.85/6.97 => ~ ! [D2: int] :
% 12.85/6.97 ( B
% 12.85/6.97 != ( plus_plus_int @ A @ ( times_times_int @ C @ D2 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4278_mod__eqE,axiom,
% 12.85/6.97 ! [A: code_integer,C: code_integer,B: code_integer] :
% 12.85/6.97 ( ( ( modulo364778990260209775nteger @ A @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ B @ C ) )
% 12.85/6.97 => ~ ! [D2: code_integer] :
% 12.85/6.97 ( B
% 12.85/6.97 != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D2 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4279_mod__eq__0__iff__dvd,axiom,
% 12.85/6.97 ! [A: nat,B: nat] :
% 12.85/6.97 ( ( ( modulo_modulo_nat @ A @ B )
% 12.85/6.97 = zero_zero_nat )
% 12.85/6.97 = ( dvd_dvd_nat @ B @ A ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4280_mod__eq__0__iff__dvd,axiom,
% 12.85/6.97 ! [A: int,B: int] :
% 12.85/6.97 ( ( ( modulo_modulo_int @ A @ B )
% 12.85/6.97 = zero_zero_int )
% 12.85/6.97 = ( dvd_dvd_int @ B @ A ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4281_mod__eq__0__iff__dvd,axiom,
% 12.85/6.97 ! [A: code_integer,B: code_integer] :
% 12.85/6.97 ( ( ( modulo364778990260209775nteger @ A @ B )
% 12.85/6.97 = zero_z3403309356797280102nteger )
% 12.85/6.97 = ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4282_dvd__eq__mod__eq__0,axiom,
% 12.85/6.97 ( dvd_dvd_nat
% 12.85/6.97 = ( ^ [A3: nat,B2: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ B2 @ A3 )
% 12.85/6.97 = zero_zero_nat ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4283_dvd__eq__mod__eq__0,axiom,
% 12.85/6.97 ( dvd_dvd_int
% 12.85/6.97 = ( ^ [A3: int,B2: int] :
% 12.85/6.97 ( ( modulo_modulo_int @ B2 @ A3 )
% 12.85/6.97 = zero_zero_int ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4284_dvd__eq__mod__eq__0,axiom,
% 12.85/6.97 ( dvd_dvd_Code_integer
% 12.85/6.97 = ( ^ [A3: code_integer,B2: code_integer] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ B2 @ A3 )
% 12.85/6.97 = zero_z3403309356797280102nteger ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4285_mod__0__imp__dvd,axiom,
% 12.85/6.97 ! [A: nat,B: nat] :
% 12.85/6.97 ( ( ( modulo_modulo_nat @ A @ B )
% 12.85/6.97 = zero_zero_nat )
% 12.85/6.97 => ( dvd_dvd_nat @ B @ A ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4286_mod__0__imp__dvd,axiom,
% 12.85/6.97 ! [A: int,B: int] :
% 12.85/6.97 ( ( ( modulo_modulo_int @ A @ B )
% 12.85/6.97 = zero_zero_int )
% 12.85/6.97 => ( dvd_dvd_int @ B @ A ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4287_mod__0__imp__dvd,axiom,
% 12.85/6.97 ! [A: code_integer,B: code_integer] :
% 12.85/6.97 ( ( ( modulo364778990260209775nteger @ A @ B )
% 12.85/6.97 = zero_z3403309356797280102nteger )
% 12.85/6.97 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4288_div__add1__eq,axiom,
% 12.85/6.97 ! [A: nat,B: nat,C: nat] :
% 12.85/6.97 ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 12.85/6.97 = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4289_div__add1__eq,axiom,
% 12.85/6.97 ! [A: int,B: int,C: int] :
% 12.85/6.97 ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 12.85/6.97 = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4290_div__add1__eq,axiom,
% 12.85/6.97 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.97 ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 12.85/6.97 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4291_of__bool__less__eq__one,axiom,
% 12.85/6.97 ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4292_of__bool__less__eq__one,axiom,
% 12.85/6.97 ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4293_of__bool__less__eq__one,axiom,
% 12.85/6.97 ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4294_of__bool__less__eq__one,axiom,
% 12.85/6.97 ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4295_of__bool__less__eq__one,axiom,
% 12.85/6.97 ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4296_split__of__bool__asm,axiom,
% 12.85/6.97 ! [P: complex > $o,P5: $o] :
% 12.85/6.97 ( ( P @ ( zero_n1201886186963655149omplex @ P5 ) )
% 12.85/6.97 = ( ~ ( ( P5
% 12.85/6.97 & ~ ( P @ one_one_complex ) )
% 12.85/6.97 | ( ~ P5
% 12.85/6.97 & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4297_split__of__bool__asm,axiom,
% 12.85/6.97 ! [P: real > $o,P5: $o] :
% 12.85/6.97 ( ( P @ ( zero_n3304061248610475627l_real @ P5 ) )
% 12.85/6.97 = ( ~ ( ( P5
% 12.85/6.97 & ~ ( P @ one_one_real ) )
% 12.85/6.97 | ( ~ P5
% 12.85/6.97 & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4298_split__of__bool__asm,axiom,
% 12.85/6.97 ! [P: rat > $o,P5: $o] :
% 12.85/6.97 ( ( P @ ( zero_n2052037380579107095ol_rat @ P5 ) )
% 12.85/6.97 = ( ~ ( ( P5
% 12.85/6.97 & ~ ( P @ one_one_rat ) )
% 12.85/6.97 | ( ~ P5
% 12.85/6.97 & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4299_split__of__bool__asm,axiom,
% 12.85/6.97 ! [P: nat > $o,P5: $o] :
% 12.85/6.97 ( ( P @ ( zero_n2687167440665602831ol_nat @ P5 ) )
% 12.85/6.97 = ( ~ ( ( P5
% 12.85/6.97 & ~ ( P @ one_one_nat ) )
% 12.85/6.97 | ( ~ P5
% 12.85/6.97 & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4300_split__of__bool__asm,axiom,
% 12.85/6.97 ! [P: int > $o,P5: $o] :
% 12.85/6.97 ( ( P @ ( zero_n2684676970156552555ol_int @ P5 ) )
% 12.85/6.97 = ( ~ ( ( P5
% 12.85/6.97 & ~ ( P @ one_one_int ) )
% 12.85/6.97 | ( ~ P5
% 12.85/6.97 & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4301_split__of__bool__asm,axiom,
% 12.85/6.97 ! [P: code_integer > $o,P5: $o] :
% 12.85/6.97 ( ( P @ ( zero_n356916108424825756nteger @ P5 ) )
% 12.85/6.97 = ( ~ ( ( P5
% 12.85/6.97 & ~ ( P @ one_one_Code_integer ) )
% 12.85/6.97 | ( ~ P5
% 12.85/6.97 & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4302_split__of__bool,axiom,
% 12.85/6.97 ! [P: complex > $o,P5: $o] :
% 12.85/6.97 ( ( P @ ( zero_n1201886186963655149omplex @ P5 ) )
% 12.85/6.97 = ( ( P5
% 12.85/6.97 => ( P @ one_one_complex ) )
% 12.85/6.97 & ( ~ P5
% 12.85/6.97 => ( P @ zero_zero_complex ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4303_split__of__bool,axiom,
% 12.85/6.97 ! [P: real > $o,P5: $o] :
% 12.85/6.97 ( ( P @ ( zero_n3304061248610475627l_real @ P5 ) )
% 12.85/6.97 = ( ( P5
% 12.85/6.97 => ( P @ one_one_real ) )
% 12.85/6.97 & ( ~ P5
% 12.85/6.97 => ( P @ zero_zero_real ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4304_split__of__bool,axiom,
% 12.85/6.97 ! [P: rat > $o,P5: $o] :
% 12.85/6.97 ( ( P @ ( zero_n2052037380579107095ol_rat @ P5 ) )
% 12.85/6.97 = ( ( P5
% 12.85/6.97 => ( P @ one_one_rat ) )
% 12.85/6.97 & ( ~ P5
% 12.85/6.97 => ( P @ zero_zero_rat ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4305_split__of__bool,axiom,
% 12.85/6.97 ! [P: nat > $o,P5: $o] :
% 12.85/6.97 ( ( P @ ( zero_n2687167440665602831ol_nat @ P5 ) )
% 12.85/6.97 = ( ( P5
% 12.85/6.97 => ( P @ one_one_nat ) )
% 12.85/6.97 & ( ~ P5
% 12.85/6.97 => ( P @ zero_zero_nat ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4306_split__of__bool,axiom,
% 12.85/6.97 ! [P: int > $o,P5: $o] :
% 12.85/6.97 ( ( P @ ( zero_n2684676970156552555ol_int @ P5 ) )
% 12.85/6.97 = ( ( P5
% 12.85/6.97 => ( P @ one_one_int ) )
% 12.85/6.97 & ( ~ P5
% 12.85/6.97 => ( P @ zero_zero_int ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4307_split__of__bool,axiom,
% 12.85/6.97 ! [P: code_integer > $o,P5: $o] :
% 12.85/6.97 ( ( P @ ( zero_n356916108424825756nteger @ P5 ) )
% 12.85/6.97 = ( ( P5
% 12.85/6.97 => ( P @ one_one_Code_integer ) )
% 12.85/6.97 & ( ~ P5
% 12.85/6.97 => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4308_of__bool__def,axiom,
% 12.85/6.97 ( zero_n1201886186963655149omplex
% 12.85/6.97 = ( ^ [P6: $o] : ( if_complex @ P6 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4309_of__bool__def,axiom,
% 12.85/6.97 ( zero_n3304061248610475627l_real
% 12.85/6.97 = ( ^ [P6: $o] : ( if_real @ P6 @ one_one_real @ zero_zero_real ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4310_of__bool__def,axiom,
% 12.85/6.97 ( zero_n2052037380579107095ol_rat
% 12.85/6.97 = ( ^ [P6: $o] : ( if_rat @ P6 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4311_of__bool__def,axiom,
% 12.85/6.97 ( zero_n2687167440665602831ol_nat
% 12.85/6.97 = ( ^ [P6: $o] : ( if_nat @ P6 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4312_of__bool__def,axiom,
% 12.85/6.97 ( zero_n2684676970156552555ol_int
% 12.85/6.97 = ( ^ [P6: $o] : ( if_int @ P6 @ one_one_int @ zero_zero_int ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4313_of__bool__def,axiom,
% 12.85/6.97 ( zero_n356916108424825756nteger
% 12.85/6.97 = ( ^ [P6: $o] : ( if_Code_integer @ P6 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4314_mod__eq__dvd__iff,axiom,
% 12.85/6.97 ! [A: int,C: int,B: int] :
% 12.85/6.97 ( ( ( modulo_modulo_int @ A @ C )
% 12.85/6.97 = ( modulo_modulo_int @ B @ C ) )
% 12.85/6.97 = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4315_mod__eq__dvd__iff,axiom,
% 12.85/6.97 ! [A: code_integer,C: code_integer,B: code_integer] :
% 12.85/6.97 ( ( ( modulo364778990260209775nteger @ A @ C )
% 12.85/6.97 = ( modulo364778990260209775nteger @ B @ C ) )
% 12.85/6.97 = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4316_dvd__minus__mod,axiom,
% 12.85/6.97 ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4317_dvd__minus__mod,axiom,
% 12.85/6.97 ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4318_dvd__minus__mod,axiom,
% 12.85/6.97 ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4319_mod__Suc,axiom,
% 12.85/6.97 ! [M: nat,N: nat] :
% 12.85/6.97 ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 12.85/6.97 = N )
% 12.85/6.97 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 12.85/6.97 = zero_zero_nat ) )
% 12.85/6.97 & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 12.85/6.97 != N )
% 12.85/6.97 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 12.85/6.97 = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4320_mod__induct,axiom,
% 12.85/6.97 ! [P: nat > $o,N: nat,P5: nat,M: nat] :
% 12.85/6.97 ( ( P @ N )
% 12.85/6.97 => ( ( ord_less_nat @ N @ P5 )
% 12.85/6.97 => ( ( ord_less_nat @ M @ P5 )
% 12.85/6.97 => ( ! [N2: nat] :
% 12.85/6.97 ( ( ord_less_nat @ N2 @ P5 )
% 12.85/6.97 => ( ( P @ N2 )
% 12.85/6.97 => ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P5 ) ) ) )
% 12.85/6.97 => ( P @ M ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4321_mod__less__divisor,axiom,
% 12.85/6.97 ! [N: nat,M: nat] :
% 12.85/6.97 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.97 => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4322_gcd__nat__induct,axiom,
% 12.85/6.97 ! [P: nat > nat > $o,M: nat,N: nat] :
% 12.85/6.97 ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
% 12.85/6.97 => ( ! [M4: nat,N2: nat] :
% 12.85/6.97 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 12.85/6.97 => ( ( P @ N2 @ ( modulo_modulo_nat @ M4 @ N2 ) )
% 12.85/6.97 => ( P @ M4 @ N2 ) ) )
% 12.85/6.97 => ( P @ M @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4323_mod__Suc__le__divisor,axiom,
% 12.85/6.97 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4324_mod__eq__0D,axiom,
% 12.85/6.97 ! [M: nat,D: nat] :
% 12.85/6.97 ( ( ( modulo_modulo_nat @ M @ D )
% 12.85/6.97 = zero_zero_nat )
% 12.85/6.97 => ? [Q4: nat] :
% 12.85/6.97 ( M
% 12.85/6.97 = ( times_times_nat @ D @ Q4 ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4325_mod__if,axiom,
% 12.85/6.97 ( modulo_modulo_nat
% 12.85/6.97 = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( ord_less_nat @ M5 @ N4 ) @ M5 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M5 @ N4 ) @ N4 ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4326_mod__geq,axiom,
% 12.85/6.97 ! [M: nat,N: nat] :
% 12.85/6.97 ( ~ ( ord_less_nat @ M @ N )
% 12.85/6.97 => ( ( modulo_modulo_nat @ M @ N )
% 12.85/6.97 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4327_le__mod__geq,axiom,
% 12.85/6.97 ! [N: nat,M: nat] :
% 12.85/6.97 ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.97 => ( ( modulo_modulo_nat @ M @ N )
% 12.85/6.97 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4328_nat__mod__eq__iff,axiom,
% 12.85/6.97 ! [X: nat,N: nat,Y4: nat] :
% 12.85/6.97 ( ( ( modulo_modulo_nat @ X @ N )
% 12.85/6.97 = ( modulo_modulo_nat @ Y4 @ N ) )
% 12.85/6.97 = ( ? [Q1: nat,Q22: nat] :
% 12.85/6.97 ( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
% 12.85/6.97 = ( plus_plus_nat @ Y4 @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4329_pochhammer__pos,axiom,
% 12.85/6.97 ! [X: real,N: nat] :
% 12.85/6.97 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.97 => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4330_pochhammer__pos,axiom,
% 12.85/6.97 ! [X: rat,N: nat] :
% 12.85/6.97 ( ( ord_less_rat @ zero_zero_rat @ X )
% 12.85/6.97 => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4331_pochhammer__pos,axiom,
% 12.85/6.97 ! [X: nat,N: nat] :
% 12.85/6.97 ( ( ord_less_nat @ zero_zero_nat @ X )
% 12.85/6.97 => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4332_pochhammer__pos,axiom,
% 12.85/6.97 ! [X: int,N: nat] :
% 12.85/6.97 ( ( ord_less_int @ zero_zero_int @ X )
% 12.85/6.97 => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4333_pochhammer__eq__0__mono,axiom,
% 12.85/6.97 ! [A: complex,N: nat,M: nat] :
% 12.85/6.97 ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 12.85/6.97 = zero_zero_complex )
% 12.85/6.97 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.97 => ( ( comm_s2602460028002588243omplex @ A @ M )
% 12.85/6.97 = zero_zero_complex ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4334_pochhammer__eq__0__mono,axiom,
% 12.85/6.97 ! [A: real,N: nat,M: nat] :
% 12.85/6.97 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 12.85/6.97 = zero_zero_real )
% 12.85/6.97 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.97 => ( ( comm_s7457072308508201937r_real @ A @ M )
% 12.85/6.97 = zero_zero_real ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4335_pochhammer__eq__0__mono,axiom,
% 12.85/6.97 ! [A: rat,N: nat,M: nat] :
% 12.85/6.97 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 12.85/6.97 = zero_zero_rat )
% 12.85/6.97 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.97 => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 12.85/6.97 = zero_zero_rat ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4336_pochhammer__neq__0__mono,axiom,
% 12.85/6.97 ! [A: complex,M: nat,N: nat] :
% 12.85/6.97 ( ( ( comm_s2602460028002588243omplex @ A @ M )
% 12.85/6.97 != zero_zero_complex )
% 12.85/6.97 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.97 => ( ( comm_s2602460028002588243omplex @ A @ N )
% 12.85/6.97 != zero_zero_complex ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4337_pochhammer__neq__0__mono,axiom,
% 12.85/6.97 ! [A: real,M: nat,N: nat] :
% 12.85/6.97 ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 12.85/6.97 != zero_zero_real )
% 12.85/6.97 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.97 => ( ( comm_s7457072308508201937r_real @ A @ N )
% 12.85/6.97 != zero_zero_real ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4338_pochhammer__neq__0__mono,axiom,
% 12.85/6.97 ! [A: rat,M: nat,N: nat] :
% 12.85/6.97 ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 12.85/6.97 != zero_zero_rat )
% 12.85/6.97 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.97 => ( ( comm_s4028243227959126397er_rat @ A @ N )
% 12.85/6.97 != zero_zero_rat ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4339_exp__mod__exp,axiom,
% 12.85/6.97 ! [M: nat,N: nat] :
% 12.85/6.97 ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.97 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4340_exp__mod__exp,axiom,
% 12.85/6.97 ! [M: nat,N: nat] :
% 12.85/6.97 ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.97 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 12.85/6.97
% 12.85/6.97 thf(fact_4341_exp__mod__exp,axiom,
% 12.85/6.97 ! [M: nat,N: nat] :
% 12.85/6.97 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.98 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4342_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 12.85/6.98 ! [B: code_integer,A: code_integer] :
% 12.85/6.98 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 12.85/6.98 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4343_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 12.85/6.98 ! [B: nat,A: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ B )
% 12.85/6.98 => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4344_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 12.85/6.98 ! [B: int,A: int] :
% 12.85/6.98 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.98 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4345_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 12.85/6.98 => ( ( ord_le6747313008572928689nteger @ A @ B )
% 12.85/6.98 => ( ( modulo364778990260209775nteger @ A @ B )
% 12.85/6.98 = A ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4346_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 12.85/6.98 ! [A: nat,B: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.98 => ( ( ord_less_nat @ A @ B )
% 12.85/6.98 => ( ( modulo_modulo_nat @ A @ B )
% 12.85/6.98 = A ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4347_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.98 => ( ( ord_less_int @ A @ B )
% 12.85/6.98 => ( ( modulo_modulo_int @ A @ B )
% 12.85/6.98 = A ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4348_cong__exp__iff__simps_I2_J,axiom,
% 12.85/6.98 ! [N: num,Q2: num] :
% 12.85/6.98 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 = zero_zero_nat )
% 12.85/6.98 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 12.85/6.98 = zero_zero_nat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4349_cong__exp__iff__simps_I2_J,axiom,
% 12.85/6.98 ! [N: num,Q2: num] :
% 12.85/6.98 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 = zero_zero_int )
% 12.85/6.98 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 12.85/6.98 = zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4350_cong__exp__iff__simps_I2_J,axiom,
% 12.85/6.98 ! [N: num,Q2: num] :
% 12.85/6.98 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 = zero_z3403309356797280102nteger )
% 12.85/6.98 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 12.85/6.98 = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4351_cong__exp__iff__simps_I1_J,axiom,
% 12.85/6.98 ! [N: num] :
% 12.85/6.98 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
% 12.85/6.98 = zero_zero_nat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4352_cong__exp__iff__simps_I1_J,axiom,
% 12.85/6.98 ! [N: num] :
% 12.85/6.98 ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
% 12.85/6.98 = zero_zero_int ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4353_cong__exp__iff__simps_I1_J,axiom,
% 12.85/6.98 ! [N: num] :
% 12.85/6.98 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
% 12.85/6.98 = zero_z3403309356797280102nteger ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4354_cong__exp__iff__simps_I6_J,axiom,
% 12.85/6.98 ! [Q2: num,N: num] :
% 12.85/6.98 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4355_cong__exp__iff__simps_I6_J,axiom,
% 12.85/6.98 ! [Q2: num,N: num] :
% 12.85/6.98 ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4356_cong__exp__iff__simps_I6_J,axiom,
% 12.85/6.98 ! [Q2: num,N: num] :
% 12.85/6.98 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4357_cong__exp__iff__simps_I8_J,axiom,
% 12.85/6.98 ! [M: num,Q2: num] :
% 12.85/6.98 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4358_cong__exp__iff__simps_I8_J,axiom,
% 12.85/6.98 ! [M: num,Q2: num] :
% 12.85/6.98 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4359_cong__exp__iff__simps_I8_J,axiom,
% 12.85/6.98 ! [M: num,Q2: num] :
% 12.85/6.98 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4360_div__mult1__eq,axiom,
% 12.85/6.98 ! [A: nat,B: nat,C: nat] :
% 12.85/6.98 ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
% 12.85/6.98 = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4361_div__mult1__eq,axiom,
% 12.85/6.98 ! [A: int,B: int,C: int] :
% 12.85/6.98 ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
% 12.85/6.98 = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4362_div__mult1__eq,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.98 ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 12.85/6.98 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4363_mult__div__mod__eq,axiom,
% 12.85/6.98 ! [B: nat,A: nat] :
% 12.85/6.98 ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
% 12.85/6.98 = A ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4364_mult__div__mod__eq,axiom,
% 12.85/6.98 ! [B: int,A: int] :
% 12.85/6.98 ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
% 12.85/6.98 = A ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4365_mult__div__mod__eq,axiom,
% 12.85/6.98 ! [B: code_integer,A: code_integer] :
% 12.85/6.98 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 12.85/6.98 = A ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4366_mod__mult__div__eq,axiom,
% 12.85/6.98 ! [A: nat,B: nat] :
% 12.85/6.98 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 12.85/6.98 = A ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4367_mod__mult__div__eq,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 12.85/6.98 = A ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4368_mod__mult__div__eq,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 12.85/6.98 = A ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4369_mod__div__mult__eq,axiom,
% 12.85/6.98 ! [A: nat,B: nat] :
% 12.85/6.98 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 12.85/6.98 = A ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4370_mod__div__mult__eq,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 12.85/6.98 = A ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4371_mod__div__mult__eq,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 12.85/6.98 = A ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4372_div__mult__mod__eq,axiom,
% 12.85/6.98 ! [A: nat,B: nat] :
% 12.85/6.98 ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
% 12.85/6.98 = A ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4373_div__mult__mod__eq,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
% 12.85/6.98 = A ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4374_div__mult__mod__eq,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 12.85/6.98 = A ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4375_mod__div__decomp,axiom,
% 12.85/6.98 ! [A: nat,B: nat] :
% 12.85/6.98 ( A
% 12.85/6.98 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4376_mod__div__decomp,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( A
% 12.85/6.98 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4377_mod__div__decomp,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( A
% 12.85/6.98 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4378_cancel__div__mod__rules_I1_J,axiom,
% 12.85/6.98 ! [A: nat,B: nat,C: nat] :
% 12.85/6.98 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 12.85/6.98 = ( plus_plus_nat @ A @ C ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4379_cancel__div__mod__rules_I1_J,axiom,
% 12.85/6.98 ! [A: int,B: int,C: int] :
% 12.85/6.98 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 12.85/6.98 = ( plus_plus_int @ A @ C ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4380_cancel__div__mod__rules_I1_J,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer,C: code_integer] :
% 12.85/6.98 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 12.85/6.98 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4381_cancel__div__mod__rules_I2_J,axiom,
% 12.85/6.98 ! [B: nat,A: nat,C: nat] :
% 12.85/6.98 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 12.85/6.98 = ( plus_plus_nat @ A @ C ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4382_cancel__div__mod__rules_I2_J,axiom,
% 12.85/6.98 ! [B: int,A: int,C: int] :
% 12.85/6.98 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 12.85/6.98 = ( plus_plus_int @ A @ C ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4383_cancel__div__mod__rules_I2_J,axiom,
% 12.85/6.98 ! [B: code_integer,A: code_integer,C: code_integer] :
% 12.85/6.98 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 12.85/6.98 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4384_unit__imp__mod__eq__0,axiom,
% 12.85/6.98 ! [B: nat,A: nat] :
% 12.85/6.98 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 12.85/6.98 => ( ( modulo_modulo_nat @ A @ B )
% 12.85/6.98 = zero_zero_nat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4385_unit__imp__mod__eq__0,axiom,
% 12.85/6.98 ! [B: int,A: int] :
% 12.85/6.98 ( ( dvd_dvd_int @ B @ one_one_int )
% 12.85/6.98 => ( ( modulo_modulo_int @ A @ B )
% 12.85/6.98 = zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4386_unit__imp__mod__eq__0,axiom,
% 12.85/6.98 ! [B: code_integer,A: code_integer] :
% 12.85/6.98 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 12.85/6.98 => ( ( modulo364778990260209775nteger @ A @ B )
% 12.85/6.98 = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4387_minus__mult__div__eq__mod,axiom,
% 12.85/6.98 ! [A: nat,B: nat] :
% 12.85/6.98 ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 12.85/6.98 = ( modulo_modulo_nat @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4388_minus__mult__div__eq__mod,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 12.85/6.98 = ( modulo_modulo_int @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4389_minus__mult__div__eq__mod,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 12.85/6.98 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4390_minus__mod__eq__mult__div,axiom,
% 12.85/6.98 ! [A: nat,B: nat] :
% 12.85/6.98 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 12.85/6.98 = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4391_minus__mod__eq__mult__div,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 12.85/6.98 = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4392_minus__mod__eq__mult__div,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 12.85/6.98 = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4393_minus__mod__eq__div__mult,axiom,
% 12.85/6.98 ! [A: nat,B: nat] :
% 12.85/6.98 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 12.85/6.98 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4394_minus__mod__eq__div__mult,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 12.85/6.98 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4395_minus__mod__eq__div__mult,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 12.85/6.98 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4396_minus__div__mult__eq__mod,axiom,
% 12.85/6.98 ! [A: nat,B: nat] :
% 12.85/6.98 ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 12.85/6.98 = ( modulo_modulo_nat @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4397_minus__div__mult__eq__mod,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 12.85/6.98 = ( modulo_modulo_int @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4398_minus__div__mult__eq__mod,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 12.85/6.98 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4399_cong__exp__iff__simps_I10_J,axiom,
% 12.85/6.98 ! [M: num,Q2: num,N: num] :
% 12.85/6.98 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4400_cong__exp__iff__simps_I10_J,axiom,
% 12.85/6.98 ! [M: num,Q2: num,N: num] :
% 12.85/6.98 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4401_cong__exp__iff__simps_I10_J,axiom,
% 12.85/6.98 ! [M: num,Q2: num,N: num] :
% 12.85/6.98 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4402_cong__exp__iff__simps_I12_J,axiom,
% 12.85/6.98 ! [M: num,Q2: num,N: num] :
% 12.85/6.98 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4403_cong__exp__iff__simps_I12_J,axiom,
% 12.85/6.98 ! [M: num,Q2: num,N: num] :
% 12.85/6.98 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4404_cong__exp__iff__simps_I12_J,axiom,
% 12.85/6.98 ! [M: num,Q2: num,N: num] :
% 12.85/6.98 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4405_cong__exp__iff__simps_I13_J,axiom,
% 12.85/6.98 ! [M: num,Q2: num,N: num] :
% 12.85/6.98 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 12.85/6.98 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 12.85/6.98 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4406_cong__exp__iff__simps_I13_J,axiom,
% 12.85/6.98 ! [M: num,Q2: num,N: num] :
% 12.85/6.98 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 12.85/6.98 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 12.85/6.98 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4407_cong__exp__iff__simps_I13_J,axiom,
% 12.85/6.98 ! [M: num,Q2: num,N: num] :
% 12.85/6.98 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 12.85/6.98 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 12.85/6.98 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4408_mod__le__divisor,axiom,
% 12.85/6.98 ! [N: nat,M: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.98 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4409_mod__greater__zero__iff__not__dvd,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
% 12.85/6.98 = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4410_div__less__mono,axiom,
% 12.85/6.98 ! [A2: nat,B4: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_nat @ A2 @ B4 )
% 12.85/6.98 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.98 => ( ( ( modulo_modulo_nat @ A2 @ N )
% 12.85/6.98 = zero_zero_nat )
% 12.85/6.98 => ( ( ( modulo_modulo_nat @ B4 @ N )
% 12.85/6.98 = zero_zero_nat )
% 12.85/6.98 => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B4 @ N ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4411_mod__eq__nat1E,axiom,
% 12.85/6.98 ! [M: nat,Q2: nat,N: nat] :
% 12.85/6.98 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 12.85/6.98 = ( modulo_modulo_nat @ N @ Q2 ) )
% 12.85/6.98 => ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.98 => ~ ! [S3: nat] :
% 12.85/6.98 ( M
% 12.85/6.98 != ( plus_plus_nat @ N @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4412_mod__eq__nat2E,axiom,
% 12.85/6.98 ! [M: nat,Q2: nat,N: nat] :
% 12.85/6.98 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 12.85/6.98 = ( modulo_modulo_nat @ N @ Q2 ) )
% 12.85/6.98 => ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.98 => ~ ! [S3: nat] :
% 12.85/6.98 ( N
% 12.85/6.98 != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4413_nat__mod__eq__lemma,axiom,
% 12.85/6.98 ! [X: nat,N: nat,Y4: nat] :
% 12.85/6.98 ( ( ( modulo_modulo_nat @ X @ N )
% 12.85/6.98 = ( modulo_modulo_nat @ Y4 @ N ) )
% 12.85/6.98 => ( ( ord_less_eq_nat @ Y4 @ X )
% 12.85/6.98 => ? [Q4: nat] :
% 12.85/6.98 ( X
% 12.85/6.98 = ( plus_plus_nat @ Y4 @ ( times_times_nat @ N @ Q4 ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4414_pochhammer__nonneg,axiom,
% 12.85/6.98 ! [X: rat,N: nat] :
% 12.85/6.98 ( ( ord_less_rat @ zero_zero_rat @ X )
% 12.85/6.98 => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4415_pochhammer__nonneg,axiom,
% 12.85/6.98 ! [X: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ X )
% 12.85/6.98 => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4416_pochhammer__nonneg,axiom,
% 12.85/6.98 ! [X: int,N: nat] :
% 12.85/6.98 ( ( ord_less_int @ zero_zero_int @ X )
% 12.85/6.98 => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4417_pochhammer__nonneg,axiom,
% 12.85/6.98 ! [X: real,N: nat] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4418_mod__eq__dvd__iff__nat,axiom,
% 12.85/6.98 ! [N: nat,M: nat,Q2: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.98 => ( ( ( modulo_modulo_nat @ M @ Q2 )
% 12.85/6.98 = ( modulo_modulo_nat @ N @ Q2 ) )
% 12.85/6.98 = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4419_mod__mult2__eq,axiom,
% 12.85/6.98 ! [M: nat,N: nat,Q2: nat] :
% 12.85/6.98 ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 12.85/6.98 = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4420_div__mod__decomp,axiom,
% 12.85/6.98 ! [A2: nat,N: nat] :
% 12.85/6.98 ( A2
% 12.85/6.98 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N ) @ N ) @ ( modulo_modulo_nat @ A2 @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4421_modulo__nat__def,axiom,
% 12.85/6.98 ( modulo_modulo_nat
% 12.85/6.98 = ( ^ [M5: nat,N4: nat] : ( minus_minus_nat @ M5 @ ( times_times_nat @ ( divide_divide_nat @ M5 @ N4 ) @ N4 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4422_pochhammer__0__left,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( ( N = zero_zero_nat )
% 12.85/6.98 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 12.85/6.98 = one_one_complex ) )
% 12.85/6.98 & ( ( N != zero_zero_nat )
% 12.85/6.98 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 12.85/6.98 = zero_zero_complex ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4423_pochhammer__0__left,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( ( N = zero_zero_nat )
% 12.85/6.98 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 12.85/6.98 = one_one_real ) )
% 12.85/6.98 & ( ( N != zero_zero_nat )
% 12.85/6.98 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 12.85/6.98 = zero_zero_real ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4424_pochhammer__0__left,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( ( N = zero_zero_nat )
% 12.85/6.98 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 12.85/6.98 = one_one_rat ) )
% 12.85/6.98 & ( ( N != zero_zero_nat )
% 12.85/6.98 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 12.85/6.98 = zero_zero_rat ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4425_pochhammer__0__left,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( ( N = zero_zero_nat )
% 12.85/6.98 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 12.85/6.98 = one_one_nat ) )
% 12.85/6.98 & ( ( N != zero_zero_nat )
% 12.85/6.98 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 12.85/6.98 = zero_zero_nat ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4426_pochhammer__0__left,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( ( N = zero_zero_nat )
% 12.85/6.98 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 12.85/6.98 = one_one_int ) )
% 12.85/6.98 & ( ( N != zero_zero_nat )
% 12.85/6.98 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 12.85/6.98 = zero_zero_int ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4427_cong__exp__iff__simps_I3_J,axiom,
% 12.85/6.98 ! [N: num,Q2: num] :
% 12.85/6.98 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 != zero_zero_nat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4428_cong__exp__iff__simps_I3_J,axiom,
% 12.85/6.98 ! [N: num,Q2: num] :
% 12.85/6.98 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 != zero_zero_int ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4429_cong__exp__iff__simps_I3_J,axiom,
% 12.85/6.98 ! [N: num,Q2: num] :
% 12.85/6.98 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 != zero_z3403309356797280102nteger ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4430_mod__mult2__eq_H,axiom,
% 12.85/6.98 ! [A: code_integer,M: nat,N: nat] :
% 12.85/6.98 ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
% 12.85/6.98 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4431_mod__mult2__eq_H,axiom,
% 12.85/6.98 ! [A: nat,M: nat,N: nat] :
% 12.85/6.98 ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 12.85/6.98 = ( 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 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4432_mod__mult2__eq_H,axiom,
% 12.85/6.98 ! [A: int,M: nat,N: nat] :
% 12.85/6.98 ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 12.85/6.98 = ( 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 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4433_even__even__mod__4__iff,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.98 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4434_field__char__0__class_Oof__nat__div,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
% 12.85/6.98 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4435_field__char__0__class_Oof__nat__div,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
% 12.85/6.98 = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4436_field__char__0__class_Oof__nat__div,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
% 12.85/6.98 = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4437_split__mod,axiom,
% 12.85/6.98 ! [P: nat > $o,M: nat,N: nat] :
% 12.85/6.98 ( ( P @ ( modulo_modulo_nat @ M @ N ) )
% 12.85/6.98 = ( ( ( N = zero_zero_nat )
% 12.85/6.98 => ( P @ M ) )
% 12.85/6.98 & ( ( N != zero_zero_nat )
% 12.85/6.98 => ! [I2: nat,J3: nat] :
% 12.85/6.98 ( ( ord_less_nat @ J3 @ N )
% 12.85/6.98 => ( ( M
% 12.85/6.98 = ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J3 ) )
% 12.85/6.98 => ( P @ J3 ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4438_mod__nat__eqI,axiom,
% 12.85/6.98 ! [R2: nat,N: nat,M: nat] :
% 12.85/6.98 ( ( ord_less_nat @ R2 @ N )
% 12.85/6.98 => ( ( ord_less_eq_nat @ R2 @ M )
% 12.85/6.98 => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R2 ) )
% 12.85/6.98 => ( ( modulo_modulo_nat @ M @ N )
% 12.85/6.98 = R2 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4439_pochhammer__rec,axiom,
% 12.85/6.98 ! [A: complex,N: nat] :
% 12.85/6.98 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4440_pochhammer__rec,axiom,
% 12.85/6.98 ! [A: real,N: nat] :
% 12.85/6.98 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4441_pochhammer__rec,axiom,
% 12.85/6.98 ! [A: rat,N: nat] :
% 12.85/6.98 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4442_pochhammer__rec,axiom,
% 12.85/6.98 ! [A: nat,N: nat] :
% 12.85/6.98 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4443_pochhammer__rec,axiom,
% 12.85/6.98 ! [A: int,N: nat] :
% 12.85/6.98 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4444_pochhammer__rec_H,axiom,
% 12.85/6.98 ! [Z: complex,N: nat] :
% 12.85/6.98 ( ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_complex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4445_pochhammer__rec_H,axiom,
% 12.85/6.98 ! [Z: real,N: nat] :
% 12.85/6.98 ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4446_pochhammer__rec_H,axiom,
% 12.85/6.98 ! [Z: rat,N: nat] :
% 12.85/6.98 ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4447_pochhammer__rec_H,axiom,
% 12.85/6.98 ! [Z: nat,N: nat] :
% 12.85/6.98 ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4448_pochhammer__rec_H,axiom,
% 12.85/6.98 ! [Z: int,N: nat] :
% 12.85/6.98 ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4449_pochhammer__Suc,axiom,
% 12.85/6.98 ! [A: complex,N: nat] :
% 12.85/6.98 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4450_pochhammer__Suc,axiom,
% 12.85/6.98 ! [A: real,N: nat] :
% 12.85/6.98 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4451_pochhammer__Suc,axiom,
% 12.85/6.98 ! [A: rat,N: nat] :
% 12.85/6.98 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4452_pochhammer__Suc,axiom,
% 12.85/6.98 ! [A: nat,N: nat] :
% 12.85/6.98 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4453_pochhammer__Suc,axiom,
% 12.85/6.98 ! [A: int,N: nat] :
% 12.85/6.98 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4454_real__of__nat__div__aux,axiom,
% 12.85/6.98 ! [X: nat,D: nat] :
% 12.85/6.98 ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
% 12.85/6.98 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4455_pochhammer__product_H,axiom,
% 12.85/6.98 ! [Z: complex,N: nat,M: nat] :
% 12.85/6.98 ( ( comm_s2602460028002588243omplex @ Z @ ( plus_plus_nat @ N @ M ) )
% 12.85/6.98 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ N ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ M ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4456_pochhammer__product_H,axiom,
% 12.85/6.98 ! [Z: real,N: nat,M: nat] :
% 12.85/6.98 ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N @ M ) )
% 12.85/6.98 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ M ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4457_pochhammer__product_H,axiom,
% 12.85/6.98 ! [Z: rat,N: nat,M: nat] :
% 12.85/6.98 ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N @ M ) )
% 12.85/6.98 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4458_pochhammer__product_H,axiom,
% 12.85/6.98 ! [Z: nat,N: nat,M: nat] :
% 12.85/6.98 ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N @ M ) )
% 12.85/6.98 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4459_pochhammer__product_H,axiom,
% 12.85/6.98 ! [Z: int,N: nat,M: nat] :
% 12.85/6.98 ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N @ M ) )
% 12.85/6.98 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4460_binomial__maximum_H,axiom,
% 12.85/6.98 ! [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 ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4461_binomial__mono,axiom,
% 12.85/6.98 ! [K: nat,K6: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ K6 )
% 12.85/6.98 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 12.85/6.98 => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4462_binomial__antimono,axiom,
% 12.85/6.98 ! [K: nat,K6: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ K6 )
% 12.85/6.98 => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 12.85/6.98 => ( ( ord_less_eq_nat @ K6 @ N )
% 12.85/6.98 => ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4463_binomial__maximum,axiom,
% 12.85/6.98 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4464_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 12.85/6.98 ! [C: code_integer,A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 12.85/6.98 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 12.85/6.98 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4465_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 12.85/6.98 ! [C: nat,A: nat,B: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 12.85/6.98 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
% 12.85/6.98 = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4466_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 12.85/6.98 ! [C: int,A: int,B: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.98 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 12.85/6.98 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4467_even__iff__mod__2__eq__zero,axiom,
% 12.85/6.98 ! [A: nat] :
% 12.85/6.98 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.98 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.98 = zero_zero_nat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4468_even__iff__mod__2__eq__zero,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.98 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.98 = zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4469_even__iff__mod__2__eq__zero,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.98 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.98 = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4470_cong__exp__iff__simps_I7_J,axiom,
% 12.85/6.98 ! [Q2: num,N: num] :
% 12.85/6.98 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 12.85/6.98 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 12.85/6.98 = zero_zero_nat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4471_cong__exp__iff__simps_I7_J,axiom,
% 12.85/6.98 ! [Q2: num,N: num] :
% 12.85/6.98 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 12.85/6.98 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 12.85/6.98 = zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4472_cong__exp__iff__simps_I7_J,axiom,
% 12.85/6.98 ! [Q2: num,N: num] :
% 12.85/6.98 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 12.85/6.98 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 12.85/6.98 = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4473_cong__exp__iff__simps_I11_J,axiom,
% 12.85/6.98 ! [M: num,Q2: num] :
% 12.85/6.98 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 12.85/6.98 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 12.85/6.98 = zero_zero_nat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4474_cong__exp__iff__simps_I11_J,axiom,
% 12.85/6.98 ! [M: num,Q2: num] :
% 12.85/6.98 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 12.85/6.98 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 12.85/6.98 = zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4475_cong__exp__iff__simps_I11_J,axiom,
% 12.85/6.98 ! [M: num,Q2: num] :
% 12.85/6.98 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 12.85/6.98 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 12.85/6.98 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 12.85/6.98 = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4476_odd__iff__mod__2__eq__one,axiom,
% 12.85/6.98 ! [A: nat] :
% 12.85/6.98 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 12.85/6.98 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.98 = one_one_nat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4477_odd__iff__mod__2__eq__one,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 12.85/6.98 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.98 = one_one_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4478_odd__iff__mod__2__eq__one,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 12.85/6.98 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.98 = one_one_Code_integer ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4479_Suc__mod__eq__add3__mod,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 12.85/6.98 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4480_Suc__times__mod__eq,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 12.85/6.98 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
% 12.85/6.98 = one_one_nat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4481_pochhammer__product,axiom,
% 12.85/6.98 ! [M: nat,N: nat,Z: complex] :
% 12.85/6.98 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.98 => ( ( comm_s2602460028002588243omplex @ Z @ N )
% 12.85/6.98 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4482_pochhammer__product,axiom,
% 12.85/6.98 ! [M: nat,N: nat,Z: real] :
% 12.85/6.98 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.98 => ( ( comm_s7457072308508201937r_real @ Z @ N )
% 12.85/6.98 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4483_pochhammer__product,axiom,
% 12.85/6.98 ! [M: nat,N: nat,Z: rat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.98 => ( ( comm_s4028243227959126397er_rat @ Z @ N )
% 12.85/6.98 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4484_pochhammer__product,axiom,
% 12.85/6.98 ! [M: nat,N: nat,Z: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.98 => ( ( comm_s4663373288045622133er_nat @ Z @ N )
% 12.85/6.98 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4485_pochhammer__product,axiom,
% 12.85/6.98 ! [M: nat,N: nat,Z: int] :
% 12.85/6.98 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.98 => ( ( comm_s4660882817536571857er_int @ Z @ N )
% 12.85/6.98 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4486_binomial__strict__antimono,axiom,
% 12.85/6.98 ! [K: nat,K6: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_nat @ K @ K6 )
% 12.85/6.98 => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 12.85/6.98 => ( ( ord_less_eq_nat @ K6 @ N )
% 12.85/6.98 => ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4487_binomial__strict__mono,axiom,
% 12.85/6.98 ! [K: nat,K6: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_nat @ K @ K6 )
% 12.85/6.98 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 12.85/6.98 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4488_binomial__less__binomial__Suc,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.98 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4489_central__binomial__odd,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.98 => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/6.98 = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4490_divmod__digit__0_I2_J,axiom,
% 12.85/6.98 ! [B: nat,A: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ B )
% 12.85/6.98 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 12.85/6.98 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
% 12.85/6.98 = ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4491_divmod__digit__0_I2_J,axiom,
% 12.85/6.98 ! [B: int,A: int] :
% 12.85/6.98 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.98 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 12.85/6.98 => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
% 12.85/6.98 = ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4492_divmod__digit__0_I2_J,axiom,
% 12.85/6.98 ! [B: code_integer,A: code_integer] :
% 12.85/6.98 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 12.85/6.98 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 12.85/6.98 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
% 12.85/6.98 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4493_bits__stable__imp__add__self,axiom,
% 12.85/6.98 ! [A: nat] :
% 12.85/6.98 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.98 = A )
% 12.85/6.98 => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.98 = zero_zero_nat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4494_bits__stable__imp__add__self,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.98 = A )
% 12.85/6.98 => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 12.85/6.98 = zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4495_bits__stable__imp__add__self,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.98 = A )
% 12.85/6.98 => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 12.85/6.98 = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4496_parity__cases,axiom,
% 12.85/6.98 ! [A: nat] :
% 12.85/6.98 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.98 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.98 != zero_zero_nat ) )
% 12.85/6.98 => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.98 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.98 != one_one_nat ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4497_parity__cases,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.98 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.98 != zero_zero_int ) )
% 12.85/6.98 => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.98 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.98 != one_one_int ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4498_parity__cases,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.98 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.98 != zero_z3403309356797280102nteger ) )
% 12.85/6.98 => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.98 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.98 != one_one_Code_integer ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4499_mod2__eq__if,axiom,
% 12.85/6.98 ! [A: nat] :
% 12.85/6.98 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.98 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.98 = zero_zero_nat ) )
% 12.85/6.98 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/6.98 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.98 = one_one_nat ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4500_mod2__eq__if,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.98 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.98 = zero_zero_int ) )
% 12.85/6.98 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/6.98 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.98 = one_one_int ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4501_mod2__eq__if,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.98 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.98 = zero_z3403309356797280102nteger ) )
% 12.85/6.98 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/6.98 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.98 = one_one_Code_integer ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4502_div__exp__mod__exp__eq,axiom,
% 12.85/6.98 ! [A: nat,N: nat,M: nat] :
% 12.85/6.98 ( ( 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 ) )
% 12.85/6.98 = ( 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 ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4503_div__exp__mod__exp__eq,axiom,
% 12.85/6.98 ! [A: int,N: nat,M: nat] :
% 12.85/6.98 ( ( 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 ) )
% 12.85/6.98 = ( 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 ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4504_div__exp__mod__exp__eq,axiom,
% 12.85/6.98 ! [A: code_integer,N: nat,M: nat] :
% 12.85/6.98 ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 12.85/6.98 = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4505_bits__induct,axiom,
% 12.85/6.98 ! [P: nat > $o,A: nat] :
% 12.85/6.98 ( ! [A4: nat] :
% 12.85/6.98 ( ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.98 = A4 )
% 12.85/6.98 => ( P @ A4 ) )
% 12.85/6.98 => ( ! [A4: nat,B3: $o] :
% 12.85/6.98 ( ( P @ A4 )
% 12.85/6.98 => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.98 = A4 )
% 12.85/6.98 => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 12.85/6.98 => ( P @ A ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4506_bits__induct,axiom,
% 12.85/6.98 ! [P: int > $o,A: int] :
% 12.85/6.98 ( ! [A4: int] :
% 12.85/6.98 ( ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.98 = A4 )
% 12.85/6.98 => ( P @ A4 ) )
% 12.85/6.98 => ( ! [A4: int,B3: $o] :
% 12.85/6.98 ( ( P @ A4 )
% 12.85/6.98 => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.98 = A4 )
% 12.85/6.98 => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 12.85/6.98 => ( P @ A ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4507_bits__induct,axiom,
% 12.85/6.98 ! [P: code_integer > $o,A: code_integer] :
% 12.85/6.98 ( ! [A4: code_integer] :
% 12.85/6.98 ( ( ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.98 = A4 )
% 12.85/6.98 => ( P @ A4 ) )
% 12.85/6.98 => ( ! [A4: code_integer,B3: $o] :
% 12.85/6.98 ( ( P @ A4 )
% 12.85/6.98 => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.98 = A4 )
% 12.85/6.98 => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 12.85/6.98 => ( P @ A ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4508_verit__le__mono__div,axiom,
% 12.85/6.98 ! [A2: nat,B4: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_nat @ A2 @ B4 )
% 12.85/6.98 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.98 => ( ord_less_eq_nat
% 12.85/6.98 @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N )
% 12.85/6.98 @ ( if_nat
% 12.85/6.98 @ ( ( modulo_modulo_nat @ B4 @ N )
% 12.85/6.98 = zero_zero_nat )
% 12.85/6.98 @ one_one_nat
% 12.85/6.98 @ zero_zero_nat ) )
% 12.85/6.98 @ ( divide_divide_nat @ B4 @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4509_divmod__digit__0_I1_J,axiom,
% 12.85/6.98 ! [B: nat,A: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ B )
% 12.85/6.98 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 12.85/6.98 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 12.85/6.98 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4510_divmod__digit__0_I1_J,axiom,
% 12.85/6.98 ! [B: int,A: int] :
% 12.85/6.98 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.98 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 12.85/6.98 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 12.85/6.98 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4511_divmod__digit__0_I1_J,axiom,
% 12.85/6.98 ! [B: code_integer,A: code_integer] :
% 12.85/6.98 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 12.85/6.98 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 12.85/6.98 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 12.85/6.98 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4512_mult__exp__mod__exp__eq,axiom,
% 12.85/6.98 ! [M: nat,N: nat,A: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.98 => ( ( 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 ) )
% 12.85/6.98 = ( 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 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4513_mult__exp__mod__exp__eq,axiom,
% 12.85/6.98 ! [M: nat,N: nat,A: int] :
% 12.85/6.98 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.98 => ( ( 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 ) )
% 12.85/6.98 = ( 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 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4514_mult__exp__mod__exp__eq,axiom,
% 12.85/6.98 ! [M: nat,N: nat,A: code_integer] :
% 12.85/6.98 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.98 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.98 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4515_mod__double__modulus,axiom,
% 12.85/6.98 ! [M: code_integer,X: code_integer] :
% 12.85/6.98 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 12.85/6.98 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 12.85/6.98 => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 12.85/6.98 = ( modulo364778990260209775nteger @ X @ M ) )
% 12.85/6.98 | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 12.85/6.98 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4516_mod__double__modulus,axiom,
% 12.85/6.98 ! [M: nat,X: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/6.98 => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 12.85/6.98 => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.85/6.98 = ( modulo_modulo_nat @ X @ M ) )
% 12.85/6.98 | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.85/6.98 = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4517_mod__double__modulus,axiom,
% 12.85/6.98 ! [M: int,X: int] :
% 12.85/6.98 ( ( ord_less_int @ zero_zero_int @ M )
% 12.85/6.98 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/6.98 => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 12.85/6.98 = ( modulo_modulo_int @ X @ M ) )
% 12.85/6.98 | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 12.85/6.98 = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4518_divmod__digit__1_I2_J,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 12.85/6.98 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 12.85/6.98 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 12.85/6.98 => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 12.85/6.98 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4519_divmod__digit__1_I2_J,axiom,
% 12.85/6.98 ! [A: nat,B: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.98 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 12.85/6.98 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 12.85/6.98 => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 12.85/6.98 = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4520_divmod__digit__1_I2_J,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.98 => ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.98 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 12.85/6.98 => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 12.85/6.98 = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4521_unset__bit__Suc,axiom,
% 12.85/6.98 ! [N: nat,A: int] :
% 12.85/6.98 ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
% 12.85/6.98 = ( 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 ) ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4522_unset__bit__Suc,axiom,
% 12.85/6.98 ! [N: nat,A: nat] :
% 12.85/6.98 ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
% 12.85/6.98 = ( 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 ) ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4523_unset__bit__Suc,axiom,
% 12.85/6.98 ! [N: nat,A: code_integer] :
% 12.85/6.98 ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
% 12.85/6.98 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4524_set__bit__Suc,axiom,
% 12.85/6.98 ! [N: nat,A: int] :
% 12.85/6.98 ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
% 12.85/6.98 = ( 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 ) ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4525_set__bit__Suc,axiom,
% 12.85/6.98 ! [N: nat,A: nat] :
% 12.85/6.98 ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
% 12.85/6.98 = ( 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 ) ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4526_set__bit__Suc,axiom,
% 12.85/6.98 ! [N: nat,A: code_integer] :
% 12.85/6.98 ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
% 12.85/6.98 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4527_flip__bit__Suc,axiom,
% 12.85/6.98 ! [N: nat,A: int] :
% 12.85/6.98 ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
% 12.85/6.98 = ( 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 ) ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4528_flip__bit__Suc,axiom,
% 12.85/6.98 ! [N: nat,A: nat] :
% 12.85/6.98 ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
% 12.85/6.98 = ( 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 ) ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4529_flip__bit__Suc,axiom,
% 12.85/6.98 ! [N: nat,A: code_integer] :
% 12.85/6.98 ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
% 12.85/6.98 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4530_even__mod__4__div__2,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.85/6.98 = ( suc @ zero_zero_nat ) )
% 12.85/6.98 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4531_exp__div__exp__eq,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.98 = ( times_times_nat
% 12.85/6.98 @ ( zero_n2687167440665602831ol_nat
% 12.85/6.98 @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 12.85/6.98 != zero_zero_nat )
% 12.85/6.98 & ( ord_less_eq_nat @ N @ M ) ) )
% 12.85/6.98 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4532_exp__div__exp__eq,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.98 = ( times_times_int
% 12.85/6.98 @ ( zero_n2684676970156552555ol_int
% 12.85/6.98 @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 12.85/6.98 != zero_zero_int )
% 12.85/6.98 & ( ord_less_eq_nat @ N @ M ) ) )
% 12.85/6.98 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4533_exp__div__exp__eq,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.98 = ( times_3573771949741848930nteger
% 12.85/6.98 @ ( zero_n356916108424825756nteger
% 12.85/6.98 @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 12.85/6.98 != zero_z3403309356797280102nteger )
% 12.85/6.98 & ( ord_less_eq_nat @ N @ M ) ) )
% 12.85/6.98 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4534_odd__mod__4__div__2,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.85/6.98 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 12.85/6.98 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4535_divmod__digit__1_I1_J,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 12.85/6.98 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 12.85/6.98 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 12.85/6.98 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
% 12.85/6.98 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4536_divmod__digit__1_I1_J,axiom,
% 12.85/6.98 ! [A: nat,B: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 12.85/6.98 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 12.85/6.98 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 12.85/6.98 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
% 12.85/6.98 = ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4537_divmod__digit__1_I1_J,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.98 => ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.98 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 12.85/6.98 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
% 12.85/6.98 = ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4538_zero__less__binomial__iff,axiom,
% 12.85/6.98 ! [N: nat,K: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
% 12.85/6.98 = ( ord_less_eq_nat @ K @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4539_choose__two,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.98 = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4540_binomial__n__0,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( binomial @ N @ zero_zero_nat )
% 12.85/6.98 = one_one_nat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4541_binomial__Suc__Suc,axiom,
% 12.85/6.98 ! [N: nat,K: nat] :
% 12.85/6.98 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 12.85/6.98 = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4542_binomial__eq__0__iff,axiom,
% 12.85/6.98 ! [N: nat,K: nat] :
% 12.85/6.98 ( ( ( binomial @ N @ K )
% 12.85/6.98 = zero_zero_nat )
% 12.85/6.98 = ( ord_less_nat @ N @ K ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4543_binomial__1,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
% 12.85/6.98 = N ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4544_binomial__0__Suc,axiom,
% 12.85/6.98 ! [K: nat] :
% 12.85/6.98 ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
% 12.85/6.98 = zero_zero_nat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4545_binomial__addition__formula,axiom,
% 12.85/6.98 ! [N: nat,K: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.98 => ( ( binomial @ N @ ( suc @ K ) )
% 12.85/6.98 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4546_binomial__Suc__n,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( binomial @ ( suc @ N ) @ N )
% 12.85/6.98 = ( suc @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4547_mod__neg__neg__trivial,axiom,
% 12.85/6.98 ! [K: int,L: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 12.85/6.98 => ( ( ord_less_int @ L @ K )
% 12.85/6.98 => ( ( modulo_modulo_int @ K @ L )
% 12.85/6.98 = K ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4548_mod__pos__pos__trivial,axiom,
% 12.85/6.98 ! [K: int,L: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.85/6.98 => ( ( ord_less_int @ K @ L )
% 12.85/6.98 => ( ( modulo_modulo_int @ K @ L )
% 12.85/6.98 = K ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4549_zmod__numeral__Bit0,axiom,
% 12.85/6.98 ! [V: num,W: num] :
% 12.85/6.98 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 12.85/6.98 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4550_zmod__numeral__Bit1,axiom,
% 12.85/6.98 ! [V: num,W: num] :
% 12.85/6.98 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 12.85/6.98 = ( 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 ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4551_zmod__le__nonneg__dividend,axiom,
% 12.85/6.98 ! [M: int,K: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 12.85/6.98 => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4552_neg__mod__bound,axiom,
% 12.85/6.98 ! [L: int,K: int] :
% 12.85/6.98 ( ( ord_less_int @ L @ zero_zero_int )
% 12.85/6.98 => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4553_Euclidean__Division_Opos__mod__bound,axiom,
% 12.85/6.98 ! [L: int,K: int] :
% 12.85/6.98 ( ( ord_less_int @ zero_zero_int @ L )
% 12.85/6.98 => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4554_zmod__eq__0D,axiom,
% 12.85/6.98 ! [M: int,D: int] :
% 12.85/6.98 ( ( ( modulo_modulo_int @ M @ D )
% 12.85/6.98 = zero_zero_int )
% 12.85/6.98 => ? [Q4: int] :
% 12.85/6.98 ( M
% 12.85/6.98 = ( times_times_int @ D @ Q4 ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4555_zmod__eq__0__iff,axiom,
% 12.85/6.98 ! [M: int,D: int] :
% 12.85/6.98 ( ( ( modulo_modulo_int @ M @ D )
% 12.85/6.98 = zero_zero_int )
% 12.85/6.98 = ( ? [Q3: int] :
% 12.85/6.98 ( M
% 12.85/6.98 = ( times_times_int @ D @ Q3 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4556_zmod__int,axiom,
% 12.85/6.98 ! [A: nat,B: nat] :
% 12.85/6.98 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
% 12.85/6.98 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4557_neg__mod__sign,axiom,
% 12.85/6.98 ! [L: int,K: int] :
% 12.85/6.98 ( ( ord_less_int @ L @ zero_zero_int )
% 12.85/6.98 => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4558_Euclidean__Division_Opos__mod__sign,axiom,
% 12.85/6.98 ! [L: int,K: int] :
% 12.85/6.98 ( ( ord_less_int @ zero_zero_int @ L )
% 12.85/6.98 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4559_zmod__trivial__iff,axiom,
% 12.85/6.98 ! [I: int,K: int] :
% 12.85/6.98 ( ( ( modulo_modulo_int @ I @ K )
% 12.85/6.98 = I )
% 12.85/6.98 = ( ( K = zero_zero_int )
% 12.85/6.98 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 12.85/6.98 & ( ord_less_int @ I @ K ) )
% 12.85/6.98 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 12.85/6.98 & ( ord_less_int @ K @ I ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4560_pos__mod__conj,axiom,
% 12.85/6.98 ! [B: int,A: int] :
% 12.85/6.98 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.98 => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
% 12.85/6.98 & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4561_neg__mod__conj,axiom,
% 12.85/6.98 ! [B: int,A: int] :
% 12.85/6.98 ( ( ord_less_int @ B @ zero_zero_int )
% 12.85/6.98 => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
% 12.85/6.98 & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4562_zdiv__mono__strict,axiom,
% 12.85/6.98 ! [A2: int,B4: int,N: int] :
% 12.85/6.98 ( ( ord_less_int @ A2 @ B4 )
% 12.85/6.98 => ( ( ord_less_int @ zero_zero_int @ N )
% 12.85/6.98 => ( ( ( modulo_modulo_int @ A2 @ N )
% 12.85/6.98 = zero_zero_int )
% 12.85/6.98 => ( ( ( modulo_modulo_int @ B4 @ N )
% 12.85/6.98 = zero_zero_int )
% 12.85/6.98 => ( ord_less_int @ ( divide_divide_int @ A2 @ N ) @ ( divide_divide_int @ B4 @ N ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4563_div__mod__decomp__int,axiom,
% 12.85/6.98 ! [A2: int,N: int] :
% 12.85/6.98 ( A2
% 12.85/6.98 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N ) @ N ) @ ( modulo_modulo_int @ A2 @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4564_mod__pos__neg__trivial,axiom,
% 12.85/6.98 ! [K: int,L: int] :
% 12.85/6.98 ( ( ord_less_int @ zero_zero_int @ K )
% 12.85/6.98 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 12.85/6.98 => ( ( modulo_modulo_int @ K @ L )
% 12.85/6.98 = ( plus_plus_int @ K @ L ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4565_mod__pos__geq,axiom,
% 12.85/6.98 ! [L: int,K: int] :
% 12.85/6.98 ( ( ord_less_int @ zero_zero_int @ L )
% 12.85/6.98 => ( ( ord_less_eq_int @ L @ K )
% 12.85/6.98 => ( ( modulo_modulo_int @ K @ L )
% 12.85/6.98 = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4566_mod__int__pos__iff,axiom,
% 12.85/6.98 ! [K: int,L: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
% 12.85/6.98 = ( ( dvd_dvd_int @ L @ K )
% 12.85/6.98 | ( ( L = zero_zero_int )
% 12.85/6.98 & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 12.85/6.98 | ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4567_split__zmod,axiom,
% 12.85/6.98 ! [P: int > $o,N: int,K: int] :
% 12.85/6.98 ( ( P @ ( modulo_modulo_int @ N @ K ) )
% 12.85/6.98 = ( ( ( K = zero_zero_int )
% 12.85/6.98 => ( P @ N ) )
% 12.85/6.98 & ( ( ord_less_int @ zero_zero_int @ K )
% 12.85/6.98 => ! [I2: int,J3: int] :
% 12.85/6.98 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 12.85/6.98 & ( ord_less_int @ J3 @ K )
% 12.85/6.98 & ( N
% 12.85/6.98 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 12.85/6.98 => ( P @ J3 ) ) )
% 12.85/6.98 & ( ( ord_less_int @ K @ zero_zero_int )
% 12.85/6.98 => ! [I2: int,J3: int] :
% 12.85/6.98 ( ( ( ord_less_int @ K @ J3 )
% 12.85/6.98 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 12.85/6.98 & ( N
% 12.85/6.98 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 12.85/6.98 => ( P @ J3 ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4568_int__mod__neg__eq,axiom,
% 12.85/6.98 ! [A: int,B: int,Q2: int,R2: int] :
% 12.85/6.98 ( ( A
% 12.85/6.98 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
% 12.85/6.98 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 12.85/6.98 => ( ( ord_less_int @ B @ R2 )
% 12.85/6.98 => ( ( modulo_modulo_int @ A @ B )
% 12.85/6.98 = R2 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4569_int__mod__pos__eq,axiom,
% 12.85/6.98 ! [A: int,B: int,Q2: int,R2: int] :
% 12.85/6.98 ( ( A
% 12.85/6.98 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
% 12.85/6.98 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 12.85/6.98 => ( ( ord_less_int @ R2 @ B )
% 12.85/6.98 => ( ( modulo_modulo_int @ A @ B )
% 12.85/6.98 = R2 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4570_zmod__zmult2__eq,axiom,
% 12.85/6.98 ! [C: int,A: int,B: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 12.85/6.98 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 12.85/6.98 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4571_verit__le__mono__div__int,axiom,
% 12.85/6.98 ! [A2: int,B4: int,N: int] :
% 12.85/6.98 ( ( ord_less_int @ A2 @ B4 )
% 12.85/6.98 => ( ( ord_less_int @ zero_zero_int @ N )
% 12.85/6.98 => ( ord_less_eq_int
% 12.85/6.98 @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N )
% 12.85/6.98 @ ( if_int
% 12.85/6.98 @ ( ( modulo_modulo_int @ B4 @ N )
% 12.85/6.98 = zero_zero_int )
% 12.85/6.98 @ one_one_int
% 12.85/6.98 @ zero_zero_int ) )
% 12.85/6.98 @ ( divide_divide_int @ B4 @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4572_split__neg__lemma,axiom,
% 12.85/6.98 ! [K: int,P: int > int > $o,N: int] :
% 12.85/6.98 ( ( ord_less_int @ K @ zero_zero_int )
% 12.85/6.98 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 12.85/6.98 = ( ! [I2: int,J3: int] :
% 12.85/6.98 ( ( ( ord_less_int @ K @ J3 )
% 12.85/6.98 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 12.85/6.98 & ( N
% 12.85/6.98 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 12.85/6.98 => ( P @ I2 @ J3 ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4573_split__pos__lemma,axiom,
% 12.85/6.98 ! [K: int,P: int > int > $o,N: int] :
% 12.85/6.98 ( ( ord_less_int @ zero_zero_int @ K )
% 12.85/6.98 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 12.85/6.98 = ( ! [I2: int,J3: int] :
% 12.85/6.98 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 12.85/6.98 & ( ord_less_int @ J3 @ K )
% 12.85/6.98 & ( N
% 12.85/6.98 = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 12.85/6.98 => ( P @ I2 @ J3 ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4574_pos__zmod__mult__2,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.98 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 12.85/6.98 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4575_neg__zmod__mult__2,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.85/6.98 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 12.85/6.98 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4576_binomial__eq__0,axiom,
% 12.85/6.98 ! [N: nat,K: nat] :
% 12.85/6.98 ( ( ord_less_nat @ N @ K )
% 12.85/6.98 => ( ( binomial @ N @ K )
% 12.85/6.98 = zero_zero_nat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4577_Suc__times__binomial__eq,axiom,
% 12.85/6.98 ! [N: nat,K: nat] :
% 12.85/6.98 ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
% 12.85/6.98 = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4578_Suc__times__binomial,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
% 12.85/6.98 = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4579_binomial__symmetric,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.98 => ( ( binomial @ N @ K )
% 12.85/6.98 = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4580_choose__mult__lemma,axiom,
% 12.85/6.98 ! [M: nat,R2: nat,K: nat] :
% 12.85/6.98 ( ( 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 ) )
% 12.85/6.98 = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4581_binomial__le__pow,axiom,
% 12.85/6.98 ! [R2: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ R2 @ N )
% 12.85/6.98 => ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4582_zero__less__binomial,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.98 => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4583_Suc__times__binomial__add,axiom,
% 12.85/6.98 ! [A: nat,B: nat] :
% 12.85/6.98 ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
% 12.85/6.98 = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4584_choose__mult,axiom,
% 12.85/6.98 ! [K: nat,M: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ M )
% 12.85/6.98 => ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.98 => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 12.85/6.98 = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4585_binomial__Suc__Suc__eq__times,axiom,
% 12.85/6.98 ! [N: nat,K: nat] :
% 12.85/6.98 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 12.85/6.98 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4586_binomial__absorb__comp,axiom,
% 12.85/6.98 ! [N: nat,K: nat] :
% 12.85/6.98 ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
% 12.85/6.98 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4587_binomial__absorption,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
% 12.85/6.98 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4588_binomial__ge__n__over__k__pow__k,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.98 => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4589_binomial__ge__n__over__k__pow__k,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.98 => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4590_binomial__le__pow2,axiom,
% 12.85/6.98 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4591_choose__reduce__nat,axiom,
% 12.85/6.98 ! [N: nat,K: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.98 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/6.98 => ( ( binomial @ N @ K )
% 12.85/6.98 = ( 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 ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4592_times__binomial__minus1__eq,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/6.98 => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
% 12.85/6.98 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4593_mod__exhaust__less__4,axiom,
% 12.85/6.98 ! [M: nat] :
% 12.85/6.98 ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.85/6.98 = zero_zero_nat )
% 12.85/6.98 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.85/6.98 = one_one_nat )
% 12.85/6.98 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.85/6.98 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.98 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.85/6.98 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4594_artanh__def,axiom,
% 12.85/6.98 ( artanh_real
% 12.85/6.98 = ( ^ [X3: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4595_signed__take__bit__Suc,axiom,
% 12.85/6.98 ! [N: nat,A: code_integer] :
% 12.85/6.98 ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
% 12.85/6.98 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4596_signed__take__bit__Suc,axiom,
% 12.85/6.98 ! [N: nat,A: int] :
% 12.85/6.98 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
% 12.85/6.98 = ( 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 ) ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4597__C10_C,axiom,
% 12.85/6.98 ! [L: nat] :
% 12.85/6.98 ( ( foldr_real_real @ plus_plus_real @ ( replicate_real @ L @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ zero_zero_real )
% 12.85/6.98 = ( times_times_real @ ( semiri5074537144036343181t_real @ L ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4598_add__scale__eq__noteq,axiom,
% 12.85/6.98 ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
% 12.85/6.98 ( ( R2 != zero_zero_complex )
% 12.85/6.98 => ( ( ( A = B )
% 12.85/6.98 & ( C != D ) )
% 12.85/6.98 => ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C ) )
% 12.85/6.98 != ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4599_add__scale__eq__noteq,axiom,
% 12.85/6.98 ! [R2: real,A: real,B: real,C: real,D: real] :
% 12.85/6.98 ( ( R2 != zero_zero_real )
% 12.85/6.98 => ( ( ( A = B )
% 12.85/6.98 & ( C != D ) )
% 12.85/6.98 => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
% 12.85/6.98 != ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4600_add__scale__eq__noteq,axiom,
% 12.85/6.98 ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
% 12.85/6.98 ( ( R2 != zero_zero_rat )
% 12.85/6.98 => ( ( ( A = B )
% 12.85/6.98 & ( C != D ) )
% 12.85/6.98 => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
% 12.85/6.98 != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4601_add__scale__eq__noteq,axiom,
% 12.85/6.98 ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
% 12.85/6.98 ( ( R2 != zero_zero_nat )
% 12.85/6.98 => ( ( ( A = B )
% 12.85/6.98 & ( C != D ) )
% 12.85/6.98 => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
% 12.85/6.98 != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4602_add__scale__eq__noteq,axiom,
% 12.85/6.98 ! [R2: int,A: int,B: int,C: int,D: int] :
% 12.85/6.98 ( ( R2 != zero_zero_int )
% 12.85/6.98 => ( ( ( A = B )
% 12.85/6.98 & ( C != D ) )
% 12.85/6.98 => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
% 12.85/6.98 != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4603_fact__double,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.98 = ( 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 ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4604_fact__double,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.98 = ( 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 ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4605_fact__double,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.98 = ( 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 ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4606_num_Osize__gen_I3_J,axiom,
% 12.85/6.98 ! [X33: num] :
% 12.85/6.98 ( ( size_num @ ( bit1 @ X33 ) )
% 12.85/6.98 = ( plus_plus_nat @ ( size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4607_foldr0,axiom,
% 12.85/6.98 ! [Xs: list_real,C: real,D: real] :
% 12.85/6.98 ( ( foldr_real_real @ plus_plus_real @ Xs @ ( plus_plus_real @ C @ D ) )
% 12.85/6.98 = ( plus_plus_real @ ( foldr_real_real @ plus_plus_real @ Xs @ D ) @ C ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4608_ln__less__cancel__iff,axiom,
% 12.85/6.98 ! [X: real,Y4: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/6.98 => ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y4 ) )
% 12.85/6.98 = ( ord_less_real @ X @ Y4 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4609_ln__inj__iff,axiom,
% 12.85/6.98 ! [X: real,Y4: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/6.98 => ( ( ( ln_ln_real @ X )
% 12.85/6.98 = ( ln_ln_real @ Y4 ) )
% 12.85/6.98 = ( X = Y4 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4610_of__nat__fact,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( semiri681578069525770553at_rat @ ( semiri1408675320244567234ct_nat @ N ) )
% 12.85/6.98 = ( semiri773545260158071498ct_rat @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4611_of__nat__fact,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( semiri1314217659103216013at_int @ ( semiri1408675320244567234ct_nat @ N ) )
% 12.85/6.98 = ( semiri1406184849735516958ct_int @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4612_of__nat__fact,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( semiri1316708129612266289at_nat @ ( semiri1408675320244567234ct_nat @ N ) )
% 12.85/6.98 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4613_of__nat__fact,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( semiri5074537144036343181t_real @ ( semiri1408675320244567234ct_nat @ N ) )
% 12.85/6.98 = ( semiri2265585572941072030t_real @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4614_of__nat__fact,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( semiri8010041392384452111omplex @ ( semiri1408675320244567234ct_nat @ N ) )
% 12.85/6.98 = ( semiri5044797733671781792omplex @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4615_signed__take__bit__of__0,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
% 12.85/6.98 = zero_zero_int ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4616_replicate__eq__replicate,axiom,
% 12.85/6.98 ! [M: nat,X: real,N: nat,Y4: real] :
% 12.85/6.98 ( ( ( replicate_real @ M @ X )
% 12.85/6.98 = ( replicate_real @ N @ Y4 ) )
% 12.85/6.98 = ( ( M = N )
% 12.85/6.98 & ( ( M != zero_zero_nat )
% 12.85/6.98 => ( X = Y4 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4617_replicate__eq__replicate,axiom,
% 12.85/6.98 ! [M: nat,X: vEBT_VEBT,N: nat,Y4: vEBT_VEBT] :
% 12.85/6.98 ( ( ( replicate_VEBT_VEBT @ M @ X )
% 12.85/6.98 = ( replicate_VEBT_VEBT @ N @ Y4 ) )
% 12.85/6.98 = ( ( M = N )
% 12.85/6.98 & ( ( M != zero_zero_nat )
% 12.85/6.98 => ( X = Y4 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4618_ln__le__cancel__iff,axiom,
% 12.85/6.98 ! [X: real,Y4: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/6.98 => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y4 ) )
% 12.85/6.98 = ( ord_less_eq_real @ X @ Y4 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4619_ln__less__zero__iff,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 12.85/6.98 = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4620_ln__gt__zero__iff,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 12.85/6.98 = ( ord_less_real @ one_one_real @ X ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4621_ln__eq__zero__iff,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ( ( ln_ln_real @ X )
% 12.85/6.98 = zero_zero_real )
% 12.85/6.98 = ( X = one_one_real ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4622_ln__one,axiom,
% 12.85/6.98 ( ( ln_ln_real @ one_one_real )
% 12.85/6.98 = zero_zero_real ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4623_fact__0,axiom,
% 12.85/6.98 ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
% 12.85/6.98 = one_one_rat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4624_fact__0,axiom,
% 12.85/6.98 ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
% 12.85/6.98 = one_one_int ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4625_fact__0,axiom,
% 12.85/6.98 ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
% 12.85/6.98 = one_one_nat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4626_fact__0,axiom,
% 12.85/6.98 ( ( semiri2265585572941072030t_real @ zero_zero_nat )
% 12.85/6.98 = one_one_real ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4627_fact__0,axiom,
% 12.85/6.98 ( ( semiri5044797733671781792omplex @ zero_zero_nat )
% 12.85/6.98 = one_one_complex ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4628_signed__take__bit__Suc__1,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
% 12.85/6.98 = one_one_int ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4629_signed__take__bit__numeral__of__1,axiom,
% 12.85/6.98 ! [K: num] :
% 12.85/6.98 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
% 12.85/6.98 = one_one_int ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4630_ln__ge__zero__iff,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 12.85/6.98 = ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4631_ln__le__zero__iff,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 12.85/6.98 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4632_fact__Suc__0,axiom,
% 12.85/6.98 ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
% 12.85/6.98 = one_one_rat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4633_fact__Suc__0,axiom,
% 12.85/6.98 ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
% 12.85/6.98 = one_one_int ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4634_fact__Suc__0,axiom,
% 12.85/6.98 ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
% 12.85/6.98 = one_one_nat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4635_fact__Suc__0,axiom,
% 12.85/6.98 ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
% 12.85/6.98 = one_one_real ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4636_fact__Suc__0,axiom,
% 12.85/6.98 ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
% 12.85/6.98 = one_one_complex ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4637_fact__Suc,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4638_fact__Suc,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4639_fact__Suc,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4640_fact__Suc,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4641_fact__Suc,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( semiri5044797733671781792omplex @ ( suc @ N ) )
% 12.85/6.98 = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4642_fact__2,axiom,
% 12.85/6.98 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.98 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4643_fact__2,axiom,
% 12.85/6.98 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.98 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4644_fact__2,axiom,
% 12.85/6.98 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.98 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4645_fact__2,axiom,
% 12.85/6.98 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.98 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4646_fact__2,axiom,
% 12.85/6.98 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.98 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4647_signed__take__bit__Suc__bit0,axiom,
% 12.85/6.98 ! [N: nat,K: num] :
% 12.85/6.98 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 12.85/6.98 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4648_signed__take__bit__Suc__bit1,axiom,
% 12.85/6.98 ! [N: nat,K: num] :
% 12.85/6.98 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 12.85/6.98 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4649_fact__mono__nat,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.98 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4650_fact__ge__self,axiom,
% 12.85/6.98 ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4651_fact__nonzero,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( semiri773545260158071498ct_rat @ N )
% 12.85/6.98 != zero_zero_rat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4652_fact__nonzero,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( semiri1406184849735516958ct_int @ N )
% 12.85/6.98 != zero_zero_int ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4653_fact__nonzero,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( semiri1408675320244567234ct_nat @ N )
% 12.85/6.98 != zero_zero_nat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4654_fact__nonzero,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( semiri2265585572941072030t_real @ N )
% 12.85/6.98 != zero_zero_real ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4655_fact__nonzero,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( semiri5044797733671781792omplex @ N )
% 12.85/6.98 != zero_zero_complex ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4656_signed__take__bit__mult,axiom,
% 12.85/6.98 ! [N: nat,K: int,L: int] :
% 12.85/6.98 ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 12.85/6.98 = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4657_signed__take__bit__add,axiom,
% 12.85/6.98 ! [N: nat,K: int,L: int] :
% 12.85/6.98 ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 12.85/6.98 = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4658_signed__take__bit__diff,axiom,
% 12.85/6.98 ! [N: nat,K: int,L: int] :
% 12.85/6.98 ( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 12.85/6.98 = ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4659_ln__less__self,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4660_fact__less__mono__nat,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/6.98 => ( ( ord_less_nat @ M @ N )
% 12.85/6.98 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4661_fact__ge__zero,axiom,
% 12.85/6.98 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4662_fact__ge__zero,axiom,
% 12.85/6.98 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4663_fact__ge__zero,axiom,
% 12.85/6.98 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4664_fact__ge__zero,axiom,
% 12.85/6.98 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4665_fact__gt__zero,axiom,
% 12.85/6.98 ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4666_fact__gt__zero,axiom,
% 12.85/6.98 ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4667_fact__gt__zero,axiom,
% 12.85/6.98 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4668_fact__gt__zero,axiom,
% 12.85/6.98 ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4669_fact__not__neg,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4670_fact__not__neg,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4671_fact__not__neg,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4672_fact__not__neg,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4673_fact__ge__1,axiom,
% 12.85/6.98 ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4674_fact__ge__1,axiom,
% 12.85/6.98 ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4675_fact__ge__1,axiom,
% 12.85/6.98 ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4676_fact__ge__1,axiom,
% 12.85/6.98 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4677_fact__mono,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.98 => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4678_fact__mono,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.98 => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4679_fact__mono,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.98 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4680_fact__mono,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.98 => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4681_fact__dvd,axiom,
% 12.85/6.98 ! [N: nat,M: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.98 => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4682_fact__dvd,axiom,
% 12.85/6.98 ! [N: nat,M: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.98 => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4683_fact__dvd,axiom,
% 12.85/6.98 ! [N: nat,M: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.98 => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4684_fact__dvd,axiom,
% 12.85/6.98 ! [N: nat,M: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ N @ M )
% 12.85/6.98 => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4685_fact__diff__Suc,axiom,
% 12.85/6.98 ! [N: nat,M: nat] :
% 12.85/6.98 ( ( ord_less_nat @ N @ ( suc @ M ) )
% 12.85/6.98 => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
% 12.85/6.98 = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4686_ln__bound,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4687_ln__gt__zero,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ord_less_real @ one_one_real @ X )
% 12.85/6.98 => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4688_ln__less__zero,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ( ord_less_real @ X @ one_one_real )
% 12.85/6.98 => ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4689_ln__gt__zero__imp__gt__one,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 12.85/6.98 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ord_less_real @ one_one_real @ X ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4690_ln__ge__zero,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ord_less_eq_real @ one_one_real @ X )
% 12.85/6.98 => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4691_fact__ge__Suc__0__nat,axiom,
% 12.85/6.98 ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4692_dvd__fact,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 12.85/6.98 => ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.98 => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4693_fact__less__mono,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/6.98 => ( ( ord_less_nat @ M @ N )
% 12.85/6.98 => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4694_fact__less__mono,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/6.98 => ( ( ord_less_nat @ M @ N )
% 12.85/6.98 => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4695_fact__less__mono,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/6.98 => ( ( ord_less_nat @ M @ N )
% 12.85/6.98 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4696_fact__less__mono,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/6.98 => ( ( ord_less_nat @ M @ N )
% 12.85/6.98 => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4697_fact__fact__dvd__fact,axiom,
% 12.85/6.98 ! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4698_fact__fact__dvd__fact,axiom,
% 12.85/6.98 ! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4699_fact__fact__dvd__fact,axiom,
% 12.85/6.98 ! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4700_fact__fact__dvd__fact,axiom,
% 12.85/6.98 ! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4701_fact__fact__dvd__fact,axiom,
% 12.85/6.98 ! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4702_fact__mod,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.98 => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
% 12.85/6.98 = zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4703_fact__mod,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.98 => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
% 12.85/6.98 = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4704_fact__mod,axiom,
% 12.85/6.98 ! [M: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ M @ N )
% 12.85/6.98 => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
% 12.85/6.98 = zero_zero_nat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4705_fact__le__power,axiom,
% 12.85/6.98 ! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4706_fact__le__power,axiom,
% 12.85/6.98 ! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4707_fact__le__power,axiom,
% 12.85/6.98 ! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4708_fact__le__power,axiom,
% 12.85/6.98 ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4709_ln__2__less__1,axiom,
% 12.85/6.98 ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4710_ln__ge__zero__imp__ge__one,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 12.85/6.98 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4711_ln__add__one__self__le__self,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4712_ln__mult,axiom,
% 12.85/6.98 ! [X: real,Y4: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/6.98 => ( ( ln_ln_real @ ( times_times_real @ X @ Y4 ) )
% 12.85/6.98 = ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y4 ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4713_ln__eq__minus__one,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ( ( ln_ln_real @ X )
% 12.85/6.98 = ( minus_minus_real @ X @ one_one_real ) )
% 12.85/6.98 => ( X = one_one_real ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4714_ln__div,axiom,
% 12.85/6.98 ! [X: real,Y4: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/6.98 => ( ( ln_ln_real @ ( divide_divide_real @ X @ Y4 ) )
% 12.85/6.98 = ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y4 ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4715_fact__div__fact__le__pow,axiom,
% 12.85/6.98 ! [R2: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ R2 @ N )
% 12.85/6.98 => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4716_binomial__fact__lemma,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.98 => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 12.85/6.98 = ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4717_choose__dvd,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.98 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4718_choose__dvd,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.98 => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4719_choose__dvd,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.98 => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4720_choose__dvd,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.98 => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4721_choose__dvd,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.98 => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4722_ln__le__minus__one,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4723_ln__diff__le,axiom,
% 12.85/6.98 ! [X: real,Y4: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/6.98 => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y4 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y4 ) @ Y4 ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4724_ln__realpow,axiom,
% 12.85/6.98 ! [X: real,N: nat] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
% 12.85/6.98 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4725_binomial__altdef__nat,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.98 => ( ( binomial @ N @ K )
% 12.85/6.98 = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4726_signed__take__bit__int__less__exp,axiom,
% 12.85/6.98 ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4727_num_Osize__gen_I1_J,axiom,
% 12.85/6.98 ( ( size_num @ one )
% 12.85/6.98 = zero_zero_nat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4728_even__signed__take__bit__iff,axiom,
% 12.85/6.98 ! [M: nat,A: code_integer] :
% 12.85/6.98 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
% 12.85/6.98 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4729_even__signed__take__bit__iff,axiom,
% 12.85/6.98 ! [M: nat,A: int] :
% 12.85/6.98 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
% 12.85/6.98 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4730_square__fact__le__2__fact,axiom,
% 12.85/6.98 ! [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 ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4731_fact__num__eq__if,axiom,
% 12.85/6.98 ( semiri773545260158071498ct_rat
% 12.85/6.98 = ( ^ [M5: nat] : ( if_rat @ ( M5 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M5 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4732_fact__num__eq__if,axiom,
% 12.85/6.98 ( semiri1406184849735516958ct_int
% 12.85/6.98 = ( ^ [M5: nat] : ( if_int @ ( M5 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4733_fact__num__eq__if,axiom,
% 12.85/6.98 ( semiri1408675320244567234ct_nat
% 12.85/6.98 = ( ^ [M5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M5 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4734_fact__num__eq__if,axiom,
% 12.85/6.98 ( semiri2265585572941072030t_real
% 12.85/6.98 = ( ^ [M5: nat] : ( if_real @ ( M5 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4735_fact__num__eq__if,axiom,
% 12.85/6.98 ( semiri5044797733671781792omplex
% 12.85/6.98 = ( ^ [M5: nat] : ( if_complex @ ( M5 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M5 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4736_fact__reduce,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.98 => ( ( semiri773545260158071498ct_rat @ N )
% 12.85/6.98 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4737_fact__reduce,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.98 => ( ( semiri1406184849735516958ct_int @ N )
% 12.85/6.98 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4738_fact__reduce,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.98 => ( ( semiri1408675320244567234ct_nat @ N )
% 12.85/6.98 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4739_fact__reduce,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.98 => ( ( semiri2265585572941072030t_real @ N )
% 12.85/6.98 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4740_fact__reduce,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/6.98 => ( ( semiri5044797733671781792omplex @ N )
% 12.85/6.98 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4741_signed__take__bit__int__less__self__iff,axiom,
% 12.85/6.98 ! [N: nat,K: int] :
% 12.85/6.98 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 12.85/6.98 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4742_signed__take__bit__int__greater__eq__self__iff,axiom,
% 12.85/6.98 ! [K: int,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 12.85/6.98 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4743_fact__binomial,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.98 => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
% 12.85/6.98 = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4744_fact__binomial,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.98 => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
% 12.85/6.98 = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4745_fact__binomial,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.98 => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
% 12.85/6.98 = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4746_binomial__fact,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.98 => ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 12.85/6.98 = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4747_binomial__fact,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.98 => ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 12.85/6.98 = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4748_binomial__fact,axiom,
% 12.85/6.98 ! [K: nat,N: nat] :
% 12.85/6.98 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.98 => ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 12.85/6.98 = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4749_add__0__iff,axiom,
% 12.85/6.98 ! [B: complex,A: complex] :
% 12.85/6.98 ( ( B
% 12.85/6.98 = ( plus_plus_complex @ B @ A ) )
% 12.85/6.98 = ( A = zero_zero_complex ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4750_add__0__iff,axiom,
% 12.85/6.98 ! [B: real,A: real] :
% 12.85/6.98 ( ( B
% 12.85/6.98 = ( plus_plus_real @ B @ A ) )
% 12.85/6.98 = ( A = zero_zero_real ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4751_add__0__iff,axiom,
% 12.85/6.98 ! [B: rat,A: rat] :
% 12.85/6.98 ( ( B
% 12.85/6.98 = ( plus_plus_rat @ B @ A ) )
% 12.85/6.98 = ( A = zero_zero_rat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4752_add__0__iff,axiom,
% 12.85/6.98 ! [B: nat,A: nat] :
% 12.85/6.98 ( ( B
% 12.85/6.98 = ( plus_plus_nat @ B @ A ) )
% 12.85/6.98 = ( A = zero_zero_nat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4753_add__0__iff,axiom,
% 12.85/6.98 ! [B: int,A: int] :
% 12.85/6.98 ( ( B
% 12.85/6.98 = ( plus_plus_int @ B @ A ) )
% 12.85/6.98 = ( A = zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4754_crossproduct__eq,axiom,
% 12.85/6.98 ! [W: real,Y4: real,X: real,Z: real] :
% 12.85/6.98 ( ( ( plus_plus_real @ ( times_times_real @ W @ Y4 ) @ ( times_times_real @ X @ Z ) )
% 12.85/6.98 = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X @ Y4 ) ) )
% 12.85/6.98 = ( ( W = X )
% 12.85/6.98 | ( Y4 = Z ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4755_crossproduct__eq,axiom,
% 12.85/6.98 ! [W: rat,Y4: rat,X: rat,Z: rat] :
% 12.85/6.98 ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y4 ) @ ( times_times_rat @ X @ Z ) )
% 12.85/6.98 = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y4 ) ) )
% 12.85/6.98 = ( ( W = X )
% 12.85/6.98 | ( Y4 = Z ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4756_crossproduct__eq,axiom,
% 12.85/6.98 ! [W: nat,Y4: nat,X: nat,Z: nat] :
% 12.85/6.98 ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y4 ) @ ( times_times_nat @ X @ Z ) )
% 12.85/6.98 = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y4 ) ) )
% 12.85/6.98 = ( ( W = X )
% 12.85/6.98 | ( Y4 = Z ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4757_crossproduct__eq,axiom,
% 12.85/6.98 ! [W: int,Y4: int,X: int,Z: int] :
% 12.85/6.98 ( ( ( plus_plus_int @ ( times_times_int @ W @ Y4 ) @ ( times_times_int @ X @ Z ) )
% 12.85/6.98 = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y4 ) ) )
% 12.85/6.98 = ( ( W = X )
% 12.85/6.98 | ( Y4 = Z ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4758_crossproduct__noteq,axiom,
% 12.85/6.98 ! [A: real,B: real,C: real,D: real] :
% 12.85/6.98 ( ( ( A != B )
% 12.85/6.98 & ( C != D ) )
% 12.85/6.98 = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
% 12.85/6.98 != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4759_crossproduct__noteq,axiom,
% 12.85/6.98 ! [A: rat,B: rat,C: rat,D: rat] :
% 12.85/6.98 ( ( ( A != B )
% 12.85/6.98 & ( C != D ) )
% 12.85/6.98 = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
% 12.85/6.98 != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4760_crossproduct__noteq,axiom,
% 12.85/6.98 ! [A: nat,B: nat,C: nat,D: nat] :
% 12.85/6.98 ( ( ( A != B )
% 12.85/6.98 & ( C != D ) )
% 12.85/6.98 = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
% 12.85/6.98 != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4761_crossproduct__noteq,axiom,
% 12.85/6.98 ! [A: int,B: int,C: int,D: int] :
% 12.85/6.98 ( ( ( A != B )
% 12.85/6.98 & ( C != D ) )
% 12.85/6.98 = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
% 12.85/6.98 != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4762_signed__take__bit__int__less__eq,axiom,
% 12.85/6.98 ! [N: nat,K: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 12.85/6.98 => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4763_eq__diff__eq_H,axiom,
% 12.85/6.98 ! [X: real,Y4: real,Z: real] :
% 12.85/6.98 ( ( X
% 12.85/6.98 = ( minus_minus_real @ Y4 @ Z ) )
% 12.85/6.98 = ( Y4
% 12.85/6.98 = ( plus_plus_real @ X @ Z ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4764_ln__one__plus__pos__lower__bound,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ( ord_less_eq_real @ X @ one_one_real )
% 12.85/6.98 => ( ord_less_eq_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4765_num_Osize__gen_I2_J,axiom,
% 12.85/6.98 ! [X2: num] :
% 12.85/6.98 ( ( size_num @ ( bit0 @ X2 ) )
% 12.85/6.98 = ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4766_binomial__code,axiom,
% 12.85/6.98 ( binomial
% 12.85/6.98 = ( ^ [N4: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N4 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N4 @ ( minus_minus_nat @ N4 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N4 @ K3 ) @ one_one_nat ) @ N4 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4767_tanh__ln__real,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ( tanh_real @ ( ln_ln_real @ X ) )
% 12.85/6.98 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4768_ln__one__minus__pos__lower__bound,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/6.98 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/6.98 => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4769_signed__take__bit__Suc__minus__bit1,axiom,
% 12.85/6.98 ! [N: nat,K: num] :
% 12.85/6.98 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 12.85/6.98 = ( 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 ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4770_signed__take__bit__rec,axiom,
% 12.85/6.98 ( bit_ri6519982836138164636nteger
% 12.85/6.98 = ( ^ [N4: nat,A3: code_integer] : ( if_Code_integer @ ( N4 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4771_signed__take__bit__rec,axiom,
% 12.85/6.98 ( bit_ri631733984087533419it_int
% 12.85/6.98 = ( ^ [N4: nat,A3: int] : ( if_int @ ( N4 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4772_map__cnt,axiom,
% 12.85/6.98 ( ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ zero_zero_real )
% 12.85/6.98 = ( times_times_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4773_abs__ln__one__plus__x__minus__x__bound,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/6.98 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4774_add_Oinverse__inverse,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 12.85/6.98 = A ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4775_add_Oinverse__inverse,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 12.85/6.98 = A ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4776_add_Oinverse__inverse,axiom,
% 12.85/6.98 ! [A: complex] :
% 12.85/6.98 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
% 12.85/6.98 = A ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4777_add_Oinverse__inverse,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
% 12.85/6.98 = A ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4778_add_Oinverse__inverse,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
% 12.85/6.98 = A ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4779_neg__equal__iff__equal,axiom,
% 12.85/6.98 ! [A: real,B: real] :
% 12.85/6.98 ( ( ( uminus_uminus_real @ A )
% 12.85/6.98 = ( uminus_uminus_real @ B ) )
% 12.85/6.98 = ( A = B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4780_neg__equal__iff__equal,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( ( uminus_uminus_int @ A )
% 12.85/6.98 = ( uminus_uminus_int @ B ) )
% 12.85/6.98 = ( A = B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4781_neg__equal__iff__equal,axiom,
% 12.85/6.98 ! [A: complex,B: complex] :
% 12.85/6.98 ( ( ( uminus1482373934393186551omplex @ A )
% 12.85/6.98 = ( uminus1482373934393186551omplex @ B ) )
% 12.85/6.98 = ( A = B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4782_neg__equal__iff__equal,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( ( uminus1351360451143612070nteger @ A )
% 12.85/6.98 = ( uminus1351360451143612070nteger @ B ) )
% 12.85/6.98 = ( A = B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4783_neg__equal__iff__equal,axiom,
% 12.85/6.98 ! [A: rat,B: rat] :
% 12.85/6.98 ( ( ( uminus_uminus_rat @ A )
% 12.85/6.98 = ( uminus_uminus_rat @ B ) )
% 12.85/6.98 = ( A = B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4784_verit__minus__simplify_I4_J,axiom,
% 12.85/6.98 ! [B: real] :
% 12.85/6.98 ( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
% 12.85/6.98 = B ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4785_verit__minus__simplify_I4_J,axiom,
% 12.85/6.98 ! [B: int] :
% 12.85/6.98 ( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
% 12.85/6.98 = B ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4786_verit__minus__simplify_I4_J,axiom,
% 12.85/6.98 ! [B: complex] :
% 12.85/6.98 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B ) )
% 12.85/6.98 = B ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4787_verit__minus__simplify_I4_J,axiom,
% 12.85/6.98 ! [B: code_integer] :
% 12.85/6.98 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B ) )
% 12.85/6.98 = B ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4788_verit__minus__simplify_I4_J,axiom,
% 12.85/6.98 ! [B: rat] :
% 12.85/6.98 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B ) )
% 12.85/6.98 = B ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4789_Compl__subset__Compl__iff,axiom,
% 12.85/6.98 ! [A2: set_int,B4: set_int] :
% 12.85/6.98 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B4 ) )
% 12.85/6.98 = ( ord_less_eq_set_int @ B4 @ A2 ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4790_Compl__anti__mono,axiom,
% 12.85/6.98 ! [A2: set_int,B4: set_int] :
% 12.85/6.98 ( ( ord_less_eq_set_int @ A2 @ B4 )
% 12.85/6.98 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B4 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4791_abs__idempotent,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 12.85/6.98 = ( abs_abs_real @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4792_abs__idempotent,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 12.85/6.98 = ( abs_abs_int @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4793_abs__idempotent,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 12.85/6.98 = ( abs_abs_Code_integer @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4794_abs__idempotent,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 12.85/6.98 = ( abs_abs_rat @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4795_abs__abs,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 12.85/6.98 = ( abs_abs_real @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4796_abs__abs,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 12.85/6.98 = ( abs_abs_int @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4797_abs__abs,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 12.85/6.98 = ( abs_abs_Code_integer @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4798_abs__abs,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 12.85/6.98 = ( abs_abs_rat @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4799_neg__le__iff__le,axiom,
% 12.85/6.98 ! [B: code_integer,A: code_integer] :
% 12.85/6.98 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 12.85/6.98 = ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4800_neg__le__iff__le,axiom,
% 12.85/6.98 ! [B: rat,A: rat] :
% 12.85/6.98 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 12.85/6.98 = ( ord_less_eq_rat @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4801_neg__le__iff__le,axiom,
% 12.85/6.98 ! [B: int,A: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 12.85/6.98 = ( ord_less_eq_int @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4802_neg__le__iff__le,axiom,
% 12.85/6.98 ! [B: real,A: real] :
% 12.85/6.98 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 12.85/6.98 = ( ord_less_eq_real @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4803_neg__equal__zero,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( ( uminus_uminus_real @ A )
% 12.85/6.98 = A )
% 12.85/6.98 = ( A = zero_zero_real ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4804_neg__equal__zero,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( ( uminus_uminus_int @ A )
% 12.85/6.98 = A )
% 12.85/6.98 = ( A = zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4805_neg__equal__zero,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( ( uminus1351360451143612070nteger @ A )
% 12.85/6.98 = A )
% 12.85/6.98 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4806_neg__equal__zero,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( ( uminus_uminus_rat @ A )
% 12.85/6.98 = A )
% 12.85/6.98 = ( A = zero_zero_rat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4807_equal__neg__zero,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( A
% 12.85/6.98 = ( uminus_uminus_real @ A ) )
% 12.85/6.98 = ( A = zero_zero_real ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4808_equal__neg__zero,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( A
% 12.85/6.98 = ( uminus_uminus_int @ A ) )
% 12.85/6.98 = ( A = zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4809_equal__neg__zero,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( A
% 12.85/6.98 = ( uminus1351360451143612070nteger @ A ) )
% 12.85/6.98 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4810_equal__neg__zero,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( A
% 12.85/6.98 = ( uminus_uminus_rat @ A ) )
% 12.85/6.98 = ( A = zero_zero_rat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4811_neg__equal__0__iff__equal,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( ( uminus_uminus_real @ A )
% 12.85/6.98 = zero_zero_real )
% 12.85/6.98 = ( A = zero_zero_real ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4812_neg__equal__0__iff__equal,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( ( uminus_uminus_int @ A )
% 12.85/6.98 = zero_zero_int )
% 12.85/6.98 = ( A = zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4813_neg__equal__0__iff__equal,axiom,
% 12.85/6.98 ! [A: complex] :
% 12.85/6.98 ( ( ( uminus1482373934393186551omplex @ A )
% 12.85/6.98 = zero_zero_complex )
% 12.85/6.98 = ( A = zero_zero_complex ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4814_neg__equal__0__iff__equal,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( ( uminus1351360451143612070nteger @ A )
% 12.85/6.98 = zero_z3403309356797280102nteger )
% 12.85/6.98 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4815_neg__equal__0__iff__equal,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( ( uminus_uminus_rat @ A )
% 12.85/6.98 = zero_zero_rat )
% 12.85/6.98 = ( A = zero_zero_rat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4816_neg__0__equal__iff__equal,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( zero_zero_real
% 12.85/6.98 = ( uminus_uminus_real @ A ) )
% 12.85/6.98 = ( zero_zero_real = A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4817_neg__0__equal__iff__equal,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( zero_zero_int
% 12.85/6.98 = ( uminus_uminus_int @ A ) )
% 12.85/6.98 = ( zero_zero_int = A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4818_neg__0__equal__iff__equal,axiom,
% 12.85/6.98 ! [A: complex] :
% 12.85/6.98 ( ( zero_zero_complex
% 12.85/6.98 = ( uminus1482373934393186551omplex @ A ) )
% 12.85/6.98 = ( zero_zero_complex = A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4819_neg__0__equal__iff__equal,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( zero_z3403309356797280102nteger
% 12.85/6.98 = ( uminus1351360451143612070nteger @ A ) )
% 12.85/6.98 = ( zero_z3403309356797280102nteger = A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4820_neg__0__equal__iff__equal,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( zero_zero_rat
% 12.85/6.98 = ( uminus_uminus_rat @ A ) )
% 12.85/6.98 = ( zero_zero_rat = A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4821_add_Oinverse__neutral,axiom,
% 12.85/6.98 ( ( uminus_uminus_real @ zero_zero_real )
% 12.85/6.98 = zero_zero_real ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4822_add_Oinverse__neutral,axiom,
% 12.85/6.98 ( ( uminus_uminus_int @ zero_zero_int )
% 12.85/6.98 = zero_zero_int ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4823_add_Oinverse__neutral,axiom,
% 12.85/6.98 ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 12.85/6.98 = zero_zero_complex ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4824_add_Oinverse__neutral,axiom,
% 12.85/6.98 ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 12.85/6.98 = zero_z3403309356797280102nteger ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4825_add_Oinverse__neutral,axiom,
% 12.85/6.98 ( ( uminus_uminus_rat @ zero_zero_rat )
% 12.85/6.98 = zero_zero_rat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4826_neg__less__iff__less,axiom,
% 12.85/6.98 ! [B: real,A: real] :
% 12.85/6.98 ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 12.85/6.98 = ( ord_less_real @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4827_neg__less__iff__less,axiom,
% 12.85/6.98 ! [B: int,A: int] :
% 12.85/6.98 ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 12.85/6.98 = ( ord_less_int @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4828_neg__less__iff__less,axiom,
% 12.85/6.98 ! [B: code_integer,A: code_integer] :
% 12.85/6.98 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 12.85/6.98 = ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4829_neg__less__iff__less,axiom,
% 12.85/6.98 ! [B: rat,A: rat] :
% 12.85/6.98 ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 12.85/6.98 = ( ord_less_rat @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4830_neg__numeral__eq__iff,axiom,
% 12.85/6.98 ! [M: num,N: num] :
% 12.85/6.98 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 12.85/6.98 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.85/6.98 = ( M = N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4831_neg__numeral__eq__iff,axiom,
% 12.85/6.98 ! [M: num,N: num] :
% 12.85/6.98 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 12.85/6.98 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/6.98 = ( M = N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4832_neg__numeral__eq__iff,axiom,
% 12.85/6.98 ! [M: num,N: num] :
% 12.85/6.98 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 12.85/6.98 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 12.85/6.98 = ( M = N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4833_neg__numeral__eq__iff,axiom,
% 12.85/6.98 ! [M: num,N: num] :
% 12.85/6.98 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 12.85/6.98 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.85/6.98 = ( M = N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4834_neg__numeral__eq__iff,axiom,
% 12.85/6.98 ! [M: num,N: num] :
% 12.85/6.98 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 12.85/6.98 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.85/6.98 = ( M = N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4835_mult__minus__left,axiom,
% 12.85/6.98 ! [A: real,B: real] :
% 12.85/6.98 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 12.85/6.98 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4836_mult__minus__left,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 12.85/6.98 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4837_mult__minus__left,axiom,
% 12.85/6.98 ! [A: complex,B: complex] :
% 12.85/6.98 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 12.85/6.98 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4838_mult__minus__left,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 12.85/6.98 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4839_mult__minus__left,axiom,
% 12.85/6.98 ! [A: rat,B: rat] :
% 12.85/6.98 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 12.85/6.98 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4840_minus__mult__minus,axiom,
% 12.85/6.98 ! [A: real,B: real] :
% 12.85/6.98 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 12.85/6.98 = ( times_times_real @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4841_minus__mult__minus,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 12.85/6.98 = ( times_times_int @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4842_minus__mult__minus,axiom,
% 12.85/6.98 ! [A: complex,B: complex] :
% 12.85/6.98 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 12.85/6.98 = ( times_times_complex @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4843_minus__mult__minus,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 12.85/6.98 = ( times_3573771949741848930nteger @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4844_minus__mult__minus,axiom,
% 12.85/6.98 ! [A: rat,B: rat] :
% 12.85/6.98 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 12.85/6.98 = ( times_times_rat @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4845_mult__minus__right,axiom,
% 12.85/6.98 ! [A: real,B: real] :
% 12.85/6.98 ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
% 12.85/6.98 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4846_mult__minus__right,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
% 12.85/6.98 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4847_mult__minus__right,axiom,
% 12.85/6.98 ! [A: complex,B: complex] :
% 12.85/6.98 ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 12.85/6.98 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4848_mult__minus__right,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 12.85/6.98 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4849_mult__minus__right,axiom,
% 12.85/6.98 ! [A: rat,B: rat] :
% 12.85/6.98 ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
% 12.85/6.98 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4850_add__minus__cancel,axiom,
% 12.85/6.98 ! [A: real,B: real] :
% 12.85/6.98 ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
% 12.85/6.98 = B ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4851_add__minus__cancel,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 12.85/6.98 = B ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4852_add__minus__cancel,axiom,
% 12.85/6.98 ! [A: complex,B: complex] :
% 12.85/6.98 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
% 12.85/6.98 = B ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4853_add__minus__cancel,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
% 12.85/6.98 = B ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4854_add__minus__cancel,axiom,
% 12.85/6.98 ! [A: rat,B: rat] :
% 12.85/6.98 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
% 12.85/6.98 = B ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4855_minus__add__cancel,axiom,
% 12.85/6.98 ! [A: real,B: real] :
% 12.85/6.98 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
% 12.85/6.98 = B ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4856_minus__add__cancel,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 12.85/6.98 = B ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4857_minus__add__cancel,axiom,
% 12.85/6.98 ! [A: complex,B: complex] :
% 12.85/6.98 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
% 12.85/6.98 = B ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4858_minus__add__cancel,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 12.85/6.98 = B ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4859_minus__add__cancel,axiom,
% 12.85/6.98 ! [A: rat,B: rat] :
% 12.85/6.98 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
% 12.85/6.98 = B ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4860_minus__add__distrib,axiom,
% 12.85/6.98 ! [A: real,B: real] :
% 12.85/6.98 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 12.85/6.98 = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4861_minus__add__distrib,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 12.85/6.98 = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4862_minus__add__distrib,axiom,
% 12.85/6.98 ! [A: complex,B: complex] :
% 12.85/6.98 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 12.85/6.98 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4863_minus__add__distrib,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 12.85/6.98 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4864_minus__add__distrib,axiom,
% 12.85/6.98 ! [A: rat,B: rat] :
% 12.85/6.98 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 12.85/6.98 = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4865_minus__diff__eq,axiom,
% 12.85/6.98 ! [A: real,B: real] :
% 12.85/6.98 ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
% 12.85/6.98 = ( minus_minus_real @ B @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4866_minus__diff__eq,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 12.85/6.98 = ( minus_minus_int @ B @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4867_minus__diff__eq,axiom,
% 12.85/6.98 ! [A: complex,B: complex] :
% 12.85/6.98 ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
% 12.85/6.98 = ( minus_minus_complex @ B @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4868_minus__diff__eq,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
% 12.85/6.98 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4869_minus__diff__eq,axiom,
% 12.85/6.98 ! [A: rat,B: rat] :
% 12.85/6.98 ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
% 12.85/6.98 = ( minus_minus_rat @ B @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4870_div__minus__minus,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 12.85/6.98 = ( divide_divide_int @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4871_div__minus__minus,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 12.85/6.98 = ( divide6298287555418463151nteger @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4872_abs__0,axiom,
% 12.85/6.98 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 12.85/6.98 = zero_z3403309356797280102nteger ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4873_abs__0,axiom,
% 12.85/6.98 ( ( abs_abs_complex @ zero_zero_complex )
% 12.85/6.98 = zero_zero_complex ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4874_abs__0,axiom,
% 12.85/6.98 ( ( abs_abs_real @ zero_zero_real )
% 12.85/6.98 = zero_zero_real ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4875_abs__0,axiom,
% 12.85/6.98 ( ( abs_abs_rat @ zero_zero_rat )
% 12.85/6.98 = zero_zero_rat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4876_abs__0,axiom,
% 12.85/6.98 ( ( abs_abs_int @ zero_zero_int )
% 12.85/6.98 = zero_zero_int ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4877_abs__zero,axiom,
% 12.85/6.98 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 12.85/6.98 = zero_z3403309356797280102nteger ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4878_abs__zero,axiom,
% 12.85/6.98 ( ( abs_abs_real @ zero_zero_real )
% 12.85/6.98 = zero_zero_real ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4879_abs__zero,axiom,
% 12.85/6.98 ( ( abs_abs_rat @ zero_zero_rat )
% 12.85/6.98 = zero_zero_rat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4880_abs__zero,axiom,
% 12.85/6.98 ( ( abs_abs_int @ zero_zero_int )
% 12.85/6.98 = zero_zero_int ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4881_abs__eq__0,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( ( abs_abs_Code_integer @ A )
% 12.85/6.98 = zero_z3403309356797280102nteger )
% 12.85/6.98 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4882_abs__eq__0,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( ( abs_abs_real @ A )
% 12.85/6.98 = zero_zero_real )
% 12.85/6.98 = ( A = zero_zero_real ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4883_abs__eq__0,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( ( abs_abs_rat @ A )
% 12.85/6.98 = zero_zero_rat )
% 12.85/6.98 = ( A = zero_zero_rat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4884_abs__eq__0,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( ( abs_abs_int @ A )
% 12.85/6.98 = zero_zero_int )
% 12.85/6.98 = ( A = zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4885_abs__0__eq,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( zero_z3403309356797280102nteger
% 12.85/6.98 = ( abs_abs_Code_integer @ A ) )
% 12.85/6.98 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4886_abs__0__eq,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( zero_zero_real
% 12.85/6.98 = ( abs_abs_real @ A ) )
% 12.85/6.98 = ( A = zero_zero_real ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4887_abs__0__eq,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( zero_zero_rat
% 12.85/6.98 = ( abs_abs_rat @ A ) )
% 12.85/6.98 = ( A = zero_zero_rat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4888_abs__0__eq,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( zero_zero_int
% 12.85/6.98 = ( abs_abs_int @ A ) )
% 12.85/6.98 = ( A = zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4889_minus__dvd__iff,axiom,
% 12.85/6.98 ! [X: real,Y4: real] :
% 12.85/6.98 ( ( dvd_dvd_real @ ( uminus_uminus_real @ X ) @ Y4 )
% 12.85/6.98 = ( dvd_dvd_real @ X @ Y4 ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4890_minus__dvd__iff,axiom,
% 12.85/6.98 ! [X: int,Y4: int] :
% 12.85/6.98 ( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y4 )
% 12.85/6.98 = ( dvd_dvd_int @ X @ Y4 ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4891_minus__dvd__iff,axiom,
% 12.85/6.98 ! [X: complex,Y4: complex] :
% 12.85/6.98 ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X ) @ Y4 )
% 12.85/6.98 = ( dvd_dvd_complex @ X @ Y4 ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4892_minus__dvd__iff,axiom,
% 12.85/6.98 ! [X: code_integer,Y4: code_integer] :
% 12.85/6.98 ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ Y4 )
% 12.85/6.98 = ( dvd_dvd_Code_integer @ X @ Y4 ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4893_minus__dvd__iff,axiom,
% 12.85/6.98 ! [X: rat,Y4: rat] :
% 12.85/6.98 ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X ) @ Y4 )
% 12.85/6.98 = ( dvd_dvd_rat @ X @ Y4 ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4894_dvd__minus__iff,axiom,
% 12.85/6.98 ! [X: real,Y4: real] :
% 12.85/6.98 ( ( dvd_dvd_real @ X @ ( uminus_uminus_real @ Y4 ) )
% 12.85/6.98 = ( dvd_dvd_real @ X @ Y4 ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4895_dvd__minus__iff,axiom,
% 12.85/6.98 ! [X: int,Y4: int] :
% 12.85/6.98 ( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y4 ) )
% 12.85/6.98 = ( dvd_dvd_int @ X @ Y4 ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4896_dvd__minus__iff,axiom,
% 12.85/6.98 ! [X: complex,Y4: complex] :
% 12.85/6.98 ( ( dvd_dvd_complex @ X @ ( uminus1482373934393186551omplex @ Y4 ) )
% 12.85/6.98 = ( dvd_dvd_complex @ X @ Y4 ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4897_dvd__minus__iff,axiom,
% 12.85/6.98 ! [X: code_integer,Y4: code_integer] :
% 12.85/6.98 ( ( dvd_dvd_Code_integer @ X @ ( uminus1351360451143612070nteger @ Y4 ) )
% 12.85/6.98 = ( dvd_dvd_Code_integer @ X @ Y4 ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4898_dvd__minus__iff,axiom,
% 12.85/6.98 ! [X: rat,Y4: rat] :
% 12.85/6.98 ( ( dvd_dvd_rat @ X @ ( uminus_uminus_rat @ Y4 ) )
% 12.85/6.98 = ( dvd_dvd_rat @ X @ Y4 ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4899_abs__numeral,axiom,
% 12.85/6.98 ! [N: num] :
% 12.85/6.98 ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
% 12.85/6.98 = ( numera6620942414471956472nteger @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4900_abs__numeral,axiom,
% 12.85/6.98 ! [N: num] :
% 12.85/6.98 ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
% 12.85/6.98 = ( numeral_numeral_real @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4901_abs__numeral,axiom,
% 12.85/6.98 ! [N: num] :
% 12.85/6.98 ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
% 12.85/6.98 = ( numeral_numeral_rat @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4902_abs__numeral,axiom,
% 12.85/6.98 ! [N: num] :
% 12.85/6.98 ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
% 12.85/6.98 = ( numeral_numeral_int @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4903_abs__mult__self__eq,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 12.85/6.98 = ( times_3573771949741848930nteger @ A @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4904_abs__mult__self__eq,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
% 12.85/6.98 = ( times_times_real @ A @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4905_abs__mult__self__eq,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
% 12.85/6.98 = ( times_times_rat @ A @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4906_abs__mult__self__eq,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
% 12.85/6.98 = ( times_times_int @ A @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4907_abs__add__abs,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
% 12.85/6.98 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4908_abs__add__abs,axiom,
% 12.85/6.98 ! [A: real,B: real] :
% 12.85/6.98 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 12.85/6.98 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4909_abs__add__abs,axiom,
% 12.85/6.98 ! [A: rat,B: rat] :
% 12.85/6.98 ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
% 12.85/6.98 = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4910_abs__add__abs,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
% 12.85/6.98 = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4911_abs__1,axiom,
% 12.85/6.98 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 12.85/6.98 = one_one_Code_integer ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4912_abs__1,axiom,
% 12.85/6.98 ( ( abs_abs_complex @ one_one_complex )
% 12.85/6.98 = one_one_complex ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4913_abs__1,axiom,
% 12.85/6.98 ( ( abs_abs_real @ one_one_real )
% 12.85/6.98 = one_one_real ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4914_abs__1,axiom,
% 12.85/6.98 ( ( abs_abs_rat @ one_one_rat )
% 12.85/6.98 = one_one_rat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4915_abs__1,axiom,
% 12.85/6.98 ( ( abs_abs_int @ one_one_int )
% 12.85/6.98 = one_one_int ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4916_abs__divide,axiom,
% 12.85/6.98 ! [A: complex,B: complex] :
% 12.85/6.98 ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B ) )
% 12.85/6.98 = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4917_abs__divide,axiom,
% 12.85/6.98 ! [A: real,B: real] :
% 12.85/6.98 ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 12.85/6.98 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4918_abs__divide,axiom,
% 12.85/6.98 ! [A: rat,B: rat] :
% 12.85/6.98 ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 12.85/6.98 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4919_abs__minus__cancel,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 12.85/6.98 = ( abs_abs_real @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4920_abs__minus__cancel,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 12.85/6.98 = ( abs_abs_int @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4921_abs__minus__cancel,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 12.85/6.98 = ( abs_abs_Code_integer @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4922_abs__minus__cancel,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 12.85/6.98 = ( abs_abs_rat @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4923_abs__minus,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 12.85/6.98 = ( abs_abs_real @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4924_abs__minus,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 12.85/6.98 = ( abs_abs_int @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4925_abs__minus,axiom,
% 12.85/6.98 ! [A: complex] :
% 12.85/6.98 ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
% 12.85/6.98 = ( abs_abs_complex @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4926_abs__minus,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 12.85/6.98 = ( abs_abs_Code_integer @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4927_abs__minus,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 12.85/6.98 = ( abs_abs_rat @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4928_mod__minus__minus,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 12.85/6.98 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4929_mod__minus__minus,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 12.85/6.98 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4930_dvd__abs__iff,axiom,
% 12.85/6.98 ! [M: real,K: real] :
% 12.85/6.98 ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
% 12.85/6.98 = ( dvd_dvd_real @ M @ K ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4931_dvd__abs__iff,axiom,
% 12.85/6.98 ! [M: int,K: int] :
% 12.85/6.98 ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
% 12.85/6.98 = ( dvd_dvd_int @ M @ K ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4932_dvd__abs__iff,axiom,
% 12.85/6.98 ! [M: code_integer,K: code_integer] :
% 12.85/6.98 ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
% 12.85/6.98 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4933_dvd__abs__iff,axiom,
% 12.85/6.98 ! [M: rat,K: rat] :
% 12.85/6.98 ( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
% 12.85/6.98 = ( dvd_dvd_rat @ M @ K ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4934_abs__dvd__iff,axiom,
% 12.85/6.98 ! [M: real,K: real] :
% 12.85/6.98 ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
% 12.85/6.98 = ( dvd_dvd_real @ M @ K ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4935_abs__dvd__iff,axiom,
% 12.85/6.98 ! [M: int,K: int] :
% 12.85/6.98 ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
% 12.85/6.98 = ( dvd_dvd_int @ M @ K ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4936_abs__dvd__iff,axiom,
% 12.85/6.98 ! [M: code_integer,K: code_integer] :
% 12.85/6.98 ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
% 12.85/6.98 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4937_abs__dvd__iff,axiom,
% 12.85/6.98 ! [M: rat,K: rat] :
% 12.85/6.98 ( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
% 12.85/6.98 = ( dvd_dvd_rat @ M @ K ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4938_abs__of__nat,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 12.85/6.98 = ( semiri4939895301339042750nteger @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4939_abs__of__nat,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
% 12.85/6.98 = ( semiri5074537144036343181t_real @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4940_abs__of__nat,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
% 12.85/6.98 = ( semiri681578069525770553at_rat @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4941_abs__of__nat,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
% 12.85/6.98 = ( semiri1314217659103216013at_int @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4942_abs__bool__eq,axiom,
% 12.85/6.98 ! [P: $o] :
% 12.85/6.98 ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
% 12.85/6.98 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4943_abs__bool__eq,axiom,
% 12.85/6.98 ! [P: $o] :
% 12.85/6.98 ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 12.85/6.98 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4944_abs__bool__eq,axiom,
% 12.85/6.98 ! [P: $o] :
% 12.85/6.98 ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 12.85/6.98 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4945_abs__bool__eq,axiom,
% 12.85/6.98 ! [P: $o] :
% 12.85/6.98 ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
% 12.85/6.98 = ( zero_n356916108424825756nteger @ P ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4946_tanh__0,axiom,
% 12.85/6.98 ( ( tanh_complex @ zero_zero_complex )
% 12.85/6.98 = zero_zero_complex ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4947_tanh__0,axiom,
% 12.85/6.98 ( ( tanh_real @ zero_zero_real )
% 12.85/6.98 = zero_zero_real ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4948_tanh__real__zero__iff,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( ( tanh_real @ X )
% 12.85/6.98 = zero_zero_real )
% 12.85/6.98 = ( X = zero_zero_real ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4949_tanh__real__le__iff,axiom,
% 12.85/6.98 ! [X: real,Y4: real] :
% 12.85/6.98 ( ( ord_less_eq_real @ ( tanh_real @ X ) @ ( tanh_real @ Y4 ) )
% 12.85/6.98 = ( ord_less_eq_real @ X @ Y4 ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4950_neg__less__eq__nonneg,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 12.85/6.98 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4951_neg__less__eq__nonneg,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 12.85/6.98 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4952_neg__less__eq__nonneg,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 12.85/6.98 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4953_neg__less__eq__nonneg,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 12.85/6.98 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4954_less__eq__neg__nonpos,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 12.85/6.98 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4955_less__eq__neg__nonpos,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 12.85/6.98 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4956_less__eq__neg__nonpos,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 12.85/6.98 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4957_less__eq__neg__nonpos,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 12.85/6.98 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4958_neg__le__0__iff__le,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 12.85/6.98 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4959_neg__le__0__iff__le,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 12.85/6.98 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4960_neg__le__0__iff__le,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 12.85/6.98 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4961_neg__le__0__iff__le,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 12.85/6.98 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4962_neg__0__le__iff__le,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 12.85/6.98 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4963_neg__0__le__iff__le,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 12.85/6.98 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4964_neg__0__le__iff__le,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 12.85/6.98 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4965_neg__0__le__iff__le,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 12.85/6.98 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4966_less__neg__neg,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 12.85/6.98 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4967_less__neg__neg,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 12.85/6.98 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4968_less__neg__neg,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 12.85/6.98 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4969_less__neg__neg,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 12.85/6.98 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4970_neg__less__pos,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 12.85/6.98 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4971_neg__less__pos,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 12.85/6.98 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4972_neg__less__pos,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 12.85/6.98 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4973_neg__less__pos,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 12.85/6.98 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4974_neg__0__less__iff__less,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 12.85/6.98 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4975_neg__0__less__iff__less,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 12.85/6.98 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4976_neg__0__less__iff__less,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 12.85/6.98 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4977_neg__0__less__iff__less,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 12.85/6.98 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4978_neg__less__0__iff__less,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 12.85/6.98 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4979_neg__less__0__iff__less,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 12.85/6.98 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4980_neg__less__0__iff__less,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 12.85/6.98 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4981_neg__less__0__iff__less,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 12.85/6.98 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4982_ab__left__minus,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 12.85/6.98 = zero_zero_real ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4983_ab__left__minus,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 12.85/6.98 = zero_zero_int ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4984_ab__left__minus,axiom,
% 12.85/6.98 ! [A: complex] :
% 12.85/6.98 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 12.85/6.98 = zero_zero_complex ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4985_ab__left__minus,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 12.85/6.98 = zero_z3403309356797280102nteger ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4986_ab__left__minus,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 12.85/6.98 = zero_zero_rat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4987_add_Oright__inverse,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 12.85/6.98 = zero_zero_real ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4988_add_Oright__inverse,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 12.85/6.98 = zero_zero_int ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4989_add_Oright__inverse,axiom,
% 12.85/6.98 ! [A: complex] :
% 12.85/6.98 ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 12.85/6.98 = zero_zero_complex ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4990_add_Oright__inverse,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 12.85/6.98 = zero_z3403309356797280102nteger ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4991_add_Oright__inverse,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 12.85/6.98 = zero_zero_rat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4992_verit__minus__simplify_I3_J,axiom,
% 12.85/6.98 ! [B: real] :
% 12.85/6.98 ( ( minus_minus_real @ zero_zero_real @ B )
% 12.85/6.98 = ( uminus_uminus_real @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4993_verit__minus__simplify_I3_J,axiom,
% 12.85/6.98 ! [B: int] :
% 12.85/6.98 ( ( minus_minus_int @ zero_zero_int @ B )
% 12.85/6.98 = ( uminus_uminus_int @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4994_verit__minus__simplify_I3_J,axiom,
% 12.85/6.98 ! [B: complex] :
% 12.85/6.98 ( ( minus_minus_complex @ zero_zero_complex @ B )
% 12.85/6.98 = ( uminus1482373934393186551omplex @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4995_verit__minus__simplify_I3_J,axiom,
% 12.85/6.98 ! [B: code_integer] :
% 12.85/6.98 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
% 12.85/6.98 = ( uminus1351360451143612070nteger @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4996_verit__minus__simplify_I3_J,axiom,
% 12.85/6.98 ! [B: rat] :
% 12.85/6.98 ( ( minus_minus_rat @ zero_zero_rat @ B )
% 12.85/6.98 = ( uminus_uminus_rat @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4997_diff__0,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( minus_minus_real @ zero_zero_real @ A )
% 12.85/6.98 = ( uminus_uminus_real @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4998_diff__0,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( minus_minus_int @ zero_zero_int @ A )
% 12.85/6.98 = ( uminus_uminus_int @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_4999_diff__0,axiom,
% 12.85/6.98 ! [A: complex] :
% 12.85/6.98 ( ( minus_minus_complex @ zero_zero_complex @ A )
% 12.85/6.98 = ( uminus1482373934393186551omplex @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5000_diff__0,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 12.85/6.98 = ( uminus1351360451143612070nteger @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5001_diff__0,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( minus_minus_rat @ zero_zero_rat @ A )
% 12.85/6.98 = ( uminus_uminus_rat @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5002_add__neg__numeral__simps_I3_J,axiom,
% 12.85/6.98 ! [M: num,N: num] :
% 12.85/6.98 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.85/6.98 = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5003_add__neg__numeral__simps_I3_J,axiom,
% 12.85/6.98 ! [M: num,N: num] :
% 12.85/6.98 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/6.98 = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5004_add__neg__numeral__simps_I3_J,axiom,
% 12.85/6.98 ! [M: num,N: num] :
% 12.85/6.98 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 12.85/6.98 = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5005_add__neg__numeral__simps_I3_J,axiom,
% 12.85/6.98 ! [M: num,N: num] :
% 12.85/6.98 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.85/6.98 = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5006_add__neg__numeral__simps_I3_J,axiom,
% 12.85/6.98 ! [M: num,N: num] :
% 12.85/6.98 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.85/6.98 = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5007_mult__minus1__right,axiom,
% 12.85/6.98 ! [Z: real] :
% 12.85/6.98 ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
% 12.85/6.98 = ( uminus_uminus_real @ Z ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5008_mult__minus1__right,axiom,
% 12.85/6.98 ! [Z: int] :
% 12.85/6.98 ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/6.98 = ( uminus_uminus_int @ Z ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5009_mult__minus1__right,axiom,
% 12.85/6.98 ! [Z: complex] :
% 12.85/6.98 ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.85/6.98 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5010_mult__minus1__right,axiom,
% 12.85/6.98 ! [Z: code_integer] :
% 12.85/6.98 ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/6.98 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5011_mult__minus1__right,axiom,
% 12.85/6.98 ! [Z: rat] :
% 12.85/6.98 ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.85/6.98 = ( uminus_uminus_rat @ Z ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5012_mult__minus1,axiom,
% 12.85/6.98 ! [Z: real] :
% 12.85/6.98 ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
% 12.85/6.98 = ( uminus_uminus_real @ Z ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5013_mult__minus1,axiom,
% 12.85/6.98 ! [Z: int] :
% 12.85/6.98 ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
% 12.85/6.98 = ( uminus_uminus_int @ Z ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5014_mult__minus1,axiom,
% 12.85/6.98 ! [Z: complex] :
% 12.85/6.98 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
% 12.85/6.98 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5015_mult__minus1,axiom,
% 12.85/6.98 ! [Z: code_integer] :
% 12.85/6.98 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
% 12.85/6.98 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5016_mult__minus1,axiom,
% 12.85/6.98 ! [Z: rat] :
% 12.85/6.98 ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
% 12.85/6.98 = ( uminus_uminus_rat @ Z ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5017_abs__le__zero__iff,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 12.85/6.98 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5018_abs__le__zero__iff,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 12.85/6.98 = ( A = zero_zero_rat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5019_abs__le__zero__iff,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 12.85/6.98 = ( A = zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5020_abs__le__zero__iff,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 12.85/6.98 = ( A = zero_zero_real ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5021_abs__le__self__iff,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 12.85/6.98 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5022_abs__le__self__iff,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 12.85/6.98 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5023_abs__le__self__iff,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 12.85/6.98 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5024_abs__le__self__iff,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 12.85/6.98 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5025_abs__of__nonneg,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 12.85/6.98 => ( ( abs_abs_Code_integer @ A )
% 12.85/6.98 = A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5026_abs__of__nonneg,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.98 => ( ( abs_abs_rat @ A )
% 12.85/6.98 = A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5027_abs__of__nonneg,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.98 => ( ( abs_abs_int @ A )
% 12.85/6.98 = A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5028_abs__of__nonneg,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.98 => ( ( abs_abs_real @ A )
% 12.85/6.98 = A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5029_diff__minus__eq__add,axiom,
% 12.85/6.98 ! [A: real,B: real] :
% 12.85/6.98 ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
% 12.85/6.98 = ( plus_plus_real @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5030_diff__minus__eq__add,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 12.85/6.98 = ( plus_plus_int @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5031_diff__minus__eq__add,axiom,
% 12.85/6.98 ! [A: complex,B: complex] :
% 12.85/6.98 ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 12.85/6.98 = ( plus_plus_complex @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5032_diff__minus__eq__add,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 12.85/6.98 = ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5033_diff__minus__eq__add,axiom,
% 12.85/6.98 ! [A: rat,B: rat] :
% 12.85/6.98 ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
% 12.85/6.98 = ( plus_plus_rat @ A @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5034_uminus__add__conv__diff,axiom,
% 12.85/6.98 ! [A: real,B: real] :
% 12.85/6.98 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
% 12.85/6.98 = ( minus_minus_real @ B @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5035_uminus__add__conv__diff,axiom,
% 12.85/6.98 ! [A: int,B: int] :
% 12.85/6.98 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 12.85/6.98 = ( minus_minus_int @ B @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5036_uminus__add__conv__diff,axiom,
% 12.85/6.98 ! [A: complex,B: complex] :
% 12.85/6.98 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 12.85/6.98 = ( minus_minus_complex @ B @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5037_uminus__add__conv__diff,axiom,
% 12.85/6.98 ! [A: code_integer,B: code_integer] :
% 12.85/6.98 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 12.85/6.98 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5038_uminus__add__conv__diff,axiom,
% 12.85/6.98 ! [A: rat,B: rat] :
% 12.85/6.98 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
% 12.85/6.98 = ( minus_minus_rat @ B @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5039_zero__less__abs__iff,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 12.85/6.98 = ( A != zero_z3403309356797280102nteger ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5040_zero__less__abs__iff,axiom,
% 12.85/6.98 ! [A: real] :
% 12.85/6.98 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 12.85/6.98 = ( A != zero_zero_real ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5041_zero__less__abs__iff,axiom,
% 12.85/6.98 ! [A: rat] :
% 12.85/6.98 ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 12.85/6.98 = ( A != zero_zero_rat ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5042_zero__less__abs__iff,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 12.85/6.98 = ( A != zero_zero_int ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5043_div__minus1__right,axiom,
% 12.85/6.98 ! [A: int] :
% 12.85/6.98 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/6.98 = ( uminus_uminus_int @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5044_div__minus1__right,axiom,
% 12.85/6.98 ! [A: code_integer] :
% 12.85/6.98 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/6.98 = ( uminus1351360451143612070nteger @ A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5045_divide__minus1,axiom,
% 12.85/6.98 ! [X: real] :
% 12.85/6.98 ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 12.85/6.98 = ( uminus_uminus_real @ X ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5046_divide__minus1,axiom,
% 12.85/6.98 ! [X: complex] :
% 12.85/6.98 ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.85/6.98 = ( uminus1482373934393186551omplex @ X ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5047_divide__minus1,axiom,
% 12.85/6.98 ! [X: rat] :
% 12.85/6.98 ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.85/6.98 = ( uminus_uminus_rat @ X ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5048_abs__neg__numeral,axiom,
% 12.85/6.98 ! [N: num] :
% 12.85/6.98 ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.85/6.98 = ( numeral_numeral_real @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5049_abs__neg__numeral,axiom,
% 12.85/6.98 ! [N: num] :
% 12.85/6.98 ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/6.98 = ( numeral_numeral_int @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5050_abs__neg__numeral,axiom,
% 12.85/6.98 ! [N: num] :
% 12.85/6.98 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.85/6.98 = ( numera6620942414471956472nteger @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5051_abs__neg__numeral,axiom,
% 12.85/6.98 ! [N: num] :
% 12.85/6.98 ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.85/6.98 = ( numeral_numeral_rat @ N ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5052_abs__neg__one,axiom,
% 12.85/6.98 ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
% 12.85/6.98 = one_one_real ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5053_abs__neg__one,axiom,
% 12.85/6.98 ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/6.98 = one_one_int ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5054_abs__neg__one,axiom,
% 12.85/6.98 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/6.98 = one_one_Code_integer ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5055_abs__neg__one,axiom,
% 12.85/6.98 ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.85/6.98 = one_one_rat ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5056_minus__mod__self1,axiom,
% 12.85/6.98 ! [B: int,A: int] :
% 12.85/6.98 ( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
% 12.85/6.98 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5057_minus__mod__self1,axiom,
% 12.85/6.98 ! [B: code_integer,A: code_integer] :
% 12.85/6.98 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ B )
% 12.85/6.98 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5058_negative__eq__positive,axiom,
% 12.85/6.98 ! [N: nat,M: nat] :
% 12.85/6.98 ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
% 12.85/6.98 = ( semiri1314217659103216013at_int @ M ) )
% 12.85/6.98 = ( ( N = zero_zero_nat )
% 12.85/6.98 & ( M = zero_zero_nat ) ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5059_real__add__minus__iff,axiom,
% 12.85/6.98 ! [X: real,A: real] :
% 12.85/6.98 ( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
% 12.85/6.98 = zero_zero_real )
% 12.85/6.98 = ( X = A ) ) ).
% 12.85/6.98
% 12.85/6.98 thf(fact_5060_signed__take__bit__of__minus__1,axiom,
% 12.85/6.98 ! [N: nat] :
% 12.85/6.98 ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5061_signed__take__bit__of__minus__1,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/6.99 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5062_negative__zle,axiom,
% 12.85/6.99 ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5063_tanh__real__neg__iff,axiom,
% 12.85/6.99 ! [X: real] :
% 12.85/6.99 ( ( ord_less_real @ ( tanh_real @ X ) @ zero_zero_real )
% 12.85/6.99 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5064_tanh__real__pos__iff,axiom,
% 12.85/6.99 ! [X: real] :
% 12.85/6.99 ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X ) )
% 12.85/6.99 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5065_tanh__real__nonpos__iff,axiom,
% 12.85/6.99 ! [X: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ ( tanh_real @ X ) @ zero_zero_real )
% 12.85/6.99 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5066_tanh__real__nonneg__iff,axiom,
% 12.85/6.99 ! [X: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X ) )
% 12.85/6.99 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5067_dbl__simps_I1_J,axiom,
% 12.85/6.99 ! [K: num] :
% 12.85/6.99 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 12.85/6.99 = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5068_dbl__simps_I1_J,axiom,
% 12.85/6.99 ! [K: num] :
% 12.85/6.99 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.85/6.99 = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5069_dbl__simps_I1_J,axiom,
% 12.85/6.99 ! [K: num] :
% 12.85/6.99 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5070_dbl__simps_I1_J,axiom,
% 12.85/6.99 ! [K: num] :
% 12.85/6.99 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5071_dbl__simps_I1_J,axiom,
% 12.85/6.99 ! [K: num] :
% 12.85/6.99 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 12.85/6.99 = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5072_dbl__inc__simps_I4_J,axiom,
% 12.85/6.99 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
% 12.85/6.99 = ( uminus_uminus_real @ one_one_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5073_dbl__inc__simps_I4_J,axiom,
% 12.85/6.99 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/6.99 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5074_dbl__inc__simps_I4_J,axiom,
% 12.85/6.99 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5075_dbl__inc__simps_I4_J,axiom,
% 12.85/6.99 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5076_dbl__inc__simps_I4_J,axiom,
% 12.85/6.99 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.85/6.99 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5077_add__neg__numeral__special_I8_J,axiom,
% 12.85/6.99 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 12.85/6.99 = zero_zero_real ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5078_add__neg__numeral__special_I8_J,axiom,
% 12.85/6.99 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 12.85/6.99 = zero_zero_int ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5079_add__neg__numeral__special_I8_J,axiom,
% 12.85/6.99 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 12.85/6.99 = zero_zero_complex ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5080_add__neg__numeral__special_I8_J,axiom,
% 12.85/6.99 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 12.85/6.99 = zero_z3403309356797280102nteger ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5081_add__neg__numeral__special_I8_J,axiom,
% 12.85/6.99 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 12.85/6.99 = zero_zero_rat ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5082_add__neg__numeral__special_I7_J,axiom,
% 12.85/6.99 ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 12.85/6.99 = zero_zero_real ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5083_add__neg__numeral__special_I7_J,axiom,
% 12.85/6.99 ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/6.99 = zero_zero_int ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5084_add__neg__numeral__special_I7_J,axiom,
% 12.85/6.99 ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.85/6.99 = zero_zero_complex ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5085_add__neg__numeral__special_I7_J,axiom,
% 12.85/6.99 ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/6.99 = zero_z3403309356797280102nteger ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5086_add__neg__numeral__special_I7_J,axiom,
% 12.85/6.99 ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.85/6.99 = zero_zero_rat ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5087_diff__numeral__special_I12_J,axiom,
% 12.85/6.99 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 12.85/6.99 = zero_zero_real ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5088_diff__numeral__special_I12_J,axiom,
% 12.85/6.99 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/6.99 = zero_zero_int ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5089_diff__numeral__special_I12_J,axiom,
% 12.85/6.99 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.85/6.99 = zero_zero_complex ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5090_diff__numeral__special_I12_J,axiom,
% 12.85/6.99 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/6.99 = zero_z3403309356797280102nteger ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5091_diff__numeral__special_I12_J,axiom,
% 12.85/6.99 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.85/6.99 = zero_zero_rat ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5092_neg__one__eq__numeral__iff,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( ( uminus_uminus_real @ one_one_real )
% 12.85/6.99 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.85/6.99 = ( N = one ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5093_neg__one__eq__numeral__iff,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( ( uminus_uminus_int @ one_one_int )
% 12.85/6.99 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/6.99 = ( N = one ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5094_neg__one__eq__numeral__iff,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( ( uminus1482373934393186551omplex @ one_one_complex )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 12.85/6.99 = ( N = one ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5095_neg__one__eq__numeral__iff,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.85/6.99 = ( N = one ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5096_neg__one__eq__numeral__iff,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( ( uminus_uminus_rat @ one_one_rat )
% 12.85/6.99 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.85/6.99 = ( N = one ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5097_numeral__eq__neg__one__iff,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
% 12.85/6.99 = ( uminus_uminus_real @ one_one_real ) )
% 12.85/6.99 = ( N = one ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5098_numeral__eq__neg__one__iff,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
% 12.85/6.99 = ( uminus_uminus_int @ one_one_int ) )
% 12.85/6.99 = ( N = one ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5099_numeral__eq__neg__one__iff,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.85/6.99 = ( N = one ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5100_numeral__eq__neg__one__iff,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/6.99 = ( N = one ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5101_numeral__eq__neg__one__iff,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
% 12.85/6.99 = ( uminus_uminus_rat @ one_one_rat ) )
% 12.85/6.99 = ( N = one ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5102_divide__le__0__abs__iff,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
% 12.85/6.99 = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.85/6.99 | ( B = zero_zero_rat ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5103_divide__le__0__abs__iff,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
% 12.85/6.99 = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.85/6.99 | ( B = zero_zero_real ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5104_zero__le__divide__abs__iff,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
% 12.85/6.99 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.99 | ( B = zero_zero_rat ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5105_zero__le__divide__abs__iff,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
% 12.85/6.99 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.99 | ( B = zero_zero_real ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5106_abs__of__nonpos,axiom,
% 12.85/6.99 ! [A: code_integer] :
% 12.85/6.99 ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
% 12.85/6.99 => ( ( abs_abs_Code_integer @ A )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5107_abs__of__nonpos,axiom,
% 12.85/6.99 ! [A: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 12.85/6.99 => ( ( abs_abs_rat @ A )
% 12.85/6.99 = ( uminus_uminus_rat @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5108_abs__of__nonpos,axiom,
% 12.85/6.99 ! [A: int] :
% 12.85/6.99 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 12.85/6.99 => ( ( abs_abs_int @ A )
% 12.85/6.99 = ( uminus_uminus_int @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5109_abs__of__nonpos,axiom,
% 12.85/6.99 ! [A: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.85/6.99 => ( ( abs_abs_real @ A )
% 12.85/6.99 = ( uminus_uminus_real @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5110_minus__one__mult__self,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
% 12.85/6.99 = one_one_real ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5111_minus__one__mult__self,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
% 12.85/6.99 = one_one_int ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5112_minus__one__mult__self,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
% 12.85/6.99 = one_one_complex ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5113_minus__one__mult__self,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
% 12.85/6.99 = one_one_Code_integer ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5114_minus__one__mult__self,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
% 12.85/6.99 = one_one_rat ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5115_left__minus__one__mult__self,axiom,
% 12.85/6.99 ! [N: nat,A: real] :
% 12.85/6.99 ( ( 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 ) )
% 12.85/6.99 = A ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5116_left__minus__one__mult__self,axiom,
% 12.85/6.99 ! [N: nat,A: int] :
% 12.85/6.99 ( ( 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 ) )
% 12.85/6.99 = A ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5117_left__minus__one__mult__self,axiom,
% 12.85/6.99 ! [N: nat,A: complex] :
% 12.85/6.99 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
% 12.85/6.99 = A ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5118_left__minus__one__mult__self,axiom,
% 12.85/6.99 ! [N: nat,A: code_integer] :
% 12.85/6.99 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
% 12.85/6.99 = A ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5119_left__minus__one__mult__self,axiom,
% 12.85/6.99 ! [N: nat,A: rat] :
% 12.85/6.99 ( ( 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 ) )
% 12.85/6.99 = A ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5120_mod__minus1__right,axiom,
% 12.85/6.99 ! [A: int] :
% 12.85/6.99 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/6.99 = zero_zero_int ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5121_mod__minus1__right,axiom,
% 12.85/6.99 ! [A: code_integer] :
% 12.85/6.99 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/6.99 = zero_z3403309356797280102nteger ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5122_norm__neg__numeral,axiom,
% 12.85/6.99 ! [W: num] :
% 12.85/6.99 ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.85/6.99 = ( numeral_numeral_real @ W ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5123_norm__neg__numeral,axiom,
% 12.85/6.99 ! [W: num] :
% 12.85/6.99 ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 12.85/6.99 = ( numeral_numeral_real @ W ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5124_semiring__norm_I168_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: real] :
% 12.85/6.99 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y4 ) )
% 12.85/6.99 = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5125_semiring__norm_I168_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: int] :
% 12.85/6.99 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y4 ) )
% 12.85/6.99 = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5126_semiring__norm_I168_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: complex] :
% 12.85/6.99 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y4 ) )
% 12.85/6.99 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5127_semiring__norm_I168_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: code_integer] :
% 12.85/6.99 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y4 ) )
% 12.85/6.99 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5128_semiring__norm_I168_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: rat] :
% 12.85/6.99 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y4 ) )
% 12.85/6.99 = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5129_diff__numeral__simps_I3_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 12.85/6.99 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5130_diff__numeral__simps_I3_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.85/6.99 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5131_diff__numeral__simps_I3_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5132_diff__numeral__simps_I3_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5133_diff__numeral__simps_I3_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 12.85/6.99 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5134_diff__numeral__simps_I2_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.85/6.99 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5135_diff__numeral__simps_I2_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/6.99 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5136_diff__numeral__simps_I2_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 12.85/6.99 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5137_diff__numeral__simps_I2_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.85/6.99 = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5138_diff__numeral__simps_I2_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.85/6.99 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5139_semiring__norm_I172_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: real] :
% 12.85/6.99 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y4 ) )
% 12.85/6.99 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5140_semiring__norm_I172_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: int] :
% 12.85/6.99 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y4 ) )
% 12.85/6.99 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5141_semiring__norm_I172_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: complex] :
% 12.85/6.99 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y4 ) )
% 12.85/6.99 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5142_semiring__norm_I172_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: code_integer] :
% 12.85/6.99 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y4 ) )
% 12.85/6.99 = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5143_semiring__norm_I172_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: rat] :
% 12.85/6.99 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y4 ) )
% 12.85/6.99 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5144_semiring__norm_I171_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: real] :
% 12.85/6.99 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y4 ) )
% 12.85/6.99 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5145_semiring__norm_I171_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: int] :
% 12.85/6.99 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y4 ) )
% 12.85/6.99 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5146_semiring__norm_I171_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: complex] :
% 12.85/6.99 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y4 ) )
% 12.85/6.99 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5147_semiring__norm_I171_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: code_integer] :
% 12.85/6.99 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y4 ) )
% 12.85/6.99 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5148_semiring__norm_I171_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: rat] :
% 12.85/6.99 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y4 ) )
% 12.85/6.99 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5149_semiring__norm_I170_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: real] :
% 12.85/6.99 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y4 ) )
% 12.85/6.99 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5150_semiring__norm_I170_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: int] :
% 12.85/6.99 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y4 ) )
% 12.85/6.99 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5151_semiring__norm_I170_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: complex] :
% 12.85/6.99 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y4 ) )
% 12.85/6.99 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5152_semiring__norm_I170_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: code_integer] :
% 12.85/6.99 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y4 ) )
% 12.85/6.99 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5153_semiring__norm_I170_J,axiom,
% 12.85/6.99 ! [V: num,W: num,Y4: rat] :
% 12.85/6.99 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y4 ) )
% 12.85/6.99 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5154_mult__neg__numeral__simps_I3_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.85/6.99 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5155_mult__neg__numeral__simps_I3_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/6.99 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5156_mult__neg__numeral__simps_I3_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5157_mult__neg__numeral__simps_I3_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5158_mult__neg__numeral__simps_I3_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.85/6.99 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5159_mult__neg__numeral__simps_I2_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 12.85/6.99 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5160_mult__neg__numeral__simps_I2_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.85/6.99 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5161_mult__neg__numeral__simps_I2_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5162_mult__neg__numeral__simps_I2_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5163_mult__neg__numeral__simps_I2_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 12.85/6.99 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5164_mult__neg__numeral__simps_I1_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.85/6.99 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5165_mult__neg__numeral__simps_I1_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/6.99 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5166_mult__neg__numeral__simps_I1_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 12.85/6.99 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5167_mult__neg__numeral__simps_I1_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.85/6.99 = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5168_mult__neg__numeral__simps_I1_J,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.85/6.99 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5169_negative__zless,axiom,
% 12.85/6.99 ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5170_neg__numeral__le__iff,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.85/6.99 = ( ord_less_eq_num @ N @ M ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5171_neg__numeral__le__iff,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.85/6.99 = ( ord_less_eq_num @ N @ M ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5172_neg__numeral__le__iff,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/6.99 = ( ord_less_eq_num @ N @ M ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5173_neg__numeral__le__iff,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.85/6.99 = ( ord_less_eq_num @ N @ M ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5174_neg__numeral__less__iff,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.85/6.99 = ( ord_less_num @ N @ M ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5175_neg__numeral__less__iff,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/6.99 = ( ord_less_num @ N @ M ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5176_neg__numeral__less__iff,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.85/6.99 = ( ord_less_num @ N @ M ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5177_neg__numeral__less__iff,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.85/6.99 = ( ord_less_num @ N @ M ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5178_not__neg__one__le__neg__numeral__iff,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 12.85/6.99 = ( M != one ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5179_not__neg__one__le__neg__numeral__iff,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 12.85/6.99 = ( M != one ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5180_not__neg__one__le__neg__numeral__iff,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 12.85/6.99 = ( M != one ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5181_not__neg__one__le__neg__numeral__iff,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 12.85/6.99 = ( M != one ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5182_divide__le__eq__numeral1_I2_J,axiom,
% 12.85/6.99 ! [B: rat,W: num,A: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 12.85/6.99 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5183_divide__le__eq__numeral1_I2_J,axiom,
% 12.85/6.99 ! [B: real,W: num,A: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 12.85/6.99 = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5184_le__divide__eq__numeral1_I2_J,axiom,
% 12.85/6.99 ! [A: rat,B: rat,W: num] :
% 12.85/6.99 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 12.85/6.99 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5185_le__divide__eq__numeral1_I2_J,axiom,
% 12.85/6.99 ! [A: real,B: real,W: num] :
% 12.85/6.99 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 12.85/6.99 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5186_divide__eq__eq__numeral1_I2_J,axiom,
% 12.85/6.99 ! [B: real,W: num,A: real] :
% 12.85/6.99 ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.85/6.99 = A )
% 12.85/6.99 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 12.85/6.99 != zero_zero_real )
% 12.85/6.99 => ( B
% 12.85/6.99 = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
% 12.85/6.99 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 12.85/6.99 = zero_zero_real )
% 12.85/6.99 => ( A = zero_zero_real ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5187_divide__eq__eq__numeral1_I2_J,axiom,
% 12.85/6.99 ! [B: complex,W: num,A: complex] :
% 12.85/6.99 ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 12.85/6.99 = A )
% 12.85/6.99 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 12.85/6.99 != zero_zero_complex )
% 12.85/6.99 => ( B
% 12.85/6.99 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
% 12.85/6.99 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 12.85/6.99 = zero_zero_complex )
% 12.85/6.99 => ( A = zero_zero_complex ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5188_divide__eq__eq__numeral1_I2_J,axiom,
% 12.85/6.99 ! [B: rat,W: num,A: rat] :
% 12.85/6.99 ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 12.85/6.99 = A )
% 12.85/6.99 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 12.85/6.99 != zero_zero_rat )
% 12.85/6.99 => ( B
% 12.85/6.99 = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
% 12.85/6.99 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 12.85/6.99 = zero_zero_rat )
% 12.85/6.99 => ( A = zero_zero_rat ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5189_eq__divide__eq__numeral1_I2_J,axiom,
% 12.85/6.99 ! [A: real,B: real,W: num] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 12.85/6.99 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 12.85/6.99 != zero_zero_real )
% 12.85/6.99 => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.85/6.99 = B ) )
% 12.85/6.99 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 12.85/6.99 = zero_zero_real )
% 12.85/6.99 => ( A = zero_zero_real ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5190_eq__divide__eq__numeral1_I2_J,axiom,
% 12.85/6.99 ! [A: complex,B: complex,W: num] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
% 12.85/6.99 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 12.85/6.99 != zero_zero_complex )
% 12.85/6.99 => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 12.85/6.99 = B ) )
% 12.85/6.99 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 12.85/6.99 = zero_zero_complex )
% 12.85/6.99 => ( A = zero_zero_complex ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5191_eq__divide__eq__numeral1_I2_J,axiom,
% 12.85/6.99 ! [A: rat,B: rat,W: num] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 12.85/6.99 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 12.85/6.99 != zero_zero_rat )
% 12.85/6.99 => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 12.85/6.99 = B ) )
% 12.85/6.99 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 12.85/6.99 = zero_zero_rat )
% 12.85/6.99 => ( A = zero_zero_rat ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5192_neg__numeral__less__neg__one__iff,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 12.85/6.99 = ( M != one ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5193_neg__numeral__less__neg__one__iff,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/6.99 = ( M != one ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5194_neg__numeral__less__neg__one__iff,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/6.99 = ( M != one ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5195_neg__numeral__less__neg__one__iff,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.85/6.99 = ( M != one ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5196_divide__less__eq__numeral1_I2_J,axiom,
% 12.85/6.99 ! [B: real,W: num,A: real] :
% 12.85/6.99 ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 12.85/6.99 = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5197_divide__less__eq__numeral1_I2_J,axiom,
% 12.85/6.99 ! [B: rat,W: num,A: rat] :
% 12.85/6.99 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 12.85/6.99 = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5198_less__divide__eq__numeral1_I2_J,axiom,
% 12.85/6.99 ! [A: real,B: real,W: num] :
% 12.85/6.99 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 12.85/6.99 = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5199_less__divide__eq__numeral1_I2_J,axiom,
% 12.85/6.99 ! [A: rat,B: rat,W: num] :
% 12.85/6.99 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 12.85/6.99 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5200_power2__minus,axiom,
% 12.85/6.99 ! [A: real] :
% 12.85/6.99 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5201_power2__minus,axiom,
% 12.85/6.99 ! [A: int] :
% 12.85/6.99 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5202_power2__minus,axiom,
% 12.85/6.99 ! [A: complex] :
% 12.85/6.99 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5203_power2__minus,axiom,
% 12.85/6.99 ! [A: code_integer] :
% 12.85/6.99 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5204_power2__minus,axiom,
% 12.85/6.99 ! [A: rat] :
% 12.85/6.99 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5205_zero__less__power__abs__iff,axiom,
% 12.85/6.99 ! [A: code_integer,N: nat] :
% 12.85/6.99 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
% 12.85/6.99 = ( ( A != zero_z3403309356797280102nteger )
% 12.85/6.99 | ( N = zero_zero_nat ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5206_zero__less__power__abs__iff,axiom,
% 12.85/6.99 ! [A: real,N: nat] :
% 12.85/6.99 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 12.85/6.99 = ( ( A != zero_zero_real )
% 12.85/6.99 | ( N = zero_zero_nat ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5207_zero__less__power__abs__iff,axiom,
% 12.85/6.99 ! [A: rat,N: nat] :
% 12.85/6.99 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
% 12.85/6.99 = ( ( A != zero_zero_rat )
% 12.85/6.99 | ( N = zero_zero_nat ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5208_zero__less__power__abs__iff,axiom,
% 12.85/6.99 ! [A: int,N: nat] :
% 12.85/6.99 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
% 12.85/6.99 = ( ( A != zero_zero_int )
% 12.85/6.99 | ( N = zero_zero_nat ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5209_abs__power2,axiom,
% 12.85/6.99 ! [A: code_integer] :
% 12.85/6.99 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.99 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5210_abs__power2,axiom,
% 12.85/6.99 ! [A: rat] :
% 12.85/6.99 ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.99 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5211_abs__power2,axiom,
% 12.85/6.99 ! [A: int] :
% 12.85/6.99 ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.99 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5212_abs__power2,axiom,
% 12.85/6.99 ! [A: real] :
% 12.85/6.99 ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.99 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5213_power2__abs,axiom,
% 12.85/6.99 ! [A: code_integer] :
% 12.85/6.99 ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5214_power2__abs,axiom,
% 12.85/6.99 ! [A: rat] :
% 12.85/6.99 ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5215_power2__abs,axiom,
% 12.85/6.99 ! [A: int] :
% 12.85/6.99 ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5216_power2__abs,axiom,
% 12.85/6.99 ! [A: real] :
% 12.85/6.99 ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5217_add__neg__numeral__special_I9_J,axiom,
% 12.85/6.99 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 12.85/6.99 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5218_add__neg__numeral__special_I9_J,axiom,
% 12.85/6.99 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/6.99 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5219_add__neg__numeral__special_I9_J,axiom,
% 12.85/6.99 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5220_add__neg__numeral__special_I9_J,axiom,
% 12.85/6.99 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5221_add__neg__numeral__special_I9_J,axiom,
% 12.85/6.99 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.85/6.99 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5222_diff__numeral__special_I11_J,axiom,
% 12.85/6.99 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 12.85/6.99 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5223_diff__numeral__special_I11_J,axiom,
% 12.85/6.99 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/6.99 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5224_diff__numeral__special_I11_J,axiom,
% 12.85/6.99 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.85/6.99 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5225_diff__numeral__special_I11_J,axiom,
% 12.85/6.99 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/6.99 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5226_diff__numeral__special_I11_J,axiom,
% 12.85/6.99 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.85/6.99 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5227_diff__numeral__special_I10_J,axiom,
% 12.85/6.99 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 12.85/6.99 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5228_diff__numeral__special_I10_J,axiom,
% 12.85/6.99 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 12.85/6.99 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5229_diff__numeral__special_I10_J,axiom,
% 12.85/6.99 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5230_diff__numeral__special_I10_J,axiom,
% 12.85/6.99 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5231_diff__numeral__special_I10_J,axiom,
% 12.85/6.99 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 12.85/6.99 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5232_minus__1__div__2__eq,axiom,
% 12.85/6.99 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.99 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5233_minus__1__div__2__eq,axiom,
% 12.85/6.99 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5234_minus__1__mod__2__eq,axiom,
% 12.85/6.99 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.99 = one_one_int ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5235_minus__1__mod__2__eq,axiom,
% 12.85/6.99 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.99 = one_one_Code_integer ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5236_bits__minus__1__mod__2__eq,axiom,
% 12.85/6.99 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/6.99 = one_one_int ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5237_bits__minus__1__mod__2__eq,axiom,
% 12.85/6.99 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/6.99 = one_one_Code_integer ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5238_Power_Oring__1__class_Opower__minus__even,axiom,
% 12.85/6.99 ! [A: real,N: nat] :
% 12.85/6.99 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.99 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5239_Power_Oring__1__class_Opower__minus__even,axiom,
% 12.85/6.99 ! [A: int,N: nat] :
% 12.85/6.99 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.99 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5240_Power_Oring__1__class_Opower__minus__even,axiom,
% 12.85/6.99 ! [A: complex,N: nat] :
% 12.85/6.99 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.99 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5241_Power_Oring__1__class_Opower__minus__even,axiom,
% 12.85/6.99 ! [A: code_integer,N: nat] :
% 12.85/6.99 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.99 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5242_Power_Oring__1__class_Opower__minus__even,axiom,
% 12.85/6.99 ! [A: rat,N: nat] :
% 12.85/6.99 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.99 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5243_power__minus__odd,axiom,
% 12.85/6.99 ! [N: nat,A: real] :
% 12.85/6.99 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 12.85/6.99 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5244_power__minus__odd,axiom,
% 12.85/6.99 ! [N: nat,A: int] :
% 12.85/6.99 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 12.85/6.99 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5245_power__minus__odd,axiom,
% 12.85/6.99 ! [N: nat,A: complex] :
% 12.85/6.99 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5246_power__minus__odd,axiom,
% 12.85/6.99 ! [N: nat,A: code_integer] :
% 12.85/6.99 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5247_power__minus__odd,axiom,
% 12.85/6.99 ! [N: nat,A: rat] :
% 12.85/6.99 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 12.85/6.99 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5248_Parity_Oring__1__class_Opower__minus__even,axiom,
% 12.85/6.99 ! [N: nat,A: real] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 12.85/6.99 = ( power_power_real @ A @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5249_Parity_Oring__1__class_Opower__minus__even,axiom,
% 12.85/6.99 ! [N: nat,A: int] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 12.85/6.99 = ( power_power_int @ A @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5250_Parity_Oring__1__class_Opower__minus__even,axiom,
% 12.85/6.99 ! [N: nat,A: complex] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 12.85/6.99 = ( power_power_complex @ A @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5251_Parity_Oring__1__class_Opower__minus__even,axiom,
% 12.85/6.99 ! [N: nat,A: code_integer] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 12.85/6.99 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5252_Parity_Oring__1__class_Opower__minus__even,axiom,
% 12.85/6.99 ! [N: nat,A: rat] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 12.85/6.99 = ( power_power_rat @ A @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5253_power__even__abs__numeral,axiom,
% 12.85/6.99 ! [W: num,A: code_integer] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.99 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.99 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5254_power__even__abs__numeral,axiom,
% 12.85/6.99 ! [W: num,A: rat] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.99 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.99 = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5255_power__even__abs__numeral,axiom,
% 12.85/6.99 ! [W: num,A: int] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.99 => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.99 = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5256_power__even__abs__numeral,axiom,
% 12.85/6.99 ! [W: num,A: real] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.99 => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
% 12.85/6.99 = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5257_diff__numeral__special_I3_J,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.85/6.99 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5258_diff__numeral__special_I3_J,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/6.99 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5259_diff__numeral__special_I3_J,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 12.85/6.99 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5260_diff__numeral__special_I3_J,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.85/6.99 = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5261_diff__numeral__special_I3_J,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.85/6.99 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5262_diff__numeral__special_I4_J,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
% 12.85/6.99 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5263_diff__numeral__special_I4_J,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
% 12.85/6.99 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5264_diff__numeral__special_I4_J,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5265_diff__numeral__special_I4_J,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5266_diff__numeral__special_I4_J,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
% 12.85/6.99 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5267_dbl__simps_I4_J,axiom,
% 12.85/6.99 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
% 12.85/6.99 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5268_dbl__simps_I4_J,axiom,
% 12.85/6.99 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/6.99 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5269_dbl__simps_I4_J,axiom,
% 12.85/6.99 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5270_dbl__simps_I4_J,axiom,
% 12.85/6.99 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5271_dbl__simps_I4_J,axiom,
% 12.85/6.99 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.85/6.99 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5272_power__minus1__even,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.99 = one_one_real ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5273_power__minus1__even,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.99 = one_one_int ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5274_power__minus1__even,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.99 = one_one_complex ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5275_power__minus1__even,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.99 = one_one_Code_integer ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5276_power__minus1__even,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/6.99 = one_one_rat ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5277_neg__one__odd__power,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 12.85/6.99 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5278_neg__one__odd__power,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 12.85/6.99 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5279_neg__one__odd__power,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5280_neg__one__odd__power,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5281_neg__one__odd__power,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 12.85/6.99 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5282_neg__one__even__power,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 12.85/6.99 = one_one_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5283_neg__one__even__power,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 12.85/6.99 = one_one_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5284_neg__one__even__power,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 12.85/6.99 = one_one_complex ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5285_neg__one__even__power,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 12.85/6.99 = one_one_Code_integer ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5286_neg__one__even__power,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 12.85/6.99 = one_one_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5287_signed__take__bit__0,axiom,
% 12.85/6.99 ! [A: code_integer] :
% 12.85/6.99 ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5288_signed__take__bit__0,axiom,
% 12.85/6.99 ! [A: int] :
% 12.85/6.99 ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 12.85/6.99 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5289_signed__take__bit__Suc__minus__bit0,axiom,
% 12.85/6.99 ! [N: nat,K: num] :
% 12.85/6.99 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 12.85/6.99 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5290_signed__take__bit__minus,axiom,
% 12.85/6.99 ! [N: nat,K: int] :
% 12.85/6.99 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
% 12.85/6.99 = ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5291_equation__minus__iff,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( uminus_uminus_real @ B ) )
% 12.85/6.99 = ( B
% 12.85/6.99 = ( uminus_uminus_real @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5292_equation__minus__iff,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( uminus_uminus_int @ B ) )
% 12.85/6.99 = ( B
% 12.85/6.99 = ( uminus_uminus_int @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5293_equation__minus__iff,axiom,
% 12.85/6.99 ! [A: complex,B: complex] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( uminus1482373934393186551omplex @ B ) )
% 12.85/6.99 = ( B
% 12.85/6.99 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5294_equation__minus__iff,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( uminus1351360451143612070nteger @ B ) )
% 12.85/6.99 = ( B
% 12.85/6.99 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5295_equation__minus__iff,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( uminus_uminus_rat @ B ) )
% 12.85/6.99 = ( B
% 12.85/6.99 = ( uminus_uminus_rat @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5296_minus__equation__iff,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ( uminus_uminus_real @ A )
% 12.85/6.99 = B )
% 12.85/6.99 = ( ( uminus_uminus_real @ B )
% 12.85/6.99 = A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5297_minus__equation__iff,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ( uminus_uminus_int @ A )
% 12.85/6.99 = B )
% 12.85/6.99 = ( ( uminus_uminus_int @ B )
% 12.85/6.99 = A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5298_minus__equation__iff,axiom,
% 12.85/6.99 ! [A: complex,B: complex] :
% 12.85/6.99 ( ( ( uminus1482373934393186551omplex @ A )
% 12.85/6.99 = B )
% 12.85/6.99 = ( ( uminus1482373934393186551omplex @ B )
% 12.85/6.99 = A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5299_minus__equation__iff,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( ( uminus1351360451143612070nteger @ A )
% 12.85/6.99 = B )
% 12.85/6.99 = ( ( uminus1351360451143612070nteger @ B )
% 12.85/6.99 = A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5300_minus__equation__iff,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ( uminus_uminus_rat @ A )
% 12.85/6.99 = B )
% 12.85/6.99 = ( ( uminus_uminus_rat @ B )
% 12.85/6.99 = A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5301_abs__eq__iff,axiom,
% 12.85/6.99 ! [X: real,Y4: real] :
% 12.85/6.99 ( ( ( abs_abs_real @ X )
% 12.85/6.99 = ( abs_abs_real @ Y4 ) )
% 12.85/6.99 = ( ( X = Y4 )
% 12.85/6.99 | ( X
% 12.85/6.99 = ( uminus_uminus_real @ Y4 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5302_abs__eq__iff,axiom,
% 12.85/6.99 ! [X: int,Y4: int] :
% 12.85/6.99 ( ( ( abs_abs_int @ X )
% 12.85/6.99 = ( abs_abs_int @ Y4 ) )
% 12.85/6.99 = ( ( X = Y4 )
% 12.85/6.99 | ( X
% 12.85/6.99 = ( uminus_uminus_int @ Y4 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5303_abs__eq__iff,axiom,
% 12.85/6.99 ! [X: code_integer,Y4: code_integer] :
% 12.85/6.99 ( ( ( abs_abs_Code_integer @ X )
% 12.85/6.99 = ( abs_abs_Code_integer @ Y4 ) )
% 12.85/6.99 = ( ( X = Y4 )
% 12.85/6.99 | ( X
% 12.85/6.99 = ( uminus1351360451143612070nteger @ Y4 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5304_abs__eq__iff,axiom,
% 12.85/6.99 ! [X: rat,Y4: rat] :
% 12.85/6.99 ( ( ( abs_abs_rat @ X )
% 12.85/6.99 = ( abs_abs_rat @ Y4 ) )
% 12.85/6.99 = ( ( X = Y4 )
% 12.85/6.99 | ( X
% 12.85/6.99 = ( uminus_uminus_rat @ Y4 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5305_verit__negate__coefficient_I3_J,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( A = B )
% 12.85/6.99 => ( ( uminus_uminus_real @ A )
% 12.85/6.99 = ( uminus_uminus_real @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5306_verit__negate__coefficient_I3_J,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( A = B )
% 12.85/6.99 => ( ( uminus_uminus_int @ A )
% 12.85/6.99 = ( uminus_uminus_int @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5307_verit__negate__coefficient_I3_J,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( A = B )
% 12.85/6.99 => ( ( uminus1351360451143612070nteger @ A )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5308_verit__negate__coefficient_I3_J,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( A = B )
% 12.85/6.99 => ( ( uminus_uminus_rat @ A )
% 12.85/6.99 = ( uminus_uminus_rat @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5309_abs__ge__minus__self,axiom,
% 12.85/6.99 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5310_abs__ge__minus__self,axiom,
% 12.85/6.99 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5311_abs__ge__minus__self,axiom,
% 12.85/6.99 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5312_abs__ge__minus__self,axiom,
% 12.85/6.99 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5313_abs__le__iff,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 12.85/6.99 = ( ( ord_le3102999989581377725nteger @ A @ B )
% 12.85/6.99 & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5314_abs__le__iff,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 12.85/6.99 = ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.99 & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5315_abs__le__iff,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 12.85/6.99 = ( ( ord_less_eq_int @ A @ B )
% 12.85/6.99 & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5316_abs__le__iff,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 12.85/6.99 = ( ( ord_less_eq_real @ A @ B )
% 12.85/6.99 & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5317_abs__le__D2,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 12.85/6.99 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5318_abs__le__D2,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 12.85/6.99 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5319_abs__le__D2,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 12.85/6.99 => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5320_abs__le__D2,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 12.85/6.99 => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5321_abs__leI,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( ord_le3102999989581377725nteger @ A @ B )
% 12.85/6.99 => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 12.85/6.99 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5322_abs__leI,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.99 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 12.85/6.99 => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5323_abs__leI,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.99 => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 12.85/6.99 => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5324_abs__leI,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.99 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 12.85/6.99 => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5325_abs__less__iff,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
% 12.85/6.99 = ( ( ord_less_real @ A @ B )
% 12.85/6.99 & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5326_abs__less__iff,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
% 12.85/6.99 = ( ( ord_less_int @ A @ B )
% 12.85/6.99 & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5327_abs__less__iff,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 12.85/6.99 = ( ( ord_le6747313008572928689nteger @ A @ B )
% 12.85/6.99 & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5328_abs__less__iff,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
% 12.85/6.99 = ( ( ord_less_rat @ A @ B )
% 12.85/6.99 & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5329_eq__abs__iff_H,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( abs_abs_Code_integer @ B ) )
% 12.85/6.99 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 12.85/6.99 & ( ( B = A )
% 12.85/6.99 | ( B
% 12.85/6.99 = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5330_eq__abs__iff_H,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( abs_abs_rat @ B ) )
% 12.85/6.99 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.99 & ( ( B = A )
% 12.85/6.99 | ( B
% 12.85/6.99 = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5331_eq__abs__iff_H,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( abs_abs_int @ B ) )
% 12.85/6.99 = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.99 & ( ( B = A )
% 12.85/6.99 | ( B
% 12.85/6.99 = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5332_eq__abs__iff_H,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( abs_abs_real @ B ) )
% 12.85/6.99 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.99 & ( ( B = A )
% 12.85/6.99 | ( B
% 12.85/6.99 = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5333_abs__eq__iff_H,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( ( abs_abs_Code_integer @ A )
% 12.85/6.99 = B )
% 12.85/6.99 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 12.85/6.99 & ( ( A = B )
% 12.85/6.99 | ( A
% 12.85/6.99 = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5334_abs__eq__iff_H,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ( abs_abs_rat @ A )
% 12.85/6.99 = B )
% 12.85/6.99 = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 12.85/6.99 & ( ( A = B )
% 12.85/6.99 | ( A
% 12.85/6.99 = ( uminus_uminus_rat @ B ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5335_abs__eq__iff_H,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ( abs_abs_int @ A )
% 12.85/6.99 = B )
% 12.85/6.99 = ( ( ord_less_eq_int @ zero_zero_int @ B )
% 12.85/6.99 & ( ( A = B )
% 12.85/6.99 | ( A
% 12.85/6.99 = ( uminus_uminus_int @ B ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5336_abs__eq__iff_H,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ( abs_abs_real @ A )
% 12.85/6.99 = B )
% 12.85/6.99 = ( ( ord_less_eq_real @ zero_zero_real @ B )
% 12.85/6.99 & ( ( A = B )
% 12.85/6.99 | ( A
% 12.85/6.99 = ( uminus_uminus_real @ B ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5337_abs__minus__le__zero,axiom,
% 12.85/6.99 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5338_abs__minus__le__zero,axiom,
% 12.85/6.99 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5339_abs__minus__le__zero,axiom,
% 12.85/6.99 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5340_abs__minus__le__zero,axiom,
% 12.85/6.99 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5341_abs__if__raw,axiom,
% 12.85/6.99 ( abs_abs_real
% 12.85/6.99 = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5342_abs__if__raw,axiom,
% 12.85/6.99 ( abs_abs_int
% 12.85/6.99 = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5343_abs__if__raw,axiom,
% 12.85/6.99 ( abs_abs_Code_integer
% 12.85/6.99 = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5344_abs__if__raw,axiom,
% 12.85/6.99 ( abs_abs_rat
% 12.85/6.99 = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5345_abs__of__neg,axiom,
% 12.85/6.99 ! [A: real] :
% 12.85/6.99 ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/6.99 => ( ( abs_abs_real @ A )
% 12.85/6.99 = ( uminus_uminus_real @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5346_abs__of__neg,axiom,
% 12.85/6.99 ! [A: int] :
% 12.85/6.99 ( ( ord_less_int @ A @ zero_zero_int )
% 12.85/6.99 => ( ( abs_abs_int @ A )
% 12.85/6.99 = ( uminus_uminus_int @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5347_abs__of__neg,axiom,
% 12.85/6.99 ! [A: code_integer] :
% 12.85/6.99 ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 12.85/6.99 => ( ( abs_abs_Code_integer @ A )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5348_abs__of__neg,axiom,
% 12.85/6.99 ! [A: rat] :
% 12.85/6.99 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/6.99 => ( ( abs_abs_rat @ A )
% 12.85/6.99 = ( uminus_uminus_rat @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5349_abs__if,axiom,
% 12.85/6.99 ( abs_abs_real
% 12.85/6.99 = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5350_abs__if,axiom,
% 12.85/6.99 ( abs_abs_int
% 12.85/6.99 = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5351_abs__if,axiom,
% 12.85/6.99 ( abs_abs_Code_integer
% 12.85/6.99 = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5352_abs__if,axiom,
% 12.85/6.99 ( abs_abs_rat
% 12.85/6.99 = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5353_abs__real__def,axiom,
% 12.85/6.99 ( abs_abs_real
% 12.85/6.99 = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5354_abs__le__D1,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 12.85/6.99 => ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5355_abs__le__D1,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 12.85/6.99 => ( ord_less_eq_rat @ A @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5356_abs__le__D1,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 12.85/6.99 => ( ord_less_eq_int @ A @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5357_abs__le__D1,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 12.85/6.99 => ( ord_less_eq_real @ A @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5358_abs__ge__self,axiom,
% 12.85/6.99 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5359_abs__ge__self,axiom,
% 12.85/6.99 ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5360_abs__ge__self,axiom,
% 12.85/6.99 ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5361_abs__ge__self,axiom,
% 12.85/6.99 ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5362_abs__eq__0__iff,axiom,
% 12.85/6.99 ! [A: code_integer] :
% 12.85/6.99 ( ( ( abs_abs_Code_integer @ A )
% 12.85/6.99 = zero_z3403309356797280102nteger )
% 12.85/6.99 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5363_abs__eq__0__iff,axiom,
% 12.85/6.99 ! [A: complex] :
% 12.85/6.99 ( ( ( abs_abs_complex @ A )
% 12.85/6.99 = zero_zero_complex )
% 12.85/6.99 = ( A = zero_zero_complex ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5364_abs__eq__0__iff,axiom,
% 12.85/6.99 ! [A: real] :
% 12.85/6.99 ( ( ( abs_abs_real @ A )
% 12.85/6.99 = zero_zero_real )
% 12.85/6.99 = ( A = zero_zero_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5365_abs__eq__0__iff,axiom,
% 12.85/6.99 ! [A: rat] :
% 12.85/6.99 ( ( ( abs_abs_rat @ A )
% 12.85/6.99 = zero_zero_rat )
% 12.85/6.99 = ( A = zero_zero_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5366_abs__eq__0__iff,axiom,
% 12.85/6.99 ! [A: int] :
% 12.85/6.99 ( ( ( abs_abs_int @ A )
% 12.85/6.99 = zero_zero_int )
% 12.85/6.99 = ( A = zero_zero_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5367_abs__mult,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 12.85/6.99 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5368_abs__mult,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 12.85/6.99 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5369_abs__mult,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 12.85/6.99 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5370_abs__mult,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 12.85/6.99 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5371_abs__one,axiom,
% 12.85/6.99 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 12.85/6.99 = one_one_Code_integer ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5372_abs__one,axiom,
% 12.85/6.99 ( ( abs_abs_real @ one_one_real )
% 12.85/6.99 = one_one_real ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5373_abs__one,axiom,
% 12.85/6.99 ( ( abs_abs_rat @ one_one_rat )
% 12.85/6.99 = one_one_rat ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5374_abs__one,axiom,
% 12.85/6.99 ( ( abs_abs_int @ one_one_int )
% 12.85/6.99 = one_one_int ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5375_abs__minus__commute,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
% 12.85/6.99 = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5376_abs__minus__commute,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 12.85/6.99 = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5377_abs__minus__commute,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
% 12.85/6.99 = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5378_abs__minus__commute,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
% 12.85/6.99 = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5379_dvd__if__abs__eq,axiom,
% 12.85/6.99 ! [L: real,K: real] :
% 12.85/6.99 ( ( ( abs_abs_real @ L )
% 12.85/6.99 = ( abs_abs_real @ K ) )
% 12.85/6.99 => ( dvd_dvd_real @ L @ K ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5380_dvd__if__abs__eq,axiom,
% 12.85/6.99 ! [L: int,K: int] :
% 12.85/6.99 ( ( ( abs_abs_int @ L )
% 12.85/6.99 = ( abs_abs_int @ K ) )
% 12.85/6.99 => ( dvd_dvd_int @ L @ K ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5381_dvd__if__abs__eq,axiom,
% 12.85/6.99 ! [L: code_integer,K: code_integer] :
% 12.85/6.99 ( ( ( abs_abs_Code_integer @ L )
% 12.85/6.99 = ( abs_abs_Code_integer @ K ) )
% 12.85/6.99 => ( dvd_dvd_Code_integer @ L @ K ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5382_dvd__if__abs__eq,axiom,
% 12.85/6.99 ! [L: rat,K: rat] :
% 12.85/6.99 ( ( ( abs_abs_rat @ L )
% 12.85/6.99 = ( abs_abs_rat @ K ) )
% 12.85/6.99 => ( dvd_dvd_rat @ L @ K ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5383_le__minus__iff,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 12.85/6.99 = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5384_le__minus__iff,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
% 12.85/6.99 = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5385_le__minus__iff,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 12.85/6.99 = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5386_le__minus__iff,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 12.85/6.99 = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5387_minus__le__iff,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 12.85/6.99 = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5388_minus__le__iff,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 12.85/6.99 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5389_minus__le__iff,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 12.85/6.99 = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5390_minus__le__iff,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 12.85/6.99 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5391_le__imp__neg__le,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( ord_le3102999989581377725nteger @ A @ B )
% 12.85/6.99 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5392_le__imp__neg__le,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ A @ B )
% 12.85/6.99 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5393_le__imp__neg__le,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ord_less_eq_int @ A @ B )
% 12.85/6.99 => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5394_le__imp__neg__le,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ A @ B )
% 12.85/6.99 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5395_less__minus__iff,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 12.85/6.99 = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5396_less__minus__iff,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
% 12.85/6.99 = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5397_less__minus__iff,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 12.85/6.99 = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5398_less__minus__iff,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
% 12.85/6.99 = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5399_minus__less__iff,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 12.85/6.99 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5400_minus__less__iff,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
% 12.85/6.99 = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5401_minus__less__iff,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 12.85/6.99 = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5402_minus__less__iff,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
% 12.85/6.99 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5403_verit__negate__coefficient_I2_J,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ord_less_real @ A @ B )
% 12.85/6.99 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5404_verit__negate__coefficient_I2_J,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ord_less_int @ A @ B )
% 12.85/6.99 => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5405_verit__negate__coefficient_I2_J,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( ord_le6747313008572928689nteger @ A @ B )
% 12.85/6.99 => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5406_verit__negate__coefficient_I2_J,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ord_less_rat @ A @ B )
% 12.85/6.99 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5407_neg__numeral__neq__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 12.85/6.99 != ( numeral_numeral_real @ N ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5408_neg__numeral__neq__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 12.85/6.99 != ( numeral_numeral_int @ N ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5409_neg__numeral__neq__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 12.85/6.99 != ( numera6690914467698888265omplex @ N ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5410_neg__numeral__neq__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 12.85/6.99 != ( numera6620942414471956472nteger @ N ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5411_neg__numeral__neq__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 12.85/6.99 != ( numeral_numeral_rat @ N ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5412_numeral__neq__neg__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( numeral_numeral_real @ M )
% 12.85/6.99 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5413_numeral__neq__neg__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( numeral_numeral_int @ M )
% 12.85/6.99 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5414_numeral__neq__neg__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( numera6690914467698888265omplex @ M )
% 12.85/6.99 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5415_numeral__neq__neg__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( numera6620942414471956472nteger @ M )
% 12.85/6.99 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5416_numeral__neq__neg__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ( ( numeral_numeral_rat @ M )
% 12.85/6.99 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5417_square__eq__iff,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ( times_times_real @ A @ A )
% 12.85/6.99 = ( times_times_real @ B @ B ) )
% 12.85/6.99 = ( ( A = B )
% 12.85/6.99 | ( A
% 12.85/6.99 = ( uminus_uminus_real @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5418_square__eq__iff,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ( times_times_int @ A @ A )
% 12.85/6.99 = ( times_times_int @ B @ B ) )
% 12.85/6.99 = ( ( A = B )
% 12.85/6.99 | ( A
% 12.85/6.99 = ( uminus_uminus_int @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5419_square__eq__iff,axiom,
% 12.85/6.99 ! [A: complex,B: complex] :
% 12.85/6.99 ( ( ( times_times_complex @ A @ A )
% 12.85/6.99 = ( times_times_complex @ B @ B ) )
% 12.85/6.99 = ( ( A = B )
% 12.85/6.99 | ( A
% 12.85/6.99 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5420_square__eq__iff,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( ( times_3573771949741848930nteger @ A @ A )
% 12.85/6.99 = ( times_3573771949741848930nteger @ B @ B ) )
% 12.85/6.99 = ( ( A = B )
% 12.85/6.99 | ( A
% 12.85/6.99 = ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5421_square__eq__iff,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ( times_times_rat @ A @ A )
% 12.85/6.99 = ( times_times_rat @ B @ B ) )
% 12.85/6.99 = ( ( A = B )
% 12.85/6.99 | ( A
% 12.85/6.99 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5422_minus__mult__commute,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 12.85/6.99 = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5423_minus__mult__commute,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 12.85/6.99 = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5424_minus__mult__commute,axiom,
% 12.85/6.99 ! [A: complex,B: complex] :
% 12.85/6.99 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 12.85/6.99 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5425_minus__mult__commute,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 12.85/6.99 = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5426_minus__mult__commute,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 12.85/6.99 = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5427_group__cancel_Oneg1,axiom,
% 12.85/6.99 ! [A2: real,K: real,A: real] :
% 12.85/6.99 ( ( A2
% 12.85/6.99 = ( plus_plus_real @ K @ A ) )
% 12.85/6.99 => ( ( uminus_uminus_real @ A2 )
% 12.85/6.99 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5428_group__cancel_Oneg1,axiom,
% 12.85/6.99 ! [A2: int,K: int,A: int] :
% 12.85/6.99 ( ( A2
% 12.85/6.99 = ( plus_plus_int @ K @ A ) )
% 12.85/6.99 => ( ( uminus_uminus_int @ A2 )
% 12.85/6.99 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5429_group__cancel_Oneg1,axiom,
% 12.85/6.99 ! [A2: complex,K: complex,A: complex] :
% 12.85/6.99 ( ( A2
% 12.85/6.99 = ( plus_plus_complex @ K @ A ) )
% 12.85/6.99 => ( ( uminus1482373934393186551omplex @ A2 )
% 12.85/6.99 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5430_group__cancel_Oneg1,axiom,
% 12.85/6.99 ! [A2: code_integer,K: code_integer,A: code_integer] :
% 12.85/6.99 ( ( A2
% 12.85/6.99 = ( plus_p5714425477246183910nteger @ K @ A ) )
% 12.85/6.99 => ( ( uminus1351360451143612070nteger @ A2 )
% 12.85/6.99 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5431_group__cancel_Oneg1,axiom,
% 12.85/6.99 ! [A2: rat,K: rat,A: rat] :
% 12.85/6.99 ( ( A2
% 12.85/6.99 = ( plus_plus_rat @ K @ A ) )
% 12.85/6.99 => ( ( uminus_uminus_rat @ A2 )
% 12.85/6.99 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5432_add_Oinverse__distrib__swap,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 12.85/6.99 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5433_add_Oinverse__distrib__swap,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 12.85/6.99 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5434_add_Oinverse__distrib__swap,axiom,
% 12.85/6.99 ! [A: complex,B: complex] :
% 12.85/6.99 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 12.85/6.99 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5435_add_Oinverse__distrib__swap,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 12.85/6.99 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5436_add_Oinverse__distrib__swap,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 12.85/6.99 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5437_is__num__normalize_I8_J,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 12.85/6.99 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5438_is__num__normalize_I8_J,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 12.85/6.99 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5439_is__num__normalize_I8_J,axiom,
% 12.85/6.99 ! [A: complex,B: complex] :
% 12.85/6.99 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 12.85/6.99 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5440_is__num__normalize_I8_J,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 12.85/6.99 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5441_is__num__normalize_I8_J,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 12.85/6.99 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5442_one__neq__neg__one,axiom,
% 12.85/6.99 ( one_one_real
% 12.85/6.99 != ( uminus_uminus_real @ one_one_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5443_one__neq__neg__one,axiom,
% 12.85/6.99 ( one_one_int
% 12.85/6.99 != ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5444_one__neq__neg__one,axiom,
% 12.85/6.99 ( one_one_complex
% 12.85/6.99 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5445_one__neq__neg__one,axiom,
% 12.85/6.99 ( one_one_Code_integer
% 12.85/6.99 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5446_one__neq__neg__one,axiom,
% 12.85/6.99 ( one_one_rat
% 12.85/6.99 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5447_minus__diff__commute,axiom,
% 12.85/6.99 ! [B: real,A: real] :
% 12.85/6.99 ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
% 12.85/6.99 = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5448_minus__diff__commute,axiom,
% 12.85/6.99 ! [B: int,A: int] :
% 12.85/6.99 ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 12.85/6.99 = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5449_minus__diff__commute,axiom,
% 12.85/6.99 ! [B: complex,A: complex] :
% 12.85/6.99 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
% 12.85/6.99 = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5450_minus__diff__commute,axiom,
% 12.85/6.99 ! [B: code_integer,A: code_integer] :
% 12.85/6.99 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
% 12.85/6.99 = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5451_minus__diff__commute,axiom,
% 12.85/6.99 ! [B: rat,A: rat] :
% 12.85/6.99 ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
% 12.85/6.99 = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5452_minus__diff__minus,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 12.85/6.99 = ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5453_minus__diff__minus,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 12.85/6.99 = ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5454_minus__diff__minus,axiom,
% 12.85/6.99 ! [A: complex,B: complex] :
% 12.85/6.99 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5455_minus__diff__minus,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5456_minus__diff__minus,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 12.85/6.99 = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5457_div__minus__right,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 12.85/6.99 = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5458_div__minus__right,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 12.85/6.99 = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5459_minus__divide__right,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 12.85/6.99 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5460_minus__divide__right,axiom,
% 12.85/6.99 ! [A: complex,B: complex] :
% 12.85/6.99 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 12.85/6.99 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5461_minus__divide__right,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 12.85/6.99 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5462_minus__divide__divide,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 12.85/6.99 = ( divide_divide_real @ A @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5463_minus__divide__divide,axiom,
% 12.85/6.99 ! [A: complex,B: complex] :
% 12.85/6.99 ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 12.85/6.99 = ( divide1717551699836669952omplex @ A @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5464_minus__divide__divide,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 12.85/6.99 = ( divide_divide_rat @ A @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5465_minus__divide__left,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 12.85/6.99 = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5466_minus__divide__left,axiom,
% 12.85/6.99 ! [A: complex,B: complex] :
% 12.85/6.99 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 12.85/6.99 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5467_minus__divide__left,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 12.85/6.99 = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5468_mod__minus__eq,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
% 12.85/6.99 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5469_mod__minus__eq,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) @ B )
% 12.85/6.99 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5470_mod__minus__cong,axiom,
% 12.85/6.99 ! [A: int,B: int,A6: int] :
% 12.85/6.99 ( ( ( modulo_modulo_int @ A @ B )
% 12.85/6.99 = ( modulo_modulo_int @ A6 @ B ) )
% 12.85/6.99 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 12.85/6.99 = ( modulo_modulo_int @ ( uminus_uminus_int @ A6 ) @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5471_mod__minus__cong,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer,A6: code_integer] :
% 12.85/6.99 ( ( ( modulo364778990260209775nteger @ A @ B )
% 12.85/6.99 = ( modulo364778990260209775nteger @ A6 @ B ) )
% 12.85/6.99 => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 12.85/6.99 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A6 ) @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5472_mod__minus__right,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 12.85/6.99 = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5473_mod__minus__right,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5474_uminus__int__code_I1_J,axiom,
% 12.85/6.99 ( ( uminus_uminus_int @ zero_zero_int )
% 12.85/6.99 = zero_zero_int ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5475_int__cases2,axiom,
% 12.85/6.99 ! [Z: int] :
% 12.85/6.99 ( ! [N2: nat] :
% 12.85/6.99 ( Z
% 12.85/6.99 != ( semiri1314217659103216013at_int @ N2 ) )
% 12.85/6.99 => ~ ! [N2: nat] :
% 12.85/6.99 ( Z
% 12.85/6.99 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5476_uminus__dvd__conv_I1_J,axiom,
% 12.85/6.99 ( dvd_dvd_int
% 12.85/6.99 = ( ^ [D4: int] : ( dvd_dvd_int @ ( uminus_uminus_int @ D4 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5477_uminus__dvd__conv_I2_J,axiom,
% 12.85/6.99 ( dvd_dvd_int
% 12.85/6.99 = ( ^ [D4: int,T2: int] : ( dvd_dvd_int @ D4 @ ( uminus_uminus_int @ T2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5478_abs__ge__zero,axiom,
% 12.85/6.99 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5479_abs__ge__zero,axiom,
% 12.85/6.99 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5480_abs__ge__zero,axiom,
% 12.85/6.99 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5481_abs__ge__zero,axiom,
% 12.85/6.99 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5482_abs__of__pos,axiom,
% 12.85/6.99 ! [A: code_integer] :
% 12.85/6.99 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 12.85/6.99 => ( ( abs_abs_Code_integer @ A )
% 12.85/6.99 = A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5483_abs__of__pos,axiom,
% 12.85/6.99 ! [A: real] :
% 12.85/6.99 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/6.99 => ( ( abs_abs_real @ A )
% 12.85/6.99 = A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5484_abs__of__pos,axiom,
% 12.85/6.99 ! [A: rat] :
% 12.85/6.99 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/6.99 => ( ( abs_abs_rat @ A )
% 12.85/6.99 = A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5485_abs__of__pos,axiom,
% 12.85/6.99 ! [A: int] :
% 12.85/6.99 ( ( ord_less_int @ zero_zero_int @ A )
% 12.85/6.99 => ( ( abs_abs_int @ A )
% 12.85/6.99 = A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5486_abs__not__less__zero,axiom,
% 12.85/6.99 ! [A: code_integer] :
% 12.85/6.99 ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5487_abs__not__less__zero,axiom,
% 12.85/6.99 ! [A: real] :
% 12.85/6.99 ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5488_abs__not__less__zero,axiom,
% 12.85/6.99 ! [A: rat] :
% 12.85/6.99 ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5489_abs__not__less__zero,axiom,
% 12.85/6.99 ! [A: int] :
% 12.85/6.99 ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5490_abs__triangle__ineq,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5491_abs__triangle__ineq,axiom,
% 12.85/6.99 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5492_abs__triangle__ineq,axiom,
% 12.85/6.99 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5493_abs__triangle__ineq,axiom,
% 12.85/6.99 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5494_abs__mult__less,axiom,
% 12.85/6.99 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 12.85/6.99 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 12.85/6.99 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
% 12.85/6.99 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5495_abs__mult__less,axiom,
% 12.85/6.99 ! [A: real,C: real,B: real,D: real] :
% 12.85/6.99 ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 12.85/6.99 => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
% 12.85/6.99 => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5496_abs__mult__less,axiom,
% 12.85/6.99 ! [A: rat,C: rat,B: rat,D: rat] :
% 12.85/6.99 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 12.85/6.99 => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
% 12.85/6.99 => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5497_abs__mult__less,axiom,
% 12.85/6.99 ! [A: int,C: int,B: int,D: int] :
% 12.85/6.99 ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 12.85/6.99 => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
% 12.85/6.99 => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5498_abs__triangle__ineq2,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5499_abs__triangle__ineq2,axiom,
% 12.85/6.99 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5500_abs__triangle__ineq2,axiom,
% 12.85/6.99 ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5501_abs__triangle__ineq2,axiom,
% 12.85/6.99 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5502_abs__triangle__ineq3,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5503_abs__triangle__ineq3,axiom,
% 12.85/6.99 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5504_abs__triangle__ineq3,axiom,
% 12.85/6.99 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5505_abs__triangle__ineq3,axiom,
% 12.85/6.99 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5506_abs__triangle__ineq2__sym,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5507_abs__triangle__ineq2__sym,axiom,
% 12.85/6.99 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5508_abs__triangle__ineq2__sym,axiom,
% 12.85/6.99 ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5509_abs__triangle__ineq2__sym,axiom,
% 12.85/6.99 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5510_nonzero__abs__divide,axiom,
% 12.85/6.99 ! [B: real,A: real] :
% 12.85/6.99 ( ( B != zero_zero_real )
% 12.85/6.99 => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 12.85/6.99 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5511_nonzero__abs__divide,axiom,
% 12.85/6.99 ! [B: rat,A: rat] :
% 12.85/6.99 ( ( B != zero_zero_rat )
% 12.85/6.99 => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 12.85/6.99 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5512_neg__numeral__le__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5513_neg__numeral__le__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5514_neg__numeral__le__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5515_neg__numeral__le__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5516_not__numeral__le__neg__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5517_not__numeral__le__neg__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5518_not__numeral__le__neg__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5519_not__numeral__le__neg__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5520_zero__neq__neg__numeral,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( zero_zero_real
% 12.85/6.99 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5521_zero__neq__neg__numeral,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( zero_zero_int
% 12.85/6.99 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5522_zero__neq__neg__numeral,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( zero_zero_complex
% 12.85/6.99 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5523_zero__neq__neg__numeral,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( zero_z3403309356797280102nteger
% 12.85/6.99 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5524_zero__neq__neg__numeral,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( zero_zero_rat
% 12.85/6.99 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5525_not__numeral__less__neg__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5526_not__numeral__less__neg__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5527_not__numeral__less__neg__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5528_not__numeral__less__neg__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] :
% 12.85/6.99 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5529_neg__numeral__less__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5530_neg__numeral__less__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5531_neg__numeral__less__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5532_neg__numeral__less__numeral,axiom,
% 12.85/6.99 ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5533_le__minus__one__simps_I2_J,axiom,
% 12.85/6.99 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5534_le__minus__one__simps_I2_J,axiom,
% 12.85/6.99 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5535_le__minus__one__simps_I2_J,axiom,
% 12.85/6.99 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5536_le__minus__one__simps_I2_J,axiom,
% 12.85/6.99 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5537_le__minus__one__simps_I4_J,axiom,
% 12.85/6.99 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5538_le__minus__one__simps_I4_J,axiom,
% 12.85/6.99 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5539_le__minus__one__simps_I4_J,axiom,
% 12.85/6.99 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5540_le__minus__one__simps_I4_J,axiom,
% 12.85/6.99 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5541_neg__eq__iff__add__eq__0,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ( uminus_uminus_real @ A )
% 12.85/6.99 = B )
% 12.85/6.99 = ( ( plus_plus_real @ A @ B )
% 12.85/6.99 = zero_zero_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5542_neg__eq__iff__add__eq__0,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ( uminus_uminus_int @ A )
% 12.85/6.99 = B )
% 12.85/6.99 = ( ( plus_plus_int @ A @ B )
% 12.85/6.99 = zero_zero_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5543_neg__eq__iff__add__eq__0,axiom,
% 12.85/6.99 ! [A: complex,B: complex] :
% 12.85/6.99 ( ( ( uminus1482373934393186551omplex @ A )
% 12.85/6.99 = B )
% 12.85/6.99 = ( ( plus_plus_complex @ A @ B )
% 12.85/6.99 = zero_zero_complex ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5544_neg__eq__iff__add__eq__0,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( ( uminus1351360451143612070nteger @ A )
% 12.85/6.99 = B )
% 12.85/6.99 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 12.85/6.99 = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5545_neg__eq__iff__add__eq__0,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ( uminus_uminus_rat @ A )
% 12.85/6.99 = B )
% 12.85/6.99 = ( ( plus_plus_rat @ A @ B )
% 12.85/6.99 = zero_zero_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5546_eq__neg__iff__add__eq__0,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( uminus_uminus_real @ B ) )
% 12.85/6.99 = ( ( plus_plus_real @ A @ B )
% 12.85/6.99 = zero_zero_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5547_eq__neg__iff__add__eq__0,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( uminus_uminus_int @ B ) )
% 12.85/6.99 = ( ( plus_plus_int @ A @ B )
% 12.85/6.99 = zero_zero_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5548_eq__neg__iff__add__eq__0,axiom,
% 12.85/6.99 ! [A: complex,B: complex] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( uminus1482373934393186551omplex @ B ) )
% 12.85/6.99 = ( ( plus_plus_complex @ A @ B )
% 12.85/6.99 = zero_zero_complex ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5549_eq__neg__iff__add__eq__0,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( uminus1351360451143612070nteger @ B ) )
% 12.85/6.99 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 12.85/6.99 = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5550_eq__neg__iff__add__eq__0,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( uminus_uminus_rat @ B ) )
% 12.85/6.99 = ( ( plus_plus_rat @ A @ B )
% 12.85/6.99 = zero_zero_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5551_add_Oinverse__unique,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ( plus_plus_real @ A @ B )
% 12.85/6.99 = zero_zero_real )
% 12.85/6.99 => ( ( uminus_uminus_real @ A )
% 12.85/6.99 = B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5552_add_Oinverse__unique,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ( plus_plus_int @ A @ B )
% 12.85/6.99 = zero_zero_int )
% 12.85/6.99 => ( ( uminus_uminus_int @ A )
% 12.85/6.99 = B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5553_add_Oinverse__unique,axiom,
% 12.85/6.99 ! [A: complex,B: complex] :
% 12.85/6.99 ( ( ( plus_plus_complex @ A @ B )
% 12.85/6.99 = zero_zero_complex )
% 12.85/6.99 => ( ( uminus1482373934393186551omplex @ A )
% 12.85/6.99 = B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5554_add_Oinverse__unique,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 12.85/6.99 = zero_z3403309356797280102nteger )
% 12.85/6.99 => ( ( uminus1351360451143612070nteger @ A )
% 12.85/6.99 = B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5555_add_Oinverse__unique,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ( plus_plus_rat @ A @ B )
% 12.85/6.99 = zero_zero_rat )
% 12.85/6.99 => ( ( uminus_uminus_rat @ A )
% 12.85/6.99 = B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5556_ab__group__add__class_Oab__left__minus,axiom,
% 12.85/6.99 ! [A: real] :
% 12.85/6.99 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 12.85/6.99 = zero_zero_real ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5557_ab__group__add__class_Oab__left__minus,axiom,
% 12.85/6.99 ! [A: int] :
% 12.85/6.99 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 12.85/6.99 = zero_zero_int ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5558_ab__group__add__class_Oab__left__minus,axiom,
% 12.85/6.99 ! [A: complex] :
% 12.85/6.99 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 12.85/6.99 = zero_zero_complex ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5559_ab__group__add__class_Oab__left__minus,axiom,
% 12.85/6.99 ! [A: code_integer] :
% 12.85/6.99 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 12.85/6.99 = zero_z3403309356797280102nteger ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5560_ab__group__add__class_Oab__left__minus,axiom,
% 12.85/6.99 ! [A: rat] :
% 12.85/6.99 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 12.85/6.99 = zero_zero_rat ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5561_add__eq__0__iff,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ( plus_plus_real @ A @ B )
% 12.85/6.99 = zero_zero_real )
% 12.85/6.99 = ( B
% 12.85/6.99 = ( uminus_uminus_real @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5562_add__eq__0__iff,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ( plus_plus_int @ A @ B )
% 12.85/6.99 = zero_zero_int )
% 12.85/6.99 = ( B
% 12.85/6.99 = ( uminus_uminus_int @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5563_add__eq__0__iff,axiom,
% 12.85/6.99 ! [A: complex,B: complex] :
% 12.85/6.99 ( ( ( plus_plus_complex @ A @ B )
% 12.85/6.99 = zero_zero_complex )
% 12.85/6.99 = ( B
% 12.85/6.99 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5564_add__eq__0__iff,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 12.85/6.99 = zero_z3403309356797280102nteger )
% 12.85/6.99 = ( B
% 12.85/6.99 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5565_add__eq__0__iff,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ( plus_plus_rat @ A @ B )
% 12.85/6.99 = zero_zero_rat )
% 12.85/6.99 = ( B
% 12.85/6.99 = ( uminus_uminus_rat @ A ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5566_zero__neq__neg__one,axiom,
% 12.85/6.99 ( zero_zero_real
% 12.85/6.99 != ( uminus_uminus_real @ one_one_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5567_zero__neq__neg__one,axiom,
% 12.85/6.99 ( zero_zero_int
% 12.85/6.99 != ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5568_zero__neq__neg__one,axiom,
% 12.85/6.99 ( zero_zero_complex
% 12.85/6.99 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5569_zero__neq__neg__one,axiom,
% 12.85/6.99 ( zero_z3403309356797280102nteger
% 12.85/6.99 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5570_zero__neq__neg__one,axiom,
% 12.85/6.99 ( zero_zero_rat
% 12.85/6.99 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5571_less__minus__one__simps_I4_J,axiom,
% 12.85/6.99 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5572_less__minus__one__simps_I4_J,axiom,
% 12.85/6.99 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5573_less__minus__one__simps_I4_J,axiom,
% 12.85/6.99 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5574_less__minus__one__simps_I4_J,axiom,
% 12.85/6.99 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5575_less__minus__one__simps_I2_J,axiom,
% 12.85/6.99 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5576_less__minus__one__simps_I2_J,axiom,
% 12.85/6.99 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5577_less__minus__one__simps_I2_J,axiom,
% 12.85/6.99 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5578_less__minus__one__simps_I2_J,axiom,
% 12.85/6.99 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5579_numeral__times__minus__swap,axiom,
% 12.85/6.99 ! [W: num,X: real] :
% 12.85/6.99 ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X ) )
% 12.85/6.99 = ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5580_numeral__times__minus__swap,axiom,
% 12.85/6.99 ! [W: num,X: int] :
% 12.85/6.99 ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
% 12.85/6.99 = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5581_numeral__times__minus__swap,axiom,
% 12.85/6.99 ! [W: num,X: complex] :
% 12.85/6.99 ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X ) )
% 12.85/6.99 = ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5582_numeral__times__minus__swap,axiom,
% 12.85/6.99 ! [W: num,X: code_integer] :
% 12.85/6.99 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X ) )
% 12.85/6.99 = ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5583_numeral__times__minus__swap,axiom,
% 12.85/6.99 ! [W: num,X: rat] :
% 12.85/6.99 ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
% 12.85/6.99 = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5584_nonzero__minus__divide__right,axiom,
% 12.85/6.99 ! [B: real,A: real] :
% 12.85/6.99 ( ( B != zero_zero_real )
% 12.85/6.99 => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 12.85/6.99 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5585_nonzero__minus__divide__right,axiom,
% 12.85/6.99 ! [B: complex,A: complex] :
% 12.85/6.99 ( ( B != zero_zero_complex )
% 12.85/6.99 => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 12.85/6.99 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5586_nonzero__minus__divide__right,axiom,
% 12.85/6.99 ! [B: rat,A: rat] :
% 12.85/6.99 ( ( B != zero_zero_rat )
% 12.85/6.99 => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 12.85/6.99 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5587_nonzero__minus__divide__divide,axiom,
% 12.85/6.99 ! [B: real,A: real] :
% 12.85/6.99 ( ( B != zero_zero_real )
% 12.85/6.99 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 12.85/6.99 = ( divide_divide_real @ A @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5588_nonzero__minus__divide__divide,axiom,
% 12.85/6.99 ! [B: complex,A: complex] :
% 12.85/6.99 ( ( B != zero_zero_complex )
% 12.85/6.99 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 12.85/6.99 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5589_nonzero__minus__divide__divide,axiom,
% 12.85/6.99 ! [B: rat,A: rat] :
% 12.85/6.99 ( ( B != zero_zero_rat )
% 12.85/6.99 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 12.85/6.99 = ( divide_divide_rat @ A @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5590_one__neq__neg__numeral,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( one_one_real
% 12.85/6.99 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5591_one__neq__neg__numeral,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( one_one_int
% 12.85/6.99 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5592_one__neq__neg__numeral,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( one_one_complex
% 12.85/6.99 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5593_one__neq__neg__numeral,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( one_one_Code_integer
% 12.85/6.99 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5594_one__neq__neg__numeral,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( one_one_rat
% 12.85/6.99 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5595_numeral__neq__neg__one,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( numeral_numeral_real @ N )
% 12.85/6.99 != ( uminus_uminus_real @ one_one_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5596_numeral__neq__neg__one,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( numeral_numeral_int @ N )
% 12.85/6.99 != ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5597_numeral__neq__neg__one,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( numera6690914467698888265omplex @ N )
% 12.85/6.99 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5598_numeral__neq__neg__one,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( numera6620942414471956472nteger @ N )
% 12.85/6.99 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5599_numeral__neq__neg__one,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ( ( numeral_numeral_rat @ N )
% 12.85/6.99 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5600_square__eq__1__iff,axiom,
% 12.85/6.99 ! [X: real] :
% 12.85/6.99 ( ( ( times_times_real @ X @ X )
% 12.85/6.99 = one_one_real )
% 12.85/6.99 = ( ( X = one_one_real )
% 12.85/6.99 | ( X
% 12.85/6.99 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5601_square__eq__1__iff,axiom,
% 12.85/6.99 ! [X: int] :
% 12.85/6.99 ( ( ( times_times_int @ X @ X )
% 12.85/6.99 = one_one_int )
% 12.85/6.99 = ( ( X = one_one_int )
% 12.85/6.99 | ( X
% 12.85/6.99 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5602_square__eq__1__iff,axiom,
% 12.85/6.99 ! [X: complex] :
% 12.85/6.99 ( ( ( times_times_complex @ X @ X )
% 12.85/6.99 = one_one_complex )
% 12.85/6.99 = ( ( X = one_one_complex )
% 12.85/6.99 | ( X
% 12.85/6.99 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5603_square__eq__1__iff,axiom,
% 12.85/6.99 ! [X: code_integer] :
% 12.85/6.99 ( ( ( times_3573771949741848930nteger @ X @ X )
% 12.85/6.99 = one_one_Code_integer )
% 12.85/6.99 = ( ( X = one_one_Code_integer )
% 12.85/6.99 | ( X
% 12.85/6.99 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5604_square__eq__1__iff,axiom,
% 12.85/6.99 ! [X: rat] :
% 12.85/6.99 ( ( ( times_times_rat @ X @ X )
% 12.85/6.99 = one_one_rat )
% 12.85/6.99 = ( ( X = one_one_rat )
% 12.85/6.99 | ( X
% 12.85/6.99 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5605_group__cancel_Osub2,axiom,
% 12.85/6.99 ! [B4: real,K: real,B: real,A: real] :
% 12.85/6.99 ( ( B4
% 12.85/6.99 = ( plus_plus_real @ K @ B ) )
% 12.85/6.99 => ( ( minus_minus_real @ A @ B4 )
% 12.85/6.99 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5606_group__cancel_Osub2,axiom,
% 12.85/6.99 ! [B4: int,K: int,B: int,A: int] :
% 12.85/6.99 ( ( B4
% 12.85/6.99 = ( plus_plus_int @ K @ B ) )
% 12.85/6.99 => ( ( minus_minus_int @ A @ B4 )
% 12.85/6.99 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5607_group__cancel_Osub2,axiom,
% 12.85/6.99 ! [B4: complex,K: complex,B: complex,A: complex] :
% 12.85/6.99 ( ( B4
% 12.85/6.99 = ( plus_plus_complex @ K @ B ) )
% 12.85/6.99 => ( ( minus_minus_complex @ A @ B4 )
% 12.85/6.99 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5608_group__cancel_Osub2,axiom,
% 12.85/6.99 ! [B4: code_integer,K: code_integer,B: code_integer,A: code_integer] :
% 12.85/6.99 ( ( B4
% 12.85/6.99 = ( plus_p5714425477246183910nteger @ K @ B ) )
% 12.85/6.99 => ( ( minus_8373710615458151222nteger @ A @ B4 )
% 12.85/6.99 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5609_group__cancel_Osub2,axiom,
% 12.85/6.99 ! [B4: rat,K: rat,B: rat,A: rat] :
% 12.85/6.99 ( ( B4
% 12.85/6.99 = ( plus_plus_rat @ K @ B ) )
% 12.85/6.99 => ( ( minus_minus_rat @ A @ B4 )
% 12.85/6.99 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5610_diff__conv__add__uminus,axiom,
% 12.85/6.99 ( minus_minus_real
% 12.85/6.99 = ( ^ [A3: real,B2: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5611_diff__conv__add__uminus,axiom,
% 12.85/6.99 ( minus_minus_int
% 12.85/6.99 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5612_diff__conv__add__uminus,axiom,
% 12.85/6.99 ( minus_minus_complex
% 12.85/6.99 = ( ^ [A3: complex,B2: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5613_diff__conv__add__uminus,axiom,
% 12.85/6.99 ( minus_8373710615458151222nteger
% 12.85/6.99 = ( ^ [A3: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5614_diff__conv__add__uminus,axiom,
% 12.85/6.99 ( minus_minus_rat
% 12.85/6.99 = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5615_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 12.85/6.99 ( minus_minus_real
% 12.85/6.99 = ( ^ [A3: real,B2: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5616_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 12.85/6.99 ( minus_minus_int
% 12.85/6.99 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5617_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 12.85/6.99 ( minus_minus_complex
% 12.85/6.99 = ( ^ [A3: complex,B2: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5618_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 12.85/6.99 ( minus_8373710615458151222nteger
% 12.85/6.99 = ( ^ [A3: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5619_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 12.85/6.99 ( minus_minus_rat
% 12.85/6.99 = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5620_dvd__div__neg,axiom,
% 12.85/6.99 ! [B: real,A: real] :
% 12.85/6.99 ( ( dvd_dvd_real @ B @ A )
% 12.85/6.99 => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
% 12.85/6.99 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5621_dvd__div__neg,axiom,
% 12.85/6.99 ! [B: int,A: int] :
% 12.85/6.99 ( ( dvd_dvd_int @ B @ A )
% 12.85/6.99 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 12.85/6.99 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5622_dvd__div__neg,axiom,
% 12.85/6.99 ! [B: complex,A: complex] :
% 12.85/6.99 ( ( dvd_dvd_complex @ B @ A )
% 12.85/6.99 => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5623_dvd__div__neg,axiom,
% 12.85/6.99 ! [B: code_integer,A: code_integer] :
% 12.85/6.99 ( ( dvd_dvd_Code_integer @ B @ A )
% 12.85/6.99 => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5624_dvd__div__neg,axiom,
% 12.85/6.99 ! [B: rat,A: rat] :
% 12.85/6.99 ( ( dvd_dvd_rat @ B @ A )
% 12.85/6.99 => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
% 12.85/6.99 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5625_dvd__neg__div,axiom,
% 12.85/6.99 ! [B: real,A: real] :
% 12.85/6.99 ( ( dvd_dvd_real @ B @ A )
% 12.85/6.99 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
% 12.85/6.99 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5626_dvd__neg__div,axiom,
% 12.85/6.99 ! [B: int,A: int] :
% 12.85/6.99 ( ( dvd_dvd_int @ B @ A )
% 12.85/6.99 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 12.85/6.99 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5627_dvd__neg__div,axiom,
% 12.85/6.99 ! [B: complex,A: complex] :
% 12.85/6.99 ( ( dvd_dvd_complex @ B @ A )
% 12.85/6.99 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5628_dvd__neg__div,axiom,
% 12.85/6.99 ! [B: code_integer,A: code_integer] :
% 12.85/6.99 ( ( dvd_dvd_Code_integer @ B @ A )
% 12.85/6.99 => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5629_dvd__neg__div,axiom,
% 12.85/6.99 ! [B: rat,A: rat] :
% 12.85/6.99 ( ( dvd_dvd_rat @ B @ A )
% 12.85/6.99 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
% 12.85/6.99 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5630_real__minus__mult__self__le,axiom,
% 12.85/6.99 ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5631_int__of__nat__induct,axiom,
% 12.85/6.99 ! [P: int > $o,Z: int] :
% 12.85/6.99 ( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
% 12.85/6.99 => ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
% 12.85/6.99 => ( P @ Z ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5632_int__cases,axiom,
% 12.85/6.99 ! [Z: int] :
% 12.85/6.99 ( ! [N2: nat] :
% 12.85/6.99 ( Z
% 12.85/6.99 != ( semiri1314217659103216013at_int @ N2 ) )
% 12.85/6.99 => ~ ! [N2: nat] :
% 12.85/6.99 ( Z
% 12.85/6.99 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5633_zmult__eq__1__iff,axiom,
% 12.85/6.99 ! [M: int,N: int] :
% 12.85/6.99 ( ( ( times_times_int @ M @ N )
% 12.85/6.99 = one_one_int )
% 12.85/6.99 = ( ( ( M = one_one_int )
% 12.85/6.99 & ( N = one_one_int ) )
% 12.85/6.99 | ( ( M
% 12.85/6.99 = ( uminus_uminus_int @ one_one_int ) )
% 12.85/6.99 & ( N
% 12.85/6.99 = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5634_pos__zmult__eq__1__iff__lemma,axiom,
% 12.85/6.99 ! [M: int,N: int] :
% 12.85/6.99 ( ( ( times_times_int @ M @ N )
% 12.85/6.99 = one_one_int )
% 12.85/6.99 => ( ( M = one_one_int )
% 12.85/6.99 | ( M
% 12.85/6.99 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5635_minus__int__code_I2_J,axiom,
% 12.85/6.99 ! [L: int] :
% 12.85/6.99 ( ( minus_minus_int @ zero_zero_int @ L )
% 12.85/6.99 = ( uminus_uminus_int @ L ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5636_not__int__zless__negative,axiom,
% 12.85/6.99 ! [N: nat,M: nat] :
% 12.85/6.99 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5637_zmod__zminus1__not__zero,axiom,
% 12.85/6.99 ! [K: int,L: int] :
% 12.85/6.99 ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
% 12.85/6.99 != zero_zero_int )
% 12.85/6.99 => ( ( modulo_modulo_int @ K @ L )
% 12.85/6.99 != zero_zero_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5638_zmod__zminus2__not__zero,axiom,
% 12.85/6.99 ! [K: int,L: int] :
% 12.85/6.99 ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L ) )
% 12.85/6.99 != zero_zero_int )
% 12.85/6.99 => ( ( modulo_modulo_int @ K @ L )
% 12.85/6.99 != zero_zero_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5639_minus__real__def,axiom,
% 12.85/6.99 ( minus_minus_real
% 12.85/6.99 = ( ^ [X3: real,Y6: real] : ( plus_plus_real @ X3 @ ( uminus_uminus_real @ Y6 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5640_dense__eq0__I,axiom,
% 12.85/6.99 ! [X: rat] :
% 12.85/6.99 ( ! [E: rat] :
% 12.85/6.99 ( ( ord_less_rat @ zero_zero_rat @ E )
% 12.85/6.99 => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E ) )
% 12.85/6.99 => ( X = zero_zero_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5641_dense__eq0__I,axiom,
% 12.85/6.99 ! [X: real] :
% 12.85/6.99 ( ! [E: real] :
% 12.85/6.99 ( ( ord_less_real @ zero_zero_real @ E )
% 12.85/6.99 => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E ) )
% 12.85/6.99 => ( X = zero_zero_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5642_abs__mult__pos,axiom,
% 12.85/6.99 ! [X: code_integer,Y4: code_integer] :
% 12.85/6.99 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 12.85/6.99 => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y4 ) @ X )
% 12.85/6.99 = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y4 @ X ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5643_abs__mult__pos,axiom,
% 12.85/6.99 ! [X: rat,Y4: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 12.85/6.99 => ( ( times_times_rat @ ( abs_abs_rat @ Y4 ) @ X )
% 12.85/6.99 = ( abs_abs_rat @ ( times_times_rat @ Y4 @ X ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5644_abs__mult__pos,axiom,
% 12.85/6.99 ! [X: int,Y4: int] :
% 12.85/6.99 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/6.99 => ( ( times_times_int @ ( abs_abs_int @ Y4 ) @ X )
% 12.85/6.99 = ( abs_abs_int @ ( times_times_int @ Y4 @ X ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5645_abs__mult__pos,axiom,
% 12.85/6.99 ! [X: real,Y4: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/6.99 => ( ( times_times_real @ ( abs_abs_real @ Y4 ) @ X )
% 12.85/6.99 = ( abs_abs_real @ ( times_times_real @ Y4 @ X ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5646_abs__eq__mult,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 12.85/6.99 | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 12.85/6.99 & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 12.85/6.99 | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
% 12.85/6.99 => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 12.85/6.99 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5647_abs__eq__mult,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/6.99 | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 12.85/6.99 & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 12.85/6.99 | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 12.85/6.99 => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 12.85/6.99 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5648_abs__eq__mult,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 12.85/6.99 | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 12.85/6.99 & ( ( ord_less_eq_int @ zero_zero_int @ B )
% 12.85/6.99 | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 12.85/6.99 => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 12.85/6.99 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5649_abs__eq__mult,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/6.99 | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 12.85/6.99 & ( ( ord_less_eq_real @ zero_zero_real @ B )
% 12.85/6.99 | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 12.85/6.99 => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 12.85/6.99 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5650_zero__le__power__abs,axiom,
% 12.85/6.99 ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5651_zero__le__power__abs,axiom,
% 12.85/6.99 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5652_zero__le__power__abs,axiom,
% 12.85/6.99 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5653_zero__le__power__abs,axiom,
% 12.85/6.99 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5654_abs__div__pos,axiom,
% 12.85/6.99 ! [Y4: real,X: real] :
% 12.85/6.99 ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/6.99 => ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y4 )
% 12.85/6.99 = ( abs_abs_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5655_abs__div__pos,axiom,
% 12.85/6.99 ! [Y4: rat,X: rat] :
% 12.85/6.99 ( ( ord_less_rat @ zero_zero_rat @ Y4 )
% 12.85/6.99 => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y4 )
% 12.85/6.99 = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5656_abs__diff__le__iff,axiom,
% 12.85/6.99 ! [X: code_integer,A: code_integer,R2: code_integer] :
% 12.85/6.99 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
% 12.85/6.99 = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
% 12.85/6.99 & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5657_abs__diff__le__iff,axiom,
% 12.85/6.99 ! [X: rat,A: rat,R2: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
% 12.85/6.99 = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
% 12.85/6.99 & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5658_abs__diff__le__iff,axiom,
% 12.85/6.99 ! [X: int,A: int,R2: int] :
% 12.85/6.99 ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
% 12.85/6.99 = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X )
% 12.85/6.99 & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5659_abs__diff__le__iff,axiom,
% 12.85/6.99 ! [X: real,A: real,R2: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
% 12.85/6.99 = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X )
% 12.85/6.99 & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5660_abs__triangle__ineq4,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5661_abs__triangle__ineq4,axiom,
% 12.85/6.99 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5662_abs__triangle__ineq4,axiom,
% 12.85/6.99 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5663_abs__triangle__ineq4,axiom,
% 12.85/6.99 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5664_abs__diff__triangle__ineq,axiom,
% 12.85/6.99 ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5665_abs__diff__triangle__ineq,axiom,
% 12.85/6.99 ! [A: rat,B: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5666_abs__diff__triangle__ineq,axiom,
% 12.85/6.99 ! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5667_abs__diff__triangle__ineq,axiom,
% 12.85/6.99 ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5668_abs__diff__less__iff,axiom,
% 12.85/6.99 ! [X: code_integer,A: code_integer,R2: code_integer] :
% 12.85/6.99 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
% 12.85/6.99 = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
% 12.85/6.99 & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5669_abs__diff__less__iff,axiom,
% 12.85/6.99 ! [X: real,A: real,R2: real] :
% 12.85/6.99 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
% 12.85/6.99 = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X )
% 12.85/6.99 & ( ord_less_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5670_abs__diff__less__iff,axiom,
% 12.85/6.99 ! [X: rat,A: rat,R2: rat] :
% 12.85/6.99 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
% 12.85/6.99 = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
% 12.85/6.99 & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5671_abs__diff__less__iff,axiom,
% 12.85/6.99 ! [X: int,A: int,R2: int] :
% 12.85/6.99 ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
% 12.85/6.99 = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X )
% 12.85/6.99 & ( ord_less_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5672_neg__numeral__le__zero,axiom,
% 12.85/6.99 ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5673_neg__numeral__le__zero,axiom,
% 12.85/6.99 ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5674_neg__numeral__le__zero,axiom,
% 12.85/6.99 ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5675_neg__numeral__le__zero,axiom,
% 12.85/6.99 ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5676_not__zero__le__neg__numeral,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5677_not__zero__le__neg__numeral,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5678_not__zero__le__neg__numeral,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5679_not__zero__le__neg__numeral,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5680_neg__numeral__less__zero,axiom,
% 12.85/6.99 ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5681_neg__numeral__less__zero,axiom,
% 12.85/6.99 ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5682_neg__numeral__less__zero,axiom,
% 12.85/6.99 ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5683_neg__numeral__less__zero,axiom,
% 12.85/6.99 ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5684_not__zero__less__neg__numeral,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5685_not__zero__less__neg__numeral,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5686_not__zero__less__neg__numeral,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5687_not__zero__less__neg__numeral,axiom,
% 12.85/6.99 ! [N: num] :
% 12.85/6.99 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5688_le__minus__one__simps_I1_J,axiom,
% 12.85/6.99 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5689_le__minus__one__simps_I1_J,axiom,
% 12.85/6.99 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5690_le__minus__one__simps_I1_J,axiom,
% 12.85/6.99 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5691_le__minus__one__simps_I1_J,axiom,
% 12.85/6.99 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5692_le__minus__one__simps_I3_J,axiom,
% 12.85/6.99 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5693_le__minus__one__simps_I3_J,axiom,
% 12.85/6.99 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5694_le__minus__one__simps_I3_J,axiom,
% 12.85/6.99 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5695_le__minus__one__simps_I3_J,axiom,
% 12.85/6.99 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5696_less__minus__one__simps_I3_J,axiom,
% 12.85/6.99 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5697_less__minus__one__simps_I3_J,axiom,
% 12.85/6.99 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5698_less__minus__one__simps_I3_J,axiom,
% 12.85/6.99 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5699_less__minus__one__simps_I3_J,axiom,
% 12.85/6.99 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5700_less__minus__one__simps_I1_J,axiom,
% 12.85/6.99 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5701_less__minus__one__simps_I1_J,axiom,
% 12.85/6.99 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5702_less__minus__one__simps_I1_J,axiom,
% 12.85/6.99 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5703_less__minus__one__simps_I1_J,axiom,
% 12.85/6.99 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5704_neg__numeral__le__one,axiom,
% 12.85/6.99 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5705_neg__numeral__le__one,axiom,
% 12.85/6.99 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5706_neg__numeral__le__one,axiom,
% 12.85/6.99 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5707_neg__numeral__le__one,axiom,
% 12.85/6.99 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5708_neg__one__le__numeral,axiom,
% 12.85/6.99 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5709_neg__one__le__numeral,axiom,
% 12.85/6.99 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5710_neg__one__le__numeral,axiom,
% 12.85/6.99 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5711_neg__one__le__numeral,axiom,
% 12.85/6.99 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5712_neg__numeral__le__neg__one,axiom,
% 12.85/6.99 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5713_neg__numeral__le__neg__one,axiom,
% 12.85/6.99 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5714_neg__numeral__le__neg__one,axiom,
% 12.85/6.99 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5715_neg__numeral__le__neg__one,axiom,
% 12.85/6.99 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5716_not__numeral__le__neg__one,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5717_not__numeral__le__neg__one,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5718_not__numeral__le__neg__one,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5719_not__numeral__le__neg__one,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5720_not__one__le__neg__numeral,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5721_not__one__le__neg__numeral,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5722_not__one__le__neg__numeral,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5723_not__one__le__neg__numeral,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5724_neg__numeral__less__one,axiom,
% 12.85/6.99 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5725_neg__numeral__less__one,axiom,
% 12.85/6.99 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5726_neg__numeral__less__one,axiom,
% 12.85/6.99 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5727_neg__numeral__less__one,axiom,
% 12.85/6.99 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5728_neg__one__less__numeral,axiom,
% 12.85/6.99 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5729_neg__one__less__numeral,axiom,
% 12.85/6.99 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5730_neg__one__less__numeral,axiom,
% 12.85/6.99 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5731_neg__one__less__numeral,axiom,
% 12.85/6.99 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5732_not__numeral__less__neg__one,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5733_not__numeral__less__neg__one,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5734_not__numeral__less__neg__one,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5735_not__numeral__less__neg__one,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5736_not__one__less__neg__numeral,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5737_not__one__less__neg__numeral,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5738_not__one__less__neg__numeral,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5739_not__one__less__neg__numeral,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5740_not__neg__one__less__neg__numeral,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5741_not__neg__one__less__neg__numeral,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5742_not__neg__one__less__neg__numeral,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5743_not__neg__one__less__neg__numeral,axiom,
% 12.85/6.99 ! [M: num] :
% 12.85/6.99 ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5744_eq__minus__divide__eq,axiom,
% 12.85/6.99 ! [A: real,B: real,C: real] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 12.85/6.99 = ( ( ( C != zero_zero_real )
% 12.85/6.99 => ( ( times_times_real @ A @ C )
% 12.85/6.99 = ( uminus_uminus_real @ B ) ) )
% 12.85/6.99 & ( ( C = zero_zero_real )
% 12.85/6.99 => ( A = zero_zero_real ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5745_eq__minus__divide__eq,axiom,
% 12.85/6.99 ! [A: complex,B: complex,C: complex] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
% 12.85/6.99 = ( ( ( C != zero_zero_complex )
% 12.85/6.99 => ( ( times_times_complex @ A @ C )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ B ) ) )
% 12.85/6.99 & ( ( C = zero_zero_complex )
% 12.85/6.99 => ( A = zero_zero_complex ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5746_eq__minus__divide__eq,axiom,
% 12.85/6.99 ! [A: rat,B: rat,C: rat] :
% 12.85/6.99 ( ( A
% 12.85/6.99 = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 12.85/6.99 = ( ( ( C != zero_zero_rat )
% 12.85/6.99 => ( ( times_times_rat @ A @ C )
% 12.85/6.99 = ( uminus_uminus_rat @ B ) ) )
% 12.85/6.99 & ( ( C = zero_zero_rat )
% 12.85/6.99 => ( A = zero_zero_rat ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5747_minus__divide__eq__eq,axiom,
% 12.85/6.99 ! [B: real,C: real,A: real] :
% 12.85/6.99 ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
% 12.85/6.99 = A )
% 12.85/6.99 = ( ( ( C != zero_zero_real )
% 12.85/6.99 => ( ( uminus_uminus_real @ B )
% 12.85/6.99 = ( times_times_real @ A @ C ) ) )
% 12.85/6.99 & ( ( C = zero_zero_real )
% 12.85/6.99 => ( A = zero_zero_real ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5748_minus__divide__eq__eq,axiom,
% 12.85/6.99 ! [B: complex,C: complex,A: complex] :
% 12.85/6.99 ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
% 12.85/6.99 = A )
% 12.85/6.99 = ( ( ( C != zero_zero_complex )
% 12.85/6.99 => ( ( uminus1482373934393186551omplex @ B )
% 12.85/6.99 = ( times_times_complex @ A @ C ) ) )
% 12.85/6.99 & ( ( C = zero_zero_complex )
% 12.85/6.99 => ( A = zero_zero_complex ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5749_minus__divide__eq__eq,axiom,
% 12.85/6.99 ! [B: rat,C: rat,A: rat] :
% 12.85/6.99 ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
% 12.85/6.99 = A )
% 12.85/6.99 = ( ( ( C != zero_zero_rat )
% 12.85/6.99 => ( ( uminus_uminus_rat @ B )
% 12.85/6.99 = ( times_times_rat @ A @ C ) ) )
% 12.85/6.99 & ( ( C = zero_zero_rat )
% 12.85/6.99 => ( A = zero_zero_rat ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5750_nonzero__neg__divide__eq__eq,axiom,
% 12.85/6.99 ! [B: real,A: real,C: real] :
% 12.85/6.99 ( ( B != zero_zero_real )
% 12.85/6.99 => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 12.85/6.99 = C )
% 12.85/6.99 = ( ( uminus_uminus_real @ A )
% 12.85/6.99 = ( times_times_real @ C @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5751_nonzero__neg__divide__eq__eq,axiom,
% 12.85/6.99 ! [B: complex,A: complex,C: complex] :
% 12.85/6.99 ( ( B != zero_zero_complex )
% 12.85/6.99 => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 12.85/6.99 = C )
% 12.85/6.99 = ( ( uminus1482373934393186551omplex @ A )
% 12.85/6.99 = ( times_times_complex @ C @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5752_nonzero__neg__divide__eq__eq,axiom,
% 12.85/6.99 ! [B: rat,A: rat,C: rat] :
% 12.85/6.99 ( ( B != zero_zero_rat )
% 12.85/6.99 => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 12.85/6.99 = C )
% 12.85/6.99 = ( ( uminus_uminus_rat @ A )
% 12.85/6.99 = ( times_times_rat @ C @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5753_nonzero__neg__divide__eq__eq2,axiom,
% 12.85/6.99 ! [B: real,C: real,A: real] :
% 12.85/6.99 ( ( B != zero_zero_real )
% 12.85/6.99 => ( ( C
% 12.85/6.99 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
% 12.85/6.99 = ( ( times_times_real @ C @ B )
% 12.85/6.99 = ( uminus_uminus_real @ A ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5754_nonzero__neg__divide__eq__eq2,axiom,
% 12.85/6.99 ! [B: complex,C: complex,A: complex] :
% 12.85/6.99 ( ( B != zero_zero_complex )
% 12.85/6.99 => ( ( C
% 12.85/6.99 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 12.85/6.99 = ( ( times_times_complex @ C @ B )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5755_nonzero__neg__divide__eq__eq2,axiom,
% 12.85/6.99 ! [B: rat,C: rat,A: rat] :
% 12.85/6.99 ( ( B != zero_zero_rat )
% 12.85/6.99 => ( ( C
% 12.85/6.99 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
% 12.85/6.99 = ( ( times_times_rat @ C @ B )
% 12.85/6.99 = ( uminus_uminus_rat @ A ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5756_mult__1s__ring__1_I2_J,axiom,
% 12.85/6.99 ! [B: real] :
% 12.85/6.99 ( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
% 12.85/6.99 = ( uminus_uminus_real @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5757_mult__1s__ring__1_I2_J,axiom,
% 12.85/6.99 ! [B: int] :
% 12.85/6.99 ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
% 12.85/6.99 = ( uminus_uminus_int @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5758_mult__1s__ring__1_I2_J,axiom,
% 12.85/6.99 ! [B: complex] :
% 12.85/6.99 ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5759_mult__1s__ring__1_I2_J,axiom,
% 12.85/6.99 ! [B: code_integer] :
% 12.85/6.99 ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5760_mult__1s__ring__1_I2_J,axiom,
% 12.85/6.99 ! [B: rat] :
% 12.85/6.99 ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
% 12.85/6.99 = ( uminus_uminus_rat @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5761_mult__1s__ring__1_I1_J,axiom,
% 12.85/6.99 ! [B: real] :
% 12.85/6.99 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
% 12.85/6.99 = ( uminus_uminus_real @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5762_mult__1s__ring__1_I1_J,axiom,
% 12.85/6.99 ! [B: int] :
% 12.85/6.99 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
% 12.85/6.99 = ( uminus_uminus_int @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5763_mult__1s__ring__1_I1_J,axiom,
% 12.85/6.99 ! [B: complex] :
% 12.85/6.99 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5764_mult__1s__ring__1_I1_J,axiom,
% 12.85/6.99 ! [B: code_integer] :
% 12.85/6.99 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5765_mult__1s__ring__1_I1_J,axiom,
% 12.85/6.99 ! [B: rat] :
% 12.85/6.99 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
% 12.85/6.99 = ( uminus_uminus_rat @ B ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5766_divide__eq__minus__1__iff,axiom,
% 12.85/6.99 ! [A: real,B: real] :
% 12.85/6.99 ( ( ( divide_divide_real @ A @ B )
% 12.85/6.99 = ( uminus_uminus_real @ one_one_real ) )
% 12.85/6.99 = ( ( B != zero_zero_real )
% 12.85/6.99 & ( A
% 12.85/6.99 = ( uminus_uminus_real @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5767_divide__eq__minus__1__iff,axiom,
% 12.85/6.99 ! [A: complex,B: complex] :
% 12.85/6.99 ( ( ( divide1717551699836669952omplex @ A @ B )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.85/6.99 = ( ( B != zero_zero_complex )
% 12.85/6.99 & ( A
% 12.85/6.99 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5768_divide__eq__minus__1__iff,axiom,
% 12.85/6.99 ! [A: rat,B: rat] :
% 12.85/6.99 ( ( ( divide_divide_rat @ A @ B )
% 12.85/6.99 = ( uminus_uminus_rat @ one_one_rat ) )
% 12.85/6.99 = ( ( B != zero_zero_rat )
% 12.85/6.99 & ( A
% 12.85/6.99 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5769_uminus__numeral__One,axiom,
% 12.85/6.99 ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
% 12.85/6.99 = ( uminus_uminus_real @ one_one_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5770_uminus__numeral__One,axiom,
% 12.85/6.99 ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
% 12.85/6.99 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5771_uminus__numeral__One,axiom,
% 12.85/6.99 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5772_uminus__numeral__One,axiom,
% 12.85/6.99 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5773_uminus__numeral__One,axiom,
% 12.85/6.99 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
% 12.85/6.99 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5774_power__minus,axiom,
% 12.85/6.99 ! [A: real,N: nat] :
% 12.85/6.99 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 12.85/6.99 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5775_power__minus,axiom,
% 12.85/6.99 ! [A: int,N: nat] :
% 12.85/6.99 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 12.85/6.99 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5776_power__minus,axiom,
% 12.85/6.99 ! [A: complex,N: nat] :
% 12.85/6.99 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 12.85/6.99 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5777_power__minus,axiom,
% 12.85/6.99 ! [A: code_integer,N: nat] :
% 12.85/6.99 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 12.85/6.99 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5778_power__minus,axiom,
% 12.85/6.99 ! [A: rat,N: nat] :
% 12.85/6.99 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 12.85/6.99 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5779_power__minus__Bit0,axiom,
% 12.85/6.99 ! [X: real,K: num] :
% 12.85/6.99 ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 12.85/6.99 = ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5780_power__minus__Bit0,axiom,
% 12.85/6.99 ! [X: int,K: num] :
% 12.85/6.99 ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 12.85/6.99 = ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5781_power__minus__Bit0,axiom,
% 12.85/6.99 ! [X: complex,K: num] :
% 12.85/6.99 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 12.85/6.99 = ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5782_power__minus__Bit0,axiom,
% 12.85/6.99 ! [X: code_integer,K: num] :
% 12.85/6.99 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 12.85/6.99 = ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5783_power__minus__Bit0,axiom,
% 12.85/6.99 ! [X: rat,K: num] :
% 12.85/6.99 ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 12.85/6.99 = ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5784_lemma__interval__lt,axiom,
% 12.85/6.99 ! [A: real,X: real,B: real] :
% 12.85/6.99 ( ( ord_less_real @ A @ X )
% 12.85/6.99 => ( ( ord_less_real @ X @ B )
% 12.85/6.99 => ? [D2: real] :
% 12.85/6.99 ( ( ord_less_real @ zero_zero_real @ D2 )
% 12.85/6.99 & ! [Y3: real] :
% 12.85/6.99 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D2 )
% 12.85/6.99 => ( ( ord_less_real @ A @ Y3 )
% 12.85/6.99 & ( ord_less_real @ Y3 @ B ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5785_norm__uminus__minus,axiom,
% 12.85/6.99 ! [X: real,Y4: real] :
% 12.85/6.99 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y4 ) )
% 12.85/6.99 = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y4 ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5786_norm__uminus__minus,axiom,
% 12.85/6.99 ! [X: complex,Y4: complex] :
% 12.85/6.99 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y4 ) )
% 12.85/6.99 = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y4 ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5787_power__minus__Bit1,axiom,
% 12.85/6.99 ! [X: real,K: num] :
% 12.85/6.99 ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 12.85/6.99 = ( uminus_uminus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5788_power__minus__Bit1,axiom,
% 12.85/6.99 ! [X: int,K: num] :
% 12.85/6.99 ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 12.85/6.99 = ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5789_power__minus__Bit1,axiom,
% 12.85/6.99 ! [X: complex,K: num] :
% 12.85/6.99 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5790_power__minus__Bit1,axiom,
% 12.85/6.99 ! [X: code_integer,K: num] :
% 12.85/6.99 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5791_power__minus__Bit1,axiom,
% 12.85/6.99 ! [X: rat,K: num] :
% 12.85/6.99 ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 12.85/6.99 = ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5792_sin__bound__lemma,axiom,
% 12.85/6.99 ! [X: real,Y4: real,U: real,V: real] :
% 12.85/6.99 ( ( X = Y4 )
% 12.85/6.99 => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 12.85/6.99 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y4 ) ) @ V ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5793_int__cases4,axiom,
% 12.85/6.99 ! [M: int] :
% 12.85/6.99 ( ! [N2: nat] :
% 12.85/6.99 ( M
% 12.85/6.99 != ( semiri1314217659103216013at_int @ N2 ) )
% 12.85/6.99 => ~ ! [N2: nat] :
% 12.85/6.99 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 12.85/6.99 => ( M
% 12.85/6.99 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5794_real__0__less__add__iff,axiom,
% 12.85/6.99 ! [X: real,Y4: real] :
% 12.85/6.99 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y4 ) )
% 12.85/6.99 = ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5795_real__add__less__0__iff,axiom,
% 12.85/6.99 ! [X: real,Y4: real] :
% 12.85/6.99 ( ( ord_less_real @ ( plus_plus_real @ X @ Y4 ) @ zero_zero_real )
% 12.85/6.99 = ( ord_less_real @ Y4 @ ( uminus_uminus_real @ X ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5796_real__add__le__0__iff,axiom,
% 12.85/6.99 ! [X: real,Y4: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y4 ) @ zero_zero_real )
% 12.85/6.99 = ( ord_less_eq_real @ Y4 @ ( uminus_uminus_real @ X ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5797_real__0__le__add__iff,axiom,
% 12.85/6.99 ! [X: real,Y4: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y4 ) )
% 12.85/6.99 = ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y4 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5798_int__zle__neg,axiom,
% 12.85/6.99 ! [N: nat,M: nat] :
% 12.85/6.99 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
% 12.85/6.99 = ( ( N = zero_zero_nat )
% 12.85/6.99 & ( M = zero_zero_nat ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5799_nonpos__int__cases,axiom,
% 12.85/6.99 ! [K: int] :
% 12.85/6.99 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 12.85/6.99 => ~ ! [N2: nat] :
% 12.85/6.99 ( K
% 12.85/6.99 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5800_negative__zle__0,axiom,
% 12.85/6.99 ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5801_zmod__zminus1__eq__if,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ( ( modulo_modulo_int @ A @ B )
% 12.85/6.99 = zero_zero_int )
% 12.85/6.99 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 12.85/6.99 = zero_zero_int ) )
% 12.85/6.99 & ( ( ( modulo_modulo_int @ A @ B )
% 12.85/6.99 != zero_zero_int )
% 12.85/6.99 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 12.85/6.99 = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5802_zmod__zminus2__eq__if,axiom,
% 12.85/6.99 ! [A: int,B: int] :
% 12.85/6.99 ( ( ( ( modulo_modulo_int @ A @ B )
% 12.85/6.99 = zero_zero_int )
% 12.85/6.99 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 12.85/6.99 = zero_zero_int ) )
% 12.85/6.99 & ( ( ( modulo_modulo_int @ A @ B )
% 12.85/6.99 != zero_zero_int )
% 12.85/6.99 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 12.85/6.99 = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5803_abs__add__one__gt__zero,axiom,
% 12.85/6.99 ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5804_abs__add__one__gt__zero,axiom,
% 12.85/6.99 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5805_abs__add__one__gt__zero,axiom,
% 12.85/6.99 ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5806_abs__add__one__gt__zero,axiom,
% 12.85/6.99 ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5807_less__minus__divide__eq,axiom,
% 12.85/6.99 ! [A: real,B: real,C: real] :
% 12.85/6.99 ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 12.85/6.99 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 12.85/6.99 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 12.85/6.99 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5808_less__minus__divide__eq,axiom,
% 12.85/6.99 ! [A: rat,B: rat,C: rat] :
% 12.85/6.99 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 12.85/6.99 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 12.85/6.99 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 12.85/6.99 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5809_minus__divide__less__eq,axiom,
% 12.85/6.99 ! [B: real,C: real,A: real] :
% 12.85/6.99 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 12.85/6.99 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 12.85/6.99 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 12.85/6.99 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5810_minus__divide__less__eq,axiom,
% 12.85/6.99 ! [B: rat,C: rat,A: rat] :
% 12.85/6.99 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 12.85/6.99 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 12.85/6.99 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 12.85/6.99 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5811_neg__less__minus__divide__eq,axiom,
% 12.85/6.99 ! [C: real,A: real,B: real] :
% 12.85/6.99 ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 12.85/6.99 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5812_neg__less__minus__divide__eq,axiom,
% 12.85/6.99 ! [C: rat,A: rat,B: rat] :
% 12.85/6.99 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 12.85/6.99 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5813_neg__minus__divide__less__eq,axiom,
% 12.85/6.99 ! [C: real,B: real,A: real] :
% 12.85/6.99 ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 12.85/6.99 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5814_neg__minus__divide__less__eq,axiom,
% 12.85/6.99 ! [C: rat,B: rat,A: rat] :
% 12.85/6.99 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 12.85/6.99 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5815_pos__less__minus__divide__eq,axiom,
% 12.85/6.99 ! [C: real,A: real,B: real] :
% 12.85/6.99 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 12.85/6.99 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5816_pos__less__minus__divide__eq,axiom,
% 12.85/6.99 ! [C: rat,A: rat,B: rat] :
% 12.85/6.99 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 12.85/6.99 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5817_pos__minus__divide__less__eq,axiom,
% 12.85/6.99 ! [C: real,B: real,A: real] :
% 12.85/6.99 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 12.85/6.99 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5818_pos__minus__divide__less__eq,axiom,
% 12.85/6.99 ! [C: rat,B: rat,A: rat] :
% 12.85/6.99 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 12.85/6.99 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5819_divide__eq__eq__numeral_I2_J,axiom,
% 12.85/6.99 ! [B: real,C: real,W: num] :
% 12.85/6.99 ( ( ( divide_divide_real @ B @ C )
% 12.85/6.99 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.85/6.99 = ( ( ( C != zero_zero_real )
% 12.85/6.99 => ( B
% 12.85/6.99 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 12.85/6.99 & ( ( C = zero_zero_real )
% 12.85/6.99 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 12.85/6.99 = zero_zero_real ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5820_divide__eq__eq__numeral_I2_J,axiom,
% 12.85/6.99 ! [B: complex,C: complex,W: num] :
% 12.85/6.99 ( ( ( divide1717551699836669952omplex @ B @ C )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 12.85/6.99 = ( ( ( C != zero_zero_complex )
% 12.85/6.99 => ( B
% 12.85/6.99 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
% 12.85/6.99 & ( ( C = zero_zero_complex )
% 12.85/6.99 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 12.85/6.99 = zero_zero_complex ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5821_divide__eq__eq__numeral_I2_J,axiom,
% 12.85/6.99 ! [B: rat,C: rat,W: num] :
% 12.85/6.99 ( ( ( divide_divide_rat @ B @ C )
% 12.85/6.99 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 12.85/6.99 = ( ( ( C != zero_zero_rat )
% 12.85/6.99 => ( B
% 12.85/6.99 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 12.85/6.99 & ( ( C = zero_zero_rat )
% 12.85/6.99 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 12.85/6.99 = zero_zero_rat ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5822_eq__divide__eq__numeral_I2_J,axiom,
% 12.85/6.99 ! [W: num,B: real,C: real] :
% 12.85/6.99 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 12.85/6.99 = ( divide_divide_real @ B @ C ) )
% 12.85/6.99 = ( ( ( C != zero_zero_real )
% 12.85/6.99 => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
% 12.85/6.99 = B ) )
% 12.85/6.99 & ( ( C = zero_zero_real )
% 12.85/6.99 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 12.85/6.99 = zero_zero_real ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5823_eq__divide__eq__numeral_I2_J,axiom,
% 12.85/6.99 ! [W: num,B: complex,C: complex] :
% 12.85/6.99 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 12.85/6.99 = ( divide1717551699836669952omplex @ B @ C ) )
% 12.85/6.99 = ( ( ( C != zero_zero_complex )
% 12.85/6.99 => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
% 12.85/6.99 = B ) )
% 12.85/6.99 & ( ( C = zero_zero_complex )
% 12.85/6.99 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 12.85/6.99 = zero_zero_complex ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5824_eq__divide__eq__numeral_I2_J,axiom,
% 12.85/6.99 ! [W: num,B: rat,C: rat] :
% 12.85/6.99 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 12.85/6.99 = ( divide_divide_rat @ B @ C ) )
% 12.85/6.99 = ( ( ( C != zero_zero_rat )
% 12.85/6.99 => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
% 12.85/6.99 = B ) )
% 12.85/6.99 & ( ( C = zero_zero_rat )
% 12.85/6.99 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 12.85/6.99 = zero_zero_rat ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5825_add__divide__eq__if__simps_I3_J,axiom,
% 12.85/6.99 ! [Z: real,A: real,B: real] :
% 12.85/6.99 ( ( ( Z = zero_zero_real )
% 12.85/6.99 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 12.85/6.99 = B ) )
% 12.85/6.99 & ( ( Z != zero_zero_real )
% 12.85/6.99 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 12.85/6.99 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5826_add__divide__eq__if__simps_I3_J,axiom,
% 12.85/6.99 ! [Z: complex,A: complex,B: complex] :
% 12.85/6.99 ( ( ( Z = zero_zero_complex )
% 12.85/6.99 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 12.85/6.99 = B ) )
% 12.85/6.99 & ( ( Z != zero_zero_complex )
% 12.85/6.99 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 12.85/6.99 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5827_add__divide__eq__if__simps_I3_J,axiom,
% 12.85/6.99 ! [Z: rat,A: rat,B: rat] :
% 12.85/6.99 ( ( ( Z = zero_zero_rat )
% 12.85/6.99 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 12.85/6.99 = B ) )
% 12.85/6.99 & ( ( Z != zero_zero_rat )
% 12.85/6.99 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 12.85/6.99 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5828_minus__divide__add__eq__iff,axiom,
% 12.85/6.99 ! [Z: real,X: real,Y4: real] :
% 12.85/6.99 ( ( Z != zero_zero_real )
% 12.85/6.99 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y4 )
% 12.85/6.99 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y4 @ Z ) ) @ Z ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5829_minus__divide__add__eq__iff,axiom,
% 12.85/6.99 ! [Z: complex,X: complex,Y4: complex] :
% 12.85/6.99 ( ( Z != zero_zero_complex )
% 12.85/6.99 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y4 )
% 12.85/6.99 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y4 @ Z ) ) @ Z ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5830_minus__divide__add__eq__iff,axiom,
% 12.85/6.99 ! [Z: rat,X: rat,Y4: rat] :
% 12.85/6.99 ( ( Z != zero_zero_rat )
% 12.85/6.99 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y4 )
% 12.85/6.99 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y4 @ Z ) ) @ Z ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5831_add__divide__eq__if__simps_I6_J,axiom,
% 12.85/6.99 ! [Z: real,A: real,B: real] :
% 12.85/6.99 ( ( ( Z = zero_zero_real )
% 12.85/6.99 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 12.85/6.99 = ( uminus_uminus_real @ B ) ) )
% 12.85/6.99 & ( ( Z != zero_zero_real )
% 12.85/6.99 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 12.85/6.99 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5832_add__divide__eq__if__simps_I6_J,axiom,
% 12.85/6.99 ! [Z: complex,A: complex,B: complex] :
% 12.85/6.99 ( ( ( Z = zero_zero_complex )
% 12.85/6.99 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ B ) ) )
% 12.85/6.99 & ( ( Z != zero_zero_complex )
% 12.85/6.99 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 12.85/6.99 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5833_add__divide__eq__if__simps_I6_J,axiom,
% 12.85/6.99 ! [Z: rat,A: rat,B: rat] :
% 12.85/6.99 ( ( ( Z = zero_zero_rat )
% 12.85/6.99 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 12.85/6.99 = ( uminus_uminus_rat @ B ) ) )
% 12.85/6.99 & ( ( Z != zero_zero_rat )
% 12.85/6.99 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 12.85/6.99 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5834_add__divide__eq__if__simps_I5_J,axiom,
% 12.85/6.99 ! [Z: real,A: real,B: real] :
% 12.85/6.99 ( ( ( Z = zero_zero_real )
% 12.85/6.99 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 12.85/6.99 = ( uminus_uminus_real @ B ) ) )
% 12.85/6.99 & ( ( Z != zero_zero_real )
% 12.85/6.99 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 12.85/6.99 = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5835_add__divide__eq__if__simps_I5_J,axiom,
% 12.85/6.99 ! [Z: complex,A: complex,B: complex] :
% 12.85/6.99 ( ( ( Z = zero_zero_complex )
% 12.85/6.99 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ B ) ) )
% 12.85/6.99 & ( ( Z != zero_zero_complex )
% 12.85/6.99 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 12.85/6.99 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5836_add__divide__eq__if__simps_I5_J,axiom,
% 12.85/6.99 ! [Z: rat,A: rat,B: rat] :
% 12.85/6.99 ( ( ( Z = zero_zero_rat )
% 12.85/6.99 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 12.85/6.99 = ( uminus_uminus_rat @ B ) ) )
% 12.85/6.99 & ( ( Z != zero_zero_rat )
% 12.85/6.99 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 12.85/6.99 = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5837_minus__divide__diff__eq__iff,axiom,
% 12.85/6.99 ! [Z: real,X: real,Y4: real] :
% 12.85/6.99 ( ( Z != zero_zero_real )
% 12.85/6.99 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y4 )
% 12.85/6.99 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y4 @ Z ) ) @ Z ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5838_minus__divide__diff__eq__iff,axiom,
% 12.85/6.99 ! [Z: complex,X: complex,Y4: complex] :
% 12.85/6.99 ( ( Z != zero_zero_complex )
% 12.85/6.99 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y4 )
% 12.85/6.99 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y4 @ Z ) ) @ Z ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5839_minus__divide__diff__eq__iff,axiom,
% 12.85/6.99 ! [Z: rat,X: rat,Y4: rat] :
% 12.85/6.99 ( ( Z != zero_zero_rat )
% 12.85/6.99 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y4 )
% 12.85/6.99 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y4 @ Z ) ) @ Z ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5840_even__minus,axiom,
% 12.85/6.99 ! [A: int] :
% 12.85/6.99 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
% 12.85/6.99 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5841_even__minus,axiom,
% 12.85/6.99 ! [A: code_integer] :
% 12.85/6.99 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
% 12.85/6.99 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5842_lemma__interval,axiom,
% 12.85/6.99 ! [A: real,X: real,B: real] :
% 12.85/6.99 ( ( ord_less_real @ A @ X )
% 12.85/6.99 => ( ( ord_less_real @ X @ B )
% 12.85/6.99 => ? [D2: real] :
% 12.85/6.99 ( ( ord_less_real @ zero_zero_real @ D2 )
% 12.85/6.99 & ! [Y3: real] :
% 12.85/6.99 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D2 )
% 12.85/6.99 => ( ( ord_less_eq_real @ A @ Y3 )
% 12.85/6.99 & ( ord_less_eq_real @ Y3 @ B ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5843_power2__eq__iff,axiom,
% 12.85/6.99 ! [X: real,Y4: real] :
% 12.85/6.99 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.99 = ( ( X = Y4 )
% 12.85/6.99 | ( X
% 12.85/6.99 = ( uminus_uminus_real @ Y4 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5844_power2__eq__iff,axiom,
% 12.85/6.99 ! [X: int,Y4: int] :
% 12.85/6.99 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.99 = ( ( X = Y4 )
% 12.85/6.99 | ( X
% 12.85/6.99 = ( uminus_uminus_int @ Y4 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5845_power2__eq__iff,axiom,
% 12.85/6.99 ! [X: complex,Y4: complex] :
% 12.85/6.99 ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = ( power_power_complex @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.99 = ( ( X = Y4 )
% 12.85/6.99 | ( X
% 12.85/6.99 = ( uminus1482373934393186551omplex @ Y4 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5846_power2__eq__iff,axiom,
% 12.85/6.99 ! [X: code_integer,Y4: code_integer] :
% 12.85/6.99 ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = ( power_8256067586552552935nteger @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.99 = ( ( X = Y4 )
% 12.85/6.99 | ( X
% 12.85/6.99 = ( uminus1351360451143612070nteger @ Y4 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5847_power2__eq__iff,axiom,
% 12.85/6.99 ! [X: rat,Y4: rat] :
% 12.85/6.99 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.99 = ( ( X = Y4 )
% 12.85/6.99 | ( X
% 12.85/6.99 = ( uminus_uminus_rat @ Y4 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5848_norm__triangle__ineq3,axiom,
% 12.85/6.99 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5849_norm__triangle__ineq3,axiom,
% 12.85/6.99 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5850_int__cases3,axiom,
% 12.85/6.99 ! [K: int] :
% 12.85/6.99 ( ( K != zero_zero_int )
% 12.85/6.99 => ( ! [N2: nat] :
% 12.85/6.99 ( ( K
% 12.85/6.99 = ( semiri1314217659103216013at_int @ N2 ) )
% 12.85/6.99 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 12.85/6.99 => ~ ! [N2: nat] :
% 12.85/6.99 ( ( K
% 12.85/6.99 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 12.85/6.99 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5851_pochhammer__eq__0__iff,axiom,
% 12.85/6.99 ! [A: complex,N: nat] :
% 12.85/6.99 ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 12.85/6.99 = zero_zero_complex )
% 12.85/6.99 = ( ? [K3: nat] :
% 12.85/6.99 ( ( ord_less_nat @ K3 @ N )
% 12.85/6.99 & ( A
% 12.85/6.99 = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5852_pochhammer__eq__0__iff,axiom,
% 12.85/6.99 ! [A: real,N: nat] :
% 12.85/6.99 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 12.85/6.99 = zero_zero_real )
% 12.85/6.99 = ( ? [K3: nat] :
% 12.85/6.99 ( ( ord_less_nat @ K3 @ N )
% 12.85/6.99 & ( A
% 12.85/6.99 = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5853_pochhammer__eq__0__iff,axiom,
% 12.85/6.99 ! [A: rat,N: nat] :
% 12.85/6.99 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 12.85/6.99 = zero_zero_rat )
% 12.85/6.99 = ( ? [K3: nat] :
% 12.85/6.99 ( ( ord_less_nat @ K3 @ N )
% 12.85/6.99 & ( A
% 12.85/6.99 = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5854_pochhammer__of__nat__eq__0__iff,axiom,
% 12.85/6.99 ! [N: nat,K: nat] :
% 12.85/6.99 ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 12.85/6.99 = zero_z3403309356797280102nteger )
% 12.85/6.99 = ( ord_less_nat @ N @ K ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5855_pochhammer__of__nat__eq__0__iff,axiom,
% 12.85/6.99 ! [N: nat,K: nat] :
% 12.85/6.99 ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 12.85/6.99 = zero_zero_complex )
% 12.85/6.99 = ( ord_less_nat @ N @ K ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5856_pochhammer__of__nat__eq__0__iff,axiom,
% 12.85/6.99 ! [N: nat,K: nat] :
% 12.85/6.99 ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 12.85/6.99 = zero_zero_real )
% 12.85/6.99 = ( ord_less_nat @ N @ K ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5857_pochhammer__of__nat__eq__0__iff,axiom,
% 12.85/6.99 ! [N: nat,K: nat] :
% 12.85/6.99 ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 12.85/6.99 = zero_zero_rat )
% 12.85/6.99 = ( ord_less_nat @ N @ K ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5858_pochhammer__of__nat__eq__0__iff,axiom,
% 12.85/6.99 ! [N: nat,K: nat] :
% 12.85/6.99 ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 12.85/6.99 = zero_zero_int )
% 12.85/6.99 = ( ord_less_nat @ N @ K ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5859_pochhammer__of__nat__eq__0__lemma,axiom,
% 12.85/6.99 ! [N: nat,K: nat] :
% 12.85/6.99 ( ( ord_less_nat @ N @ K )
% 12.85/6.99 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 12.85/6.99 = zero_z3403309356797280102nteger ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5860_pochhammer__of__nat__eq__0__lemma,axiom,
% 12.85/6.99 ! [N: nat,K: nat] :
% 12.85/6.99 ( ( ord_less_nat @ N @ K )
% 12.85/6.99 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 12.85/6.99 = zero_zero_complex ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5861_pochhammer__of__nat__eq__0__lemma,axiom,
% 12.85/6.99 ! [N: nat,K: nat] :
% 12.85/6.99 ( ( ord_less_nat @ N @ K )
% 12.85/6.99 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 12.85/6.99 = zero_zero_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5862_pochhammer__of__nat__eq__0__lemma,axiom,
% 12.85/6.99 ! [N: nat,K: nat] :
% 12.85/6.99 ( ( ord_less_nat @ N @ K )
% 12.85/6.99 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 12.85/6.99 = zero_zero_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5863_pochhammer__of__nat__eq__0__lemma,axiom,
% 12.85/6.99 ! [N: nat,K: nat] :
% 12.85/6.99 ( ( ord_less_nat @ N @ K )
% 12.85/6.99 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 12.85/6.99 = zero_zero_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5864_ln__add__one__self__le__self2,axiom,
% 12.85/6.99 ! [X: real] :
% 12.85/6.99 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 12.85/6.99 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5865_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 12.85/6.99 ! [K: nat,N: nat] :
% 12.85/6.99 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.99 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 12.85/6.99 != zero_z3403309356797280102nteger ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5866_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 12.85/6.99 ! [K: nat,N: nat] :
% 12.85/6.99 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.99 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 12.85/6.99 != zero_zero_complex ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5867_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 12.85/6.99 ! [K: nat,N: nat] :
% 12.85/6.99 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.99 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 12.85/6.99 != zero_zero_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5868_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 12.85/6.99 ! [K: nat,N: nat] :
% 12.85/6.99 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.99 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 12.85/6.99 != zero_zero_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5869_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 12.85/6.99 ! [K: nat,N: nat] :
% 12.85/6.99 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.99 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 12.85/6.99 != zero_zero_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5870_not__zle__0__negative,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5871_negD,axiom,
% 12.85/6.99 ! [X: int] :
% 12.85/6.99 ( ( ord_less_int @ X @ zero_zero_int )
% 12.85/6.99 => ? [N2: nat] :
% 12.85/6.99 ( X
% 12.85/6.99 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5872_negative__zless__0,axiom,
% 12.85/6.99 ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5873_verit__less__mono__div__int2,axiom,
% 12.85/6.99 ! [A2: int,B4: int,N: int] :
% 12.85/6.99 ( ( ord_less_eq_int @ A2 @ B4 )
% 12.85/6.99 => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
% 12.85/6.99 => ( ord_less_eq_int @ ( divide_divide_int @ B4 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5874_div__eq__minus1,axiom,
% 12.85/6.99 ! [B: int] :
% 12.85/6.99 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.99 => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 12.85/6.99 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5875_abs__le__square__iff,axiom,
% 12.85/6.99 ! [X: code_integer,Y4: code_integer] :
% 12.85/6.99 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y4 ) )
% 12.85/6.99 = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5876_abs__le__square__iff,axiom,
% 12.85/6.99 ! [X: rat,Y4: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y4 ) )
% 12.85/6.99 = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5877_abs__le__square__iff,axiom,
% 12.85/6.99 ! [X: int,Y4: int] :
% 12.85/6.99 ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y4 ) )
% 12.85/6.99 = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5878_abs__le__square__iff,axiom,
% 12.85/6.99 ! [X: real,Y4: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y4 ) )
% 12.85/6.99 = ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5879_abs__square__eq__1,axiom,
% 12.85/6.99 ! [X: code_integer] :
% 12.85/6.99 ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = one_one_Code_integer )
% 12.85/6.99 = ( ( abs_abs_Code_integer @ X )
% 12.85/6.99 = one_one_Code_integer ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5880_abs__square__eq__1,axiom,
% 12.85/6.99 ! [X: rat] :
% 12.85/6.99 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = one_one_rat )
% 12.85/6.99 = ( ( abs_abs_rat @ X )
% 12.85/6.99 = one_one_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5881_abs__square__eq__1,axiom,
% 12.85/6.99 ! [X: int] :
% 12.85/6.99 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = one_one_int )
% 12.85/6.99 = ( ( abs_abs_int @ X )
% 12.85/6.99 = one_one_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5882_abs__square__eq__1,axiom,
% 12.85/6.99 ! [X: real] :
% 12.85/6.99 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = one_one_real )
% 12.85/6.99 = ( ( abs_abs_real @ X )
% 12.85/6.99 = one_one_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5883_power__even__abs,axiom,
% 12.85/6.99 ! [N: nat,A: code_integer] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
% 12.85/6.99 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5884_power__even__abs,axiom,
% 12.85/6.99 ! [N: nat,A: rat] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N )
% 12.85/6.99 = ( power_power_rat @ A @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5885_power__even__abs,axiom,
% 12.85/6.99 ! [N: nat,A: int] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
% 12.85/6.99 = ( power_power_int @ A @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5886_power__even__abs,axiom,
% 12.85/6.99 ! [N: nat,A: real] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
% 12.85/6.99 = ( power_power_real @ A @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5887_le__minus__divide__eq,axiom,
% 12.85/6.99 ! [A: rat,B: rat,C: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 12.85/6.99 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 12.85/6.99 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 12.85/6.99 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5888_le__minus__divide__eq,axiom,
% 12.85/6.99 ! [A: real,B: real,C: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 12.85/6.99 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 12.85/6.99 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 12.85/6.99 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5889_minus__divide__le__eq,axiom,
% 12.85/6.99 ! [B: rat,C: rat,A: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 12.85/6.99 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 12.85/6.99 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 12.85/6.99 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5890_minus__divide__le__eq,axiom,
% 12.85/6.99 ! [B: real,C: real,A: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 12.85/6.99 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 12.85/6.99 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 12.85/6.99 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5891_neg__le__minus__divide__eq,axiom,
% 12.85/6.99 ! [C: rat,A: rat,B: rat] :
% 12.85/6.99 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 12.85/6.99 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5892_neg__le__minus__divide__eq,axiom,
% 12.85/6.99 ! [C: real,A: real,B: real] :
% 12.85/6.99 ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 12.85/6.99 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5893_neg__minus__divide__le__eq,axiom,
% 12.85/6.99 ! [C: rat,B: rat,A: rat] :
% 12.85/6.99 ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 12.85/6.99 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5894_neg__minus__divide__le__eq,axiom,
% 12.85/6.99 ! [C: real,B: real,A: real] :
% 12.85/6.99 ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 12.85/6.99 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5895_pos__le__minus__divide__eq,axiom,
% 12.85/6.99 ! [C: rat,A: rat,B: rat] :
% 12.85/6.99 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 12.85/6.99 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5896_pos__le__minus__divide__eq,axiom,
% 12.85/6.99 ! [C: real,A: real,B: real] :
% 12.85/6.99 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 12.85/6.99 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5897_pos__minus__divide__le__eq,axiom,
% 12.85/6.99 ! [C: rat,B: rat,A: rat] :
% 12.85/6.99 ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 12.85/6.99 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5898_pos__minus__divide__le__eq,axiom,
% 12.85/6.99 ! [C: real,B: real,A: real] :
% 12.85/6.99 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 12.85/6.99 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5899_divide__less__eq__numeral_I2_J,axiom,
% 12.85/6.99 ! [B: real,C: real,W: num] :
% 12.85/6.99 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.85/6.99 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 12.85/6.99 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 12.85/6.99 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5900_divide__less__eq__numeral_I2_J,axiom,
% 12.85/6.99 ! [B: rat,C: rat,W: num] :
% 12.85/6.99 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 12.85/6.99 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 12.85/6.99 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 12.85/6.99 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5901_less__divide__eq__numeral_I2_J,axiom,
% 12.85/6.99 ! [W: num,B: real,C: real] :
% 12.85/6.99 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 12.85/6.99 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 12.85/6.99 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 12.85/6.99 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5902_less__divide__eq__numeral_I2_J,axiom,
% 12.85/6.99 ! [W: num,B: rat,C: rat] :
% 12.85/6.99 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 12.85/6.99 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 12.85/6.99 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 12.85/6.99 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5903_power2__eq__1__iff,axiom,
% 12.85/6.99 ! [A: real] :
% 12.85/6.99 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = one_one_real )
% 12.85/6.99 = ( ( A = one_one_real )
% 12.85/6.99 | ( A
% 12.85/6.99 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5904_power2__eq__1__iff,axiom,
% 12.85/6.99 ! [A: int] :
% 12.85/6.99 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = one_one_int )
% 12.85/6.99 = ( ( A = one_one_int )
% 12.85/6.99 | ( A
% 12.85/6.99 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5905_power2__eq__1__iff,axiom,
% 12.85/6.99 ! [A: complex] :
% 12.85/6.99 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = one_one_complex )
% 12.85/6.99 = ( ( A = one_one_complex )
% 12.85/6.99 | ( A
% 12.85/6.99 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5906_power2__eq__1__iff,axiom,
% 12.85/6.99 ! [A: code_integer] :
% 12.85/6.99 ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = one_one_Code_integer )
% 12.85/6.99 = ( ( A = one_one_Code_integer )
% 12.85/6.99 | ( A
% 12.85/6.99 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5907_power2__eq__1__iff,axiom,
% 12.85/6.99 ! [A: rat] :
% 12.85/6.99 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/6.99 = one_one_rat )
% 12.85/6.99 = ( ( A = one_one_rat )
% 12.85/6.99 | ( A
% 12.85/6.99 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5908_uminus__power__if,axiom,
% 12.85/6.99 ! [N: nat,A: real] :
% 12.85/6.99 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 12.85/6.99 = ( power_power_real @ A @ N ) ) )
% 12.85/6.99 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 12.85/6.99 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5909_uminus__power__if,axiom,
% 12.85/6.99 ! [N: nat,A: int] :
% 12.85/6.99 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 12.85/6.99 = ( power_power_int @ A @ N ) ) )
% 12.85/6.99 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 12.85/6.99 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5910_uminus__power__if,axiom,
% 12.85/6.99 ! [N: nat,A: complex] :
% 12.85/6.99 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 12.85/6.99 = ( power_power_complex @ A @ N ) ) )
% 12.85/6.99 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5911_uminus__power__if,axiom,
% 12.85/6.99 ! [N: nat,A: code_integer] :
% 12.85/6.99 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 12.85/6.99 = ( power_8256067586552552935nteger @ A @ N ) ) )
% 12.85/6.99 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5912_uminus__power__if,axiom,
% 12.85/6.99 ! [N: nat,A: rat] :
% 12.85/6.99 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 12.85/6.99 = ( power_power_rat @ A @ N ) ) )
% 12.85/6.99 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 12.85/6.99 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5913_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 12.85/6.99 ! [K: nat,N: nat] :
% 12.85/6.99 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.99 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
% 12.85/6.99 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5914_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 12.85/6.99 ! [K: nat,N: nat] :
% 12.85/6.99 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.99 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
% 12.85/6.99 = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5915_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 12.85/6.99 ! [K: nat,N: nat] :
% 12.85/6.99 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.99 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
% 12.85/6.99 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5916_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 12.85/6.99 ! [K: nat,N: nat] :
% 12.85/6.99 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.99 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
% 12.85/6.99 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5917_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 12.85/6.99 ! [K: nat,N: nat] :
% 12.85/6.99 ( ( ord_less_eq_nat @ K @ N )
% 12.85/6.99 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
% 12.85/6.99 = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5918_realpow__square__minus__le,axiom,
% 12.85/6.99 ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5919_neg__int__cases,axiom,
% 12.85/6.99 ! [K: int] :
% 12.85/6.99 ( ( ord_less_int @ K @ zero_zero_int )
% 12.85/6.99 => ~ ! [N2: nat] :
% 12.85/6.99 ( ( K
% 12.85/6.99 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 12.85/6.99 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5920_ln__one__minus__pos__upper__bound,axiom,
% 12.85/6.99 ! [X: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/6.99 => ( ( ord_less_real @ X @ one_one_real )
% 12.85/6.99 => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5921_signed__take__bit__int__less__eq__self__iff,axiom,
% 12.85/6.99 ! [N: nat,K: int] :
% 12.85/6.99 ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 12.85/6.99 = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5922_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 12.85/6.99 ! [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 ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5923_signed__take__bit__int__greater__self__iff,axiom,
% 12.85/6.99 ! [K: int,N: nat] :
% 12.85/6.99 ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 12.85/6.99 = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5924_minus__mod__int__eq,axiom,
% 12.85/6.99 ! [L: int,K: int] :
% 12.85/6.99 ( ( ord_less_eq_int @ zero_zero_int @ L )
% 12.85/6.99 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
% 12.85/6.99 = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5925_zmod__minus1,axiom,
% 12.85/6.99 ! [B: int] :
% 12.85/6.99 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/6.99 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 12.85/6.99 = ( minus_minus_int @ B @ one_one_int ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5926_zdiv__zminus2__eq__if,axiom,
% 12.85/6.99 ! [B: int,A: int] :
% 12.85/6.99 ( ( B != zero_zero_int )
% 12.85/6.99 => ( ( ( ( modulo_modulo_int @ A @ B )
% 12.85/6.99 = zero_zero_int )
% 12.85/6.99 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 12.85/6.99 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 12.85/6.99 & ( ( ( modulo_modulo_int @ A @ B )
% 12.85/6.99 != zero_zero_int )
% 12.85/6.99 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 12.85/6.99 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5927_zdiv__zminus1__eq__if,axiom,
% 12.85/6.99 ! [B: int,A: int] :
% 12.85/6.99 ( ( B != zero_zero_int )
% 12.85/6.99 => ( ( ( ( modulo_modulo_int @ A @ B )
% 12.85/6.99 = zero_zero_int )
% 12.85/6.99 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 12.85/6.99 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 12.85/6.99 & ( ( ( modulo_modulo_int @ A @ B )
% 12.85/6.99 != zero_zero_int )
% 12.85/6.99 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 12.85/6.99 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5928_power2__le__iff__abs__le,axiom,
% 12.85/6.99 ! [Y4: code_integer,X: code_integer] :
% 12.85/6.99 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y4 )
% 12.85/6.99 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.99 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y4 ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5929_power2__le__iff__abs__le,axiom,
% 12.85/6.99 ! [Y4: rat,X: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
% 12.85/6.99 => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.99 = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y4 ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5930_power2__le__iff__abs__le,axiom,
% 12.85/6.99 ! [Y4: int,X: int] :
% 12.85/6.99 ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/6.99 => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.99 = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y4 ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5931_power2__le__iff__abs__le,axiom,
% 12.85/6.99 ! [Y4: real,X: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/6.99 => ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/6.99 = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y4 ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5932_abs__square__le__1,axiom,
% 12.85/6.99 ! [X: code_integer] :
% 12.85/6.99 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 12.85/6.99 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5933_abs__square__le__1,axiom,
% 12.85/6.99 ! [X: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 12.85/6.99 = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5934_abs__square__le__1,axiom,
% 12.85/6.99 ! [X: int] :
% 12.85/6.99 ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 12.85/6.99 = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5935_abs__square__le__1,axiom,
% 12.85/6.99 ! [X: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 12.85/6.99 = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5936_abs__square__less__1,axiom,
% 12.85/6.99 ! [X: code_integer] :
% 12.85/6.99 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 12.85/6.99 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5937_abs__square__less__1,axiom,
% 12.85/6.99 ! [X: real] :
% 12.85/6.99 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 12.85/6.99 = ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5938_abs__square__less__1,axiom,
% 12.85/6.99 ! [X: rat] :
% 12.85/6.99 ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 12.85/6.99 = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5939_abs__square__less__1,axiom,
% 12.85/6.99 ! [X: int] :
% 12.85/6.99 ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 12.85/6.99 = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5940_power__mono__even,axiom,
% 12.85/6.99 ! [N: nat,A: code_integer,B: code_integer] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
% 12.85/6.99 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5941_power__mono__even,axiom,
% 12.85/6.99 ! [N: nat,A: rat,B: rat] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
% 12.85/6.99 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5942_power__mono__even,axiom,
% 12.85/6.99 ! [N: nat,A: int,B: int] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
% 12.85/6.99 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5943_power__mono__even,axiom,
% 12.85/6.99 ! [N: nat,A: real,B: real] :
% 12.85/6.99 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
% 12.85/6.99 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5944_divide__le__eq__numeral_I2_J,axiom,
% 12.85/6.99 ! [B: rat,C: rat,W: num] :
% 12.85/6.99 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 12.85/6.99 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 12.85/6.99 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 12.85/6.99 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5945_divide__le__eq__numeral_I2_J,axiom,
% 12.85/6.99 ! [B: real,C: real,W: num] :
% 12.85/6.99 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.85/6.99 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 12.85/6.99 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 12.85/6.99 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5946_le__divide__eq__numeral_I2_J,axiom,
% 12.85/6.99 ! [W: num,B: rat,C: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 12.85/6.99 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 12.85/6.99 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 12.85/6.99 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 12.85/6.99 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 12.85/6.99 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5947_le__divide__eq__numeral_I2_J,axiom,
% 12.85/6.99 ! [W: num,B: real,C: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 12.85/6.99 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 12.85/6.99 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 12.85/6.99 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 12.85/6.99 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 12.85/6.99 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5948_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
% 12.85/6.99 ! [X: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 12.85/6.99 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 12.85/6.99 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5949_square__le__1,axiom,
% 12.85/6.99 ! [X: code_integer] :
% 12.85/6.99 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 12.85/6.99 => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
% 12.85/6.99 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5950_square__le__1,axiom,
% 12.85/6.99 ! [X: rat] :
% 12.85/6.99 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
% 12.85/6.99 => ( ( ord_less_eq_rat @ X @ one_one_rat )
% 12.85/6.99 => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5951_square__le__1,axiom,
% 12.85/6.99 ! [X: int] :
% 12.85/6.99 ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 12.85/6.99 => ( ( ord_less_eq_int @ X @ one_one_int )
% 12.85/6.99 => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5952_square__le__1,axiom,
% 12.85/6.99 ! [X: real] :
% 12.85/6.99 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 12.85/6.99 => ( ( ord_less_eq_real @ X @ one_one_real )
% 12.85/6.99 => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5953_minus__power__mult__self,axiom,
% 12.85/6.99 ! [A: real,N: nat] :
% 12.85/6.99 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 12.85/6.99 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5954_minus__power__mult__self,axiom,
% 12.85/6.99 ! [A: int,N: nat] :
% 12.85/6.99 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 12.85/6.99 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5955_minus__power__mult__self,axiom,
% 12.85/6.99 ! [A: complex,N: nat] :
% 12.85/6.99 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
% 12.85/6.99 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5956_minus__power__mult__self,axiom,
% 12.85/6.99 ! [A: code_integer,N: nat] :
% 12.85/6.99 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 12.85/6.99 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5957_minus__power__mult__self,axiom,
% 12.85/6.99 ! [A: rat,N: nat] :
% 12.85/6.99 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 12.85/6.99 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5958_minus__one__power__iff,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 12.85/6.99 = one_one_real ) )
% 12.85/6.99 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 12.85/6.99 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5959_minus__one__power__iff,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 12.85/6.99 = one_one_int ) )
% 12.85/6.99 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 12.85/6.99 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5960_minus__one__power__iff,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 12.85/6.99 = one_one_complex ) )
% 12.85/6.99 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 12.85/6.99 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5961_minus__one__power__iff,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 12.85/6.99 = one_one_Code_integer ) )
% 12.85/6.99 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 12.85/6.99 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5962_minus__one__power__iff,axiom,
% 12.85/6.99 ! [N: nat] :
% 12.85/6.99 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 12.85/6.99 = one_one_rat ) )
% 12.85/6.99 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/6.99 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 12.85/6.99 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5963_pochhammer__absorb__comp,axiom,
% 12.85/6.99 ! [R2: code_integer,K: nat] :
% 12.85/6.99 ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K ) )
% 12.85/6.99 = ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5964_pochhammer__absorb__comp,axiom,
% 12.85/6.99 ! [R2: complex,K: nat] :
% 12.85/6.99 ( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K ) )
% 12.85/6.99 = ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5965_pochhammer__absorb__comp,axiom,
% 12.85/6.99 ! [R2: real,K: nat] :
% 12.85/6.99 ( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K ) )
% 12.85/6.99 = ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5966_pochhammer__absorb__comp,axiom,
% 12.85/6.99 ! [R2: rat,K: nat] :
% 12.85/6.99 ( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K ) )
% 12.85/6.99 = ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K ) ) ) ).
% 12.85/6.99
% 12.85/6.99 thf(fact_5967_pochhammer__absorb__comp,axiom,
% 12.85/6.99 ! [R2: int,K: nat] :
% 12.85/6.99 ( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K ) )
% 12.85/7.00 = ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5968_pochhammer__same,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ N )
% 12.85/7.00 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5969_pochhammer__same,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ N )
% 12.85/7.00 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5970_pochhammer__same,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ N )
% 12.85/7.00 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5971_pochhammer__same,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ N )
% 12.85/7.00 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5972_pochhammer__same,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ N )
% 12.85/7.00 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5973_signed__take__bit__int__eq__self__iff,axiom,
% 12.85/7.00 ! [N: nat,K: int] :
% 12.85/7.00 ( ( ( bit_ri631733984087533419it_int @ N @ K )
% 12.85/7.00 = K )
% 12.85/7.00 = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 12.85/7.00 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5974_signed__take__bit__int__eq__self,axiom,
% 12.85/7.00 ! [N: nat,K: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 12.85/7.00 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.00 => ( ( bit_ri631733984087533419it_int @ N @ K )
% 12.85/7.00 = K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5975_Bernoulli__inequality,axiom,
% 12.85/7.00 ! [X: real,N: nat] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 12.85/7.00 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5976_minus__1__div__exp__eq__int,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.00 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5977_div__pos__neg__trivial,axiom,
% 12.85/7.00 ! [K: int,L: int] :
% 12.85/7.00 ( ( ord_less_int @ zero_zero_int @ K )
% 12.85/7.00 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 12.85/7.00 => ( ( divide_divide_int @ K @ L )
% 12.85/7.00 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5978_power__minus1__odd,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.85/7.00 = ( uminus_uminus_real @ one_one_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5979_power__minus1__odd,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.85/7.00 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5980_power__minus1__odd,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.85/7.00 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5981_power__minus1__odd,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.85/7.00 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5982_power__minus1__odd,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.85/7.00 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5983_pochhammer__minus,axiom,
% 12.85/7.00 ! [B: code_integer,K: nat] :
% 12.85/7.00 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
% 12.85/7.00 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5984_pochhammer__minus,axiom,
% 12.85/7.00 ! [B: complex,K: nat] :
% 12.85/7.00 ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
% 12.85/7.00 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5985_pochhammer__minus,axiom,
% 12.85/7.00 ! [B: real,K: nat] :
% 12.85/7.00 ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
% 12.85/7.00 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5986_pochhammer__minus,axiom,
% 12.85/7.00 ! [B: rat,K: nat] :
% 12.85/7.00 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
% 12.85/7.00 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5987_pochhammer__minus,axiom,
% 12.85/7.00 ! [B: int,K: nat] :
% 12.85/7.00 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
% 12.85/7.00 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5988_pochhammer__minus_H,axiom,
% 12.85/7.00 ! [B: code_integer,K: nat] :
% 12.85/7.00 ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
% 12.85/7.00 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5989_pochhammer__minus_H,axiom,
% 12.85/7.00 ! [B: complex,K: nat] :
% 12.85/7.00 ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
% 12.85/7.00 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5990_pochhammer__minus_H,axiom,
% 12.85/7.00 ! [B: real,K: nat] :
% 12.85/7.00 ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
% 12.85/7.00 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5991_pochhammer__minus_H,axiom,
% 12.85/7.00 ! [B: rat,K: nat] :
% 12.85/7.00 ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
% 12.85/7.00 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5992_pochhammer__minus_H,axiom,
% 12.85/7.00 ! [B: int,K: nat] :
% 12.85/7.00 ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
% 12.85/7.00 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5993_int__bit__induct,axiom,
% 12.85/7.00 ! [P: int > $o,K: int] :
% 12.85/7.00 ( ( P @ zero_zero_int )
% 12.85/7.00 => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.00 => ( ! [K2: int] :
% 12.85/7.00 ( ( P @ K2 )
% 12.85/7.00 => ( ( K2 != zero_zero_int )
% 12.85/7.00 => ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
% 12.85/7.00 => ( ! [K2: int] :
% 12.85/7.00 ( ( P @ K2 )
% 12.85/7.00 => ( ( K2
% 12.85/7.00 != ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.00 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
% 12.85/7.00 => ( P @ K ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5994_signed__take__bit__int__greater__eq,axiom,
% 12.85/7.00 ! [K: int,N: nat] :
% 12.85/7.00 ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 12.85/7.00 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5995_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ one_one_real )
% 12.85/7.00 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5996_fact__code,axiom,
% 12.85/7.00 ( semiri773545260158071498ct_rat
% 12.85/7.00 = ( ^ [N4: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5997_fact__code,axiom,
% 12.85/7.00 ( semiri1406184849735516958ct_int
% 12.85/7.00 = ( ^ [N4: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5998_fact__code,axiom,
% 12.85/7.00 ( semiri1408675320244567234ct_nat
% 12.85/7.00 = ( ^ [N4: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_5999_fact__code,axiom,
% 12.85/7.00 ( semiri2265585572941072030t_real
% 12.85/7.00 = ( ^ [N4: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6000_fact__code,axiom,
% 12.85/7.00 ( semiri5044797733671781792omplex
% 12.85/7.00 = ( ^ [N4: nat] : ( semiri8010041392384452111omplex @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6001_abs__sqrt__wlog,axiom,
% 12.85/7.00 ! [P: code_integer > code_integer > $o,X: code_integer] :
% 12.85/7.00 ( ! [X4: code_integer] :
% 12.85/7.00 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
% 12.85/7.00 => ( P @ X4 @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.00 => ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6002_abs__sqrt__wlog,axiom,
% 12.85/7.00 ! [P: rat > rat > $o,X: rat] :
% 12.85/7.00 ( ! [X4: rat] :
% 12.85/7.00 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 12.85/7.00 => ( P @ X4 @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.00 => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6003_abs__sqrt__wlog,axiom,
% 12.85/7.00 ! [P: int > int > $o,X: int] :
% 12.85/7.00 ( ! [X4: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 12.85/7.00 => ( P @ X4 @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.00 => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6004_abs__sqrt__wlog,axiom,
% 12.85/7.00 ! [P: real > real > $o,X: real] :
% 12.85/7.00 ( ! [X4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.85/7.00 => ( P @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.00 => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6005_sin__coeff__def,axiom,
% 12.85/7.00 ( sin_coeff
% 12.85/7.00 = ( ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N4 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6006_cos__coeff__def,axiom,
% 12.85/7.00 ( cos_coeff
% 12.85/7.00 = ( ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N4 ) ) @ zero_zero_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6007_compl__le__compl__iff,axiom,
% 12.85/7.00 ! [X: set_int,Y4: set_int] :
% 12.85/7.00 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( uminus1532241313380277803et_int @ Y4 ) )
% 12.85/7.00 = ( ord_less_eq_set_int @ Y4 @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6008_listsum__bound,axiom,
% 12.85/7.00 ! [Xs: list_complex,F: complex > real,Y4: real] :
% 12.85/7.00 ( ! [X4: complex] :
% 12.85/7.00 ( ( member_complex @ X4 @ ( set_complex2 @ Xs ) )
% 12.85/7.00 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 12.85/7.00 => ( ord_less_eq_real @ Y4 @ ( foldr_real_real @ plus_plus_real @ ( map_complex_real @ F @ Xs ) @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6009_listsum__bound,axiom,
% 12.85/7.00 ! [Xs: list_set_nat,F: set_nat > real,Y4: real] :
% 12.85/7.00 ( ! [X4: set_nat] :
% 12.85/7.00 ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
% 12.85/7.00 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 12.85/7.00 => ( ord_less_eq_real @ Y4 @ ( foldr_real_real @ plus_plus_real @ ( map_set_nat_real @ F @ Xs ) @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6010_listsum__bound,axiom,
% 12.85/7.00 ! [Xs: list_VEBT_VEBT,F: vEBT_VEBT > real,Y4: real] :
% 12.85/7.00 ( ! [X4: vEBT_VEBT] :
% 12.85/7.00 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 12.85/7.00 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 12.85/7.00 => ( ord_less_eq_real @ Y4 @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ F @ Xs ) @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6011_listsum__bound,axiom,
% 12.85/7.00 ! [Xs: list_nat,F: nat > real,Y4: real] :
% 12.85/7.00 ( ! [X4: nat] :
% 12.85/7.00 ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 12.85/7.00 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 12.85/7.00 => ( ord_less_eq_real @ Y4 @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ F @ Xs ) @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6012_listsum__bound,axiom,
% 12.85/7.00 ! [Xs: list_real,F: real > real,Y4: real] :
% 12.85/7.00 ( ! [X4: real] :
% 12.85/7.00 ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
% 12.85/7.00 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 12.85/7.00 => ( ord_less_eq_real @ Y4 @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ F @ Xs ) @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6013_listsum__bound,axiom,
% 12.85/7.00 ! [Xs: list_int,F: int > real,Y4: real] :
% 12.85/7.00 ( ! [X4: int] :
% 12.85/7.00 ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 12.85/7.00 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 12.85/7.00 => ( ord_less_eq_real @ Y4 @ ( foldr_real_real @ plus_plus_real @ ( map_int_real @ F @ Xs ) @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6014_signed__take__bit__numeral__minus__bit1,axiom,
% 12.85/7.00 ! [L: num,K: num] :
% 12.85/7.00 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 12.85/7.00 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6015_dbl__dec__simps_I4_J,axiom,
% 12.85/7.00 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
% 12.85/7.00 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6016_dbl__dec__simps_I4_J,axiom,
% 12.85/7.00 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.00 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6017_dbl__dec__simps_I4_J,axiom,
% 12.85/7.00 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.85/7.00 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6018_dbl__dec__simps_I4_J,axiom,
% 12.85/7.00 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.00 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6019_dbl__dec__simps_I4_J,axiom,
% 12.85/7.00 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.85/7.00 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6020_ComplI,axiom,
% 12.85/7.00 ! [C: complex,A2: set_complex] :
% 12.85/7.00 ( ~ ( member_complex @ C @ A2 )
% 12.85/7.00 => ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6021_ComplI,axiom,
% 12.85/7.00 ! [C: real,A2: set_real] :
% 12.85/7.00 ( ~ ( member_real @ C @ A2 )
% 12.85/7.00 => ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6022_ComplI,axiom,
% 12.85/7.00 ! [C: set_nat,A2: set_set_nat] :
% 12.85/7.00 ( ~ ( member_set_nat @ C @ A2 )
% 12.85/7.00 => ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6023_ComplI,axiom,
% 12.85/7.00 ! [C: nat,A2: set_nat] :
% 12.85/7.00 ( ~ ( member_nat @ C @ A2 )
% 12.85/7.00 => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6024_ComplI,axiom,
% 12.85/7.00 ! [C: int,A2: set_int] :
% 12.85/7.00 ( ~ ( member_int @ C @ A2 )
% 12.85/7.00 => ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6025_Compl__iff,axiom,
% 12.85/7.00 ! [C: complex,A2: set_complex] :
% 12.85/7.00 ( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) )
% 12.85/7.00 = ( ~ ( member_complex @ C @ A2 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6026_Compl__iff,axiom,
% 12.85/7.00 ! [C: real,A2: set_real] :
% 12.85/7.00 ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
% 12.85/7.00 = ( ~ ( member_real @ C @ A2 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6027_Compl__iff,axiom,
% 12.85/7.00 ! [C: set_nat,A2: set_set_nat] :
% 12.85/7.00 ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
% 12.85/7.00 = ( ~ ( member_set_nat @ C @ A2 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6028_Compl__iff,axiom,
% 12.85/7.00 ! [C: nat,A2: set_nat] :
% 12.85/7.00 ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
% 12.85/7.00 = ( ~ ( member_nat @ C @ A2 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6029_Compl__iff,axiom,
% 12.85/7.00 ! [C: int,A2: set_int] :
% 12.85/7.00 ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
% 12.85/7.00 = ( ~ ( member_int @ C @ A2 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6030_zdvd1__eq,axiom,
% 12.85/7.00 ! [X: int] :
% 12.85/7.00 ( ( dvd_dvd_int @ X @ one_one_int )
% 12.85/7.00 = ( ( abs_abs_int @ X )
% 12.85/7.00 = one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6031_dbl__dec__simps_I3_J,axiom,
% 12.85/7.00 ( ( neg_nu6511756317524482435omplex @ one_one_complex )
% 12.85/7.00 = one_one_complex ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6032_dbl__dec__simps_I3_J,axiom,
% 12.85/7.00 ( ( neg_nu6075765906172075777c_real @ one_one_real )
% 12.85/7.00 = one_one_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6033_dbl__dec__simps_I3_J,axiom,
% 12.85/7.00 ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
% 12.85/7.00 = one_one_rat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6034_dbl__dec__simps_I3_J,axiom,
% 12.85/7.00 ( ( neg_nu3811975205180677377ec_int @ one_one_int )
% 12.85/7.00 = one_one_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6035_zabs__less__one__iff,axiom,
% 12.85/7.00 ! [Z: int] :
% 12.85/7.00 ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 12.85/7.00 = ( Z = zero_zero_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6036_in__set__replicate,axiom,
% 12.85/7.00 ! [X: complex,N: nat,Y4: complex] :
% 12.85/7.00 ( ( member_complex @ X @ ( set_complex2 @ ( replicate_complex @ N @ Y4 ) ) )
% 12.85/7.00 = ( ( X = Y4 )
% 12.85/7.00 & ( N != zero_zero_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6037_in__set__replicate,axiom,
% 12.85/7.00 ! [X: set_nat,N: nat,Y4: set_nat] :
% 12.85/7.00 ( ( member_set_nat @ X @ ( set_set_nat2 @ ( replicate_set_nat @ N @ Y4 ) ) )
% 12.85/7.00 = ( ( X = Y4 )
% 12.85/7.00 & ( N != zero_zero_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6038_in__set__replicate,axiom,
% 12.85/7.00 ! [X: nat,N: nat,Y4: nat] :
% 12.85/7.00 ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ Y4 ) ) )
% 12.85/7.00 = ( ( X = Y4 )
% 12.85/7.00 & ( N != zero_zero_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6039_in__set__replicate,axiom,
% 12.85/7.00 ! [X: int,N: nat,Y4: int] :
% 12.85/7.00 ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ Y4 ) ) )
% 12.85/7.00 = ( ( X = Y4 )
% 12.85/7.00 & ( N != zero_zero_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6040_in__set__replicate,axiom,
% 12.85/7.00 ! [X: real,N: nat,Y4: real] :
% 12.85/7.00 ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N @ Y4 ) ) )
% 12.85/7.00 = ( ( X = Y4 )
% 12.85/7.00 & ( N != zero_zero_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6041_in__set__replicate,axiom,
% 12.85/7.00 ! [X: vEBT_VEBT,N: nat,Y4: vEBT_VEBT] :
% 12.85/7.00 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y4 ) ) )
% 12.85/7.00 = ( ( X = Y4 )
% 12.85/7.00 & ( N != zero_zero_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6042_Bex__set__replicate,axiom,
% 12.85/7.00 ! [N: nat,A: nat,P: nat > $o] :
% 12.85/7.00 ( ( ? [X3: nat] :
% 12.85/7.00 ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
% 12.85/7.00 & ( P @ X3 ) ) )
% 12.85/7.00 = ( ( P @ A )
% 12.85/7.00 & ( N != zero_zero_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6043_Bex__set__replicate,axiom,
% 12.85/7.00 ! [N: nat,A: int,P: int > $o] :
% 12.85/7.00 ( ( ? [X3: int] :
% 12.85/7.00 ( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
% 12.85/7.00 & ( P @ X3 ) ) )
% 12.85/7.00 = ( ( P @ A )
% 12.85/7.00 & ( N != zero_zero_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6044_Bex__set__replicate,axiom,
% 12.85/7.00 ! [N: nat,A: real,P: real > $o] :
% 12.85/7.00 ( ( ? [X3: real] :
% 12.85/7.00 ( ( member_real @ X3 @ ( set_real2 @ ( replicate_real @ N @ A ) ) )
% 12.85/7.00 & ( P @ X3 ) ) )
% 12.85/7.00 = ( ( P @ A )
% 12.85/7.00 & ( N != zero_zero_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6045_Bex__set__replicate,axiom,
% 12.85/7.00 ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 12.85/7.00 ( ( ? [X3: vEBT_VEBT] :
% 12.85/7.00 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
% 12.85/7.00 & ( P @ X3 ) ) )
% 12.85/7.00 = ( ( P @ A )
% 12.85/7.00 & ( N != zero_zero_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6046_Ball__set__replicate,axiom,
% 12.85/7.00 ! [N: nat,A: nat,P: nat > $o] :
% 12.85/7.00 ( ( ! [X3: nat] :
% 12.85/7.00 ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
% 12.85/7.00 => ( P @ X3 ) ) )
% 12.85/7.00 = ( ( P @ A )
% 12.85/7.00 | ( N = zero_zero_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6047_Ball__set__replicate,axiom,
% 12.85/7.00 ! [N: nat,A: int,P: int > $o] :
% 12.85/7.00 ( ( ! [X3: int] :
% 12.85/7.00 ( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
% 12.85/7.00 => ( P @ X3 ) ) )
% 12.85/7.00 = ( ( P @ A )
% 12.85/7.00 | ( N = zero_zero_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6048_Ball__set__replicate,axiom,
% 12.85/7.00 ! [N: nat,A: real,P: real > $o] :
% 12.85/7.00 ( ( ! [X3: real] :
% 12.85/7.00 ( ( member_real @ X3 @ ( set_real2 @ ( replicate_real @ N @ A ) ) )
% 12.85/7.00 => ( P @ X3 ) ) )
% 12.85/7.00 = ( ( P @ A )
% 12.85/7.00 | ( N = zero_zero_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6049_Ball__set__replicate,axiom,
% 12.85/7.00 ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 12.85/7.00 ( ( ! [X3: vEBT_VEBT] :
% 12.85/7.00 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
% 12.85/7.00 => ( P @ X3 ) ) )
% 12.85/7.00 = ( ( P @ A )
% 12.85/7.00 | ( N = zero_zero_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6050_pred__numeral__simps_I1_J,axiom,
% 12.85/7.00 ( ( pred_numeral @ one )
% 12.85/7.00 = zero_zero_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6051_eq__numeral__Suc,axiom,
% 12.85/7.00 ! [K: num,N: nat] :
% 12.85/7.00 ( ( ( numeral_numeral_nat @ K )
% 12.85/7.00 = ( suc @ N ) )
% 12.85/7.00 = ( ( pred_numeral @ K )
% 12.85/7.00 = N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6052_Suc__eq__numeral,axiom,
% 12.85/7.00 ! [N: nat,K: num] :
% 12.85/7.00 ( ( ( suc @ N )
% 12.85/7.00 = ( numeral_numeral_nat @ K ) )
% 12.85/7.00 = ( N
% 12.85/7.00 = ( pred_numeral @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6053_sin__coeff__0,axiom,
% 12.85/7.00 ( ( sin_coeff @ zero_zero_nat )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6054_cos__coeff__0,axiom,
% 12.85/7.00 ( ( cos_coeff @ zero_zero_nat )
% 12.85/7.00 = one_one_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6055_pred__numeral__simps_I3_J,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( pred_numeral @ ( bit1 @ K ) )
% 12.85/7.00 = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6056_less__numeral__Suc,axiom,
% 12.85/7.00 ! [K: num,N: nat] :
% 12.85/7.00 ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 12.85/7.00 = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6057_less__Suc__numeral,axiom,
% 12.85/7.00 ! [N: nat,K: num] :
% 12.85/7.00 ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 12.85/7.00 = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6058_le__numeral__Suc,axiom,
% 12.85/7.00 ! [K: num,N: nat] :
% 12.85/7.00 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 12.85/7.00 = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6059_le__Suc__numeral,axiom,
% 12.85/7.00 ! [N: nat,K: num] :
% 12.85/7.00 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 12.85/7.00 = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6060_diff__numeral__Suc,axiom,
% 12.85/7.00 ! [K: num,N: nat] :
% 12.85/7.00 ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 12.85/7.00 = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6061_diff__Suc__numeral,axiom,
% 12.85/7.00 ! [N: nat,K: num] :
% 12.85/7.00 ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 12.85/7.00 = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6062_dbl__dec__simps_I2_J,axiom,
% 12.85/7.00 ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 12.85/7.00 = ( uminus_uminus_real @ one_one_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6063_dbl__dec__simps_I2_J,axiom,
% 12.85/7.00 ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 12.85/7.00 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6064_dbl__dec__simps_I2_J,axiom,
% 12.85/7.00 ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 12.85/7.00 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6065_dbl__dec__simps_I2_J,axiom,
% 12.85/7.00 ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 12.85/7.00 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6066_dbl__dec__simps_I2_J,axiom,
% 12.85/7.00 ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 12.85/7.00 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6067_dbl__inc__simps_I1_J,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 12.85/7.00 = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6068_dbl__inc__simps_I1_J,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.85/7.00 = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6069_dbl__inc__simps_I1_J,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 12.85/7.00 = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6070_dbl__inc__simps_I1_J,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 12.85/7.00 = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6071_dbl__inc__simps_I1_J,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 12.85/7.00 = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6072_dbl__dec__simps_I1_J,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 12.85/7.00 = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6073_dbl__dec__simps_I1_J,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.85/7.00 = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6074_dbl__dec__simps_I1_J,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 12.85/7.00 = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6075_dbl__dec__simps_I1_J,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 12.85/7.00 = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6076_dbl__dec__simps_I1_J,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 12.85/7.00 = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6077_signed__take__bit__numeral__bit0,axiom,
% 12.85/7.00 ! [L: num,K: num] :
% 12.85/7.00 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 12.85/7.00 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6078_signed__take__bit__numeral__minus__bit0,axiom,
% 12.85/7.00 ! [L: num,K: num] :
% 12.85/7.00 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 12.85/7.00 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6079_signed__take__bit__numeral__bit1,axiom,
% 12.85/7.00 ! [L: num,K: num] :
% 12.85/7.00 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 12.85/7.00 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6080_ComplD,axiom,
% 12.85/7.00 ! [C: complex,A2: set_complex] :
% 12.85/7.00 ( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) )
% 12.85/7.00 => ~ ( member_complex @ C @ A2 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6081_ComplD,axiom,
% 12.85/7.00 ! [C: real,A2: set_real] :
% 12.85/7.00 ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
% 12.85/7.00 => ~ ( member_real @ C @ A2 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6082_ComplD,axiom,
% 12.85/7.00 ! [C: set_nat,A2: set_set_nat] :
% 12.85/7.00 ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
% 12.85/7.00 => ~ ( member_set_nat @ C @ A2 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6083_ComplD,axiom,
% 12.85/7.00 ! [C: nat,A2: set_nat] :
% 12.85/7.00 ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
% 12.85/7.00 => ~ ( member_nat @ C @ A2 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6084_ComplD,axiom,
% 12.85/7.00 ! [C: int,A2: set_int] :
% 12.85/7.00 ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
% 12.85/7.00 => ~ ( member_int @ C @ A2 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6085_subset__code_I1_J,axiom,
% 12.85/7.00 ! [Xs: list_complex,B4: set_complex] :
% 12.85/7.00 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ B4 )
% 12.85/7.00 = ( ! [X3: complex] :
% 12.85/7.00 ( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
% 12.85/7.00 => ( member_complex @ X3 @ B4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6086_subset__code_I1_J,axiom,
% 12.85/7.00 ! [Xs: list_set_nat,B4: set_set_nat] :
% 12.85/7.00 ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B4 )
% 12.85/7.00 = ( ! [X3: set_nat] :
% 12.85/7.00 ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
% 12.85/7.00 => ( member_set_nat @ X3 @ B4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6087_subset__code_I1_J,axiom,
% 12.85/7.00 ! [Xs: list_VEBT_VEBT,B4: set_VEBT_VEBT] :
% 12.85/7.00 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B4 )
% 12.85/7.00 = ( ! [X3: vEBT_VEBT] :
% 12.85/7.00 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 12.85/7.00 => ( member_VEBT_VEBT @ X3 @ B4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6088_subset__code_I1_J,axiom,
% 12.85/7.00 ! [Xs: list_nat,B4: set_nat] :
% 12.85/7.00 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B4 )
% 12.85/7.00 = ( ! [X3: nat] :
% 12.85/7.00 ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 12.85/7.00 => ( member_nat @ X3 @ B4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6089_subset__code_I1_J,axiom,
% 12.85/7.00 ! [Xs: list_real,B4: set_real] :
% 12.85/7.00 ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B4 )
% 12.85/7.00 = ( ! [X3: real] :
% 12.85/7.00 ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 12.85/7.00 => ( member_real @ X3 @ B4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6090_subset__code_I1_J,axiom,
% 12.85/7.00 ! [Xs: list_int,B4: set_int] :
% 12.85/7.00 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B4 )
% 12.85/7.00 = ( ! [X3: int] :
% 12.85/7.00 ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 12.85/7.00 => ( member_int @ X3 @ B4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6091_zdvd__antisym__abs,axiom,
% 12.85/7.00 ! [A: int,B: int] :
% 12.85/7.00 ( ( dvd_dvd_int @ A @ B )
% 12.85/7.00 => ( ( dvd_dvd_int @ B @ A )
% 12.85/7.00 => ( ( abs_abs_int @ A )
% 12.85/7.00 = ( abs_abs_int @ B ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6092_abs__zmult__eq__1,axiom,
% 12.85/7.00 ! [M: int,N: int] :
% 12.85/7.00 ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
% 12.85/7.00 = one_one_int )
% 12.85/7.00 => ( ( abs_abs_int @ M )
% 12.85/7.00 = one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6093_abs__div,axiom,
% 12.85/7.00 ! [Y4: int,X: int] :
% 12.85/7.00 ( ( dvd_dvd_int @ Y4 @ X )
% 12.85/7.00 => ( ( abs_abs_int @ ( divide_divide_int @ X @ Y4 ) )
% 12.85/7.00 = ( divide_divide_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6094_numeral__eq__Suc,axiom,
% 12.85/7.00 ( numeral_numeral_nat
% 12.85/7.00 = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6095_sin__coeff__Suc,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( sin_coeff @ ( suc @ N ) )
% 12.85/7.00 = ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6096_zabs__def,axiom,
% 12.85/7.00 ( abs_abs_int
% 12.85/7.00 = ( ^ [I2: int] : ( if_int @ ( ord_less_int @ I2 @ zero_zero_int ) @ ( uminus_uminus_int @ I2 ) @ I2 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6097_dvd__imp__le__int,axiom,
% 12.85/7.00 ! [I: int,D: int] :
% 12.85/7.00 ( ( I != zero_zero_int )
% 12.85/7.00 => ( ( dvd_dvd_int @ D @ I )
% 12.85/7.00 => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6098_abs__mod__less,axiom,
% 12.85/7.00 ! [L: int,K: int] :
% 12.85/7.00 ( ( L != zero_zero_int )
% 12.85/7.00 => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6099_cos__coeff__Suc,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( cos_coeff @ ( suc @ N ) )
% 12.85/7.00 = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6100_pred__numeral__def,axiom,
% 12.85/7.00 ( pred_numeral
% 12.85/7.00 = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6101_zdvd__mult__cancel1,axiom,
% 12.85/7.00 ! [M: int,N: int] :
% 12.85/7.00 ( ( M != zero_zero_int )
% 12.85/7.00 => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
% 12.85/7.00 = ( ( abs_abs_int @ N )
% 12.85/7.00 = one_one_int ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6102_fact__numeral,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
% 12.85/7.00 = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6103_fact__numeral,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
% 12.85/7.00 = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6104_fact__numeral,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
% 12.85/7.00 = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6105_fact__numeral,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
% 12.85/7.00 = ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6106_fact__numeral,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
% 12.85/7.00 = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6107_even__abs__add__iff,axiom,
% 12.85/7.00 ! [K: int,L: int] :
% 12.85/7.00 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L ) )
% 12.85/7.00 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6108_even__add__abs__iff,axiom,
% 12.85/7.00 ! [K: int,L: int] :
% 12.85/7.00 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L ) ) )
% 12.85/7.00 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6109_nat__intermed__int__val,axiom,
% 12.85/7.00 ! [M: nat,N: nat,F: nat > int,K: int] :
% 12.85/7.00 ( ! [I3: nat] :
% 12.85/7.00 ( ( ( ord_less_eq_nat @ M @ I3 )
% 12.85/7.00 & ( ord_less_nat @ I3 @ N ) )
% 12.85/7.00 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 12.85/7.00 => ( ( ord_less_eq_nat @ M @ N )
% 12.85/7.00 => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 12.85/7.00 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 12.85/7.00 => ? [I3: nat] :
% 12.85/7.00 ( ( ord_less_eq_nat @ M @ I3 )
% 12.85/7.00 & ( ord_less_eq_nat @ I3 @ N )
% 12.85/7.00 & ( ( F @ I3 )
% 12.85/7.00 = K ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6110_dbl__dec__def,axiom,
% 12.85/7.00 ( neg_nu6511756317524482435omplex
% 12.85/7.00 = ( ^ [X3: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X3 @ X3 ) @ one_one_complex ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6111_dbl__dec__def,axiom,
% 12.85/7.00 ( neg_nu6075765906172075777c_real
% 12.85/7.00 = ( ^ [X3: real] : ( minus_minus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6112_dbl__dec__def,axiom,
% 12.85/7.00 ( neg_nu3179335615603231917ec_rat
% 12.85/7.00 = ( ^ [X3: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6113_dbl__dec__def,axiom,
% 12.85/7.00 ( neg_nu3811975205180677377ec_int
% 12.85/7.00 = ( ^ [X3: int] : ( minus_minus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6114_incr__lemma,axiom,
% 12.85/7.00 ! [D: int,Z: int,X: int] :
% 12.85/7.00 ( ( ord_less_int @ zero_zero_int @ D )
% 12.85/7.00 => ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6115_decr__lemma,axiom,
% 12.85/7.00 ! [D: int,X: int,Z: int] :
% 12.85/7.00 ( ( ord_less_int @ zero_zero_int @ D )
% 12.85/7.00 => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6116_compl__mono,axiom,
% 12.85/7.00 ! [X: set_int,Y4: set_int] :
% 12.85/7.00 ( ( ord_less_eq_set_int @ X @ Y4 )
% 12.85/7.00 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y4 ) @ ( uminus1532241313380277803et_int @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6117_compl__le__swap1,axiom,
% 12.85/7.00 ! [Y4: set_int,X: set_int] :
% 12.85/7.00 ( ( ord_less_eq_set_int @ Y4 @ ( uminus1532241313380277803et_int @ X ) )
% 12.85/7.00 => ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6118_compl__le__swap2,axiom,
% 12.85/7.00 ! [Y4: set_int,X: set_int] :
% 12.85/7.00 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y4 ) @ X )
% 12.85/7.00 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6119_nat__ivt__aux,axiom,
% 12.85/7.00 ! [N: nat,F: nat > int,K: int] :
% 12.85/7.00 ( ! [I3: nat] :
% 12.85/7.00 ( ( ord_less_nat @ I3 @ N )
% 12.85/7.00 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 12.85/7.00 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 12.85/7.00 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 12.85/7.00 => ? [I3: nat] :
% 12.85/7.00 ( ( ord_less_eq_nat @ I3 @ N )
% 12.85/7.00 & ( ( F @ I3 )
% 12.85/7.00 = K ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6120_complex__mod__minus__le__complex__mod,axiom,
% 12.85/7.00 ! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6121_complex__mod__triangle__ineq2,axiom,
% 12.85/7.00 ! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6122_nat0__intermed__int__val,axiom,
% 12.85/7.00 ! [N: nat,F: nat > int,K: int] :
% 12.85/7.00 ( ! [I3: nat] :
% 12.85/7.00 ( ( ord_less_nat @ I3 @ N )
% 12.85/7.00 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 12.85/7.00 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 12.85/7.00 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 12.85/7.00 => ? [I3: nat] :
% 12.85/7.00 ( ( ord_less_eq_nat @ I3 @ N )
% 12.85/7.00 & ( ( F @ I3 )
% 12.85/7.00 = K ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6123_set__n__deg__not__0,axiom,
% 12.85/7.00 ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
% 12.85/7.00 ( ! [X4: vEBT_VEBT] :
% 12.85/7.00 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.85/7.00 => ( vEBT_invar_vebt @ X4 @ N ) )
% 12.85/7.00 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 12.85/7.00 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.85/7.00 => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6124_arctan__double,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 12.85/7.00 => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
% 12.85/7.00 = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6125_concat__bit__Suc,axiom,
% 12.85/7.00 ! [N: nat,K: int,L: int] :
% 12.85/7.00 ( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
% 12.85/7.00 = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6126_modulo__int__unfold,axiom,
% 12.85/7.00 ! [L: int,K: int,N: nat,M: nat] :
% 12.85/7.00 ( ( ( ( ( sgn_sgn_int @ L )
% 12.85/7.00 = zero_zero_int )
% 12.85/7.00 | ( ( sgn_sgn_int @ K )
% 12.85/7.00 = zero_zero_int )
% 12.85/7.00 | ( N = zero_zero_nat ) )
% 12.85/7.00 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 12.85/7.00 = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 12.85/7.00 & ( ~ ( ( ( sgn_sgn_int @ L )
% 12.85/7.00 = zero_zero_int )
% 12.85/7.00 | ( ( sgn_sgn_int @ K )
% 12.85/7.00 = zero_zero_int )
% 12.85/7.00 | ( N = zero_zero_nat ) )
% 12.85/7.00 => ( ( ( ( sgn_sgn_int @ K )
% 12.85/7.00 = ( sgn_sgn_int @ L ) )
% 12.85/7.00 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 12.85/7.00 = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
% 12.85/7.00 & ( ( ( sgn_sgn_int @ K )
% 12.85/7.00 != ( sgn_sgn_int @ L ) )
% 12.85/7.00 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 12.85/7.00 = ( times_times_int @ ( sgn_sgn_int @ L )
% 12.85/7.00 @ ( minus_minus_int
% 12.85/7.00 @ ( semiri1314217659103216013at_int
% 12.85/7.00 @ ( times_times_nat @ N
% 12.85/7.00 @ ( zero_n2687167440665602831ol_nat
% 12.85/7.00 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
% 12.85/7.00 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6127_mask__numeral,axiom,
% 12.85/7.00 ! [N: num] :
% 12.85/7.00 ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N ) )
% 12.85/7.00 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6128_mask__numeral,axiom,
% 12.85/7.00 ! [N: num] :
% 12.85/7.00 ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N ) )
% 12.85/7.00 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6129_sgn__sgn,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( sgn_sgn_int @ ( sgn_sgn_int @ A ) )
% 12.85/7.00 = ( sgn_sgn_int @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6130_sgn__sgn,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( sgn_sgn_real @ ( sgn_sgn_real @ A ) )
% 12.85/7.00 = ( sgn_sgn_real @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6131_sgn__sgn,axiom,
% 12.85/7.00 ! [A: complex] :
% 12.85/7.00 ( ( sgn_sgn_complex @ ( sgn_sgn_complex @ A ) )
% 12.85/7.00 = ( sgn_sgn_complex @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6132_sgn__sgn,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( sgn_sgn_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 12.85/7.00 = ( sgn_sgn_Code_integer @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6133_sgn__sgn,axiom,
% 12.85/7.00 ! [A: rat] :
% 12.85/7.00 ( ( sgn_sgn_rat @ ( sgn_sgn_rat @ A ) )
% 12.85/7.00 = ( sgn_sgn_rat @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6134_mask__nat__positive__iff,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 12.85/7.00 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6135_sgn__zero,axiom,
% 12.85/7.00 ( ( sgn_sgn_complex @ zero_zero_complex )
% 12.85/7.00 = zero_zero_complex ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6136_sgn__zero,axiom,
% 12.85/7.00 ( ( sgn_sgn_real @ zero_zero_real )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6137_sgn__0,axiom,
% 12.85/7.00 ( ( sgn_sgn_Code_integer @ zero_z3403309356797280102nteger )
% 12.85/7.00 = zero_z3403309356797280102nteger ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6138_sgn__0,axiom,
% 12.85/7.00 ( ( sgn_sgn_complex @ zero_zero_complex )
% 12.85/7.00 = zero_zero_complex ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6139_sgn__0,axiom,
% 12.85/7.00 ( ( sgn_sgn_real @ zero_zero_real )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6140_sgn__0,axiom,
% 12.85/7.00 ( ( sgn_sgn_rat @ zero_zero_rat )
% 12.85/7.00 = zero_zero_rat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6141_sgn__0,axiom,
% 12.85/7.00 ( ( sgn_sgn_int @ zero_zero_int )
% 12.85/7.00 = zero_zero_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6142_sgn__1,axiom,
% 12.85/7.00 ( ( sgn_sgn_int @ one_one_int )
% 12.85/7.00 = one_one_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6143_sgn__1,axiom,
% 12.85/7.00 ( ( sgn_sgn_real @ one_one_real )
% 12.85/7.00 = one_one_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6144_sgn__1,axiom,
% 12.85/7.00 ( ( sgn_sgn_complex @ one_one_complex )
% 12.85/7.00 = one_one_complex ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6145_sgn__1,axiom,
% 12.85/7.00 ( ( sgn_sgn_Code_integer @ one_one_Code_integer )
% 12.85/7.00 = one_one_Code_integer ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6146_sgn__1,axiom,
% 12.85/7.00 ( ( sgn_sgn_rat @ one_one_rat )
% 12.85/7.00 = one_one_rat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6147_sgn__divide,axiom,
% 12.85/7.00 ! [A: complex,B: complex] :
% 12.85/7.00 ( ( sgn_sgn_complex @ ( divide1717551699836669952omplex @ A @ B ) )
% 12.85/7.00 = ( divide1717551699836669952omplex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6148_sgn__divide,axiom,
% 12.85/7.00 ! [A: real,B: real] :
% 12.85/7.00 ( ( sgn_sgn_real @ ( divide_divide_real @ A @ B ) )
% 12.85/7.00 = ( divide_divide_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6149_sgn__divide,axiom,
% 12.85/7.00 ! [A: rat,B: rat] :
% 12.85/7.00 ( ( sgn_sgn_rat @ ( divide_divide_rat @ A @ B ) )
% 12.85/7.00 = ( divide_divide_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6150_idom__abs__sgn__class_Osgn__minus,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( sgn_sgn_real @ ( uminus_uminus_real @ A ) )
% 12.85/7.00 = ( uminus_uminus_real @ ( sgn_sgn_real @ A ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6151_idom__abs__sgn__class_Osgn__minus,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( sgn_sgn_int @ ( uminus_uminus_int @ A ) )
% 12.85/7.00 = ( uminus_uminus_int @ ( sgn_sgn_int @ A ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6152_idom__abs__sgn__class_Osgn__minus,axiom,
% 12.85/7.00 ! [A: complex] :
% 12.85/7.00 ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ A ) )
% 12.85/7.00 = ( uminus1482373934393186551omplex @ ( sgn_sgn_complex @ A ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6153_idom__abs__sgn__class_Osgn__minus,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 12.85/7.00 = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ A ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6154_idom__abs__sgn__class_Osgn__minus,axiom,
% 12.85/7.00 ! [A: rat] :
% 12.85/7.00 ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ A ) )
% 12.85/7.00 = ( uminus_uminus_rat @ ( sgn_sgn_rat @ A ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6155_arctan__eq__zero__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ( arctan @ X )
% 12.85/7.00 = zero_zero_real )
% 12.85/7.00 = ( X = zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6156_arctan__zero__zero,axiom,
% 12.85/7.00 ( ( arctan @ zero_zero_real )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6157_concat__bit__0,axiom,
% 12.85/7.00 ! [K: int,L: int] :
% 12.85/7.00 ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
% 12.85/7.00 = L ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6158_sgn__less,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 12.85/7.00 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6159_sgn__less,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( ord_less_real @ ( sgn_sgn_real @ A ) @ zero_zero_real )
% 12.85/7.00 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6160_sgn__less,axiom,
% 12.85/7.00 ! [A: rat] :
% 12.85/7.00 ( ( ord_less_rat @ ( sgn_sgn_rat @ A ) @ zero_zero_rat )
% 12.85/7.00 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6161_sgn__less,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( ord_less_int @ ( sgn_sgn_int @ A ) @ zero_zero_int )
% 12.85/7.00 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6162_sgn__greater,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A ) )
% 12.85/7.00 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6163_sgn__greater,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A ) )
% 12.85/7.00 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6164_sgn__greater,axiom,
% 12.85/7.00 ! [A: rat] :
% 12.85/7.00 ( ( ord_less_rat @ zero_zero_rat @ ( sgn_sgn_rat @ A ) )
% 12.85/7.00 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6165_sgn__greater,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A ) )
% 12.85/7.00 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6166_divide__sgn,axiom,
% 12.85/7.00 ! [A: real,B: real] :
% 12.85/7.00 ( ( divide_divide_real @ A @ ( sgn_sgn_real @ B ) )
% 12.85/7.00 = ( times_times_real @ A @ ( sgn_sgn_real @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6167_divide__sgn,axiom,
% 12.85/7.00 ! [A: rat,B: rat] :
% 12.85/7.00 ( ( divide_divide_rat @ A @ ( sgn_sgn_rat @ B ) )
% 12.85/7.00 = ( times_times_rat @ A @ ( sgn_sgn_rat @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6168_mask__0,axiom,
% 12.85/7.00 ( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
% 12.85/7.00 = zero_zero_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6169_mask__0,axiom,
% 12.85/7.00 ( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
% 12.85/7.00 = zero_zero_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6170_mask__eq__0__iff,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( ( bit_se2002935070580805687sk_nat @ N )
% 12.85/7.00 = zero_zero_nat )
% 12.85/7.00 = ( N = zero_zero_nat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6171_mask__eq__0__iff,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( ( bit_se2000444600071755411sk_int @ N )
% 12.85/7.00 = zero_zero_int )
% 12.85/7.00 = ( N = zero_zero_nat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6172_arctan__less__zero__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ ( arctan @ X ) @ zero_zero_real )
% 12.85/7.00 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6173_zero__less__arctan__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ ( arctan @ X ) )
% 12.85/7.00 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6174_zero__le__arctan__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X ) )
% 12.85/7.00 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6175_arctan__le__zero__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( arctan @ X ) @ zero_zero_real )
% 12.85/7.00 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6176_concat__bit__nonnegative__iff,axiom,
% 12.85/7.00 ! [N: nat,K: int,L: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
% 12.85/7.00 = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6177_concat__bit__negative__iff,axiom,
% 12.85/7.00 ! [N: nat,K: int,L: int] :
% 12.85/7.00 ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
% 12.85/7.00 = ( ord_less_int @ L @ zero_zero_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6178_sgn__pos,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 12.85/7.00 => ( ( sgn_sgn_Code_integer @ A )
% 12.85/7.00 = one_one_Code_integer ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6179_sgn__pos,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.00 => ( ( sgn_sgn_real @ A )
% 12.85/7.00 = one_one_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6180_sgn__pos,axiom,
% 12.85/7.00 ! [A: rat] :
% 12.85/7.00 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/7.00 => ( ( sgn_sgn_rat @ A )
% 12.85/7.00 = one_one_rat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6181_sgn__pos,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( ord_less_int @ zero_zero_int @ A )
% 12.85/7.00 => ( ( sgn_sgn_int @ A )
% 12.85/7.00 = one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6182_abs__sgn__eq__1,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( A != zero_z3403309356797280102nteger )
% 12.85/7.00 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 12.85/7.00 = one_one_Code_integer ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6183_abs__sgn__eq__1,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( A != zero_zero_real )
% 12.85/7.00 => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 12.85/7.00 = one_one_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6184_abs__sgn__eq__1,axiom,
% 12.85/7.00 ! [A: rat] :
% 12.85/7.00 ( ( A != zero_zero_rat )
% 12.85/7.00 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 12.85/7.00 = one_one_rat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6185_abs__sgn__eq__1,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( A != zero_zero_int )
% 12.85/7.00 => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 12.85/7.00 = one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6186_sgn__mult__self__eq,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ A ) )
% 12.85/7.00 = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6187_sgn__mult__self__eq,axiom,
% 12.85/7.00 ! [A: rat] :
% 12.85/7.00 ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ A ) )
% 12.85/7.00 = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6188_sgn__mult__self__eq,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ A ) )
% 12.85/7.00 = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6189_sgn__mult__self__eq,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
% 12.85/7.00 = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6190_mask__Suc__0,axiom,
% 12.85/7.00 ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
% 12.85/7.00 = one_one_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6191_mask__Suc__0,axiom,
% 12.85/7.00 ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
% 12.85/7.00 = one_one_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6192_sgn__abs,axiom,
% 12.85/7.00 ! [A: complex] :
% 12.85/7.00 ( ( abs_abs_complex @ ( sgn_sgn_complex @ A ) )
% 12.85/7.00 = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6193_sgn__abs,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 12.85/7.00 = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6194_sgn__abs,axiom,
% 12.85/7.00 ! [A: rat] :
% 12.85/7.00 ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 12.85/7.00 = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6195_sgn__abs,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 12.85/7.00 = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6196_sgn__abs,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 12.85/7.00 = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6197_idom__abs__sgn__class_Oabs__sgn,axiom,
% 12.85/7.00 ! [A: complex] :
% 12.85/7.00 ( ( sgn_sgn_complex @ ( abs_abs_complex @ A ) )
% 12.85/7.00 = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6198_idom__abs__sgn__class_Oabs__sgn,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( sgn_sgn_real @ ( abs_abs_real @ A ) )
% 12.85/7.00 = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6199_idom__abs__sgn__class_Oabs__sgn,axiom,
% 12.85/7.00 ! [A: rat] :
% 12.85/7.00 ( ( sgn_sgn_rat @ ( abs_abs_rat @ A ) )
% 12.85/7.00 = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6200_idom__abs__sgn__class_Oabs__sgn,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( sgn_sgn_int @ ( abs_abs_int @ A ) )
% 12.85/7.00 = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6201_idom__abs__sgn__class_Oabs__sgn,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( sgn_sgn_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 12.85/7.00 = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6202_sgn__mult__dvd__iff,axiom,
% 12.85/7.00 ! [R2: int,L: int,K: int] :
% 12.85/7.00 ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K )
% 12.85/7.00 = ( ( dvd_dvd_int @ L @ K )
% 12.85/7.00 & ( ( R2 = zero_zero_int )
% 12.85/7.00 => ( K = zero_zero_int ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6203_mult__sgn__dvd__iff,axiom,
% 12.85/7.00 ! [L: int,R2: int,K: int] :
% 12.85/7.00 ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K )
% 12.85/7.00 = ( ( dvd_dvd_int @ L @ K )
% 12.85/7.00 & ( ( R2 = zero_zero_int )
% 12.85/7.00 => ( K = zero_zero_int ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6204_dvd__sgn__mult__iff,axiom,
% 12.85/7.00 ! [L: int,R2: int,K: int] :
% 12.85/7.00 ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
% 12.85/7.00 = ( ( dvd_dvd_int @ L @ K )
% 12.85/7.00 | ( R2 = zero_zero_int ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6205_dvd__mult__sgn__iff,axiom,
% 12.85/7.00 ! [L: int,K: int,R2: int] :
% 12.85/7.00 ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
% 12.85/7.00 = ( ( dvd_dvd_int @ L @ K )
% 12.85/7.00 | ( R2 = zero_zero_int ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6206_sgn__neg,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/7.00 => ( ( sgn_sgn_real @ A )
% 12.85/7.00 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6207_sgn__neg,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( ord_less_int @ A @ zero_zero_int )
% 12.85/7.00 => ( ( sgn_sgn_int @ A )
% 12.85/7.00 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6208_sgn__neg,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 12.85/7.00 => ( ( sgn_sgn_Code_integer @ A )
% 12.85/7.00 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6209_sgn__neg,axiom,
% 12.85/7.00 ! [A: rat] :
% 12.85/7.00 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/7.00 => ( ( sgn_sgn_rat @ A )
% 12.85/7.00 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6210_sgn__of__nat,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( sgn_sgn_real @ ( semiri5074537144036343181t_real @ N ) )
% 12.85/7.00 = ( zero_n3304061248610475627l_real @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6211_sgn__of__nat,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( sgn_sgn_rat @ ( semiri681578069525770553at_rat @ N ) )
% 12.85/7.00 = ( zero_n2052037380579107095ol_rat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6212_sgn__of__nat,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( sgn_sgn_int @ ( semiri1314217659103216013at_int @ N ) )
% 12.85/7.00 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6213_sgn__of__nat,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( sgn_sgn_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 12.85/7.00 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6214_of__nat__mask__eq,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 12.85/7.00 = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6215_of__nat__mask__eq,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N ) )
% 12.85/7.00 = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6216_size__neq__size__imp__neq,axiom,
% 12.85/7.00 ! [X: list_real,Y4: list_real] :
% 12.85/7.00 ( ( ( size_size_list_real @ X )
% 12.85/7.00 != ( size_size_list_real @ Y4 ) )
% 12.85/7.00 => ( X != Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6217_size__neq__size__imp__neq,axiom,
% 12.85/7.00 ! [X: list_o,Y4: list_o] :
% 12.85/7.00 ( ( ( size_size_list_o @ X )
% 12.85/7.00 != ( size_size_list_o @ Y4 ) )
% 12.85/7.00 => ( X != Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6218_size__neq__size__imp__neq,axiom,
% 12.85/7.00 ! [X: list_nat,Y4: list_nat] :
% 12.85/7.00 ( ( ( size_size_list_nat @ X )
% 12.85/7.00 != ( size_size_list_nat @ Y4 ) )
% 12.85/7.00 => ( X != Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6219_size__neq__size__imp__neq,axiom,
% 12.85/7.00 ! [X: list_int,Y4: list_int] :
% 12.85/7.00 ( ( ( size_size_list_int @ X )
% 12.85/7.00 != ( size_size_list_int @ Y4 ) )
% 12.85/7.00 => ( X != Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6220_size__neq__size__imp__neq,axiom,
% 12.85/7.00 ! [X: num,Y4: num] :
% 12.85/7.00 ( ( ( size_size_num @ X )
% 12.85/7.00 != ( size_size_num @ Y4 ) )
% 12.85/7.00 => ( X != Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6221_sgn__zero__iff,axiom,
% 12.85/7.00 ! [X: complex] :
% 12.85/7.00 ( ( ( sgn_sgn_complex @ X )
% 12.85/7.00 = zero_zero_complex )
% 12.85/7.00 = ( X = zero_zero_complex ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6222_sgn__zero__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ( sgn_sgn_real @ X )
% 12.85/7.00 = zero_zero_real )
% 12.85/7.00 = ( X = zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6223_sgn__0__0,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( ( sgn_sgn_Code_integer @ A )
% 12.85/7.00 = zero_z3403309356797280102nteger )
% 12.85/7.00 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6224_sgn__0__0,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( ( sgn_sgn_real @ A )
% 12.85/7.00 = zero_zero_real )
% 12.85/7.00 = ( A = zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6225_sgn__0__0,axiom,
% 12.85/7.00 ! [A: rat] :
% 12.85/7.00 ( ( ( sgn_sgn_rat @ A )
% 12.85/7.00 = zero_zero_rat )
% 12.85/7.00 = ( A = zero_zero_rat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6226_sgn__0__0,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( ( sgn_sgn_int @ A )
% 12.85/7.00 = zero_zero_int )
% 12.85/7.00 = ( A = zero_zero_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6227_sgn__eq__0__iff,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( ( sgn_sgn_Code_integer @ A )
% 12.85/7.00 = zero_z3403309356797280102nteger )
% 12.85/7.00 = ( A = zero_z3403309356797280102nteger ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6228_sgn__eq__0__iff,axiom,
% 12.85/7.00 ! [A: complex] :
% 12.85/7.00 ( ( ( sgn_sgn_complex @ A )
% 12.85/7.00 = zero_zero_complex )
% 12.85/7.00 = ( A = zero_zero_complex ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6229_sgn__eq__0__iff,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( ( sgn_sgn_real @ A )
% 12.85/7.00 = zero_zero_real )
% 12.85/7.00 = ( A = zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6230_sgn__eq__0__iff,axiom,
% 12.85/7.00 ! [A: rat] :
% 12.85/7.00 ( ( ( sgn_sgn_rat @ A )
% 12.85/7.00 = zero_zero_rat )
% 12.85/7.00 = ( A = zero_zero_rat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6231_sgn__eq__0__iff,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( ( sgn_sgn_int @ A )
% 12.85/7.00 = zero_zero_int )
% 12.85/7.00 = ( A = zero_zero_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6232_Real__Vector__Spaces_Osgn__mult,axiom,
% 12.85/7.00 ! [X: complex,Y4: complex] :
% 12.85/7.00 ( ( sgn_sgn_complex @ ( times_times_complex @ X @ Y4 ) )
% 12.85/7.00 = ( times_times_complex @ ( sgn_sgn_complex @ X ) @ ( sgn_sgn_complex @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6233_Real__Vector__Spaces_Osgn__mult,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( sgn_sgn_real @ ( times_times_real @ X @ Y4 ) )
% 12.85/7.00 = ( times_times_real @ ( sgn_sgn_real @ X ) @ ( sgn_sgn_real @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6234_sgn__mult,axiom,
% 12.85/7.00 ! [A: complex,B: complex] :
% 12.85/7.00 ( ( sgn_sgn_complex @ ( times_times_complex @ A @ B ) )
% 12.85/7.00 = ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6235_sgn__mult,axiom,
% 12.85/7.00 ! [A: code_integer,B: code_integer] :
% 12.85/7.00 ( ( sgn_sgn_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 12.85/7.00 = ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6236_sgn__mult,axiom,
% 12.85/7.00 ! [A: real,B: real] :
% 12.85/7.00 ( ( sgn_sgn_real @ ( times_times_real @ A @ B ) )
% 12.85/7.00 = ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6237_sgn__mult,axiom,
% 12.85/7.00 ! [A: rat,B: rat] :
% 12.85/7.00 ( ( sgn_sgn_rat @ ( times_times_rat @ A @ B ) )
% 12.85/7.00 = ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6238_sgn__mult,axiom,
% 12.85/7.00 ! [A: int,B: int] :
% 12.85/7.00 ( ( sgn_sgn_int @ ( times_times_int @ A @ B ) )
% 12.85/7.00 = ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6239_same__sgn__sgn__add,axiom,
% 12.85/7.00 ! [B: code_integer,A: code_integer] :
% 12.85/7.00 ( ( ( sgn_sgn_Code_integer @ B )
% 12.85/7.00 = ( sgn_sgn_Code_integer @ A ) )
% 12.85/7.00 => ( ( sgn_sgn_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 12.85/7.00 = ( sgn_sgn_Code_integer @ A ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6240_same__sgn__sgn__add,axiom,
% 12.85/7.00 ! [B: real,A: real] :
% 12.85/7.00 ( ( ( sgn_sgn_real @ B )
% 12.85/7.00 = ( sgn_sgn_real @ A ) )
% 12.85/7.00 => ( ( sgn_sgn_real @ ( plus_plus_real @ A @ B ) )
% 12.85/7.00 = ( sgn_sgn_real @ A ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6241_same__sgn__sgn__add,axiom,
% 12.85/7.00 ! [B: rat,A: rat] :
% 12.85/7.00 ( ( ( sgn_sgn_rat @ B )
% 12.85/7.00 = ( sgn_sgn_rat @ A ) )
% 12.85/7.00 => ( ( sgn_sgn_rat @ ( plus_plus_rat @ A @ B ) )
% 12.85/7.00 = ( sgn_sgn_rat @ A ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6242_same__sgn__sgn__add,axiom,
% 12.85/7.00 ! [B: int,A: int] :
% 12.85/7.00 ( ( ( sgn_sgn_int @ B )
% 12.85/7.00 = ( sgn_sgn_int @ A ) )
% 12.85/7.00 => ( ( sgn_sgn_int @ ( plus_plus_int @ A @ B ) )
% 12.85/7.00 = ( sgn_sgn_int @ A ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6243_div__eq__sgn__abs,axiom,
% 12.85/7.00 ! [K: int,L: int] :
% 12.85/7.00 ( ( ( sgn_sgn_int @ K )
% 12.85/7.00 = ( sgn_sgn_int @ L ) )
% 12.85/7.00 => ( ( divide_divide_int @ K @ L )
% 12.85/7.00 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6244_arctan__monotone_H,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/7.00 => ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6245_arctan__le__iff,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y4 ) )
% 12.85/7.00 = ( ord_less_eq_real @ X @ Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6246_less__eq__mask,axiom,
% 12.85/7.00 ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6247_concat__bit__assoc,axiom,
% 12.85/7.00 ! [N: nat,K: int,M: nat,L: int,R2: int] :
% 12.85/7.00 ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L @ R2 ) )
% 12.85/7.00 = ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R2 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6248_sgn__not__eq__imp,axiom,
% 12.85/7.00 ! [B: real,A: real] :
% 12.85/7.00 ( ( ( sgn_sgn_real @ B )
% 12.85/7.00 != ( sgn_sgn_real @ A ) )
% 12.85/7.00 => ( ( ( sgn_sgn_real @ A )
% 12.85/7.00 != zero_zero_real )
% 12.85/7.00 => ( ( ( sgn_sgn_real @ B )
% 12.85/7.00 != zero_zero_real )
% 12.85/7.00 => ( ( sgn_sgn_real @ A )
% 12.85/7.00 = ( uminus_uminus_real @ ( sgn_sgn_real @ B ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6249_sgn__not__eq__imp,axiom,
% 12.85/7.00 ! [B: int,A: int] :
% 12.85/7.00 ( ( ( sgn_sgn_int @ B )
% 12.85/7.00 != ( sgn_sgn_int @ A ) )
% 12.85/7.00 => ( ( ( sgn_sgn_int @ A )
% 12.85/7.00 != zero_zero_int )
% 12.85/7.00 => ( ( ( sgn_sgn_int @ B )
% 12.85/7.00 != zero_zero_int )
% 12.85/7.00 => ( ( sgn_sgn_int @ A )
% 12.85/7.00 = ( uminus_uminus_int @ ( sgn_sgn_int @ B ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6250_sgn__not__eq__imp,axiom,
% 12.85/7.00 ! [B: code_integer,A: code_integer] :
% 12.85/7.00 ( ( ( sgn_sgn_Code_integer @ B )
% 12.85/7.00 != ( sgn_sgn_Code_integer @ A ) )
% 12.85/7.00 => ( ( ( sgn_sgn_Code_integer @ A )
% 12.85/7.00 != zero_z3403309356797280102nteger )
% 12.85/7.00 => ( ( ( sgn_sgn_Code_integer @ B )
% 12.85/7.00 != zero_z3403309356797280102nteger )
% 12.85/7.00 => ( ( sgn_sgn_Code_integer @ A )
% 12.85/7.00 = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ B ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6251_sgn__not__eq__imp,axiom,
% 12.85/7.00 ! [B: rat,A: rat] :
% 12.85/7.00 ( ( ( sgn_sgn_rat @ B )
% 12.85/7.00 != ( sgn_sgn_rat @ A ) )
% 12.85/7.00 => ( ( ( sgn_sgn_rat @ A )
% 12.85/7.00 != zero_zero_rat )
% 12.85/7.00 => ( ( ( sgn_sgn_rat @ B )
% 12.85/7.00 != zero_zero_rat )
% 12.85/7.00 => ( ( sgn_sgn_rat @ A )
% 12.85/7.00 = ( uminus_uminus_rat @ ( sgn_sgn_rat @ B ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6252_sgn__minus__1,axiom,
% 12.85/7.00 ( ( sgn_sgn_real @ ( uminus_uminus_real @ one_one_real ) )
% 12.85/7.00 = ( uminus_uminus_real @ one_one_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6253_sgn__minus__1,axiom,
% 12.85/7.00 ( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.00 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6254_sgn__minus__1,axiom,
% 12.85/7.00 ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.85/7.00 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6255_sgn__minus__1,axiom,
% 12.85/7.00 ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.00 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6256_sgn__minus__1,axiom,
% 12.85/7.00 ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.85/7.00 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6257_linordered__idom__class_Oabs__sgn,axiom,
% 12.85/7.00 ( abs_abs_Code_integer
% 12.85/7.00 = ( ^ [K3: code_integer] : ( times_3573771949741848930nteger @ K3 @ ( sgn_sgn_Code_integer @ K3 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6258_linordered__idom__class_Oabs__sgn,axiom,
% 12.85/7.00 ( abs_abs_real
% 12.85/7.00 = ( ^ [K3: real] : ( times_times_real @ K3 @ ( sgn_sgn_real @ K3 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6259_linordered__idom__class_Oabs__sgn,axiom,
% 12.85/7.00 ( abs_abs_rat
% 12.85/7.00 = ( ^ [K3: rat] : ( times_times_rat @ K3 @ ( sgn_sgn_rat @ K3 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6260_linordered__idom__class_Oabs__sgn,axiom,
% 12.85/7.00 ( abs_abs_int
% 12.85/7.00 = ( ^ [K3: int] : ( times_times_int @ K3 @ ( sgn_sgn_int @ K3 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6261_abs__mult__sgn,axiom,
% 12.85/7.00 ! [A: complex] :
% 12.85/7.00 ( ( times_times_complex @ ( abs_abs_complex @ A ) @ ( sgn_sgn_complex @ A ) )
% 12.85/7.00 = A ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6262_abs__mult__sgn,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
% 12.85/7.00 = A ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6263_abs__mult__sgn,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( sgn_sgn_real @ A ) )
% 12.85/7.00 = A ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6264_abs__mult__sgn,axiom,
% 12.85/7.00 ! [A: rat] :
% 12.85/7.00 ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( sgn_sgn_rat @ A ) )
% 12.85/7.00 = A ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6265_abs__mult__sgn,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( times_times_int @ ( abs_abs_int @ A ) @ ( sgn_sgn_int @ A ) )
% 12.85/7.00 = A ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6266_sgn__mult__abs,axiom,
% 12.85/7.00 ! [A: complex] :
% 12.85/7.00 ( ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( abs_abs_complex @ A ) )
% 12.85/7.00 = A ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6267_sgn__mult__abs,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 12.85/7.00 = A ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6268_sgn__mult__abs,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( abs_abs_real @ A ) )
% 12.85/7.00 = A ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6269_sgn__mult__abs,axiom,
% 12.85/7.00 ! [A: rat] :
% 12.85/7.00 ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( abs_abs_rat @ A ) )
% 12.85/7.00 = A ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6270_sgn__mult__abs,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( abs_abs_int @ A ) )
% 12.85/7.00 = A ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6271_mult__sgn__abs,axiom,
% 12.85/7.00 ! [X: code_integer] :
% 12.85/7.00 ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ X ) @ ( abs_abs_Code_integer @ X ) )
% 12.85/7.00 = X ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6272_mult__sgn__abs,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( times_times_real @ ( sgn_sgn_real @ X ) @ ( abs_abs_real @ X ) )
% 12.85/7.00 = X ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6273_mult__sgn__abs,axiom,
% 12.85/7.00 ! [X: rat] :
% 12.85/7.00 ( ( times_times_rat @ ( sgn_sgn_rat @ X ) @ ( abs_abs_rat @ X ) )
% 12.85/7.00 = X ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6274_mult__sgn__abs,axiom,
% 12.85/7.00 ! [X: int] :
% 12.85/7.00 ( ( times_times_int @ ( sgn_sgn_int @ X ) @ ( abs_abs_int @ X ) )
% 12.85/7.00 = X ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6275_int__sgnE,axiom,
% 12.85/7.00 ! [K: int] :
% 12.85/7.00 ~ ! [N2: nat,L2: int] :
% 12.85/7.00 ( K
% 12.85/7.00 != ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6276_same__sgn__abs__add,axiom,
% 12.85/7.00 ! [B: code_integer,A: code_integer] :
% 12.85/7.00 ( ( ( sgn_sgn_Code_integer @ B )
% 12.85/7.00 = ( sgn_sgn_Code_integer @ A ) )
% 12.85/7.00 => ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 12.85/7.00 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6277_same__sgn__abs__add,axiom,
% 12.85/7.00 ! [B: real,A: real] :
% 12.85/7.00 ( ( ( sgn_sgn_real @ B )
% 12.85/7.00 = ( sgn_sgn_real @ A ) )
% 12.85/7.00 => ( ( abs_abs_real @ ( plus_plus_real @ A @ B ) )
% 12.85/7.00 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6278_same__sgn__abs__add,axiom,
% 12.85/7.00 ! [B: rat,A: rat] :
% 12.85/7.00 ( ( ( sgn_sgn_rat @ B )
% 12.85/7.00 = ( sgn_sgn_rat @ A ) )
% 12.85/7.00 => ( ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) )
% 12.85/7.00 = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6279_same__sgn__abs__add,axiom,
% 12.85/7.00 ! [B: int,A: int] :
% 12.85/7.00 ( ( ( sgn_sgn_int @ B )
% 12.85/7.00 = ( sgn_sgn_int @ A ) )
% 12.85/7.00 => ( ( abs_abs_int @ ( plus_plus_int @ A @ B ) )
% 12.85/7.00 = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6280_mask__nonnegative__int,axiom,
% 12.85/7.00 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6281_not__mask__negative__int,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6282_sgn__1__pos,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( ( sgn_sgn_Code_integer @ A )
% 12.85/7.00 = one_one_Code_integer )
% 12.85/7.00 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6283_sgn__1__pos,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( ( sgn_sgn_real @ A )
% 12.85/7.00 = one_one_real )
% 12.85/7.00 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6284_sgn__1__pos,axiom,
% 12.85/7.00 ! [A: rat] :
% 12.85/7.00 ( ( ( sgn_sgn_rat @ A )
% 12.85/7.00 = one_one_rat )
% 12.85/7.00 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6285_sgn__1__pos,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( ( sgn_sgn_int @ A )
% 12.85/7.00 = one_one_int )
% 12.85/7.00 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6286_abs__sgn__eq,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( ( A = zero_z3403309356797280102nteger )
% 12.85/7.00 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 12.85/7.00 = zero_z3403309356797280102nteger ) )
% 12.85/7.00 & ( ( A != zero_z3403309356797280102nteger )
% 12.85/7.00 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 12.85/7.00 = one_one_Code_integer ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6287_abs__sgn__eq,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( ( A = zero_zero_real )
% 12.85/7.00 => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 12.85/7.00 = zero_zero_real ) )
% 12.85/7.00 & ( ( A != zero_zero_real )
% 12.85/7.00 => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 12.85/7.00 = one_one_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6288_abs__sgn__eq,axiom,
% 12.85/7.00 ! [A: rat] :
% 12.85/7.00 ( ( ( A = zero_zero_rat )
% 12.85/7.00 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 12.85/7.00 = zero_zero_rat ) )
% 12.85/7.00 & ( ( A != zero_zero_rat )
% 12.85/7.00 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 12.85/7.00 = one_one_rat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6289_abs__sgn__eq,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( ( A = zero_zero_int )
% 12.85/7.00 => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 12.85/7.00 = zero_zero_int ) )
% 12.85/7.00 & ( ( A != zero_zero_int )
% 12.85/7.00 => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 12.85/7.00 = one_one_int ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6290_div__sgn__abs__cancel,axiom,
% 12.85/7.00 ! [V: int,K: int,L: int] :
% 12.85/7.00 ( ( V != zero_zero_int )
% 12.85/7.00 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
% 12.85/7.00 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6291_div__dvd__sgn__abs,axiom,
% 12.85/7.00 ! [L: int,K: int] :
% 12.85/7.00 ( ( dvd_dvd_int @ L @ K )
% 12.85/7.00 => ( ( divide_divide_int @ K @ L )
% 12.85/7.00 = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6292_sgn__mod,axiom,
% 12.85/7.00 ! [L: int,K: int] :
% 12.85/7.00 ( ( L != zero_zero_int )
% 12.85/7.00 => ( ~ ( dvd_dvd_int @ L @ K )
% 12.85/7.00 => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
% 12.85/7.00 = ( sgn_sgn_int @ L ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6293_length__pos__if__in__set,axiom,
% 12.85/7.00 ! [X: complex,Xs: list_complex] :
% 12.85/7.00 ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
% 12.85/7.00 => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6294_length__pos__if__in__set,axiom,
% 12.85/7.00 ! [X: set_nat,Xs: list_set_nat] :
% 12.85/7.00 ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 12.85/7.00 => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6295_length__pos__if__in__set,axiom,
% 12.85/7.00 ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 12.85/7.00 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 12.85/7.00 => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6296_length__pos__if__in__set,axiom,
% 12.85/7.00 ! [X: real,Xs: list_real] :
% 12.85/7.00 ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 12.85/7.00 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6297_length__pos__if__in__set,axiom,
% 12.85/7.00 ! [X: $o,Xs: list_o] :
% 12.85/7.00 ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 12.85/7.00 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6298_length__pos__if__in__set,axiom,
% 12.85/7.00 ! [X: nat,Xs: list_nat] :
% 12.85/7.00 ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 12.85/7.00 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6299_length__pos__if__in__set,axiom,
% 12.85/7.00 ! [X: int,Xs: list_int] :
% 12.85/7.00 ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 12.85/7.00 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6300_less__mask,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.85/7.00 => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6301_sgn__if,axiom,
% 12.85/7.00 ( sgn_sgn_real
% 12.85/7.00 = ( ^ [X3: real] : ( if_real @ ( X3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6302_sgn__if,axiom,
% 12.85/7.00 ( sgn_sgn_int
% 12.85/7.00 = ( ^ [X3: int] : ( if_int @ ( X3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6303_sgn__if,axiom,
% 12.85/7.00 ( sgn_sgn_Code_integer
% 12.85/7.00 = ( ^ [X3: code_integer] : ( if_Code_integer @ ( X3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X3 ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6304_sgn__if,axiom,
% 12.85/7.00 ( sgn_sgn_rat
% 12.85/7.00 = ( ^ [X3: rat] : ( if_rat @ ( X3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6305_sgn__1__neg,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( ( sgn_sgn_real @ A )
% 12.85/7.00 = ( uminus_uminus_real @ one_one_real ) )
% 12.85/7.00 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6306_sgn__1__neg,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( ( sgn_sgn_int @ A )
% 12.85/7.00 = ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.00 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6307_sgn__1__neg,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( ( sgn_sgn_Code_integer @ A )
% 12.85/7.00 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.00 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6308_sgn__1__neg,axiom,
% 12.85/7.00 ! [A: rat] :
% 12.85/7.00 ( ( ( sgn_sgn_rat @ A )
% 12.85/7.00 = ( uminus_uminus_rat @ one_one_rat ) )
% 12.85/7.00 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6309_zsgn__def,axiom,
% 12.85/7.00 ( sgn_sgn_int
% 12.85/7.00 = ( ^ [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 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6310_norm__sgn,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ( X = zero_zero_real )
% 12.85/7.00 => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
% 12.85/7.00 = zero_zero_real ) )
% 12.85/7.00 & ( ( X != zero_zero_real )
% 12.85/7.00 => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
% 12.85/7.00 = one_one_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6311_norm__sgn,axiom,
% 12.85/7.00 ! [X: complex] :
% 12.85/7.00 ( ( ( X = zero_zero_complex )
% 12.85/7.00 => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
% 12.85/7.00 = zero_zero_real ) )
% 12.85/7.00 & ( ( X != zero_zero_complex )
% 12.85/7.00 => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
% 12.85/7.00 = one_one_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6312_div__noneq__sgn__abs,axiom,
% 12.85/7.00 ! [L: int,K: int] :
% 12.85/7.00 ( ( L != zero_zero_int )
% 12.85/7.00 => ( ( ( sgn_sgn_int @ K )
% 12.85/7.00 != ( sgn_sgn_int @ L ) )
% 12.85/7.00 => ( ( divide_divide_int @ K @ L )
% 12.85/7.00 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) )
% 12.85/7.00 @ ( zero_n2684676970156552555ol_int
% 12.85/7.00 @ ~ ( dvd_dvd_int @ L @ K ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6313_Suc__mask__eq__exp,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
% 12.85/7.00 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6314_mask__nat__less__exp,axiom,
% 12.85/7.00 ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6315_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N ) )
% 12.85/7.00 = ( N = zero_zero_nat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6316_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) )
% 12.85/7.00 = ( N = zero_zero_nat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6317_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) )
% 12.85/7.00 = ( N = zero_zero_nat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6318_mask__nat__def,axiom,
% 12.85/7.00 ( bit_se2002935070580805687sk_nat
% 12.85/7.00 = ( ^ [N4: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6319_mask__half__int,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/7.00 = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6320_mask__int__def,axiom,
% 12.85/7.00 ( bit_se2000444600071755411sk_int
% 12.85/7.00 = ( ^ [N4: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) @ one_one_int ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6321_mask__eq__exp__minus__1,axiom,
% 12.85/7.00 ( bit_se2002935070580805687sk_nat
% 12.85/7.00 = ( ^ [N4: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6322_mask__eq__exp__minus__1,axiom,
% 12.85/7.00 ( bit_se2000444600071755411sk_int
% 12.85/7.00 = ( ^ [N4: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) @ one_one_int ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6323_arctan__add,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 12.85/7.00 => ( ( ord_less_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
% 12.85/7.00 => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y4 ) )
% 12.85/7.00 = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y4 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y4 ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6324_divide__int__unfold,axiom,
% 12.85/7.00 ! [L: int,K: int,N: nat,M: nat] :
% 12.85/7.00 ( ( ( ( ( sgn_sgn_int @ L )
% 12.85/7.00 = zero_zero_int )
% 12.85/7.00 | ( ( sgn_sgn_int @ K )
% 12.85/7.00 = zero_zero_int )
% 12.85/7.00 | ( N = zero_zero_nat ) )
% 12.85/7.00 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 12.85/7.00 = zero_zero_int ) )
% 12.85/7.00 & ( ~ ( ( ( sgn_sgn_int @ L )
% 12.85/7.00 = zero_zero_int )
% 12.85/7.00 | ( ( sgn_sgn_int @ K )
% 12.85/7.00 = zero_zero_int )
% 12.85/7.00 | ( N = zero_zero_nat ) )
% 12.85/7.00 => ( ( ( ( sgn_sgn_int @ K )
% 12.85/7.00 = ( sgn_sgn_int @ L ) )
% 12.85/7.00 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 12.85/7.00 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
% 12.85/7.00 & ( ( ( sgn_sgn_int @ K )
% 12.85/7.00 != ( sgn_sgn_int @ L ) )
% 12.85/7.00 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 12.85/7.00 = ( uminus_uminus_int
% 12.85/7.00 @ ( semiri1314217659103216013at_int
% 12.85/7.00 @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
% 12.85/7.00 @ ( zero_n2687167440665602831ol_nat
% 12.85/7.00 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6325_divide__int__def,axiom,
% 12.85/7.00 ( divide_divide_int
% 12.85/7.00 = ( ^ [K3: int,L3: int] :
% 12.85/7.00 ( if_int @ ( L3 = zero_zero_int ) @ zero_zero_int
% 12.85/7.00 @ ( if_int
% 12.85/7.00 @ ( ( sgn_sgn_int @ K3 )
% 12.85/7.00 = ( sgn_sgn_int @ L3 ) )
% 12.85/7.00 @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) )
% 12.85/7.00 @ ( uminus_uminus_int
% 12.85/7.00 @ ( semiri1314217659103216013at_int
% 12.85/7.00 @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) )
% 12.85/7.00 @ ( zero_n2687167440665602831ol_nat
% 12.85/7.00 @ ~ ( dvd_dvd_int @ L3 @ K3 ) ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6326_take__bit__rec,axiom,
% 12.85/7.00 ( bit_se1745604003318907178nteger
% 12.85/7.00 = ( ^ [N4: nat,A3: code_integer] : ( if_Code_integer @ ( N4 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6327_take__bit__rec,axiom,
% 12.85/7.00 ( bit_se2923211474154528505it_int
% 12.85/7.00 = ( ^ [N4: nat,A3: int] : ( if_int @ ( N4 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6328_take__bit__rec,axiom,
% 12.85/7.00 ( bit_se2925701944663578781it_nat
% 12.85/7.00 = ( ^ [N4: nat,A3: nat] : ( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6329_tanh__real__altdef,axiom,
% 12.85/7.00 ( tanh_real
% 12.85/7.00 = ( ^ [X3: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6330_power__numeral,axiom,
% 12.85/7.00 ! [K: num,L: num] :
% 12.85/7.00 ( ( power_power_complex @ ( numera6690914467698888265omplex @ K ) @ ( numeral_numeral_nat @ L ) )
% 12.85/7.00 = ( numera6690914467698888265omplex @ ( pow @ K @ L ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6331_power__numeral,axiom,
% 12.85/7.00 ! [K: num,L: num] :
% 12.85/7.00 ( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L ) )
% 12.85/7.00 = ( numeral_numeral_real @ ( pow @ K @ L ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6332_power__numeral,axiom,
% 12.85/7.00 ! [K: num,L: num] :
% 12.85/7.00 ( ( power_power_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_nat @ L ) )
% 12.85/7.00 = ( numeral_numeral_rat @ ( pow @ K @ L ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6333_power__numeral,axiom,
% 12.85/7.00 ! [K: num,L: num] :
% 12.85/7.00 ( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
% 12.85/7.00 = ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6334_power__numeral,axiom,
% 12.85/7.00 ! [K: num,L: num] :
% 12.85/7.00 ( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
% 12.85/7.00 = ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6335_and__int__unfold,axiom,
% 12.85/7.00 ( bit_se725231765392027082nd_int
% 12.85/7.00 = ( ^ [K3: int,L3: int] :
% 12.85/7.00 ( if_int
% 12.85/7.00 @ ( ( K3 = zero_zero_int )
% 12.85/7.00 | ( L3 = zero_zero_int ) )
% 12.85/7.00 @ zero_zero_int
% 12.85/7.00 @ ( if_int
% 12.85/7.00 @ ( K3
% 12.85/7.00 = ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.00 @ L3
% 12.85/7.00 @ ( if_int
% 12.85/7.00 @ ( L3
% 12.85/7.00 = ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.00 @ K3
% 12.85/7.00 @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6336_arctan__half,axiom,
% 12.85/7.00 ( arctan
% 12.85/7.00 = ( ^ [X3: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X3 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6337_and_Oright__idem,axiom,
% 12.85/7.00 ! [A: int,B: int] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ B )
% 12.85/7.00 = ( bit_se725231765392027082nd_int @ A @ B ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6338_and_Oright__idem,axiom,
% 12.85/7.00 ! [A: nat,B: nat] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ B )
% 12.85/7.00 = ( bit_se727722235901077358nd_nat @ A @ B ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6339_and_Oleft__idem,axiom,
% 12.85/7.00 ! [A: int,B: int] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 12.85/7.00 = ( bit_se725231765392027082nd_int @ A @ B ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6340_and_Oleft__idem,axiom,
% 12.85/7.00 ! [A: nat,B: nat] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 12.85/7.00 = ( bit_se727722235901077358nd_nat @ A @ B ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6341_and_Oidem,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ A @ A )
% 12.85/7.00 = A ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6342_and_Oidem,axiom,
% 12.85/7.00 ! [A: nat] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ A @ A )
% 12.85/7.00 = A ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6343_real__sqrt__eq__iff,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ( sqrt @ X )
% 12.85/7.00 = ( sqrt @ Y4 ) )
% 12.85/7.00 = ( X = Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6344_sgn__le__0__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( sgn_sgn_real @ X ) @ zero_zero_real )
% 12.85/7.00 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6345_zero__le__sgn__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X ) )
% 12.85/7.00 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6346_take__bit__of__0,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
% 12.85/7.00 = zero_zero_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6347_take__bit__of__0,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
% 12.85/7.00 = zero_zero_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6348_bit_Oconj__zero__right,axiom,
% 12.85/7.00 ! [X: int] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
% 12.85/7.00 = zero_zero_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6349_bit_Oconj__zero__left,axiom,
% 12.85/7.00 ! [X: int] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
% 12.85/7.00 = zero_zero_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6350_zero__and__eq,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
% 12.85/7.00 = zero_zero_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6351_zero__and__eq,axiom,
% 12.85/7.00 ! [A: nat] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
% 12.85/7.00 = zero_zero_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6352_and__zero__eq,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
% 12.85/7.00 = zero_zero_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6353_and__zero__eq,axiom,
% 12.85/7.00 ! [A: nat] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
% 12.85/7.00 = zero_zero_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6354_real__sqrt__eq__zero__cancel__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ( sqrt @ X )
% 12.85/7.00 = zero_zero_real )
% 12.85/7.00 = ( X = zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6355_real__sqrt__zero,axiom,
% 12.85/7.00 ( ( sqrt @ zero_zero_real )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6356_real__sqrt__less__iff,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y4 ) )
% 12.85/7.00 = ( ord_less_real @ X @ Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6357_real__sqrt__le__iff,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y4 ) )
% 12.85/7.00 = ( ord_less_eq_real @ X @ Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6358_real__sqrt__eq__1__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ( sqrt @ X )
% 12.85/7.00 = one_one_real )
% 12.85/7.00 = ( X = one_one_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6359_real__sqrt__one,axiom,
% 12.85/7.00 ( ( sqrt @ one_one_real )
% 12.85/7.00 = one_one_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6360_nat__int,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
% 12.85/7.00 = N ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6361_take__bit__and,axiom,
% 12.85/7.00 ! [N: nat,A: int,B: int] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 12.85/7.00 = ( bit_se725231765392027082nd_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6362_take__bit__and,axiom,
% 12.85/7.00 ! [N: nat,A: nat,B: nat] :
% 12.85/7.00 ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 12.85/7.00 = ( bit_se727722235901077358nd_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6363_exp__le__cancel__iff,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( exp_real @ X ) @ ( exp_real @ Y4 ) )
% 12.85/7.00 = ( ord_less_eq_real @ X @ Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6364_concat__bit__of__zero__2,axiom,
% 12.85/7.00 ! [N: nat,K: int] :
% 12.85/7.00 ( ( bit_concat_bit @ N @ K @ zero_zero_int )
% 12.85/7.00 = ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6365_exp__zero,axiom,
% 12.85/7.00 ( ( exp_complex @ zero_zero_complex )
% 12.85/7.00 = one_one_complex ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6366_exp__zero,axiom,
% 12.85/7.00 ( ( exp_real @ zero_zero_real )
% 12.85/7.00 = one_one_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6367_take__bit__0,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
% 12.85/7.00 = zero_zero_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6368_take__bit__0,axiom,
% 12.85/7.00 ! [A: nat] :
% 12.85/7.00 ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
% 12.85/7.00 = zero_zero_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6369_take__bit__Suc__1,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
% 12.85/7.00 = one_one_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6370_take__bit__Suc__1,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
% 12.85/7.00 = one_one_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6371_bit_Oconj__one__right,axiom,
% 12.85/7.00 ! [X: code_integer] :
% 12.85/7.00 ( ( bit_se3949692690581998587nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.00 = X ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6372_bit_Oconj__one__right,axiom,
% 12.85/7.00 ! [X: int] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ X @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.00 = X ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6373_and_Oright__neutral,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.00 = A ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6374_and_Oright__neutral,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.00 = A ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6375_and_Oleft__neutral,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
% 12.85/7.00 = A ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6376_and_Oleft__neutral,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
% 12.85/7.00 = A ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6377_take__bit__numeral__1,axiom,
% 12.85/7.00 ! [L: num] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ one_one_int )
% 12.85/7.00 = one_one_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6378_take__bit__numeral__1,axiom,
% 12.85/7.00 ! [L: num] :
% 12.85/7.00 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ one_one_nat )
% 12.85/7.00 = one_one_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6379_nat__numeral,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 12.85/7.00 = ( numeral_numeral_nat @ K ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6380_real__sqrt__lt__0__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
% 12.85/7.00 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6381_real__sqrt__gt__0__iff,axiom,
% 12.85/7.00 ! [Y4: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y4 ) )
% 12.85/7.00 = ( ord_less_real @ zero_zero_real @ Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6382_real__sqrt__ge__0__iff,axiom,
% 12.85/7.00 ! [Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y4 ) )
% 12.85/7.00 = ( ord_less_eq_real @ zero_zero_real @ Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6383_real__sqrt__le__0__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
% 12.85/7.00 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6384_real__sqrt__lt__1__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
% 12.85/7.00 = ( ord_less_real @ X @ one_one_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6385_real__sqrt__gt__1__iff,axiom,
% 12.85/7.00 ! [Y4: real] :
% 12.85/7.00 ( ( ord_less_real @ one_one_real @ ( sqrt @ Y4 ) )
% 12.85/7.00 = ( ord_less_real @ one_one_real @ Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6386_real__sqrt__ge__1__iff,axiom,
% 12.85/7.00 ! [Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y4 ) )
% 12.85/7.00 = ( ord_less_eq_real @ one_one_real @ Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6387_real__sqrt__le__1__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
% 12.85/7.00 = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6388_exp__eq__one__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ( exp_real @ X )
% 12.85/7.00 = one_one_real )
% 12.85/7.00 = ( X = zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6389_and__nonnegative__int__iff,axiom,
% 12.85/7.00 ! [K: int,L: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 12.85/7.00 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.85/7.00 | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6390_and__negative__int__iff,axiom,
% 12.85/7.00 ! [K: int,L: int] :
% 12.85/7.00 ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
% 12.85/7.00 = ( ( ord_less_int @ K @ zero_zero_int )
% 12.85/7.00 & ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6391_real__sqrt__mult__self,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
% 12.85/7.00 = ( abs_abs_real @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6392_real__sqrt__abs2,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( sqrt @ ( times_times_real @ X @ X ) )
% 12.85/7.00 = ( abs_abs_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6393_nat__of__bool,axiom,
% 12.85/7.00 ! [P: $o] :
% 12.85/7.00 ( ( nat2 @ ( zero_n2684676970156552555ol_int @ P ) )
% 12.85/7.00 = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6394_take__bit__of__1__eq__0__iff,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 12.85/7.00 = zero_zero_int )
% 12.85/7.00 = ( N = zero_zero_nat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6395_take__bit__of__1__eq__0__iff,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 12.85/7.00 = zero_zero_nat )
% 12.85/7.00 = ( N = zero_zero_nat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6396_and__numerals_I2_J,axiom,
% 12.85/7.00 ! [Y4: num] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y4 ) ) )
% 12.85/7.00 = one_one_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6397_and__numerals_I2_J,axiom,
% 12.85/7.00 ! [Y4: num] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 12.85/7.00 = one_one_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6398_and__numerals_I8_J,axiom,
% 12.85/7.00 ! [X: num] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
% 12.85/7.00 = one_one_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6399_and__numerals_I8_J,axiom,
% 12.85/7.00 ! [X: num] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
% 12.85/7.00 = one_one_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6400_real__sqrt__four,axiom,
% 12.85/7.00 ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.85/7.00 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6401_nat__1,axiom,
% 12.85/7.00 ( ( nat2 @ one_one_int )
% 12.85/7.00 = ( suc @ zero_zero_nat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6402_nat__0__iff,axiom,
% 12.85/7.00 ! [I: int] :
% 12.85/7.00 ( ( ( nat2 @ I )
% 12.85/7.00 = zero_zero_nat )
% 12.85/7.00 = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6403_nat__le__0,axiom,
% 12.85/7.00 ! [Z: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 12.85/7.00 => ( ( nat2 @ Z )
% 12.85/7.00 = zero_zero_nat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6404_zless__nat__conj,axiom,
% 12.85/7.00 ! [W: int,Z: int] :
% 12.85/7.00 ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 12.85/7.00 = ( ( ord_less_int @ zero_zero_int @ Z )
% 12.85/7.00 & ( ord_less_int @ W @ Z ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6405_nat__neg__numeral,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.85/7.00 = zero_zero_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6406_one__less__exp__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
% 12.85/7.00 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6407_exp__less__one__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
% 12.85/7.00 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6408_exp__le__one__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
% 12.85/7.00 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6409_one__le__exp__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
% 12.85/7.00 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6410_nat__zminus__int,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
% 12.85/7.00 = zero_zero_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6411_int__nat__eq,axiom,
% 12.85/7.00 ! [Z: int] :
% 12.85/7.00 ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.85/7.00 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 12.85/7.00 = Z ) )
% 12.85/7.00 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.85/7.00 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 12.85/7.00 = zero_zero_int ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6412_take__bit__minus__one__eq__mask,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( bit_se1745604003318907178nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.00 = ( bit_se2119862282449309892nteger @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6413_take__bit__minus__one__eq__mask,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.00 = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6414_exp__ln,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( exp_real @ ( ln_ln_real @ X ) )
% 12.85/7.00 = X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6415_exp__ln__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ( exp_real @ ( ln_ln_real @ X ) )
% 12.85/7.00 = X )
% 12.85/7.00 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6416_take__bit__of__Suc__0,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
% 12.85/7.00 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6417_and__numerals_I1_J,axiom,
% 12.85/7.00 ! [Y4: num] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y4 ) ) )
% 12.85/7.00 = zero_zero_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6418_and__numerals_I1_J,axiom,
% 12.85/7.00 ! [Y4: num] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 12.85/7.00 = zero_zero_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6419_and__numerals_I5_J,axiom,
% 12.85/7.00 ! [X: num] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
% 12.85/7.00 = zero_zero_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6420_and__numerals_I5_J,axiom,
% 12.85/7.00 ! [X: num] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
% 12.85/7.00 = zero_zero_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6421_and__numerals_I3_J,axiom,
% 12.85/7.00 ! [X: num,Y4: num] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y4 ) ) )
% 12.85/7.00 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6422_and__numerals_I3_J,axiom,
% 12.85/7.00 ! [X: num,Y4: num] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 12.85/7.00 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6423_zero__less__nat__eq,axiom,
% 12.85/7.00 ! [Z: int] :
% 12.85/7.00 ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
% 12.85/7.00 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6424_take__bit__of__1,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
% 12.85/7.00 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6425_take__bit__of__1,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 12.85/7.00 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6426_take__bit__of__1,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 12.85/7.00 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6427_diff__nat__numeral,axiom,
% 12.85/7.00 ! [V: num,V2: num] :
% 12.85/7.00 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V2 ) )
% 12.85/7.00 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V2 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6428_and__minus__numerals_I6_J,axiom,
% 12.85/7.00 ! [N: num] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 12.85/7.00 = one_one_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6429_and__minus__numerals_I2_J,axiom,
% 12.85/7.00 ! [N: num] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.85/7.00 = one_one_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6430_nat__eq__numeral__power__cancel__iff,axiom,
% 12.85/7.00 ! [Y4: int,X: num,N: nat] :
% 12.85/7.00 ( ( ( nat2 @ Y4 )
% 12.85/7.00 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 12.85/7.00 = ( Y4
% 12.85/7.00 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6431_numeral__power__eq__nat__cancel__iff,axiom,
% 12.85/7.00 ! [X: num,N: nat,Y4: int] :
% 12.85/7.00 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 12.85/7.00 = ( nat2 @ Y4 ) )
% 12.85/7.00 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 12.85/7.00 = Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6432_dvd__nat__abs__iff,axiom,
% 12.85/7.00 ! [N: nat,K: int] :
% 12.85/7.00 ( ( dvd_dvd_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 12.85/7.00 = ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6433_nat__abs__dvd__iff,axiom,
% 12.85/7.00 ! [K: int,N: nat] :
% 12.85/7.00 ( ( dvd_dvd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 12.85/7.00 = ( dvd_dvd_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6434_and__numerals_I4_J,axiom,
% 12.85/7.00 ! [X: num,Y4: num] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y4 ) ) )
% 12.85/7.00 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6435_and__numerals_I4_J,axiom,
% 12.85/7.00 ! [X: num,Y4: num] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 12.85/7.00 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6436_and__numerals_I6_J,axiom,
% 12.85/7.00 ! [X: num,Y4: num] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y4 ) ) )
% 12.85/7.00 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6437_and__numerals_I6_J,axiom,
% 12.85/7.00 ! [X: num,Y4: num] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 12.85/7.00 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6438_even__take__bit__eq,axiom,
% 12.85/7.00 ! [N: nat,A: code_integer] :
% 12.85/7.00 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N @ A ) )
% 12.85/7.00 = ( ( N = zero_zero_nat )
% 12.85/7.00 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6439_even__take__bit__eq,axiom,
% 12.85/7.00 ! [N: nat,A: int] :
% 12.85/7.00 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 12.85/7.00 = ( ( N = zero_zero_nat )
% 12.85/7.00 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6440_even__take__bit__eq,axiom,
% 12.85/7.00 ! [N: nat,A: nat] :
% 12.85/7.00 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 12.85/7.00 = ( ( N = zero_zero_nat )
% 12.85/7.00 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6441_one__less__nat__eq,axiom,
% 12.85/7.00 ! [Z: int] :
% 12.85/7.00 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
% 12.85/7.00 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6442_real__sqrt__abs,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( sqrt @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.00 = ( abs_abs_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6443_and__minus__numerals_I5_J,axiom,
% 12.85/7.00 ! [N: num] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 12.85/7.00 = zero_zero_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6444_and__minus__numerals_I1_J,axiom,
% 12.85/7.00 ! [N: num] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.85/7.00 = zero_zero_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6445_take__bit__Suc__0,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
% 12.85/7.00 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6446_take__bit__Suc__0,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
% 12.85/7.00 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6447_take__bit__Suc__0,axiom,
% 12.85/7.00 ! [A: nat] :
% 12.85/7.00 ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
% 12.85/7.00 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6448_real__sqrt__pow2__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.00 = X )
% 12.85/7.00 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6449_real__sqrt__pow2,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.00 = X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6450_real__sqrt__sum__squares__mult__squared__eq,axiom,
% 12.85/7.00 ! [X: real,Y4: real,Xa: real,Ya: real] :
% 12.85/7.00 ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( 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 ) ) )
% 12.85/7.00 = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( 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 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6451_nat__numeral__diff__1,axiom,
% 12.85/7.00 ! [V: num] :
% 12.85/7.00 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
% 12.85/7.00 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6452_numeral__power__less__nat__cancel__iff,axiom,
% 12.85/7.00 ! [X: num,N: nat,A: int] :
% 12.85/7.00 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 12.85/7.00 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6453_nat__less__numeral__power__cancel__iff,axiom,
% 12.85/7.00 ! [A: int,X: num,N: nat] :
% 12.85/7.00 ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 12.85/7.00 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6454_nat__le__numeral__power__cancel__iff,axiom,
% 12.85/7.00 ! [A: int,X: num,N: nat] :
% 12.85/7.00 ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 12.85/7.00 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6455_numeral__power__le__nat__cancel__iff,axiom,
% 12.85/7.00 ! [X: num,N: nat,A: int] :
% 12.85/7.00 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 12.85/7.00 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6456_and__numerals_I7_J,axiom,
% 12.85/7.00 ! [X: num,Y4: num] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y4 ) ) )
% 12.85/7.00 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6457_and__numerals_I7_J,axiom,
% 12.85/7.00 ! [X: num,Y4: num] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 12.85/7.00 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6458_take__bit__of__exp,axiom,
% 12.85/7.00 ! [M: nat,N: nat] :
% 12.85/7.00 ( ( bit_se1745604003318907178nteger @ M @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.00 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6459_take__bit__of__exp,axiom,
% 12.85/7.00 ! [M: nat,N: nat] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.00 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6460_take__bit__of__exp,axiom,
% 12.85/7.00 ! [M: nat,N: nat] :
% 12.85/7.00 ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.00 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6461_take__bit__of__2,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/7.00 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6462_take__bit__of__2,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/7.00 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6463_take__bit__of__2,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.00 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6464_take__bit__eq__mask,axiom,
% 12.85/7.00 ( bit_se2923211474154528505it_int
% 12.85/7.00 = ( ^ [N4: nat,A3: int] : ( bit_se725231765392027082nd_int @ A3 @ ( bit_se2000444600071755411sk_int @ N4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6465_take__bit__eq__mask,axiom,
% 12.85/7.00 ( bit_se2925701944663578781it_nat
% 12.85/7.00 = ( ^ [N4: nat,A3: nat] : ( bit_se727722235901077358nd_nat @ A3 @ ( bit_se2002935070580805687sk_nat @ N4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6466_nat__mask__eq,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
% 12.85/7.00 = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6467_and_Oleft__commute,axiom,
% 12.85/7.00 ! [B: int,A: int,C: int] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ B @ ( bit_se725231765392027082nd_int @ A @ C ) )
% 12.85/7.00 = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6468_and_Oleft__commute,axiom,
% 12.85/7.00 ! [B: nat,A: nat,C: nat] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ B @ ( bit_se727722235901077358nd_nat @ A @ C ) )
% 12.85/7.00 = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6469_and_Ocommute,axiom,
% 12.85/7.00 ( bit_se725231765392027082nd_int
% 12.85/7.00 = ( ^ [A3: int,B2: int] : ( bit_se725231765392027082nd_int @ B2 @ A3 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6470_and_Ocommute,axiom,
% 12.85/7.00 ( bit_se727722235901077358nd_nat
% 12.85/7.00 = ( ^ [A3: nat,B2: nat] : ( bit_se727722235901077358nd_nat @ B2 @ A3 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6471_and_Oassoc,axiom,
% 12.85/7.00 ! [A: int,B: int,C: int] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ C )
% 12.85/7.00 = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6472_and_Oassoc,axiom,
% 12.85/7.00 ! [A: nat,B: nat,C: nat] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ C )
% 12.85/7.00 = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6473_of__nat__and__eq,axiom,
% 12.85/7.00 ! [M: nat,N: nat] :
% 12.85/7.00 ( ( semiri1314217659103216013at_int @ ( bit_se727722235901077358nd_nat @ M @ N ) )
% 12.85/7.00 = ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6474_of__nat__and__eq,axiom,
% 12.85/7.00 ! [M: nat,N: nat] :
% 12.85/7.00 ( ( semiri1316708129612266289at_nat @ ( bit_se727722235901077358nd_nat @ M @ N ) )
% 12.85/7.00 = ( bit_se727722235901077358nd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6475_take__bit__of__nat,axiom,
% 12.85/7.00 ! [N: nat,M: nat] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 12.85/7.00 = ( semiri1314217659103216013at_int @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6476_take__bit__of__nat,axiom,
% 12.85/7.00 ! [N: nat,M: nat] :
% 12.85/7.00 ( ( bit_se2925701944663578781it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 12.85/7.00 = ( semiri1316708129612266289at_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6477_nat__take__bit__eq,axiom,
% 12.85/7.00 ! [K: int,N: nat] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.85/7.00 => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 12.85/7.00 = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6478_take__bit__nat__eq,axiom,
% 12.85/7.00 ! [K: int,N: nat] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.85/7.00 => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
% 12.85/7.00 = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6479_norm__exp,axiom,
% 12.85/7.00 ! [X: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6480_norm__exp,axiom,
% 12.85/7.00 ! [X: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6481_take__bit__add,axiom,
% 12.85/7.00 ! [N: nat,A: int,B: int] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) )
% 12.85/7.00 = ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ A @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6482_take__bit__add,axiom,
% 12.85/7.00 ! [N: nat,A: nat,B: nat] :
% 12.85/7.00 ( ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) )
% 12.85/7.00 = ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ A @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6483_take__bit__tightened,axiom,
% 12.85/7.00 ! [N: nat,A: int,B: int,M: nat] :
% 12.85/7.00 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 12.85/7.00 = ( bit_se2923211474154528505it_int @ N @ B ) )
% 12.85/7.00 => ( ( ord_less_eq_nat @ M @ N )
% 12.85/7.00 => ( ( bit_se2923211474154528505it_int @ M @ A )
% 12.85/7.00 = ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6484_take__bit__tightened,axiom,
% 12.85/7.00 ! [N: nat,A: nat,B: nat,M: nat] :
% 12.85/7.00 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 12.85/7.00 = ( bit_se2925701944663578781it_nat @ N @ B ) )
% 12.85/7.00 => ( ( ord_less_eq_nat @ M @ N )
% 12.85/7.00 => ( ( bit_se2925701944663578781it_nat @ M @ A )
% 12.85/7.00 = ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6485_take__bit__nat__less__eq__self,axiom,
% 12.85/7.00 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6486_take__bit__tightened__less__eq__nat,axiom,
% 12.85/7.00 ! [M: nat,N: nat,Q2: nat] :
% 12.85/7.00 ( ( ord_less_eq_nat @ M @ N )
% 12.85/7.00 => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N @ Q2 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6487_real__sqrt__less__mono,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_real @ X @ Y4 )
% 12.85/7.00 => ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6488_real__sqrt__le__mono,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/7.00 => ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6489_real__sqrt__divide,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( sqrt @ ( divide_divide_real @ X @ Y4 ) )
% 12.85/7.00 = ( divide_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6490_real__sqrt__mult,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( sqrt @ ( times_times_real @ X @ Y4 ) )
% 12.85/7.00 = ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6491_real__sqrt__power,axiom,
% 12.85/7.00 ! [X: real,K: nat] :
% 12.85/7.00 ( ( sqrt @ ( power_power_real @ X @ K ) )
% 12.85/7.00 = ( power_power_real @ ( sqrt @ X ) @ K ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6492_real__sqrt__minus,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( sqrt @ ( uminus_uminus_real @ X ) )
% 12.85/7.00 = ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6493_exp__not__eq__zero,axiom,
% 12.85/7.00 ! [X: complex] :
% 12.85/7.00 ( ( exp_complex @ X )
% 12.85/7.00 != zero_zero_complex ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6494_exp__not__eq__zero,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( exp_real @ X )
% 12.85/7.00 != zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6495_take__bit__minus,axiom,
% 12.85/7.00 ! [N: nat,K: int] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 12.85/7.00 = ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6496_exp__times__arg__commute,axiom,
% 12.85/7.00 ! [A2: complex] :
% 12.85/7.00 ( ( times_times_complex @ ( exp_complex @ A2 ) @ A2 )
% 12.85/7.00 = ( times_times_complex @ A2 @ ( exp_complex @ A2 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6497_exp__times__arg__commute,axiom,
% 12.85/7.00 ! [A2: real] :
% 12.85/7.00 ( ( times_times_real @ ( exp_real @ A2 ) @ A2 )
% 12.85/7.00 = ( times_times_real @ A2 @ ( exp_real @ A2 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6498_take__bit__mult,axiom,
% 12.85/7.00 ! [N: nat,K: int,L: int] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
% 12.85/7.00 = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6499_take__bit__diff,axiom,
% 12.85/7.00 ! [N: nat,K: int,L: int] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
% 12.85/7.00 = ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6500_concat__bit__take__bit__eq,axiom,
% 12.85/7.00 ! [N: nat,B: int] :
% 12.85/7.00 ( ( bit_concat_bit @ N @ ( bit_se2923211474154528505it_int @ N @ B ) )
% 12.85/7.00 = ( bit_concat_bit @ N @ B ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6501_concat__bit__eq__iff,axiom,
% 12.85/7.00 ! [N: nat,K: int,L: int,R2: int,S2: int] :
% 12.85/7.00 ( ( ( bit_concat_bit @ N @ K @ L )
% 12.85/7.00 = ( bit_concat_bit @ N @ R2 @ S2 ) )
% 12.85/7.00 = ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 12.85/7.00 = ( bit_se2923211474154528505it_int @ N @ R2 ) )
% 12.85/7.00 & ( L = S2 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6502_and__eq__minus__1__iff,axiom,
% 12.85/7.00 ! [A: code_integer,B: code_integer] :
% 12.85/7.00 ( ( ( bit_se3949692690581998587nteger @ A @ B )
% 12.85/7.00 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.00 = ( ( A
% 12.85/7.00 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.00 & ( B
% 12.85/7.00 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6503_and__eq__minus__1__iff,axiom,
% 12.85/7.00 ! [A: int,B: int] :
% 12.85/7.00 ( ( ( bit_se725231765392027082nd_int @ A @ B )
% 12.85/7.00 = ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.00 = ( ( A
% 12.85/7.00 = ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.00 & ( B
% 12.85/7.00 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6504_real__sqrt__gt__zero,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6505_exp__total,axiom,
% 12.85/7.00 ! [Y4: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ? [X4: real] :
% 12.85/7.00 ( ( exp_real @ X4 )
% 12.85/7.00 = Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6506_exp__gt__zero,axiom,
% 12.85/7.00 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6507_not__exp__less__zero,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6508_real__sqrt__ge__zero,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6509_real__sqrt__eq__zero__cancel,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ( sqrt @ X )
% 12.85/7.00 = zero_zero_real )
% 12.85/7.00 => ( X = zero_zero_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6510_exp__ge__zero,axiom,
% 12.85/7.00 ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6511_not__exp__le__zero,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6512_real__sqrt__ge__one,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ one_one_real @ X )
% 12.85/7.00 => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6513_nat__zero__as__int,axiom,
% 12.85/7.00 ( zero_zero_nat
% 12.85/7.00 = ( nat2 @ zero_zero_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6514_AND__upper2_H,axiom,
% 12.85/7.00 ! [Y4: int,Z: int,X: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/7.00 => ( ( ord_less_eq_int @ Y4 @ Z )
% 12.85/7.00 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y4 ) @ Z ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6515_AND__upper1_H,axiom,
% 12.85/7.00 ! [Y4: int,Z: int,Ya: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/7.00 => ( ( ord_less_eq_int @ Y4 @ Z )
% 12.85/7.00 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y4 @ Ya ) @ Z ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6516_AND__upper2,axiom,
% 12.85/7.00 ! [Y4: int,X: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/7.00 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y4 ) @ Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6517_AND__upper1,axiom,
% 12.85/7.00 ! [X: int,Y4: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/7.00 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y4 ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6518_AND__lower,axiom,
% 12.85/7.00 ! [X: int,Y4: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/7.00 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6519_take__bit__tightened__less__eq__int,axiom,
% 12.85/7.00 ! [M: nat,N: nat,K: int] :
% 12.85/7.00 ( ( ord_less_eq_nat @ M @ N )
% 12.85/7.00 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6520_nat__numeral__as__int,axiom,
% 12.85/7.00 ( numeral_numeral_nat
% 12.85/7.00 = ( ^ [I2: num] : ( nat2 @ ( numeral_numeral_int @ I2 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6521_take__bit__int__less__eq__self__iff,axiom,
% 12.85/7.00 ! [N: nat,K: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 12.85/7.00 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6522_take__bit__nonnegative,axiom,
% 12.85/7.00 ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6523_take__bit__int__greater__self__iff,axiom,
% 12.85/7.00 ! [K: int,N: nat] :
% 12.85/7.00 ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 12.85/7.00 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6524_not__take__bit__negative,axiom,
% 12.85/7.00 ! [N: nat,K: int] :
% 12.85/7.00 ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6525_signed__take__bit__eq__iff__take__bit__eq,axiom,
% 12.85/7.00 ! [N: nat,A: int,B: int] :
% 12.85/7.00 ( ( ( bit_ri631733984087533419it_int @ N @ A )
% 12.85/7.00 = ( bit_ri631733984087533419it_int @ N @ B ) )
% 12.85/7.00 = ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 12.85/7.00 = ( bit_se2923211474154528505it_int @ ( suc @ N ) @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6526_real__sgn__eq,axiom,
% 12.85/7.00 ( sgn_sgn_real
% 12.85/7.00 = ( ^ [X3: real] : ( divide_divide_real @ X3 @ ( abs_abs_real @ X3 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6527_nat__mono,axiom,
% 12.85/7.00 ! [X: int,Y4: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ X @ Y4 )
% 12.85/7.00 => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6528_signed__take__bit__take__bit,axiom,
% 12.85/7.00 ! [M: nat,N: nat,A: int] :
% 12.85/7.00 ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 12.85/7.00 = ( if_int_int @ ( ord_less_eq_nat @ N @ M ) @ ( bit_se2923211474154528505it_int @ N ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6529_eq__nat__nat__iff,axiom,
% 12.85/7.00 ! [Z: int,Z6: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.85/7.00 => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 12.85/7.00 => ( ( ( nat2 @ Z )
% 12.85/7.00 = ( nat2 @ Z6 ) )
% 12.85/7.00 = ( Z = Z6 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6530_all__nat,axiom,
% 12.85/7.00 ( ( ^ [P2: nat > $o] :
% 12.85/7.00 ! [X6: nat] : ( P2 @ X6 ) )
% 12.85/7.00 = ( ^ [P3: nat > $o] :
% 12.85/7.00 ! [X3: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 12.85/7.00 => ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6531_ex__nat,axiom,
% 12.85/7.00 ( ( ^ [P2: nat > $o] :
% 12.85/7.00 ? [X6: nat] : ( P2 @ X6 ) )
% 12.85/7.00 = ( ^ [P3: nat > $o] :
% 12.85/7.00 ? [X3: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 12.85/7.00 & ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6532_mult__exp__exp,axiom,
% 12.85/7.00 ! [X: complex,Y4: complex] :
% 12.85/7.00 ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y4 ) )
% 12.85/7.00 = ( exp_complex @ ( plus_plus_complex @ X @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6533_mult__exp__exp,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y4 ) )
% 12.85/7.00 = ( exp_real @ ( plus_plus_real @ X @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6534_exp__add__commuting,axiom,
% 12.85/7.00 ! [X: complex,Y4: complex] :
% 12.85/7.00 ( ( ( times_times_complex @ X @ Y4 )
% 12.85/7.00 = ( times_times_complex @ Y4 @ X ) )
% 12.85/7.00 => ( ( exp_complex @ ( plus_plus_complex @ X @ Y4 ) )
% 12.85/7.00 = ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6535_exp__add__commuting,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ( times_times_real @ X @ Y4 )
% 12.85/7.00 = ( times_times_real @ Y4 @ X ) )
% 12.85/7.00 => ( ( exp_real @ ( plus_plus_real @ X @ Y4 ) )
% 12.85/7.00 = ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6536_exp__diff,axiom,
% 12.85/7.00 ! [X: complex,Y4: complex] :
% 12.85/7.00 ( ( exp_complex @ ( minus_minus_complex @ X @ Y4 ) )
% 12.85/7.00 = ( divide1717551699836669952omplex @ ( exp_complex @ X ) @ ( exp_complex @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6537_exp__diff,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( exp_real @ ( minus_minus_real @ X @ Y4 ) )
% 12.85/7.00 = ( divide_divide_real @ ( exp_real @ X ) @ ( exp_real @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6538_nat__one__as__int,axiom,
% 12.85/7.00 ( one_one_nat
% 12.85/7.00 = ( nat2 @ one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6539_take__bit__unset__bit__eq,axiom,
% 12.85/7.00 ! [N: nat,M: nat,A: int] :
% 12.85/7.00 ( ( ( ord_less_eq_nat @ N @ M )
% 12.85/7.00 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 12.85/7.00 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 12.85/7.00 & ( ~ ( ord_less_eq_nat @ N @ M )
% 12.85/7.00 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 12.85/7.00 = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6540_take__bit__unset__bit__eq,axiom,
% 12.85/7.00 ! [N: nat,M: nat,A: nat] :
% 12.85/7.00 ( ( ( ord_less_eq_nat @ N @ M )
% 12.85/7.00 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 12.85/7.00 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 12.85/7.00 & ( ~ ( ord_less_eq_nat @ N @ M )
% 12.85/7.00 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 12.85/7.00 = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6541_take__bit__unset__bit__eq,axiom,
% 12.85/7.00 ! [N: nat,M: nat,A: code_integer] :
% 12.85/7.00 ( ( ( ord_less_eq_nat @ N @ M )
% 12.85/7.00 => ( ( bit_se1745604003318907178nteger @ N @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 12.85/7.00 = ( bit_se1745604003318907178nteger @ N @ A ) ) )
% 12.85/7.00 & ( ~ ( ord_less_eq_nat @ N @ M )
% 12.85/7.00 => ( ( bit_se1745604003318907178nteger @ N @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 12.85/7.00 = ( bit_se8260200283734997820nteger @ M @ ( bit_se1745604003318907178nteger @ N @ A ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6542_take__bit__set__bit__eq,axiom,
% 12.85/7.00 ! [N: nat,M: nat,A: code_integer] :
% 12.85/7.00 ( ( ( ord_less_eq_nat @ N @ M )
% 12.85/7.00 => ( ( bit_se1745604003318907178nteger @ N @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 12.85/7.00 = ( bit_se1745604003318907178nteger @ N @ A ) ) )
% 12.85/7.00 & ( ~ ( ord_less_eq_nat @ N @ M )
% 12.85/7.00 => ( ( bit_se1745604003318907178nteger @ N @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 12.85/7.00 = ( bit_se2793503036327961859nteger @ M @ ( bit_se1745604003318907178nteger @ N @ A ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6543_take__bit__set__bit__eq,axiom,
% 12.85/7.00 ! [N: nat,M: nat,A: int] :
% 12.85/7.00 ( ( ( ord_less_eq_nat @ N @ M )
% 12.85/7.00 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 12.85/7.00 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 12.85/7.00 & ( ~ ( ord_less_eq_nat @ N @ M )
% 12.85/7.00 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 12.85/7.00 = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6544_take__bit__set__bit__eq,axiom,
% 12.85/7.00 ! [N: nat,M: nat,A: nat] :
% 12.85/7.00 ( ( ( ord_less_eq_nat @ N @ M )
% 12.85/7.00 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 12.85/7.00 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 12.85/7.00 & ( ~ ( ord_less_eq_nat @ N @ M )
% 12.85/7.00 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 12.85/7.00 = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6545_take__bit__flip__bit__eq,axiom,
% 12.85/7.00 ! [N: nat,M: nat,A: code_integer] :
% 12.85/7.00 ( ( ( ord_less_eq_nat @ N @ M )
% 12.85/7.00 => ( ( bit_se1745604003318907178nteger @ N @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 12.85/7.00 = ( bit_se1745604003318907178nteger @ N @ A ) ) )
% 12.85/7.00 & ( ~ ( ord_less_eq_nat @ N @ M )
% 12.85/7.00 => ( ( bit_se1745604003318907178nteger @ N @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 12.85/7.00 = ( bit_se1345352211410354436nteger @ M @ ( bit_se1745604003318907178nteger @ N @ A ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6546_take__bit__flip__bit__eq,axiom,
% 12.85/7.00 ! [N: nat,M: nat,A: int] :
% 12.85/7.00 ( ( ( ord_less_eq_nat @ N @ M )
% 12.85/7.00 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 12.85/7.00 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 12.85/7.00 & ( ~ ( ord_less_eq_nat @ N @ M )
% 12.85/7.00 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 12.85/7.00 = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6547_take__bit__flip__bit__eq,axiom,
% 12.85/7.00 ! [N: nat,M: nat,A: nat] :
% 12.85/7.00 ( ( ( ord_less_eq_nat @ N @ M )
% 12.85/7.00 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 12.85/7.00 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 12.85/7.00 & ( ~ ( ord_less_eq_nat @ N @ M )
% 12.85/7.00 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 12.85/7.00 = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6548_unset__bit__nat__def,axiom,
% 12.85/7.00 ( bit_se4205575877204974255it_nat
% 12.85/7.00 = ( ^ [M5: nat,N4: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M5 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6549_pow_Osimps_I1_J,axiom,
% 12.85/7.00 ! [X: num] :
% 12.85/7.00 ( ( pow @ X @ one )
% 12.85/7.00 = X ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6550_exp__gt__one,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6551_real__div__sqrt,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( divide_divide_real @ X @ ( sqrt @ X ) )
% 12.85/7.00 = ( sqrt @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6552_sqrt__add__le__add__sqrt,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y4 ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y4 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6553_take__bit__signed__take__bit,axiom,
% 12.85/7.00 ! [M: nat,N: nat,A: int] :
% 12.85/7.00 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 12.85/7.00 => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N @ A ) )
% 12.85/7.00 = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6554_exp__ge__add__one__self,axiom,
% 12.85/7.00 ! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6555_le__real__sqrt__sumsq,axiom,
% 12.85/7.00 ! [X: real,Y4: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6556_and__less__eq,axiom,
% 12.85/7.00 ! [L: int,K: int] :
% 12.85/7.00 ( ( ord_less_int @ L @ zero_zero_int )
% 12.85/7.00 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6557_AND__upper1_H_H,axiom,
% 12.85/7.00 ! [Y4: int,Z: int,Ya: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/7.00 => ( ( ord_less_int @ Y4 @ Z )
% 12.85/7.00 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y4 @ Ya ) @ Z ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6558_AND__upper2_H_H,axiom,
% 12.85/7.00 ! [Y4: int,Z: int,X: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/7.00 => ( ( ord_less_int @ Y4 @ Z )
% 12.85/7.00 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y4 ) @ Z ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6559_nat__mono__iff,axiom,
% 12.85/7.00 ! [Z: int,W: int] :
% 12.85/7.00 ( ( ord_less_int @ zero_zero_int @ Z )
% 12.85/7.00 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 12.85/7.00 = ( ord_less_int @ W @ Z ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6560_take__bit__eq__mask__iff,axiom,
% 12.85/7.00 ! [N: nat,K: int] :
% 12.85/7.00 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 12.85/7.00 = ( bit_se2000444600071755411sk_int @ N ) )
% 12.85/7.00 = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 12.85/7.00 = zero_zero_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6561_exp__minus__inverse,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) )
% 12.85/7.00 = one_one_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6562_exp__minus__inverse,axiom,
% 12.85/7.00 ! [X: complex] :
% 12.85/7.00 ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) )
% 12.85/7.00 = one_one_complex ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6563_take__bit__decr__eq,axiom,
% 12.85/7.00 ! [N: nat,K: int] :
% 12.85/7.00 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 12.85/7.00 != zero_zero_int )
% 12.85/7.00 => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
% 12.85/7.00 = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6564_zless__nat__eq__int__zless,axiom,
% 12.85/7.00 ! [M: nat,Z: int] :
% 12.85/7.00 ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
% 12.85/7.00 = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6565_nat__le__iff,axiom,
% 12.85/7.00 ! [X: int,N: nat] :
% 12.85/7.00 ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
% 12.85/7.00 = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6566_nat__0__le,axiom,
% 12.85/7.00 ! [Z: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.85/7.00 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 12.85/7.00 = Z ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6567_int__eq__iff,axiom,
% 12.85/7.00 ! [M: nat,Z: int] :
% 12.85/7.00 ( ( ( semiri1314217659103216013at_int @ M )
% 12.85/7.00 = Z )
% 12.85/7.00 = ( ( M
% 12.85/7.00 = ( nat2 @ Z ) )
% 12.85/7.00 & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6568_nat__int__add,axiom,
% 12.85/7.00 ! [A: nat,B: nat] :
% 12.85/7.00 ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
% 12.85/7.00 = ( plus_plus_nat @ A @ B ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6569_exp__of__nat2__mult,axiom,
% 12.85/7.00 ! [X: complex,N: nat] :
% 12.85/7.00 ( ( exp_complex @ ( times_times_complex @ X @ ( semiri8010041392384452111omplex @ N ) ) )
% 12.85/7.00 = ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6570_exp__of__nat2__mult,axiom,
% 12.85/7.00 ! [X: real,N: nat] :
% 12.85/7.00 ( ( exp_real @ ( times_times_real @ X @ ( semiri5074537144036343181t_real @ N ) ) )
% 12.85/7.00 = ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6571_exp__of__nat__mult,axiom,
% 12.85/7.00 ! [N: nat,X: complex] :
% 12.85/7.00 ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ X ) )
% 12.85/7.00 = ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6572_exp__of__nat__mult,axiom,
% 12.85/7.00 ! [N: nat,X: real] :
% 12.85/7.00 ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) )
% 12.85/7.00 = ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6573_int__minus,axiom,
% 12.85/7.00 ! [N: nat,M: nat] :
% 12.85/7.00 ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
% 12.85/7.00 = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6574_nat__abs__mult__distrib,axiom,
% 12.85/7.00 ! [W: int,Z: int] :
% 12.85/7.00 ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
% 12.85/7.00 = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6575_even__and__iff,axiom,
% 12.85/7.00 ! [A: code_integer,B: code_integer] :
% 12.85/7.00 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ A @ B ) )
% 12.85/7.00 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/7.00 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6576_even__and__iff,axiom,
% 12.85/7.00 ! [A: int,B: int] :
% 12.85/7.00 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 12.85/7.00 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/7.00 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6577_even__and__iff,axiom,
% 12.85/7.00 ! [A: nat,B: nat] :
% 12.85/7.00 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 12.85/7.00 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/7.00 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6578_sqrt2__less__2,axiom,
% 12.85/7.00 ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6579_sgn__real__def,axiom,
% 12.85/7.00 ( sgn_sgn_real
% 12.85/7.00 = ( ^ [A3: real] : ( if_real @ ( A3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6580_exp__ge__add__one__self__aux,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6581_even__and__iff__int,axiom,
% 12.85/7.00 ! [K: int,L: int] :
% 12.85/7.00 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 12.85/7.00 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 12.85/7.00 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6582_lemma__exp__total,axiom,
% 12.85/7.00 ! [Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ one_one_real @ Y4 )
% 12.85/7.00 => ? [X4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.85/7.00 & ( ord_less_eq_real @ X4 @ ( minus_minus_real @ Y4 @ one_one_real ) )
% 12.85/7.00 & ( ( exp_real @ X4 )
% 12.85/7.00 = Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6583_ln__ge__iff,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ Y4 @ ( ln_ln_real @ X ) )
% 12.85/7.00 = ( ord_less_eq_real @ ( exp_real @ Y4 ) @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6584_ln__x__over__x__mono,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/7.00 => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y4 ) @ Y4 ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6585_nat__less__eq__zless,axiom,
% 12.85/7.00 ! [W: int,Z: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 12.85/7.00 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 12.85/7.00 = ( ord_less_int @ W @ Z ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6586_nat__le__eq__zle,axiom,
% 12.85/7.00 ! [W: int,Z: int] :
% 12.85/7.00 ( ( ( ord_less_int @ zero_zero_int @ W )
% 12.85/7.00 | ( ord_less_eq_int @ zero_zero_int @ Z ) )
% 12.85/7.00 => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 12.85/7.00 = ( ord_less_eq_int @ W @ Z ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6587_nat__eq__iff2,axiom,
% 12.85/7.00 ! [M: nat,W: int] :
% 12.85/7.00 ( ( M
% 12.85/7.00 = ( nat2 @ W ) )
% 12.85/7.00 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 12.85/7.00 => ( W
% 12.85/7.00 = ( semiri1314217659103216013at_int @ M ) ) )
% 12.85/7.00 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 12.85/7.00 => ( M = zero_zero_nat ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6588_nat__eq__iff,axiom,
% 12.85/7.00 ! [W: int,M: nat] :
% 12.85/7.00 ( ( ( nat2 @ W )
% 12.85/7.00 = M )
% 12.85/7.00 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 12.85/7.00 => ( W
% 12.85/7.00 = ( semiri1314217659103216013at_int @ M ) ) )
% 12.85/7.00 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 12.85/7.00 => ( M = zero_zero_nat ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6589_split__nat,axiom,
% 12.85/7.00 ! [P: nat > $o,I: int] :
% 12.85/7.00 ( ( P @ ( nat2 @ I ) )
% 12.85/7.00 = ( ! [N4: nat] :
% 12.85/7.00 ( ( I
% 12.85/7.00 = ( semiri1314217659103216013at_int @ N4 ) )
% 12.85/7.00 => ( P @ N4 ) )
% 12.85/7.00 & ( ( ord_less_int @ I @ zero_zero_int )
% 12.85/7.00 => ( P @ zero_zero_nat ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6590_le__nat__iff,axiom,
% 12.85/7.00 ! [K: int,N: nat] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.85/7.00 => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 12.85/7.00 = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6591_nat__add__distrib,axiom,
% 12.85/7.00 ! [Z: int,Z6: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.85/7.00 => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 12.85/7.00 => ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
% 12.85/7.00 = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6592_nat__mult__distrib,axiom,
% 12.85/7.00 ! [Z: int,Z6: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.85/7.00 => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
% 12.85/7.00 = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6593_Suc__as__int,axiom,
% 12.85/7.00 ( suc
% 12.85/7.00 = ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6594_nat__diff__distrib,axiom,
% 12.85/7.00 ! [Z6: int,Z: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 12.85/7.00 => ( ( ord_less_eq_int @ Z6 @ Z )
% 12.85/7.00 => ( ( nat2 @ ( minus_minus_int @ Z @ Z6 ) )
% 12.85/7.00 = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6595_nat__diff__distrib_H,axiom,
% 12.85/7.00 ! [X: int,Y4: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/7.00 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/7.00 => ( ( nat2 @ ( minus_minus_int @ X @ Y4 ) )
% 12.85/7.00 = ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y4 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6596_nat__abs__triangle__ineq,axiom,
% 12.85/7.00 ! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6597_nat__div__distrib,axiom,
% 12.85/7.00 ! [X: int,Y4: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/7.00 => ( ( nat2 @ ( divide_divide_int @ X @ Y4 ) )
% 12.85/7.00 = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6598_nat__div__distrib_H,axiom,
% 12.85/7.00 ! [Y4: int,X: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/7.00 => ( ( nat2 @ ( divide_divide_int @ X @ Y4 ) )
% 12.85/7.00 = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6599_nat__power__eq,axiom,
% 12.85/7.00 ! [Z: int,N: nat] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.85/7.00 => ( ( nat2 @ ( power_power_int @ Z @ N ) )
% 12.85/7.00 = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6600_nat__mod__distrib,axiom,
% 12.85/7.00 ! [X: int,Y4: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/7.00 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/7.00 => ( ( nat2 @ ( modulo_modulo_int @ X @ Y4 ) )
% 12.85/7.00 = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y4 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6601_div__abs__eq__div__nat,axiom,
% 12.85/7.00 ! [K: int,L: int] :
% 12.85/7.00 ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
% 12.85/7.00 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6602_num_Osize_I4_J,axiom,
% 12.85/7.00 ( ( size_size_num @ one )
% 12.85/7.00 = zero_zero_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6603_mod__abs__eq__div__nat,axiom,
% 12.85/7.00 ! [K: int,L: int] :
% 12.85/7.00 ( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
% 12.85/7.00 = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6604_take__bit__Suc__bit0,axiom,
% 12.85/7.00 ! [N: nat,K: num] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 12.85/7.00 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6605_take__bit__Suc__bit0,axiom,
% 12.85/7.00 ! [N: nat,K: num] :
% 12.85/7.00 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 12.85/7.00 = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6606_take__bit__eq__mod,axiom,
% 12.85/7.00 ( bit_se1745604003318907178nteger
% 12.85/7.00 = ( ^ [N4: nat,A3: code_integer] : ( modulo364778990260209775nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6607_take__bit__eq__mod,axiom,
% 12.85/7.00 ( bit_se2923211474154528505it_int
% 12.85/7.00 = ( ^ [N4: nat,A3: int] : ( modulo_modulo_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6608_take__bit__eq__mod,axiom,
% 12.85/7.00 ( bit_se2925701944663578781it_nat
% 12.85/7.00 = ( ^ [N4: nat,A3: nat] : ( modulo_modulo_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6609_one__and__eq,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
% 12.85/7.00 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6610_one__and__eq,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ one_one_int @ A )
% 12.85/7.00 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6611_one__and__eq,axiom,
% 12.85/7.00 ! [A: nat] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ one_one_nat @ A )
% 12.85/7.00 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6612_and__one__eq,axiom,
% 12.85/7.00 ! [A: code_integer] :
% 12.85/7.00 ( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
% 12.85/7.00 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6613_and__one__eq,axiom,
% 12.85/7.00 ! [A: int] :
% 12.85/7.00 ( ( bit_se725231765392027082nd_int @ A @ one_one_int )
% 12.85/7.00 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6614_and__one__eq,axiom,
% 12.85/7.00 ! [A: nat] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ A @ one_one_nat )
% 12.85/7.00 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6615_take__bit__nat__eq__self__iff,axiom,
% 12.85/7.00 ! [N: nat,M: nat] :
% 12.85/7.00 ( ( ( bit_se2925701944663578781it_nat @ N @ M )
% 12.85/7.00 = M )
% 12.85/7.00 = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6616_take__bit__nat__less__exp,axiom,
% 12.85/7.00 ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6617_take__bit__nat__eq__self,axiom,
% 12.85/7.00 ! [M: nat,N: nat] :
% 12.85/7.00 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.00 => ( ( bit_se2925701944663578781it_nat @ N @ M )
% 12.85/7.00 = M ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6618_real__less__rsqrt,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y4 )
% 12.85/7.00 => ( ord_less_real @ X @ ( sqrt @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6619_real__le__rsqrt,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y4 )
% 12.85/7.00 => ( ord_less_eq_real @ X @ ( sqrt @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6620_sqrt__le__D,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( sqrt @ X ) @ Y4 )
% 12.85/7.00 => ( ord_less_eq_real @ X @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6621_exp__le,axiom,
% 12.85/7.00 ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6622_take__bit__nat__def,axiom,
% 12.85/7.00 ( bit_se2925701944663578781it_nat
% 12.85/7.00 = ( ^ [N4: nat,M5: nat] : ( modulo_modulo_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6623_nat__2,axiom,
% 12.85/7.00 ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/7.00 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6624_sgn__power__injE,axiom,
% 12.85/7.00 ! [A: real,N: nat,X: real,B: real] :
% 12.85/7.00 ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 12.85/7.00 = X )
% 12.85/7.00 => ( ( X
% 12.85/7.00 = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
% 12.85/7.00 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.00 => ( A = B ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6625_take__bit__int__less__exp,axiom,
% 12.85/7.00 ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6626_Suc__nat__eq__nat__zadd1,axiom,
% 12.85/7.00 ! [Z: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.85/7.00 => ( ( suc @ ( nat2 @ Z ) )
% 12.85/7.00 = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6627_nat__less__iff,axiom,
% 12.85/7.00 ! [W: int,M: nat] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 12.85/7.00 => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
% 12.85/7.00 = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6628_take__bit__int__def,axiom,
% 12.85/7.00 ( bit_se2923211474154528505it_int
% 12.85/7.00 = ( ^ [N4: nat,K3: int] : ( modulo_modulo_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6629_nat__mult__distrib__neg,axiom,
% 12.85/7.00 ! [Z: int,Z6: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 12.85/7.00 => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
% 12.85/7.00 = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6630_exp__divide__power__eq,axiom,
% 12.85/7.00 ! [N: nat,X: complex] :
% 12.85/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.00 => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
% 12.85/7.00 = ( exp_complex @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6631_exp__divide__power__eq,axiom,
% 12.85/7.00 ! [N: nat,X: real] :
% 12.85/7.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.00 => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
% 12.85/7.00 = ( exp_real @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6632_nat__abs__int__diff,axiom,
% 12.85/7.00 ! [A: nat,B: nat] :
% 12.85/7.00 ( ( ( ord_less_eq_nat @ A @ B )
% 12.85/7.00 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 12.85/7.00 = ( minus_minus_nat @ B @ A ) ) )
% 12.85/7.00 & ( ~ ( ord_less_eq_nat @ A @ B )
% 12.85/7.00 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 12.85/7.00 = ( minus_minus_nat @ A @ B ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6633_tanh__altdef,axiom,
% 12.85/7.00 ( tanh_real
% 12.85/7.00 = ( ^ [X3: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) @ ( plus_plus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6634_tanh__altdef,axiom,
% 12.85/7.00 ( tanh_complex
% 12.85/7.00 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6635_take__bit__eq__0__iff,axiom,
% 12.85/7.00 ! [N: nat,A: code_integer] :
% 12.85/7.00 ( ( ( bit_se1745604003318907178nteger @ N @ A )
% 12.85/7.00 = zero_z3403309356797280102nteger )
% 12.85/7.00 = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6636_take__bit__eq__0__iff,axiom,
% 12.85/7.00 ! [N: nat,A: int] :
% 12.85/7.00 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 12.85/7.00 = zero_zero_int )
% 12.85/7.00 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6637_take__bit__eq__0__iff,axiom,
% 12.85/7.00 ! [N: nat,A: nat] :
% 12.85/7.00 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 12.85/7.00 = zero_zero_nat )
% 12.85/7.00 = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6638_take__bit__numeral__bit0,axiom,
% 12.85/7.00 ! [L: num,K: num] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 12.85/7.00 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6639_take__bit__numeral__bit0,axiom,
% 12.85/7.00 ! [L: num,K: num] :
% 12.85/7.00 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 12.85/7.00 = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6640_take__bit__nat__less__self__iff,axiom,
% 12.85/7.00 ! [N: nat,M: nat] :
% 12.85/7.00 ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
% 12.85/7.00 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6641_real__sqrt__unique,axiom,
% 12.85/7.00 ! [Y4: real,X: real] :
% 12.85/7.00 ( ( ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.00 = X )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ( sqrt @ X )
% 12.85/7.00 = Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6642_real__le__lsqrt,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ord_less_eq_real @ ( sqrt @ X ) @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6643_lemma__real__divide__sqrt__less,axiom,
% 12.85/7.00 ! [U: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ U )
% 12.85/7.00 => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6644_real__sqrt__sum__squares__eq__cancel2,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.00 = Y4 )
% 12.85/7.00 => ( X = zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6645_real__sqrt__sum__squares__eq__cancel,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.00 = X )
% 12.85/7.00 => ( Y4 = zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6646_real__sqrt__sum__squares__triangle__ineq,axiom,
% 12.85/7.00 ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6647_real__sqrt__sum__squares__ge2,axiom,
% 12.85/7.00 ! [Y4: real,X: real] : ( ord_less_eq_real @ Y4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6648_real__sqrt__sum__squares__ge1,axiom,
% 12.85/7.00 ! [X: real,Y4: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6649_exp__half__le2,axiom,
% 12.85/7.00 ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6650_sqrt__ge__absD,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y4 ) )
% 12.85/7.00 => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6651_take__bit__Suc__minus__bit0,axiom,
% 12.85/7.00 ! [N: nat,K: num] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 12.85/7.00 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6652_take__bit__int__greater__eq__self__iff,axiom,
% 12.85/7.00 ! [K: int,N: nat] :
% 12.85/7.00 ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 12.85/7.00 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6653_take__bit__int__less__self__iff,axiom,
% 12.85/7.00 ! [N: nat,K: int] :
% 12.85/7.00 ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 12.85/7.00 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6654_exp__double,axiom,
% 12.85/7.00 ! [Z: complex] :
% 12.85/7.00 ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) )
% 12.85/7.00 = ( power_power_complex @ ( exp_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6655_exp__double,axiom,
% 12.85/7.00 ! [Z: real] :
% 12.85/7.00 ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) )
% 12.85/7.00 = ( power_power_real @ ( exp_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6656_nat__dvd__iff,axiom,
% 12.85/7.00 ! [Z: int,M: nat] :
% 12.85/7.00 ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
% 12.85/7.00 = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.85/7.00 => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
% 12.85/7.00 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.85/7.00 => ( M = zero_zero_nat ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6657_real__less__lsqrt,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ( ord_less_real @ X @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ord_less_real @ ( sqrt @ X ) @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6658_sqrt__sum__squares__le__sum,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6659_sqrt__sum__squares__le__sum__abs,axiom,
% 12.85/7.00 ! [X: real,Y4: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6660_real__sqrt__ge__abs2,axiom,
% 12.85/7.00 ! [Y4: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6661_real__sqrt__ge__abs1,axiom,
% 12.85/7.00 ! [X: real,Y4: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6662_sqrt__even__pow2,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/7.00 => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.00 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6663_ln__sqrt,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ln_ln_real @ ( sqrt @ X ) )
% 12.85/7.00 = ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6664_arsinh__real__def,axiom,
% 12.85/7.00 ( arsinh_real
% 12.85/7.00 = ( ^ [X3: real] : ( ln_ln_real @ ( plus_plus_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6665_take__bit__int__eq__self,axiom,
% 12.85/7.00 ! [K: int,N: nat] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.85/7.00 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.00 => ( ( bit_se2923211474154528505it_int @ N @ K )
% 12.85/7.00 = K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6666_take__bit__int__eq__self__iff,axiom,
% 12.85/7.00 ! [N: nat,K: int] :
% 12.85/7.00 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 12.85/7.00 = K )
% 12.85/7.00 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.85/7.00 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6667_and__int__rec,axiom,
% 12.85/7.00 ( bit_se725231765392027082nd_int
% 12.85/7.00 = ( ^ [K3: int,L3: int] :
% 12.85/7.00 ( plus_plus_int
% 12.85/7.00 @ ( zero_n2684676970156552555ol_int
% 12.85/7.00 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 12.85/7.00 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
% 12.85/7.00 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6668_take__bit__numeral__minus__bit0,axiom,
% 12.85/7.00 ! [L: num,K: num] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 12.85/7.00 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6669_take__bit__incr__eq,axiom,
% 12.85/7.00 ! [N: nat,K: int] :
% 12.85/7.00 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 12.85/7.00 != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 12.85/7.00 => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 12.85/7.00 = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6670_take__bit__eq__mask__iff__exp__dvd,axiom,
% 12.85/7.00 ! [N: nat,K: int] :
% 12.85/7.00 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 12.85/7.00 = ( bit_se2000444600071755411sk_int @ N ) )
% 12.85/7.00 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6671_exp__bound__half,axiom,
% 12.85/7.00 ! [Z: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6672_exp__bound__half,axiom,
% 12.85/7.00 ! [Z: complex] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6673_take__bit__Suc__minus__1__eq,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.00 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_Code_integer ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6674_take__bit__Suc__minus__1__eq,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.00 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6675_take__bit__Suc__bit1,axiom,
% 12.85/7.00 ! [N: nat,K: num] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 12.85/7.00 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6676_take__bit__Suc__bit1,axiom,
% 12.85/7.00 ! [N: nat,K: num] :
% 12.85/7.00 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 12.85/7.00 = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6677_take__bit__numeral__minus__1__eq,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.00 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_Code_integer ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6678_take__bit__numeral__minus__1__eq,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.00 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6679_take__bit__Suc,axiom,
% 12.85/7.00 ! [N: nat,A: code_integer] :
% 12.85/7.00 ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A )
% 12.85/7.00 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6680_take__bit__Suc,axiom,
% 12.85/7.00 ! [N: nat,A: int] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 12.85/7.00 = ( 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 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6681_take__bit__Suc,axiom,
% 12.85/7.00 ! [N: nat,A: nat] :
% 12.85/7.00 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A )
% 12.85/7.00 = ( 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 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6682_arsinh__real__aux,axiom,
% 12.85/7.00 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6683_exp__bound,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ one_one_real )
% 12.85/7.00 => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6684_real__sqrt__sum__squares__mult__ge__zero,axiom,
% 12.85/7.00 ! [X: real,Y4: real,Xa: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( 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 ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6685_real__sqrt__power__even,axiom,
% 12.85/7.00 ! [N: nat,X: real] :
% 12.85/7.00 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( power_power_real @ ( sqrt @ X ) @ N )
% 12.85/7.00 = ( power_power_real @ X @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6686_arith__geo__mean__sqrt,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y4 ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6687_take__bit__int__less__eq,axiom,
% 12.85/7.00 ! [N: nat,K: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 12.85/7.00 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.00 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6688_take__bit__int__greater__eq,axiom,
% 12.85/7.00 ! [K: int,N: nat] :
% 12.85/7.00 ( ( ord_less_int @ K @ zero_zero_int )
% 12.85/7.00 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6689_even__nat__iff,axiom,
% 12.85/7.00 ! [K: int] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.85/7.00 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
% 12.85/7.00 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6690_signed__take__bit__eq__take__bit__shift,axiom,
% 12.85/7.00 ( bit_ri631733984087533419it_int
% 12.85/7.00 = ( ^ [N4: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6691_num_Osize_I5_J,axiom,
% 12.85/7.00 ! [X2: num] :
% 12.85/7.00 ( ( size_size_num @ ( bit0 @ X2 ) )
% 12.85/7.00 = ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6692_stable__imp__take__bit__eq,axiom,
% 12.85/7.00 ! [A: code_integer,N: nat] :
% 12.85/7.00 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/7.00 = A )
% 12.85/7.00 => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/7.00 => ( ( bit_se1745604003318907178nteger @ N @ A )
% 12.85/7.00 = zero_z3403309356797280102nteger ) )
% 12.85/7.00 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/7.00 => ( ( bit_se1745604003318907178nteger @ N @ A )
% 12.85/7.00 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6693_stable__imp__take__bit__eq,axiom,
% 12.85/7.00 ! [A: int,N: nat] :
% 12.85/7.00 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/7.00 = A )
% 12.85/7.00 => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/7.00 => ( ( bit_se2923211474154528505it_int @ N @ A )
% 12.85/7.00 = zero_zero_int ) )
% 12.85/7.00 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/7.00 => ( ( bit_se2923211474154528505it_int @ N @ A )
% 12.85/7.00 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6694_stable__imp__take__bit__eq,axiom,
% 12.85/7.00 ! [A: nat,N: nat] :
% 12.85/7.00 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.00 = A )
% 12.85/7.00 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/7.00 => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 12.85/7.00 = zero_zero_nat ) )
% 12.85/7.00 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/7.00 => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 12.85/7.00 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6695_num_Osize_I6_J,axiom,
% 12.85/7.00 ! [X33: num] :
% 12.85/7.00 ( ( size_size_num @ ( bit1 @ X33 ) )
% 12.85/7.00 = ( plus_plus_nat @ ( size_size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6696_take__bit__numeral__bit1,axiom,
% 12.85/7.00 ! [L: num,K: num] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 12.85/7.00 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6697_take__bit__numeral__bit1,axiom,
% 12.85/7.00 ! [L: num,K: num] :
% 12.85/7.00 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 12.85/7.00 = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6698_real__exp__bound__lemma,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6699_cos__x__y__le__one,axiom,
% 12.85/7.00 ! [X: real,Y4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6700_real__sqrt__sum__squares__less,axiom,
% 12.85/7.00 ! [X: real,U: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 12.85/7.00 => ( ( ord_less_real @ ( abs_abs_real @ Y4 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 12.85/7.00 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6701_arcosh__real__def,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ one_one_real @ X )
% 12.85/7.00 => ( ( arcosh_real @ X )
% 12.85/7.00 = ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6702_exp__ge__one__plus__x__over__n__power__n,axiom,
% 12.85/7.00 ! [N: nat,X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
% 12.85/7.00 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.00 => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6703_exp__ge__one__minus__x__over__n__power__n,axiom,
% 12.85/7.00 ! [X: real,N: nat] :
% 12.85/7.00 ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 12.85/7.00 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.00 => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6704_take__bit__minus__small__eq,axiom,
% 12.85/7.00 ! [K: int,N: nat] :
% 12.85/7.00 ( ( ord_less_int @ zero_zero_int @ K )
% 12.85/7.00 => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.00 => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
% 12.85/7.00 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6705_exp__bound__lemma,axiom,
% 12.85/7.00 ! [Z: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( 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 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6706_exp__bound__lemma,axiom,
% 12.85/7.00 ! [Z: complex] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( 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 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6707_modulo__int__def,axiom,
% 12.85/7.00 ( modulo_modulo_int
% 12.85/7.00 = ( ^ [K3: int,L3: int] :
% 12.85/7.00 ( if_int @ ( L3 = zero_zero_int ) @ K3
% 12.85/7.00 @ ( if_int
% 12.85/7.00 @ ( ( sgn_sgn_int @ K3 )
% 12.85/7.00 = ( sgn_sgn_int @ L3 ) )
% 12.85/7.00 @ ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) ) )
% 12.85/7.00 @ ( times_times_int @ ( sgn_sgn_int @ L3 )
% 12.85/7.00 @ ( minus_minus_int
% 12.85/7.00 @ ( times_times_int @ ( abs_abs_int @ L3 )
% 12.85/7.00 @ ( zero_n2684676970156552555ol_int
% 12.85/7.00 @ ~ ( dvd_dvd_int @ L3 @ K3 ) ) )
% 12.85/7.00 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6708_sqrt__sum__squares__half__less,axiom,
% 12.85/7.00 ! [X: real,U: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ( ord_less_real @ Y4 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6709_exp__lower__Taylor__quadratic,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( divide_divide_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6710_take__bit__numeral__minus__bit1,axiom,
% 12.85/7.00 ! [L: num,K: num] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 12.85/7.00 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6711_take__bit__Suc__minus__bit1,axiom,
% 12.85/7.00 ! [N: nat,K: num] :
% 12.85/7.00 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 12.85/7.00 = ( 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 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6712_foldr__same__int,axiom,
% 12.85/7.00 ! [Xs: list_nat,Y4: nat] :
% 12.85/7.00 ( ! [X4: nat,Y: nat] :
% 12.85/7.00 ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 12.85/7.00 => ( ( member_nat @ Y @ ( set_nat2 @ Xs ) )
% 12.85/7.00 => ( X4 = Y ) ) )
% 12.85/7.00 => ( ! [X4: nat] :
% 12.85/7.00 ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 12.85/7.00 => ( X4 = Y4 ) )
% 12.85/7.00 => ( ( foldr_nat_nat @ plus_plus_nat @ Xs @ zero_zero_nat )
% 12.85/7.00 = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6713_log__base__10__eq1,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 12.85/7.00 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6714_log__base__10__eq2,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 12.85/7.00 = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6715_machin,axiom,
% 12.85/7.00 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.85/7.00 = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6716_foldr__one,axiom,
% 12.85/7.00 ! [D: nat,Ys: list_nat] : ( ord_less_eq_nat @ D @ ( foldr_nat_nat @ plus_plus_nat @ Ys @ D ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6717_log__one,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( log @ A @ one_one_real )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6718_pred__numeral__inc,axiom,
% 12.85/7.00 ! [K: num] :
% 12.85/7.00 ( ( pred_numeral @ ( inc @ K ) )
% 12.85/7.00 = ( numeral_numeral_nat @ K ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6719_log__eq__one,axiom,
% 12.85/7.00 ! [A: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.00 => ( ( A != one_one_real )
% 12.85/7.00 => ( ( log @ A @ A )
% 12.85/7.00 = one_one_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6720_log__less__cancel__iff,axiom,
% 12.85/7.00 ! [A: real,X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_real @ one_one_real @ A )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y4 ) )
% 12.85/7.00 = ( ord_less_real @ X @ Y4 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6721_log__less__one__cancel__iff,axiom,
% 12.85/7.00 ! [A: real,X: real] :
% 12.85/7.00 ( ( ord_less_real @ one_one_real @ A )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_real @ ( log @ A @ X ) @ one_one_real )
% 12.85/7.00 = ( ord_less_real @ X @ A ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6722_one__less__log__cancel__iff,axiom,
% 12.85/7.00 ! [A: real,X: real] :
% 12.85/7.00 ( ( ord_less_real @ one_one_real @ A )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_real @ one_one_real @ ( log @ A @ X ) )
% 12.85/7.00 = ( ord_less_real @ A @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6723_log__less__zero__cancel__iff,axiom,
% 12.85/7.00 ! [A: real,X: real] :
% 12.85/7.00 ( ( ord_less_real @ one_one_real @ A )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_real @ ( log @ A @ X ) @ zero_zero_real )
% 12.85/7.00 = ( ord_less_real @ X @ one_one_real ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6724_zero__less__log__cancel__iff,axiom,
% 12.85/7.00 ! [A: real,X: real] :
% 12.85/7.00 ( ( ord_less_real @ one_one_real @ A )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X ) )
% 12.85/7.00 = ( ord_less_real @ one_one_real @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6725_log__le__cancel__iff,axiom,
% 12.85/7.00 ! [A: real,X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_real @ one_one_real @ A )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y4 ) )
% 12.85/7.00 = ( ord_less_eq_real @ X @ Y4 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6726_log__le__one__cancel__iff,axiom,
% 12.85/7.00 ! [A: real,X: real] :
% 12.85/7.00 ( ( ord_less_real @ one_one_real @ A )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
% 12.85/7.00 = ( ord_less_eq_real @ X @ A ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6727_one__le__log__cancel__iff,axiom,
% 12.85/7.00 ! [A: real,X: real] :
% 12.85/7.00 ( ( ord_less_real @ one_one_real @ A )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
% 12.85/7.00 = ( ord_less_eq_real @ A @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6728_log__le__zero__cancel__iff,axiom,
% 12.85/7.00 ! [A: real,X: real] :
% 12.85/7.00 ( ( ord_less_real @ one_one_real @ A )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
% 12.85/7.00 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6729_zero__le__log__cancel__iff,axiom,
% 12.85/7.00 ! [A: real,X: real] :
% 12.85/7.00 ( ( ord_less_real @ one_one_real @ A )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
% 12.85/7.00 = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6730_add__neg__numeral__special_I6_J,axiom,
% 12.85/7.00 ! [M: num] :
% 12.85/7.00 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 12.85/7.00 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6731_add__neg__numeral__special_I6_J,axiom,
% 12.85/7.00 ! [M: num] :
% 12.85/7.00 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.00 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6732_add__neg__numeral__special_I6_J,axiom,
% 12.85/7.00 ! [M: num] :
% 12.85/7.00 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.85/7.00 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6733_add__neg__numeral__special_I6_J,axiom,
% 12.85/7.00 ! [M: num] :
% 12.85/7.00 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.00 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6734_add__neg__numeral__special_I6_J,axiom,
% 12.85/7.00 ! [M: num] :
% 12.85/7.00 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.85/7.00 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6735_add__neg__numeral__special_I5_J,axiom,
% 12.85/7.00 ! [N: num] :
% 12.85/7.00 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.85/7.00 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6736_add__neg__numeral__special_I5_J,axiom,
% 12.85/7.00 ! [N: num] :
% 12.85/7.00 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.00 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6737_add__neg__numeral__special_I5_J,axiom,
% 12.85/7.00 ! [N: num] :
% 12.85/7.00 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 12.85/7.00 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6738_add__neg__numeral__special_I5_J,axiom,
% 12.85/7.00 ! [N: num] :
% 12.85/7.00 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.85/7.00 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6739_add__neg__numeral__special_I5_J,axiom,
% 12.85/7.00 ! [N: num] :
% 12.85/7.00 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.85/7.00 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6740_diff__numeral__special_I6_J,axiom,
% 12.85/7.00 ! [M: num] :
% 12.85/7.00 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) )
% 12.85/7.00 = ( numeral_numeral_real @ ( inc @ M ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6741_diff__numeral__special_I6_J,axiom,
% 12.85/7.00 ! [M: num] :
% 12.85/7.00 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.00 = ( numeral_numeral_int @ ( inc @ M ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6742_diff__numeral__special_I6_J,axiom,
% 12.85/7.00 ! [M: num] :
% 12.85/7.00 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.85/7.00 = ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6743_diff__numeral__special_I6_J,axiom,
% 12.85/7.00 ! [M: num] :
% 12.85/7.00 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.00 = ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6744_diff__numeral__special_I6_J,axiom,
% 12.85/7.00 ! [M: num] :
% 12.85/7.00 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 12.85/7.00 = ( numeral_numeral_rat @ ( inc @ M ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6745_diff__numeral__special_I5_J,axiom,
% 12.85/7.00 ! [N: num] :
% 12.85/7.00 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N ) )
% 12.85/7.00 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6746_diff__numeral__special_I5_J,axiom,
% 12.85/7.00 ! [N: num] :
% 12.85/7.00 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 12.85/7.00 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6747_diff__numeral__special_I5_J,axiom,
% 12.85/7.00 ! [N: num] :
% 12.85/7.00 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N ) )
% 12.85/7.00 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6748_diff__numeral__special_I5_J,axiom,
% 12.85/7.00 ! [N: num] :
% 12.85/7.00 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
% 12.85/7.00 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6749_diff__numeral__special_I5_J,axiom,
% 12.85/7.00 ! [N: num] :
% 12.85/7.00 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
% 12.85/7.00 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6750_and__nat__numerals_I3_J,axiom,
% 12.85/7.00 ! [X: num] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 12.85/7.00 = zero_zero_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6751_and__nat__numerals_I1_J,axiom,
% 12.85/7.00 ! [Y4: num] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 12.85/7.00 = zero_zero_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6752_log__pow__cancel,axiom,
% 12.85/7.00 ! [A: real,B: nat] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.00 => ( ( A != one_one_real )
% 12.85/7.00 => ( ( log @ A @ ( power_power_real @ A @ B ) )
% 12.85/7.00 = ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6753_and__nat__numerals_I4_J,axiom,
% 12.85/7.00 ! [X: num] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 12.85/7.00 = one_one_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6754_and__nat__numerals_I2_J,axiom,
% 12.85/7.00 ! [Y4: num] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 12.85/7.00 = one_one_nat ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6755_Suc__0__and__eq,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.85/7.00 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6756_and__Suc__0__eq,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
% 12.85/7.00 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6757_pi__neq__zero,axiom,
% 12.85/7.00 pi != zero_zero_real ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6758_num__induct,axiom,
% 12.85/7.00 ! [P: num > $o,X: num] :
% 12.85/7.00 ( ( P @ one )
% 12.85/7.00 => ( ! [X4: num] :
% 12.85/7.00 ( ( P @ X4 )
% 12.85/7.00 => ( P @ ( inc @ X4 ) ) )
% 12.85/7.00 => ( P @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6759_add__inc,axiom,
% 12.85/7.00 ! [X: num,Y4: num] :
% 12.85/7.00 ( ( plus_plus_num @ X @ ( inc @ Y4 ) )
% 12.85/7.00 = ( inc @ ( plus_plus_num @ X @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6760_log__def,axiom,
% 12.85/7.00 ( log
% 12.85/7.00 = ( ^ [A3: real,X3: real] : ( divide_divide_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ A3 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6761_pi__gt__zero,axiom,
% 12.85/7.00 ord_less_real @ zero_zero_real @ pi ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6762_pi__not__less__zero,axiom,
% 12.85/7.00 ~ ( ord_less_real @ pi @ zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6763_pi__ge__zero,axiom,
% 12.85/7.00 ord_less_eq_real @ zero_zero_real @ pi ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6764_inc_Osimps_I1_J,axiom,
% 12.85/7.00 ( ( inc @ one )
% 12.85/7.00 = ( bit0 @ one ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6765_inc_Osimps_I3_J,axiom,
% 12.85/7.00 ! [X: num] :
% 12.85/7.00 ( ( inc @ ( bit1 @ X ) )
% 12.85/7.00 = ( bit0 @ ( inc @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6766_inc_Osimps_I2_J,axiom,
% 12.85/7.00 ! [X: num] :
% 12.85/7.00 ( ( inc @ ( bit0 @ X ) )
% 12.85/7.00 = ( bit1 @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6767_add__One,axiom,
% 12.85/7.00 ! [X: num] :
% 12.85/7.00 ( ( plus_plus_num @ X @ one )
% 12.85/7.00 = ( inc @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6768_and__nat__def,axiom,
% 12.85/7.00 ( bit_se727722235901077358nd_nat
% 12.85/7.00 = ( ^ [M5: nat,N4: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6769_mult__inc,axiom,
% 12.85/7.00 ! [X: num,Y4: num] :
% 12.85/7.00 ( ( times_times_num @ X @ ( inc @ Y4 ) )
% 12.85/7.00 = ( plus_plus_num @ ( times_times_num @ X @ Y4 ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6770_log__base__change,axiom,
% 12.85/7.00 ! [A: real,B: real,X: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.00 => ( ( A != one_one_real )
% 12.85/7.00 => ( ( log @ B @ X )
% 12.85/7.00 = ( divide_divide_real @ ( log @ A @ X ) @ ( log @ A @ B ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6771_log__of__power__eq,axiom,
% 12.85/7.00 ! [M: nat,B: real,N: nat] :
% 12.85/7.00 ( ( ( semiri5074537144036343181t_real @ M )
% 12.85/7.00 = ( power_power_real @ B @ N ) )
% 12.85/7.00 => ( ( ord_less_real @ one_one_real @ B )
% 12.85/7.00 => ( ( semiri5074537144036343181t_real @ N )
% 12.85/7.00 = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6772_less__log__of__power,axiom,
% 12.85/7.00 ! [B: real,N: nat,M: real] :
% 12.85/7.00 ( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
% 12.85/7.00 => ( ( ord_less_real @ one_one_real @ B )
% 12.85/7.00 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6773_numeral__inc,axiom,
% 12.85/7.00 ! [X: num] :
% 12.85/7.00 ( ( numera6690914467698888265omplex @ ( inc @ X ) )
% 12.85/7.00 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6774_numeral__inc,axiom,
% 12.85/7.00 ! [X: num] :
% 12.85/7.00 ( ( numeral_numeral_real @ ( inc @ X ) )
% 12.85/7.00 = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6775_numeral__inc,axiom,
% 12.85/7.00 ! [X: num] :
% 12.85/7.00 ( ( numeral_numeral_rat @ ( inc @ X ) )
% 12.85/7.00 = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6776_numeral__inc,axiom,
% 12.85/7.00 ! [X: num] :
% 12.85/7.00 ( ( numeral_numeral_nat @ ( inc @ X ) )
% 12.85/7.00 = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6777_numeral__inc,axiom,
% 12.85/7.00 ! [X: num] :
% 12.85/7.00 ( ( numeral_numeral_int @ ( inc @ X ) )
% 12.85/7.00 = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6778_pi__less__4,axiom,
% 12.85/7.00 ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6779_pi__ge__two,axiom,
% 12.85/7.00 ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6780_pi__half__neq__two,axiom,
% 12.85/7.00 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.85/7.00 != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6781_log__mult,axiom,
% 12.85/7.00 ! [A: real,X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.00 => ( ( A != one_one_real )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ( log @ A @ ( times_times_real @ X @ Y4 ) )
% 12.85/7.00 = ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y4 ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6782_log__divide,axiom,
% 12.85/7.00 ! [A: real,X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.00 => ( ( A != one_one_real )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ( log @ A @ ( divide_divide_real @ X @ Y4 ) )
% 12.85/7.00 = ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y4 ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6783_le__log__of__power,axiom,
% 12.85/7.00 ! [B: real,N: nat,M: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
% 12.85/7.00 => ( ( ord_less_real @ one_one_real @ B )
% 12.85/7.00 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6784_log__base__pow,axiom,
% 12.85/7.00 ! [A: real,N: nat,X: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.00 => ( ( log @ ( power_power_real @ A @ N ) @ X )
% 12.85/7.00 = ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6785_log__nat__power,axiom,
% 12.85/7.00 ! [X: real,B: real,N: nat] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( log @ B @ ( power_power_real @ X @ N ) )
% 12.85/7.00 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6786_pi__half__neq__zero,axiom,
% 12.85/7.00 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.85/7.00 != zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6787_pi__half__less__two,axiom,
% 12.85/7.00 ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6788_pi__half__le__two,axiom,
% 12.85/7.00 ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6789_log2__of__power__eq,axiom,
% 12.85/7.00 ! [M: nat,N: nat] :
% 12.85/7.00 ( ( M
% 12.85/7.00 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.00 => ( ( semiri5074537144036343181t_real @ N )
% 12.85/7.00 = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6790_log__of__power__less,axiom,
% 12.85/7.00 ! [M: nat,B: real,N: nat] :
% 12.85/7.00 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 12.85/7.00 => ( ( ord_less_real @ one_one_real @ B )
% 12.85/7.00 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/7.00 => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6791_log__eq__div__ln__mult__log,axiom,
% 12.85/7.00 ! [A: real,B: real,X: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.00 => ( ( A != one_one_real )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ B )
% 12.85/7.00 => ( ( B != one_one_real )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( log @ A @ X )
% 12.85/7.00 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6792_pi__half__gt__zero,axiom,
% 12.85/7.00 ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6793_pi__half__ge__zero,axiom,
% 12.85/7.00 ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6794_m2pi__less__pi,axiom,
% 12.85/7.00 ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6795_arctan__ubound,axiom,
% 12.85/7.00 ! [Y4: real] : ( ord_less_real @ ( arctan @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6796_arctan__one,axiom,
% 12.85/7.00 ( ( arctan @ one_one_real )
% 12.85/7.00 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6797_log__of__power__le,axiom,
% 12.85/7.00 ! [M: nat,B: real,N: nat] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 12.85/7.00 => ( ( ord_less_real @ one_one_real @ B )
% 12.85/7.00 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/7.00 => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6798_minus__pi__half__less__zero,axiom,
% 12.85/7.00 ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6799_arctan__lbound,axiom,
% 12.85/7.00 ! [Y4: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6800_arctan__bounded,axiom,
% 12.85/7.00 ! [Y4: real] :
% 12.85/7.00 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y4 ) )
% 12.85/7.00 & ( ord_less_real @ ( arctan @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6801_and__nat__unfold,axiom,
% 12.85/7.00 ( bit_se727722235901077358nd_nat
% 12.85/7.00 = ( ^ [M5: nat,N4: nat] :
% 12.85/7.00 ( if_nat
% 12.85/7.00 @ ( ( M5 = zero_zero_nat )
% 12.85/7.00 | ( N4 = zero_zero_nat ) )
% 12.85/7.00 @ zero_zero_nat
% 12.85/7.00 @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6802_less__log2__of__power,axiom,
% 12.85/7.00 ! [N: nat,M: nat] :
% 12.85/7.00 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 12.85/7.00 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6803_le__log2__of__power,axiom,
% 12.85/7.00 ! [N: nat,M: nat] :
% 12.85/7.00 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 12.85/7.00 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6804_and__nat__rec,axiom,
% 12.85/7.00 ( bit_se727722235901077358nd_nat
% 12.85/7.00 = ( ^ [M5: nat,N4: nat] :
% 12.85/7.00 ( plus_plus_nat
% 12.85/7.00 @ ( zero_n2687167440665602831ol_nat
% 12.85/7.00 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 )
% 12.85/7.00 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
% 12.85/7.00 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6805_log2__of__power__less,axiom,
% 12.85/7.00 ! [M: nat,N: nat] :
% 12.85/7.00 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.00 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/7.00 => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6806_log2__of__power__le,axiom,
% 12.85/7.00 ! [M: nat,N: nat] :
% 12.85/7.00 ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.00 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/7.00 => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6807_machin__Euler,axiom,
% 12.85/7.00 ( ( 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 ) ) ) ) ) ) ) ) ) ) )
% 12.85/7.00 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6808_arctan__inverse,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( X != zero_zero_real )
% 12.85/7.00 => ( ( arctan @ ( divide_divide_real @ one_one_real @ X ) )
% 12.85/7.00 = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6809_sin__cos__npi,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( 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 ) ) ) )
% 12.85/7.00 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6810_ceiling__log__nat__eq__powr__iff,axiom,
% 12.85/7.00 ! [B: nat,K: nat,N: nat] :
% 12.85/7.00 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 12.85/7.00 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/7.00 => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 12.85/7.00 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
% 12.85/7.00 = ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 12.85/7.00 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6811_cos__pi__eq__zero,axiom,
% 12.85/7.00 ! [M: nat] :
% 12.85/7.00 ( ( 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 ) ) ) )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6812_ceiling__log__nat__eq__if,axiom,
% 12.85/7.00 ! [B: nat,N: nat,K: nat] :
% 12.85/7.00 ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 12.85/7.00 => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 12.85/7.00 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 12.85/7.00 => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 12.85/7.00 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6813_ceiling__log2__div2,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/7.00 => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 12.85/7.00 = ( 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 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6814_floor__log__nat__eq__powr__iff,axiom,
% 12.85/7.00 ! [B: nat,K: nat,N: nat] :
% 12.85/7.00 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 12.85/7.00 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/7.00 => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 12.85/7.00 = ( semiri1314217659103216013at_int @ N ) )
% 12.85/7.00 = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 12.85/7.00 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6815_sin__zero,axiom,
% 12.85/7.00 ( ( sin_complex @ zero_zero_complex )
% 12.85/7.00 = zero_zero_complex ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6816_sin__zero,axiom,
% 12.85/7.00 ( ( sin_real @ zero_zero_real )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6817_cos__zero,axiom,
% 12.85/7.00 ( ( cos_complex @ zero_zero_complex )
% 12.85/7.00 = one_one_complex ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6818_cos__zero,axiom,
% 12.85/7.00 ( ( cos_real @ zero_zero_real )
% 12.85/7.00 = one_one_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6819_floor__zero,axiom,
% 12.85/7.00 ( ( archim6058952711729229775r_real @ zero_zero_real )
% 12.85/7.00 = zero_zero_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6820_floor__zero,axiom,
% 12.85/7.00 ( ( archim3151403230148437115or_rat @ zero_zero_rat )
% 12.85/7.00 = zero_zero_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6821_floor__numeral,axiom,
% 12.85/7.00 ! [V: num] :
% 12.85/7.00 ( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V ) )
% 12.85/7.00 = ( numeral_numeral_int @ V ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6822_floor__numeral,axiom,
% 12.85/7.00 ! [V: num] :
% 12.85/7.00 ( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
% 12.85/7.00 = ( numeral_numeral_int @ V ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6823_ceiling__zero,axiom,
% 12.85/7.00 ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
% 12.85/7.00 = zero_zero_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6824_ceiling__zero,axiom,
% 12.85/7.00 ( ( archim7802044766580827645g_real @ zero_zero_real )
% 12.85/7.00 = zero_zero_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6825_ceiling__numeral,axiom,
% 12.85/7.00 ! [V: num] :
% 12.85/7.00 ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
% 12.85/7.00 = ( numeral_numeral_int @ V ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6826_ceiling__numeral,axiom,
% 12.85/7.00 ! [V: num] :
% 12.85/7.00 ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
% 12.85/7.00 = ( numeral_numeral_int @ V ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6827_sin__pi,axiom,
% 12.85/7.00 ( ( sin_real @ pi )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6828_floor__of__nat,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( archim6058952711729229775r_real @ ( semiri5074537144036343181t_real @ N ) )
% 12.85/7.00 = ( semiri1314217659103216013at_int @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6829_floor__of__nat,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( archim3151403230148437115or_rat @ ( semiri681578069525770553at_rat @ N ) )
% 12.85/7.00 = ( semiri1314217659103216013at_int @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6830_ceiling__of__nat,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( archim7802044766580827645g_real @ ( semiri5074537144036343181t_real @ N ) )
% 12.85/7.00 = ( semiri1314217659103216013at_int @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6831_ceiling__of__nat,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( archim2889992004027027881ng_rat @ ( semiri681578069525770553at_rat @ N ) )
% 12.85/7.00 = ( semiri1314217659103216013at_int @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6832_sin__pi__minus,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( sin_real @ ( minus_minus_real @ pi @ X ) )
% 12.85/7.00 = ( sin_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6833_cos__minus__pi,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( cos_real @ ( minus_minus_real @ X @ pi ) )
% 12.85/7.00 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6834_cos__pi__minus,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( cos_real @ ( minus_minus_real @ pi @ X ) )
% 12.85/7.00 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6835_sin__minus__pi,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( sin_real @ ( minus_minus_real @ X @ pi ) )
% 12.85/7.00 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6836_zero__le__floor,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
% 12.85/7.00 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6837_zero__le__floor,axiom,
% 12.85/7.00 ! [X: rat] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
% 12.85/7.00 = ( ord_less_eq_rat @ zero_zero_rat @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6838_sin__cos__squared__add3,axiom,
% 12.85/7.00 ! [X: complex] :
% 12.85/7.00 ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ X ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ X ) ) )
% 12.85/7.00 = one_one_complex ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6839_sin__cos__squared__add3,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ X ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ X ) ) )
% 12.85/7.00 = one_one_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6840_floor__less__zero,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
% 12.85/7.00 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6841_floor__less__zero,axiom,
% 12.85/7.00 ! [X: rat] :
% 12.85/7.00 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
% 12.85/7.00 = ( ord_less_rat @ X @ zero_zero_rat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6842_numeral__le__floor,axiom,
% 12.85/7.00 ! [V: num,X: real] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
% 12.85/7.00 = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6843_numeral__le__floor,axiom,
% 12.85/7.00 ! [V: num,X: rat] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
% 12.85/7.00 = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6844_zero__less__floor,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
% 12.85/7.00 = ( ord_less_eq_real @ one_one_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6845_zero__less__floor,axiom,
% 12.85/7.00 ! [X: rat] :
% 12.85/7.00 ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
% 12.85/7.00 = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6846_floor__le__zero,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
% 12.85/7.00 = ( ord_less_real @ X @ one_one_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6847_floor__le__zero,axiom,
% 12.85/7.00 ! [X: rat] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
% 12.85/7.00 = ( ord_less_rat @ X @ one_one_rat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6848_ceiling__le__zero,axiom,
% 12.85/7.00 ! [X: rat] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
% 12.85/7.00 = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6849_ceiling__le__zero,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
% 12.85/7.00 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6850_floor__less__numeral,axiom,
% 12.85/7.00 ! [X: real,V: num] :
% 12.85/7.00 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
% 12.85/7.00 = ( ord_less_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6851_floor__less__numeral,axiom,
% 12.85/7.00 ! [X: rat,V: num] :
% 12.85/7.00 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 12.85/7.00 = ( ord_less_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6852_zero__less__ceiling,axiom,
% 12.85/7.00 ! [X: rat] :
% 12.85/7.00 ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
% 12.85/7.00 = ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6853_zero__less__ceiling,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
% 12.85/7.00 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6854_one__le__floor,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
% 12.85/7.00 = ( ord_less_eq_real @ one_one_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6855_one__le__floor,axiom,
% 12.85/7.00 ! [X: rat] :
% 12.85/7.00 ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
% 12.85/7.00 = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6856_ceiling__le__numeral,axiom,
% 12.85/7.00 ! [X: rat,V: num] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 12.85/7.00 = ( ord_less_eq_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6857_ceiling__le__numeral,axiom,
% 12.85/7.00 ! [X: real,V: num] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
% 12.85/7.00 = ( ord_less_eq_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6858_ceiling__less__one,axiom,
% 12.85/7.00 ! [X: rat] :
% 12.85/7.00 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
% 12.85/7.00 = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6859_ceiling__less__one,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
% 12.85/7.00 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6860_one__le__ceiling,axiom,
% 12.85/7.00 ! [X: rat] :
% 12.85/7.00 ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
% 12.85/7.00 = ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6861_one__le__ceiling,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
% 12.85/7.00 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6862_numeral__less__ceiling,axiom,
% 12.85/7.00 ! [V: num,X: real] :
% 12.85/7.00 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
% 12.85/7.00 = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6863_numeral__less__ceiling,axiom,
% 12.85/7.00 ! [V: num,X: rat] :
% 12.85/7.00 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
% 12.85/7.00 = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6864_floor__neg__numeral,axiom,
% 12.85/7.00 ! [V: num] :
% 12.85/7.00 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 12.85/7.00 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6865_floor__neg__numeral,axiom,
% 12.85/7.00 ! [V: num] :
% 12.85/7.00 ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 12.85/7.00 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6866_ceiling__le__one,axiom,
% 12.85/7.00 ! [X: rat] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
% 12.85/7.00 = ( ord_less_eq_rat @ X @ one_one_rat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6867_ceiling__le__one,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
% 12.85/7.00 = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6868_ceiling__add__numeral,axiom,
% 12.85/7.00 ! [X: real,V: num] :
% 12.85/7.00 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ V ) ) )
% 12.85/7.00 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6869_ceiling__add__numeral,axiom,
% 12.85/7.00 ! [X: rat,V: num] :
% 12.85/7.00 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
% 12.85/7.00 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6870_floor__diff__numeral,axiom,
% 12.85/7.00 ! [X: real,V: num] :
% 12.85/7.00 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
% 12.85/7.00 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6871_floor__diff__numeral,axiom,
% 12.85/7.00 ! [X: rat,V: num] :
% 12.85/7.00 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
% 12.85/7.00 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6872_ceiling__neg__numeral,axiom,
% 12.85/7.00 ! [V: num] :
% 12.85/7.00 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 12.85/7.00 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6873_ceiling__neg__numeral,axiom,
% 12.85/7.00 ! [V: num] :
% 12.85/7.00 ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 12.85/7.00 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6874_floor__diff__one,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ one_one_real ) )
% 12.85/7.00 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6875_floor__diff__one,axiom,
% 12.85/7.00 ! [X: rat] :
% 12.85/7.00 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
% 12.85/7.00 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6876_ceiling__diff__numeral,axiom,
% 12.85/7.00 ! [X: real,V: num] :
% 12.85/7.00 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
% 12.85/7.00 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6877_ceiling__diff__numeral,axiom,
% 12.85/7.00 ! [X: rat,V: num] :
% 12.85/7.00 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
% 12.85/7.00 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6878_ceiling__diff__one,axiom,
% 12.85/7.00 ! [X: rat] :
% 12.85/7.00 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
% 12.85/7.00 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6879_ceiling__diff__one,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ one_one_real ) )
% 12.85/7.00 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6880_floor__numeral__power,axiom,
% 12.85/7.00 ! [X: num,N: nat] :
% 12.85/7.00 ( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 12.85/7.00 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6881_floor__numeral__power,axiom,
% 12.85/7.00 ! [X: num,N: nat] :
% 12.85/7.00 ( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 12.85/7.00 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6882_ceiling__numeral__power,axiom,
% 12.85/7.00 ! [X: num,N: nat] :
% 12.85/7.00 ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 12.85/7.00 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6883_ceiling__numeral__power,axiom,
% 12.85/7.00 ! [X: num,N: nat] :
% 12.85/7.00 ( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 12.85/7.00 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6884_sin__npi2,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6885_sin__npi,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6886_floor__divide__eq__div__numeral,axiom,
% 12.85/7.00 ! [A: num,B: num] :
% 12.85/7.00 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 12.85/7.00 = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6887_nat__ceiling__le__eq,axiom,
% 12.85/7.00 ! [X: real,A: nat] :
% 12.85/7.00 ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
% 12.85/7.00 = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6888_ceiling__less__zero,axiom,
% 12.85/7.00 ! [X: rat] :
% 12.85/7.00 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
% 12.85/7.00 = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6889_ceiling__less__zero,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
% 12.85/7.00 = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6890_zero__le__ceiling,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
% 12.85/7.00 = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6891_zero__le__ceiling,axiom,
% 12.85/7.00 ! [X: rat] :
% 12.85/7.00 ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
% 12.85/7.00 = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6892_ceiling__divide__eq__div__numeral,axiom,
% 12.85/7.00 ! [A: num,B: num] :
% 12.85/7.00 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 12.85/7.00 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6893_numeral__less__floor,axiom,
% 12.85/7.00 ! [V: num,X: real] :
% 12.85/7.00 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
% 12.85/7.00 = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6894_numeral__less__floor,axiom,
% 12.85/7.00 ! [V: num,X: rat] :
% 12.85/7.00 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
% 12.85/7.00 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6895_floor__le__numeral,axiom,
% 12.85/7.00 ! [X: real,V: num] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
% 12.85/7.00 = ( ord_less_real @ X @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6896_floor__le__numeral,axiom,
% 12.85/7.00 ! [X: rat,V: num] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 12.85/7.00 = ( ord_less_rat @ X @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6897_one__less__floor,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
% 12.85/7.00 = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6898_one__less__floor,axiom,
% 12.85/7.00 ! [X: rat] :
% 12.85/7.00 ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
% 12.85/7.00 = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6899_floor__le__one,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
% 12.85/7.00 = ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6900_floor__le__one,axiom,
% 12.85/7.00 ! [X: rat] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
% 12.85/7.00 = ( ord_less_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6901_ceiling__less__numeral,axiom,
% 12.85/7.00 ! [X: rat,V: num] :
% 12.85/7.00 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 12.85/7.00 = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6902_ceiling__less__numeral,axiom,
% 12.85/7.00 ! [X: real,V: num] :
% 12.85/7.00 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
% 12.85/7.00 = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6903_numeral__le__ceiling,axiom,
% 12.85/7.00 ! [V: num,X: real] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
% 12.85/7.00 = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6904_numeral__le__ceiling,axiom,
% 12.85/7.00 ! [V: num,X: rat] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
% 12.85/7.00 = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6905_neg__numeral__le__floor,axiom,
% 12.85/7.00 ! [V: num,X: real] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
% 12.85/7.00 = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6906_neg__numeral__le__floor,axiom,
% 12.85/7.00 ! [V: num,X: rat] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
% 12.85/7.00 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6907_floor__less__neg__numeral,axiom,
% 12.85/7.00 ! [X: real,V: num] :
% 12.85/7.00 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.85/7.00 = ( ord_less_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6908_floor__less__neg__numeral,axiom,
% 12.85/7.00 ! [X: rat,V: num] :
% 12.85/7.00 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.85/7.00 = ( ord_less_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6909_ceiling__le__neg__numeral,axiom,
% 12.85/7.00 ! [X: rat,V: num] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.85/7.00 = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6910_ceiling__le__neg__numeral,axiom,
% 12.85/7.00 ! [X: real,V: num] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.85/7.00 = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6911_neg__numeral__less__ceiling,axiom,
% 12.85/7.00 ! [V: num,X: real] :
% 12.85/7.00 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
% 12.85/7.00 = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6912_neg__numeral__less__ceiling,axiom,
% 12.85/7.00 ! [V: num,X: rat] :
% 12.85/7.00 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
% 12.85/7.00 = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6913_cos__pi__half,axiom,
% 12.85/7.00 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6914_sin__two__pi,axiom,
% 12.85/7.00 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6915_sin__pi__half,axiom,
% 12.85/7.00 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 = one_one_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6916_cos__two__pi,axiom,
% 12.85/7.00 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 12.85/7.00 = one_one_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6917_cos__periodic,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( cos_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 12.85/7.00 = ( cos_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6918_sin__periodic,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( sin_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 12.85/7.00 = ( sin_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6919_cos__2pi__minus,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 12.85/7.00 = ( cos_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6920_cos__npi2,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 12.85/7.00 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6921_cos__npi,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 12.85/7.00 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6922_floor__one__divide__eq__div__numeral,axiom,
% 12.85/7.00 ! [B: num] :
% 12.85/7.00 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
% 12.85/7.00 = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6923_floor__minus__divide__eq__div__numeral,axiom,
% 12.85/7.00 ! [A: num,B: num] :
% 12.85/7.00 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 12.85/7.00 = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6924_ceiling__minus__divide__eq__div__numeral,axiom,
% 12.85/7.00 ! [A: num,B: num] :
% 12.85/7.00 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 12.85/7.00 = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6925_sin__cos__squared__add2,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.00 = one_one_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6926_sin__cos__squared__add2,axiom,
% 12.85/7.00 ! [X: complex] :
% 12.85/7.00 ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.00 = one_one_complex ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6927_sin__cos__squared__add,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.00 = one_one_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6928_sin__cos__squared__add,axiom,
% 12.85/7.00 ! [X: complex] :
% 12.85/7.00 ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.00 = one_one_complex ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6929_sin__2npi,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6930_cos__2npi,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 12.85/7.00 = one_one_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6931_sin__2pi__minus,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 12.85/7.00 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6932_neg__numeral__less__floor,axiom,
% 12.85/7.00 ! [V: num,X: real] :
% 12.85/7.00 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
% 12.85/7.00 = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6933_neg__numeral__less__floor,axiom,
% 12.85/7.00 ! [V: num,X: rat] :
% 12.85/7.00 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
% 12.85/7.00 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6934_floor__le__neg__numeral,axiom,
% 12.85/7.00 ! [X: real,V: num] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.85/7.00 = ( ord_less_real @ X @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6935_floor__le__neg__numeral,axiom,
% 12.85/7.00 ! [X: rat,V: num] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.85/7.00 = ( ord_less_rat @ X @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6936_ceiling__less__neg__numeral,axiom,
% 12.85/7.00 ! [X: rat,V: num] :
% 12.85/7.00 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.85/7.00 = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6937_ceiling__less__neg__numeral,axiom,
% 12.85/7.00 ! [X: real,V: num] :
% 12.85/7.00 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.85/7.00 = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6938_neg__numeral__le__ceiling,axiom,
% 12.85/7.00 ! [V: num,X: real] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
% 12.85/7.00 = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6939_neg__numeral__le__ceiling,axiom,
% 12.85/7.00 ! [V: num,X: rat] :
% 12.85/7.00 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
% 12.85/7.00 = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6940_cos__3over2__pi,axiom,
% 12.85/7.00 ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6941_floor__minus__one__divide__eq__div__numeral,axiom,
% 12.85/7.00 ! [B: num] :
% 12.85/7.00 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
% 12.85/7.00 = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6942_sin__3over2__pi,axiom,
% 12.85/7.00 ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 12.85/7.00 = ( uminus_uminus_real @ one_one_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6943_sin__add,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( sin_real @ ( plus_plus_real @ X @ Y4 ) )
% 12.85/7.00 = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y4 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6944_floor__le__ceiling,axiom,
% 12.85/7.00 ! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6945_floor__le__ceiling,axiom,
% 12.85/7.00 ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim2889992004027027881ng_rat @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6946_polar__Ex,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ? [R3: real,A4: real] :
% 12.85/7.00 ( ( X
% 12.85/7.00 = ( times_times_real @ R3 @ ( cos_real @ A4 ) ) )
% 12.85/7.00 & ( Y4
% 12.85/7.00 = ( times_times_real @ R3 @ ( sin_real @ A4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6947_sin__diff,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( sin_real @ ( minus_minus_real @ X @ Y4 ) )
% 12.85/7.00 = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y4 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6948_cos__one__sin__zero,axiom,
% 12.85/7.00 ! [X: complex] :
% 12.85/7.00 ( ( ( cos_complex @ X )
% 12.85/7.00 = one_one_complex )
% 12.85/7.00 => ( ( sin_complex @ X )
% 12.85/7.00 = zero_zero_complex ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6949_cos__one__sin__zero,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ( cos_real @ X )
% 12.85/7.00 = one_one_real )
% 12.85/7.00 => ( ( sin_real @ X )
% 12.85/7.00 = zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6950_cos__add,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( cos_real @ ( plus_plus_real @ X @ Y4 ) )
% 12.85/7.00 = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6951_cos__diff,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( cos_real @ ( minus_minus_real @ X @ Y4 ) )
% 12.85/7.00 = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6952_sin__zero__norm__cos__one,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ( sin_real @ X )
% 12.85/7.00 = zero_zero_real )
% 12.85/7.00 => ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
% 12.85/7.00 = one_one_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6953_sin__zero__norm__cos__one,axiom,
% 12.85/7.00 ! [X: complex] :
% 12.85/7.00 ( ( ( sin_complex @ X )
% 12.85/7.00 = zero_zero_complex )
% 12.85/7.00 => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
% 12.85/7.00 = one_one_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6954_ceiling__diff__floor__le__1,axiom,
% 12.85/7.00 ! [X: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim6058952711729229775r_real @ X ) ) @ one_one_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6955_ceiling__diff__floor__le__1,axiom,
% 12.85/7.00 ! [X: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim3151403230148437115or_rat @ X ) ) @ one_one_int ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6956_sin__zero__abs__cos__one,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ( sin_real @ X )
% 12.85/7.00 = zero_zero_real )
% 12.85/7.00 => ( ( abs_abs_real @ ( cos_real @ X ) )
% 12.85/7.00 = one_one_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6957_sin__double,axiom,
% 12.85/7.00 ! [X: complex] :
% 12.85/7.00 ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 12.85/7.00 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X ) ) @ ( cos_complex @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6958_sin__double,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 12.85/7.00 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X ) ) @ ( cos_real @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6959_sincos__principal__value,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ? [Y: real] :
% 12.85/7.00 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y )
% 12.85/7.00 & ( ord_less_eq_real @ Y @ pi )
% 12.85/7.00 & ( ( sin_real @ Y )
% 12.85/7.00 = ( sin_real @ X ) )
% 12.85/7.00 & ( ( cos_real @ Y )
% 12.85/7.00 = ( cos_real @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6960_floor__mono,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/7.00 => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6961_floor__mono,axiom,
% 12.85/7.00 ! [X: rat,Y4: rat] :
% 12.85/7.00 ( ( ord_less_eq_rat @ X @ Y4 )
% 12.85/7.00 => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6962_ceiling__mono,axiom,
% 12.85/7.00 ! [Y4: rat,X: rat] :
% 12.85/7.00 ( ( ord_less_eq_rat @ Y4 @ X )
% 12.85/7.00 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y4 ) @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6963_ceiling__mono,axiom,
% 12.85/7.00 ! [Y4: real,X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ Y4 @ X )
% 12.85/7.00 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y4 ) @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6964_sin__x__le__x,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ord_less_eq_real @ ( sin_real @ X ) @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6965_sin__le__one,axiom,
% 12.85/7.00 ! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6966_cos__le__one,axiom,
% 12.85/7.00 ! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6967_abs__sin__x__le__abs__x,axiom,
% 12.85/7.00 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ ( abs_abs_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6968_cos__arctan__not__zero,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( cos_real @ ( arctan @ X ) )
% 12.85/7.00 != zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6969_sin__cos__le1,axiom,
% 12.85/7.00 ! [X: real,Y4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y4 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) ) ) @ one_one_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6970_sin__squared__eq,axiom,
% 12.85/7.00 ! [X: complex] :
% 12.85/7.00 ( ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.00 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6971_sin__squared__eq,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.00 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6972_cos__squared__eq,axiom,
% 12.85/7.00 ! [X: complex] :
% 12.85/7.00 ( ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.00 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6973_cos__squared__eq,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.00 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6974_le__floor__add,axiom,
% 12.85/7.00 ! [X: real,Y4: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y4 ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6975_le__floor__add,axiom,
% 12.85/7.00 ! [X: rat,Y4: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y4 ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6976_of__nat__ceiling,axiom,
% 12.85/7.00 ! [R2: rat] : ( ord_less_eq_rat @ R2 @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R2 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6977_of__nat__ceiling,axiom,
% 12.85/7.00 ! [R2: real] : ( ord_less_eq_real @ R2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R2 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6978_sin__gt__zero,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_real @ X @ pi )
% 12.85/7.00 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6979_sin__x__ge__neg__x,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ ( sin_real @ X ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6980_sin__ge__zero,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ pi )
% 12.85/7.00 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6981_real__nat__ceiling__ge,axiom,
% 12.85/7.00 ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6982_ceiling__add__le,axiom,
% 12.85/7.00 ! [X: rat,Y4: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ Y4 ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6983_ceiling__add__le,axiom,
% 12.85/7.00 ! [X: real,Y4: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ Y4 ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6984_sin__ge__minus__one,axiom,
% 12.85/7.00 ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6985_cos__monotone__0__pi__le,axiom,
% 12.85/7.00 ! [Y4: real,X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ( ord_less_eq_real @ Y4 @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ pi )
% 12.85/7.00 => ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6986_cos__mono__le__eq,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ pi )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ( ord_less_eq_real @ Y4 @ pi )
% 12.85/7.00 => ( ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) )
% 12.85/7.00 = ( ord_less_eq_real @ Y4 @ X ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6987_cos__inj__pi,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ pi )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ( ord_less_eq_real @ Y4 @ pi )
% 12.85/7.00 => ( ( ( cos_real @ X )
% 12.85/7.00 = ( cos_real @ Y4 ) )
% 12.85/7.00 => ( X = Y4 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6988_cos__ge__minus__one,axiom,
% 12.85/7.00 ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6989_abs__sin__le__one,axiom,
% 12.85/7.00 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6990_abs__cos__le__one,axiom,
% 12.85/7.00 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6991_sin__times__sin,axiom,
% 12.85/7.00 ! [W: complex,Z: complex] :
% 12.85/7.00 ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 12.85/7.00 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6992_sin__times__sin,axiom,
% 12.85/7.00 ! [W: real,Z: real] :
% 12.85/7.00 ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 12.85/7.00 = ( 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 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6993_sin__times__cos,axiom,
% 12.85/7.00 ! [W: complex,Z: complex] :
% 12.85/7.00 ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
% 12.85/7.00 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6994_sin__times__cos,axiom,
% 12.85/7.00 ! [W: real,Z: real] :
% 12.85/7.00 ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
% 12.85/7.00 = ( 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 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6995_cos__times__sin,axiom,
% 12.85/7.00 ! [W: complex,Z: complex] :
% 12.85/7.00 ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
% 12.85/7.00 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6996_cos__times__sin,axiom,
% 12.85/7.00 ! [W: real,Z: real] :
% 12.85/7.00 ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
% 12.85/7.00 = ( 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 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6997_sin__plus__sin,axiom,
% 12.85/7.00 ! [W: complex,Z: complex] :
% 12.85/7.00 ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 12.85/7.00 = ( 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 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6998_sin__plus__sin,axiom,
% 12.85/7.00 ! [W: real,Z: real] :
% 12.85/7.00 ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 12.85/7.00 = ( 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 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_6999_sin__diff__sin,axiom,
% 12.85/7.00 ! [W: complex,Z: complex] :
% 12.85/7.00 ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 12.85/7.00 = ( 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 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7000_sin__diff__sin,axiom,
% 12.85/7.00 ! [W: real,Z: real] :
% 12.85/7.00 ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 12.85/7.00 = ( 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 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7001_cos__diff__cos,axiom,
% 12.85/7.00 ! [W: complex,Z: complex] :
% 12.85/7.00 ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 12.85/7.00 = ( 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 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7002_cos__diff__cos,axiom,
% 12.85/7.00 ! [W: real,Z: real] :
% 12.85/7.00 ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 12.85/7.00 = ( 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 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7003_cos__double,axiom,
% 12.85/7.00 ! [X: complex] :
% 12.85/7.00 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 12.85/7.00 = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7004_cos__double,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 12.85/7.00 = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7005_cos__double__sin,axiom,
% 12.85/7.00 ! [W: complex] :
% 12.85/7.00 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 12.85/7.00 = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7006_cos__double__sin,axiom,
% 12.85/7.00 ! [W: real] :
% 12.85/7.00 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 12.85/7.00 = ( 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 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7007_of__nat__floor,axiom,
% 12.85/7.00 ! [R2: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 12.85/7.00 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R2 ) ) ) @ R2 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7008_of__nat__floor,axiom,
% 12.85/7.00 ! [R2: rat] :
% 12.85/7.00 ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 12.85/7.00 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R2 ) ) ) @ R2 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7009_le__mult__nat__floor,axiom,
% 12.85/7.00 ! [A: real,B: real] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ B ) ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7010_le__mult__nat__floor,axiom,
% 12.85/7.00 ! [A: rat,B: rat] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim3151403230148437115or_rat @ A ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7011_floor__divide__of__nat__eq,axiom,
% 12.85/7.00 ! [M: nat,N: nat] :
% 12.85/7.00 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 12.85/7.00 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7012_floor__divide__of__nat__eq,axiom,
% 12.85/7.00 ! [M: nat,N: nat] :
% 12.85/7.00 ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) )
% 12.85/7.00 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7013_nat__floor__neg,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 12.85/7.00 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 12.85/7.00 = zero_zero_nat ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7014_cos__two__neq__zero,axiom,
% 12.85/7.00 ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.85/7.00 != zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7015_floor__eq3,axiom,
% 12.85/7.00 ! [N: nat,X: real] :
% 12.85/7.00 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 12.85/7.00 => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 12.85/7.00 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 12.85/7.00 = N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7016_le__nat__floor,axiom,
% 12.85/7.00 ! [X: nat,A: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
% 12.85/7.00 => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7017_cos__monotone__0__pi,axiom,
% 12.85/7.00 ! [Y4: real,X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ( ord_less_real @ Y4 @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ pi )
% 12.85/7.00 => ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7018_cos__mono__less__eq,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ pi )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ( ord_less_eq_real @ Y4 @ pi )
% 12.85/7.00 => ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) )
% 12.85/7.00 = ( ord_less_real @ Y4 @ X ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7019_sin__eq__0__pi,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 12.85/7.00 => ( ( ord_less_real @ X @ pi )
% 12.85/7.00 => ( ( ( sin_real @ X )
% 12.85/7.00 = zero_zero_real )
% 12.85/7.00 => ( X = zero_zero_real ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7020_sin__zero__pi__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ ( abs_abs_real @ X ) @ pi )
% 12.85/7.00 => ( ( ( sin_real @ X )
% 12.85/7.00 = zero_zero_real )
% 12.85/7.00 = ( X = zero_zero_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7021_cos__monotone__minus__pi__0_H,axiom,
% 12.85/7.00 ! [Y4: real,X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y4 )
% 12.85/7.00 => ( ( ord_less_eq_real @ Y4 @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 12.85/7.00 => ( ord_less_eq_real @ ( cos_real @ Y4 ) @ ( cos_real @ X ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7022_sincos__total__pi,axiom,
% 12.85/7.00 ! [Y4: real,X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.00 = one_one_real )
% 12.85/7.00 => ? [T3: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 12.85/7.00 & ( ord_less_eq_real @ T3 @ pi )
% 12.85/7.00 & ( X
% 12.85/7.00 = ( cos_real @ T3 ) )
% 12.85/7.00 & ( Y4
% 12.85/7.00 = ( sin_real @ T3 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7023_sin__cos__sqrt,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
% 12.85/7.00 => ( ( sin_real @ X )
% 12.85/7.00 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7024_sin__expansion__lemma,axiom,
% 12.85/7.00 ! [X: real,M: nat] :
% 12.85/7.00 ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 12.85/7.00 = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7025_le__mult__floor,axiom,
% 12.85/7.00 ! [A: real,B: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 12.85/7.00 => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7026_le__mult__floor,axiom,
% 12.85/7.00 ! [A: rat,B: rat] :
% 12.85/7.00 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/7.00 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 12.85/7.00 => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7027_cos__expansion__lemma,axiom,
% 12.85/7.00 ! [X: real,M: nat] :
% 12.85/7.00 ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 12.85/7.00 = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7028_sin__gt__zero__02,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.85/7.00 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7029_mult__ceiling__le,axiom,
% 12.85/7.00 ! [A: rat,B: rat] :
% 12.85/7.00 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/7.00 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 12.85/7.00 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7030_mult__ceiling__le,axiom,
% 12.85/7.00 ! [A: real,B: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 12.85/7.00 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7031_floor__eq4,axiom,
% 12.85/7.00 ! [N: nat,X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 12.85/7.00 => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 12.85/7.00 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 12.85/7.00 = N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7032_cos__two__less__zero,axiom,
% 12.85/7.00 ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7033_cos__two__le__zero,axiom,
% 12.85/7.00 ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7034_cos__is__zero,axiom,
% 12.85/7.00 ? [X4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.85/7.00 & ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.85/7.00 & ( ( cos_real @ X4 )
% 12.85/7.00 = zero_zero_real )
% 12.85/7.00 & ! [Y3: real] :
% 12.85/7.00 ( ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.85/7.00 & ( ord_less_eq_real @ Y3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.85/7.00 & ( ( cos_real @ Y3 )
% 12.85/7.00 = zero_zero_real ) )
% 12.85/7.00 => ( Y3 = X4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7035_cos__monotone__minus__pi__0,axiom,
% 12.85/7.00 ! [Y4: real,X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y4 )
% 12.85/7.00 => ( ( ord_less_real @ Y4 @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 12.85/7.00 => ( ord_less_real @ ( cos_real @ Y4 ) @ ( cos_real @ X ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7036_cos__total,axiom,
% 12.85/7.00 ! [Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 12.85/7.00 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/7.00 => ? [X4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.85/7.00 & ( ord_less_eq_real @ X4 @ pi )
% 12.85/7.00 & ( ( cos_real @ X4 )
% 12.85/7.00 = Y4 )
% 12.85/7.00 & ! [Y3: real] :
% 12.85/7.00 ( ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 12.85/7.00 & ( ord_less_eq_real @ Y3 @ pi )
% 12.85/7.00 & ( ( cos_real @ Y3 )
% 12.85/7.00 = Y4 ) )
% 12.85/7.00 => ( Y3 = X4 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7037_sincos__total__pi__half,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.00 = one_one_real )
% 12.85/7.00 => ? [T3: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 12.85/7.00 & ( ord_less_eq_real @ T3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 & ( X
% 12.85/7.00 = ( cos_real @ T3 ) )
% 12.85/7.00 & ( Y4
% 12.85/7.00 = ( sin_real @ T3 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7038_sincos__total__2pi__le,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.00 = one_one_real )
% 12.85/7.00 => ? [T3: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 12.85/7.00 & ( ord_less_eq_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 12.85/7.00 & ( X
% 12.85/7.00 = ( cos_real @ T3 ) )
% 12.85/7.00 & ( Y4
% 12.85/7.00 = ( sin_real @ T3 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7039_sincos__total__2pi,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.00 = one_one_real )
% 12.85/7.00 => ~ ! [T3: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 12.85/7.00 => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 12.85/7.00 => ( ( X
% 12.85/7.00 = ( cos_real @ T3 ) )
% 12.85/7.00 => ( Y4
% 12.85/7.00 != ( sin_real @ T3 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7040_sin__pi__divide__n__ge__0,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( N != zero_zero_nat )
% 12.85/7.00 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7041_sin__45,axiom,
% 12.85/7.00 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 12.85/7.00 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7042_cos__45,axiom,
% 12.85/7.00 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 12.85/7.00 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7043_cos__times__cos,axiom,
% 12.85/7.00 ! [W: complex,Z: complex] :
% 12.85/7.00 ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 12.85/7.00 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7044_cos__times__cos,axiom,
% 12.85/7.00 ! [W: real,Z: real] :
% 12.85/7.00 ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 12.85/7.00 = ( 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 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7045_cos__plus__cos,axiom,
% 12.85/7.00 ! [W: complex,Z: complex] :
% 12.85/7.00 ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 12.85/7.00 = ( 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 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7046_cos__plus__cos,axiom,
% 12.85/7.00 ! [W: real,Z: real] :
% 12.85/7.00 ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 12.85/7.00 = ( 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 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7047_sin__gt__zero2,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7048_sin__lt__zero,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ pi @ X )
% 12.85/7.00 => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 12.85/7.00 => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7049_cos__double__less__one,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.85/7.00 => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7050_sin__30,axiom,
% 12.85/7.00 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 12.85/7.00 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7051_cos__gt__zero,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7052_sin__monotone__2pi__le,axiom,
% 12.85/7.00 ! [Y4: real,X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 12.85/7.00 => ( ( ord_less_eq_real @ Y4 @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ord_less_eq_real @ ( sin_real @ Y4 ) @ ( sin_real @ X ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7053_sin__mono__le__eq,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 12.85/7.00 => ( ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y4 ) )
% 12.85/7.00 = ( ord_less_eq_real @ X @ Y4 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7054_sin__inj__pi,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 12.85/7.00 => ( ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ( ( sin_real @ X )
% 12.85/7.00 = ( sin_real @ Y4 ) )
% 12.85/7.00 => ( X = Y4 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7055_cos__60,axiom,
% 12.85/7.00 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 12.85/7.00 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7056_sin__60,axiom,
% 12.85/7.00 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 12.85/7.00 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7057_cos__30,axiom,
% 12.85/7.00 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 12.85/7.00 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7058_cos__double__cos,axiom,
% 12.85/7.00 ! [W: complex] :
% 12.85/7.00 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 12.85/7.00 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7059_cos__double__cos,axiom,
% 12.85/7.00 ! [W: real] :
% 12.85/7.00 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 12.85/7.00 = ( 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 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7060_cos__treble__cos,axiom,
% 12.85/7.00 ! [X: complex] :
% 12.85/7.00 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X ) )
% 12.85/7.00 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7061_cos__treble__cos,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X ) )
% 12.85/7.00 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7062_sin__le__zero,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ pi @ X )
% 12.85/7.00 => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 12.85/7.00 => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7063_sin__less__zero,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 12.85/7.00 => ( ( ord_less_real @ X @ zero_zero_real )
% 12.85/7.00 => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7064_sin__monotone__2pi,axiom,
% 12.85/7.00 ! [Y4: real,X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 12.85/7.00 => ( ( ord_less_real @ Y4 @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ord_less_real @ ( sin_real @ Y4 ) @ ( sin_real @ X ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7065_sin__mono__less__eq,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 12.85/7.00 => ( ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y4 ) )
% 12.85/7.00 = ( ord_less_real @ X @ Y4 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7066_sin__total,axiom,
% 12.85/7.00 ! [Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 12.85/7.00 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/7.00 => ? [X4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 12.85/7.00 & ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 & ( ( sin_real @ X4 )
% 12.85/7.00 = Y4 )
% 12.85/7.00 & ! [Y3: real] :
% 12.85/7.00 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 12.85/7.00 & ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 & ( ( sin_real @ Y3 )
% 12.85/7.00 = Y4 ) )
% 12.85/7.00 => ( Y3 = X4 ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7067_cos__gt__zero__pi,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 12.85/7.00 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7068_cos__ge__zero,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7069_cos__one__2pi,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ( cos_real @ X )
% 12.85/7.00 = one_one_real )
% 12.85/7.00 = ( ? [X3: nat] :
% 12.85/7.00 ( X
% 12.85/7.00 = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 12.85/7.00 | ? [X3: nat] :
% 12.85/7.00 ( X
% 12.85/7.00 = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7070_sin__pi__divide__n__gt__0,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/7.00 => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7071_sin__arctan,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( sin_real @ ( arctan @ X ) )
% 12.85/7.00 = ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7072_cos__arctan,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( cos_real @ ( arctan @ X ) )
% 12.85/7.00 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7073_floor__log2__div2,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/7.00 => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 12.85/7.00 = ( 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 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7074_floor__log__nat__eq__if,axiom,
% 12.85/7.00 ! [B: nat,N: nat,K: nat] :
% 12.85/7.00 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 12.85/7.00 => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 12.85/7.00 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 12.85/7.00 => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 12.85/7.00 = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7075_sin__zero__lemma,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ( sin_real @ X )
% 12.85/7.00 = zero_zero_real )
% 12.85/7.00 => ? [N2: nat] :
% 12.85/7.00 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 12.85/7.00 & ( X
% 12.85/7.00 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7076_sin__zero__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ( sin_real @ X )
% 12.85/7.00 = zero_zero_real )
% 12.85/7.00 = ( ? [N4: nat] :
% 12.85/7.00 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 12.85/7.00 & ( X
% 12.85/7.00 = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 12.85/7.00 | ? [N4: nat] :
% 12.85/7.00 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 12.85/7.00 & ( X
% 12.85/7.00 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7077_cos__zero__lemma,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ( cos_real @ X )
% 12.85/7.00 = zero_zero_real )
% 12.85/7.00 => ? [N2: nat] :
% 12.85/7.00 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 12.85/7.00 & ( X
% 12.85/7.00 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7078_cos__zero__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ( cos_real @ X )
% 12.85/7.00 = zero_zero_real )
% 12.85/7.00 = ( ? [N4: nat] :
% 12.85/7.00 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 12.85/7.00 & ( X
% 12.85/7.00 = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 12.85/7.00 | ? [N4: nat] :
% 12.85/7.00 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 12.85/7.00 & ( X
% 12.85/7.00 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7079_tan__double,axiom,
% 12.85/7.00 ! [X: complex] :
% 12.85/7.00 ( ( ( cos_complex @ X )
% 12.85/7.00 != zero_zero_complex )
% 12.85/7.00 => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 12.85/7.00 != zero_zero_complex )
% 12.85/7.00 => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 12.85/7.00 = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7080_tan__double,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ( cos_real @ X )
% 12.85/7.00 != zero_zero_real )
% 12.85/7.00 => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 12.85/7.00 != zero_zero_real )
% 12.85/7.00 => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 12.85/7.00 = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7081_sin__tan,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ( sin_real @ X )
% 12.85/7.00 = ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7082_cos__tan,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 => ( ( cos_real @ X )
% 12.85/7.00 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7083_complex__unimodular__polar,axiom,
% 12.85/7.00 ! [Z: complex] :
% 12.85/7.00 ( ( ( real_V1022390504157884413omplex @ Z )
% 12.85/7.00 = one_one_real )
% 12.85/7.00 => ~ ! [T3: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 12.85/7.00 => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 12.85/7.00 => ( Z
% 12.85/7.00 != ( complex2 @ ( cos_real @ T3 ) @ ( sin_real @ T3 ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7084_ceiling__log__eq__powr__iff,axiom,
% 12.85/7.00 ! [X: real,B: real,K: nat] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_real @ one_one_real @ B )
% 12.85/7.00 => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
% 12.85/7.00 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 12.85/7.00 = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
% 12.85/7.00 & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7085_cos__arcsin,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ one_one_real )
% 12.85/7.00 => ( ( cos_real @ ( arcsin @ X ) )
% 12.85/7.00 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7086_powr__eq__0__iff,axiom,
% 12.85/7.00 ! [W: real,Z: real] :
% 12.85/7.00 ( ( ( powr_real @ W @ Z )
% 12.85/7.00 = zero_zero_real )
% 12.85/7.00 = ( W = zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7087_powr__0,axiom,
% 12.85/7.00 ! [Z: real] :
% 12.85/7.00 ( ( powr_real @ zero_zero_real @ Z )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7088_tan__zero,axiom,
% 12.85/7.00 ( ( tan_complex @ zero_zero_complex )
% 12.85/7.00 = zero_zero_complex ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7089_tan__zero,axiom,
% 12.85/7.00 ( ( tan_real @ zero_zero_real )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7090_arcsin__0,axiom,
% 12.85/7.00 ( ( arcsin @ zero_zero_real )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7091_powr__zero__eq__one,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ( X = zero_zero_real )
% 12.85/7.00 => ( ( powr_real @ X @ zero_zero_real )
% 12.85/7.00 = zero_zero_real ) )
% 12.85/7.00 & ( ( X != zero_zero_real )
% 12.85/7.00 => ( ( powr_real @ X @ zero_zero_real )
% 12.85/7.00 = one_one_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7092_powr__gt__zero,axiom,
% 12.85/7.00 ! [X: real,A: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
% 12.85/7.00 = ( X != zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7093_powr__nonneg__iff,axiom,
% 12.85/7.00 ! [A: real,X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
% 12.85/7.00 = ( A = zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7094_tan__pi,axiom,
% 12.85/7.00 ( ( tan_real @ pi )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7095_powr__eq__one__iff,axiom,
% 12.85/7.00 ! [A: real,X: real] :
% 12.85/7.00 ( ( ord_less_real @ one_one_real @ A )
% 12.85/7.00 => ( ( ( powr_real @ A @ X )
% 12.85/7.00 = one_one_real )
% 12.85/7.00 = ( X = zero_zero_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7096_powr__one__gt__zero__iff,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ( powr_real @ X @ one_one_real )
% 12.85/7.00 = X )
% 12.85/7.00 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7097_powr__one,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( powr_real @ X @ one_one_real )
% 12.85/7.00 = X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7098_powr__le__cancel__iff,axiom,
% 12.85/7.00 ! [X: real,A: real,B: real] :
% 12.85/7.00 ( ( ord_less_real @ one_one_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 12.85/7.00 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7099_numeral__powr__numeral__real,axiom,
% 12.85/7.00 ! [M: num,N: num] :
% 12.85/7.00 ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 12.85/7.00 = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7100_powr__log__cancel,axiom,
% 12.85/7.00 ! [A: real,X: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.00 => ( ( A != one_one_real )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( powr_real @ A @ ( log @ A @ X ) )
% 12.85/7.00 = X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7101_log__powr__cancel,axiom,
% 12.85/7.00 ! [A: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.00 => ( ( A != one_one_real )
% 12.85/7.00 => ( ( log @ A @ ( powr_real @ A @ Y4 ) )
% 12.85/7.00 = Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7102_tan__npi,axiom,
% 12.85/7.00 ! [N: nat] :
% 12.85/7.00 ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 12.85/7.00 = zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7103_tan__periodic__n,axiom,
% 12.85/7.00 ! [X: real,N: num] :
% 12.85/7.00 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
% 12.85/7.00 = ( tan_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7104_tan__periodic__nat,axiom,
% 12.85/7.00 ! [X: real,N: nat] :
% 12.85/7.00 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
% 12.85/7.00 = ( tan_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7105_sin__arcsin,axiom,
% 12.85/7.00 ! [Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 12.85/7.00 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/7.00 => ( ( sin_real @ ( arcsin @ Y4 ) )
% 12.85/7.00 = Y4 ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7106_powr__numeral,axiom,
% 12.85/7.00 ! [X: real,N: num] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( powr_real @ X @ ( numeral_numeral_real @ N ) )
% 12.85/7.00 = ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7107_arcsin__1,axiom,
% 12.85/7.00 ( ( arcsin @ one_one_real )
% 12.85/7.00 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7108_tan__periodic,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 12.85/7.00 = ( tan_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7109_arcsin__minus__1,axiom,
% 12.85/7.00 ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
% 12.85/7.00 = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7110_square__powr__half,axiom,
% 12.85/7.00 ! [X: real] :
% 12.85/7.00 ( ( powr_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.00 = ( abs_abs_real @ X ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7111_powr__powr,axiom,
% 12.85/7.00 ! [X: real,A: real,B: real] :
% 12.85/7.00 ( ( powr_real @ ( powr_real @ X @ A ) @ B )
% 12.85/7.00 = ( powr_real @ X @ ( times_times_real @ A @ B ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7112_Complex__eq__0,axiom,
% 12.85/7.00 ! [A: real,B: real] :
% 12.85/7.00 ( ( ( complex2 @ A @ B )
% 12.85/7.00 = zero_zero_complex )
% 12.85/7.00 = ( ( A = zero_zero_real )
% 12.85/7.00 & ( B = zero_zero_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7113_zero__complex_Ocode,axiom,
% 12.85/7.00 ( zero_zero_complex
% 12.85/7.00 = ( complex2 @ zero_zero_real @ zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7114_complex__diff,axiom,
% 12.85/7.00 ! [A: real,B: real,C: real,D: real] :
% 12.85/7.00 ( ( minus_minus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 12.85/7.00 = ( complex2 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7115_powr__less__mono2__neg,axiom,
% 12.85/7.00 ! [A: real,X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_real @ X @ Y4 )
% 12.85/7.00 => ( ord_less_real @ ( powr_real @ Y4 @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7116_powr__non__neg,axiom,
% 12.85/7.00 ! [A: real,X: real] :
% 12.85/7.00 ~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7117_powr__ge__pzero,axiom,
% 12.85/7.00 ! [X: real,Y4: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y4 ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7118_powr__mono2,axiom,
% 12.85/7.00 ! [A: real,X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/7.00 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y4 @ A ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7119_powr__mono,axiom,
% 12.85/7.00 ! [A: real,B: real,X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ A @ B )
% 12.85/7.00 => ( ( ord_less_eq_real @ one_one_real @ X )
% 12.85/7.00 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7120_Complex__eq__1,axiom,
% 12.85/7.00 ! [A: real,B: real] :
% 12.85/7.00 ( ( ( complex2 @ A @ B )
% 12.85/7.00 = one_one_complex )
% 12.85/7.00 = ( ( A = one_one_real )
% 12.85/7.00 & ( B = zero_zero_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7121_one__complex_Ocode,axiom,
% 12.85/7.00 ( one_one_complex
% 12.85/7.00 = ( complex2 @ one_one_real @ zero_zero_real ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7122_Complex__eq__numeral,axiom,
% 12.85/7.00 ! [A: real,B: real,W: num] :
% 12.85/7.00 ( ( ( complex2 @ A @ B )
% 12.85/7.00 = ( numera6690914467698888265omplex @ W ) )
% 12.85/7.00 = ( ( A
% 12.85/7.00 = ( numeral_numeral_real @ W ) )
% 12.85/7.00 & ( B = zero_zero_real ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7123_powr__less__mono2,axiom,
% 12.85/7.00 ! [A: real,X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_real @ X @ Y4 )
% 12.85/7.00 => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y4 @ A ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7124_powr__mono2_H,axiom,
% 12.85/7.00 ! [A: real,X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/7.00 => ( ord_less_eq_real @ ( powr_real @ Y4 @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7125_gr__one__powr,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_real @ one_one_real @ X )
% 12.85/7.00 => ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7126_powr__inj,axiom,
% 12.85/7.00 ! [A: real,X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.00 => ( ( A != one_one_real )
% 12.85/7.00 => ( ( ( powr_real @ A @ X )
% 12.85/7.00 = ( powr_real @ A @ Y4 ) )
% 12.85/7.00 = ( X = Y4 ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7127_ge__one__powr__ge__zero,axiom,
% 12.85/7.00 ! [X: real,A: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ one_one_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/7.00 => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7128_powr__mono__both,axiom,
% 12.85/7.00 ! [A: real,B: real,X: real,Y4: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/7.00 => ( ( ord_less_eq_real @ A @ B )
% 12.85/7.00 => ( ( ord_less_eq_real @ one_one_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/7.00 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y4 @ B ) ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7129_powr__le1,axiom,
% 12.85/7.00 ! [A: real,X: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ X @ one_one_real )
% 12.85/7.00 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7130_powr__divide,axiom,
% 12.85/7.00 ! [X: real,Y4: real,A: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ( powr_real @ ( divide_divide_real @ X @ Y4 ) @ A )
% 12.85/7.00 = ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y4 @ A ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7131_powr__mult,axiom,
% 12.85/7.00 ! [X: real,Y4: real,A: real] :
% 12.85/7.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.00 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.00 => ( ( powr_real @ ( times_times_real @ X @ Y4 ) @ A )
% 12.85/7.00 = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y4 @ A ) ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7132_divide__powr__uminus,axiom,
% 12.85/7.00 ! [A: real,B: real,C: real] :
% 12.85/7.00 ( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
% 12.85/7.00 = ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7133_ln__powr,axiom,
% 12.85/7.00 ! [X: real,Y4: real] :
% 12.85/7.00 ( ( X != zero_zero_real )
% 12.85/7.00 => ( ( ln_ln_real @ ( powr_real @ X @ Y4 ) )
% 12.85/7.00 = ( times_times_real @ Y4 @ ( ln_ln_real @ X ) ) ) ) ).
% 12.85/7.00
% 12.85/7.00 thf(fact_7134_log__base__powr,axiom,
% 12.85/7.01 ! [A: real,B: real,X: real] :
% 12.85/7.01 ( ( A != zero_zero_real )
% 12.85/7.01 => ( ( log @ ( powr_real @ A @ B ) @ X )
% 12.85/7.01 = ( divide_divide_real @ ( log @ A @ X ) @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7135_log__powr,axiom,
% 12.85/7.01 ! [X: real,B: real,Y4: real] :
% 12.85/7.01 ( ( X != zero_zero_real )
% 12.85/7.01 => ( ( log @ B @ ( powr_real @ X @ Y4 ) )
% 12.85/7.01 = ( times_times_real @ Y4 @ ( log @ B @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7136_Complex__eq__neg__1,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ( complex2 @ A @ B )
% 12.85/7.01 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 12.85/7.01 = ( ( A
% 12.85/7.01 = ( uminus_uminus_real @ one_one_real ) )
% 12.85/7.01 & ( B = zero_zero_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7137_Complex__eq__neg__numeral,axiom,
% 12.85/7.01 ! [A: real,B: real,W: num] :
% 12.85/7.01 ( ( ( complex2 @ A @ B )
% 12.85/7.01 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 12.85/7.01 = ( ( A
% 12.85/7.01 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.85/7.01 & ( B = zero_zero_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7138_powr__add,axiom,
% 12.85/7.01 ! [X: real,A: real,B: real] :
% 12.85/7.01 ( ( powr_real @ X @ ( plus_plus_real @ A @ B ) )
% 12.85/7.01 = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7139_powr__diff,axiom,
% 12.85/7.01 ! [W: real,Z1: real,Z22: real] :
% 12.85/7.01 ( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z22 ) )
% 12.85/7.01 = ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z22 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7140_complex__mult,axiom,
% 12.85/7.01 ! [A: real,B: real,C: real,D: real] :
% 12.85/7.01 ( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 12.85/7.01 = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7141_arcsin__le__arcsin,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 12.85/7.01 => ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/7.01 => ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y4 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7142_arcsin__minus,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 12.85/7.01 => ( ( ord_less_eq_real @ X @ one_one_real )
% 12.85/7.01 => ( ( arcsin @ ( uminus_uminus_real @ X ) )
% 12.85/7.01 = ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7143_arcsin__le__mono,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y4 ) )
% 12.85/7.01 = ( ord_less_eq_real @ X @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7144_arcsin__eq__iff,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
% 12.85/7.01 => ( ( ( arcsin @ X )
% 12.85/7.01 = ( arcsin @ Y4 ) )
% 12.85/7.01 = ( X = Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7145_tan__def,axiom,
% 12.85/7.01 ( tan_complex
% 12.85/7.01 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X3 ) @ ( cos_complex @ X3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7146_tan__def,axiom,
% 12.85/7.01 ( tan_real
% 12.85/7.01 = ( ^ [X3: real] : ( divide_divide_real @ ( sin_real @ X3 ) @ ( cos_real @ X3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7147_powr__realpow,axiom,
% 12.85/7.01 ! [X: real,N: nat] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 12.85/7.01 = ( power_power_real @ X @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7148_powr__less__iff,axiom,
% 12.85/7.01 ! [B: real,X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ one_one_real @ B )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( ord_less_real @ ( powr_real @ B @ Y4 ) @ X )
% 12.85/7.01 = ( ord_less_real @ Y4 @ ( log @ B @ X ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7149_less__powr__iff,axiom,
% 12.85/7.01 ! [B: real,X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ one_one_real @ B )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ ( powr_real @ B @ Y4 ) )
% 12.85/7.01 = ( ord_less_real @ ( log @ B @ X ) @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7150_log__less__iff,axiom,
% 12.85/7.01 ! [B: real,X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ one_one_real @ B )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( ord_less_real @ ( log @ B @ X ) @ Y4 )
% 12.85/7.01 = ( ord_less_real @ X @ ( powr_real @ B @ Y4 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7151_less__log__iff,axiom,
% 12.85/7.01 ! [B: real,X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ one_one_real @ B )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( ord_less_real @ Y4 @ ( log @ B @ X ) )
% 12.85/7.01 = ( ord_less_real @ ( powr_real @ B @ Y4 ) @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7152_powr__minus__divide,axiom,
% 12.85/7.01 ! [X: real,A: real] :
% 12.85/7.01 ( ( powr_real @ X @ ( uminus_uminus_real @ A ) )
% 12.85/7.01 = ( divide_divide_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7153_powr__neg__one,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 12.85/7.01 = ( divide_divide_real @ one_one_real @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7154_arcsin__less__arcsin,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ Y4 )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/7.01 => ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y4 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7155_powr__mult__base,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( times_times_real @ X @ ( powr_real @ X @ Y4 ) )
% 12.85/7.01 = ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7156_arcsin__less__mono,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
% 12.85/7.01 => ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y4 ) )
% 12.85/7.01 = ( ord_less_real @ X @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7157_le__log__iff,axiom,
% 12.85/7.01 ! [B: real,X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ one_one_real @ B )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ ( log @ B @ X ) )
% 12.85/7.01 = ( ord_less_eq_real @ ( powr_real @ B @ Y4 ) @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7158_log__le__iff,axiom,
% 12.85/7.01 ! [B: real,X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ one_one_real @ B )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y4 )
% 12.85/7.01 = ( ord_less_eq_real @ X @ ( powr_real @ B @ Y4 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7159_le__powr__iff,axiom,
% 12.85/7.01 ! [B: real,X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ one_one_real @ B )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y4 ) )
% 12.85/7.01 = ( ord_less_eq_real @ ( log @ B @ X ) @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7160_powr__le__iff,axiom,
% 12.85/7.01 ! [B: real,X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ one_one_real @ B )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( powr_real @ B @ Y4 ) @ X )
% 12.85/7.01 = ( ord_less_eq_real @ Y4 @ ( log @ B @ X ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7161_ln__powr__bound,axiom,
% 12.85/7.01 ! [X: real,A: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ one_one_real @ X )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.01 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7162_ln__powr__bound2,axiom,
% 12.85/7.01 ! [X: real,A: real] :
% 12.85/7.01 ( ( ord_less_real @ one_one_real @ X )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.01 => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7163_tan__45,axiom,
% 12.85/7.01 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 12.85/7.01 = one_one_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7164_log__add__eq__powr,axiom,
% 12.85/7.01 ! [B: real,X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ B )
% 12.85/7.01 => ( ( B != one_one_real )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( plus_plus_real @ ( log @ B @ X ) @ Y4 )
% 12.85/7.01 = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y4 ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7165_add__log__eq__powr,axiom,
% 12.85/7.01 ! [B: real,X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ B )
% 12.85/7.01 => ( ( B != one_one_real )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( plus_plus_real @ Y4 @ ( log @ B @ X ) )
% 12.85/7.01 = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y4 ) @ X ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7166_minus__log__eq__powr,axiom,
% 12.85/7.01 ! [B: real,X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ B )
% 12.85/7.01 => ( ( B != one_one_real )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( minus_minus_real @ Y4 @ ( log @ B @ X ) )
% 12.85/7.01 = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y4 ) @ X ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7167_tan__60,axiom,
% 12.85/7.01 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 12.85/7.01 = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7168_cos__arcsin__nonzero,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ one_one_real )
% 12.85/7.01 => ( ( cos_real @ ( arcsin @ X ) )
% 12.85/7.01 != zero_zero_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7169_powr__def,axiom,
% 12.85/7.01 ( powr_real
% 12.85/7.01 = ( ^ [X3: real,A3: real] : ( if_real @ ( X3 = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A3 @ ( ln_ln_real @ X3 ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7170_lemma__tan__total,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ Y4 )
% 12.85/7.01 => ? [X4: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ X4 )
% 12.85/7.01 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 & ( ord_less_real @ Y4 @ ( tan_real @ X4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7171_tan__gt__zero,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 => ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7172_lemma__tan__total1,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ? [X4: real] :
% 12.85/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 12.85/7.01 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 & ( ( tan_real @ X4 )
% 12.85/7.01 = Y4 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7173_tan__mono__lt__eq,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 12.85/7.01 => ( ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 => ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) )
% 12.85/7.01 = ( ord_less_real @ X @ Y4 ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7174_tan__monotone_H,axiom,
% 12.85/7.01 ! [Y4: real,X: real] :
% 12.85/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 12.85/7.01 => ( ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 => ( ( ord_less_real @ Y4 @ X )
% 12.85/7.01 = ( ord_less_real @ ( tan_real @ Y4 ) @ ( tan_real @ X ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7175_tan__monotone,axiom,
% 12.85/7.01 ! [Y4: real,X: real] :
% 12.85/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 12.85/7.01 => ( ( ord_less_real @ Y4 @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 => ( ord_less_real @ ( tan_real @ Y4 ) @ ( tan_real @ X ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7176_tan__total,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ? [X4: real] :
% 12.85/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 12.85/7.01 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 & ( ( tan_real @ X4 )
% 12.85/7.01 = Y4 )
% 12.85/7.01 & ! [Y3: real] :
% 12.85/7.01 ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 12.85/7.01 & ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 & ( ( tan_real @ Y3 )
% 12.85/7.01 = Y4 ) )
% 12.85/7.01 => ( Y3 = X4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7177_tan__minus__45,axiom,
% 12.85/7.01 ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
% 12.85/7.01 = ( uminus_uminus_real @ one_one_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7178_tan__inverse,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y4 ) )
% 12.85/7.01 = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7179_log__minus__eq__powr,axiom,
% 12.85/7.01 ! [B: real,X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ B )
% 12.85/7.01 => ( ( B != one_one_real )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( minus_minus_real @ ( log @ B @ X ) @ Y4 )
% 12.85/7.01 = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y4 ) ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7180_complex__norm,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( real_V1022390504157884413omplex @ ( complex2 @ X @ Y4 ) )
% 12.85/7.01 = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7181_add__tan__eq,axiom,
% 12.85/7.01 ! [X: complex,Y4: complex] :
% 12.85/7.01 ( ( ( cos_complex @ X )
% 12.85/7.01 != zero_zero_complex )
% 12.85/7.01 => ( ( ( cos_complex @ Y4 )
% 12.85/7.01 != zero_zero_complex )
% 12.85/7.01 => ( ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y4 ) )
% 12.85/7.01 = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X @ Y4 ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y4 ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7182_add__tan__eq,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ( cos_real @ X )
% 12.85/7.01 != zero_zero_real )
% 12.85/7.01 => ( ( ( cos_real @ Y4 )
% 12.85/7.01 != zero_zero_real )
% 12.85/7.01 => ( ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) )
% 12.85/7.01 = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X @ Y4 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7183_powr__half__sqrt,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 = ( sqrt @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7184_powr__neg__numeral,axiom,
% 12.85/7.01 ! [X: real,N: num] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.85/7.01 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7185_tan__total__pos,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.01 => ? [X4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 12.85/7.01 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 & ( ( tan_real @ X4 )
% 12.85/7.01 = Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7186_tan__pos__pi2__le,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7187_tan__less__zero,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ zero_zero_real )
% 12.85/7.01 => ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7188_tan__mono__le,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 12.85/7.01 => ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/7.01 => ( ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 => ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7189_tan__mono__le__eq,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 12.85/7.01 => ( ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) )
% 12.85/7.01 = ( ord_less_eq_real @ X @ Y4 ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7190_tan__bound__pi2,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 12.85/7.01 => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7191_tan__30,axiom,
% 12.85/7.01 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 12.85/7.01 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7192_arctan__unique,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 => ( ( ( tan_real @ X )
% 12.85/7.01 = Y4 )
% 12.85/7.01 => ( ( arctan @ Y4 )
% 12.85/7.01 = X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7193_arctan__tan,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 => ( ( arctan @ ( tan_real @ X ) )
% 12.85/7.01 = X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7194_arctan,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y4 ) )
% 12.85/7.01 & ( ord_less_real @ ( arctan @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 & ( ( tan_real @ ( arctan @ Y4 ) )
% 12.85/7.01 = Y4 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7195_lemma__tan__add1,axiom,
% 12.85/7.01 ! [X: complex,Y4: complex] :
% 12.85/7.01 ( ( ( cos_complex @ X )
% 12.85/7.01 != zero_zero_complex )
% 12.85/7.01 => ( ( ( cos_complex @ Y4 )
% 12.85/7.01 != zero_zero_complex )
% 12.85/7.01 => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y4 ) ) )
% 12.85/7.01 = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X @ Y4 ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y4 ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7196_lemma__tan__add1,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ( cos_real @ X )
% 12.85/7.01 != zero_zero_real )
% 12.85/7.01 => ( ( ( cos_real @ Y4 )
% 12.85/7.01 != zero_zero_real )
% 12.85/7.01 => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) ) )
% 12.85/7.01 = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X @ Y4 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7197_tan__diff,axiom,
% 12.85/7.01 ! [X: complex,Y4: complex] :
% 12.85/7.01 ( ( ( cos_complex @ X )
% 12.85/7.01 != zero_zero_complex )
% 12.85/7.01 => ( ( ( cos_complex @ Y4 )
% 12.85/7.01 != zero_zero_complex )
% 12.85/7.01 => ( ( ( cos_complex @ ( minus_minus_complex @ X @ Y4 ) )
% 12.85/7.01 != zero_zero_complex )
% 12.85/7.01 => ( ( tan_complex @ ( minus_minus_complex @ X @ Y4 ) )
% 12.85/7.01 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y4 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y4 ) ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7198_tan__diff,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ( cos_real @ X )
% 12.85/7.01 != zero_zero_real )
% 12.85/7.01 => ( ( ( cos_real @ Y4 )
% 12.85/7.01 != zero_zero_real )
% 12.85/7.01 => ( ( ( cos_real @ ( minus_minus_real @ X @ Y4 ) )
% 12.85/7.01 != zero_zero_real )
% 12.85/7.01 => ( ( tan_real @ ( minus_minus_real @ X @ Y4 ) )
% 12.85/7.01 = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7199_tan__add,axiom,
% 12.85/7.01 ! [X: complex,Y4: complex] :
% 12.85/7.01 ( ( ( cos_complex @ X )
% 12.85/7.01 != zero_zero_complex )
% 12.85/7.01 => ( ( ( cos_complex @ Y4 )
% 12.85/7.01 != zero_zero_complex )
% 12.85/7.01 => ( ( ( cos_complex @ ( plus_plus_complex @ X @ Y4 ) )
% 12.85/7.01 != zero_zero_complex )
% 12.85/7.01 => ( ( tan_complex @ ( plus_plus_complex @ X @ Y4 ) )
% 12.85/7.01 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y4 ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y4 ) ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7200_tan__add,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ( cos_real @ X )
% 12.85/7.01 != zero_zero_real )
% 12.85/7.01 => ( ( ( cos_real @ Y4 )
% 12.85/7.01 != zero_zero_real )
% 12.85/7.01 => ( ( ( cos_real @ ( plus_plus_real @ X @ Y4 ) )
% 12.85/7.01 != zero_zero_real )
% 12.85/7.01 => ( ( tan_real @ ( plus_plus_real @ X @ Y4 ) )
% 12.85/7.01 = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7201_tan__total__pi4,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 12.85/7.01 => ? [Z2: real] :
% 12.85/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z2 )
% 12.85/7.01 & ( ord_less_real @ Z2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 12.85/7.01 & ( ( tan_real @ Z2 )
% 12.85/7.01 = X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7202_arcsin__lt__bounded,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 12.85/7.01 => ( ( ord_less_real @ Y4 @ one_one_real )
% 12.85/7.01 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y4 ) )
% 12.85/7.01 & ( ord_less_real @ ( arcsin @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7203_arcsin__lbound,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/7.01 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7204_arcsin__ubound,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/7.01 => ( ord_less_eq_real @ ( arcsin @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7205_arcsin__bounded,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y4 ) )
% 12.85/7.01 & ( ord_less_eq_real @ ( arcsin @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7206_arcsin__sin,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 12.85/7.01 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 => ( ( arcsin @ ( sin_real @ X ) )
% 12.85/7.01 = X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7207_tan__half,axiom,
% 12.85/7.01 ( tan_complex
% 12.85/7.01 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_complex ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7208_tan__half,axiom,
% 12.85/7.01 ( tan_real
% 12.85/7.01 = ( ^ [X3: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_real ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7209_arcsin,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y4 ) )
% 12.85/7.01 & ( ord_less_eq_real @ ( arcsin @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 & ( ( sin_real @ ( arcsin @ Y4 ) )
% 12.85/7.01 = Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7210_arcsin__pi,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y4 ) )
% 12.85/7.01 & ( ord_less_eq_real @ ( arcsin @ Y4 ) @ pi )
% 12.85/7.01 & ( ( sin_real @ ( arcsin @ Y4 ) )
% 12.85/7.01 = Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7211_arcsin__le__iff,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 12.85/7.01 => ( ( ord_less_eq_real @ X @ one_one_real )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y4 )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y4 )
% 12.85/7.01 = ( ord_less_eq_real @ X @ ( sin_real @ Y4 ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7212_le__arcsin__iff,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 12.85/7.01 => ( ( ord_less_eq_real @ X @ one_one_real )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y4 )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ ( arcsin @ X ) )
% 12.85/7.01 = ( ord_less_eq_real @ ( sin_real @ Y4 ) @ X ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7213_arcosh__def,axiom,
% 12.85/7.01 ( arcosh_real
% 12.85/7.01 = ( ^ [X3: real] : ( ln_ln_real @ ( plus_plus_real @ X3 @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7214_sin__arccos__abs,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
% 12.85/7.01 => ( ( sin_real @ ( arccos @ Y4 ) )
% 12.85/7.01 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7215_sin__arccos,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 12.85/7.01 => ( ( ord_less_eq_real @ X @ one_one_real )
% 12.85/7.01 => ( ( sin_real @ ( arccos @ X ) )
% 12.85/7.01 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7216_arsinh__def,axiom,
% 12.85/7.01 ( arsinh_real
% 12.85/7.01 = ( ^ [X3: real] : ( ln_ln_real @ ( plus_plus_real @ X3 @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7217_cos__npi__int,axiom,
% 12.85/7.01 ! [N: int] :
% 12.85/7.01 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 12.85/7.01 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 12.85/7.01 = one_one_real ) )
% 12.85/7.01 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 12.85/7.01 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 12.85/7.01 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7218_of__int__eq__iff,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ( ring_1_of_int_real @ W )
% 12.85/7.01 = ( ring_1_of_int_real @ Z ) )
% 12.85/7.01 = ( W = Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7219_of__int__eq__iff,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ( ring_17405671764205052669omplex @ W )
% 12.85/7.01 = ( ring_17405671764205052669omplex @ Z ) )
% 12.85/7.01 = ( W = Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7220_of__int__eq__iff,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ( ring_1_of_int_rat @ W )
% 12.85/7.01 = ( ring_1_of_int_rat @ Z ) )
% 12.85/7.01 = ( W = Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7221_of__int__floor__cancel,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) )
% 12.85/7.01 = X )
% 12.85/7.01 = ( ? [N4: int] :
% 12.85/7.01 ( X
% 12.85/7.01 = ( ring_1_of_int_real @ N4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7222_of__int__floor__cancel,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ( ( ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) )
% 12.85/7.01 = X )
% 12.85/7.01 = ( ? [N4: int] :
% 12.85/7.01 ( X
% 12.85/7.01 = ( ring_1_of_int_rat @ N4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7223_of__int__ceiling__cancel,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) )
% 12.85/7.01 = X )
% 12.85/7.01 = ( ? [N4: int] :
% 12.85/7.01 ( X
% 12.85/7.01 = ( ring_1_of_int_rat @ N4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7224_of__int__ceiling__cancel,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) )
% 12.85/7.01 = X )
% 12.85/7.01 = ( ? [N4: int] :
% 12.85/7.01 ( X
% 12.85/7.01 = ( ring_1_of_int_real @ N4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7225_of__int__eq__0__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ( ring_1_of_int_int @ Z )
% 12.85/7.01 = zero_zero_int )
% 12.85/7.01 = ( Z = zero_zero_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7226_of__int__eq__0__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ( ring_1_of_int_real @ Z )
% 12.85/7.01 = zero_zero_real )
% 12.85/7.01 = ( Z = zero_zero_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7227_of__int__eq__0__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ( ring_17405671764205052669omplex @ Z )
% 12.85/7.01 = zero_zero_complex )
% 12.85/7.01 = ( Z = zero_zero_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7228_of__int__eq__0__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ( ring_1_of_int_rat @ Z )
% 12.85/7.01 = zero_zero_rat )
% 12.85/7.01 = ( Z = zero_zero_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7229_of__int__0__eq__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( zero_zero_int
% 12.85/7.01 = ( ring_1_of_int_int @ Z ) )
% 12.85/7.01 = ( Z = zero_zero_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7230_of__int__0__eq__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( zero_zero_real
% 12.85/7.01 = ( ring_1_of_int_real @ Z ) )
% 12.85/7.01 = ( Z = zero_zero_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7231_of__int__0__eq__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( zero_zero_complex
% 12.85/7.01 = ( ring_17405671764205052669omplex @ Z ) )
% 12.85/7.01 = ( Z = zero_zero_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7232_of__int__0__eq__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( zero_zero_rat
% 12.85/7.01 = ( ring_1_of_int_rat @ Z ) )
% 12.85/7.01 = ( Z = zero_zero_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7233_of__int__0,axiom,
% 12.85/7.01 ( ( ring_1_of_int_int @ zero_zero_int )
% 12.85/7.01 = zero_zero_int ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7234_of__int__0,axiom,
% 12.85/7.01 ( ( ring_1_of_int_real @ zero_zero_int )
% 12.85/7.01 = zero_zero_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7235_of__int__0,axiom,
% 12.85/7.01 ( ( ring_17405671764205052669omplex @ zero_zero_int )
% 12.85/7.01 = zero_zero_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7236_of__int__0,axiom,
% 12.85/7.01 ( ( ring_1_of_int_rat @ zero_zero_int )
% 12.85/7.01 = zero_zero_rat ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7237_of__real__0,axiom,
% 12.85/7.01 ( ( real_V1803761363581548252l_real @ zero_zero_real )
% 12.85/7.01 = zero_zero_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7238_of__real__0,axiom,
% 12.85/7.01 ( ( real_V4546457046886955230omplex @ zero_zero_real )
% 12.85/7.01 = zero_zero_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7239_of__real__eq__0__iff,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ( real_V1803761363581548252l_real @ X )
% 12.85/7.01 = zero_zero_real )
% 12.85/7.01 = ( X = zero_zero_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7240_of__real__eq__0__iff,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ( real_V4546457046886955230omplex @ X )
% 12.85/7.01 = zero_zero_complex )
% 12.85/7.01 = ( X = zero_zero_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7241_of__int__le__iff,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 12.85/7.01 = ( ord_less_eq_int @ W @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7242_of__int__le__iff,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 12.85/7.01 = ( ord_less_eq_int @ W @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7243_of__int__le__iff,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 12.85/7.01 = ( ord_less_eq_int @ W @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7244_of__int__eq__numeral__iff,axiom,
% 12.85/7.01 ! [Z: int,N: num] :
% 12.85/7.01 ( ( ( ring_17405671764205052669omplex @ Z )
% 12.85/7.01 = ( numera6690914467698888265omplex @ N ) )
% 12.85/7.01 = ( Z
% 12.85/7.01 = ( numeral_numeral_int @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7245_of__int__eq__numeral__iff,axiom,
% 12.85/7.01 ! [Z: int,N: num] :
% 12.85/7.01 ( ( ( ring_1_of_int_real @ Z )
% 12.85/7.01 = ( numeral_numeral_real @ N ) )
% 12.85/7.01 = ( Z
% 12.85/7.01 = ( numeral_numeral_int @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7246_of__int__eq__numeral__iff,axiom,
% 12.85/7.01 ! [Z: int,N: num] :
% 12.85/7.01 ( ( ( ring_1_of_int_rat @ Z )
% 12.85/7.01 = ( numeral_numeral_rat @ N ) )
% 12.85/7.01 = ( Z
% 12.85/7.01 = ( numeral_numeral_int @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7247_of__int__eq__numeral__iff,axiom,
% 12.85/7.01 ! [Z: int,N: num] :
% 12.85/7.01 ( ( ( ring_1_of_int_int @ Z )
% 12.85/7.01 = ( numeral_numeral_int @ N ) )
% 12.85/7.01 = ( Z
% 12.85/7.01 = ( numeral_numeral_int @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7248_of__int__numeral,axiom,
% 12.85/7.01 ! [K: num] :
% 12.85/7.01 ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
% 12.85/7.01 = ( numera6690914467698888265omplex @ K ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7249_of__int__numeral,axiom,
% 12.85/7.01 ! [K: num] :
% 12.85/7.01 ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 12.85/7.01 = ( numeral_numeral_real @ K ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7250_of__int__numeral,axiom,
% 12.85/7.01 ! [K: num] :
% 12.85/7.01 ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
% 12.85/7.01 = ( numeral_numeral_rat @ K ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7251_of__int__numeral,axiom,
% 12.85/7.01 ! [K: num] :
% 12.85/7.01 ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 12.85/7.01 = ( numeral_numeral_int @ K ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7252_of__int__less__iff,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 12.85/7.01 = ( ord_less_int @ W @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7253_of__int__less__iff,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 12.85/7.01 = ( ord_less_int @ W @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7254_of__int__less__iff,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 12.85/7.01 = ( ord_less_int @ W @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7255_of__int__eq__1__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ( ring_1_of_int_int @ Z )
% 12.85/7.01 = one_one_int )
% 12.85/7.01 = ( Z = one_one_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7256_of__int__eq__1__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ( ring_1_of_int_real @ Z )
% 12.85/7.01 = one_one_real )
% 12.85/7.01 = ( Z = one_one_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7257_of__int__eq__1__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ( ring_17405671764205052669omplex @ Z )
% 12.85/7.01 = one_one_complex )
% 12.85/7.01 = ( Z = one_one_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7258_of__int__eq__1__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ( ring_1_of_int_rat @ Z )
% 12.85/7.01 = one_one_rat )
% 12.85/7.01 = ( Z = one_one_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7259_of__int__1,axiom,
% 12.85/7.01 ( ( ring_1_of_int_int @ one_one_int )
% 12.85/7.01 = one_one_int ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7260_of__int__1,axiom,
% 12.85/7.01 ( ( ring_1_of_int_real @ one_one_int )
% 12.85/7.01 = one_one_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7261_of__int__1,axiom,
% 12.85/7.01 ( ( ring_17405671764205052669omplex @ one_one_int )
% 12.85/7.01 = one_one_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7262_of__int__1,axiom,
% 12.85/7.01 ( ( ring_1_of_int_rat @ one_one_int )
% 12.85/7.01 = one_one_rat ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7263_of__int__mult,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ring_17405671764205052669omplex @ ( times_times_int @ W @ Z ) )
% 12.85/7.01 = ( times_times_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7264_of__int__mult,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
% 12.85/7.01 = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7265_of__int__mult,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
% 12.85/7.01 = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7266_of__int__mult,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
% 12.85/7.01 = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7267_of__int__add,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
% 12.85/7.01 = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7268_of__int__add,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
% 12.85/7.01 = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7269_of__int__add,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ring_17405671764205052669omplex @ ( plus_plus_int @ W @ Z ) )
% 12.85/7.01 = ( plus_plus_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7270_of__int__add,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
% 12.85/7.01 = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7271_of__int__minus,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z ) )
% 12.85/7.01 = ( uminus_uminus_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7272_of__int__minus,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z ) )
% 12.85/7.01 = ( uminus_uminus_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7273_of__int__minus,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ Z ) )
% 12.85/7.01 = ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7274_of__int__minus,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ Z ) )
% 12.85/7.01 = ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7275_of__int__minus,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ Z ) )
% 12.85/7.01 = ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7276_of__real__mult,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( real_V1803761363581548252l_real @ ( times_times_real @ X @ Y4 ) )
% 12.85/7.01 = ( times_times_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7277_of__real__mult,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( real_V4546457046886955230omplex @ ( times_times_real @ X @ Y4 ) )
% 12.85/7.01 = ( times_times_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7278_of__real__numeral,axiom,
% 12.85/7.01 ! [W: num] :
% 12.85/7.01 ( ( real_V1803761363581548252l_real @ ( numeral_numeral_real @ W ) )
% 12.85/7.01 = ( numeral_numeral_real @ W ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7279_of__real__numeral,axiom,
% 12.85/7.01 ! [W: num] :
% 12.85/7.01 ( ( real_V4546457046886955230omplex @ ( numeral_numeral_real @ W ) )
% 12.85/7.01 = ( numera6690914467698888265omplex @ W ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7280_of__real__divide,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X @ Y4 ) )
% 12.85/7.01 = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7281_of__real__divide,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X @ Y4 ) )
% 12.85/7.01 = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7282_of__int__diff,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ring_17405671764205052669omplex @ ( minus_minus_int @ W @ Z ) )
% 12.85/7.01 = ( minus_minus_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7283_of__int__diff,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
% 12.85/7.01 = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7284_of__int__diff,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
% 12.85/7.01 = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7285_of__int__diff,axiom,
% 12.85/7.01 ! [W: int,Z: int] :
% 12.85/7.01 ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
% 12.85/7.01 = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7286_of__int__of__nat__eq,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( ring_17405671764205052669omplex @ ( semiri1314217659103216013at_int @ N ) )
% 12.85/7.01 = ( semiri8010041392384452111omplex @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7287_of__int__of__nat__eq,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N ) )
% 12.85/7.01 = ( semiri5074537144036343181t_real @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7288_of__int__of__nat__eq,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( ring_1_of_int_rat @ ( semiri1314217659103216013at_int @ N ) )
% 12.85/7.01 = ( semiri681578069525770553at_rat @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7289_of__int__of__nat__eq,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
% 12.85/7.01 = ( semiri1314217659103216013at_int @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7290_of__real__diff,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( real_V1803761363581548252l_real @ ( minus_minus_real @ X @ Y4 ) )
% 12.85/7.01 = ( minus_minus_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7291_of__real__diff,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( real_V4546457046886955230omplex @ ( minus_minus_real @ X @ Y4 ) )
% 12.85/7.01 = ( minus_minus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7292_of__real__of__nat__eq,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( real_V4546457046886955230omplex @ ( semiri5074537144036343181t_real @ N ) )
% 12.85/7.01 = ( semiri8010041392384452111omplex @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7293_of__real__of__nat__eq,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( real_V1803761363581548252l_real @ ( semiri5074537144036343181t_real @ N ) )
% 12.85/7.01 = ( semiri5074537144036343181t_real @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7294_of__int__abs,axiom,
% 12.85/7.01 ! [X: int] :
% 12.85/7.01 ( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
% 12.85/7.01 = ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7295_of__int__abs,axiom,
% 12.85/7.01 ! [X: int] :
% 12.85/7.01 ( ( ring_18347121197199848620nteger @ ( abs_abs_int @ X ) )
% 12.85/7.01 = ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7296_of__int__abs,axiom,
% 12.85/7.01 ! [X: int] :
% 12.85/7.01 ( ( ring_1_of_int_real @ ( abs_abs_int @ X ) )
% 12.85/7.01 = ( abs_abs_real @ ( ring_1_of_int_real @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7297_of__int__abs,axiom,
% 12.85/7.01 ! [X: int] :
% 12.85/7.01 ( ( ring_1_of_int_rat @ ( abs_abs_int @ X ) )
% 12.85/7.01 = ( abs_abs_rat @ ( ring_1_of_int_rat @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7298_of__int__power__eq__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: int,B: int,W: nat] :
% 12.85/7.01 ( ( ( ring_1_of_int_rat @ X )
% 12.85/7.01 = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 12.85/7.01 = ( X
% 12.85/7.01 = ( power_power_int @ B @ W ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7299_of__int__power__eq__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: int,B: int,W: nat] :
% 12.85/7.01 ( ( ( ring_1_of_int_int @ X )
% 12.85/7.01 = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 12.85/7.01 = ( X
% 12.85/7.01 = ( power_power_int @ B @ W ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7300_of__int__power__eq__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: int,B: int,W: nat] :
% 12.85/7.01 ( ( ( ring_1_of_int_real @ X )
% 12.85/7.01 = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 12.85/7.01 = ( X
% 12.85/7.01 = ( power_power_int @ B @ W ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7301_of__int__power__eq__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: int,B: int,W: nat] :
% 12.85/7.01 ( ( ( ring_17405671764205052669omplex @ X )
% 12.85/7.01 = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
% 12.85/7.01 = ( X
% 12.85/7.01 = ( power_power_int @ B @ W ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7302_of__int__eq__of__int__power__cancel__iff,axiom,
% 12.85/7.01 ! [B: int,W: nat,X: int] :
% 12.85/7.01 ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
% 12.85/7.01 = ( ring_1_of_int_rat @ X ) )
% 12.85/7.01 = ( ( power_power_int @ B @ W )
% 12.85/7.01 = X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7303_of__int__eq__of__int__power__cancel__iff,axiom,
% 12.85/7.01 ! [B: int,W: nat,X: int] :
% 12.85/7.01 ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
% 12.85/7.01 = ( ring_1_of_int_int @ X ) )
% 12.85/7.01 = ( ( power_power_int @ B @ W )
% 12.85/7.01 = X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7304_of__int__eq__of__int__power__cancel__iff,axiom,
% 12.85/7.01 ! [B: int,W: nat,X: int] :
% 12.85/7.01 ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
% 12.85/7.01 = ( ring_1_of_int_real @ X ) )
% 12.85/7.01 = ( ( power_power_int @ B @ W )
% 12.85/7.01 = X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7305_of__int__eq__of__int__power__cancel__iff,axiom,
% 12.85/7.01 ! [B: int,W: nat,X: int] :
% 12.85/7.01 ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
% 12.85/7.01 = ( ring_17405671764205052669omplex @ X ) )
% 12.85/7.01 = ( ( power_power_int @ B @ W )
% 12.85/7.01 = X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7306_of__int__power,axiom,
% 12.85/7.01 ! [Z: int,N: nat] :
% 12.85/7.01 ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N ) )
% 12.85/7.01 = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7307_of__int__power,axiom,
% 12.85/7.01 ! [Z: int,N: nat] :
% 12.85/7.01 ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
% 12.85/7.01 = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7308_of__int__power,axiom,
% 12.85/7.01 ! [Z: int,N: nat] :
% 12.85/7.01 ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
% 12.85/7.01 = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7309_of__int__power,axiom,
% 12.85/7.01 ! [Z: int,N: nat] :
% 12.85/7.01 ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N ) )
% 12.85/7.01 = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7310_arccos__1,axiom,
% 12.85/7.01 ( ( arccos @ one_one_real )
% 12.85/7.01 = zero_zero_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7311_of__int__of__bool,axiom,
% 12.85/7.01 ! [P: $o] :
% 12.85/7.01 ( ( ring_1_of_int_real @ ( zero_n2684676970156552555ol_int @ P ) )
% 12.85/7.01 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7312_of__int__of__bool,axiom,
% 12.85/7.01 ! [P: $o] :
% 12.85/7.01 ( ( ring_17405671764205052669omplex @ ( zero_n2684676970156552555ol_int @ P ) )
% 12.85/7.01 = ( zero_n1201886186963655149omplex @ P ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7313_of__int__of__bool,axiom,
% 12.85/7.01 ! [P: $o] :
% 12.85/7.01 ( ( ring_1_of_int_rat @ ( zero_n2684676970156552555ol_int @ P ) )
% 12.85/7.01 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7314_of__int__of__bool,axiom,
% 12.85/7.01 ! [P: $o] :
% 12.85/7.01 ( ( ring_1_of_int_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 12.85/7.01 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7315_of__int__of__bool,axiom,
% 12.85/7.01 ! [P: $o] :
% 12.85/7.01 ( ( ring_18347121197199848620nteger @ ( zero_n2684676970156552555ol_int @ P ) )
% 12.85/7.01 = ( zero_n356916108424825756nteger @ P ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7316_sin__of__real__pi,axiom,
% 12.85/7.01 ( ( sin_real @ ( real_V1803761363581548252l_real @ pi ) )
% 12.85/7.01 = zero_zero_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7317_sin__of__real__pi,axiom,
% 12.85/7.01 ( ( sin_complex @ ( real_V4546457046886955230omplex @ pi ) )
% 12.85/7.01 = zero_zero_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7318_floor__diff__of__int,axiom,
% 12.85/7.01 ! [X: real,Z: int] :
% 12.85/7.01 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
% 12.85/7.01 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7319_floor__diff__of__int,axiom,
% 12.85/7.01 ! [X: rat,Z: int] :
% 12.85/7.01 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
% 12.85/7.01 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7320_ceiling__diff__of__int,axiom,
% 12.85/7.01 ! [X: rat,Z: int] :
% 12.85/7.01 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
% 12.85/7.01 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7321_ceiling__diff__of__int,axiom,
% 12.85/7.01 ! [X: real,Z: int] :
% 12.85/7.01 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
% 12.85/7.01 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7322_of__nat__nat__take__bit__eq,axiom,
% 12.85/7.01 ! [N: nat,K: int] :
% 12.85/7.01 ( ( semiri8010041392384452111omplex @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 12.85/7.01 = ( ring_17405671764205052669omplex @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7323_of__nat__nat__take__bit__eq,axiom,
% 12.85/7.01 ! [N: nat,K: int] :
% 12.85/7.01 ( ( semiri5074537144036343181t_real @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 12.85/7.01 = ( ring_1_of_int_real @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7324_of__nat__nat__take__bit__eq,axiom,
% 12.85/7.01 ! [N: nat,K: int] :
% 12.85/7.01 ( ( semiri681578069525770553at_rat @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 12.85/7.01 = ( ring_1_of_int_rat @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7325_of__nat__nat__take__bit__eq,axiom,
% 12.85/7.01 ! [N: nat,K: int] :
% 12.85/7.01 ( ( semiri1314217659103216013at_int @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 12.85/7.01 = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7326_of__int__le__0__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 12.85/7.01 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7327_of__int__le__0__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 12.85/7.01 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7328_of__int__le__0__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 12.85/7.01 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7329_of__int__0__le__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 12.85/7.01 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7330_of__int__0__le__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 12.85/7.01 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7331_of__int__0__le__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 12.85/7.01 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7332_of__int__less__0__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 12.85/7.01 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7333_of__int__less__0__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 12.85/7.01 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7334_of__int__less__0__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 12.85/7.01 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7335_of__int__0__less__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 12.85/7.01 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7336_of__int__0__less__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 12.85/7.01 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7337_of__int__0__less__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 12.85/7.01 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7338_of__int__numeral__le__iff,axiom,
% 12.85/7.01 ! [N: num,Z: int] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 12.85/7.01 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7339_of__int__numeral__le__iff,axiom,
% 12.85/7.01 ! [N: num,Z: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 12.85/7.01 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7340_of__int__numeral__le__iff,axiom,
% 12.85/7.01 ! [N: num,Z: int] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 12.85/7.01 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7341_of__int__le__numeral__iff,axiom,
% 12.85/7.01 ! [Z: int,N: num] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 12.85/7.01 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7342_of__int__le__numeral__iff,axiom,
% 12.85/7.01 ! [Z: int,N: num] :
% 12.85/7.01 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 12.85/7.01 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7343_of__int__le__numeral__iff,axiom,
% 12.85/7.01 ! [Z: int,N: num] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 12.85/7.01 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7344_of__int__less__numeral__iff,axiom,
% 12.85/7.01 ! [Z: int,N: num] :
% 12.85/7.01 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 12.85/7.01 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7345_of__int__less__numeral__iff,axiom,
% 12.85/7.01 ! [Z: int,N: num] :
% 12.85/7.01 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 12.85/7.01 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7346_of__int__less__numeral__iff,axiom,
% 12.85/7.01 ! [Z: int,N: num] :
% 12.85/7.01 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 12.85/7.01 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7347_of__int__numeral__less__iff,axiom,
% 12.85/7.01 ! [N: num,Z: int] :
% 12.85/7.01 ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 12.85/7.01 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7348_of__int__numeral__less__iff,axiom,
% 12.85/7.01 ! [N: num,Z: int] :
% 12.85/7.01 ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 12.85/7.01 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7349_of__int__numeral__less__iff,axiom,
% 12.85/7.01 ! [N: num,Z: int] :
% 12.85/7.01 ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 12.85/7.01 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7350_of__int__le__1__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 12.85/7.01 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7351_of__int__le__1__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 12.85/7.01 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7352_of__int__le__1__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 12.85/7.01 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7353_of__int__1__le__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 12.85/7.01 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7354_of__int__1__le__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 12.85/7.01 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7355_of__int__1__le__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 12.85/7.01 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7356_of__int__less__1__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 12.85/7.01 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7357_of__int__less__1__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 12.85/7.01 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7358_of__int__less__1__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 12.85/7.01 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7359_of__int__1__less__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 12.85/7.01 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7360_of__int__1__less__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 12.85/7.01 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7361_of__int__1__less__iff,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 12.85/7.01 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7362_of__real__neg__numeral,axiom,
% 12.85/7.01 ! [W: num] :
% 12.85/7.01 ( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.85/7.01 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7363_of__real__neg__numeral,axiom,
% 12.85/7.01 ! [W: num] :
% 12.85/7.01 ( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.85/7.01 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7364_of__int__power__le__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: int,B: int,W: nat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 12.85/7.01 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7365_of__int__power__le__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: int,B: int,W: nat] :
% 12.85/7.01 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 12.85/7.01 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7366_of__int__power__le__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: int,B: int,W: nat] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 12.85/7.01 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7367_of__int__le__of__int__power__cancel__iff,axiom,
% 12.85/7.01 ! [B: int,W: nat,X: int] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 12.85/7.01 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7368_of__int__le__of__int__power__cancel__iff,axiom,
% 12.85/7.01 ! [B: int,W: nat,X: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 12.85/7.01 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7369_of__int__le__of__int__power__cancel__iff,axiom,
% 12.85/7.01 ! [B: int,W: nat,X: int] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 12.85/7.01 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7370_numeral__power__eq__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,Y4: int] :
% 12.85/7.01 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
% 12.85/7.01 = ( ring_17405671764205052669omplex @ Y4 ) )
% 12.85/7.01 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 12.85/7.01 = Y4 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7371_numeral__power__eq__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,Y4: int] :
% 12.85/7.01 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 12.85/7.01 = ( ring_1_of_int_real @ Y4 ) )
% 12.85/7.01 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 12.85/7.01 = Y4 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7372_numeral__power__eq__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,Y4: int] :
% 12.85/7.01 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
% 12.85/7.01 = ( ring_1_of_int_rat @ Y4 ) )
% 12.85/7.01 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 12.85/7.01 = Y4 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7373_numeral__power__eq__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,Y4: int] :
% 12.85/7.01 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 12.85/7.01 = ( ring_1_of_int_int @ Y4 ) )
% 12.85/7.01 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 12.85/7.01 = Y4 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7374_of__int__eq__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [Y4: int,X: num,N: nat] :
% 12.85/7.01 ( ( ( ring_17405671764205052669omplex @ Y4 )
% 12.85/7.01 = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
% 12.85/7.01 = ( Y4
% 12.85/7.01 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7375_of__int__eq__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [Y4: int,X: num,N: nat] :
% 12.85/7.01 ( ( ( ring_1_of_int_real @ Y4 )
% 12.85/7.01 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 12.85/7.01 = ( Y4
% 12.85/7.01 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7376_of__int__eq__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [Y4: int,X: num,N: nat] :
% 12.85/7.01 ( ( ( ring_1_of_int_rat @ Y4 )
% 12.85/7.01 = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 12.85/7.01 = ( Y4
% 12.85/7.01 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7377_of__int__eq__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [Y4: int,X: num,N: nat] :
% 12.85/7.01 ( ( ( ring_1_of_int_int @ Y4 )
% 12.85/7.01 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 12.85/7.01 = ( Y4
% 12.85/7.01 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7378_of__int__power__less__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: int,B: int,W: nat] :
% 12.85/7.01 ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 12.85/7.01 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7379_of__int__power__less__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: int,B: int,W: nat] :
% 12.85/7.01 ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 12.85/7.01 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7380_of__int__power__less__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: int,B: int,W: nat] :
% 12.85/7.01 ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 12.85/7.01 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7381_of__int__less__of__int__power__cancel__iff,axiom,
% 12.85/7.01 ! [B: int,W: nat,X: int] :
% 12.85/7.01 ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 12.85/7.01 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7382_of__int__less__of__int__power__cancel__iff,axiom,
% 12.85/7.01 ! [B: int,W: nat,X: int] :
% 12.85/7.01 ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 12.85/7.01 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7383_of__int__less__of__int__power__cancel__iff,axiom,
% 12.85/7.01 ! [B: int,W: nat,X: int] :
% 12.85/7.01 ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 12.85/7.01 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7384_of__nat__nat,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.85/7.01 => ( ( semiri8010041392384452111omplex @ ( nat2 @ Z ) )
% 12.85/7.01 = ( ring_17405671764205052669omplex @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7385_of__nat__nat,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.85/7.01 => ( ( semiri5074537144036343181t_real @ ( nat2 @ Z ) )
% 12.85/7.01 = ( ring_1_of_int_real @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7386_of__nat__nat,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.85/7.01 => ( ( semiri681578069525770553at_rat @ ( nat2 @ Z ) )
% 12.85/7.01 = ( ring_1_of_int_rat @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7387_of__nat__nat,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.85/7.01 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 12.85/7.01 = ( ring_1_of_int_int @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7388_sin__npi__int,axiom,
% 12.85/7.01 ! [N: int] :
% 12.85/7.01 ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 12.85/7.01 = zero_zero_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7389_cos__arccos,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/7.01 => ( ( cos_real @ ( arccos @ Y4 ) )
% 12.85/7.01 = Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7390_tan__periodic__int,axiom,
% 12.85/7.01 ! [X: real,I: int] :
% 12.85/7.01 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
% 12.85/7.01 = ( tan_real @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7391_of__int__le__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [A: int,X: num,N: nat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 12.85/7.01 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7392_of__int__le__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [A: int,X: num,N: nat] :
% 12.85/7.01 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 12.85/7.01 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7393_of__int__le__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [A: int,X: num,N: nat] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 12.85/7.01 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7394_numeral__power__le__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,A: int] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 12.85/7.01 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7395_numeral__power__le__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,A: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 12.85/7.01 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7396_numeral__power__le__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,A: int] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 12.85/7.01 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7397_of__int__less__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [A: int,X: num,N: nat] :
% 12.85/7.01 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 12.85/7.01 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7398_of__int__less__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [A: int,X: num,N: nat] :
% 12.85/7.01 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 12.85/7.01 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7399_of__int__less__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [A: int,X: num,N: nat] :
% 12.85/7.01 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 12.85/7.01 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7400_numeral__power__less__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,A: int] :
% 12.85/7.01 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 12.85/7.01 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7401_numeral__power__less__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,A: int] :
% 12.85/7.01 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 12.85/7.01 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7402_numeral__power__less__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,A: int] :
% 12.85/7.01 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 12.85/7.01 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7403_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [Y4: int,X: num,N: nat] :
% 12.85/7.01 ( ( ( ring_1_of_int_real @ Y4 )
% 12.85/7.01 = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 12.85/7.01 = ( Y4
% 12.85/7.01 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7404_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [Y4: int,X: num,N: nat] :
% 12.85/7.01 ( ( ( ring_1_of_int_int @ Y4 )
% 12.85/7.01 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 12.85/7.01 = ( Y4
% 12.85/7.01 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7405_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [Y4: int,X: num,N: nat] :
% 12.85/7.01 ( ( ( ring_17405671764205052669omplex @ Y4 )
% 12.85/7.01 = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N ) )
% 12.85/7.01 = ( Y4
% 12.85/7.01 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7406_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [Y4: int,X: num,N: nat] :
% 12.85/7.01 ( ( ( ring_18347121197199848620nteger @ Y4 )
% 12.85/7.01 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 12.85/7.01 = ( Y4
% 12.85/7.01 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7407_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [Y4: int,X: num,N: nat] :
% 12.85/7.01 ( ( ( ring_1_of_int_rat @ Y4 )
% 12.85/7.01 = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 12.85/7.01 = ( Y4
% 12.85/7.01 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7408_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,Y4: int] :
% 12.85/7.01 ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N )
% 12.85/7.01 = ( ring_1_of_int_real @ Y4 ) )
% 12.85/7.01 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 12.85/7.01 = Y4 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7409_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,Y4: int] :
% 12.85/7.01 ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 12.85/7.01 = ( ring_1_of_int_int @ Y4 ) )
% 12.85/7.01 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 12.85/7.01 = Y4 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7410_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,Y4: int] :
% 12.85/7.01 ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N )
% 12.85/7.01 = ( ring_17405671764205052669omplex @ Y4 ) )
% 12.85/7.01 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 12.85/7.01 = Y4 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7411_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,Y4: int] :
% 12.85/7.01 ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N )
% 12.85/7.01 = ( ring_18347121197199848620nteger @ Y4 ) )
% 12.85/7.01 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 12.85/7.01 = Y4 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7412_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,Y4: int] :
% 12.85/7.01 ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N )
% 12.85/7.01 = ( ring_1_of_int_rat @ Y4 ) )
% 12.85/7.01 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 12.85/7.01 = Y4 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7413_norm__of__real__addn,axiom,
% 12.85/7.01 ! [X: real,B: num] :
% 12.85/7.01 ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( numeral_numeral_real @ B ) ) )
% 12.85/7.01 = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7414_norm__of__real__addn,axiom,
% 12.85/7.01 ! [X: real,B: num] :
% 12.85/7.01 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( numera6690914467698888265omplex @ B ) ) )
% 12.85/7.01 = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7415_arccos__0,axiom,
% 12.85/7.01 ( ( arccos @ zero_zero_real )
% 12.85/7.01 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7416_cos__of__real__pi__half,axiom,
% 12.85/7.01 ( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 = zero_zero_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7417_cos__of__real__pi__half,axiom,
% 12.85/7.01 ( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 12.85/7.01 = zero_zero_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7418_sin__of__real__pi__half,axiom,
% 12.85/7.01 ( ( sin_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 = one_one_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7419_sin__of__real__pi__half,axiom,
% 12.85/7.01 ( ( sin_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 12.85/7.01 = one_one_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7420_sin__int__2pin,axiom,
% 12.85/7.01 ! [N: int] :
% 12.85/7.01 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 12.85/7.01 = zero_zero_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7421_cos__int__2pin,axiom,
% 12.85/7.01 ! [N: int] :
% 12.85/7.01 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 12.85/7.01 = one_one_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7422_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,A: int] :
% 12.85/7.01 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 12.85/7.01 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7423_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,A: int] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 12.85/7.01 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7424_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,A: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 12.85/7.01 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7425_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,A: int] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 12.85/7.01 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7426_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [A: int,X: num,N: nat] :
% 12.85/7.01 ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 12.85/7.01 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7427_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [A: int,X: num,N: nat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 12.85/7.01 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7428_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [A: int,X: num,N: nat] :
% 12.85/7.01 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 12.85/7.01 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7429_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [A: int,X: num,N: nat] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 12.85/7.01 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7430_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [A: int,X: num,N: nat] :
% 12.85/7.01 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 12.85/7.01 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7431_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [A: int,X: num,N: nat] :
% 12.85/7.01 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 12.85/7.01 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7432_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [A: int,X: num,N: nat] :
% 12.85/7.01 ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 12.85/7.01 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7433_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 12.85/7.01 ! [A: int,X: num,N: nat] :
% 12.85/7.01 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 12.85/7.01 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7434_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,A: int] :
% 12.85/7.01 ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 12.85/7.01 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7435_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,A: int] :
% 12.85/7.01 ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 12.85/7.01 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7436_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,A: int] :
% 12.85/7.01 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 12.85/7.01 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7437_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 12.85/7.01 ! [X: num,N: nat,A: int] :
% 12.85/7.01 ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 12.85/7.01 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7438_ex__le__of__int,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ? [Z2: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7439_ex__le__of__int,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ? [Z2: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7440_mult__of__int__commute,axiom,
% 12.85/7.01 ! [X: int,Y4: complex] :
% 12.85/7.01 ( ( times_times_complex @ ( ring_17405671764205052669omplex @ X ) @ Y4 )
% 12.85/7.01 = ( times_times_complex @ Y4 @ ( ring_17405671764205052669omplex @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7441_mult__of__int__commute,axiom,
% 12.85/7.01 ! [X: int,Y4: real] :
% 12.85/7.01 ( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y4 )
% 12.85/7.01 = ( times_times_real @ Y4 @ ( ring_1_of_int_real @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7442_mult__of__int__commute,axiom,
% 12.85/7.01 ! [X: int,Y4: rat] :
% 12.85/7.01 ( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y4 )
% 12.85/7.01 = ( times_times_rat @ Y4 @ ( ring_1_of_int_rat @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7443_mult__of__int__commute,axiom,
% 12.85/7.01 ! [X: int,Y4: int] :
% 12.85/7.01 ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y4 )
% 12.85/7.01 = ( times_times_int @ Y4 @ ( ring_1_of_int_int @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7444_cos__int__times__real,axiom,
% 12.85/7.01 ! [M: int,X: real] :
% 12.85/7.01 ( ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X ) ) )
% 12.85/7.01 = ( real_V1803761363581548252l_real @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7445_cos__int__times__real,axiom,
% 12.85/7.01 ! [M: int,X: real] :
% 12.85/7.01 ( ( cos_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X ) ) )
% 12.85/7.01 = ( real_V4546457046886955230omplex @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7446_sin__int__times__real,axiom,
% 12.85/7.01 ! [M: int,X: real] :
% 12.85/7.01 ( ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X ) ) )
% 12.85/7.01 = ( real_V1803761363581548252l_real @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7447_sin__int__times__real,axiom,
% 12.85/7.01 ! [M: int,X: real] :
% 12.85/7.01 ( ( sin_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X ) ) )
% 12.85/7.01 = ( real_V4546457046886955230omplex @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7448_of__int__floor__le,axiom,
% 12.85/7.01 ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7449_of__int__floor__le,axiom,
% 12.85/7.01 ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7450_le__of__int__ceiling,axiom,
% 12.85/7.01 ! [X: rat] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7451_le__of__int__ceiling,axiom,
% 12.85/7.01 ! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7452_complex__of__real__def,axiom,
% 12.85/7.01 ( real_V4546457046886955230omplex
% 12.85/7.01 = ( ^ [R5: real] : ( complex2 @ R5 @ zero_zero_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7453_complex__of__real__code,axiom,
% 12.85/7.01 ( real_V4546457046886955230omplex
% 12.85/7.01 = ( ^ [X3: real] : ( complex2 @ X3 @ zero_zero_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7454_complex__eq__cancel__iff2,axiom,
% 12.85/7.01 ! [X: real,Y4: real,Xa: real] :
% 12.85/7.01 ( ( ( complex2 @ X @ Y4 )
% 12.85/7.01 = ( real_V4546457046886955230omplex @ Xa ) )
% 12.85/7.01 = ( ( X = Xa )
% 12.85/7.01 & ( Y4 = zero_zero_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7455_complex__of__real__mult__Complex,axiom,
% 12.85/7.01 ! [R2: real,X: real,Y4: real] :
% 12.85/7.01 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X @ Y4 ) )
% 12.85/7.01 = ( complex2 @ ( times_times_real @ R2 @ X ) @ ( times_times_real @ R2 @ Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7456_Complex__mult__complex__of__real,axiom,
% 12.85/7.01 ! [X: real,Y4: real,R2: real] :
% 12.85/7.01 ( ( times_times_complex @ ( complex2 @ X @ Y4 ) @ ( real_V4546457046886955230omplex @ R2 ) )
% 12.85/7.01 = ( complex2 @ ( times_times_real @ X @ R2 ) @ ( times_times_real @ Y4 @ R2 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7457_take__bit__of__int,axiom,
% 12.85/7.01 ! [N: nat,K: int] :
% 12.85/7.01 ( ( bit_se2923211474154528505it_int @ N @ ( ring_1_of_int_int @ K ) )
% 12.85/7.01 = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7458_of__int__and__eq,axiom,
% 12.85/7.01 ! [K: int,L: int] :
% 12.85/7.01 ( ( ring_1_of_int_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 12.85/7.01 = ( bit_se725231765392027082nd_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7459_of__int__mask__eq,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( ring_1_of_int_int @ ( bit_se2000444600071755411sk_int @ N ) )
% 12.85/7.01 = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7460_nonzero__of__real__divide,axiom,
% 12.85/7.01 ! [Y4: real,X: real] :
% 12.85/7.01 ( ( Y4 != zero_zero_real )
% 12.85/7.01 => ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X @ Y4 ) )
% 12.85/7.01 = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7461_nonzero__of__real__divide,axiom,
% 12.85/7.01 ! [Y4: real,X: real] :
% 12.85/7.01 ( ( Y4 != zero_zero_real )
% 12.85/7.01 => ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X @ Y4 ) )
% 12.85/7.01 = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7462_le__floor__iff,axiom,
% 12.85/7.01 ! [Z: int,X: real] :
% 12.85/7.01 ( ( ord_less_eq_int @ Z @ ( archim6058952711729229775r_real @ X ) )
% 12.85/7.01 = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7463_le__floor__iff,axiom,
% 12.85/7.01 ! [Z: int,X: rat] :
% 12.85/7.01 ( ( ord_less_eq_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
% 12.85/7.01 = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7464_ceiling__le__iff,axiom,
% 12.85/7.01 ! [X: rat,Z: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
% 12.85/7.01 = ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7465_ceiling__le__iff,axiom,
% 12.85/7.01 ! [X: real,Z: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z )
% 12.85/7.01 = ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7466_ceiling__le,axiom,
% 12.85/7.01 ! [X: rat,A: int] :
% 12.85/7.01 ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) )
% 12.85/7.01 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7467_ceiling__le,axiom,
% 12.85/7.01 ! [X: real,A: int] :
% 12.85/7.01 ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
% 12.85/7.01 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7468_real__of__int__div4,axiom,
% 12.85/7.01 ! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7469_floor__divide__of__int__eq,axiom,
% 12.85/7.01 ! [K: int,L: int] :
% 12.85/7.01 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( ring_1_of_int_real @ K ) @ ( ring_1_of_int_real @ L ) ) )
% 12.85/7.01 = ( divide_divide_int @ K @ L ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7470_floor__divide__of__int__eq,axiom,
% 12.85/7.01 ! [K: int,L: int] :
% 12.85/7.01 ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ K ) @ ( ring_1_of_int_rat @ L ) ) )
% 12.85/7.01 = ( divide_divide_int @ K @ L ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7471_floor__power,axiom,
% 12.85/7.01 ! [X: real,N: nat] :
% 12.85/7.01 ( ( X
% 12.85/7.01 = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) )
% 12.85/7.01 => ( ( archim6058952711729229775r_real @ ( power_power_real @ X @ N ) )
% 12.85/7.01 = ( power_power_int @ ( archim6058952711729229775r_real @ X ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7472_floor__power,axiom,
% 12.85/7.01 ! [X: rat,N: nat] :
% 12.85/7.01 ( ( X
% 12.85/7.01 = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) )
% 12.85/7.01 => ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X @ N ) )
% 12.85/7.01 = ( power_power_int @ ( archim3151403230148437115or_rat @ X ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7473_real__of__int__div,axiom,
% 12.85/7.01 ! [D: int,N: int] :
% 12.85/7.01 ( ( dvd_dvd_int @ D @ N )
% 12.85/7.01 => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
% 12.85/7.01 = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7474_arccos__le__arccos,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 12.85/7.01 => ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/7.01 => ( ord_less_eq_real @ ( arccos @ Y4 ) @ ( arccos @ X ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7475_arccos__eq__iff,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 12.85/7.01 & ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real ) )
% 12.85/7.01 => ( ( ( arccos @ X )
% 12.85/7.01 = ( arccos @ Y4 ) )
% 12.85/7.01 = ( X = Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7476_arccos__le__mono,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y4 ) )
% 12.85/7.01 = ( ord_less_eq_real @ Y4 @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7477_of__int__nonneg,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.85/7.01 => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7478_of__int__nonneg,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.85/7.01 => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7479_of__int__nonneg,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 12.85/7.01 => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7480_of__int__leD,axiom,
% 12.85/7.01 ! [N: int,X: code_integer] :
% 12.85/7.01 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 12.85/7.01 => ( ( N = zero_zero_int )
% 12.85/7.01 | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7481_of__int__leD,axiom,
% 12.85/7.01 ! [N: int,X: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 12.85/7.01 => ( ( N = zero_zero_int )
% 12.85/7.01 | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7482_of__int__leD,axiom,
% 12.85/7.01 ! [N: int,X: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 12.85/7.01 => ( ( N = zero_zero_int )
% 12.85/7.01 | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7483_of__int__leD,axiom,
% 12.85/7.01 ! [N: int,X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 12.85/7.01 => ( ( N = zero_zero_int )
% 12.85/7.01 | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7484_of__int__pos,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_int @ zero_zero_int @ Z )
% 12.85/7.01 => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7485_of__int__pos,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_int @ zero_zero_int @ Z )
% 12.85/7.01 => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7486_of__int__pos,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( ord_less_int @ zero_zero_int @ Z )
% 12.85/7.01 => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7487_of__int__lessD,axiom,
% 12.85/7.01 ! [N: int,X: code_integer] :
% 12.85/7.01 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 12.85/7.01 => ( ( N = zero_zero_int )
% 12.85/7.01 | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7488_of__int__lessD,axiom,
% 12.85/7.01 ! [N: int,X: real] :
% 12.85/7.01 ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 12.85/7.01 => ( ( N = zero_zero_int )
% 12.85/7.01 | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7489_of__int__lessD,axiom,
% 12.85/7.01 ! [N: int,X: rat] :
% 12.85/7.01 ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 12.85/7.01 => ( ( N = zero_zero_int )
% 12.85/7.01 | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7490_of__int__lessD,axiom,
% 12.85/7.01 ! [N: int,X: int] :
% 12.85/7.01 ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 12.85/7.01 => ( ( N = zero_zero_int )
% 12.85/7.01 | ( ord_less_int @ one_one_int @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7491_floor__exists1,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ? [X4: int] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ X )
% 12.85/7.01 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 12.85/7.01 & ! [Y3: int] :
% 12.85/7.01 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y3 ) @ X )
% 12.85/7.01 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y3 @ one_one_int ) ) ) )
% 12.85/7.01 => ( Y3 = X4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7492_floor__exists1,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ? [X4: int] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ X )
% 12.85/7.01 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 12.85/7.01 & ! [Y3: int] :
% 12.85/7.01 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y3 ) @ X )
% 12.85/7.01 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y3 @ one_one_int ) ) ) )
% 12.85/7.01 => ( Y3 = X4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7493_floor__exists,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ? [Z2: int] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
% 12.85/7.01 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7494_floor__exists,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ? [Z2: int] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
% 12.85/7.01 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7495_of__int__ceiling__le__add__one,axiom,
% 12.85/7.01 ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7496_of__int__ceiling__le__add__one,axiom,
% 12.85/7.01 ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7497_of__int__neg__numeral,axiom,
% 12.85/7.01 ! [K: num] :
% 12.85/7.01 ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.85/7.01 = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7498_of__int__neg__numeral,axiom,
% 12.85/7.01 ! [K: num] :
% 12.85/7.01 ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.85/7.01 = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7499_of__int__neg__numeral,axiom,
% 12.85/7.01 ! [K: num] :
% 12.85/7.01 ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.85/7.01 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7500_of__int__neg__numeral,axiom,
% 12.85/7.01 ! [K: num] :
% 12.85/7.01 ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.85/7.01 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7501_of__int__neg__numeral,axiom,
% 12.85/7.01 ! [K: num] :
% 12.85/7.01 ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.85/7.01 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7502_of__int__ceiling__diff__one__le,axiom,
% 12.85/7.01 ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7503_of__int__ceiling__diff__one__le,axiom,
% 12.85/7.01 ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7504_of__nat__less__of__int__iff,axiom,
% 12.85/7.01 ! [N: nat,X: int] :
% 12.85/7.01 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
% 12.85/7.01 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7505_of__nat__less__of__int__iff,axiom,
% 12.85/7.01 ! [N: nat,X: int] :
% 12.85/7.01 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X ) )
% 12.85/7.01 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7506_of__nat__less__of__int__iff,axiom,
% 12.85/7.01 ! [N: nat,X: int] :
% 12.85/7.01 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
% 12.85/7.01 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7507_int__le__real__less,axiom,
% 12.85/7.01 ( ord_less_eq_int
% 12.85/7.01 = ( ^ [N4: int,M5: int] : ( ord_less_real @ ( ring_1_of_int_real @ N4 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M5 ) @ one_one_real ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7508_int__less__real__le,axiom,
% 12.85/7.01 ( ord_less_int
% 12.85/7.01 = ( ^ [N4: int,M5: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N4 ) @ one_one_real ) @ ( ring_1_of_int_real @ M5 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7509_ceiling__divide__eq__div,axiom,
% 12.85/7.01 ! [A: int,B: int] :
% 12.85/7.01 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) )
% 12.85/7.01 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7510_ceiling__divide__eq__div,axiom,
% 12.85/7.01 ! [A: int,B: int] :
% 12.85/7.01 ( ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) )
% 12.85/7.01 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7511_sin__zero__iff__int2,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ( sin_real @ X )
% 12.85/7.01 = zero_zero_real )
% 12.85/7.01 = ( ? [I2: int] :
% 12.85/7.01 ( X
% 12.85/7.01 = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ pi ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7512_arccos__cos__eq__abs__2pi,axiom,
% 12.85/7.01 ! [Theta: real] :
% 12.85/7.01 ~ ! [K2: int] :
% 12.85/7.01 ( ( arccos @ ( cos_real @ Theta ) )
% 12.85/7.01 != ( 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 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7513_real__of__int__div__aux,axiom,
% 12.85/7.01 ! [X: int,D: int] :
% 12.85/7.01 ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
% 12.85/7.01 = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7514_floor__eq,axiom,
% 12.85/7.01 ! [N: int,X: real] :
% 12.85/7.01 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 12.85/7.01 => ( ( archim6058952711729229775r_real @ X )
% 12.85/7.01 = N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7515_real__of__int__floor__add__one__gt,axiom,
% 12.85/7.01 ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7516_real__of__int__floor__add__one__ge,axiom,
% 12.85/7.01 ! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7517_real__of__int__floor__gt__diff__one,axiom,
% 12.85/7.01 ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7518_real__of__int__floor__ge__diff__one,axiom,
% 12.85/7.01 ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7519_arccos__lbound,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/7.01 => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7520_arccos__less__arccos,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ Y4 )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/7.01 => ( ord_less_real @ ( arccos @ Y4 ) @ ( arccos @ X ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7521_arccos__less__mono,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
% 12.85/7.01 => ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y4 ) )
% 12.85/7.01 = ( ord_less_real @ Y4 @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7522_arccos__ubound,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/7.01 => ( ord_less_eq_real @ ( arccos @ Y4 ) @ pi ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7523_arccos__cos,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( ord_less_eq_real @ X @ pi )
% 12.85/7.01 => ( ( arccos @ ( cos_real @ X ) )
% 12.85/7.01 = X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7524_floor__split,axiom,
% 12.85/7.01 ! [P: int > $o,T: real] :
% 12.85/7.01 ( ( P @ ( archim6058952711729229775r_real @ T ) )
% 12.85/7.01 = ( ! [I2: int] :
% 12.85/7.01 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I2 ) @ T )
% 12.85/7.01 & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) ) )
% 12.85/7.01 => ( P @ I2 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7525_floor__split,axiom,
% 12.85/7.01 ! [P: int > $o,T: rat] :
% 12.85/7.01 ( ( P @ ( archim3151403230148437115or_rat @ T ) )
% 12.85/7.01 = ( ! [I2: int] :
% 12.85/7.01 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I2 ) @ T )
% 12.85/7.01 & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) ) )
% 12.85/7.01 => ( P @ I2 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7526_floor__eq__iff,axiom,
% 12.85/7.01 ! [X: real,A: int] :
% 12.85/7.01 ( ( ( archim6058952711729229775r_real @ X )
% 12.85/7.01 = A )
% 12.85/7.01 = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X )
% 12.85/7.01 & ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7527_floor__eq__iff,axiom,
% 12.85/7.01 ! [X: rat,A: int] :
% 12.85/7.01 ( ( ( archim3151403230148437115or_rat @ X )
% 12.85/7.01 = A )
% 12.85/7.01 = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X )
% 12.85/7.01 & ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7528_floor__unique,axiom,
% 12.85/7.01 ! [Z: int,X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) )
% 12.85/7.01 => ( ( archim6058952711729229775r_real @ X )
% 12.85/7.01 = Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7529_floor__unique,axiom,
% 12.85/7.01 ! [Z: int,X: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X )
% 12.85/7.01 => ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
% 12.85/7.01 => ( ( archim3151403230148437115or_rat @ X )
% 12.85/7.01 = Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7530_cos__arccos__abs,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
% 12.85/7.01 => ( ( cos_real @ ( arccos @ Y4 ) )
% 12.85/7.01 = Y4 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7531_norm__of__real__diff,axiom,
% 12.85/7.01 ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( real_V1803761363581548252l_real @ B ) @ ( real_V1803761363581548252l_real @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7532_norm__of__real__diff,axiom,
% 12.85/7.01 ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( real_V4546457046886955230omplex @ B ) @ ( real_V4546457046886955230omplex @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7533_ceiling__split,axiom,
% 12.85/7.01 ! [P: int > $o,T: rat] :
% 12.85/7.01 ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
% 12.85/7.01 = ( ! [I2: int] :
% 12.85/7.01 ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) @ T )
% 12.85/7.01 & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I2 ) ) )
% 12.85/7.01 => ( P @ I2 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7534_ceiling__split,axiom,
% 12.85/7.01 ! [P: int > $o,T: real] :
% 12.85/7.01 ( ( P @ ( archim7802044766580827645g_real @ T ) )
% 12.85/7.01 = ( ! [I2: int] :
% 12.85/7.01 ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) @ T )
% 12.85/7.01 & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I2 ) ) )
% 12.85/7.01 => ( P @ I2 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7535_ceiling__eq__iff,axiom,
% 12.85/7.01 ! [X: rat,A: int] :
% 12.85/7.01 ( ( ( archim2889992004027027881ng_rat @ X )
% 12.85/7.01 = A )
% 12.85/7.01 = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X )
% 12.85/7.01 & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7536_ceiling__eq__iff,axiom,
% 12.85/7.01 ! [X: real,A: int] :
% 12.85/7.01 ( ( ( archim7802044766580827645g_real @ X )
% 12.85/7.01 = A )
% 12.85/7.01 = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X )
% 12.85/7.01 & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7537_ceiling__unique,axiom,
% 12.85/7.01 ! [Z: int,X: rat] :
% 12.85/7.01 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X )
% 12.85/7.01 => ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) )
% 12.85/7.01 => ( ( archim2889992004027027881ng_rat @ X )
% 12.85/7.01 = Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7538_ceiling__unique,axiom,
% 12.85/7.01 ! [Z: int,X: real] :
% 12.85/7.01 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X )
% 12.85/7.01 => ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) )
% 12.85/7.01 => ( ( archim7802044766580827645g_real @ X )
% 12.85/7.01 = Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7539_ceiling__correct,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) @ one_one_rat ) @ X )
% 12.85/7.01 & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7540_ceiling__correct,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) @ one_one_real ) @ X )
% 12.85/7.01 & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7541_arccos__cos__eq__abs,axiom,
% 12.85/7.01 ! [Theta: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
% 12.85/7.01 => ( ( arccos @ ( cos_real @ Theta ) )
% 12.85/7.01 = ( abs_abs_real @ Theta ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7542_less__floor__iff,axiom,
% 12.85/7.01 ! [Z: int,X: real] :
% 12.85/7.01 ( ( ord_less_int @ Z @ ( archim6058952711729229775r_real @ X ) )
% 12.85/7.01 = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7543_less__floor__iff,axiom,
% 12.85/7.01 ! [Z: int,X: rat] :
% 12.85/7.01 ( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
% 12.85/7.01 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7544_floor__le__iff,axiom,
% 12.85/7.01 ! [X: real,Z: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ Z )
% 12.85/7.01 = ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7545_floor__le__iff,axiom,
% 12.85/7.01 ! [X: rat,Z: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
% 12.85/7.01 = ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7546_ceiling__less__iff,axiom,
% 12.85/7.01 ! [X: rat,Z: int] :
% 12.85/7.01 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
% 12.85/7.01 = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7547_ceiling__less__iff,axiom,
% 12.85/7.01 ! [X: real,Z: int] :
% 12.85/7.01 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ Z )
% 12.85/7.01 = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7548_le__ceiling__iff,axiom,
% 12.85/7.01 ! [Z: int,X: rat] :
% 12.85/7.01 ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
% 12.85/7.01 = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7549_le__ceiling__iff,axiom,
% 12.85/7.01 ! [Z: int,X: real] :
% 12.85/7.01 ( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X ) )
% 12.85/7.01 = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7550_floor__correct,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X )
% 12.85/7.01 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7551_floor__correct,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X )
% 12.85/7.01 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7552_real__of__int__div2,axiom,
% 12.85/7.01 ! [N: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7553_real__of__int__div3,axiom,
% 12.85/7.01 ! [N: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) @ one_one_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7554_floor__eq2,axiom,
% 12.85/7.01 ! [N: int,X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 12.85/7.01 => ( ( archim6058952711729229775r_real @ X )
% 12.85/7.01 = N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7555_floor__divide__real__eq__div,axiom,
% 12.85/7.01 ! [B: int,A: real] :
% 12.85/7.01 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 12.85/7.01 => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
% 12.85/7.01 = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7556_floor__divide__lower,axiom,
% 12.85/7.01 ! [Q2: real,P5: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 12.85/7.01 => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P5 @ Q2 ) ) ) @ Q2 ) @ P5 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7557_floor__divide__lower,axiom,
% 12.85/7.01 ! [Q2: rat,P5: rat] :
% 12.85/7.01 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 12.85/7.01 => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P5 @ Q2 ) ) ) @ Q2 ) @ P5 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7558_ceiling__divide__upper,axiom,
% 12.85/7.01 ! [Q2: rat,P5: rat] :
% 12.85/7.01 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 12.85/7.01 => ( ord_less_eq_rat @ P5 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P5 @ Q2 ) ) ) @ Q2 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7559_ceiling__divide__upper,axiom,
% 12.85/7.01 ! [Q2: real,P5: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 12.85/7.01 => ( ord_less_eq_real @ P5 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P5 @ Q2 ) ) ) @ Q2 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7560_arccos__lt__bounded,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 12.85/7.01 => ( ( ord_less_real @ Y4 @ one_one_real )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y4 ) )
% 12.85/7.01 & ( ord_less_real @ ( arccos @ Y4 ) @ pi ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7561_even__of__int__iff,axiom,
% 12.85/7.01 ! [K: int] :
% 12.85/7.01 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
% 12.85/7.01 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7562_even__of__int__iff,axiom,
% 12.85/7.01 ! [K: int] :
% 12.85/7.01 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
% 12.85/7.01 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7563_arccos__bounded,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/7.01 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y4 ) )
% 12.85/7.01 & ( ord_less_eq_real @ ( arccos @ Y4 ) @ pi ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7564_sin__arccos__nonzero,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ one_one_real )
% 12.85/7.01 => ( ( sin_real @ ( arccos @ X ) )
% 12.85/7.01 != zero_zero_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7565_arccos__cos2,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X )
% 12.85/7.01 => ( ( arccos @ ( cos_real @ X ) )
% 12.85/7.01 = ( uminus_uminus_real @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7566_arccos__minus,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 12.85/7.01 => ( ( ord_less_eq_real @ X @ one_one_real )
% 12.85/7.01 => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 12.85/7.01 = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7567_of__int__of__nat,axiom,
% 12.85/7.01 ( ring_18347121197199848620nteger
% 12.85/7.01 = ( ^ [K3: int] : ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri4939895301339042750nteger @ ( nat2 @ K3 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7568_of__int__of__nat,axiom,
% 12.85/7.01 ( ring_17405671764205052669omplex
% 12.85/7.01 = ( ^ [K3: int] : ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri8010041392384452111omplex @ ( nat2 @ K3 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7569_of__int__of__nat,axiom,
% 12.85/7.01 ( ring_1_of_int_real
% 12.85/7.01 = ( ^ [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 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7570_of__int__of__nat,axiom,
% 12.85/7.01 ( ring_1_of_int_rat
% 12.85/7.01 = ( ^ [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 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7571_of__int__of__nat,axiom,
% 12.85/7.01 ( ring_1_of_int_int
% 12.85/7.01 = ( ^ [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 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7572_floor__divide__upper,axiom,
% 12.85/7.01 ! [Q2: real,P5: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 12.85/7.01 => ( ord_less_real @ P5 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P5 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7573_floor__divide__upper,axiom,
% 12.85/7.01 ! [Q2: rat,P5: rat] :
% 12.85/7.01 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 12.85/7.01 => ( ord_less_rat @ P5 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P5 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7574_ceiling__divide__lower,axiom,
% 12.85/7.01 ! [Q2: real,P5: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 12.85/7.01 => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P5 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) @ P5 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7575_ceiling__divide__lower,axiom,
% 12.85/7.01 ! [Q2: rat,P5: rat] :
% 12.85/7.01 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 12.85/7.01 => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P5 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) @ P5 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7576_ceiling__eq,axiom,
% 12.85/7.01 ! [N: int,X: rat] :
% 12.85/7.01 ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X )
% 12.85/7.01 => ( ( ord_less_eq_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
% 12.85/7.01 => ( ( archim2889992004027027881ng_rat @ X )
% 12.85/7.01 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7577_ceiling__eq,axiom,
% 12.85/7.01 ! [N: int,X: real] :
% 12.85/7.01 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
% 12.85/7.01 => ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 12.85/7.01 => ( ( archim7802044766580827645g_real @ X )
% 12.85/7.01 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7578_cos__one__2pi__int,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ( cos_real @ X )
% 12.85/7.01 = one_one_real )
% 12.85/7.01 = ( ? [X3: int] :
% 12.85/7.01 ( X
% 12.85/7.01 = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7579_arccos,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/7.01 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y4 ) )
% 12.85/7.01 & ( ord_less_eq_real @ ( arccos @ Y4 ) @ pi )
% 12.85/7.01 & ( ( cos_real @ ( arccos @ Y4 ) )
% 12.85/7.01 = Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7580_arccos__minus__abs,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 12.85/7.01 => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 12.85/7.01 = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7581_sin__cos__eq,axiom,
% 12.85/7.01 ( sin_real
% 12.85/7.01 = ( ^ [X3: real] : ( cos_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7582_sin__cos__eq,axiom,
% 12.85/7.01 ( sin_complex
% 12.85/7.01 = ( ^ [X3: complex] : ( cos_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7583_cos__sin__eq,axiom,
% 12.85/7.01 ( cos_real
% 12.85/7.01 = ( ^ [X3: real] : ( sin_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7584_cos__sin__eq,axiom,
% 12.85/7.01 ( cos_complex
% 12.85/7.01 = ( ^ [X3: complex] : ( sin_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7585_minus__sin__cos__eq,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( uminus_uminus_real @ ( sin_real @ X ) )
% 12.85/7.01 = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7586_minus__sin__cos__eq,axiom,
% 12.85/7.01 ! [X: complex] :
% 12.85/7.01 ( ( uminus1482373934393186551omplex @ ( sin_complex @ X ) )
% 12.85/7.01 = ( cos_complex @ ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7587_arccos__le__pi2,axiom,
% 12.85/7.01 ! [Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ one_one_real )
% 12.85/7.01 => ( ord_less_eq_real @ ( arccos @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7588_floor__log__eq__powr__iff,axiom,
% 12.85/7.01 ! [X: real,B: real,K: int] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( ord_less_real @ one_one_real @ B )
% 12.85/7.01 => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
% 12.85/7.01 = K )
% 12.85/7.01 = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
% 12.85/7.01 & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7589_cos__zero__iff__int,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ( cos_real @ X )
% 12.85/7.01 = zero_zero_real )
% 12.85/7.01 = ( ? [I2: int] :
% 12.85/7.01 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I2 )
% 12.85/7.01 & ( X
% 12.85/7.01 = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7590_sin__zero__iff__int,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ( sin_real @ X )
% 12.85/7.01 = zero_zero_real )
% 12.85/7.01 = ( ? [I2: int] :
% 12.85/7.01 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I2 )
% 12.85/7.01 & ( X
% 12.85/7.01 = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7591_powr__int,axiom,
% 12.85/7.01 ! [X: real,I: int] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
% 12.85/7.01 => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 12.85/7.01 = ( power_power_real @ X @ ( nat2 @ I ) ) ) )
% 12.85/7.01 & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
% 12.85/7.01 => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 12.85/7.01 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7592_round__unique,axiom,
% 12.85/7.01 ! [X: rat,Y4: int] :
% 12.85/7.01 ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y4 ) )
% 12.85/7.01 => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y4 ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.01 => ( ( archim7778729529865785530nd_rat @ X )
% 12.85/7.01 = Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7593_round__unique,axiom,
% 12.85/7.01 ! [X: real,Y4: int] :
% 12.85/7.01 ( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y4 ) )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y4 ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 12.85/7.01 => ( ( archim8280529875227126926d_real @ X )
% 12.85/7.01 = Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7594_round__unique_H,axiom,
% 12.85/7.01 ! [X: real,N: int] :
% 12.85/7.01 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 => ( ( archim8280529875227126926d_real @ X )
% 12.85/7.01 = N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7595_round__unique_H,axiom,
% 12.85/7.01 ! [X: rat,N: int] :
% 12.85/7.01 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 12.85/7.01 => ( ( archim7778729529865785530nd_rat @ X )
% 12.85/7.01 = N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7596_of__int__round__abs__le,axiom,
% 12.85/7.01 ! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7597_of__int__round__abs__le,axiom,
% 12.85/7.01 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ X ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7598_of__int__round__gt,axiom,
% 12.85/7.01 ! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7599_of__int__round__gt,axiom,
% 12.85/7.01 ! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7600_of__int__round__ge,axiom,
% 12.85/7.01 ! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7601_of__int__round__ge,axiom,
% 12.85/7.01 ! [X: real] : ( ord_less_eq_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7602_round__0,axiom,
% 12.85/7.01 ( ( archim8280529875227126926d_real @ zero_zero_real )
% 12.85/7.01 = zero_zero_int ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7603_round__0,axiom,
% 12.85/7.01 ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
% 12.85/7.01 = zero_zero_int ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7604_round__numeral,axiom,
% 12.85/7.01 ! [N: num] :
% 12.85/7.01 ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
% 12.85/7.01 = ( numeral_numeral_int @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7605_round__numeral,axiom,
% 12.85/7.01 ! [N: num] :
% 12.85/7.01 ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
% 12.85/7.01 = ( numeral_numeral_int @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7606_round__of__nat,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( archim8280529875227126926d_real @ ( semiri5074537144036343181t_real @ N ) )
% 12.85/7.01 = ( semiri1314217659103216013at_int @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7607_round__of__nat,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( archim7778729529865785530nd_rat @ ( semiri681578069525770553at_rat @ N ) )
% 12.85/7.01 = ( semiri1314217659103216013at_int @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7608_round__neg__numeral,axiom,
% 12.85/7.01 ! [N: num] :
% 12.85/7.01 ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 12.85/7.01 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7609_round__neg__numeral,axiom,
% 12.85/7.01 ! [N: num] :
% 12.85/7.01 ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 12.85/7.01 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7610_round__mono,axiom,
% 12.85/7.01 ! [X: rat,Y4: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ X @ Y4 )
% 12.85/7.01 => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7611_round__mono,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/7.01 => ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim8280529875227126926d_real @ Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7612_floor__le__round,axiom,
% 12.85/7.01 ! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim8280529875227126926d_real @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7613_floor__le__round,axiom,
% 12.85/7.01 ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim7778729529865785530nd_rat @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7614_ceiling__ge__round,axiom,
% 12.85/7.01 ! [X: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7615_round__diff__minimal,axiom,
% 12.85/7.01 ! [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 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7616_round__diff__minimal,axiom,
% 12.85/7.01 ! [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 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7617_round__def,axiom,
% 12.85/7.01 ( archim8280529875227126926d_real
% 12.85/7.01 = ( ^ [X3: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7618_round__def,axiom,
% 12.85/7.01 ( archim7778729529865785530nd_rat
% 12.85/7.01 = ( ^ [X3: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7619_of__int__round__le,axiom,
% 12.85/7.01 ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7620_of__int__round__le,axiom,
% 12.85/7.01 ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7621_round__altdef,axiom,
% 12.85/7.01 ( archim8280529875227126926d_real
% 12.85/7.01 = ( ^ [X3: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X3 ) ) @ ( archim7802044766580827645g_real @ X3 ) @ ( archim6058952711729229775r_real @ X3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7622_round__altdef,axiom,
% 12.85/7.01 ( archim7778729529865785530nd_rat
% 12.85/7.01 = ( ^ [X3: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X3 ) ) @ ( archim2889992004027027881ng_rat @ X3 ) @ ( archim3151403230148437115or_rat @ X3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7623_exp__two__pi__i,axiom,
% 12.85/7.01 ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
% 12.85/7.01 = one_one_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7624_exp__two__pi__i_H,axiom,
% 12.85/7.01 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
% 12.85/7.01 = one_one_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7625_cot__less__zero,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ zero_zero_real )
% 12.85/7.01 => ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7626_cot__periodic,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( cot_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 12.85/7.01 = ( cot_real @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7627_cot__zero,axiom,
% 12.85/7.01 ( ( cot_complex @ zero_zero_complex )
% 12.85/7.01 = zero_zero_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7628_cot__zero,axiom,
% 12.85/7.01 ( ( cot_real @ zero_zero_real )
% 12.85/7.01 = zero_zero_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7629_frac__of__int,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z ) )
% 12.85/7.01 = zero_zero_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7630_frac__of__int,axiom,
% 12.85/7.01 ! [Z: int] :
% 12.85/7.01 ( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z ) )
% 12.85/7.01 = zero_zero_rat ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7631_cot__pi,axiom,
% 12.85/7.01 ( ( cot_real @ pi )
% 12.85/7.01 = zero_zero_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7632_cot__npi,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 12.85/7.01 = zero_zero_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7633_power2__i,axiom,
% 12.85/7.01 ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.01 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7634_i__even__power,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.01 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7635_complex__i__not__zero,axiom,
% 12.85/7.01 imaginary_unit != zero_zero_complex ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7636_frac__ge__0,axiom,
% 12.85/7.01 ! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7637_frac__ge__0,axiom,
% 12.85/7.01 ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7638_imaginary__unit_Ocode,axiom,
% 12.85/7.01 ( imaginary_unit
% 12.85/7.01 = ( complex2 @ zero_zero_real @ one_one_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7639_Complex__eq__i,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ( complex2 @ X @ Y4 )
% 12.85/7.01 = imaginary_unit )
% 12.85/7.01 = ( ( X = zero_zero_real )
% 12.85/7.01 & ( Y4 = one_one_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7640_frac__def,axiom,
% 12.85/7.01 ( archim2898591450579166408c_real
% 12.85/7.01 = ( ^ [X3: real] : ( minus_minus_real @ X3 @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X3 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7641_frac__def,axiom,
% 12.85/7.01 ( archimedean_frac_rat
% 12.85/7.01 = ( ^ [X3: rat] : ( minus_minus_rat @ X3 @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X3 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7642_cot__def,axiom,
% 12.85/7.01 ( cot_complex
% 12.85/7.01 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( cos_complex @ X3 ) @ ( sin_complex @ X3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7643_cot__def,axiom,
% 12.85/7.01 ( cot_real
% 12.85/7.01 = ( ^ [X3: real] : ( divide_divide_real @ ( cos_real @ X3 ) @ ( sin_real @ X3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7644_complex__of__real__i,axiom,
% 12.85/7.01 ! [R2: real] :
% 12.85/7.01 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ imaginary_unit )
% 12.85/7.01 = ( complex2 @ zero_zero_real @ R2 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7645_i__complex__of__real,axiom,
% 12.85/7.01 ! [R2: real] :
% 12.85/7.01 ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R2 ) )
% 12.85/7.01 = ( complex2 @ zero_zero_real @ R2 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7646_frac__eq,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ( ( ( archimedean_frac_rat @ X )
% 12.85/7.01 = X )
% 12.85/7.01 = ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 12.85/7.01 & ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7647_frac__eq,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ( archim2898591450579166408c_real @ X )
% 12.85/7.01 = X )
% 12.85/7.01 = ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.01 & ( ord_less_real @ X @ one_one_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7648_frac__add,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y4 ) ) @ one_one_real )
% 12.85/7.01 => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y4 ) )
% 12.85/7.01 = ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y4 ) ) ) )
% 12.85/7.01 & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y4 ) ) @ one_one_real )
% 12.85/7.01 => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y4 ) )
% 12.85/7.01 = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y4 ) ) @ one_one_real ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7649_frac__add,axiom,
% 12.85/7.01 ! [X: rat,Y4: rat] :
% 12.85/7.01 ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y4 ) ) @ one_one_rat )
% 12.85/7.01 => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y4 ) )
% 12.85/7.01 = ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y4 ) ) ) )
% 12.85/7.01 & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y4 ) ) @ one_one_rat )
% 12.85/7.01 => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y4 ) )
% 12.85/7.01 = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y4 ) ) @ one_one_rat ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7650_cot__gt__zero,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 => ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7651_tan__cot_H,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 12.85/7.01 = ( cot_real @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7652_Arg__minus__ii,axiom,
% 12.85/7.01 ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
% 12.85/7.01 = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7653_csqrt__ii,axiom,
% 12.85/7.01 ( ( csqrt @ imaginary_unit )
% 12.85/7.01 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7654_Arg__ii,axiom,
% 12.85/7.01 ( ( arg @ imaginary_unit )
% 12.85/7.01 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7655_cis__minus__pi__half,axiom,
% 12.85/7.01 ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 12.85/7.01 = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7656_signed__take__bit__eq__take__bit__minus,axiom,
% 12.85/7.01 ( bit_ri631733984087533419it_int
% 12.85/7.01 = ( ^ [N4: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N4 ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7657_csqrt__eq__0,axiom,
% 12.85/7.01 ! [Z: complex] :
% 12.85/7.01 ( ( ( csqrt @ Z )
% 12.85/7.01 = zero_zero_complex )
% 12.85/7.01 = ( Z = zero_zero_complex ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7658_csqrt__0,axiom,
% 12.85/7.01 ( ( csqrt @ zero_zero_complex )
% 12.85/7.01 = zero_zero_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7659_cis__zero,axiom,
% 12.85/7.01 ( ( cis @ zero_zero_real )
% 12.85/7.01 = one_one_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7660_bit__numeral__Bit0__Suc__iff,axiom,
% 12.85/7.01 ! [M: num,N: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 12.85/7.01 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7661_bit__numeral__Bit0__Suc__iff,axiom,
% 12.85/7.01 ! [M: num,N: nat] :
% 12.85/7.01 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 12.85/7.01 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7662_bit__numeral__Bit1__Suc__iff,axiom,
% 12.85/7.01 ! [M: num,N: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 12.85/7.01 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7663_bit__numeral__Bit1__Suc__iff,axiom,
% 12.85/7.01 ! [M: num,N: nat] :
% 12.85/7.01 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 12.85/7.01 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7664_signed__take__bit__nonnegative__iff,axiom,
% 12.85/7.01 ! [N: nat,K: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 12.85/7.01 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7665_signed__take__bit__negative__iff,axiom,
% 12.85/7.01 ! [N: nat,K: int] :
% 12.85/7.01 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
% 12.85/7.01 = ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7666_bit__numeral__simps_I2_J,axiom,
% 12.85/7.01 ! [W: num,N: num] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 12.85/7.01 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7667_bit__numeral__simps_I2_J,axiom,
% 12.85/7.01 ! [W: num,N: num] :
% 12.85/7.01 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 12.85/7.01 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7668_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 12.85/7.01 ! [W: num,N: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
% 12.85/7.01 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7669_bit__numeral__simps_I3_J,axiom,
% 12.85/7.01 ! [W: num,N: num] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 12.85/7.01 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7670_bit__numeral__simps_I3_J,axiom,
% 12.85/7.01 ! [W: num,N: num] :
% 12.85/7.01 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 12.85/7.01 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7671_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 12.85/7.01 ! [W: num,N: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
% 12.85/7.01 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7672_power2__csqrt,axiom,
% 12.85/7.01 ! [Z: complex] :
% 12.85/7.01 ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.01 = Z ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7673_bit__0,axiom,
% 12.85/7.01 ! [A: code_integer] :
% 12.85/7.01 ( ( bit_se9216721137139052372nteger @ A @ zero_zero_nat )
% 12.85/7.01 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7674_bit__0,axiom,
% 12.85/7.01 ! [A: int] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ A @ zero_zero_nat )
% 12.85/7.01 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7675_bit__0,axiom,
% 12.85/7.01 ! [A: nat] :
% 12.85/7.01 ( ( bit_se1148574629649215175it_nat @ A @ zero_zero_nat )
% 12.85/7.01 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7676_bit__minus__numeral__int_I1_J,axiom,
% 12.85/7.01 ! [W: num,N: num] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 12.85/7.01 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7677_bit__minus__numeral__int_I2_J,axiom,
% 12.85/7.01 ! [W: num,N: num] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 12.85/7.01 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7678_bit__mod__2__iff,axiom,
% 12.85/7.01 ! [A: code_integer,N: nat] :
% 12.85/7.01 ( ( bit_se9216721137139052372nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N )
% 12.85/7.01 = ( ( N = zero_zero_nat )
% 12.85/7.01 & ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7679_bit__mod__2__iff,axiom,
% 12.85/7.01 ! [A: int,N: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N )
% 12.85/7.01 = ( ( N = zero_zero_nat )
% 12.85/7.01 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7680_bit__mod__2__iff,axiom,
% 12.85/7.01 ! [A: nat,N: nat] :
% 12.85/7.01 ( ( bit_se1148574629649215175it_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 12.85/7.01 = ( ( N = zero_zero_nat )
% 12.85/7.01 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7681_cis__pi__half,axiom,
% 12.85/7.01 ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.01 = imaginary_unit ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7682_cis__2pi,axiom,
% 12.85/7.01 ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 12.85/7.01 = one_one_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7683_cis__Arg,axiom,
% 12.85/7.01 ! [Z: complex] :
% 12.85/7.01 ( ( Z != zero_zero_complex )
% 12.85/7.01 => ( ( cis @ ( arg @ Z ) )
% 12.85/7.01 = ( sgn_sgn_complex @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7684_bit__numeral__iff,axiom,
% 12.85/7.01 ! [M: num,N: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N )
% 12.85/7.01 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7685_bit__numeral__iff,axiom,
% 12.85/7.01 ! [M: num,N: nat] :
% 12.85/7.01 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N )
% 12.85/7.01 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7686_bit__of__nat__iff__bit,axiom,
% 12.85/7.01 ! [M: nat,N: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( semiri1314217659103216013at_int @ M ) @ N )
% 12.85/7.01 = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7687_bit__of__nat__iff__bit,axiom,
% 12.85/7.01 ! [M: nat,N: nat] :
% 12.85/7.01 ( ( bit_se1148574629649215175it_nat @ ( semiri1316708129612266289at_nat @ M ) @ N )
% 12.85/7.01 = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7688_bit__disjunctive__add__iff,axiom,
% 12.85/7.01 ! [A: int,B: int,N: nat] :
% 12.85/7.01 ( ! [N2: nat] :
% 12.85/7.01 ( ~ ( bit_se1146084159140164899it_int @ A @ N2 )
% 12.85/7.01 | ~ ( bit_se1146084159140164899it_int @ B @ N2 ) )
% 12.85/7.01 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B ) @ N )
% 12.85/7.01 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 12.85/7.01 | ( bit_se1146084159140164899it_int @ B @ N ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7689_bit__disjunctive__add__iff,axiom,
% 12.85/7.01 ! [A: nat,B: nat,N: nat] :
% 12.85/7.01 ( ! [N2: nat] :
% 12.85/7.01 ( ~ ( bit_se1148574629649215175it_nat @ A @ N2 )
% 12.85/7.01 | ~ ( bit_se1148574629649215175it_nat @ B @ N2 ) )
% 12.85/7.01 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B ) @ N )
% 12.85/7.01 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 12.85/7.01 | ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7690_bit__and__iff,axiom,
% 12.85/7.01 ! [A: int,B: int,N: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ N )
% 12.85/7.01 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 12.85/7.01 & ( bit_se1146084159140164899it_int @ B @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7691_bit__and__iff,axiom,
% 12.85/7.01 ! [A: nat,B: nat,N: nat] :
% 12.85/7.01 ( ( bit_se1148574629649215175it_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ N )
% 12.85/7.01 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 12.85/7.01 & ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7692_bit__and__int__iff,axiom,
% 12.85/7.01 ! [K: int,L: int,N: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ N )
% 12.85/7.01 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 12.85/7.01 & ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7693_bit__unset__bit__iff,axiom,
% 12.85/7.01 ! [M: nat,A: int,N: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( bit_se4203085406695923979it_int @ M @ A ) @ N )
% 12.85/7.01 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 12.85/7.01 & ( M != N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7694_bit__unset__bit__iff,axiom,
% 12.85/7.01 ! [M: nat,A: nat,N: nat] :
% 12.85/7.01 ( ( bit_se1148574629649215175it_nat @ ( bit_se4205575877204974255it_nat @ M @ A ) @ N )
% 12.85/7.01 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 12.85/7.01 & ( M != N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7695_bit__unset__bit__iff,axiom,
% 12.85/7.01 ! [M: nat,A: code_integer,N: nat] :
% 12.85/7.01 ( ( bit_se9216721137139052372nteger @ ( bit_se8260200283734997820nteger @ M @ A ) @ N )
% 12.85/7.01 = ( ( bit_se9216721137139052372nteger @ A @ N )
% 12.85/7.01 & ( M != N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7696_cis__divide,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B ) )
% 12.85/7.01 = ( cis @ ( minus_minus_real @ A @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7697_cis__neq__zero,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( cis @ A )
% 12.85/7.01 != zero_zero_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7698_not__bit__1__Suc,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7699_not__bit__1__Suc,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7700_bit__1__iff,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ one_one_int @ N )
% 12.85/7.01 = ( N = zero_zero_nat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7701_bit__1__iff,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N )
% 12.85/7.01 = ( N = zero_zero_nat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7702_bit__numeral__simps_I1_J,axiom,
% 12.85/7.01 ! [N: num] :
% 12.85/7.01 ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( numeral_numeral_nat @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7703_bit__numeral__simps_I1_J,axiom,
% 12.85/7.01 ! [N: num] :
% 12.85/7.01 ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7704_bit__take__bit__iff,axiom,
% 12.85/7.01 ! [M: nat,A: int,N: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ M @ A ) @ N )
% 12.85/7.01 = ( ( ord_less_nat @ N @ M )
% 12.85/7.01 & ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7705_bit__take__bit__iff,axiom,
% 12.85/7.01 ! [M: nat,A: nat,N: nat] :
% 12.85/7.01 ( ( bit_se1148574629649215175it_nat @ ( bit_se2925701944663578781it_nat @ M @ A ) @ N )
% 12.85/7.01 = ( ( ord_less_nat @ N @ M )
% 12.85/7.01 & ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7706_bit__of__bool__iff,axiom,
% 12.85/7.01 ! [B: $o,N: nat] :
% 12.85/7.01 ( ( bit_se9216721137139052372nteger @ ( zero_n356916108424825756nteger @ B ) @ N )
% 12.85/7.01 = ( B
% 12.85/7.01 & ( N = zero_zero_nat ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7707_bit__of__bool__iff,axiom,
% 12.85/7.01 ! [B: $o,N: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( zero_n2684676970156552555ol_int @ B ) @ N )
% 12.85/7.01 = ( B
% 12.85/7.01 & ( N = zero_zero_nat ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7708_bit__of__bool__iff,axiom,
% 12.85/7.01 ! [B: $o,N: nat] :
% 12.85/7.01 ( ( bit_se1148574629649215175it_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ N )
% 12.85/7.01 = ( B
% 12.85/7.01 & ( N = zero_zero_nat ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7709_signed__take__bit__eq__if__positive,axiom,
% 12.85/7.01 ! [A: int,N: nat] :
% 12.85/7.01 ( ~ ( bit_se1146084159140164899it_int @ A @ N )
% 12.85/7.01 => ( ( bit_ri631733984087533419it_int @ N @ A )
% 12.85/7.01 = ( bit_se2923211474154528505it_int @ N @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7710_DeMoivre,axiom,
% 12.85/7.01 ! [A: real,N: nat] :
% 12.85/7.01 ( ( power_power_complex @ ( cis @ A ) @ N )
% 12.85/7.01 = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7711_cis__Arg__unique,axiom,
% 12.85/7.01 ! [Z: complex,X: real] :
% 12.85/7.01 ( ( ( sgn_sgn_complex @ Z )
% 12.85/7.01 = ( cis @ X ) )
% 12.85/7.01 => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 12.85/7.01 => ( ( ord_less_eq_real @ X @ pi )
% 12.85/7.01 => ( ( arg @ Z )
% 12.85/7.01 = X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7712_bit__not__int__iff_H,axiom,
% 12.85/7.01 ! [K: int,N: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
% 12.85/7.01 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7713_flip__bit__eq__if,axiom,
% 12.85/7.01 ( bit_se2159334234014336723it_int
% 12.85/7.01 = ( ^ [N4: nat,A3: int] : ( if_nat_int_int @ ( bit_se1146084159140164899it_int @ A3 @ N4 ) @ bit_se4203085406695923979it_int @ bit_se7879613467334960850it_int @ N4 @ A3 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7714_flip__bit__eq__if,axiom,
% 12.85/7.01 ( bit_se2161824704523386999it_nat
% 12.85/7.01 = ( ^ [N4: nat,A3: nat] : ( if_nat_nat_nat @ ( bit_se1148574629649215175it_nat @ A3 @ N4 ) @ bit_se4205575877204974255it_nat @ bit_se7882103937844011126it_nat @ N4 @ A3 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7715_flip__bit__eq__if,axiom,
% 12.85/7.01 ( bit_se1345352211410354436nteger
% 12.85/7.01 = ( ^ [N4: nat,A3: code_integer] : ( if_nat5617392847756311170nteger @ ( bit_se9216721137139052372nteger @ A3 @ N4 ) @ bit_se8260200283734997820nteger @ bit_se2793503036327961859nteger @ N4 @ A3 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7716_Arg__correct,axiom,
% 12.85/7.01 ! [Z: complex] :
% 12.85/7.01 ( ( Z != zero_zero_complex )
% 12.85/7.01 => ( ( ( sgn_sgn_complex @ Z )
% 12.85/7.01 = ( cis @ ( arg @ Z ) ) )
% 12.85/7.01 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 12.85/7.01 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7717_Arg__zero,axiom,
% 12.85/7.01 ( ( arg @ zero_zero_complex )
% 12.85/7.01 = zero_zero_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7718_bit__imp__take__bit__positive,axiom,
% 12.85/7.01 ! [N: nat,M: nat,K: int] :
% 12.85/7.01 ( ( ord_less_nat @ N @ M )
% 12.85/7.01 => ( ( bit_se1146084159140164899it_int @ K @ N )
% 12.85/7.01 => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7719_bit__concat__bit__iff,axiom,
% 12.85/7.01 ! [M: nat,K: int,L: int,N: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
% 12.85/7.01 = ( ( ( ord_less_nat @ N @ M )
% 12.85/7.01 & ( bit_se1146084159140164899it_int @ K @ N ) )
% 12.85/7.01 | ( ( ord_less_eq_nat @ M @ N )
% 12.85/7.01 & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7720_signed__take__bit__eq__concat__bit,axiom,
% 12.85/7.01 ( bit_ri631733984087533419it_int
% 12.85/7.01 = ( ^ [N4: nat,K3: int] : ( bit_concat_bit @ N4 @ K3 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N4 ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7721_exp__eq__0__imp__not__bit,axiom,
% 12.85/7.01 ! [N: nat,A: int] :
% 12.85/7.01 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 12.85/7.01 = zero_zero_int )
% 12.85/7.01 => ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7722_exp__eq__0__imp__not__bit,axiom,
% 12.85/7.01 ! [N: nat,A: nat] :
% 12.85/7.01 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/7.01 = zero_zero_nat )
% 12.85/7.01 => ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7723_bit__Suc,axiom,
% 12.85/7.01 ! [A: int,N: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ A @ ( suc @ N ) )
% 12.85/7.01 = ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7724_bit__Suc,axiom,
% 12.85/7.01 ! [A: nat,N: nat] :
% 12.85/7.01 ( ( bit_se1148574629649215175it_nat @ A @ ( suc @ N ) )
% 12.85/7.01 = ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7725_bit__iff__idd__imp__stable,axiom,
% 12.85/7.01 ! [A: code_integer] :
% 12.85/7.01 ( ! [N2: nat] :
% 12.85/7.01 ( ( bit_se9216721137139052372nteger @ A @ N2 )
% 12.85/7.01 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 12.85/7.01 => ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/7.01 = A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7726_bit__iff__idd__imp__stable,axiom,
% 12.85/7.01 ! [A: int] :
% 12.85/7.01 ( ! [N2: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ A @ N2 )
% 12.85/7.01 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 12.85/7.01 => ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/7.01 = A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7727_bit__iff__idd__imp__stable,axiom,
% 12.85/7.01 ! [A: nat] :
% 12.85/7.01 ( ! [N2: nat] :
% 12.85/7.01 ( ( bit_se1148574629649215175it_nat @ A @ N2 )
% 12.85/7.01 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 12.85/7.01 => ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.01 = A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7728_stable__imp__bit__iff__odd,axiom,
% 12.85/7.01 ! [A: code_integer,N: nat] :
% 12.85/7.01 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/7.01 = A )
% 12.85/7.01 => ( ( bit_se9216721137139052372nteger @ A @ N )
% 12.85/7.01 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7729_stable__imp__bit__iff__odd,axiom,
% 12.85/7.01 ! [A: int,N: nat] :
% 12.85/7.01 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/7.01 = A )
% 12.85/7.01 => ( ( bit_se1146084159140164899it_int @ A @ N )
% 12.85/7.01 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7730_stable__imp__bit__iff__odd,axiom,
% 12.85/7.01 ! [A: nat,N: nat] :
% 12.85/7.01 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.01 = A )
% 12.85/7.01 => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 12.85/7.01 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7731_int__bit__bound,axiom,
% 12.85/7.01 ! [K: int] :
% 12.85/7.01 ~ ! [N2: nat] :
% 12.85/7.01 ( ! [M2: nat] :
% 12.85/7.01 ( ( ord_less_eq_nat @ N2 @ M2 )
% 12.85/7.01 => ( ( bit_se1146084159140164899it_int @ K @ M2 )
% 12.85/7.01 = ( bit_se1146084159140164899it_int @ K @ N2 ) ) )
% 12.85/7.01 => ~ ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 12.85/7.01 => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 12.85/7.01 = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7732_bit__iff__odd,axiom,
% 12.85/7.01 ( bit_se9216721137139052372nteger
% 12.85/7.01 = ( ^ [A3: code_integer,N4: nat] :
% 12.85/7.01 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7733_bit__iff__odd,axiom,
% 12.85/7.01 ( bit_se1146084159140164899it_int
% 12.85/7.01 = ( ^ [A3: int,N4: nat] :
% 12.85/7.01 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7734_bit__iff__odd,axiom,
% 12.85/7.01 ( bit_se1148574629649215175it_nat
% 12.85/7.01 = ( ^ [A3: nat,N4: nat] :
% 12.85/7.01 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7735_and__exp__eq__0__iff__not__bit,axiom,
% 12.85/7.01 ! [A: int,N: nat] :
% 12.85/7.01 ( ( ( bit_se725231765392027082nd_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.01 = zero_zero_int )
% 12.85/7.01 = ( ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7736_and__exp__eq__0__iff__not__bit,axiom,
% 12.85/7.01 ! [A: nat,N: nat] :
% 12.85/7.01 ( ( ( bit_se727722235901077358nd_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.01 = zero_zero_nat )
% 12.85/7.01 = ( ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7737_bit__int__def,axiom,
% 12.85/7.01 ( bit_se1146084159140164899it_int
% 12.85/7.01 = ( ^ [K3: int,N4: nat] :
% 12.85/7.01 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7738_of__real__sqrt,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( real_V4546457046886955230omplex @ ( sqrt @ X ) )
% 12.85/7.01 = ( csqrt @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7739_even__bit__succ__iff,axiom,
% 12.85/7.01 ! [A: code_integer,N: nat] :
% 12.85/7.01 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/7.01 => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ N )
% 12.85/7.01 = ( ( bit_se9216721137139052372nteger @ A @ N )
% 12.85/7.01 | ( N = zero_zero_nat ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7740_even__bit__succ__iff,axiom,
% 12.85/7.01 ! [A: int,N: nat] :
% 12.85/7.01 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/7.01 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N )
% 12.85/7.01 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 12.85/7.01 | ( N = zero_zero_nat ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7741_even__bit__succ__iff,axiom,
% 12.85/7.01 ! [A: nat,N: nat] :
% 12.85/7.01 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/7.01 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N )
% 12.85/7.01 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 12.85/7.01 | ( N = zero_zero_nat ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7742_odd__bit__iff__bit__pred,axiom,
% 12.85/7.01 ! [A: code_integer,N: nat] :
% 12.85/7.01 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/7.01 => ( ( bit_se9216721137139052372nteger @ A @ N )
% 12.85/7.01 = ( ( bit_se9216721137139052372nteger @ ( minus_8373710615458151222nteger @ A @ one_one_Code_integer ) @ N )
% 12.85/7.01 | ( N = zero_zero_nat ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7743_odd__bit__iff__bit__pred,axiom,
% 12.85/7.01 ! [A: int,N: nat] :
% 12.85/7.01 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/7.01 => ( ( bit_se1146084159140164899it_int @ A @ N )
% 12.85/7.01 = ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A @ one_one_int ) @ N )
% 12.85/7.01 | ( N = zero_zero_nat ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7744_odd__bit__iff__bit__pred,axiom,
% 12.85/7.01 ! [A: nat,N: nat] :
% 12.85/7.01 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/7.01 => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 12.85/7.01 = ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A @ one_one_nat ) @ N )
% 12.85/7.01 | ( N = zero_zero_nat ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7745_Arg__bounded,axiom,
% 12.85/7.01 ! [Z: complex] :
% 12.85/7.01 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 12.85/7.01 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7746_bit__sum__mult__2__cases,axiom,
% 12.85/7.01 ! [A: code_integer,B: code_integer,N: nat] :
% 12.85/7.01 ( ! [J2: nat] :
% 12.85/7.01 ~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J2 ) )
% 12.85/7.01 => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ N )
% 12.85/7.01 = ( ( ( N = zero_zero_nat )
% 12.85/7.01 => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 12.85/7.01 & ( ( N != zero_zero_nat )
% 12.85/7.01 => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7747_bit__sum__mult__2__cases,axiom,
% 12.85/7.01 ! [A: int,B: int,N: nat] :
% 12.85/7.01 ( ! [J2: nat] :
% 12.85/7.01 ~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J2 ) )
% 12.85/7.01 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ N )
% 12.85/7.01 = ( ( ( N = zero_zero_nat )
% 12.85/7.01 => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 12.85/7.01 & ( ( N != zero_zero_nat )
% 12.85/7.01 => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7748_bit__sum__mult__2__cases,axiom,
% 12.85/7.01 ! [A: nat,B: nat,N: nat] :
% 12.85/7.01 ( ! [J2: nat] :
% 12.85/7.01 ~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J2 ) )
% 12.85/7.01 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ N )
% 12.85/7.01 = ( ( ( N = zero_zero_nat )
% 12.85/7.01 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 12.85/7.01 & ( ( N != zero_zero_nat )
% 12.85/7.01 => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7749_bit__rec,axiom,
% 12.85/7.01 ( bit_se9216721137139052372nteger
% 12.85/7.01 = ( ^ [A3: code_integer,N4: nat] :
% 12.85/7.01 ( ( ( N4 = zero_zero_nat )
% 12.85/7.01 => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) )
% 12.85/7.01 & ( ( N4 != zero_zero_nat )
% 12.85/7.01 => ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7750_bit__rec,axiom,
% 12.85/7.01 ( bit_se1146084159140164899it_int
% 12.85/7.01 = ( ^ [A3: int,N4: nat] :
% 12.85/7.01 ( ( ( N4 = zero_zero_nat )
% 12.85/7.01 => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
% 12.85/7.01 & ( ( N4 != zero_zero_nat )
% 12.85/7.01 => ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7751_bit__rec,axiom,
% 12.85/7.01 ( bit_se1148574629649215175it_nat
% 12.85/7.01 = ( ^ [A3: nat,N4: nat] :
% 12.85/7.01 ( ( ( N4 = zero_zero_nat )
% 12.85/7.01 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) )
% 12.85/7.01 & ( ( N4 != zero_zero_nat )
% 12.85/7.01 => ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7752_set__bit__eq,axiom,
% 12.85/7.01 ( bit_se7879613467334960850it_int
% 12.85/7.01 = ( ^ [N4: nat,K3: int] :
% 12.85/7.01 ( plus_plus_int @ K3
% 12.85/7.01 @ ( times_times_int
% 12.85/7.01 @ ( zero_n2684676970156552555ol_int
% 12.85/7.01 @ ~ ( bit_se1146084159140164899it_int @ K3 @ N4 ) )
% 12.85/7.01 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7753_unset__bit__eq,axiom,
% 12.85/7.01 ( bit_se4203085406695923979it_int
% 12.85/7.01 = ( ^ [N4: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N4 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7754_take__bit__Suc__from__most,axiom,
% 12.85/7.01 ! [N: nat,K: int] :
% 12.85/7.01 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
% 12.85/7.01 = ( 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 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7755_cis__multiple__2pi,axiom,
% 12.85/7.01 ! [N: real] :
% 12.85/7.01 ( ( member_real @ N @ ring_1_Ints_real )
% 12.85/7.01 => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 12.85/7.01 = one_one_complex ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7756_powr__real__of__int,axiom,
% 12.85/7.01 ! [X: real,N: int] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
% 12.85/7.01 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 12.85/7.01 = ( power_power_real @ X @ ( nat2 @ N ) ) ) )
% 12.85/7.01 & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
% 12.85/7.01 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 12.85/7.01 = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7757_Suc__0__xor__eq,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.85/7.01 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.85/7.01 @ ( zero_n2687167440665602831ol_nat
% 12.85/7.01 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7758_xor__Suc__0__eq,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
% 12.85/7.01 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.85/7.01 @ ( zero_n2687167440665602831ol_nat
% 12.85/7.01 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7759_gbinomial__absorption_H,axiom,
% 12.85/7.01 ! [K: nat,A: complex] :
% 12.85/7.01 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/7.01 => ( ( gbinomial_complex @ A @ K )
% 12.85/7.01 = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7760_gbinomial__absorption_H,axiom,
% 12.85/7.01 ! [K: nat,A: real] :
% 12.85/7.01 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/7.01 => ( ( gbinomial_real @ A @ K )
% 12.85/7.01 = ( 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 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7761_gbinomial__absorption_H,axiom,
% 12.85/7.01 ! [K: nat,A: rat] :
% 12.85/7.01 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/7.01 => ( ( gbinomial_rat @ A @ K )
% 12.85/7.01 = ( 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 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7762_inverse__inverse__eq,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
% 12.85/7.01 = A ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7763_inverse__inverse__eq,axiom,
% 12.85/7.01 ! [A: complex] :
% 12.85/7.01 ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
% 12.85/7.01 = A ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7764_inverse__inverse__eq,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
% 12.85/7.01 = A ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7765_inverse__eq__iff__eq,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ( inverse_inverse_real @ A )
% 12.85/7.01 = ( inverse_inverse_real @ B ) )
% 12.85/7.01 = ( A = B ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7766_inverse__eq__iff__eq,axiom,
% 12.85/7.01 ! [A: complex,B: complex] :
% 12.85/7.01 ( ( ( invers8013647133539491842omplex @ A )
% 12.85/7.01 = ( invers8013647133539491842omplex @ B ) )
% 12.85/7.01 = ( A = B ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7767_inverse__eq__iff__eq,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ( inverse_inverse_rat @ A )
% 12.85/7.01 = ( inverse_inverse_rat @ B ) )
% 12.85/7.01 = ( A = B ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7768_bit_Oxor__left__self,axiom,
% 12.85/7.01 ! [X: int,Y4: int] :
% 12.85/7.01 ( ( bit_se6526347334894502574or_int @ X @ ( bit_se6526347334894502574or_int @ X @ Y4 ) )
% 12.85/7.01 = Y4 ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7769_inverse__nonzero__iff__nonzero,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( ( inverse_inverse_real @ A )
% 12.85/7.01 = zero_zero_real )
% 12.85/7.01 = ( A = zero_zero_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7770_inverse__nonzero__iff__nonzero,axiom,
% 12.85/7.01 ! [A: complex] :
% 12.85/7.01 ( ( ( invers8013647133539491842omplex @ A )
% 12.85/7.01 = zero_zero_complex )
% 12.85/7.01 = ( A = zero_zero_complex ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7771_inverse__nonzero__iff__nonzero,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( ( inverse_inverse_rat @ A )
% 12.85/7.01 = zero_zero_rat )
% 12.85/7.01 = ( A = zero_zero_rat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7772_inverse__zero,axiom,
% 12.85/7.01 ( ( inverse_inverse_real @ zero_zero_real )
% 12.85/7.01 = zero_zero_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7773_inverse__zero,axiom,
% 12.85/7.01 ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 12.85/7.01 = zero_zero_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7774_inverse__zero,axiom,
% 12.85/7.01 ( ( inverse_inverse_rat @ zero_zero_rat )
% 12.85/7.01 = zero_zero_rat ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7775_inverse__mult__distrib,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
% 12.85/7.01 = ( times_times_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7776_inverse__mult__distrib,axiom,
% 12.85/7.01 ! [A: complex,B: complex] :
% 12.85/7.01 ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
% 12.85/7.01 = ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7777_inverse__mult__distrib,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
% 12.85/7.01 = ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7778_inverse__1,axiom,
% 12.85/7.01 ( ( inverse_inverse_real @ one_one_real )
% 12.85/7.01 = one_one_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7779_inverse__1,axiom,
% 12.85/7.01 ( ( invers8013647133539491842omplex @ one_one_complex )
% 12.85/7.01 = one_one_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7780_inverse__1,axiom,
% 12.85/7.01 ( ( inverse_inverse_rat @ one_one_rat )
% 12.85/7.01 = one_one_rat ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7781_inverse__eq__1__iff,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ( inverse_inverse_real @ X )
% 12.85/7.01 = one_one_real )
% 12.85/7.01 = ( X = one_one_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7782_inverse__eq__1__iff,axiom,
% 12.85/7.01 ! [X: complex] :
% 12.85/7.01 ( ( ( invers8013647133539491842omplex @ X )
% 12.85/7.01 = one_one_complex )
% 12.85/7.01 = ( X = one_one_complex ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7783_inverse__eq__1__iff,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ( ( ( inverse_inverse_rat @ X )
% 12.85/7.01 = one_one_rat )
% 12.85/7.01 = ( X = one_one_rat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7784_inverse__divide,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( inverse_inverse_real @ ( divide_divide_real @ A @ B ) )
% 12.85/7.01 = ( divide_divide_real @ B @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7785_inverse__divide,axiom,
% 12.85/7.01 ! [A: complex,B: complex] :
% 12.85/7.01 ( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 12.85/7.01 = ( divide1717551699836669952omplex @ B @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7786_inverse__divide,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( inverse_inverse_rat @ ( divide_divide_rat @ A @ B ) )
% 12.85/7.01 = ( divide_divide_rat @ B @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7787_inverse__minus__eq,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
% 12.85/7.01 = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7788_inverse__minus__eq,axiom,
% 12.85/7.01 ! [A: complex] :
% 12.85/7.01 ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
% 12.85/7.01 = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7789_inverse__minus__eq,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
% 12.85/7.01 = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7790_xor_Oright__neutral,axiom,
% 12.85/7.01 ! [A: nat] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
% 12.85/7.01 = A ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7791_xor_Oright__neutral,axiom,
% 12.85/7.01 ! [A: int] :
% 12.85/7.01 ( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
% 12.85/7.01 = A ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7792_xor_Oleft__neutral,axiom,
% 12.85/7.01 ! [A: nat] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
% 12.85/7.01 = A ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7793_xor_Oleft__neutral,axiom,
% 12.85/7.01 ! [A: int] :
% 12.85/7.01 ( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
% 12.85/7.01 = A ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7794_xor__self__eq,axiom,
% 12.85/7.01 ! [A: nat] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ A @ A )
% 12.85/7.01 = zero_zero_nat ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7795_xor__self__eq,axiom,
% 12.85/7.01 ! [A: int] :
% 12.85/7.01 ( ( bit_se6526347334894502574or_int @ A @ A )
% 12.85/7.01 = zero_zero_int ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7796_bit_Oxor__self,axiom,
% 12.85/7.01 ! [X: int] :
% 12.85/7.01 ( ( bit_se6526347334894502574or_int @ X @ X )
% 12.85/7.01 = zero_zero_int ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7797_abs__inverse,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
% 12.85/7.01 = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7798_abs__inverse,axiom,
% 12.85/7.01 ! [A: complex] :
% 12.85/7.01 ( ( abs_abs_complex @ ( invers8013647133539491842omplex @ A ) )
% 12.85/7.01 = ( invers8013647133539491842omplex @ ( abs_abs_complex @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7799_abs__inverse,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
% 12.85/7.01 = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7800_inverse__sgn,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( inverse_inverse_real @ ( sgn_sgn_real @ A ) )
% 12.85/7.01 = ( sgn_sgn_real @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7801_inverse__sgn,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) )
% 12.85/7.01 = ( sgn_sgn_rat @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7802_sgn__inverse,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( sgn_sgn_real @ ( inverse_inverse_real @ A ) )
% 12.85/7.01 = ( inverse_inverse_real @ ( sgn_sgn_real @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7803_sgn__inverse,axiom,
% 12.85/7.01 ! [A: complex] :
% 12.85/7.01 ( ( sgn_sgn_complex @ ( invers8013647133539491842omplex @ A ) )
% 12.85/7.01 = ( invers8013647133539491842omplex @ ( sgn_sgn_complex @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7804_sgn__inverse,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( sgn_sgn_rat @ ( inverse_inverse_rat @ A ) )
% 12.85/7.01 = ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7805_take__bit__xor,axiom,
% 12.85/7.01 ! [N: nat,A: int,B: int] :
% 12.85/7.01 ( ( bit_se2923211474154528505it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B ) )
% 12.85/7.01 = ( bit_se6526347334894502574or_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7806_take__bit__xor,axiom,
% 12.85/7.01 ! [N: nat,A: nat,B: nat] :
% 12.85/7.01 ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B ) )
% 12.85/7.01 = ( bit_se6528837805403552850or_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7807_inverse__nonnegative__iff__nonnegative,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 12.85/7.01 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7808_inverse__nonnegative__iff__nonnegative,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 12.85/7.01 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7809_inverse__nonpositive__iff__nonpositive,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 12.85/7.01 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7810_inverse__nonpositive__iff__nonpositive,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 12.85/7.01 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7811_inverse__positive__iff__positive,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 12.85/7.01 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7812_inverse__positive__iff__positive,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 12.85/7.01 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7813_inverse__negative__iff__negative,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 12.85/7.01 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7814_inverse__negative__iff__negative,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 12.85/7.01 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7815_inverse__less__iff__less__neg,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/7.01 => ( ( ord_less_real @ B @ zero_zero_real )
% 12.85/7.01 => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 12.85/7.01 = ( ord_less_real @ B @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7816_inverse__less__iff__less__neg,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/7.01 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 12.85/7.01 => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 12.85/7.01 = ( ord_less_rat @ B @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7817_inverse__less__iff__less,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ B )
% 12.85/7.01 => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 12.85/7.01 = ( ord_less_real @ B @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7818_inverse__less__iff__less,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/7.01 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 12.85/7.01 => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 12.85/7.01 = ( ord_less_rat @ B @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7819_gbinomial__0_I2_J,axiom,
% 12.85/7.01 ! [K: nat] :
% 12.85/7.01 ( ( gbinomial_complex @ zero_zero_complex @ ( suc @ K ) )
% 12.85/7.01 = zero_zero_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7820_gbinomial__0_I2_J,axiom,
% 12.85/7.01 ! [K: nat] :
% 12.85/7.01 ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
% 12.85/7.01 = zero_zero_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7821_gbinomial__0_I2_J,axiom,
% 12.85/7.01 ! [K: nat] :
% 12.85/7.01 ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
% 12.85/7.01 = zero_zero_rat ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7822_gbinomial__0_I2_J,axiom,
% 12.85/7.01 ! [K: nat] :
% 12.85/7.01 ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
% 12.85/7.01 = zero_zero_nat ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7823_gbinomial__0_I2_J,axiom,
% 12.85/7.01 ! [K: nat] :
% 12.85/7.01 ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
% 12.85/7.01 = zero_zero_int ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7824_gbinomial__0_I1_J,axiom,
% 12.85/7.01 ! [A: complex] :
% 12.85/7.01 ( ( gbinomial_complex @ A @ zero_zero_nat )
% 12.85/7.01 = one_one_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7825_gbinomial__0_I1_J,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( gbinomial_real @ A @ zero_zero_nat )
% 12.85/7.01 = one_one_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7826_gbinomial__0_I1_J,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( gbinomial_rat @ A @ zero_zero_nat )
% 12.85/7.01 = one_one_rat ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7827_gbinomial__0_I1_J,axiom,
% 12.85/7.01 ! [A: nat] :
% 12.85/7.01 ( ( gbinomial_nat @ A @ zero_zero_nat )
% 12.85/7.01 = one_one_nat ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7828_gbinomial__0_I1_J,axiom,
% 12.85/7.01 ! [A: int] :
% 12.85/7.01 ( ( gbinomial_int @ A @ zero_zero_nat )
% 12.85/7.01 = one_one_int ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7829_frac__eq__0__iff,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ( archim2898591450579166408c_real @ X )
% 12.85/7.01 = zero_zero_real )
% 12.85/7.01 = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7830_frac__eq__0__iff,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ( ( ( archimedean_frac_rat @ X )
% 12.85/7.01 = zero_zero_rat )
% 12.85/7.01 = ( member_rat @ X @ ring_1_Ints_rat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7831_inverse__le__iff__le__neg,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/7.01 => ( ( ord_less_real @ B @ zero_zero_real )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 12.85/7.01 = ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7832_inverse__le__iff__le__neg,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/7.01 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 12.85/7.01 => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 12.85/7.01 = ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7833_inverse__le__iff__le,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ B )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 12.85/7.01 = ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7834_inverse__le__iff__le,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/7.01 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 12.85/7.01 => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 12.85/7.01 = ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7835_right__inverse,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( A != zero_zero_real )
% 12.85/7.01 => ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
% 12.85/7.01 = one_one_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7836_right__inverse,axiom,
% 12.85/7.01 ! [A: complex] :
% 12.85/7.01 ( ( A != zero_zero_complex )
% 12.85/7.01 => ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
% 12.85/7.01 = one_one_complex ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7837_right__inverse,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( A != zero_zero_rat )
% 12.85/7.01 => ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
% 12.85/7.01 = one_one_rat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7838_left__inverse,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( A != zero_zero_real )
% 12.85/7.01 => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 12.85/7.01 = one_one_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7839_left__inverse,axiom,
% 12.85/7.01 ! [A: complex] :
% 12.85/7.01 ( ( A != zero_zero_complex )
% 12.85/7.01 => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 12.85/7.01 = one_one_complex ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7840_left__inverse,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( A != zero_zero_rat )
% 12.85/7.01 => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 12.85/7.01 = one_one_rat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7841_inverse__eq__divide__numeral,axiom,
% 12.85/7.01 ! [W: num] :
% 12.85/7.01 ( ( inverse_inverse_real @ ( numeral_numeral_real @ W ) )
% 12.85/7.01 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ W ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7842_inverse__eq__divide__numeral,axiom,
% 12.85/7.01 ! [W: num] :
% 12.85/7.01 ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ W ) )
% 12.85/7.01 = ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7843_inverse__eq__divide__numeral,axiom,
% 12.85/7.01 ! [W: num] :
% 12.85/7.01 ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ W ) )
% 12.85/7.01 = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7844_frac__gt__0__iff,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) )
% 12.85/7.01 = ( ~ ( member_real @ X @ ring_1_Ints_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7845_frac__gt__0__iff,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ( ( ord_less_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) )
% 12.85/7.01 = ( ~ ( member_rat @ X @ ring_1_Ints_rat ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7846_inverse__eq__divide__neg__numeral,axiom,
% 12.85/7.01 ! [W: num] :
% 12.85/7.01 ( ( inverse_inverse_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 12.85/7.01 = ( divide_divide_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7847_inverse__eq__divide__neg__numeral,axiom,
% 12.85/7.01 ! [W: num] :
% 12.85/7.01 ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 12.85/7.01 = ( divide1717551699836669952omplex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7848_inverse__eq__divide__neg__numeral,axiom,
% 12.85/7.01 ! [W: num] :
% 12.85/7.01 ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 12.85/7.01 = ( divide_divide_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7849_xor__numerals_I3_J,axiom,
% 12.85/7.01 ! [X: num,Y4: num] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 12.85/7.01 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7850_xor__numerals_I3_J,axiom,
% 12.85/7.01 ! [X: num,Y4: num] :
% 12.85/7.01 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y4 ) ) )
% 12.85/7.01 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7851_xor__numerals_I1_J,axiom,
% 12.85/7.01 ! [Y4: num] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 12.85/7.01 = ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7852_xor__numerals_I1_J,axiom,
% 12.85/7.01 ! [Y4: num] :
% 12.85/7.01 ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y4 ) ) )
% 12.85/7.01 = ( numeral_numeral_int @ ( bit1 @ Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7853_xor__numerals_I2_J,axiom,
% 12.85/7.01 ! [Y4: num] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 12.85/7.01 = ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7854_xor__numerals_I2_J,axiom,
% 12.85/7.01 ! [Y4: num] :
% 12.85/7.01 ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y4 ) ) )
% 12.85/7.01 = ( numeral_numeral_int @ ( bit0 @ Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7855_xor__numerals_I5_J,axiom,
% 12.85/7.01 ! [X: num] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
% 12.85/7.01 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7856_xor__numerals_I5_J,axiom,
% 12.85/7.01 ! [X: num] :
% 12.85/7.01 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
% 12.85/7.01 = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7857_xor__numerals_I8_J,axiom,
% 12.85/7.01 ! [X: num] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
% 12.85/7.01 = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7858_xor__numerals_I8_J,axiom,
% 12.85/7.01 ! [X: num] :
% 12.85/7.01 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
% 12.85/7.01 = ( numeral_numeral_int @ ( bit0 @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7859_xor__numerals_I7_J,axiom,
% 12.85/7.01 ! [X: num,Y4: num] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 12.85/7.01 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7860_xor__numerals_I7_J,axiom,
% 12.85/7.01 ! [X: num,Y4: num] :
% 12.85/7.01 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y4 ) ) )
% 12.85/7.01 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7861_xor__nat__numerals_I1_J,axiom,
% 12.85/7.01 ! [Y4: num] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 12.85/7.01 = ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7862_xor__nat__numerals_I2_J,axiom,
% 12.85/7.01 ! [Y4: num] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 12.85/7.01 = ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7863_xor__nat__numerals_I3_J,axiom,
% 12.85/7.01 ! [X: num] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 12.85/7.01 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7864_xor__nat__numerals_I4_J,axiom,
% 12.85/7.01 ! [X: num] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 12.85/7.01 = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7865_xor__numerals_I4_J,axiom,
% 12.85/7.01 ! [X: num,Y4: num] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 12.85/7.01 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7866_xor__numerals_I4_J,axiom,
% 12.85/7.01 ! [X: num,Y4: num] :
% 12.85/7.01 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y4 ) ) )
% 12.85/7.01 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7867_xor__numerals_I6_J,axiom,
% 12.85/7.01 ! [X: num,Y4: num] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 12.85/7.01 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7868_xor__numerals_I6_J,axiom,
% 12.85/7.01 ! [X: num,Y4: num] :
% 12.85/7.01 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y4 ) ) )
% 12.85/7.01 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7869_Ints__power,axiom,
% 12.85/7.01 ! [A: int,N: nat] :
% 12.85/7.01 ( ( member_int @ A @ ring_1_Ints_int )
% 12.85/7.01 => ( member_int @ ( power_power_int @ A @ N ) @ ring_1_Ints_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7870_Ints__power,axiom,
% 12.85/7.01 ! [A: real,N: nat] :
% 12.85/7.01 ( ( member_real @ A @ ring_1_Ints_real )
% 12.85/7.01 => ( member_real @ ( power_power_real @ A @ N ) @ ring_1_Ints_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7871_Ints__power,axiom,
% 12.85/7.01 ! [A: complex,N: nat] :
% 12.85/7.01 ( ( member_complex @ A @ ring_1_Ints_complex )
% 12.85/7.01 => ( member_complex @ ( power_power_complex @ A @ N ) @ ring_1_Ints_complex ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7872_nonzero__of__real__inverse,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( X != zero_zero_real )
% 12.85/7.01 => ( ( real_V1803761363581548252l_real @ ( inverse_inverse_real @ X ) )
% 12.85/7.01 = ( inverse_inverse_real @ ( real_V1803761363581548252l_real @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7873_nonzero__of__real__inverse,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( X != zero_zero_real )
% 12.85/7.01 => ( ( real_V4546457046886955230omplex @ ( inverse_inverse_real @ X ) )
% 12.85/7.01 = ( invers8013647133539491842omplex @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7874_of__int__xor__eq,axiom,
% 12.85/7.01 ! [K: int,L: int] :
% 12.85/7.01 ( ( ring_1_of_int_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 12.85/7.01 = ( bit_se6526347334894502574or_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7875_Ints__0,axiom,
% 12.85/7.01 member_complex @ zero_zero_complex @ ring_1_Ints_complex ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7876_Ints__0,axiom,
% 12.85/7.01 member_real @ zero_zero_real @ ring_1_Ints_real ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7877_Ints__0,axiom,
% 12.85/7.01 member_rat @ zero_zero_rat @ ring_1_Ints_rat ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7878_Ints__0,axiom,
% 12.85/7.01 member_int @ zero_zero_int @ ring_1_Ints_int ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7879_Ints__numeral,axiom,
% 12.85/7.01 ! [N: num] : ( member_complex @ ( numera6690914467698888265omplex @ N ) @ ring_1_Ints_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7880_Ints__numeral,axiom,
% 12.85/7.01 ! [N: num] : ( member_real @ ( numeral_numeral_real @ N ) @ ring_1_Ints_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7881_Ints__numeral,axiom,
% 12.85/7.01 ! [N: num] : ( member_rat @ ( numeral_numeral_rat @ N ) @ ring_1_Ints_rat ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7882_Ints__numeral,axiom,
% 12.85/7.01 ! [N: num] : ( member_int @ ( numeral_numeral_int @ N ) @ ring_1_Ints_int ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7883_Ints__mult,axiom,
% 12.85/7.01 ! [A: complex,B: complex] :
% 12.85/7.01 ( ( member_complex @ A @ ring_1_Ints_complex )
% 12.85/7.01 => ( ( member_complex @ B @ ring_1_Ints_complex )
% 12.85/7.01 => ( member_complex @ ( times_times_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7884_Ints__mult,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( member_real @ A @ ring_1_Ints_real )
% 12.85/7.01 => ( ( member_real @ B @ ring_1_Ints_real )
% 12.85/7.01 => ( member_real @ ( times_times_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7885_Ints__mult,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.85/7.01 => ( ( member_rat @ B @ ring_1_Ints_rat )
% 12.85/7.01 => ( member_rat @ ( times_times_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7886_Ints__mult,axiom,
% 12.85/7.01 ! [A: int,B: int] :
% 12.85/7.01 ( ( member_int @ A @ ring_1_Ints_int )
% 12.85/7.01 => ( ( member_int @ B @ ring_1_Ints_int )
% 12.85/7.01 => ( member_int @ ( times_times_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7887_Ints__add,axiom,
% 12.85/7.01 ! [A: complex,B: complex] :
% 12.85/7.01 ( ( member_complex @ A @ ring_1_Ints_complex )
% 12.85/7.01 => ( ( member_complex @ B @ ring_1_Ints_complex )
% 12.85/7.01 => ( member_complex @ ( plus_plus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7888_Ints__add,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( member_real @ A @ ring_1_Ints_real )
% 12.85/7.01 => ( ( member_real @ B @ ring_1_Ints_real )
% 12.85/7.01 => ( member_real @ ( plus_plus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7889_Ints__add,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.85/7.01 => ( ( member_rat @ B @ ring_1_Ints_rat )
% 12.85/7.01 => ( member_rat @ ( plus_plus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7890_Ints__add,axiom,
% 12.85/7.01 ! [A: int,B: int] :
% 12.85/7.01 ( ( member_int @ A @ ring_1_Ints_int )
% 12.85/7.01 => ( ( member_int @ B @ ring_1_Ints_int )
% 12.85/7.01 => ( member_int @ ( plus_plus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7891_Ints__1,axiom,
% 12.85/7.01 member_rat @ one_one_rat @ ring_1_Ints_rat ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7892_Ints__1,axiom,
% 12.85/7.01 member_int @ one_one_int @ ring_1_Ints_int ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7893_Ints__1,axiom,
% 12.85/7.01 member_real @ one_one_real @ ring_1_Ints_real ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7894_Ints__1,axiom,
% 12.85/7.01 member_complex @ one_one_complex @ ring_1_Ints_complex ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7895_real__sqrt__inverse,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( sqrt @ ( inverse_inverse_real @ X ) )
% 12.85/7.01 = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7896_Ints__diff,axiom,
% 12.85/7.01 ! [A: complex,B: complex] :
% 12.85/7.01 ( ( member_complex @ A @ ring_1_Ints_complex )
% 12.85/7.01 => ( ( member_complex @ B @ ring_1_Ints_complex )
% 12.85/7.01 => ( member_complex @ ( minus_minus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7897_Ints__diff,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( member_real @ A @ ring_1_Ints_real )
% 12.85/7.01 => ( ( member_real @ B @ ring_1_Ints_real )
% 12.85/7.01 => ( member_real @ ( minus_minus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7898_Ints__diff,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.85/7.01 => ( ( member_rat @ B @ ring_1_Ints_rat )
% 12.85/7.01 => ( member_rat @ ( minus_minus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7899_Ints__diff,axiom,
% 12.85/7.01 ! [A: int,B: int] :
% 12.85/7.01 ( ( member_int @ A @ ring_1_Ints_int )
% 12.85/7.01 => ( ( member_int @ B @ ring_1_Ints_int )
% 12.85/7.01 => ( member_int @ ( minus_minus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7900_Ints__minus,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( member_real @ A @ ring_1_Ints_real )
% 12.85/7.01 => ( member_real @ ( uminus_uminus_real @ A ) @ ring_1_Ints_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7901_Ints__minus,axiom,
% 12.85/7.01 ! [A: int] :
% 12.85/7.01 ( ( member_int @ A @ ring_1_Ints_int )
% 12.85/7.01 => ( member_int @ ( uminus_uminus_int @ A ) @ ring_1_Ints_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7902_Ints__minus,axiom,
% 12.85/7.01 ! [A: complex] :
% 12.85/7.01 ( ( member_complex @ A @ ring_1_Ints_complex )
% 12.85/7.01 => ( member_complex @ ( uminus1482373934393186551omplex @ A ) @ ring_1_Ints_complex ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7903_Ints__minus,axiom,
% 12.85/7.01 ! [A: code_integer] :
% 12.85/7.01 ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
% 12.85/7.01 => ( member_Code_integer @ ( uminus1351360451143612070nteger @ A ) @ ring_11222124179247155820nteger ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7904_Ints__minus,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.85/7.01 => ( member_rat @ ( uminus_uminus_rat @ A ) @ ring_1_Ints_rat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7905_minus__in__Ints__iff,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( member_real @ ( uminus_uminus_real @ X ) @ ring_1_Ints_real )
% 12.85/7.01 = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7906_minus__in__Ints__iff,axiom,
% 12.85/7.01 ! [X: int] :
% 12.85/7.01 ( ( member_int @ ( uminus_uminus_int @ X ) @ ring_1_Ints_int )
% 12.85/7.01 = ( member_int @ X @ ring_1_Ints_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7907_minus__in__Ints__iff,axiom,
% 12.85/7.01 ! [X: complex] :
% 12.85/7.01 ( ( member_complex @ ( uminus1482373934393186551omplex @ X ) @ ring_1_Ints_complex )
% 12.85/7.01 = ( member_complex @ X @ ring_1_Ints_complex ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7908_minus__in__Ints__iff,axiom,
% 12.85/7.01 ! [X: code_integer] :
% 12.85/7.01 ( ( member_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ ring_11222124179247155820nteger )
% 12.85/7.01 = ( member_Code_integer @ X @ ring_11222124179247155820nteger ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7909_minus__in__Ints__iff,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ( ( member_rat @ ( uminus_uminus_rat @ X ) @ ring_1_Ints_rat )
% 12.85/7.01 = ( member_rat @ X @ ring_1_Ints_rat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7910_of__nat__gbinomial,axiom,
% 12.85/7.01 ! [N: nat,K: nat] :
% 12.85/7.01 ( ( semiri8010041392384452111omplex @ ( gbinomial_nat @ N @ K ) )
% 12.85/7.01 = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7911_of__nat__gbinomial,axiom,
% 12.85/7.01 ! [N: nat,K: nat] :
% 12.85/7.01 ( ( semiri5074537144036343181t_real @ ( gbinomial_nat @ N @ K ) )
% 12.85/7.01 = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7912_of__nat__gbinomial,axiom,
% 12.85/7.01 ! [N: nat,K: nat] :
% 12.85/7.01 ( ( semiri681578069525770553at_rat @ ( gbinomial_nat @ N @ K ) )
% 12.85/7.01 = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7913_xor_Oleft__commute,axiom,
% 12.85/7.01 ! [B: nat,A: nat,C: nat] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ B @ ( bit_se6528837805403552850or_nat @ A @ C ) )
% 12.85/7.01 = ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B @ C ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7914_xor_Oleft__commute,axiom,
% 12.85/7.01 ! [B: int,A: int,C: int] :
% 12.85/7.01 ( ( bit_se6526347334894502574or_int @ B @ ( bit_se6526347334894502574or_int @ A @ C ) )
% 12.85/7.01 = ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B @ C ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7915_xor_Ocommute,axiom,
% 12.85/7.01 ( bit_se6528837805403552850or_nat
% 12.85/7.01 = ( ^ [A3: nat,B2: nat] : ( bit_se6528837805403552850or_nat @ B2 @ A3 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7916_xor_Ocommute,axiom,
% 12.85/7.01 ( bit_se6526347334894502574or_int
% 12.85/7.01 = ( ^ [A3: int,B2: int] : ( bit_se6526347334894502574or_int @ B2 @ A3 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7917_xor_Oassoc,axiom,
% 12.85/7.01 ! [A: nat,B: nat,C: nat] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ ( bit_se6528837805403552850or_nat @ A @ B ) @ C )
% 12.85/7.01 = ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B @ C ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7918_xor_Oassoc,axiom,
% 12.85/7.01 ! [A: int,B: int,C: int] :
% 12.85/7.01 ( ( bit_se6526347334894502574or_int @ ( bit_se6526347334894502574or_int @ A @ B ) @ C )
% 12.85/7.01 = ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B @ C ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7919_inverse__eq__imp__eq,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ( inverse_inverse_real @ A )
% 12.85/7.01 = ( inverse_inverse_real @ B ) )
% 12.85/7.01 => ( A = B ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7920_inverse__eq__imp__eq,axiom,
% 12.85/7.01 ! [A: complex,B: complex] :
% 12.85/7.01 ( ( ( invers8013647133539491842omplex @ A )
% 12.85/7.01 = ( invers8013647133539491842omplex @ B ) )
% 12.85/7.01 => ( A = B ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7921_inverse__eq__imp__eq,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ( inverse_inverse_rat @ A )
% 12.85/7.01 = ( inverse_inverse_rat @ B ) )
% 12.85/7.01 => ( A = B ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7922_of__nat__xor__eq,axiom,
% 12.85/7.01 ! [M: nat,N: nat] :
% 12.85/7.01 ( ( semiri1316708129612266289at_nat @ ( bit_se6528837805403552850or_nat @ M @ N ) )
% 12.85/7.01 = ( bit_se6528837805403552850or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7923_of__nat__xor__eq,axiom,
% 12.85/7.01 ! [M: nat,N: nat] :
% 12.85/7.01 ( ( semiri1314217659103216013at_int @ ( bit_se6528837805403552850or_nat @ M @ N ) )
% 12.85/7.01 = ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7924_nonzero__imp__inverse__nonzero,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( A != zero_zero_real )
% 12.85/7.01 => ( ( inverse_inverse_real @ A )
% 12.85/7.01 != zero_zero_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7925_nonzero__imp__inverse__nonzero,axiom,
% 12.85/7.01 ! [A: complex] :
% 12.85/7.01 ( ( A != zero_zero_complex )
% 12.85/7.01 => ( ( invers8013647133539491842omplex @ A )
% 12.85/7.01 != zero_zero_complex ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7926_nonzero__imp__inverse__nonzero,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( A != zero_zero_rat )
% 12.85/7.01 => ( ( inverse_inverse_rat @ A )
% 12.85/7.01 != zero_zero_rat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7927_nonzero__inverse__inverse__eq,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( A != zero_zero_real )
% 12.85/7.01 => ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
% 12.85/7.01 = A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7928_nonzero__inverse__inverse__eq,axiom,
% 12.85/7.01 ! [A: complex] :
% 12.85/7.01 ( ( A != zero_zero_complex )
% 12.85/7.01 => ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
% 12.85/7.01 = A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7929_nonzero__inverse__inverse__eq,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( A != zero_zero_rat )
% 12.85/7.01 => ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
% 12.85/7.01 = A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7930_nonzero__inverse__eq__imp__eq,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ( inverse_inverse_real @ A )
% 12.85/7.01 = ( inverse_inverse_real @ B ) )
% 12.85/7.01 => ( ( A != zero_zero_real )
% 12.85/7.01 => ( ( B != zero_zero_real )
% 12.85/7.01 => ( A = B ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7931_nonzero__inverse__eq__imp__eq,axiom,
% 12.85/7.01 ! [A: complex,B: complex] :
% 12.85/7.01 ( ( ( invers8013647133539491842omplex @ A )
% 12.85/7.01 = ( invers8013647133539491842omplex @ B ) )
% 12.85/7.01 => ( ( A != zero_zero_complex )
% 12.85/7.01 => ( ( B != zero_zero_complex )
% 12.85/7.01 => ( A = B ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7932_nonzero__inverse__eq__imp__eq,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ( inverse_inverse_rat @ A )
% 12.85/7.01 = ( inverse_inverse_rat @ B ) )
% 12.85/7.01 => ( ( A != zero_zero_rat )
% 12.85/7.01 => ( ( B != zero_zero_rat )
% 12.85/7.01 => ( A = B ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7933_inverse__zero__imp__zero,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( ( inverse_inverse_real @ A )
% 12.85/7.01 = zero_zero_real )
% 12.85/7.01 => ( A = zero_zero_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7934_inverse__zero__imp__zero,axiom,
% 12.85/7.01 ! [A: complex] :
% 12.85/7.01 ( ( ( invers8013647133539491842omplex @ A )
% 12.85/7.01 = zero_zero_complex )
% 12.85/7.01 => ( A = zero_zero_complex ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7935_inverse__zero__imp__zero,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( ( inverse_inverse_rat @ A )
% 12.85/7.01 = zero_zero_rat )
% 12.85/7.01 => ( A = zero_zero_rat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7936_field__class_Ofield__inverse__zero,axiom,
% 12.85/7.01 ( ( inverse_inverse_real @ zero_zero_real )
% 12.85/7.01 = zero_zero_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7937_field__class_Ofield__inverse__zero,axiom,
% 12.85/7.01 ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 12.85/7.01 = zero_zero_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7938_field__class_Ofield__inverse__zero,axiom,
% 12.85/7.01 ( ( inverse_inverse_rat @ zero_zero_rat )
% 12.85/7.01 = zero_zero_rat ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7939_mult__commute__imp__mult__inverse__commute,axiom,
% 12.85/7.01 ! [Y4: real,X: real] :
% 12.85/7.01 ( ( ( times_times_real @ Y4 @ X )
% 12.85/7.01 = ( times_times_real @ X @ Y4 ) )
% 12.85/7.01 => ( ( times_times_real @ ( inverse_inverse_real @ Y4 ) @ X )
% 12.85/7.01 = ( times_times_real @ X @ ( inverse_inverse_real @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7940_mult__commute__imp__mult__inverse__commute,axiom,
% 12.85/7.01 ! [Y4: complex,X: complex] :
% 12.85/7.01 ( ( ( times_times_complex @ Y4 @ X )
% 12.85/7.01 = ( times_times_complex @ X @ Y4 ) )
% 12.85/7.01 => ( ( times_times_complex @ ( invers8013647133539491842omplex @ Y4 ) @ X )
% 12.85/7.01 = ( times_times_complex @ X @ ( invers8013647133539491842omplex @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7941_mult__commute__imp__mult__inverse__commute,axiom,
% 12.85/7.01 ! [Y4: rat,X: rat] :
% 12.85/7.01 ( ( ( times_times_rat @ Y4 @ X )
% 12.85/7.01 = ( times_times_rat @ X @ Y4 ) )
% 12.85/7.01 => ( ( times_times_rat @ ( inverse_inverse_rat @ Y4 ) @ X )
% 12.85/7.01 = ( times_times_rat @ X @ ( inverse_inverse_rat @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7942_nonzero__norm__inverse,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( A != zero_zero_real )
% 12.85/7.01 => ( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A ) )
% 12.85/7.01 = ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7943_nonzero__norm__inverse,axiom,
% 12.85/7.01 ! [A: complex] :
% 12.85/7.01 ( ( A != zero_zero_complex )
% 12.85/7.01 => ( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A ) )
% 12.85/7.01 = ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7944_Ints__of__nat,axiom,
% 12.85/7.01 ! [N: nat] : ( member_complex @ ( semiri8010041392384452111omplex @ N ) @ ring_1_Ints_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7945_Ints__of__nat,axiom,
% 12.85/7.01 ! [N: nat] : ( member_real @ ( semiri5074537144036343181t_real @ N ) @ ring_1_Ints_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7946_Ints__of__nat,axiom,
% 12.85/7.01 ! [N: nat] : ( member_rat @ ( semiri681578069525770553at_rat @ N ) @ ring_1_Ints_rat ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7947_Ints__of__nat,axiom,
% 12.85/7.01 ! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ring_1_Ints_int ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7948_bit_Oconj__xor__distrib2,axiom,
% 12.85/7.01 ! [Y4: int,Z: int,X: int] :
% 12.85/7.01 ( ( bit_se725231765392027082nd_int @ ( bit_se6526347334894502574or_int @ Y4 @ Z ) @ X )
% 12.85/7.01 = ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ Y4 @ X ) @ ( bit_se725231765392027082nd_int @ Z @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7949_bit_Oconj__xor__distrib,axiom,
% 12.85/7.01 ! [X: int,Y4: int,Z: int] :
% 12.85/7.01 ( ( bit_se725231765392027082nd_int @ X @ ( bit_se6526347334894502574or_int @ Y4 @ Z ) )
% 12.85/7.01 = ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ X @ Y4 ) @ ( bit_se725231765392027082nd_int @ X @ Z ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7950_bit__xor__iff,axiom,
% 12.85/7.01 ! [A: nat,B: nat,N: nat] :
% 12.85/7.01 ( ( bit_se1148574629649215175it_nat @ ( bit_se6528837805403552850or_nat @ A @ B ) @ N )
% 12.85/7.01 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 12.85/7.01 != ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7951_bit__xor__iff,axiom,
% 12.85/7.01 ! [A: int,B: int,N: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ A @ B ) @ N )
% 12.85/7.01 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 12.85/7.01 != ( bit_se1146084159140164899it_int @ B @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7952_Ints__abs,axiom,
% 12.85/7.01 ! [A: int] :
% 12.85/7.01 ( ( member_int @ A @ ring_1_Ints_int )
% 12.85/7.01 => ( member_int @ ( abs_abs_int @ A ) @ ring_1_Ints_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7953_Ints__abs,axiom,
% 12.85/7.01 ! [A: code_integer] :
% 12.85/7.01 ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
% 12.85/7.01 => ( member_Code_integer @ ( abs_abs_Code_integer @ A ) @ ring_11222124179247155820nteger ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7954_Ints__abs,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.85/7.01 => ( member_rat @ ( abs_abs_rat @ A ) @ ring_1_Ints_rat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7955_Ints__abs,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( member_real @ A @ ring_1_Ints_real )
% 12.85/7.01 => ( member_real @ ( abs_abs_real @ A ) @ ring_1_Ints_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7956_Ints__of__int,axiom,
% 12.85/7.01 ! [Z: int] : ( member_int @ ( ring_1_of_int_int @ Z ) @ ring_1_Ints_int ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7957_Ints__of__int,axiom,
% 12.85/7.01 ! [Z: int] : ( member_real @ ( ring_1_of_int_real @ Z ) @ ring_1_Ints_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7958_Ints__of__int,axiom,
% 12.85/7.01 ! [Z: int] : ( member_complex @ ( ring_17405671764205052669omplex @ Z ) @ ring_1_Ints_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7959_Ints__of__int,axiom,
% 12.85/7.01 ! [Z: int] : ( member_rat @ ( ring_1_of_int_rat @ Z ) @ ring_1_Ints_rat ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7960_Ints__induct,axiom,
% 12.85/7.01 ! [Q2: int,P: int > $o] :
% 12.85/7.01 ( ( member_int @ Q2 @ ring_1_Ints_int )
% 12.85/7.01 => ( ! [Z2: int] : ( P @ ( ring_1_of_int_int @ Z2 ) )
% 12.85/7.01 => ( P @ Q2 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7961_Ints__induct,axiom,
% 12.85/7.01 ! [Q2: real,P: real > $o] :
% 12.85/7.01 ( ( member_real @ Q2 @ ring_1_Ints_real )
% 12.85/7.01 => ( ! [Z2: int] : ( P @ ( ring_1_of_int_real @ Z2 ) )
% 12.85/7.01 => ( P @ Q2 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7962_Ints__induct,axiom,
% 12.85/7.01 ! [Q2: complex,P: complex > $o] :
% 12.85/7.01 ( ( member_complex @ Q2 @ ring_1_Ints_complex )
% 12.85/7.01 => ( ! [Z2: int] : ( P @ ( ring_17405671764205052669omplex @ Z2 ) )
% 12.85/7.01 => ( P @ Q2 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7963_Ints__induct,axiom,
% 12.85/7.01 ! [Q2: rat,P: rat > $o] :
% 12.85/7.01 ( ( member_rat @ Q2 @ ring_1_Ints_rat )
% 12.85/7.01 => ( ! [Z2: int] : ( P @ ( ring_1_of_int_rat @ Z2 ) )
% 12.85/7.01 => ( P @ Q2 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7964_Ints__cases,axiom,
% 12.85/7.01 ! [Q2: int] :
% 12.85/7.01 ( ( member_int @ Q2 @ ring_1_Ints_int )
% 12.85/7.01 => ~ ! [Z2: int] :
% 12.85/7.01 ( Q2
% 12.85/7.01 != ( ring_1_of_int_int @ Z2 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7965_Ints__cases,axiom,
% 12.85/7.01 ! [Q2: real] :
% 12.85/7.01 ( ( member_real @ Q2 @ ring_1_Ints_real )
% 12.85/7.01 => ~ ! [Z2: int] :
% 12.85/7.01 ( Q2
% 12.85/7.01 != ( ring_1_of_int_real @ Z2 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7966_Ints__cases,axiom,
% 12.85/7.01 ! [Q2: complex] :
% 12.85/7.01 ( ( member_complex @ Q2 @ ring_1_Ints_complex )
% 12.85/7.01 => ~ ! [Z2: int] :
% 12.85/7.01 ( Q2
% 12.85/7.01 != ( ring_17405671764205052669omplex @ Z2 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7967_Ints__cases,axiom,
% 12.85/7.01 ! [Q2: rat] :
% 12.85/7.01 ( ( member_rat @ Q2 @ ring_1_Ints_rat )
% 12.85/7.01 => ~ ! [Z2: int] :
% 12.85/7.01 ( Q2
% 12.85/7.01 != ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7968_norm__inverse__le__norm,axiom,
% 12.85/7.01 ! [R2: real,X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ R2 @ ( real_V7735802525324610683m_real @ X ) )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ R2 )
% 12.85/7.01 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7969_norm__inverse__le__norm,axiom,
% 12.85/7.01 ! [R2: real,X: complex] :
% 12.85/7.01 ( ( ord_less_eq_real @ R2 @ ( real_V1022390504157884413omplex @ X ) )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ R2 )
% 12.85/7.01 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7970_positive__imp__inverse__positive,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.01 => ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7971_positive__imp__inverse__positive,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/7.01 => ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7972_negative__imp__inverse__negative,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/7.01 => ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7973_negative__imp__inverse__negative,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( ord_less_rat @ A @ zero_zero_rat )
% 12.85/7.01 => ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7974_inverse__positive__imp__positive,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 12.85/7.01 => ( ( A != zero_zero_real )
% 12.85/7.01 => ( ord_less_real @ zero_zero_real @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7975_inverse__positive__imp__positive,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 12.85/7.01 => ( ( A != zero_zero_rat )
% 12.85/7.01 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7976_inverse__negative__imp__negative,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 12.85/7.01 => ( ( A != zero_zero_real )
% 12.85/7.01 => ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7977_inverse__negative__imp__negative,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 12.85/7.01 => ( ( A != zero_zero_rat )
% 12.85/7.01 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7978_less__imp__inverse__less__neg,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_real @ A @ B )
% 12.85/7.01 => ( ( ord_less_real @ B @ zero_zero_real )
% 12.85/7.01 => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7979_less__imp__inverse__less__neg,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_rat @ A @ B )
% 12.85/7.01 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 12.85/7.01 => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7980_inverse__less__imp__less__neg,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 12.85/7.01 => ( ( ord_less_real @ B @ zero_zero_real )
% 12.85/7.01 => ( ord_less_real @ B @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7981_inverse__less__imp__less__neg,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 12.85/7.01 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 12.85/7.01 => ( ord_less_rat @ B @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7982_less__imp__inverse__less,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_real @ A @ B )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.01 => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7983_less__imp__inverse__less,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_rat @ A @ B )
% 12.85/7.01 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/7.01 => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7984_inverse__less__imp__less,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.01 => ( ord_less_real @ B @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7985_inverse__less__imp__less,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 12.85/7.01 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/7.01 => ( ord_less_rat @ B @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7986_nonzero__inverse__mult__distrib,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( A != zero_zero_real )
% 12.85/7.01 => ( ( B != zero_zero_real )
% 12.85/7.01 => ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
% 12.85/7.01 = ( times_times_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7987_nonzero__inverse__mult__distrib,axiom,
% 12.85/7.01 ! [A: complex,B: complex] :
% 12.85/7.01 ( ( A != zero_zero_complex )
% 12.85/7.01 => ( ( B != zero_zero_complex )
% 12.85/7.01 => ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
% 12.85/7.01 = ( times_times_complex @ ( invers8013647133539491842omplex @ B ) @ ( invers8013647133539491842omplex @ A ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7988_nonzero__inverse__mult__distrib,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( A != zero_zero_rat )
% 12.85/7.01 => ( ( B != zero_zero_rat )
% 12.85/7.01 => ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
% 12.85/7.01 = ( times_times_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7989_bit__Suc__0__iff,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.85/7.01 = ( N = zero_zero_nat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7990_not__bit__Suc__0__Suc,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7991_nonzero__inverse__minus__eq,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( A != zero_zero_real )
% 12.85/7.01 => ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
% 12.85/7.01 = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7992_nonzero__inverse__minus__eq,axiom,
% 12.85/7.01 ! [A: complex] :
% 12.85/7.01 ( ( A != zero_zero_complex )
% 12.85/7.01 => ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
% 12.85/7.01 = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7993_nonzero__inverse__minus__eq,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( A != zero_zero_rat )
% 12.85/7.01 => ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
% 12.85/7.01 = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7994_inverse__numeral__1,axiom,
% 12.85/7.01 ( ( inverse_inverse_real @ ( numeral_numeral_real @ one ) )
% 12.85/7.01 = ( numeral_numeral_real @ one ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7995_inverse__numeral__1,axiom,
% 12.85/7.01 ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ one ) )
% 12.85/7.01 = ( numera6690914467698888265omplex @ one ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7996_inverse__numeral__1,axiom,
% 12.85/7.01 ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ one ) )
% 12.85/7.01 = ( numeral_numeral_rat @ one ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7997_inverse__unique,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ( times_times_real @ A @ B )
% 12.85/7.01 = one_one_real )
% 12.85/7.01 => ( ( inverse_inverse_real @ A )
% 12.85/7.01 = B ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7998_inverse__unique,axiom,
% 12.85/7.01 ! [A: complex,B: complex] :
% 12.85/7.01 ( ( ( times_times_complex @ A @ B )
% 12.85/7.01 = one_one_complex )
% 12.85/7.01 => ( ( invers8013647133539491842omplex @ A )
% 12.85/7.01 = B ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_7999_inverse__unique,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ( times_times_rat @ A @ B )
% 12.85/7.01 = one_one_rat )
% 12.85/7.01 => ( ( inverse_inverse_rat @ A )
% 12.85/7.01 = B ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8000_divide__inverse__commute,axiom,
% 12.85/7.01 ( divide_divide_real
% 12.85/7.01 = ( ^ [A3: real,B2: real] : ( times_times_real @ ( inverse_inverse_real @ B2 ) @ A3 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8001_divide__inverse__commute,axiom,
% 12.85/7.01 ( divide1717551699836669952omplex
% 12.85/7.01 = ( ^ [A3: complex,B2: complex] : ( times_times_complex @ ( invers8013647133539491842omplex @ B2 ) @ A3 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8002_divide__inverse__commute,axiom,
% 12.85/7.01 ( divide_divide_rat
% 12.85/7.01 = ( ^ [A3: rat,B2: rat] : ( times_times_rat @ ( inverse_inverse_rat @ B2 ) @ A3 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8003_divide__inverse,axiom,
% 12.85/7.01 ( divide_divide_real
% 12.85/7.01 = ( ^ [A3: real,B2: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B2 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8004_divide__inverse,axiom,
% 12.85/7.01 ( divide1717551699836669952omplex
% 12.85/7.01 = ( ^ [A3: complex,B2: complex] : ( times_times_complex @ A3 @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8005_divide__inverse,axiom,
% 12.85/7.01 ( divide_divide_rat
% 12.85/7.01 = ( ^ [A3: rat,B2: rat] : ( times_times_rat @ A3 @ ( inverse_inverse_rat @ B2 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8006_field__class_Ofield__divide__inverse,axiom,
% 12.85/7.01 ( divide_divide_real
% 12.85/7.01 = ( ^ [A3: real,B2: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B2 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8007_field__class_Ofield__divide__inverse,axiom,
% 12.85/7.01 ( divide1717551699836669952omplex
% 12.85/7.01 = ( ^ [A3: complex,B2: complex] : ( times_times_complex @ A3 @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8008_field__class_Ofield__divide__inverse,axiom,
% 12.85/7.01 ( divide_divide_rat
% 12.85/7.01 = ( ^ [A3: rat,B2: rat] : ( times_times_rat @ A3 @ ( inverse_inverse_rat @ B2 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8009_inverse__eq__divide,axiom,
% 12.85/7.01 ( inverse_inverse_real
% 12.85/7.01 = ( divide_divide_real @ one_one_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8010_inverse__eq__divide,axiom,
% 12.85/7.01 ( invers8013647133539491842omplex
% 12.85/7.01 = ( divide1717551699836669952omplex @ one_one_complex ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8011_inverse__eq__divide,axiom,
% 12.85/7.01 ( inverse_inverse_rat
% 12.85/7.01 = ( divide_divide_rat @ one_one_rat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8012_power__mult__inverse__distrib,axiom,
% 12.85/7.01 ! [X: real,M: nat] :
% 12.85/7.01 ( ( times_times_real @ ( power_power_real @ X @ M ) @ ( inverse_inverse_real @ X ) )
% 12.85/7.01 = ( times_times_real @ ( inverse_inverse_real @ X ) @ ( power_power_real @ X @ M ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8013_power__mult__inverse__distrib,axiom,
% 12.85/7.01 ! [X: complex,M: nat] :
% 12.85/7.01 ( ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( invers8013647133539491842omplex @ X ) )
% 12.85/7.01 = ( times_times_complex @ ( invers8013647133539491842omplex @ X ) @ ( power_power_complex @ X @ M ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8014_power__mult__inverse__distrib,axiom,
% 12.85/7.01 ! [X: rat,M: nat] :
% 12.85/7.01 ( ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( inverse_inverse_rat @ X ) )
% 12.85/7.01 = ( times_times_rat @ ( inverse_inverse_rat @ X ) @ ( power_power_rat @ X @ M ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8015_power__mult__power__inverse__commute,axiom,
% 12.85/7.01 ! [X: real,M: nat,N: nat] :
% 12.85/7.01 ( ( times_times_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N ) )
% 12.85/7.01 = ( times_times_real @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N ) @ ( power_power_real @ X @ M ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8016_power__mult__power__inverse__commute,axiom,
% 12.85/7.01 ! [X: complex,M: nat,N: nat] :
% 12.85/7.01 ( ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N ) )
% 12.85/7.01 = ( times_times_complex @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N ) @ ( power_power_complex @ X @ M ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8017_power__mult__power__inverse__commute,axiom,
% 12.85/7.01 ! [X: rat,M: nat,N: nat] :
% 12.85/7.01 ( ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) )
% 12.85/7.01 = ( times_times_rat @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) @ ( power_power_rat @ X @ M ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8018_mult__inverse__of__nat__commute,axiom,
% 12.85/7.01 ! [Xa: nat,X: real] :
% 12.85/7.01 ( ( times_times_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) @ X )
% 12.85/7.01 = ( times_times_real @ X @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8019_mult__inverse__of__nat__commute,axiom,
% 12.85/7.01 ! [Xa: nat,X: complex] :
% 12.85/7.01 ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) @ X )
% 12.85/7.01 = ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8020_mult__inverse__of__nat__commute,axiom,
% 12.85/7.01 ! [Xa: nat,X: rat] :
% 12.85/7.01 ( ( times_times_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) @ X )
% 12.85/7.01 = ( times_times_rat @ X @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8021_nonzero__abs__inverse,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( A != zero_zero_real )
% 12.85/7.01 => ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
% 12.85/7.01 = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8022_nonzero__abs__inverse,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( A != zero_zero_rat )
% 12.85/7.01 => ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
% 12.85/7.01 = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8023_mult__inverse__of__int__commute,axiom,
% 12.85/7.01 ! [Xa: int,X: real] :
% 12.85/7.01 ( ( times_times_real @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) @ X )
% 12.85/7.01 = ( times_times_real @ X @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8024_mult__inverse__of__int__commute,axiom,
% 12.85/7.01 ! [Xa: int,X: complex] :
% 12.85/7.01 ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) @ X )
% 12.85/7.01 = ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8025_mult__inverse__of__int__commute,axiom,
% 12.85/7.01 ! [Xa: int,X: rat] :
% 12.85/7.01 ( ( times_times_rat @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) @ X )
% 12.85/7.01 = ( times_times_rat @ X @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8026_divide__real__def,axiom,
% 12.85/7.01 ( divide_divide_real
% 12.85/7.01 = ( ^ [X3: real,Y6: real] : ( times_times_real @ X3 @ ( inverse_inverse_real @ Y6 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8027_Ints__double__eq__0__iff,axiom,
% 12.85/7.01 ! [A: complex] :
% 12.85/7.01 ( ( member_complex @ A @ ring_1_Ints_complex )
% 12.85/7.01 => ( ( ( plus_plus_complex @ A @ A )
% 12.85/7.01 = zero_zero_complex )
% 12.85/7.01 = ( A = zero_zero_complex ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8028_Ints__double__eq__0__iff,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( member_real @ A @ ring_1_Ints_real )
% 12.85/7.01 => ( ( ( plus_plus_real @ A @ A )
% 12.85/7.01 = zero_zero_real )
% 12.85/7.01 = ( A = zero_zero_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8029_Ints__double__eq__0__iff,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.85/7.01 => ( ( ( plus_plus_rat @ A @ A )
% 12.85/7.01 = zero_zero_rat )
% 12.85/7.01 = ( A = zero_zero_rat ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8030_Ints__double__eq__0__iff,axiom,
% 12.85/7.01 ! [A: int] :
% 12.85/7.01 ( ( member_int @ A @ ring_1_Ints_int )
% 12.85/7.01 => ( ( ( plus_plus_int @ A @ A )
% 12.85/7.01 = zero_zero_int )
% 12.85/7.01 = ( A = zero_zero_int ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8031_binomial__gbinomial,axiom,
% 12.85/7.01 ! [N: nat,K: nat] :
% 12.85/7.01 ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 12.85/7.01 = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8032_binomial__gbinomial,axiom,
% 12.85/7.01 ! [N: nat,K: nat] :
% 12.85/7.01 ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 12.85/7.01 = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8033_binomial__gbinomial,axiom,
% 12.85/7.01 ! [N: nat,K: nat] :
% 12.85/7.01 ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 12.85/7.01 = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8034_le__imp__inverse__le__neg,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ A @ B )
% 12.85/7.01 => ( ( ord_less_real @ B @ zero_zero_real )
% 12.85/7.01 => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8035_le__imp__inverse__le__neg,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ A @ B )
% 12.85/7.01 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 12.85/7.01 => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8036_inverse__le__imp__le__neg,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 12.85/7.01 => ( ( ord_less_real @ B @ zero_zero_real )
% 12.85/7.01 => ( ord_less_eq_real @ B @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8037_inverse__le__imp__le__neg,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 12.85/7.01 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 12.85/7.01 => ( ord_less_eq_rat @ B @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8038_le__imp__inverse__le,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ A @ B )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.01 => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8039_le__imp__inverse__le,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ A @ B )
% 12.85/7.01 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/7.01 => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8040_inverse__le__imp__le,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.01 => ( ord_less_eq_real @ B @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8041_inverse__le__imp__le,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 12.85/7.01 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/7.01 => ( ord_less_eq_rat @ B @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8042_inverse__le__1__iff,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( inverse_inverse_real @ X ) @ one_one_real )
% 12.85/7.01 = ( ( ord_less_eq_real @ X @ zero_zero_real )
% 12.85/7.01 | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8043_inverse__le__1__iff,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
% 12.85/7.01 = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 12.85/7.01 | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8044_one__less__inverse__iff,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X ) )
% 12.85/7.01 = ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 & ( ord_less_real @ X @ one_one_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8045_one__less__inverse__iff,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
% 12.85/7.01 = ( ( ord_less_rat @ zero_zero_rat @ X )
% 12.85/7.01 & ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8046_one__less__inverse,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.01 => ( ( ord_less_real @ A @ one_one_real )
% 12.85/7.01 => ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8047_one__less__inverse,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/7.01 => ( ( ord_less_rat @ A @ one_one_rat )
% 12.85/7.01 => ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8048_division__ring__inverse__add,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( A != zero_zero_real )
% 12.85/7.01 => ( ( B != zero_zero_real )
% 12.85/7.01 => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 12.85/7.01 = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( plus_plus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8049_division__ring__inverse__add,axiom,
% 12.85/7.01 ! [A: complex,B: complex] :
% 12.85/7.01 ( ( A != zero_zero_complex )
% 12.85/7.01 => ( ( B != zero_zero_complex )
% 12.85/7.01 => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 12.85/7.01 = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( plus_plus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8050_division__ring__inverse__add,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( A != zero_zero_rat )
% 12.85/7.01 => ( ( B != zero_zero_rat )
% 12.85/7.01 => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 12.85/7.01 = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( plus_plus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8051_inverse__add,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( A != zero_zero_real )
% 12.85/7.01 => ( ( B != zero_zero_real )
% 12.85/7.01 => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 12.85/7.01 = ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( inverse_inverse_real @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8052_inverse__add,axiom,
% 12.85/7.01 ! [A: complex,B: complex] :
% 12.85/7.01 ( ( A != zero_zero_complex )
% 12.85/7.01 => ( ( B != zero_zero_complex )
% 12.85/7.01 => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 12.85/7.01 = ( times_times_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( invers8013647133539491842omplex @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8053_inverse__add,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( A != zero_zero_rat )
% 12.85/7.01 => ( ( B != zero_zero_rat )
% 12.85/7.01 => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 12.85/7.01 = ( times_times_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( inverse_inverse_rat @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8054_field__class_Ofield__inverse,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( A != zero_zero_real )
% 12.85/7.01 => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 12.85/7.01 = one_one_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8055_field__class_Ofield__inverse,axiom,
% 12.85/7.01 ! [A: complex] :
% 12.85/7.01 ( ( A != zero_zero_complex )
% 12.85/7.01 => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 12.85/7.01 = one_one_complex ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8056_field__class_Ofield__inverse,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( A != zero_zero_rat )
% 12.85/7.01 => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 12.85/7.01 = one_one_rat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8057_division__ring__inverse__diff,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( A != zero_zero_real )
% 12.85/7.01 => ( ( B != zero_zero_real )
% 12.85/7.01 => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 12.85/7.01 = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ B @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8058_division__ring__inverse__diff,axiom,
% 12.85/7.01 ! [A: complex,B: complex] :
% 12.85/7.01 ( ( A != zero_zero_complex )
% 12.85/7.01 => ( ( B != zero_zero_complex )
% 12.85/7.01 => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 12.85/7.01 = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ B @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8059_division__ring__inverse__diff,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( A != zero_zero_rat )
% 12.85/7.01 => ( ( B != zero_zero_rat )
% 12.85/7.01 => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 12.85/7.01 = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ B @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8060_not__bit__Suc__0__numeral,axiom,
% 12.85/7.01 ! [N: num] :
% 12.85/7.01 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8061_nonzero__inverse__eq__divide,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( A != zero_zero_real )
% 12.85/7.01 => ( ( inverse_inverse_real @ A )
% 12.85/7.01 = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8062_nonzero__inverse__eq__divide,axiom,
% 12.85/7.01 ! [A: complex] :
% 12.85/7.01 ( ( A != zero_zero_complex )
% 12.85/7.01 => ( ( invers8013647133539491842omplex @ A )
% 12.85/7.01 = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8063_nonzero__inverse__eq__divide,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( A != zero_zero_rat )
% 12.85/7.01 => ( ( inverse_inverse_rat @ A )
% 12.85/7.01 = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8064_gbinomial__Suc__Suc,axiom,
% 12.85/7.01 ! [A: complex,K: nat] :
% 12.85/7.01 ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 12.85/7.01 = ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8065_gbinomial__Suc__Suc,axiom,
% 12.85/7.01 ! [A: real,K: nat] :
% 12.85/7.01 ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 12.85/7.01 = ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8066_gbinomial__Suc__Suc,axiom,
% 12.85/7.01 ! [A: rat,K: nat] :
% 12.85/7.01 ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 12.85/7.01 = ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8067_gbinomial__of__nat__symmetric,axiom,
% 12.85/7.01 ! [K: nat,N: nat] :
% 12.85/7.01 ( ( ord_less_eq_nat @ K @ N )
% 12.85/7.01 => ( ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K )
% 12.85/7.01 = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8068_gbinomial__of__nat__symmetric,axiom,
% 12.85/7.01 ! [K: nat,N: nat] :
% 12.85/7.01 ( ( ord_less_eq_nat @ K @ N )
% 12.85/7.01 => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
% 12.85/7.01 = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8069_gbinomial__of__nat__symmetric,axiom,
% 12.85/7.01 ! [K: nat,N: nat] :
% 12.85/7.01 ( ( ord_less_eq_nat @ K @ N )
% 12.85/7.01 => ( ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K )
% 12.85/7.01 = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8070_inverse__powr,axiom,
% 12.85/7.01 ! [Y4: real,A: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.01 => ( ( powr_real @ ( inverse_inverse_real @ Y4 ) @ A )
% 12.85/7.01 = ( inverse_inverse_real @ ( powr_real @ Y4 @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8071_Ints__odd__nonzero,axiom,
% 12.85/7.01 ! [A: complex] :
% 12.85/7.01 ( ( member_complex @ A @ ring_1_Ints_complex )
% 12.85/7.01 => ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A ) @ A )
% 12.85/7.01 != zero_zero_complex ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8072_Ints__odd__nonzero,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( member_real @ A @ ring_1_Ints_real )
% 12.85/7.01 => ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A )
% 12.85/7.01 != zero_zero_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8073_Ints__odd__nonzero,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.85/7.01 => ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A )
% 12.85/7.01 != zero_zero_rat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8074_Ints__odd__nonzero,axiom,
% 12.85/7.01 ! [A: int] :
% 12.85/7.01 ( ( member_int @ A @ ring_1_Ints_int )
% 12.85/7.01 => ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A )
% 12.85/7.01 != zero_zero_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8075_of__int__divide__in__Ints,axiom,
% 12.85/7.01 ! [B: int,A: int] :
% 12.85/7.01 ( ( dvd_dvd_int @ B @ A )
% 12.85/7.01 => ( member_complex @ ( divide1717551699836669952omplex @ ( ring_17405671764205052669omplex @ A ) @ ( ring_17405671764205052669omplex @ B ) ) @ ring_1_Ints_complex ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8076_of__int__divide__in__Ints,axiom,
% 12.85/7.01 ! [B: int,A: int] :
% 12.85/7.01 ( ( dvd_dvd_int @ B @ A )
% 12.85/7.01 => ( member_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) @ ring_1_Ints_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8077_of__int__divide__in__Ints,axiom,
% 12.85/7.01 ! [B: int,A: int] :
% 12.85/7.01 ( ( dvd_dvd_int @ B @ A )
% 12.85/7.01 => ( member_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) @ ring_1_Ints_rat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8078_of__int__divide__in__Ints,axiom,
% 12.85/7.01 ! [B: int,A: int] :
% 12.85/7.01 ( ( dvd_dvd_int @ B @ A )
% 12.85/7.01 => ( member_int @ ( divide_divide_int @ ( ring_1_of_int_int @ A ) @ ( ring_1_of_int_int @ B ) ) @ ring_1_Ints_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8079_inverse__le__iff,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 12.85/7.01 = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 12.85/7.01 => ( ord_less_eq_real @ B @ A ) )
% 12.85/7.01 & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 12.85/7.01 => ( ord_less_eq_real @ A @ B ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8080_inverse__le__iff,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 12.85/7.01 = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 12.85/7.01 => ( ord_less_eq_rat @ B @ A ) )
% 12.85/7.01 & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 12.85/7.01 => ( ord_less_eq_rat @ A @ B ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8081_inverse__less__iff,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 12.85/7.01 = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 12.85/7.01 => ( ord_less_real @ B @ A ) )
% 12.85/7.01 & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 12.85/7.01 => ( ord_less_real @ A @ B ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8082_inverse__less__iff,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 12.85/7.01 = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 12.85/7.01 => ( ord_less_rat @ B @ A ) )
% 12.85/7.01 & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 12.85/7.01 => ( ord_less_rat @ A @ B ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8083_one__le__inverse__iff,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X ) )
% 12.85/7.01 = ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 & ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8084_one__le__inverse__iff,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
% 12.85/7.01 = ( ( ord_less_rat @ zero_zero_rat @ X )
% 12.85/7.01 & ( ord_less_eq_rat @ X @ one_one_rat ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8085_inverse__less__1__iff,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ ( inverse_inverse_real @ X ) @ one_one_real )
% 12.85/7.01 = ( ( ord_less_eq_real @ X @ zero_zero_real )
% 12.85/7.01 | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8086_inverse__less__1__iff,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ( ( ord_less_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
% 12.85/7.01 = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 12.85/7.01 | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8087_one__le__inverse,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.01 => ( ( ord_less_eq_real @ A @ one_one_real )
% 12.85/7.01 => ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8088_one__le__inverse,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( ord_less_rat @ zero_zero_rat @ A )
% 12.85/7.01 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 12.85/7.01 => ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8089_inverse__diff__inverse,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( A != zero_zero_real )
% 12.85/7.01 => ( ( B != zero_zero_real )
% 12.85/7.01 => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 12.85/7.01 = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8090_inverse__diff__inverse,axiom,
% 12.85/7.01 ! [A: complex,B: complex] :
% 12.85/7.01 ( ( A != zero_zero_complex )
% 12.85/7.01 => ( ( B != zero_zero_complex )
% 12.85/7.01 => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 12.85/7.01 = ( uminus1482373934393186551omplex @ ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8091_inverse__diff__inverse,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( A != zero_zero_rat )
% 12.85/7.01 => ( ( B != zero_zero_rat )
% 12.85/7.01 => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 12.85/7.01 = ( uminus_uminus_rat @ ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8092_reals__Archimedean,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ? [N2: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8093_reals__Archimedean,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ( ( ord_less_rat @ zero_zero_rat @ X )
% 12.85/7.01 => ? [N2: nat] : ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8094_gbinomial__addition__formula,axiom,
% 12.85/7.01 ! [A: complex,K: nat] :
% 12.85/7.01 ( ( gbinomial_complex @ A @ ( suc @ K ) )
% 12.85/7.01 = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8095_gbinomial__addition__formula,axiom,
% 12.85/7.01 ! [A: real,K: nat] :
% 12.85/7.01 ( ( gbinomial_real @ A @ ( suc @ K ) )
% 12.85/7.01 = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8096_gbinomial__addition__formula,axiom,
% 12.85/7.01 ! [A: rat,K: nat] :
% 12.85/7.01 ( ( gbinomial_rat @ A @ ( suc @ K ) )
% 12.85/7.01 = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8097_gbinomial__absorb__comp,axiom,
% 12.85/7.01 ! [A: complex,K: nat] :
% 12.85/7.01 ( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A @ K ) )
% 12.85/7.01 = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8098_gbinomial__absorb__comp,axiom,
% 12.85/7.01 ! [A: real,K: nat] :
% 12.85/7.01 ( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A @ K ) )
% 12.85/7.01 = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8099_gbinomial__absorb__comp,axiom,
% 12.85/7.01 ! [A: rat,K: nat] :
% 12.85/7.01 ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
% 12.85/7.01 = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8100_gbinomial__ge__n__over__k__pow__k,axiom,
% 12.85/7.01 ! [K: nat,A: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
% 12.85/7.01 => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8101_gbinomial__ge__n__over__k__pow__k,axiom,
% 12.85/7.01 ! [K: nat,A: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
% 12.85/7.01 => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8102_gbinomial__mult__1_H,axiom,
% 12.85/7.01 ! [A: complex,K: nat] :
% 12.85/7.01 ( ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ A )
% 12.85/7.01 = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8103_gbinomial__mult__1_H,axiom,
% 12.85/7.01 ! [A: real,K: nat] :
% 12.85/7.01 ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
% 12.85/7.01 = ( 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 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8104_gbinomial__mult__1_H,axiom,
% 12.85/7.01 ! [A: rat,K: nat] :
% 12.85/7.01 ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
% 12.85/7.01 = ( 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 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8105_gbinomial__mult__1,axiom,
% 12.85/7.01 ! [A: complex,K: nat] :
% 12.85/7.01 ( ( times_times_complex @ A @ ( gbinomial_complex @ A @ K ) )
% 12.85/7.01 = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8106_gbinomial__mult__1,axiom,
% 12.85/7.01 ! [A: real,K: nat] :
% 12.85/7.01 ( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
% 12.85/7.01 = ( 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 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8107_gbinomial__mult__1,axiom,
% 12.85/7.01 ! [A: rat,K: nat] :
% 12.85/7.01 ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
% 12.85/7.01 = ( 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 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8108_forall__pos__mono__1,axiom,
% 12.85/7.01 ! [P: real > $o,E2: real] :
% 12.85/7.01 ( ! [D2: real,E: real] :
% 12.85/7.01 ( ( ord_less_real @ D2 @ E )
% 12.85/7.01 => ( ( P @ D2 )
% 12.85/7.01 => ( P @ E ) ) )
% 12.85/7.01 => ( ! [N2: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ E2 )
% 12.85/7.01 => ( P @ E2 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8109_real__arch__inverse,axiom,
% 12.85/7.01 ! [E2: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ E2 )
% 12.85/7.01 = ( ? [N4: nat] :
% 12.85/7.01 ( ( N4 != zero_zero_nat )
% 12.85/7.01 & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) )
% 12.85/7.01 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) @ E2 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8110_forall__pos__mono,axiom,
% 12.85/7.01 ! [P: real > $o,E2: real] :
% 12.85/7.01 ( ! [D2: real,E: real] :
% 12.85/7.01 ( ( ord_less_real @ D2 @ E )
% 12.85/7.01 => ( ( P @ D2 )
% 12.85/7.01 => ( P @ E ) ) )
% 12.85/7.01 => ( ! [N2: nat] :
% 12.85/7.01 ( ( N2 != zero_zero_nat )
% 12.85/7.01 => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ E2 )
% 12.85/7.01 => ( P @ E2 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8111_bit__nat__iff,axiom,
% 12.85/7.01 ! [K: int,N: nat] :
% 12.85/7.01 ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
% 12.85/7.01 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.85/7.01 & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8112_even__xor__iff,axiom,
% 12.85/7.01 ! [A: code_integer,B: code_integer] :
% 12.85/7.01 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ A @ B ) )
% 12.85/7.01 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/7.01 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8113_even__xor__iff,axiom,
% 12.85/7.01 ! [A: nat,B: nat] :
% 12.85/7.01 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ A @ B ) )
% 12.85/7.01 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/7.01 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8114_even__xor__iff,axiom,
% 12.85/7.01 ! [A: int,B: int] :
% 12.85/7.01 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ A @ B ) )
% 12.85/7.01 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/7.01 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8115_sqrt__divide__self__eq,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( divide_divide_real @ ( sqrt @ X ) @ X )
% 12.85/7.01 = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8116_ln__inverse,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( ln_ln_real @ ( inverse_inverse_real @ X ) )
% 12.85/7.01 = ( uminus_uminus_real @ ( ln_ln_real @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8117_Ints__odd__less__0,axiom,
% 12.85/7.01 ! [A: real] :
% 12.85/7.01 ( ( member_real @ A @ ring_1_Ints_real )
% 12.85/7.01 => ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A ) @ zero_zero_real )
% 12.85/7.01 = ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8118_Ints__odd__less__0,axiom,
% 12.85/7.01 ! [A: rat] :
% 12.85/7.01 ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.85/7.01 => ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A ) @ zero_zero_rat )
% 12.85/7.01 = ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8119_Ints__odd__less__0,axiom,
% 12.85/7.01 ! [A: int] :
% 12.85/7.01 ( ( member_int @ A @ ring_1_Ints_int )
% 12.85/7.01 => ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A ) @ zero_zero_int )
% 12.85/7.01 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8120_Ints__nonzero__abs__ge1,axiom,
% 12.85/7.01 ! [X: code_integer] :
% 12.85/7.01 ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
% 12.85/7.01 => ( ( X != zero_z3403309356797280102nteger )
% 12.85/7.01 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8121_Ints__nonzero__abs__ge1,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ( ( member_rat @ X @ ring_1_Ints_rat )
% 12.85/7.01 => ( ( X != zero_zero_rat )
% 12.85/7.01 => ( ord_less_eq_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8122_Ints__nonzero__abs__ge1,axiom,
% 12.85/7.01 ! [X: int] :
% 12.85/7.01 ( ( member_int @ X @ ring_1_Ints_int )
% 12.85/7.01 => ( ( X != zero_zero_int )
% 12.85/7.01 => ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8123_Ints__nonzero__abs__ge1,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( member_real @ X @ ring_1_Ints_real )
% 12.85/7.01 => ( ( X != zero_zero_real )
% 12.85/7.01 => ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8124_Ints__nonzero__abs__less1,axiom,
% 12.85/7.01 ! [X: code_integer] :
% 12.85/7.01 ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
% 12.85/7.01 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer )
% 12.85/7.01 => ( X = zero_z3403309356797280102nteger ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8125_Ints__nonzero__abs__less1,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( member_real @ X @ ring_1_Ints_real )
% 12.85/7.01 => ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 12.85/7.01 => ( X = zero_zero_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8126_Ints__nonzero__abs__less1,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ( ( member_rat @ X @ ring_1_Ints_rat )
% 12.85/7.01 => ( ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat )
% 12.85/7.01 => ( X = zero_zero_rat ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8127_Ints__nonzero__abs__less1,axiom,
% 12.85/7.01 ! [X: int] :
% 12.85/7.01 ( ( member_int @ X @ ring_1_Ints_int )
% 12.85/7.01 => ( ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int )
% 12.85/7.01 => ( X = zero_zero_int ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8128_Ints__eq__abs__less1,axiom,
% 12.85/7.01 ! [X: code_integer,Y4: code_integer] :
% 12.85/7.01 ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
% 12.85/7.01 => ( ( member_Code_integer @ Y4 @ ring_11222124179247155820nteger )
% 12.85/7.01 => ( ( X = Y4 )
% 12.85/7.01 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ Y4 ) ) @ one_one_Code_integer ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8129_Ints__eq__abs__less1,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( member_real @ X @ ring_1_Ints_real )
% 12.85/7.01 => ( ( member_real @ Y4 @ ring_1_Ints_real )
% 12.85/7.01 => ( ( X = Y4 )
% 12.85/7.01 = ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ one_one_real ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8130_Ints__eq__abs__less1,axiom,
% 12.85/7.01 ! [X: rat,Y4: rat] :
% 12.85/7.01 ( ( member_rat @ X @ ring_1_Ints_rat )
% 12.85/7.01 => ( ( member_rat @ Y4 @ ring_1_Ints_rat )
% 12.85/7.01 => ( ( X = Y4 )
% 12.85/7.01 = ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ Y4 ) ) @ one_one_rat ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8131_Ints__eq__abs__less1,axiom,
% 12.85/7.01 ! [X: int,Y4: int] :
% 12.85/7.01 ( ( member_int @ X @ ring_1_Ints_int )
% 12.85/7.01 => ( ( member_int @ Y4 @ ring_1_Ints_int )
% 12.85/7.01 => ( ( X = Y4 )
% 12.85/7.01 = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Y4 ) ) @ one_one_int ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8132_sin__times__pi__eq__0,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ( sin_real @ ( times_times_real @ X @ pi ) )
% 12.85/7.01 = zero_zero_real )
% 12.85/7.01 = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8133_ex__inverse__of__nat__less,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ? [N2: nat] :
% 12.85/7.01 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 12.85/7.01 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8134_ex__inverse__of__nat__less,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ( ( ord_less_rat @ zero_zero_rat @ X )
% 12.85/7.01 => ? [N2: nat] :
% 12.85/7.01 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 12.85/7.01 & ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8135_power__diff__conv__inverse,axiom,
% 12.85/7.01 ! [X: real,M: nat,N: nat] :
% 12.85/7.01 ( ( X != zero_zero_real )
% 12.85/7.01 => ( ( ord_less_eq_nat @ M @ N )
% 12.85/7.01 => ( ( power_power_real @ X @ ( minus_minus_nat @ N @ M ) )
% 12.85/7.01 = ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ M ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8136_power__diff__conv__inverse,axiom,
% 12.85/7.01 ! [X: complex,M: nat,N: nat] :
% 12.85/7.01 ( ( X != zero_zero_complex )
% 12.85/7.01 => ( ( ord_less_eq_nat @ M @ N )
% 12.85/7.01 => ( ( power_power_complex @ X @ ( minus_minus_nat @ N @ M ) )
% 12.85/7.01 = ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ M ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8137_power__diff__conv__inverse,axiom,
% 12.85/7.01 ! [X: rat,M: nat,N: nat] :
% 12.85/7.01 ( ( X != zero_zero_rat )
% 12.85/7.01 => ( ( ord_less_eq_nat @ M @ N )
% 12.85/7.01 => ( ( power_power_rat @ X @ ( minus_minus_nat @ N @ M ) )
% 12.85/7.01 = ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ M ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8138_Suc__times__gbinomial,axiom,
% 12.85/7.01 ! [K: nat,A: complex] :
% 12.85/7.01 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
% 12.85/7.01 = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8139_Suc__times__gbinomial,axiom,
% 12.85/7.01 ! [K: nat,A: real] :
% 12.85/7.01 ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
% 12.85/7.01 = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8140_Suc__times__gbinomial,axiom,
% 12.85/7.01 ! [K: nat,A: rat] :
% 12.85/7.01 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
% 12.85/7.01 = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8141_gbinomial__absorption,axiom,
% 12.85/7.01 ! [K: nat,A: complex] :
% 12.85/7.01 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
% 12.85/7.01 = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8142_gbinomial__absorption,axiom,
% 12.85/7.01 ! [K: nat,A: real] :
% 12.85/7.01 ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
% 12.85/7.01 = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8143_gbinomial__absorption,axiom,
% 12.85/7.01 ! [K: nat,A: rat] :
% 12.85/7.01 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
% 12.85/7.01 = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8144_gbinomial__trinomial__revision,axiom,
% 12.85/7.01 ! [K: nat,M: nat,A: complex] :
% 12.85/7.01 ( ( ord_less_eq_nat @ K @ M )
% 12.85/7.01 => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
% 12.85/7.01 = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8145_gbinomial__trinomial__revision,axiom,
% 12.85/7.01 ! [K: nat,M: nat,A: real] :
% 12.85/7.01 ( ( ord_less_eq_nat @ K @ M )
% 12.85/7.01 => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
% 12.85/7.01 = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8146_gbinomial__trinomial__revision,axiom,
% 12.85/7.01 ! [K: nat,M: nat,A: rat] :
% 12.85/7.01 ( ( ord_less_eq_nat @ K @ M )
% 12.85/7.01 => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
% 12.85/7.01 = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8147_log__inverse,axiom,
% 12.85/7.01 ! [A: real,X: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.01 => ( ( A != one_one_real )
% 12.85/7.01 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( log @ A @ ( inverse_inverse_real @ X ) )
% 12.85/7.01 = ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8148_frac__neg,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ( member_real @ X @ ring_1_Ints_real )
% 12.85/7.01 => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
% 12.85/7.01 = zero_zero_real ) )
% 12.85/7.01 & ( ~ ( member_real @ X @ ring_1_Ints_real )
% 12.85/7.01 => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
% 12.85/7.01 = ( minus_minus_real @ one_one_real @ ( archim2898591450579166408c_real @ X ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8149_frac__neg,axiom,
% 12.85/7.01 ! [X: rat] :
% 12.85/7.01 ( ( ( member_rat @ X @ ring_1_Ints_rat )
% 12.85/7.01 => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
% 12.85/7.01 = zero_zero_rat ) )
% 12.85/7.01 & ( ~ ( member_rat @ X @ ring_1_Ints_rat )
% 12.85/7.01 => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
% 12.85/7.01 = ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8150_bit__nat__def,axiom,
% 12.85/7.01 ( bit_se1148574629649215175it_nat
% 12.85/7.01 = ( ^ [M5: nat,N4: nat] :
% 12.85/7.01 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8151_gbinomial__factors,axiom,
% 12.85/7.01 ! [A: complex,K: nat] :
% 12.85/7.01 ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 12.85/7.01 = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8152_gbinomial__factors,axiom,
% 12.85/7.01 ! [A: real,K: nat] :
% 12.85/7.01 ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 12.85/7.01 = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8153_gbinomial__factors,axiom,
% 12.85/7.01 ! [A: rat,K: nat] :
% 12.85/7.01 ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 12.85/7.01 = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8154_gbinomial__rec,axiom,
% 12.85/7.01 ! [A: complex,K: nat] :
% 12.85/7.01 ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 12.85/7.01 = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8155_gbinomial__rec,axiom,
% 12.85/7.01 ! [A: real,K: nat] :
% 12.85/7.01 ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 12.85/7.01 = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8156_gbinomial__rec,axiom,
% 12.85/7.01 ! [A: rat,K: nat] :
% 12.85/7.01 ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 12.85/7.01 = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8157_gbinomial__index__swap,axiom,
% 12.85/7.01 ! [K: nat,N: nat] :
% 12.85/7.01 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ K ) )
% 12.85/7.01 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8158_gbinomial__index__swap,axiom,
% 12.85/7.01 ! [K: nat,N: nat] :
% 12.85/7.01 ( ( 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 ) )
% 12.85/7.01 = ( 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 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8159_gbinomial__index__swap,axiom,
% 12.85/7.01 ! [K: nat,N: nat] :
% 12.85/7.01 ( ( 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 ) )
% 12.85/7.01 = ( 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 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8160_gbinomial__negated__upper,axiom,
% 12.85/7.01 ( gbinomial_complex
% 12.85/7.01 = ( ^ [A3: complex,K3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A3 ) @ one_one_complex ) @ K3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8161_gbinomial__negated__upper,axiom,
% 12.85/7.01 ( gbinomial_real
% 12.85/7.01 = ( ^ [A3: 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 ) @ A3 ) @ one_one_real ) @ K3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8162_gbinomial__negated__upper,axiom,
% 12.85/7.01 ( gbinomial_rat
% 12.85/7.01 = ( ^ [A3: 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 ) @ A3 ) @ one_one_rat ) @ K3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8163_exp__plus__inverse__exp,axiom,
% 12.85/7.01 ! [X: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8164_le__mult__floor__Ints,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/7.01 => ( ( member_real @ A @ ring_1_Ints_real )
% 12.85/7.01 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_rat @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8165_le__mult__floor__Ints,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/7.01 => ( ( member_real @ A @ ring_1_Ints_real )
% 12.85/7.01 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_int @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8166_le__mult__floor__Ints,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/7.01 => ( ( member_real @ A @ ring_1_Ints_real )
% 12.85/7.01 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8167_le__mult__floor__Ints,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/7.01 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.85/7.01 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8168_le__mult__floor__Ints,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/7.01 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.85/7.01 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_int @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8169_le__mult__floor__Ints,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/7.01 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.85/7.01 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_real @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8170_frac__unique__iff,axiom,
% 12.85/7.01 ! [X: rat,A: rat] :
% 12.85/7.01 ( ( ( archimedean_frac_rat @ X )
% 12.85/7.01 = A )
% 12.85/7.01 = ( ( member_rat @ ( minus_minus_rat @ X @ A ) @ ring_1_Ints_rat )
% 12.85/7.01 & ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/7.01 & ( ord_less_rat @ A @ one_one_rat ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8171_frac__unique__iff,axiom,
% 12.85/7.01 ! [X: real,A: real] :
% 12.85/7.01 ( ( ( archim2898591450579166408c_real @ X )
% 12.85/7.01 = A )
% 12.85/7.01 = ( ( member_real @ ( minus_minus_real @ X @ A ) @ ring_1_Ints_real )
% 12.85/7.01 & ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/7.01 & ( ord_less_real @ A @ one_one_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8172_mult__ceiling__le__Ints,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/7.01 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.85/7.01 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8173_mult__ceiling__le__Ints,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/7.01 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.85/7.01 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8174_mult__ceiling__le__Ints,axiom,
% 12.85/7.01 ! [A: rat,B: rat] :
% 12.85/7.01 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 12.85/7.01 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 12.85/7.01 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8175_mult__ceiling__le__Ints,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/7.01 => ( ( member_real @ A @ ring_1_Ints_real )
% 12.85/7.01 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8176_mult__ceiling__le__Ints,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/7.01 => ( ( member_real @ A @ ring_1_Ints_real )
% 12.85/7.01 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8177_mult__ceiling__le__Ints,axiom,
% 12.85/7.01 ! [A: real,B: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/7.01 => ( ( member_real @ A @ ring_1_Ints_real )
% 12.85/7.01 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8178_gbinomial__minus,axiom,
% 12.85/7.01 ! [A: complex,K: nat] :
% 12.85/7.01 ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
% 12.85/7.01 = ( 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 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8179_gbinomial__minus,axiom,
% 12.85/7.01 ! [A: real,K: nat] :
% 12.85/7.01 ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
% 12.85/7.01 = ( 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 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8180_gbinomial__minus,axiom,
% 12.85/7.01 ! [A: rat,K: nat] :
% 12.85/7.01 ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
% 12.85/7.01 = ( 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 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8181_plus__inverse__ge__2,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8182_real__inv__sqrt__pow2,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.01 = ( inverse_inverse_real @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8183_gbinomial__reduce__nat,axiom,
% 12.85/7.01 ! [K: nat,A: complex] :
% 12.85/7.01 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/7.01 => ( ( gbinomial_complex @ A @ K )
% 12.85/7.01 = ( 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 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8184_gbinomial__reduce__nat,axiom,
% 12.85/7.01 ! [K: nat,A: real] :
% 12.85/7.01 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/7.01 => ( ( gbinomial_real @ A @ K )
% 12.85/7.01 = ( 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 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8185_gbinomial__reduce__nat,axiom,
% 12.85/7.01 ! [K: nat,A: rat] :
% 12.85/7.01 ( ( ord_less_nat @ zero_zero_nat @ K )
% 12.85/7.01 => ( ( gbinomial_rat @ A @ K )
% 12.85/7.01 = ( 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 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8186_gbinomial__pochhammer,axiom,
% 12.85/7.01 ( gbinomial_rat
% 12.85/7.01 = ( ^ [A3: 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 @ A3 ) @ K3 ) ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8187_gbinomial__pochhammer,axiom,
% 12.85/7.01 ( gbinomial_real
% 12.85/7.01 = ( ^ [A3: 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 @ A3 ) @ K3 ) ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8188_gbinomial__pochhammer,axiom,
% 12.85/7.01 ( gbinomial_complex
% 12.85/7.01 = ( ^ [A3: complex,K3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A3 ) @ K3 ) ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8189_gbinomial__pochhammer_H,axiom,
% 12.85/7.01 ( gbinomial_rat
% 12.85/7.01 = ( ^ [A3: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8190_gbinomial__pochhammer_H,axiom,
% 12.85/7.01 ( gbinomial_real
% 12.85/7.01 = ( ^ [A3: real,K3: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ K3 ) ) @ one_one_real ) @ K3 ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8191_gbinomial__pochhammer_H,axiom,
% 12.85/7.01 ( gbinomial_complex
% 12.85/7.01 = ( ^ [A3: complex,K3: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ K3 ) ) @ one_one_complex ) @ K3 ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8192_tan__cot,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 12.85/7.01 = ( inverse_inverse_real @ ( tan_real @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8193_sin__integer__2pi,axiom,
% 12.85/7.01 ! [N: real] :
% 12.85/7.01 ( ( member_real @ N @ ring_1_Ints_real )
% 12.85/7.01 => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 12.85/7.01 = zero_zero_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8194_cos__integer__2pi,axiom,
% 12.85/7.01 ! [N: real] :
% 12.85/7.01 ( ( member_real @ N @ ring_1_Ints_real )
% 12.85/7.01 => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 12.85/7.01 = one_one_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8195_xor__nat__unfold,axiom,
% 12.85/7.01 ( bit_se6528837805403552850or_nat
% 12.85/7.01 = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M5 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8196_real__le__x__sinh,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ord_less_eq_real @ X @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8197_xor__nat__rec,axiom,
% 12.85/7.01 ( bit_se6528837805403552850or_nat
% 12.85/7.01 = ( ^ [M5: nat,N4: nat] :
% 12.85/7.01 ( plus_plus_nat
% 12.85/7.01 @ ( zero_n2687167440665602831ol_nat
% 12.85/7.01 @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 12.85/7.01 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
% 12.85/7.01 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8198_one__xor__eq,axiom,
% 12.85/7.01 ! [A: code_integer] :
% 12.85/7.01 ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ A )
% 12.85/7.01 = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 12.85/7.01 @ ( zero_n356916108424825756nteger
% 12.85/7.01 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8199_one__xor__eq,axiom,
% 12.85/7.01 ! [A: nat] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ one_one_nat @ A )
% 12.85/7.01 = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 12.85/7.01 @ ( zero_n2687167440665602831ol_nat
% 12.85/7.01 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8200_one__xor__eq,axiom,
% 12.85/7.01 ! [A: int] :
% 12.85/7.01 ( ( bit_se6526347334894502574or_int @ one_one_int @ A )
% 12.85/7.01 = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 12.85/7.01 @ ( zero_n2684676970156552555ol_int
% 12.85/7.01 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8201_xor__one__eq,axiom,
% 12.85/7.01 ! [A: code_integer] :
% 12.85/7.01 ( ( bit_se3222712562003087583nteger @ A @ one_one_Code_integer )
% 12.85/7.01 = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 12.85/7.01 @ ( zero_n356916108424825756nteger
% 12.85/7.01 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8202_xor__one__eq,axiom,
% 12.85/7.01 ! [A: nat] :
% 12.85/7.01 ( ( bit_se6528837805403552850or_nat @ A @ one_one_nat )
% 12.85/7.01 = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 12.85/7.01 @ ( zero_n2687167440665602831ol_nat
% 12.85/7.01 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8203_xor__one__eq,axiom,
% 12.85/7.01 ! [A: int] :
% 12.85/7.01 ( ( bit_se6526347334894502574or_int @ A @ one_one_int )
% 12.85/7.01 = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 12.85/7.01 @ ( zero_n2684676970156552555ol_int
% 12.85/7.01 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8204_real__le__abs__sinh,axiom,
% 12.85/7.01 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8205_tan__sec,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ( cos_real @ X )
% 12.85/7.01 != zero_zero_real )
% 12.85/7.01 => ( ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.01 = ( power_power_real @ ( inverse_inverse_real @ ( cos_real @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8206_tan__sec,axiom,
% 12.85/7.01 ! [X: complex] :
% 12.85/7.01 ( ( ( cos_complex @ X )
% 12.85/7.01 != zero_zero_complex )
% 12.85/7.01 => ( ( plus_plus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.01 = ( power_power_complex @ ( invers8013647133539491842omplex @ ( cos_complex @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8207_sinh__ln__real,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( sinh_real @ ( ln_ln_real @ X ) )
% 12.85/7.01 = ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8208_cosh__ln__real,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( cosh_real @ ( ln_ln_real @ X ) )
% 12.85/7.01 = ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8209_horner__sum__of__bool__2__less,axiom,
% 12.85/7.01 ! [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 ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8210_cosh__zero__iff,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ( cosh_real @ X )
% 12.85/7.01 = zero_zero_real )
% 12.85/7.01 = ( ( power_power_real @ ( exp_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.01 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8211_cosh__zero__iff,axiom,
% 12.85/7.01 ! [X: complex] :
% 12.85/7.01 ( ( ( cosh_complex @ X )
% 12.85/7.01 = zero_zero_complex )
% 12.85/7.01 = ( ( power_power_complex @ ( exp_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.01 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8212_push__bit__numeral__minus__1,axiom,
% 12.85/7.01 ! [N: num] :
% 12.85/7.01 ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.01 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8213_push__bit__numeral__minus__1,axiom,
% 12.85/7.01 ! [N: num] :
% 12.85/7.01 ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ N ) @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.01 = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8214_push__bit__nonnegative__int__iff,axiom,
% 12.85/7.01 ! [N: nat,K: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
% 12.85/7.01 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8215_push__bit__negative__int__iff,axiom,
% 12.85/7.01 ! [N: nat,K: int] :
% 12.85/7.01 ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
% 12.85/7.01 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8216_push__bit__of__0,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( bit_se545348938243370406it_int @ N @ zero_zero_int )
% 12.85/7.01 = zero_zero_int ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8217_push__bit__of__0,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( bit_se547839408752420682it_nat @ N @ zero_zero_nat )
% 12.85/7.01 = zero_zero_nat ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8218_push__bit__eq__0__iff,axiom,
% 12.85/7.01 ! [N: nat,A: int] :
% 12.85/7.01 ( ( ( bit_se545348938243370406it_int @ N @ A )
% 12.85/7.01 = zero_zero_int )
% 12.85/7.01 = ( A = zero_zero_int ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8219_push__bit__eq__0__iff,axiom,
% 12.85/7.01 ! [N: nat,A: nat] :
% 12.85/7.01 ( ( ( bit_se547839408752420682it_nat @ N @ A )
% 12.85/7.01 = zero_zero_nat )
% 12.85/7.01 = ( A = zero_zero_nat ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8220_sinh__0,axiom,
% 12.85/7.01 ( ( sinh_complex @ zero_zero_complex )
% 12.85/7.01 = zero_zero_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8221_sinh__0,axiom,
% 12.85/7.01 ( ( sinh_real @ zero_zero_real )
% 12.85/7.01 = zero_zero_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8222_push__bit__push__bit,axiom,
% 12.85/7.01 ! [M: nat,N: nat,A: int] :
% 12.85/7.01 ( ( bit_se545348938243370406it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
% 12.85/7.01 = ( bit_se545348938243370406it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8223_push__bit__push__bit,axiom,
% 12.85/7.01 ! [M: nat,N: nat,A: nat] :
% 12.85/7.01 ( ( bit_se547839408752420682it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 12.85/7.01 = ( bit_se547839408752420682it_nat @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8224_push__bit__and,axiom,
% 12.85/7.01 ! [N: nat,A: int,B: int] :
% 12.85/7.01 ( ( bit_se545348938243370406it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 12.85/7.01 = ( bit_se725231765392027082nd_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8225_push__bit__and,axiom,
% 12.85/7.01 ! [N: nat,A: nat,B: nat] :
% 12.85/7.01 ( ( bit_se547839408752420682it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 12.85/7.01 = ( bit_se727722235901077358nd_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8226_push__bit__xor,axiom,
% 12.85/7.01 ! [N: nat,A: int,B: int] :
% 12.85/7.01 ( ( bit_se545348938243370406it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B ) )
% 12.85/7.01 = ( bit_se6526347334894502574or_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8227_push__bit__xor,axiom,
% 12.85/7.01 ! [N: nat,A: nat,B: nat] :
% 12.85/7.01 ( ( bit_se547839408752420682it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B ) )
% 12.85/7.01 = ( bit_se6528837805403552850or_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8228_concat__bit__of__zero__1,axiom,
% 12.85/7.01 ! [N: nat,L: int] :
% 12.85/7.01 ( ( bit_concat_bit @ N @ zero_zero_int @ L )
% 12.85/7.01 = ( bit_se545348938243370406it_int @ N @ L ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8229_sinh__real__zero__iff,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ( sinh_real @ X )
% 12.85/7.01 = zero_zero_real )
% 12.85/7.01 = ( X = zero_zero_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8230_sinh__real__le__iff,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( sinh_real @ X ) @ ( sinh_real @ Y4 ) )
% 12.85/7.01 = ( ord_less_eq_real @ X @ Y4 ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8231_cosh__0,axiom,
% 12.85/7.01 ( ( cosh_complex @ zero_zero_complex )
% 12.85/7.01 = one_one_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8232_cosh__0,axiom,
% 12.85/7.01 ( ( cosh_real @ zero_zero_real )
% 12.85/7.01 = one_one_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8233_sinh__real__neg__iff,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ ( sinh_real @ X ) @ zero_zero_real )
% 12.85/7.01 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8234_sinh__real__pos__iff,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X ) )
% 12.85/7.01 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8235_sinh__real__nonpos__iff,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ ( sinh_real @ X ) @ zero_zero_real )
% 12.85/7.01 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8236_sinh__real__nonneg__iff,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X ) )
% 12.85/7.01 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8237_xor__nonnegative__int__iff,axiom,
% 12.85/7.01 ! [K: int,L: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 12.85/7.01 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.85/7.01 = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8238_xor__negative__int__iff,axiom,
% 12.85/7.01 ! [K: int,L: int] :
% 12.85/7.01 ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
% 12.85/7.01 = ( ( ord_less_int @ K @ zero_zero_int )
% 12.85/7.01 != ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8239_push__bit__Suc__numeral,axiom,
% 12.85/7.01 ! [N: nat,K: num] :
% 12.85/7.01 ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( numeral_numeral_int @ K ) )
% 12.85/7.01 = ( bit_se545348938243370406it_int @ N @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8240_push__bit__Suc__numeral,axiom,
% 12.85/7.01 ! [N: nat,K: num] :
% 12.85/7.01 ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 12.85/7.01 = ( bit_se547839408752420682it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8241_push__bit__Suc__minus__numeral,axiom,
% 12.85/7.01 ! [N: nat,K: num] :
% 12.85/7.01 ( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 12.85/7.01 = ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8242_push__bit__Suc__minus__numeral,axiom,
% 12.85/7.01 ! [N: nat,K: num] :
% 12.85/7.01 ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.85/7.01 = ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8243_push__bit__numeral,axiom,
% 12.85/7.01 ! [L: num,K: num] :
% 12.85/7.01 ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ K ) )
% 12.85/7.01 = ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8244_push__bit__numeral,axiom,
% 12.85/7.01 ! [L: num,K: num] :
% 12.85/7.01 ( ( bit_se547839408752420682it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ K ) )
% 12.85/7.01 = ( bit_se547839408752420682it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8245_push__bit__Suc,axiom,
% 12.85/7.01 ! [N: nat,A: int] :
% 12.85/7.01 ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ A )
% 12.85/7.01 = ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8246_push__bit__Suc,axiom,
% 12.85/7.01 ! [N: nat,A: nat] :
% 12.85/7.01 ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ A )
% 12.85/7.01 = ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8247_push__bit__of__Suc__0,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
% 12.85/7.01 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8248_push__bit__of__1,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( bit_se545348938243370406it_int @ N @ one_one_int )
% 12.85/7.01 = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8249_push__bit__of__1,axiom,
% 12.85/7.01 ! [N: nat] :
% 12.85/7.01 ( ( bit_se547839408752420682it_nat @ N @ one_one_nat )
% 12.85/7.01 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8250_even__push__bit__iff,axiom,
% 12.85/7.01 ! [N: nat,A: code_integer] :
% 12.85/7.01 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se7788150548672797655nteger @ N @ A ) )
% 12.85/7.01 = ( ( N != zero_zero_nat )
% 12.85/7.01 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8251_even__push__bit__iff,axiom,
% 12.85/7.01 ! [N: nat,A: int] :
% 12.85/7.01 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se545348938243370406it_int @ N @ A ) )
% 12.85/7.01 = ( ( N != zero_zero_nat )
% 12.85/7.01 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8252_even__push__bit__iff,axiom,
% 12.85/7.01 ! [N: nat,A: nat] :
% 12.85/7.01 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 12.85/7.01 = ( ( N != zero_zero_nat )
% 12.85/7.01 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8253_push__bit__minus__numeral,axiom,
% 12.85/7.01 ! [L: num,K: num] :
% 12.85/7.01 ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 12.85/7.01 = ( bit_se7788150548672797655nteger @ ( pred_numeral @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8254_push__bit__minus__numeral,axiom,
% 12.85/7.01 ! [L: num,K: num] :
% 12.85/7.01 ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 12.85/7.01 = ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8255_flip__bit__int__def,axiom,
% 12.85/7.01 ( bit_se2159334234014336723it_int
% 12.85/7.01 = ( ^ [N4: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8256_bit__xor__int__iff,axiom,
% 12.85/7.01 ! [K: int,L: int,N: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ N )
% 12.85/7.01 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 12.85/7.01 != ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8257_sinh__add,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( sinh_real @ ( plus_plus_real @ X @ Y4 ) )
% 12.85/7.01 = ( plus_plus_real @ ( times_times_real @ ( sinh_real @ X ) @ ( cosh_real @ Y4 ) ) @ ( times_times_real @ ( cosh_real @ X ) @ ( sinh_real @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8258_cosh__add,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( cosh_real @ ( plus_plus_real @ X @ Y4 ) )
% 12.85/7.01 = ( plus_plus_real @ ( times_times_real @ ( cosh_real @ X ) @ ( cosh_real @ Y4 ) ) @ ( times_times_real @ ( sinh_real @ X ) @ ( sinh_real @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8259_cosh__diff,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( cosh_real @ ( minus_minus_real @ X @ Y4 ) )
% 12.85/7.01 = ( minus_minus_real @ ( times_times_real @ ( cosh_real @ X ) @ ( cosh_real @ Y4 ) ) @ ( times_times_real @ ( sinh_real @ X ) @ ( sinh_real @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8260_sinh__diff,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( sinh_real @ ( minus_minus_real @ X @ Y4 ) )
% 12.85/7.01 = ( minus_minus_real @ ( times_times_real @ ( sinh_real @ X ) @ ( cosh_real @ Y4 ) ) @ ( times_times_real @ ( cosh_real @ X ) @ ( sinh_real @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8261_cosh__real__nonzero,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( cosh_real @ X )
% 12.85/7.01 != zero_zero_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8262_push__bit__nat__eq,axiom,
% 12.85/7.01 ! [N: nat,K: int] :
% 12.85/7.01 ( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K ) )
% 12.85/7.01 = ( nat2 @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8263_sinh__le__cosh__real,axiom,
% 12.85/7.01 ! [X: real] : ( ord_less_eq_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8264_push__bit__of__nat,axiom,
% 12.85/7.01 ! [N: nat,M: nat] :
% 12.85/7.01 ( ( bit_se545348938243370406it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 12.85/7.01 = ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8265_push__bit__of__nat,axiom,
% 12.85/7.01 ! [N: nat,M: nat] :
% 12.85/7.01 ( ( bit_se547839408752420682it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 12.85/7.01 = ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8266_of__nat__push__bit,axiom,
% 12.85/7.01 ! [M: nat,N: nat] :
% 12.85/7.01 ( ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 12.85/7.01 = ( bit_se545348938243370406it_int @ M @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8267_of__nat__push__bit,axiom,
% 12.85/7.01 ! [M: nat,N: nat] :
% 12.85/7.01 ( ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 12.85/7.01 = ( bit_se547839408752420682it_nat @ M @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8268_push__bit__minus,axiom,
% 12.85/7.01 ! [N: nat,A: code_integer] :
% 12.85/7.01 ( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ A ) )
% 12.85/7.01 = ( uminus1351360451143612070nteger @ ( bit_se7788150548672797655nteger @ N @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8269_push__bit__minus,axiom,
% 12.85/7.01 ! [N: nat,A: int] :
% 12.85/7.01 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ A ) )
% 12.85/7.01 = ( uminus_uminus_int @ ( bit_se545348938243370406it_int @ N @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8270_push__bit__add,axiom,
% 12.85/7.01 ! [N: nat,A: int,B: int] :
% 12.85/7.01 ( ( bit_se545348938243370406it_int @ N @ ( plus_plus_int @ A @ B ) )
% 12.85/7.01 = ( plus_plus_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8271_push__bit__add,axiom,
% 12.85/7.01 ! [N: nat,A: nat,B: nat] :
% 12.85/7.01 ( ( bit_se547839408752420682it_nat @ N @ ( plus_plus_nat @ A @ B ) )
% 12.85/7.01 = ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8272_push__bit__of__int,axiom,
% 12.85/7.01 ! [N: nat,K: int] :
% 12.85/7.01 ( ( bit_se545348938243370406it_int @ N @ ( ring_1_of_int_int @ K ) )
% 12.85/7.01 = ( ring_1_of_int_int @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8273_tanh__def,axiom,
% 12.85/7.01 ( tanh_complex
% 12.85/7.01 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( sinh_complex @ X3 ) @ ( cosh_complex @ X3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8274_tanh__def,axiom,
% 12.85/7.01 ( tanh_real
% 12.85/7.01 = ( ^ [X3: real] : ( divide_divide_real @ ( sinh_real @ X3 ) @ ( cosh_real @ X3 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8275_sinh__minus__cosh,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( minus_minus_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) )
% 12.85/7.01 = ( uminus_uminus_real @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8276_sinh__minus__cosh,axiom,
% 12.85/7.01 ! [X: complex] :
% 12.85/7.01 ( ( minus_minus_complex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) )
% 12.85/7.01 = ( uminus1482373934393186551omplex @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8277_cosh__minus__sinh,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( minus_minus_real @ ( cosh_real @ X ) @ ( sinh_real @ X ) )
% 12.85/7.01 = ( exp_real @ ( uminus_uminus_real @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8278_cosh__minus__sinh,axiom,
% 12.85/7.01 ! [X: complex] :
% 12.85/7.01 ( ( minus_minus_complex @ ( cosh_complex @ X ) @ ( sinh_complex @ X ) )
% 12.85/7.01 = ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8279_XOR__lower,axiom,
% 12.85/7.01 ! [X: int,Y4: int] :
% 12.85/7.01 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/7.01 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/7.01 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8280_cosh__real__pos,axiom,
% 12.85/7.01 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8281_cosh__real__nonpos__le__iff,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y4 ) )
% 12.85/7.01 = ( ord_less_eq_real @ Y4 @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8282_cosh__real__nonneg__le__iff,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.01 => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y4 ) )
% 12.85/7.01 = ( ord_less_eq_real @ X @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8283_cosh__real__nonneg,axiom,
% 12.85/7.01 ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8284_cosh__real__ge__1,axiom,
% 12.85/7.01 ! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8285_sinh__double,axiom,
% 12.85/7.01 ! [X: complex] :
% 12.85/7.01 ( ( sinh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 12.85/7.01 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sinh_complex @ X ) ) @ ( cosh_complex @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8286_sinh__double,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( sinh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 12.85/7.01 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sinh_real @ X ) ) @ ( cosh_real @ X ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8287_push__bit__take__bit,axiom,
% 12.85/7.01 ! [M: nat,N: nat,A: int] :
% 12.85/7.01 ( ( bit_se545348938243370406it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 12.85/7.01 = ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ M @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8288_push__bit__take__bit,axiom,
% 12.85/7.01 ! [M: nat,N: nat,A: nat] :
% 12.85/7.01 ( ( bit_se547839408752420682it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 12.85/7.01 = ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N ) @ ( bit_se547839408752420682it_nat @ M @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8289_take__bit__push__bit,axiom,
% 12.85/7.01 ! [M: nat,N: nat,A: int] :
% 12.85/7.01 ( ( bit_se2923211474154528505it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
% 12.85/7.01 = ( bit_se545348938243370406it_int @ N @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8290_take__bit__push__bit,axiom,
% 12.85/7.01 ! [M: nat,N: nat,A: nat] :
% 12.85/7.01 ( ( bit_se2925701944663578781it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 12.85/7.01 = ( bit_se547839408752420682it_nat @ N @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8291_flip__bit__nat__def,axiom,
% 12.85/7.01 ( bit_se2161824704523386999it_nat
% 12.85/7.01 = ( ^ [M5: nat,N4: nat] : ( bit_se6528837805403552850or_nat @ N4 @ ( bit_se547839408752420682it_nat @ M5 @ one_one_nat ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8292_cosh__real__nonpos__less__iff,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 12.85/7.01 => ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
% 12.85/7.01 => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y4 ) )
% 12.85/7.01 = ( ord_less_real @ Y4 @ X ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8293_cosh__real__nonneg__less__iff,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.01 => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y4 ) )
% 12.85/7.01 = ( ord_less_real @ X @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8294_cosh__real__strict__mono,axiom,
% 12.85/7.01 ! [X: real,Y4: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( ord_less_real @ X @ Y4 )
% 12.85/7.01 => ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y4 ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8295_bit__push__bit__iff__int,axiom,
% 12.85/7.01 ! [M: nat,K: int,N: nat] :
% 12.85/7.01 ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
% 12.85/7.01 = ( ( ord_less_eq_nat @ M @ N )
% 12.85/7.01 & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8296_cosh__square__eq,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.01 = ( plus_plus_real @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8297_cosh__square__eq,axiom,
% 12.85/7.01 ! [X: complex] :
% 12.85/7.01 ( ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.01 = ( plus_plus_complex @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8298_sinh__square__eq,axiom,
% 12.85/7.01 ! [X: complex] :
% 12.85/7.01 ( ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.01 = ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8299_sinh__square__eq,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.01 = ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8300_hyperbolic__pythagoras,axiom,
% 12.85/7.01 ! [X: complex] :
% 12.85/7.01 ( ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.01 = one_one_complex ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8301_hyperbolic__pythagoras,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.01 = one_one_real ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8302_xor__nat__def,axiom,
% 12.85/7.01 ( bit_se6528837805403552850or_nat
% 12.85/7.01 = ( ^ [M5: nat,N4: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8303_bit__push__bit__iff__nat,axiom,
% 12.85/7.01 ! [M: nat,Q2: nat,N: nat] :
% 12.85/7.01 ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N )
% 12.85/7.01 = ( ( ord_less_eq_nat @ M @ N )
% 12.85/7.01 & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8304_arcosh__cosh__real,axiom,
% 12.85/7.01 ! [X: real] :
% 12.85/7.01 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.01 => ( ( arcosh_real @ ( cosh_real @ X ) )
% 12.85/7.01 = X ) ) ).
% 12.85/7.01
% 12.85/7.01 thf(fact_8305_concat__bit__eq,axiom,
% 12.85/7.01 ( bit_concat_bit
% 12.85/7.01 = ( ^ [N4: nat,K3: int,L3: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N4 @ K3 ) @ ( bit_se545348938243370406it_int @ N4 @ L3 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8306_flip__bit__eq__xor,axiom,
% 12.85/7.02 ( bit_se1345352211410354436nteger
% 12.85/7.02 = ( ^ [N4: nat,A3: code_integer] : ( bit_se3222712562003087583nteger @ A3 @ ( bit_se7788150548672797655nteger @ N4 @ one_one_Code_integer ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8307_flip__bit__eq__xor,axiom,
% 12.85/7.02 ( bit_se2159334234014336723it_int
% 12.85/7.02 = ( ^ [N4: nat,A3: int] : ( bit_se6526347334894502574or_int @ A3 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8308_flip__bit__eq__xor,axiom,
% 12.85/7.02 ( bit_se2161824704523386999it_nat
% 12.85/7.02 = ( ^ [N4: nat,A3: nat] : ( bit_se6528837805403552850or_nat @ A3 @ ( bit_se547839408752420682it_nat @ N4 @ one_one_nat ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8309_cosh__double,axiom,
% 12.85/7.02 ! [X: complex] :
% 12.85/7.02 ( ( cosh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 12.85/7.02 = ( plus_plus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8310_cosh__double,axiom,
% 12.85/7.02 ! [X: real] :
% 12.85/7.02 ( ( cosh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 12.85/7.02 = ( plus_plus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8311_push__bit__double,axiom,
% 12.85/7.02 ! [N: nat,A: int] :
% 12.85/7.02 ( ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 12.85/7.02 = ( times_times_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8312_push__bit__double,axiom,
% 12.85/7.02 ! [N: nat,A: nat] :
% 12.85/7.02 ( ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.02 = ( times_times_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8313_bit__iff__and__push__bit__not__eq__0,axiom,
% 12.85/7.02 ( bit_se1146084159140164899it_int
% 12.85/7.02 = ( ^ [A3: int,N4: nat] :
% 12.85/7.02 ( ( bit_se725231765392027082nd_int @ A3 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) )
% 12.85/7.02 != zero_zero_int ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8314_bit__iff__and__push__bit__not__eq__0,axiom,
% 12.85/7.02 ( bit_se1148574629649215175it_nat
% 12.85/7.02 = ( ^ [A3: nat,N4: nat] :
% 12.85/7.02 ( ( bit_se727722235901077358nd_nat @ A3 @ ( bit_se547839408752420682it_nat @ N4 @ one_one_nat ) )
% 12.85/7.02 != zero_zero_nat ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8315_push__bit__int__def,axiom,
% 12.85/7.02 ( bit_se545348938243370406it_int
% 12.85/7.02 = ( ^ [N4: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8316_push__bit__nat__def,axiom,
% 12.85/7.02 ( bit_se547839408752420682it_nat
% 12.85/7.02 = ( ^ [N4: nat,M5: nat] : ( times_times_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8317_push__bit__eq__mult,axiom,
% 12.85/7.02 ( bit_se545348938243370406it_int
% 12.85/7.02 = ( ^ [N4: nat,A3: int] : ( times_times_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8318_push__bit__eq__mult,axiom,
% 12.85/7.02 ( bit_se547839408752420682it_nat
% 12.85/7.02 = ( ^ [N4: nat,A3: nat] : ( times_times_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8319_exp__dvdE,axiom,
% 12.85/7.02 ! [N: nat,A: code_integer] :
% 12.85/7.02 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A )
% 12.85/7.02 => ~ ! [B3: code_integer] :
% 12.85/7.02 ( A
% 12.85/7.02 != ( bit_se7788150548672797655nteger @ N @ B3 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8320_exp__dvdE,axiom,
% 12.85/7.02 ! [N: nat,A: int] :
% 12.85/7.02 ( ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A )
% 12.85/7.02 => ~ ! [B3: int] :
% 12.85/7.02 ( A
% 12.85/7.02 != ( bit_se545348938243370406it_int @ N @ B3 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8321_exp__dvdE,axiom,
% 12.85/7.02 ! [N: nat,A: nat] :
% 12.85/7.02 ( ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A )
% 12.85/7.02 => ~ ! [B3: nat] :
% 12.85/7.02 ( A
% 12.85/7.02 != ( bit_se547839408752420682it_nat @ N @ B3 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8322_push__bit__minus__one,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.02 = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8323_tanh__add,axiom,
% 12.85/7.02 ! [X: complex,Y4: complex] :
% 12.85/7.02 ( ( ( cosh_complex @ X )
% 12.85/7.02 != zero_zero_complex )
% 12.85/7.02 => ( ( ( cosh_complex @ Y4 )
% 12.85/7.02 != zero_zero_complex )
% 12.85/7.02 => ( ( tanh_complex @ ( plus_plus_complex @ X @ Y4 ) )
% 12.85/7.02 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y4 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y4 ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8324_tanh__add,axiom,
% 12.85/7.02 ! [X: real,Y4: real] :
% 12.85/7.02 ( ( ( cosh_real @ X )
% 12.85/7.02 != zero_zero_real )
% 12.85/7.02 => ( ( ( cosh_real @ Y4 )
% 12.85/7.02 != zero_zero_real )
% 12.85/7.02 => ( ( tanh_real @ ( plus_plus_real @ X @ Y4 ) )
% 12.85/7.02 = ( divide_divide_real @ ( plus_plus_real @ ( tanh_real @ X ) @ ( tanh_real @ Y4 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tanh_real @ X ) @ ( tanh_real @ Y4 ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8325_XOR__upper,axiom,
% 12.85/7.02 ! [X: int,N: nat,Y4: int] :
% 12.85/7.02 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/7.02 => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.02 => ( ( ord_less_int @ Y4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.02 => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y4 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8326_cosh__field__def,axiom,
% 12.85/7.02 ( cosh_real
% 12.85/7.02 = ( ^ [Z5: real] : ( divide_divide_real @ ( plus_plus_real @ ( exp_real @ Z5 ) @ ( exp_real @ ( uminus_uminus_real @ Z5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8327_cosh__field__def,axiom,
% 12.85/7.02 ( cosh_complex
% 12.85/7.02 = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( exp_complex @ Z5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z5 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8328_complex__inverse,axiom,
% 12.85/7.02 ! [A: real,B: real] :
% 12.85/7.02 ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
% 12.85/7.02 = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8329_xor__int__rec,axiom,
% 12.85/7.02 ( bit_se6526347334894502574or_int
% 12.85/7.02 = ( ^ [K3: int,L3: int] :
% 12.85/7.02 ( plus_plus_int
% 12.85/7.02 @ ( zero_n2684676970156552555ol_int
% 12.85/7.02 @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) )
% 12.85/7.02 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) ) )
% 12.85/7.02 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8330_sinh__field__def,axiom,
% 12.85/7.02 ( sinh_real
% 12.85/7.02 = ( ^ [Z5: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ Z5 ) @ ( exp_real @ ( uminus_uminus_real @ Z5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8331_sinh__field__def,axiom,
% 12.85/7.02 ( sinh_complex
% 12.85/7.02 = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ Z5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z5 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8332_xor__int__unfold,axiom,
% 12.85/7.02 ( bit_se6526347334894502574or_int
% 12.85/7.02 = ( ^ [K3: int,L3: int] :
% 12.85/7.02 ( if_int
% 12.85/7.02 @ ( K3
% 12.85/7.02 = ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.02 @ ( bit_ri7919022796975470100ot_int @ L3 )
% 12.85/7.02 @ ( if_int
% 12.85/7.02 @ ( L3
% 12.85/7.02 = ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.02 @ ( bit_ri7919022796975470100ot_int @ K3 )
% 12.85/7.02 @ ( if_int @ ( K3 = zero_zero_int ) @ L3 @ ( if_int @ ( L3 = 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 @ L3 @ ( 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 @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8333_bit__horner__sum__bit__iff,axiom,
% 12.85/7.02 ! [Bs: list_o,N: nat] :
% 12.85/7.02 ( ( bit_se9216721137139052372nteger @ ( groups3417619833198082522nteger @ zero_n356916108424825756nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Bs ) @ N )
% 12.85/7.02 = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
% 12.85/7.02 & ( nth_o @ Bs @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8334_bit__horner__sum__bit__iff,axiom,
% 12.85/7.02 ! [Bs: list_o,N: nat] :
% 12.85/7.02 ( ( bit_se1148574629649215175it_nat @ ( groups9119017779487936845_o_nat @ zero_n2687167440665602831ol_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Bs ) @ N )
% 12.85/7.02 = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
% 12.85/7.02 & ( nth_o @ Bs @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8335_bit__horner__sum__bit__iff,axiom,
% 12.85/7.02 ! [Bs: list_o,N: nat] :
% 12.85/7.02 ( ( bit_se1146084159140164899it_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ N )
% 12.85/7.02 = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
% 12.85/7.02 & ( nth_o @ Bs @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8336_valid__eq2,axiom,
% 12.85/7.02 ! [T: vEBT_VEBT,D: nat] :
% 12.85/7.02 ( ( vEBT_VEBT_valid @ T @ D )
% 12.85/7.02 => ( vEBT_invar_vebt @ T @ D ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8337_valid__eq1,axiom,
% 12.85/7.02 ! [T: vEBT_VEBT,D: nat] :
% 12.85/7.02 ( ( vEBT_invar_vebt @ T @ D )
% 12.85/7.02 => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8338_valid__eq,axiom,
% 12.85/7.02 vEBT_VEBT_valid = vEBT_invar_vebt ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8339_intind,axiom,
% 12.85/7.02 ! [I: nat,N: nat,P: nat > $o,X: nat] :
% 12.85/7.02 ( ( ord_less_nat @ I @ N )
% 12.85/7.02 => ( ( P @ X )
% 12.85/7.02 => ( P @ ( nth_nat @ ( replicate_nat @ N @ X ) @ I ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8340_intind,axiom,
% 12.85/7.02 ! [I: nat,N: nat,P: int > $o,X: int] :
% 12.85/7.02 ( ( ord_less_nat @ I @ N )
% 12.85/7.02 => ( ( P @ X )
% 12.85/7.02 => ( P @ ( nth_int @ ( replicate_int @ N @ X ) @ I ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8341_intind,axiom,
% 12.85/7.02 ! [I: nat,N: nat,P: real > $o,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ I @ N )
% 12.85/7.02 => ( ( P @ X )
% 12.85/7.02 => ( P @ ( nth_real @ ( replicate_real @ N @ X ) @ I ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8342_intind,axiom,
% 12.85/7.02 ! [I: nat,N: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
% 12.85/7.02 ( ( ord_less_nat @ I @ N )
% 12.85/7.02 => ( ( P @ X )
% 12.85/7.02 => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8343_inthall,axiom,
% 12.85/7.02 ! [Xs: list_complex,P: complex > $o,N: nat] :
% 12.85/7.02 ( ! [X4: complex] :
% 12.85/7.02 ( ( member_complex @ X4 @ ( set_complex2 @ Xs ) )
% 12.85/7.02 => ( P @ X4 ) )
% 12.85/7.02 => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
% 12.85/7.02 => ( P @ ( nth_complex @ Xs @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8344_inthall,axiom,
% 12.85/7.02 ! [Xs: list_set_nat,P: set_nat > $o,N: nat] :
% 12.85/7.02 ( ! [X4: set_nat] :
% 12.85/7.02 ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
% 12.85/7.02 => ( P @ X4 ) )
% 12.85/7.02 => ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 12.85/7.02 => ( P @ ( nth_set_nat @ Xs @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8345_inthall,axiom,
% 12.85/7.02 ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
% 12.85/7.02 ( ! [X4: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 12.85/7.02 => ( P @ X4 ) )
% 12.85/7.02 => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 12.85/7.02 => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8346_inthall,axiom,
% 12.85/7.02 ! [Xs: list_real,P: real > $o,N: nat] :
% 12.85/7.02 ( ! [X4: real] :
% 12.85/7.02 ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
% 12.85/7.02 => ( P @ X4 ) )
% 12.85/7.02 => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 12.85/7.02 => ( P @ ( nth_real @ Xs @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8347_inthall,axiom,
% 12.85/7.02 ! [Xs: list_o,P: $o > $o,N: nat] :
% 12.85/7.02 ( ! [X4: $o] :
% 12.85/7.02 ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 12.85/7.02 => ( P @ X4 ) )
% 12.85/7.02 => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 12.85/7.02 => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8348_inthall,axiom,
% 12.85/7.02 ! [Xs: list_nat,P: nat > $o,N: nat] :
% 12.85/7.02 ( ! [X4: nat] :
% 12.85/7.02 ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 12.85/7.02 => ( P @ X4 ) )
% 12.85/7.02 => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 12.85/7.02 => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8349_inthall,axiom,
% 12.85/7.02 ! [Xs: list_int,P: int > $o,N: nat] :
% 12.85/7.02 ( ! [X4: int] :
% 12.85/7.02 ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 12.85/7.02 => ( P @ X4 ) )
% 12.85/7.02 => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 12.85/7.02 => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8350_bit_Ocompl__eq__compl__iff,axiom,
% 12.85/7.02 ! [X: int,Y4: int] :
% 12.85/7.02 ( ( ( bit_ri7919022796975470100ot_int @ X )
% 12.85/7.02 = ( bit_ri7919022796975470100ot_int @ Y4 ) )
% 12.85/7.02 = ( X = Y4 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8351_bit_Odouble__compl,axiom,
% 12.85/7.02 ! [X: int] :
% 12.85/7.02 ( ( bit_ri7919022796975470100ot_int @ ( bit_ri7919022796975470100ot_int @ X ) )
% 12.85/7.02 = X ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8352_bit_Oxor__compl__right,axiom,
% 12.85/7.02 ! [X: int,Y4: int] :
% 12.85/7.02 ( ( bit_se6526347334894502574or_int @ X @ ( bit_ri7919022796975470100ot_int @ Y4 ) )
% 12.85/7.02 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ X @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8353_bit_Oxor__compl__left,axiom,
% 12.85/7.02 ! [X: int,Y4: int] :
% 12.85/7.02 ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ Y4 )
% 12.85/7.02 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ X @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8354_bit_Oconj__cancel__left,axiom,
% 12.85/7.02 ! [X: int] :
% 12.85/7.02 ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
% 12.85/7.02 = zero_zero_int ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8355_bit_Oconj__cancel__right,axiom,
% 12.85/7.02 ! [X: int] :
% 12.85/7.02 ( ( bit_se725231765392027082nd_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
% 12.85/7.02 = zero_zero_int ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8356_bit_Ocompl__zero,axiom,
% 12.85/7.02 ( ( bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger )
% 12.85/7.02 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8357_bit_Ocompl__zero,axiom,
% 12.85/7.02 ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
% 12.85/7.02 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8358_bit_Ocompl__one,axiom,
% 12.85/7.02 ( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.02 = zero_z3403309356797280102nteger ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8359_bit_Ocompl__one,axiom,
% 12.85/7.02 ( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.02 = zero_zero_int ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8360_bit_Oxor__one__left,axiom,
% 12.85/7.02 ! [X: code_integer] :
% 12.85/7.02 ( ( bit_se3222712562003087583nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 12.85/7.02 = ( bit_ri7632146776885996613nteger @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8361_bit_Oxor__one__left,axiom,
% 12.85/7.02 ! [X: int] :
% 12.85/7.02 ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 12.85/7.02 = ( bit_ri7919022796975470100ot_int @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8362_bit_Oxor__one__right,axiom,
% 12.85/7.02 ! [X: code_integer] :
% 12.85/7.02 ( ( bit_se3222712562003087583nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.02 = ( bit_ri7632146776885996613nteger @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8363_bit_Oxor__one__right,axiom,
% 12.85/7.02 ! [X: int] :
% 12.85/7.02 ( ( bit_se6526347334894502574or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.02 = ( bit_ri7919022796975470100ot_int @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8364_bit_Oxor__cancel__left,axiom,
% 12.85/7.02 ! [X: code_integer] :
% 12.85/7.02 ( ( bit_se3222712562003087583nteger @ ( bit_ri7632146776885996613nteger @ X ) @ X )
% 12.85/7.02 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8365_bit_Oxor__cancel__left,axiom,
% 12.85/7.02 ! [X: int] :
% 12.85/7.02 ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
% 12.85/7.02 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8366_bit_Oxor__cancel__right,axiom,
% 12.85/7.02 ! [X: code_integer] :
% 12.85/7.02 ( ( bit_se3222712562003087583nteger @ X @ ( bit_ri7632146776885996613nteger @ X ) )
% 12.85/7.02 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8367_bit_Oxor__cancel__right,axiom,
% 12.85/7.02 ! [X: int] :
% 12.85/7.02 ( ( bit_se6526347334894502574or_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
% 12.85/7.02 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8368_not__negative__int__iff,axiom,
% 12.85/7.02 ! [K: int] :
% 12.85/7.02 ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 12.85/7.02 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8369_not__nonnegative__int__iff,axiom,
% 12.85/7.02 ! [K: int] :
% 12.85/7.02 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 12.85/7.02 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8370_minus__not__numeral__eq,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( uminus1351360451143612070nteger @ ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 12.85/7.02 = ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8371_minus__not__numeral__eq,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( uminus_uminus_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.02 = ( numeral_numeral_int @ ( inc @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8372_even__not__iff,axiom,
% 12.85/7.02 ! [A: code_integer] :
% 12.85/7.02 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri7632146776885996613nteger @ A ) )
% 12.85/7.02 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8373_even__not__iff,axiom,
% 12.85/7.02 ! [A: int] :
% 12.85/7.02 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ A ) )
% 12.85/7.02 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8374_push__bit__minus__one__eq__not__mask,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.02 = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8375_push__bit__minus__one__eq__not__mask,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.02 = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8376_not__one__eq,axiom,
% 12.85/7.02 ( ( bit_ri7632146776885996613nteger @ one_one_Code_integer )
% 12.85/7.02 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8377_not__one__eq,axiom,
% 12.85/7.02 ( ( bit_ri7919022796975470100ot_int @ one_one_int )
% 12.85/7.02 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8378_of__int__not__eq,axiom,
% 12.85/7.02 ! [K: int] :
% 12.85/7.02 ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 12.85/7.02 = ( bit_ri7919022796975470100ot_int @ ( ring_1_of_int_int @ K ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8379_take__bit__not__iff,axiom,
% 12.85/7.02 ! [N: nat,A: int,B: int] :
% 12.85/7.02 ( ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) )
% 12.85/7.02 = ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ B ) ) )
% 12.85/7.02 = ( ( bit_se2923211474154528505it_int @ N @ A )
% 12.85/7.02 = ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8380_take__bit__not__take__bit,axiom,
% 12.85/7.02 ! [N: nat,A: int] :
% 12.85/7.02 ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 12.85/7.02 = ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8381_bit__not__int__iff,axiom,
% 12.85/7.02 ! [K: int,N: nat] :
% 12.85/7.02 ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N )
% 12.85/7.02 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8382_of__int__not__numeral,axiom,
% 12.85/7.02 ! [K: num] :
% 12.85/7.02 ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) )
% 12.85/7.02 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8383_not__add__distrib,axiom,
% 12.85/7.02 ! [A: int,B: int] :
% 12.85/7.02 ( ( bit_ri7919022796975470100ot_int @ ( plus_plus_int @ A @ B ) )
% 12.85/7.02 = ( minus_minus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8384_not__diff__distrib,axiom,
% 12.85/7.02 ! [A: int,B: int] :
% 12.85/7.02 ( ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A @ B ) )
% 12.85/7.02 = ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8385_minus__eq__not__plus__1,axiom,
% 12.85/7.02 ( uminus1351360451143612070nteger
% 12.85/7.02 = ( ^ [A3: code_integer] : ( plus_p5714425477246183910nteger @ ( bit_ri7632146776885996613nteger @ A3 ) @ one_one_Code_integer ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8386_minus__eq__not__plus__1,axiom,
% 12.85/7.02 ( uminus_uminus_int
% 12.85/7.02 = ( ^ [A3: int] : ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A3 ) @ one_one_int ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8387_not__eq__complement,axiom,
% 12.85/7.02 ( bit_ri7632146776885996613nteger
% 12.85/7.02 = ( ^ [A3: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A3 ) @ one_one_Code_integer ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8388_not__eq__complement,axiom,
% 12.85/7.02 ( bit_ri7919022796975470100ot_int
% 12.85/7.02 = ( ^ [A3: int] : ( minus_minus_int @ ( uminus_uminus_int @ A3 ) @ one_one_int ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8389_minus__eq__not__minus__1,axiom,
% 12.85/7.02 ( uminus1351360451143612070nteger
% 12.85/7.02 = ( ^ [A3: code_integer] : ( bit_ri7632146776885996613nteger @ ( minus_8373710615458151222nteger @ A3 @ one_one_Code_integer ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8390_minus__eq__not__minus__1,axiom,
% 12.85/7.02 ( uminus_uminus_int
% 12.85/7.02 = ( ^ [A3: int] : ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A3 @ one_one_int ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8391_not__int__def,axiom,
% 12.85/7.02 ( bit_ri7919022796975470100ot_int
% 12.85/7.02 = ( ^ [K3: int] : ( minus_minus_int @ ( uminus_uminus_int @ K3 ) @ one_one_int ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8392_and__not__numerals_I1_J,axiom,
% 12.85/7.02 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 12.85/7.02 = zero_zero_int ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8393_disjunctive__diff,axiom,
% 12.85/7.02 ! [B: int,A: int] :
% 12.85/7.02 ( ! [N2: nat] :
% 12.85/7.02 ( ( bit_se1146084159140164899it_int @ B @ N2 )
% 12.85/7.02 => ( bit_se1146084159140164899it_int @ A @ N2 ) )
% 12.85/7.02 => ( ( minus_minus_int @ A @ B )
% 12.85/7.02 = ( bit_se725231765392027082nd_int @ A @ ( bit_ri7919022796975470100ot_int @ B ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8394_take__bit__not__eq__mask__diff,axiom,
% 12.85/7.02 ! [N: nat,A: int] :
% 12.85/7.02 ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) )
% 12.85/7.02 = ( minus_minus_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8395_minus__numeral__inc__eq,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) )
% 12.85/7.02 = ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8396_minus__numeral__inc__eq,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) )
% 12.85/7.02 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8397_unset__bit__int__def,axiom,
% 12.85/7.02 ( bit_se4203085406695923979it_int
% 12.85/7.02 = ( ^ [N4: nat,K3: int] : ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8398_not__int__div__2,axiom,
% 12.85/7.02 ! [K: int] :
% 12.85/7.02 ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/7.02 = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8399_even__not__iff__int,axiom,
% 12.85/7.02 ! [K: int] :
% 12.85/7.02 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
% 12.85/7.02 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8400_not__numeral__Bit0__eq,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) )
% 12.85/7.02 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8401_not__numeral__Bit0__eq,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) )
% 12.85/7.02 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8402_and__not__numerals_I4_J,axiom,
% 12.85/7.02 ! [M: num] :
% 12.85/7.02 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 12.85/7.02 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8403_and__not__numerals_I2_J,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.85/7.02 = one_one_int ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8404_bit__minus__int__iff,axiom,
% 12.85/7.02 ! [K: int,N: nat] :
% 12.85/7.02 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
% 12.85/7.02 = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8405_take__bit__not__mask__eq__0,axiom,
% 12.85/7.02 ! [M: nat,N: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ M @ N )
% 12.85/7.02 => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) )
% 12.85/7.02 = zero_zero_int ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8406_push__bit__mask__eq,axiom,
% 12.85/7.02 ! [M: nat,N: nat] :
% 12.85/7.02 ( ( bit_se545348938243370406it_int @ M @ ( bit_se2000444600071755411sk_int @ N ) )
% 12.85/7.02 = ( bit_se725231765392027082nd_int @ ( bit_se2000444600071755411sk_int @ ( plus_plus_nat @ N @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ M ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8407_unset__bit__eq__and__not,axiom,
% 12.85/7.02 ( bit_se4203085406695923979it_int
% 12.85/7.02 = ( ^ [N4: nat,A3: int] : ( bit_se725231765392027082nd_int @ A3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8408_unset__bit__eq__and__not,axiom,
% 12.85/7.02 ( bit_se8260200283734997820nteger
% 12.85/7.02 = ( ^ [N4: nat,A3: code_integer] : ( bit_se3949692690581998587nteger @ A3 @ ( bit_ri7632146776885996613nteger @ ( bit_se7788150548672797655nteger @ N4 @ one_one_Code_integer ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8409_and__not__numerals_I5_J,axiom,
% 12.85/7.02 ! [M: num,N: num] :
% 12.85/7.02 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.85/7.02 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8410_and__not__numerals_I7_J,axiom,
% 12.85/7.02 ! [M: num] :
% 12.85/7.02 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 12.85/7.02 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8411_and__not__numerals_I3_J,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.85/7.02 = zero_zero_int ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8412_and__not__numerals_I6_J,axiom,
% 12.85/7.02 ! [M: num,N: num] :
% 12.85/7.02 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.85/7.02 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8413_and__not__numerals_I9_J,axiom,
% 12.85/7.02 ! [M: num,N: num] :
% 12.85/7.02 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.85/7.02 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8414_bit__not__iff__eq,axiom,
% 12.85/7.02 ! [A: int,N: nat] :
% 12.85/7.02 ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ A ) @ N )
% 12.85/7.02 = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 12.85/7.02 != zero_zero_int )
% 12.85/7.02 & ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8415_minus__exp__eq__not__mask,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.02 = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8416_minus__exp__eq__not__mask,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.02 = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8417_and__not__numerals_I8_J,axiom,
% 12.85/7.02 ! [M: num,N: num] :
% 12.85/7.02 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.85/7.02 = ( 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 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8418_not__int__rec,axiom,
% 12.85/7.02 ( bit_ri7919022796975470100ot_int
% 12.85/7.02 = ( ^ [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 ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8419_Cauchy__iff2,axiom,
% 12.85/7.02 ( topolo4055970368930404560y_real
% 12.85/7.02 = ( ^ [X8: nat > real] :
% 12.85/7.02 ! [J3: nat] :
% 12.85/7.02 ? [M7: nat] :
% 12.85/7.02 ! [M5: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ M7 @ M5 )
% 12.85/7.02 => ! [N4: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ M7 @ N4 )
% 12.85/7.02 => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X8 @ M5 ) @ ( X8 @ N4 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8420_length__subseqs,axiom,
% 12.85/7.02 ! [Xs: list_real] :
% 12.85/7.02 ( ( size_s6660260683639930848t_real @ ( subseqs_real @ Xs ) )
% 12.85/7.02 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_real @ Xs ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8421_length__subseqs,axiom,
% 12.85/7.02 ! [Xs: list_o] :
% 12.85/7.02 ( ( size_s2710708370519433104list_o @ ( subseqs_o @ Xs ) )
% 12.85/7.02 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_o @ Xs ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8422_length__subseqs,axiom,
% 12.85/7.02 ! [Xs: list_nat] :
% 12.85/7.02 ( ( size_s3023201423986296836st_nat @ ( subseqs_nat @ Xs ) )
% 12.85/7.02 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_nat @ Xs ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8423_length__subseqs,axiom,
% 12.85/7.02 ! [Xs: list_int] :
% 12.85/7.02 ( ( size_s533118279054570080st_int @ ( subseqs_int @ Xs ) )
% 12.85/7.02 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_int @ Xs ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8424_csqrt_Osimps_I1_J,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( re @ ( csqrt @ Z ) )
% 12.85/7.02 = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8425_sinh__def,axiom,
% 12.85/7.02 ( sinh_complex
% 12.85/7.02 = ( ^ [X3: complex] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( minus_minus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8426_sinh__def,axiom,
% 12.85/7.02 ( sinh_real
% 12.85/7.02 = ( ^ [X3: real] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8427_scaleR__cancel__right,axiom,
% 12.85/7.02 ! [A: real,X: complex,B: real] :
% 12.85/7.02 ( ( ( real_V2046097035970521341omplex @ A @ X )
% 12.85/7.02 = ( real_V2046097035970521341omplex @ B @ X ) )
% 12.85/7.02 = ( ( A = B )
% 12.85/7.02 | ( X = zero_zero_complex ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8428_scaleR__cancel__right,axiom,
% 12.85/7.02 ! [A: real,X: real,B: real] :
% 12.85/7.02 ( ( ( real_V1485227260804924795R_real @ A @ X )
% 12.85/7.02 = ( real_V1485227260804924795R_real @ B @ X ) )
% 12.85/7.02 = ( ( A = B )
% 12.85/7.02 | ( X = zero_zero_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8429_scaleR__zero__right,axiom,
% 12.85/7.02 ! [A: real] :
% 12.85/7.02 ( ( real_V2046097035970521341omplex @ A @ zero_zero_complex )
% 12.85/7.02 = zero_zero_complex ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8430_scaleR__zero__right,axiom,
% 12.85/7.02 ! [A: real] :
% 12.85/7.02 ( ( real_V1485227260804924795R_real @ A @ zero_zero_real )
% 12.85/7.02 = zero_zero_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8431_mult__scaleR__right,axiom,
% 12.85/7.02 ! [X: complex,A: real,Y4: complex] :
% 12.85/7.02 ( ( times_times_complex @ X @ ( real_V2046097035970521341omplex @ A @ Y4 ) )
% 12.85/7.02 = ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8432_mult__scaleR__right,axiom,
% 12.85/7.02 ! [X: real,A: real,Y4: real] :
% 12.85/7.02 ( ( times_times_real @ X @ ( real_V1485227260804924795R_real @ A @ Y4 ) )
% 12.85/7.02 = ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8433_mult__scaleR__left,axiom,
% 12.85/7.02 ! [A: real,X: complex,Y4: complex] :
% 12.85/7.02 ( ( times_times_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ Y4 )
% 12.85/7.02 = ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8434_mult__scaleR__left,axiom,
% 12.85/7.02 ! [A: real,X: real,Y4: real] :
% 12.85/7.02 ( ( times_times_real @ ( real_V1485227260804924795R_real @ A @ X ) @ Y4 )
% 12.85/7.02 = ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8435_scaleR__cancel__left,axiom,
% 12.85/7.02 ! [A: real,X: complex,Y4: complex] :
% 12.85/7.02 ( ( ( real_V2046097035970521341omplex @ A @ X )
% 12.85/7.02 = ( real_V2046097035970521341omplex @ A @ Y4 ) )
% 12.85/7.02 = ( ( X = Y4 )
% 12.85/7.02 | ( A = zero_zero_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8436_scaleR__cancel__left,axiom,
% 12.85/7.02 ! [A: real,X: real,Y4: real] :
% 12.85/7.02 ( ( ( real_V1485227260804924795R_real @ A @ X )
% 12.85/7.02 = ( real_V1485227260804924795R_real @ A @ Y4 ) )
% 12.85/7.02 = ( ( X = Y4 )
% 12.85/7.02 | ( A = zero_zero_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8437_scaleR__scaleR,axiom,
% 12.85/7.02 ! [A: real,B: real,X: complex] :
% 12.85/7.02 ( ( real_V2046097035970521341omplex @ A @ ( real_V2046097035970521341omplex @ B @ X ) )
% 12.85/7.02 = ( real_V2046097035970521341omplex @ ( times_times_real @ A @ B ) @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8438_scaleR__scaleR,axiom,
% 12.85/7.02 ! [A: real,B: real,X: real] :
% 12.85/7.02 ( ( real_V1485227260804924795R_real @ A @ ( real_V1485227260804924795R_real @ B @ X ) )
% 12.85/7.02 = ( real_V1485227260804924795R_real @ ( times_times_real @ A @ B ) @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8439_complex__Re__of__nat,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( re @ ( semiri8010041392384452111omplex @ N ) )
% 12.85/7.02 = ( semiri5074537144036343181t_real @ N ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8440_scaleR__zero__left,axiom,
% 12.85/7.02 ! [X: complex] :
% 12.85/7.02 ( ( real_V2046097035970521341omplex @ zero_zero_real @ X )
% 12.85/7.02 = zero_zero_complex ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8441_scaleR__zero__left,axiom,
% 12.85/7.02 ! [X: real] :
% 12.85/7.02 ( ( real_V1485227260804924795R_real @ zero_zero_real @ X )
% 12.85/7.02 = zero_zero_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8442_scaleR__eq__0__iff,axiom,
% 12.85/7.02 ! [A: real,X: complex] :
% 12.85/7.02 ( ( ( real_V2046097035970521341omplex @ A @ X )
% 12.85/7.02 = zero_zero_complex )
% 12.85/7.02 = ( ( A = zero_zero_real )
% 12.85/7.02 | ( X = zero_zero_complex ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8443_scaleR__eq__0__iff,axiom,
% 12.85/7.02 ! [A: real,X: real] :
% 12.85/7.02 ( ( ( real_V1485227260804924795R_real @ A @ X )
% 12.85/7.02 = zero_zero_real )
% 12.85/7.02 = ( ( A = zero_zero_real )
% 12.85/7.02 | ( X = zero_zero_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8444_complex__Re__numeral,axiom,
% 12.85/7.02 ! [V: num] :
% 12.85/7.02 ( ( re @ ( numera6690914467698888265omplex @ V ) )
% 12.85/7.02 = ( numeral_numeral_real @ V ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8445_scaleR__collapse,axiom,
% 12.85/7.02 ! [U: real,A: complex] :
% 12.85/7.02 ( ( plus_plus_complex @ ( real_V2046097035970521341omplex @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V2046097035970521341omplex @ U @ A ) )
% 12.85/7.02 = A ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8446_scaleR__collapse,axiom,
% 12.85/7.02 ! [U: real,A: real] :
% 12.85/7.02 ( ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V1485227260804924795R_real @ U @ A ) )
% 12.85/7.02 = A ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8447_norm__scaleR,axiom,
% 12.85/7.02 ! [A: real,X: complex] :
% 12.85/7.02 ( ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ A @ X ) )
% 12.85/7.02 = ( times_times_real @ ( abs_abs_real @ A ) @ ( real_V1022390504157884413omplex @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8448_norm__scaleR,axiom,
% 12.85/7.02 ! [A: real,X: real] :
% 12.85/7.02 ( ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ A @ X ) )
% 12.85/7.02 = ( times_times_real @ ( abs_abs_real @ A ) @ ( real_V7735802525324610683m_real @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8449_Re__divide__of__nat,axiom,
% 12.85/7.02 ! [Z: complex,N: nat] :
% 12.85/7.02 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
% 12.85/7.02 = ( divide_divide_real @ ( re @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8450_Re__divide__of__real,axiom,
% 12.85/7.02 ! [Z: complex,R2: real] :
% 12.85/7.02 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R2 ) ) )
% 12.85/7.02 = ( divide_divide_real @ ( re @ Z ) @ R2 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8451_Re__sgn,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( re @ ( sgn_sgn_complex @ Z ) )
% 12.85/7.02 = ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8452_scaleR__times,axiom,
% 12.85/7.02 ! [U: num,W: num,A: complex] :
% 12.85/7.02 ( ( real_V2046097035970521341omplex @ ( numeral_numeral_real @ U ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 12.85/7.02 = ( real_V2046097035970521341omplex @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ A ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8453_scaleR__times,axiom,
% 12.85/7.02 ! [U: num,W: num,A: real] :
% 12.85/7.02 ( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ U ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 12.85/7.02 = ( real_V1485227260804924795R_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ A ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8454_Re__divide__numeral,axiom,
% 12.85/7.02 ! [Z: complex,W: num] :
% 12.85/7.02 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 12.85/7.02 = ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8455_cos__Arg__i__mult__zero,axiom,
% 12.85/7.02 ! [Y4: complex] :
% 12.85/7.02 ( ( Y4 != zero_zero_complex )
% 12.85/7.02 => ( ( ( re @ Y4 )
% 12.85/7.02 = zero_zero_real )
% 12.85/7.02 => ( ( cos_real @ ( arg @ Y4 ) )
% 12.85/7.02 = zero_zero_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8456_inverse__scaleR__times,axiom,
% 12.85/7.02 ! [V: num,W: num,A: complex] :
% 12.85/7.02 ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 12.85/7.02 = ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ W ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8457_inverse__scaleR__times,axiom,
% 12.85/7.02 ! [V: num,W: num,A: real] :
% 12.85/7.02 ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 12.85/7.02 = ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ W ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8458_fraction__scaleR__times,axiom,
% 12.85/7.02 ! [U: num,V: num,W: num,A: complex] :
% 12.85/7.02 ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 12.85/7.02 = ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8459_fraction__scaleR__times,axiom,
% 12.85/7.02 ! [U: num,V: num,W: num,A: real] :
% 12.85/7.02 ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 12.85/7.02 = ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8460_scaleR__half__double,axiom,
% 12.85/7.02 ! [A: complex] :
% 12.85/7.02 ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_complex @ A @ A ) )
% 12.85/7.02 = A ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8461_scaleR__half__double,axiom,
% 12.85/7.02 ! [A: real] :
% 12.85/7.02 ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ A @ A ) )
% 12.85/7.02 = A ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8462_scaleR__right__imp__eq,axiom,
% 12.85/7.02 ! [X: complex,A: real,B: real] :
% 12.85/7.02 ( ( X != zero_zero_complex )
% 12.85/7.02 => ( ( ( real_V2046097035970521341omplex @ A @ X )
% 12.85/7.02 = ( real_V2046097035970521341omplex @ B @ X ) )
% 12.85/7.02 => ( A = B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8463_scaleR__right__imp__eq,axiom,
% 12.85/7.02 ! [X: real,A: real,B: real] :
% 12.85/7.02 ( ( X != zero_zero_real )
% 12.85/7.02 => ( ( ( real_V1485227260804924795R_real @ A @ X )
% 12.85/7.02 = ( real_V1485227260804924795R_real @ B @ X ) )
% 12.85/7.02 => ( A = B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8464_scaleR__right__diff__distrib,axiom,
% 12.85/7.02 ! [A: real,X: complex,Y4: complex] :
% 12.85/7.02 ( ( real_V2046097035970521341omplex @ A @ ( minus_minus_complex @ X @ Y4 ) )
% 12.85/7.02 = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ A @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8465_scaleR__right__diff__distrib,axiom,
% 12.85/7.02 ! [A: real,X: real,Y4: real] :
% 12.85/7.02 ( ( real_V1485227260804924795R_real @ A @ ( minus_minus_real @ X @ Y4 ) )
% 12.85/7.02 = ( minus_minus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8466_scaleR__complex_Osimps_I1_J,axiom,
% 12.85/7.02 ! [R2: real,X: complex] :
% 12.85/7.02 ( ( re @ ( real_V2046097035970521341omplex @ R2 @ X ) )
% 12.85/7.02 = ( times_times_real @ R2 @ ( re @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8467_scaleR__left__imp__eq,axiom,
% 12.85/7.02 ! [A: real,X: complex,Y4: complex] :
% 12.85/7.02 ( ( A != zero_zero_real )
% 12.85/7.02 => ( ( ( real_V2046097035970521341omplex @ A @ X )
% 12.85/7.02 = ( real_V2046097035970521341omplex @ A @ Y4 ) )
% 12.85/7.02 => ( X = Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8468_scaleR__left__imp__eq,axiom,
% 12.85/7.02 ! [A: real,X: real,Y4: real] :
% 12.85/7.02 ( ( A != zero_zero_real )
% 12.85/7.02 => ( ( ( real_V1485227260804924795R_real @ A @ X )
% 12.85/7.02 = ( real_V1485227260804924795R_real @ A @ Y4 ) )
% 12.85/7.02 => ( X = Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8469_real__scaleR__def,axiom,
% 12.85/7.02 real_V1485227260804924795R_real = times_times_real ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8470_scaleR__conv__of__real,axiom,
% 12.85/7.02 ( real_V2046097035970521341omplex
% 12.85/7.02 = ( ^ [R5: real] : ( times_times_complex @ ( real_V4546457046886955230omplex @ R5 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8471_scaleR__conv__of__real,axiom,
% 12.85/7.02 ( real_V1485227260804924795R_real
% 12.85/7.02 = ( ^ [R5: real] : ( times_times_real @ ( real_V1803761363581548252l_real @ R5 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8472_scaleR__left__diff__distrib,axiom,
% 12.85/7.02 ! [A: real,B: real,X: complex] :
% 12.85/7.02 ( ( real_V2046097035970521341omplex @ ( minus_minus_real @ A @ B ) @ X )
% 12.85/7.02 = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ B @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8473_scaleR__left__diff__distrib,axiom,
% 12.85/7.02 ! [A: real,B: real,X: real] :
% 12.85/7.02 ( ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B ) @ X )
% 12.85/7.02 = ( minus_minus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8474_scaleR__left_Odiff,axiom,
% 12.85/7.02 ! [X: real,Y4: real,Xa: complex] :
% 12.85/7.02 ( ( real_V2046097035970521341omplex @ ( minus_minus_real @ X @ Y4 ) @ Xa )
% 12.85/7.02 = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ X @ Xa ) @ ( real_V2046097035970521341omplex @ Y4 @ Xa ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8475_scaleR__left_Odiff,axiom,
% 12.85/7.02 ! [X: real,Y4: real,Xa: real] :
% 12.85/7.02 ( ( real_V1485227260804924795R_real @ ( minus_minus_real @ X @ Y4 ) @ Xa )
% 12.85/7.02 = ( minus_minus_real @ ( real_V1485227260804924795R_real @ X @ Xa ) @ ( real_V1485227260804924795R_real @ Y4 @ Xa ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8476_imaginary__unit_Osimps_I1_J,axiom,
% 12.85/7.02 ( ( re @ imaginary_unit )
% 12.85/7.02 = zero_zero_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8477_complex__Re__le__cmod,axiom,
% 12.85/7.02 ! [X: complex] : ( ord_less_eq_real @ ( re @ X ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8478_zero__complex_Osimps_I1_J,axiom,
% 12.85/7.02 ( ( re @ zero_zero_complex )
% 12.85/7.02 = zero_zero_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8479_complex__scaleR,axiom,
% 12.85/7.02 ! [R2: real,A: real,B: real] :
% 12.85/7.02 ( ( real_V2046097035970521341omplex @ R2 @ ( complex2 @ A @ B ) )
% 12.85/7.02 = ( complex2 @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8480_minus__complex_Osimps_I1_J,axiom,
% 12.85/7.02 ! [X: complex,Y4: complex] :
% 12.85/7.02 ( ( re @ ( minus_minus_complex @ X @ Y4 ) )
% 12.85/7.02 = ( minus_minus_real @ ( re @ X ) @ ( re @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8481_scaleR__right__mono,axiom,
% 12.85/7.02 ! [A: real,B: real,X: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ A @ B )
% 12.85/7.02 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.02 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8482_scaleR__right__mono__neg,axiom,
% 12.85/7.02 ! [B: real,A: real,C: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ B @ A )
% 12.85/7.02 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.85/7.02 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ C ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8483_scaleR__le__cancel__left__pos,axiom,
% 12.85/7.02 ! [C: real,A: real,B: real] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/7.02 => ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) )
% 12.85/7.02 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8484_scaleR__le__cancel__left__neg,axiom,
% 12.85/7.02 ! [C: real,A: real,B: real] :
% 12.85/7.02 ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/7.02 => ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) )
% 12.85/7.02 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8485_scaleR__le__cancel__left,axiom,
% 12.85/7.02 ! [C: real,A: real,B: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) )
% 12.85/7.02 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/7.02 => ( ord_less_eq_real @ A @ B ) )
% 12.85/7.02 & ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/7.02 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8486_scaleR__left__mono,axiom,
% 12.85/7.02 ! [X: real,Y4: real,A: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/7.02 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/7.02 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8487_scaleR__left__mono__neg,axiom,
% 12.85/7.02 ! [B: real,A: real,C: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ B @ A )
% 12.85/7.02 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 12.85/7.02 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8488_vector__fraction__eq__iff,axiom,
% 12.85/7.02 ! [U: real,V: real,A: complex,X: complex] :
% 12.85/7.02 ( ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A )
% 12.85/7.02 = X )
% 12.85/7.02 = ( ( ( V = zero_zero_real )
% 12.85/7.02 => ( X = zero_zero_complex ) )
% 12.85/7.02 & ( ( V != zero_zero_real )
% 12.85/7.02 => ( ( real_V2046097035970521341omplex @ U @ A )
% 12.85/7.02 = ( real_V2046097035970521341omplex @ V @ X ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8489_vector__fraction__eq__iff,axiom,
% 12.85/7.02 ! [U: real,V: real,A: real,X: real] :
% 12.85/7.02 ( ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A )
% 12.85/7.02 = X )
% 12.85/7.02 = ( ( ( V = zero_zero_real )
% 12.85/7.02 => ( X = zero_zero_real ) )
% 12.85/7.02 & ( ( V != zero_zero_real )
% 12.85/7.02 => ( ( real_V1485227260804924795R_real @ U @ A )
% 12.85/7.02 = ( real_V1485227260804924795R_real @ V @ X ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8490_eq__vector__fraction__iff,axiom,
% 12.85/7.02 ! [X: complex,U: real,V: real,A: complex] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A ) )
% 12.85/7.02 = ( ( ( V = zero_zero_real )
% 12.85/7.02 => ( X = zero_zero_complex ) )
% 12.85/7.02 & ( ( V != zero_zero_real )
% 12.85/7.02 => ( ( real_V2046097035970521341omplex @ V @ X )
% 12.85/7.02 = ( real_V2046097035970521341omplex @ U @ A ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8491_eq__vector__fraction__iff,axiom,
% 12.85/7.02 ! [X: real,U: real,V: real,A: real] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A ) )
% 12.85/7.02 = ( ( ( V = zero_zero_real )
% 12.85/7.02 => ( X = zero_zero_real ) )
% 12.85/7.02 & ( ( V != zero_zero_real )
% 12.85/7.02 => ( ( real_V1485227260804924795R_real @ V @ X )
% 12.85/7.02 = ( real_V1485227260804924795R_real @ U @ A ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8492_Real__Vector__Spaces_Ole__add__iff1,axiom,
% 12.85/7.02 ! [A: real,E2: real,C: real,B: real,D: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E2 ) @ D ) )
% 12.85/7.02 = ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8493_Real__Vector__Spaces_Ole__add__iff2,axiom,
% 12.85/7.02 ! [A: real,E2: real,C: real,B: real,D: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E2 ) @ D ) )
% 12.85/7.02 = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8494_abs__Re__le__cmod,axiom,
% 12.85/7.02 ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8495_Re__csqrt,axiom,
% 12.85/7.02 ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8496_zero__le__scaleR__iff,axiom,
% 12.85/7.02 ! [A: real,B: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) )
% 12.85/7.02 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.02 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 12.85/7.02 | ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/7.02 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 12.85/7.02 | ( A = zero_zero_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8497_scaleR__le__0__iff,axiom,
% 12.85/7.02 ! [A: real,B: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ B ) @ zero_zero_real )
% 12.85/7.02 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.02 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 12.85/7.02 | ( ( ord_less_real @ A @ zero_zero_real )
% 12.85/7.02 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 12.85/7.02 | ( A = zero_zero_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8498_scaleR__mono,axiom,
% 12.85/7.02 ! [A: real,B: real,X: real,Y4: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ A @ B )
% 12.85/7.02 => ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/7.02 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 12.85/7.02 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.02 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ Y4 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8499_scaleR__mono_H,axiom,
% 12.85/7.02 ! [A: real,B: real,C: real,D: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ A @ B )
% 12.85/7.02 => ( ( ord_less_eq_real @ C @ D )
% 12.85/7.02 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/7.02 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 12.85/7.02 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ D ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8500_split__scaleR__neg__le,axiom,
% 12.85/7.02 ! [A: real,X: real] :
% 12.85/7.02 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/7.02 & ( ord_less_eq_real @ X @ zero_zero_real ) )
% 12.85/7.02 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.85/7.02 & ( ord_less_eq_real @ zero_zero_real @ X ) ) )
% 12.85/7.02 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8501_split__scaleR__pos__le,axiom,
% 12.85/7.02 ! [A: real,B: real] :
% 12.85/7.02 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/7.02 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 12.85/7.02 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.85/7.02 & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 12.85/7.02 => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8502_scaleR__nonneg__nonneg,axiom,
% 12.85/7.02 ! [A: real,X: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/7.02 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.02 => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8503_scaleR__nonneg__nonpos,axiom,
% 12.85/7.02 ! [A: real,X: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 12.85/7.02 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 12.85/7.02 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8504_scaleR__nonpos__nonneg,axiom,
% 12.85/7.02 ! [A: real,X: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.85/7.02 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.02 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8505_scaleR__nonpos__nonpos,axiom,
% 12.85/7.02 ! [A: real,B: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 12.85/7.02 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 12.85/7.02 => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8506_scaleR__left__le__one__le,axiom,
% 12.85/7.02 ! [X: real,A: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.02 => ( ( ord_less_eq_real @ A @ one_one_real )
% 12.85/7.02 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8507_scaleR__2,axiom,
% 12.85/7.02 ! [X: complex] :
% 12.85/7.02 ( ( real_V2046097035970521341omplex @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X )
% 12.85/7.02 = ( plus_plus_complex @ X @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8508_scaleR__2,axiom,
% 12.85/7.02 ! [X: real] :
% 12.85/7.02 ( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X )
% 12.85/7.02 = ( plus_plus_real @ X @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8509_real__vector__eq__affinity,axiom,
% 12.85/7.02 ! [M: real,Y4: complex,X: complex,C: complex] :
% 12.85/7.02 ( ( M != zero_zero_real )
% 12.85/7.02 => ( ( Y4
% 12.85/7.02 = ( plus_plus_complex @ ( real_V2046097035970521341omplex @ M @ X ) @ C ) )
% 12.85/7.02 = ( ( minus_minus_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ Y4 ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ C ) )
% 12.85/7.02 = X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8510_real__vector__eq__affinity,axiom,
% 12.85/7.02 ! [M: real,Y4: real,X: real,C: real] :
% 12.85/7.02 ( ( M != zero_zero_real )
% 12.85/7.02 => ( ( Y4
% 12.85/7.02 = ( plus_plus_real @ ( real_V1485227260804924795R_real @ M @ X ) @ C ) )
% 12.85/7.02 = ( ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ Y4 ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ C ) )
% 12.85/7.02 = X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8511_real__vector__affinity__eq,axiom,
% 12.85/7.02 ! [M: real,X: complex,C: complex,Y4: complex] :
% 12.85/7.02 ( ( M != zero_zero_real )
% 12.85/7.02 => ( ( ( plus_plus_complex @ ( real_V2046097035970521341omplex @ M @ X ) @ C )
% 12.85/7.02 = Y4 )
% 12.85/7.02 = ( X
% 12.85/7.02 = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ Y4 ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ C ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8512_real__vector__affinity__eq,axiom,
% 12.85/7.02 ! [M: real,X: real,C: real,Y4: real] :
% 12.85/7.02 ( ( M != zero_zero_real )
% 12.85/7.02 => ( ( ( plus_plus_real @ ( real_V1485227260804924795R_real @ M @ X ) @ C )
% 12.85/7.02 = Y4 )
% 12.85/7.02 = ( X
% 12.85/7.02 = ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ Y4 ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ C ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8513_pos__divideR__le__eq,axiom,
% 12.85/7.02 ! [C: real,B: real,A: real] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/7.02 => ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
% 12.85/7.02 = ( ord_less_eq_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8514_pos__le__divideR__eq,axiom,
% 12.85/7.02 ! [C: real,A: real,B: real] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/7.02 => ( ( ord_less_eq_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
% 12.85/7.02 = ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8515_neg__divideR__le__eq,axiom,
% 12.85/7.02 ! [C: real,B: real,A: real] :
% 12.85/7.02 ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/7.02 => ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
% 12.85/7.02 = ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8516_neg__le__divideR__eq,axiom,
% 12.85/7.02 ! [C: real,A: real,B: real] :
% 12.85/7.02 ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/7.02 => ( ( ord_less_eq_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
% 12.85/7.02 = ( ord_less_eq_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8517_neg__less__divideR__eq,axiom,
% 12.85/7.02 ! [C: real,A: real,B: real] :
% 12.85/7.02 ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/7.02 => ( ( ord_less_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
% 12.85/7.02 = ( ord_less_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8518_neg__divideR__less__eq,axiom,
% 12.85/7.02 ! [C: real,B: real,A: real] :
% 12.85/7.02 ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/7.02 => ( ( ord_less_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
% 12.85/7.02 = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8519_pos__less__divideR__eq,axiom,
% 12.85/7.02 ! [C: real,A: real,B: real] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/7.02 => ( ( ord_less_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
% 12.85/7.02 = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8520_pos__divideR__less__eq,axiom,
% 12.85/7.02 ! [C: real,B: real,A: real] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/7.02 => ( ( ord_less_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
% 12.85/7.02 = ( ord_less_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8521_nonzero__inverse__scaleR__distrib,axiom,
% 12.85/7.02 ! [A: real,X: complex] :
% 12.85/7.02 ( ( A != zero_zero_real )
% 12.85/7.02 => ( ( X != zero_zero_complex )
% 12.85/7.02 => ( ( invers8013647133539491842omplex @ ( real_V2046097035970521341omplex @ A @ X ) )
% 12.85/7.02 = ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ A ) @ ( invers8013647133539491842omplex @ X ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8522_nonzero__inverse__scaleR__distrib,axiom,
% 12.85/7.02 ! [A: real,X: real] :
% 12.85/7.02 ( ( A != zero_zero_real )
% 12.85/7.02 => ( ( X != zero_zero_real )
% 12.85/7.02 => ( ( inverse_inverse_real @ ( real_V1485227260804924795R_real @ A @ X ) )
% 12.85/7.02 = ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ X ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8523_pos__le__minus__divideR__eq,axiom,
% 12.85/7.02 ! [C: real,A: real,B: real] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/7.02 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
% 12.85/7.02 = ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8524_pos__minus__divideR__le__eq,axiom,
% 12.85/7.02 ! [C: real,B: real,A: real] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/7.02 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
% 12.85/7.02 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8525_neg__le__minus__divideR__eq,axiom,
% 12.85/7.02 ! [C: real,A: real,B: real] :
% 12.85/7.02 ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/7.02 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
% 12.85/7.02 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8526_neg__minus__divideR__le__eq,axiom,
% 12.85/7.02 ! [C: real,B: real,A: real] :
% 12.85/7.02 ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/7.02 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
% 12.85/7.02 = ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8527_pos__less__minus__divideR__eq,axiom,
% 12.85/7.02 ! [C: real,A: real,B: real] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/7.02 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
% 12.85/7.02 = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8528_pos__minus__divideR__less__eq,axiom,
% 12.85/7.02 ! [C: real,B: real,A: real] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/7.02 => ( ( ord_less_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
% 12.85/7.02 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8529_neg__less__minus__divideR__eq,axiom,
% 12.85/7.02 ! [C: real,A: real,B: real] :
% 12.85/7.02 ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/7.02 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
% 12.85/7.02 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8530_neg__minus__divideR__less__eq,axiom,
% 12.85/7.02 ! [C: real,B: real,A: real] :
% 12.85/7.02 ( ( ord_less_real @ C @ zero_zero_real )
% 12.85/7.02 => ( ( ord_less_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
% 12.85/7.02 = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8531_cmod__plus__Re__le__0__iff,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
% 12.85/7.02 = ( ( re @ Z )
% 12.85/7.02 = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8532_cos__n__Re__cis__pow__n,axiom,
% 12.85/7.02 ! [N: nat,A: real] :
% 12.85/7.02 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 12.85/7.02 = ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8533_CauchyD,axiom,
% 12.85/7.02 ! [X9: nat > complex,E2: real] :
% 12.85/7.02 ( ( topolo6517432010174082258omplex @ X9 )
% 12.85/7.02 => ( ( ord_less_real @ zero_zero_real @ E2 )
% 12.85/7.02 => ? [M8: nat] :
% 12.85/7.02 ! [M2: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ M8 @ M2 )
% 12.85/7.02 => ! [N6: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ M8 @ N6 )
% 12.85/7.02 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X9 @ M2 ) @ ( X9 @ N6 ) ) ) @ E2 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8534_CauchyD,axiom,
% 12.85/7.02 ! [X9: nat > real,E2: real] :
% 12.85/7.02 ( ( topolo4055970368930404560y_real @ X9 )
% 12.85/7.02 => ( ( ord_less_real @ zero_zero_real @ E2 )
% 12.85/7.02 => ? [M8: nat] :
% 12.85/7.02 ! [M2: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ M8 @ M2 )
% 12.85/7.02 => ! [N6: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ M8 @ N6 )
% 12.85/7.02 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X9 @ M2 ) @ ( X9 @ N6 ) ) ) @ E2 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8535_CauchyI,axiom,
% 12.85/7.02 ! [X9: nat > complex] :
% 12.85/7.02 ( ! [E: real] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ E )
% 12.85/7.02 => ? [M9: nat] :
% 12.85/7.02 ! [M4: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ M9 @ M4 )
% 12.85/7.02 => ! [N2: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ M9 @ N2 )
% 12.85/7.02 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X9 @ M4 ) @ ( X9 @ N2 ) ) ) @ E ) ) ) )
% 12.85/7.02 => ( topolo6517432010174082258omplex @ X9 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8536_CauchyI,axiom,
% 12.85/7.02 ! [X9: nat > real] :
% 12.85/7.02 ( ! [E: real] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ E )
% 12.85/7.02 => ? [M9: nat] :
% 12.85/7.02 ! [M4: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ M9 @ M4 )
% 12.85/7.02 => ! [N2: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ M9 @ N2 )
% 12.85/7.02 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X9 @ M4 ) @ ( X9 @ N2 ) ) ) @ E ) ) ) )
% 12.85/7.02 => ( topolo4055970368930404560y_real @ X9 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8537_Cauchy__iff,axiom,
% 12.85/7.02 ( topolo6517432010174082258omplex
% 12.85/7.02 = ( ^ [X8: nat > complex] :
% 12.85/7.02 ! [E3: real] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ E3 )
% 12.85/7.02 => ? [M7: nat] :
% 12.85/7.02 ! [M5: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ M7 @ M5 )
% 12.85/7.02 => ! [N4: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ M7 @ N4 )
% 12.85/7.02 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X8 @ M5 ) @ ( X8 @ N4 ) ) ) @ E3 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8538_Cauchy__iff,axiom,
% 12.85/7.02 ( topolo4055970368930404560y_real
% 12.85/7.02 = ( ^ [X8: nat > real] :
% 12.85/7.02 ! [E3: real] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ E3 )
% 12.85/7.02 => ? [M7: nat] :
% 12.85/7.02 ! [M5: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ M7 @ M5 )
% 12.85/7.02 => ! [N4: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ M7 @ N4 )
% 12.85/7.02 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X8 @ M5 ) @ ( X8 @ N4 ) ) ) @ E3 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8539_cosh__def,axiom,
% 12.85/7.02 ( cosh_complex
% 12.85/7.02 = ( ^ [X3: complex] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8540_cosh__def,axiom,
% 12.85/7.02 ( cosh_real
% 12.85/7.02 = ( ^ [X3: real] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8541_csqrt_Ocode,axiom,
% 12.85/7.02 ( csqrt
% 12.85/7.02 = ( ^ [Z5: complex] :
% 12.85/7.02 ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.02 @ ( times_times_real
% 12.85/7.02 @ ( if_real
% 12.85/7.02 @ ( ( im @ Z5 )
% 12.85/7.02 = zero_zero_real )
% 12.85/7.02 @ one_one_real
% 12.85/7.02 @ ( sgn_sgn_real @ ( im @ Z5 ) ) )
% 12.85/7.02 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8542_csqrt_Osimps_I2_J,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( im @ ( csqrt @ Z ) )
% 12.85/7.02 = ( times_times_real
% 12.85/7.02 @ ( if_real
% 12.85/7.02 @ ( ( im @ Z )
% 12.85/7.02 = zero_zero_real )
% 12.85/7.02 @ one_one_real
% 12.85/7.02 @ ( sgn_sgn_real @ ( im @ Z ) ) )
% 12.85/7.02 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8543_csqrt__of__real__nonpos,axiom,
% 12.85/7.02 ! [X: complex] :
% 12.85/7.02 ( ( ( im @ X )
% 12.85/7.02 = zero_zero_real )
% 12.85/7.02 => ( ( ord_less_eq_real @ ( re @ X ) @ zero_zero_real )
% 12.85/7.02 => ( ( csqrt @ X )
% 12.85/7.02 = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8544_Complex__divide,axiom,
% 12.85/7.02 ( divide1717551699836669952omplex
% 12.85/7.02 = ( ^ [X3: complex,Y6: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y6 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y6 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y6 ) ) @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y6 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8545_complex__Im__of__int,axiom,
% 12.85/7.02 ! [Z: int] :
% 12.85/7.02 ( ( im @ ( ring_17405671764205052669omplex @ Z ) )
% 12.85/7.02 = zero_zero_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8546_complex__Im__fact,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( im @ ( semiri5044797733671781792omplex @ N ) )
% 12.85/7.02 = zero_zero_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8547_Im__complex__of__real,axiom,
% 12.85/7.02 ! [Z: real] :
% 12.85/7.02 ( ( im @ ( real_V4546457046886955230omplex @ Z ) )
% 12.85/7.02 = zero_zero_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8548_Im__power__real,axiom,
% 12.85/7.02 ! [X: complex,N: nat] :
% 12.85/7.02 ( ( ( im @ X )
% 12.85/7.02 = zero_zero_real )
% 12.85/7.02 => ( ( im @ ( power_power_complex @ X @ N ) )
% 12.85/7.02 = zero_zero_real ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8549_complex__Im__numeral,axiom,
% 12.85/7.02 ! [V: num] :
% 12.85/7.02 ( ( im @ ( numera6690914467698888265omplex @ V ) )
% 12.85/7.02 = zero_zero_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8550_complex__Im__of__nat,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( im @ ( semiri8010041392384452111omplex @ N ) )
% 12.85/7.02 = zero_zero_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8551_Im__divide__of__real,axiom,
% 12.85/7.02 ! [Z: complex,R2: real] :
% 12.85/7.02 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R2 ) ) )
% 12.85/7.02 = ( divide_divide_real @ ( im @ Z ) @ R2 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8552_Im__sgn,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( im @ ( sgn_sgn_complex @ Z ) )
% 12.85/7.02 = ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8553_Re__power__real,axiom,
% 12.85/7.02 ! [X: complex,N: nat] :
% 12.85/7.02 ( ( ( im @ X )
% 12.85/7.02 = zero_zero_real )
% 12.85/7.02 => ( ( re @ ( power_power_complex @ X @ N ) )
% 12.85/7.02 = ( power_power_real @ ( re @ X ) @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8554_Im__divide__numeral,axiom,
% 12.85/7.02 ! [Z: complex,W: num] :
% 12.85/7.02 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 12.85/7.02 = ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8555_Im__divide__of__nat,axiom,
% 12.85/7.02 ! [Z: complex,N: nat] :
% 12.85/7.02 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
% 12.85/7.02 = ( divide_divide_real @ ( im @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8556_csqrt__of__real__nonneg,axiom,
% 12.85/7.02 ! [X: complex] :
% 12.85/7.02 ( ( ( im @ X )
% 12.85/7.02 = zero_zero_real )
% 12.85/7.02 => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) )
% 12.85/7.02 => ( ( csqrt @ X )
% 12.85/7.02 = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8557_csqrt__minus,axiom,
% 12.85/7.02 ! [X: complex] :
% 12.85/7.02 ( ( ( ord_less_real @ ( im @ X ) @ zero_zero_real )
% 12.85/7.02 | ( ( ( im @ X )
% 12.85/7.02 = zero_zero_real )
% 12.85/7.02 & ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) ) ) )
% 12.85/7.02 => ( ( csqrt @ ( uminus1482373934393186551omplex @ X ) )
% 12.85/7.02 = ( times_times_complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8558_scaleR__complex_Osimps_I2_J,axiom,
% 12.85/7.02 ! [R2: real,X: complex] :
% 12.85/7.02 ( ( im @ ( real_V2046097035970521341omplex @ R2 @ X ) )
% 12.85/7.02 = ( times_times_real @ R2 @ ( im @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8559_zero__complex_Osimps_I2_J,axiom,
% 12.85/7.02 ( ( im @ zero_zero_complex )
% 12.85/7.02 = zero_zero_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8560_one__complex_Osimps_I2_J,axiom,
% 12.85/7.02 ( ( im @ one_one_complex )
% 12.85/7.02 = zero_zero_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8561_minus__complex_Osimps_I2_J,axiom,
% 12.85/7.02 ! [X: complex,Y4: complex] :
% 12.85/7.02 ( ( im @ ( minus_minus_complex @ X @ Y4 ) )
% 12.85/7.02 = ( minus_minus_real @ ( im @ X ) @ ( im @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8562_complex__is__Int__iff,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( member_complex @ Z @ ring_1_Ints_complex )
% 12.85/7.02 = ( ( ( im @ Z )
% 12.85/7.02 = zero_zero_real )
% 12.85/7.02 & ? [I2: int] :
% 12.85/7.02 ( ( re @ Z )
% 12.85/7.02 = ( ring_1_of_int_real @ I2 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8563_scaleR__complex_Ocode,axiom,
% 12.85/7.02 ( real_V2046097035970521341omplex
% 12.85/7.02 = ( ^ [R5: real,X3: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X3 ) ) @ ( times_times_real @ R5 @ ( im @ X3 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8564_abs__Im__le__cmod,axiom,
% 12.85/7.02 ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8565_times__complex_Osimps_I2_J,axiom,
% 12.85/7.02 ! [X: complex,Y4: complex] :
% 12.85/7.02 ( ( im @ ( times_times_complex @ X @ Y4 ) )
% 12.85/7.02 = ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y4 ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8566_cmod__eq__Re,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( ( im @ Z )
% 12.85/7.02 = zero_zero_real )
% 12.85/7.02 => ( ( real_V1022390504157884413omplex @ Z )
% 12.85/7.02 = ( abs_abs_real @ ( re @ Z ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8567_cmod__eq__Im,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( ( re @ Z )
% 12.85/7.02 = zero_zero_real )
% 12.85/7.02 => ( ( real_V1022390504157884413omplex @ Z )
% 12.85/7.02 = ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8568_Im__eq__0,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( ( abs_abs_real @ ( re @ Z ) )
% 12.85/7.02 = ( real_V1022390504157884413omplex @ Z ) )
% 12.85/7.02 => ( ( im @ Z )
% 12.85/7.02 = zero_zero_real ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8569_cmod__Im__le__iff,axiom,
% 12.85/7.02 ! [X: complex,Y4: complex] :
% 12.85/7.02 ( ( ( re @ X )
% 12.85/7.02 = ( re @ Y4 ) )
% 12.85/7.02 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) )
% 12.85/7.02 = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( abs_abs_real @ ( im @ Y4 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8570_cmod__Re__le__iff,axiom,
% 12.85/7.02 ! [X: complex,Y4: complex] :
% 12.85/7.02 ( ( ( im @ X )
% 12.85/7.02 = ( im @ Y4 ) )
% 12.85/7.02 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) )
% 12.85/7.02 = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( abs_abs_real @ ( re @ Y4 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8571_times__complex_Osimps_I1_J,axiom,
% 12.85/7.02 ! [X: complex,Y4: complex] :
% 12.85/7.02 ( ( re @ ( times_times_complex @ X @ Y4 ) )
% 12.85/7.02 = ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y4 ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8572_minus__complex_Ocode,axiom,
% 12.85/7.02 ( minus_minus_complex
% 12.85/7.02 = ( ^ [X3: complex,Y6: complex] : ( complex2 @ ( minus_minus_real @ ( re @ X3 ) @ ( re @ Y6 ) ) @ ( minus_minus_real @ ( im @ X3 ) @ ( im @ Y6 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8573_csqrt__principal,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
% 12.85/7.02 | ( ( ( re @ ( csqrt @ Z ) )
% 12.85/7.02 = zero_zero_real )
% 12.85/7.02 & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8574_cmod__le,axiom,
% 12.85/7.02 ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8575_sin__n__Im__cis__pow__n,axiom,
% 12.85/7.02 ! [N: nat,A: real] :
% 12.85/7.02 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 12.85/7.02 = ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8576_Re__exp,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( re @ ( exp_complex @ Z ) )
% 12.85/7.02 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8577_Im__exp,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( im @ ( exp_complex @ Z ) )
% 12.85/7.02 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8578_times__complex_Ocode,axiom,
% 12.85/7.02 ( times_times_complex
% 12.85/7.02 = ( ^ [X3: complex,Y6: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y6 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y6 ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y6 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y6 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8579_cmod__power2,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.02 = ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8580_Im__power2,axiom,
% 12.85/7.02 ! [X: complex] :
% 12.85/7.02 ( ( im @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.02 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8581_Re__power2,axiom,
% 12.85/7.02 ! [X: complex] :
% 12.85/7.02 ( ( re @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.02 = ( minus_minus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8582_complex__eq__0,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( Z = zero_zero_complex )
% 12.85/7.02 = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.02 = zero_zero_real ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8583_norm__complex__def,axiom,
% 12.85/7.02 ( real_V1022390504157884413omplex
% 12.85/7.02 = ( ^ [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 ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8584_inverse__complex_Osimps_I1_J,axiom,
% 12.85/7.02 ! [X: complex] :
% 12.85/7.02 ( ( re @ ( invers8013647133539491842omplex @ X ) )
% 12.85/7.02 = ( 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 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8585_complex__neq__0,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( Z != zero_zero_complex )
% 12.85/7.02 = ( 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 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8586_Re__divide,axiom,
% 12.85/7.02 ! [X: complex,Y4: complex] :
% 12.85/7.02 ( ( re @ ( divide1717551699836669952omplex @ X @ Y4 ) )
% 12.85/7.02 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y4 ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y4 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8587_csqrt__unique,axiom,
% 12.85/7.02 ! [W: complex,Z: complex] :
% 12.85/7.02 ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.02 = Z )
% 12.85/7.02 => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
% 12.85/7.02 | ( ( ( re @ W )
% 12.85/7.02 = zero_zero_real )
% 12.85/7.02 & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
% 12.85/7.02 => ( ( csqrt @ Z )
% 12.85/7.02 = W ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8588_csqrt__square,axiom,
% 12.85/7.02 ! [B: complex] :
% 12.85/7.02 ( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
% 12.85/7.02 | ( ( ( re @ B )
% 12.85/7.02 = zero_zero_real )
% 12.85/7.02 & ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
% 12.85/7.02 => ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.02 = B ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8589_inverse__complex_Osimps_I2_J,axiom,
% 12.85/7.02 ! [X: complex] :
% 12.85/7.02 ( ( im @ ( invers8013647133539491842omplex @ X ) )
% 12.85/7.02 = ( 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 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8590_Im__divide,axiom,
% 12.85/7.02 ! [X: complex,Y4: complex] :
% 12.85/7.02 ( ( im @ ( divide1717551699836669952omplex @ X @ Y4 ) )
% 12.85/7.02 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y4 ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y4 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8591_complex__abs__le__norm,axiom,
% 12.85/7.02 ! [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 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8592_complex__unit__circle,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( Z != zero_zero_complex )
% 12.85/7.02 => ( ( 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 ) ) ) )
% 12.85/7.02 = one_one_real ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8593_inverse__complex_Ocode,axiom,
% 12.85/7.02 ( invers8013647133539491842omplex
% 12.85/7.02 = ( ^ [X3: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X3 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X3 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8594_Im__Reals__divide,axiom,
% 12.85/7.02 ! [R2: complex,Z: complex] :
% 12.85/7.02 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 12.85/7.02 => ( ( im @ ( divide1717551699836669952omplex @ R2 @ Z ) )
% 12.85/7.02 = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R2 ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8595_length__mul__elem,axiom,
% 12.85/7.02 ! [Xs: list_list_real,N: nat] :
% 12.85/7.02 ( ! [X4: list_real] :
% 12.85/7.02 ( ( member_list_real @ X4 @ ( set_list_real2 @ Xs ) )
% 12.85/7.02 => ( ( size_size_list_real @ X4 )
% 12.85/7.02 = N ) )
% 12.85/7.02 => ( ( size_size_list_real @ ( concat_real @ Xs ) )
% 12.85/7.02 = ( times_times_nat @ ( size_s6660260683639930848t_real @ Xs ) @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8596_length__mul__elem,axiom,
% 12.85/7.02 ! [Xs: list_list_o,N: nat] :
% 12.85/7.02 ( ! [X4: list_o] :
% 12.85/7.02 ( ( member_list_o @ X4 @ ( set_list_o2 @ Xs ) )
% 12.85/7.02 => ( ( size_size_list_o @ X4 )
% 12.85/7.02 = N ) )
% 12.85/7.02 => ( ( size_size_list_o @ ( concat_o @ Xs ) )
% 12.85/7.02 = ( times_times_nat @ ( size_s2710708370519433104list_o @ Xs ) @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8597_length__mul__elem,axiom,
% 12.85/7.02 ! [Xs: list_list_nat,N: nat] :
% 12.85/7.02 ( ! [X4: list_nat] :
% 12.85/7.02 ( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs ) )
% 12.85/7.02 => ( ( size_size_list_nat @ X4 )
% 12.85/7.02 = N ) )
% 12.85/7.02 => ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
% 12.85/7.02 = ( times_times_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8598_length__mul__elem,axiom,
% 12.85/7.02 ! [Xs: list_list_int,N: nat] :
% 12.85/7.02 ( ! [X4: list_int] :
% 12.85/7.02 ( ( member_list_int @ X4 @ ( set_list_int2 @ Xs ) )
% 12.85/7.02 => ( ( size_size_list_int @ X4 )
% 12.85/7.02 = N ) )
% 12.85/7.02 => ( ( size_size_list_int @ ( concat_int @ Xs ) )
% 12.85/7.02 = ( times_times_nat @ ( size_s533118279054570080st_int @ Xs ) @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8599_Re__Reals__divide,axiom,
% 12.85/7.02 ! [R2: complex,Z: complex] :
% 12.85/7.02 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 12.85/7.02 => ( ( re @ ( divide1717551699836669952omplex @ R2 @ Z ) )
% 12.85/7.02 = ( divide_divide_real @ ( times_times_real @ ( re @ R2 ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8600_complex__diff__cnj,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
% 12.85/7.02 = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8601_complex__cnj__zero__iff,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( ( cnj @ Z )
% 12.85/7.02 = zero_zero_complex )
% 12.85/7.02 = ( Z = zero_zero_complex ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8602_complex__cnj__zero,axiom,
% 12.85/7.02 ( ( cnj @ zero_zero_complex )
% 12.85/7.02 = zero_zero_complex ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8603_complex__cnj__diff,axiom,
% 12.85/7.02 ! [X: complex,Y4: complex] :
% 12.85/7.02 ( ( cnj @ ( minus_minus_complex @ X @ Y4 ) )
% 12.85/7.02 = ( minus_minus_complex @ ( cnj @ X ) @ ( cnj @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8604_complex__In__mult__cnj__zero,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( im @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 12.85/7.02 = zero_zero_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8605_Re__divide__Reals,axiom,
% 12.85/7.02 ! [R2: complex,Z: complex] :
% 12.85/7.02 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 12.85/7.02 => ( ( re @ ( divide1717551699836669952omplex @ Z @ R2 ) )
% 12.85/7.02 = ( divide_divide_real @ ( re @ Z ) @ ( re @ R2 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8606_imaginary__eq__real__iff,axiom,
% 12.85/7.02 ! [Y4: complex,X: complex] :
% 12.85/7.02 ( ( member_complex @ Y4 @ real_V2521375963428798218omplex )
% 12.85/7.02 => ( ( member_complex @ X @ real_V2521375963428798218omplex )
% 12.85/7.02 => ( ( ( times_times_complex @ imaginary_unit @ Y4 )
% 12.85/7.02 = X )
% 12.85/7.02 = ( ( X = zero_zero_complex )
% 12.85/7.02 & ( Y4 = zero_zero_complex ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8607_real__eq__imaginary__iff,axiom,
% 12.85/7.02 ! [Y4: complex,X: complex] :
% 12.85/7.02 ( ( member_complex @ Y4 @ real_V2521375963428798218omplex )
% 12.85/7.02 => ( ( member_complex @ X @ real_V2521375963428798218omplex )
% 12.85/7.02 => ( ( X
% 12.85/7.02 = ( times_times_complex @ imaginary_unit @ Y4 ) )
% 12.85/7.02 = ( ( X = zero_zero_complex )
% 12.85/7.02 & ( Y4 = zero_zero_complex ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8608_Im__divide__Reals,axiom,
% 12.85/7.02 ! [R2: complex,Z: complex] :
% 12.85/7.02 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 12.85/7.02 => ( ( im @ ( divide1717551699836669952omplex @ Z @ R2 ) )
% 12.85/7.02 = ( divide_divide_real @ ( im @ Z ) @ ( re @ R2 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8609_Reals__of__nat,axiom,
% 12.85/7.02 ! [N: nat] : ( member_complex @ ( semiri8010041392384452111omplex @ N ) @ real_V2521375963428798218omplex ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8610_Reals__of__nat,axiom,
% 12.85/7.02 ! [N: nat] : ( member_real @ ( semiri5074537144036343181t_real @ N ) @ real_V470468836141973256s_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8611_Reals__divide,axiom,
% 12.85/7.02 ! [A: complex,B: complex] :
% 12.85/7.02 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 12.85/7.02 => ( ( member_complex @ B @ real_V2521375963428798218omplex )
% 12.85/7.02 => ( member_complex @ ( divide1717551699836669952omplex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8612_Reals__divide,axiom,
% 12.85/7.02 ! [A: real,B: real] :
% 12.85/7.02 ( ( member_real @ A @ real_V470468836141973256s_real )
% 12.85/7.02 => ( ( member_real @ B @ real_V470468836141973256s_real )
% 12.85/7.02 => ( member_real @ ( divide_divide_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8613_Reals__diff,axiom,
% 12.85/7.02 ! [A: real,B: real] :
% 12.85/7.02 ( ( member_real @ A @ real_V470468836141973256s_real )
% 12.85/7.02 => ( ( member_real @ B @ real_V470468836141973256s_real )
% 12.85/7.02 => ( member_real @ ( minus_minus_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8614_Reals__diff,axiom,
% 12.85/7.02 ! [A: complex,B: complex] :
% 12.85/7.02 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 12.85/7.02 => ( ( member_complex @ B @ real_V2521375963428798218omplex )
% 12.85/7.02 => ( member_complex @ ( minus_minus_complex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8615_Reals__mult,axiom,
% 12.85/7.02 ! [A: real,B: real] :
% 12.85/7.02 ( ( member_real @ A @ real_V470468836141973256s_real )
% 12.85/7.02 => ( ( member_real @ B @ real_V470468836141973256s_real )
% 12.85/7.02 => ( member_real @ ( times_times_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8616_Reals__mult,axiom,
% 12.85/7.02 ! [A: complex,B: complex] :
% 12.85/7.02 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 12.85/7.02 => ( ( member_complex @ B @ real_V2521375963428798218omplex )
% 12.85/7.02 => ( member_complex @ ( times_times_complex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8617_Reals__numeral,axiom,
% 12.85/7.02 ! [W: num] : ( member_complex @ ( numera6690914467698888265omplex @ W ) @ real_V2521375963428798218omplex ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8618_Reals__numeral,axiom,
% 12.85/7.02 ! [W: num] : ( member_real @ ( numeral_numeral_real @ W ) @ real_V470468836141973256s_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8619_Reals__0,axiom,
% 12.85/7.02 member_real @ zero_zero_real @ real_V470468836141973256s_real ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8620_Reals__0,axiom,
% 12.85/7.02 member_complex @ zero_zero_complex @ real_V2521375963428798218omplex ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8621_complex__is__Real__iff,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( member_complex @ Z @ real_V2521375963428798218omplex )
% 12.85/7.02 = ( ( im @ Z )
% 12.85/7.02 = zero_zero_real ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8622_nonzero__Reals__divide,axiom,
% 12.85/7.02 ! [A: complex,B: complex] :
% 12.85/7.02 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 12.85/7.02 => ( ( member_complex @ B @ real_V2521375963428798218omplex )
% 12.85/7.02 => ( ( B != zero_zero_complex )
% 12.85/7.02 => ( member_complex @ ( divide1717551699836669952omplex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8623_nonzero__Reals__divide,axiom,
% 12.85/7.02 ! [A: real,B: real] :
% 12.85/7.02 ( ( member_real @ A @ real_V470468836141973256s_real )
% 12.85/7.02 => ( ( member_real @ B @ real_V470468836141973256s_real )
% 12.85/7.02 => ( ( B != zero_zero_real )
% 12.85/7.02 => ( member_real @ ( divide_divide_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8624_Complex__in__Reals,axiom,
% 12.85/7.02 ! [X: real] : ( member_complex @ ( complex2 @ X @ zero_zero_real ) @ real_V2521375963428798218omplex ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8625_nonzero__Reals__inverse,axiom,
% 12.85/7.02 ! [A: real] :
% 12.85/7.02 ( ( member_real @ A @ real_V470468836141973256s_real )
% 12.85/7.02 => ( ( A != zero_zero_real )
% 12.85/7.02 => ( member_real @ ( inverse_inverse_real @ A ) @ real_V470468836141973256s_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8626_nonzero__Reals__inverse,axiom,
% 12.85/7.02 ! [A: complex] :
% 12.85/7.02 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 12.85/7.02 => ( ( A != zero_zero_complex )
% 12.85/7.02 => ( member_complex @ ( invers8013647133539491842omplex @ A ) @ real_V2521375963428798218omplex ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8627_Re__complex__div__eq__0,axiom,
% 12.85/7.02 ! [A: complex,B: complex] :
% 12.85/7.02 ( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
% 12.85/7.02 = zero_zero_real )
% 12.85/7.02 = ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 12.85/7.02 = zero_zero_real ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8628_Im__complex__div__eq__0,axiom,
% 12.85/7.02 ! [A: complex,B: complex] :
% 12.85/7.02 ( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
% 12.85/7.02 = zero_zero_real )
% 12.85/7.02 = ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 12.85/7.02 = zero_zero_real ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8629_Re__complex__div__gt__0,axiom,
% 12.85/7.02 ! [A: complex,B: complex] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 12.85/7.02 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8630_Re__complex__div__lt__0,axiom,
% 12.85/7.02 ! [A: complex,B: complex] :
% 12.85/7.02 ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 12.85/7.02 = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8631_Re__complex__div__ge__0,axiom,
% 12.85/7.02 ! [A: complex,B: complex] :
% 12.85/7.02 ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 12.85/7.02 = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8632_Re__complex__div__le__0,axiom,
% 12.85/7.02 ! [A: complex,B: complex] :
% 12.85/7.02 ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 12.85/7.02 = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8633_Im__complex__div__gt__0,axiom,
% 12.85/7.02 ! [A: complex,B: complex] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 12.85/7.02 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8634_Im__complex__div__lt__0,axiom,
% 12.85/7.02 ! [A: complex,B: complex] :
% 12.85/7.02 ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 12.85/7.02 = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8635_Im__complex__div__ge__0,axiom,
% 12.85/7.02 ! [A: complex,B: complex] :
% 12.85/7.02 ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 12.85/7.02 = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8636_Im__complex__div__le__0,axiom,
% 12.85/7.02 ! [A: complex,B: complex] :
% 12.85/7.02 ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 12.85/7.02 = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8637_complex__mod__mult__cnj,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 12.85/7.02 = ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8638_complex__div__gt__0,axiom,
% 12.85/7.02 ! [A: complex,B: complex] :
% 12.85/7.02 ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 12.85/7.02 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
% 12.85/7.02 & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 12.85/7.02 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8639_complex__norm__square,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.02 = ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8640_complex__add__cnj,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
% 12.85/7.02 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8641_complex__div__cnj,axiom,
% 12.85/7.02 ( divide1717551699836669952omplex
% 12.85/7.02 = ( ^ [A3: complex,B2: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ ( cnj @ B2 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8642_cnj__add__mult__eq__Re,axiom,
% 12.85/7.02 ! [Z: complex,W: complex] :
% 12.85/7.02 ( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
% 12.85/7.02 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8643_complex__mult__cnj,axiom,
% 12.85/7.02 ! [Z: complex] :
% 12.85/7.02 ( ( times_times_complex @ Z @ ( cnj @ Z ) )
% 12.85/7.02 = ( 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 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8644_or__int__rec,axiom,
% 12.85/7.02 ( bit_se1409905431419307370or_int
% 12.85/7.02 = ( ^ [K3: int,L3: int] :
% 12.85/7.02 ( plus_plus_int
% 12.85/7.02 @ ( zero_n2684676970156552555ol_int
% 12.85/7.02 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 12.85/7.02 | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
% 12.85/7.02 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8645_drop__bit__numeral__minus__bit1,axiom,
% 12.85/7.02 ! [L: num,K: num] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 12.85/7.02 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8646_or__numerals_I7_J,axiom,
% 12.85/7.02 ! [X: num,Y4: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y4 ) ) )
% 12.85/7.02 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8647_or__numerals_I7_J,axiom,
% 12.85/7.02 ! [X: num,Y4: num] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 12.85/7.02 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8648_or__numerals_I6_J,axiom,
% 12.85/7.02 ! [X: num,Y4: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y4 ) ) )
% 12.85/7.02 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8649_or__numerals_I6_J,axiom,
% 12.85/7.02 ! [X: num,Y4: num] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 12.85/7.02 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8650_or_Oright__idem,axiom,
% 12.85/7.02 ! [A: int,B: int] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( bit_se1409905431419307370or_int @ A @ B ) @ B )
% 12.85/7.02 = ( bit_se1409905431419307370or_int @ A @ B ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8651_or_Oright__idem,axiom,
% 12.85/7.02 ! [A: nat,B: nat] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ B )
% 12.85/7.02 = ( bit_se1412395901928357646or_nat @ A @ B ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8652_or_Oleft__idem,axiom,
% 12.85/7.02 ! [A: int,B: int] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 12.85/7.02 = ( bit_se1409905431419307370or_int @ A @ B ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8653_or_Oleft__idem,axiom,
% 12.85/7.02 ! [A: nat,B: nat] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 12.85/7.02 = ( bit_se1412395901928357646or_nat @ A @ B ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8654_or_Oidem,axiom,
% 12.85/7.02 ! [A: int] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ A @ A )
% 12.85/7.02 = A ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8655_or_Oidem,axiom,
% 12.85/7.02 ! [A: nat] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ A @ A )
% 12.85/7.02 = A ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8656_or_Oleft__neutral,axiom,
% 12.85/7.02 ! [A: int] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ zero_zero_int @ A )
% 12.85/7.02 = A ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8657_or_Oleft__neutral,axiom,
% 12.85/7.02 ! [A: nat] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ zero_zero_nat @ A )
% 12.85/7.02 = A ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8658_or_Oright__neutral,axiom,
% 12.85/7.02 ! [A: int] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ A @ zero_zero_int )
% 12.85/7.02 = A ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8659_or_Oright__neutral,axiom,
% 12.85/7.02 ! [A: nat] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ A @ zero_zero_nat )
% 12.85/7.02 = A ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8660_drop__bit__of__0,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ N @ zero_zero_int )
% 12.85/7.02 = zero_zero_int ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8661_drop__bit__of__0,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( bit_se8570568707652914677it_nat @ N @ zero_zero_nat )
% 12.85/7.02 = zero_zero_nat ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8662_drop__bit__drop__bit,axiom,
% 12.85/7.02 ! [M: nat,N: nat,A: int] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se8568078237143864401it_int @ N @ A ) )
% 12.85/7.02 = ( bit_se8568078237143864401it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8663_drop__bit__drop__bit,axiom,
% 12.85/7.02 ! [M: nat,N: nat,A: nat] :
% 12.85/7.02 ( ( bit_se8570568707652914677it_nat @ M @ ( bit_se8570568707652914677it_nat @ N @ A ) )
% 12.85/7.02 = ( bit_se8570568707652914677it_nat @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8664_take__bit__or,axiom,
% 12.85/7.02 ! [N: nat,A: int,B: int] :
% 12.85/7.02 ( ( bit_se2923211474154528505it_int @ N @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 12.85/7.02 = ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8665_take__bit__or,axiom,
% 12.85/7.02 ! [N: nat,A: nat,B: nat] :
% 12.85/7.02 ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 12.85/7.02 = ( bit_se1412395901928357646or_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8666_push__bit__or,axiom,
% 12.85/7.02 ! [N: nat,A: int,B: int] :
% 12.85/7.02 ( ( bit_se545348938243370406it_int @ N @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 12.85/7.02 = ( bit_se1409905431419307370or_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8667_push__bit__or,axiom,
% 12.85/7.02 ! [N: nat,A: nat,B: nat] :
% 12.85/7.02 ( ( bit_se547839408752420682it_nat @ N @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 12.85/7.02 = ( bit_se1412395901928357646or_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8668_drop__bit__and,axiom,
% 12.85/7.02 ! [N: nat,A: int,B: int] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 12.85/7.02 = ( bit_se725231765392027082nd_int @ ( bit_se8568078237143864401it_int @ N @ A ) @ ( bit_se8568078237143864401it_int @ N @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8669_drop__bit__and,axiom,
% 12.85/7.02 ! [N: nat,A: nat,B: nat] :
% 12.85/7.02 ( ( bit_se8570568707652914677it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 12.85/7.02 = ( bit_se727722235901077358nd_nat @ ( bit_se8570568707652914677it_nat @ N @ A ) @ ( bit_se8570568707652914677it_nat @ N @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8670_drop__bit__or,axiom,
% 12.85/7.02 ! [N: nat,A: int,B: int] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ N @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 12.85/7.02 = ( bit_se1409905431419307370or_int @ ( bit_se8568078237143864401it_int @ N @ A ) @ ( bit_se8568078237143864401it_int @ N @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8671_drop__bit__or,axiom,
% 12.85/7.02 ! [N: nat,A: nat,B: nat] :
% 12.85/7.02 ( ( bit_se8570568707652914677it_nat @ N @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 12.85/7.02 = ( bit_se1412395901928357646or_nat @ ( bit_se8570568707652914677it_nat @ N @ A ) @ ( bit_se8570568707652914677it_nat @ N @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8672_drop__bit__xor,axiom,
% 12.85/7.02 ! [N: nat,A: int,B: int] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B ) )
% 12.85/7.02 = ( bit_se6526347334894502574or_int @ ( bit_se8568078237143864401it_int @ N @ A ) @ ( bit_se8568078237143864401it_int @ N @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8673_drop__bit__xor,axiom,
% 12.85/7.02 ! [N: nat,A: nat,B: nat] :
% 12.85/7.02 ( ( bit_se8570568707652914677it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B ) )
% 12.85/7.02 = ( bit_se6528837805403552850or_nat @ ( bit_se8570568707652914677it_nat @ N @ A ) @ ( bit_se8570568707652914677it_nat @ N @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8674_bit_Odisj__one__right,axiom,
% 12.85/7.02 ! [X: code_integer] :
% 12.85/7.02 ( ( bit_se1080825931792720795nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.02 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8675_bit_Odisj__one__right,axiom,
% 12.85/7.02 ! [X: int] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.02 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8676_bit_Odisj__one__left,axiom,
% 12.85/7.02 ! [X: code_integer] :
% 12.85/7.02 ( ( bit_se1080825931792720795nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 12.85/7.02 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8677_bit_Odisj__one__left,axiom,
% 12.85/7.02 ! [X: int] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 12.85/7.02 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8678_drop__bit__of__bool,axiom,
% 12.85/7.02 ! [N: nat,B: $o] :
% 12.85/7.02 ( ( bit_se3928097537394005634nteger @ N @ ( zero_n356916108424825756nteger @ B ) )
% 12.85/7.02 = ( zero_n356916108424825756nteger
% 12.85/7.02 @ ( ( N = zero_zero_nat )
% 12.85/7.02 & B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8679_drop__bit__of__bool,axiom,
% 12.85/7.02 ! [N: nat,B: $o] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ N @ ( zero_n2684676970156552555ol_int @ B ) )
% 12.85/7.02 = ( zero_n2684676970156552555ol_int
% 12.85/7.02 @ ( ( N = zero_zero_nat )
% 12.85/7.02 & B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8680_drop__bit__of__bool,axiom,
% 12.85/7.02 ! [N: nat,B: $o] :
% 12.85/7.02 ( ( bit_se8570568707652914677it_nat @ N @ ( zero_n2687167440665602831ol_nat @ B ) )
% 12.85/7.02 = ( zero_n2687167440665602831ol_nat
% 12.85/7.02 @ ( ( N = zero_zero_nat )
% 12.85/7.02 & B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8681_or__nonnegative__int__iff,axiom,
% 12.85/7.02 ! [K: int,L: int] :
% 12.85/7.02 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
% 12.85/7.02 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 12.85/7.02 & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8682_or__negative__int__iff,axiom,
% 12.85/7.02 ! [K: int,L: int] :
% 12.85/7.02 ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
% 12.85/7.02 = ( ( ord_less_int @ K @ zero_zero_int )
% 12.85/7.02 | ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8683_bit_Ode__Morgan__disj,axiom,
% 12.85/7.02 ! [X: int,Y4: int] :
% 12.85/7.02 ( ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ X @ Y4 ) )
% 12.85/7.02 = ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X ) @ ( bit_ri7919022796975470100ot_int @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8684_bit_Ode__Morgan__conj,axiom,
% 12.85/7.02 ! [X: int,Y4: int] :
% 12.85/7.02 ( ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ X @ Y4 ) )
% 12.85/7.02 = ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ ( bit_ri7919022796975470100ot_int @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8685_drop__bit__nonnegative__int__iff,axiom,
% 12.85/7.02 ! [N: nat,K: int] :
% 12.85/7.02 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
% 12.85/7.02 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8686_drop__bit__negative__int__iff,axiom,
% 12.85/7.02 ! [N: nat,K: int] :
% 12.85/7.02 ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
% 12.85/7.02 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8687_drop__bit__minus__one,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.02 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8688_or__nat__numerals_I2_J,axiom,
% 12.85/7.02 ! [Y4: num] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 12.85/7.02 = ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8689_or__nat__numerals_I4_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 12.85/7.02 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8690_or__numerals_I2_J,axiom,
% 12.85/7.02 ! [Y4: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y4 ) ) )
% 12.85/7.02 = ( numeral_numeral_int @ ( bit1 @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8691_or__numerals_I2_J,axiom,
% 12.85/7.02 ! [Y4: num] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 12.85/7.02 = ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8692_or__numerals_I8_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
% 12.85/7.02 = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8693_or__numerals_I8_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
% 12.85/7.02 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8694_drop__bit__Suc__bit0,axiom,
% 12.85/7.02 ! [N: nat,K: num] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 12.85/7.02 = ( bit_se8568078237143864401it_int @ N @ ( numeral_numeral_int @ K ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8695_drop__bit__Suc__bit0,axiom,
% 12.85/7.02 ! [N: nat,K: num] :
% 12.85/7.02 ( ( bit_se8570568707652914677it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 12.85/7.02 = ( bit_se8570568707652914677it_nat @ N @ ( numeral_numeral_nat @ K ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8696_drop__bit__Suc__bit1,axiom,
% 12.85/7.02 ! [N: nat,K: num] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 12.85/7.02 = ( bit_se8568078237143864401it_int @ N @ ( numeral_numeral_int @ K ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8697_drop__bit__Suc__bit1,axiom,
% 12.85/7.02 ! [N: nat,K: num] :
% 12.85/7.02 ( ( bit_se8570568707652914677it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 12.85/7.02 = ( bit_se8570568707652914677it_nat @ N @ ( numeral_numeral_nat @ K ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8698_bit_Odisj__cancel__left,axiom,
% 12.85/7.02 ! [X: code_integer] :
% 12.85/7.02 ( ( bit_se1080825931792720795nteger @ ( bit_ri7632146776885996613nteger @ X ) @ X )
% 12.85/7.02 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8699_bit_Odisj__cancel__left,axiom,
% 12.85/7.02 ! [X: int] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
% 12.85/7.02 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8700_bit_Odisj__cancel__right,axiom,
% 12.85/7.02 ! [X: code_integer] :
% 12.85/7.02 ( ( bit_se1080825931792720795nteger @ X @ ( bit_ri7632146776885996613nteger @ X ) )
% 12.85/7.02 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8701_bit_Odisj__cancel__right,axiom,
% 12.85/7.02 ! [X: int] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
% 12.85/7.02 = ( uminus_uminus_int @ one_one_int ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8702_drop__bit__of__1,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( bit_se3928097537394005634nteger @ N @ one_one_Code_integer )
% 12.85/7.02 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8703_drop__bit__of__1,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ N @ one_one_int )
% 12.85/7.02 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8704_drop__bit__of__1,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( bit_se8570568707652914677it_nat @ N @ one_one_nat )
% 12.85/7.02 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8705_or__nat__numerals_I1_J,axiom,
% 12.85/7.02 ! [Y4: num] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 12.85/7.02 = ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8706_or__nat__numerals_I3_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 12.85/7.02 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8707_or__numerals_I3_J,axiom,
% 12.85/7.02 ! [X: num,Y4: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y4 ) ) )
% 12.85/7.02 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8708_or__numerals_I3_J,axiom,
% 12.85/7.02 ! [X: num,Y4: num] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 12.85/7.02 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8709_or__numerals_I1_J,axiom,
% 12.85/7.02 ! [Y4: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y4 ) ) )
% 12.85/7.02 = ( numeral_numeral_int @ ( bit1 @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8710_or__numerals_I1_J,axiom,
% 12.85/7.02 ! [Y4: num] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
% 12.85/7.02 = ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8711_or__numerals_I5_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
% 12.85/7.02 = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8712_or__numerals_I5_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
% 12.85/7.02 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8713_drop__bit__numeral__bit0,axiom,
% 12.85/7.02 ! [L: num,K: num] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 12.85/7.02 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8714_drop__bit__numeral__bit0,axiom,
% 12.85/7.02 ! [L: num,K: num] :
% 12.85/7.02 ( ( bit_se8570568707652914677it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 12.85/7.02 = ( bit_se8570568707652914677it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8715_drop__bit__numeral__bit1,axiom,
% 12.85/7.02 ! [L: num,K: num] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 12.85/7.02 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8716_drop__bit__numeral__bit1,axiom,
% 12.85/7.02 ! [L: num,K: num] :
% 12.85/7.02 ( ( bit_se8570568707652914677it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 12.85/7.02 = ( bit_se8570568707652914677it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8717_or__minus__numerals_I6_J,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 12.85/7.02 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8718_or__minus__numerals_I2_J,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.85/7.02 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8719_drop__bit__Suc__minus__bit0,axiom,
% 12.85/7.02 ! [N: nat,K: num] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 12.85/7.02 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8720_drop__bit__numeral__minus__bit0,axiom,
% 12.85/7.02 ! [L: num,K: num] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 12.85/7.02 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8721_drop__bit__Suc__minus__bit1,axiom,
% 12.85/7.02 ! [N: nat,K: num] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 12.85/7.02 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8722_and__minus__minus__numerals,axiom,
% 12.85/7.02 ! [M: num,N: num] :
% 12.85/7.02 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.02 = ( 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 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8723_or__minus__minus__numerals,axiom,
% 12.85/7.02 ! [M: num,N: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.02 = ( 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 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8724_or__numerals_I4_J,axiom,
% 12.85/7.02 ! [X: num,Y4: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y4 ) ) )
% 12.85/7.02 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y4 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8725_or__numerals_I4_J,axiom,
% 12.85/7.02 ! [X: num,Y4: num] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
% 12.85/7.02 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y4 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8726_of__int__or__eq,axiom,
% 12.85/7.02 ! [K: int,L: int] :
% 12.85/7.02 ( ( ring_1_of_int_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
% 12.85/7.02 = ( bit_se1409905431419307370or_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8727_or_Oleft__commute,axiom,
% 12.85/7.02 ! [B: int,A: int,C: int] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ B @ ( bit_se1409905431419307370or_int @ A @ C ) )
% 12.85/7.02 = ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ B @ C ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8728_or_Oleft__commute,axiom,
% 12.85/7.02 ! [B: nat,A: nat,C: nat] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ B @ ( bit_se1412395901928357646or_nat @ A @ C ) )
% 12.85/7.02 = ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ B @ C ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8729_or_Ocommute,axiom,
% 12.85/7.02 ( bit_se1409905431419307370or_int
% 12.85/7.02 = ( ^ [A3: int,B2: int] : ( bit_se1409905431419307370or_int @ B2 @ A3 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8730_or_Ocommute,axiom,
% 12.85/7.02 ( bit_se1412395901928357646or_nat
% 12.85/7.02 = ( ^ [A3: nat,B2: nat] : ( bit_se1412395901928357646or_nat @ B2 @ A3 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8731_or_Oassoc,axiom,
% 12.85/7.02 ! [A: int,B: int,C: int] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( bit_se1409905431419307370or_int @ A @ B ) @ C )
% 12.85/7.02 = ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ B @ C ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8732_or_Oassoc,axiom,
% 12.85/7.02 ! [A: nat,B: nat,C: nat] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ C )
% 12.85/7.02 = ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ B @ C ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8733_bit_Oconj__disj__distrib,axiom,
% 12.85/7.02 ! [X: int,Y4: int,Z: int] :
% 12.85/7.02 ( ( bit_se725231765392027082nd_int @ X @ ( bit_se1409905431419307370or_int @ Y4 @ Z ) )
% 12.85/7.02 = ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ X @ Y4 ) @ ( bit_se725231765392027082nd_int @ X @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8734_bit_Odisj__conj__distrib,axiom,
% 12.85/7.02 ! [X: int,Y4: int,Z: int] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ X @ ( bit_se725231765392027082nd_int @ Y4 @ Z ) )
% 12.85/7.02 = ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ X @ Y4 ) @ ( bit_se1409905431419307370or_int @ X @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8735_bit_Oconj__disj__distrib2,axiom,
% 12.85/7.02 ! [Y4: int,Z: int,X: int] :
% 12.85/7.02 ( ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ Y4 @ Z ) @ X )
% 12.85/7.02 = ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ Y4 @ X ) @ ( bit_se725231765392027082nd_int @ Z @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8736_bit_Odisj__conj__distrib2,axiom,
% 12.85/7.02 ! [Y4: int,Z: int,X: int] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ Y4 @ Z ) @ X )
% 12.85/7.02 = ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ Y4 @ X ) @ ( bit_se1409905431419307370or_int @ Z @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8737_bit_Odisj__zero__right,axiom,
% 12.85/7.02 ! [X: int] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ X @ zero_zero_int )
% 12.85/7.02 = X ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8738_or__eq__0__iff,axiom,
% 12.85/7.02 ! [A: int,B: int] :
% 12.85/7.02 ( ( ( bit_se1409905431419307370or_int @ A @ B )
% 12.85/7.02 = zero_zero_int )
% 12.85/7.02 = ( ( A = zero_zero_int )
% 12.85/7.02 & ( B = zero_zero_int ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8739_or__eq__0__iff,axiom,
% 12.85/7.02 ! [A: nat,B: nat] :
% 12.85/7.02 ( ( ( bit_se1412395901928357646or_nat @ A @ B )
% 12.85/7.02 = zero_zero_nat )
% 12.85/7.02 = ( ( A = zero_zero_nat )
% 12.85/7.02 & ( B = zero_zero_nat ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8740_bit__or__int__iff,axiom,
% 12.85/7.02 ! [K: int,L: int,N: nat] :
% 12.85/7.02 ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ N )
% 12.85/7.02 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 12.85/7.02 | ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8741_bit__or__iff,axiom,
% 12.85/7.02 ! [A: int,B: int,N: nat] :
% 12.85/7.02 ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ A @ B ) @ N )
% 12.85/7.02 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 12.85/7.02 | ( bit_se1146084159140164899it_int @ B @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8742_bit__or__iff,axiom,
% 12.85/7.02 ! [A: nat,B: nat,N: nat] :
% 12.85/7.02 ( ( bit_se1148574629649215175it_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ N )
% 12.85/7.02 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 12.85/7.02 | ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8743_or__nat__def,axiom,
% 12.85/7.02 ( bit_se1412395901928357646or_nat
% 12.85/7.02 = ( ^ [M5: nat,N4: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8744_of__nat__drop__bit,axiom,
% 12.85/7.02 ! [M: nat,N: nat] :
% 12.85/7.02 ( ( semiri1314217659103216013at_int @ ( bit_se8570568707652914677it_nat @ M @ N ) )
% 12.85/7.02 = ( bit_se8568078237143864401it_int @ M @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8745_of__nat__drop__bit,axiom,
% 12.85/7.02 ! [M: nat,N: nat] :
% 12.85/7.02 ( ( semiri1316708129612266289at_nat @ ( bit_se8570568707652914677it_nat @ M @ N ) )
% 12.85/7.02 = ( bit_se8570568707652914677it_nat @ M @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8746_drop__bit__of__nat,axiom,
% 12.85/7.02 ! [N: nat,M: nat] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 12.85/7.02 = ( semiri1314217659103216013at_int @ ( bit_se8570568707652914677it_nat @ N @ M ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8747_drop__bit__of__nat,axiom,
% 12.85/7.02 ! [N: nat,M: nat] :
% 12.85/7.02 ( ( bit_se8570568707652914677it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 12.85/7.02 = ( semiri1316708129612266289at_nat @ ( bit_se8570568707652914677it_nat @ N @ M ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8748_of__nat__or__eq,axiom,
% 12.85/7.02 ! [M: nat,N: nat] :
% 12.85/7.02 ( ( semiri1314217659103216013at_int @ ( bit_se1412395901928357646or_nat @ M @ N ) )
% 12.85/7.02 = ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8749_of__nat__or__eq,axiom,
% 12.85/7.02 ! [M: nat,N: nat] :
% 12.85/7.02 ( ( semiri1316708129612266289at_nat @ ( bit_se1412395901928357646or_nat @ M @ N ) )
% 12.85/7.02 = ( bit_se1412395901928357646or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8750_or__greater__eq,axiom,
% 12.85/7.02 ! [L: int,K: int] :
% 12.85/7.02 ( ( ord_less_eq_int @ zero_zero_int @ L )
% 12.85/7.02 => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8751_OR__lower,axiom,
% 12.85/7.02 ! [X: int,Y4: int] :
% 12.85/7.02 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/7.02 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/7.02 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8752_disjunctive__add,axiom,
% 12.85/7.02 ! [A: int,B: int] :
% 12.85/7.02 ( ! [N2: nat] :
% 12.85/7.02 ( ~ ( bit_se1146084159140164899it_int @ A @ N2 )
% 12.85/7.02 | ~ ( bit_se1146084159140164899it_int @ B @ N2 ) )
% 12.85/7.02 => ( ( plus_plus_int @ A @ B )
% 12.85/7.02 = ( bit_se1409905431419307370or_int @ A @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8753_disjunctive__add,axiom,
% 12.85/7.02 ! [A: nat,B: nat] :
% 12.85/7.02 ( ! [N2: nat] :
% 12.85/7.02 ( ~ ( bit_se1148574629649215175it_nat @ A @ N2 )
% 12.85/7.02 | ~ ( bit_se1148574629649215175it_nat @ B @ N2 ) )
% 12.85/7.02 => ( ( plus_plus_nat @ A @ B )
% 12.85/7.02 = ( bit_se1412395901928357646or_nat @ A @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8754_plus__and__or,axiom,
% 12.85/7.02 ! [X: int,Y4: int] :
% 12.85/7.02 ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y4 ) @ ( bit_se1409905431419307370or_int @ X @ Y4 ) )
% 12.85/7.02 = ( plus_plus_int @ X @ Y4 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8755_or__eq__not__not__and,axiom,
% 12.85/7.02 ( bit_se1409905431419307370or_int
% 12.85/7.02 = ( ^ [A3: int,B2: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ A3 ) @ ( bit_ri7919022796975470100ot_int @ B2 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8756_and__eq__not__not__or,axiom,
% 12.85/7.02 ( bit_se725231765392027082nd_int
% 12.85/7.02 = ( ^ [A3: int,B2: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ A3 ) @ ( bit_ri7919022796975470100ot_int @ B2 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8757_or__int__def,axiom,
% 12.85/7.02 ( bit_se1409905431419307370or_int
% 12.85/7.02 = ( ^ [K3: int,L3: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ ( bit_ri7919022796975470100ot_int @ L3 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8758_set__bit__nat__def,axiom,
% 12.85/7.02 ( bit_se7882103937844011126it_nat
% 12.85/7.02 = ( ^ [M5: nat,N4: nat] : ( bit_se1412395901928357646or_nat @ N4 @ ( bit_se547839408752420682it_nat @ M5 @ one_one_nat ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8759_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
% 12.85/7.02 ! [N: nat,A: int] :
% 12.85/7.02 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 12.85/7.02 = A )
% 12.85/7.02 = ( ( bit_se8568078237143864401it_int @ N @ A )
% 12.85/7.02 = zero_zero_int ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8760_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
% 12.85/7.02 ! [N: nat,A: nat] :
% 12.85/7.02 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 12.85/7.02 = A )
% 12.85/7.02 = ( ( bit_se8570568707652914677it_nat @ N @ A )
% 12.85/7.02 = zero_zero_nat ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8761_drop__bit__push__bit__int,axiom,
% 12.85/7.02 ! [M: nat,N: nat,K: int] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N @ K ) )
% 12.85/7.02 = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M ) @ K ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8762_take__bit__drop__bit,axiom,
% 12.85/7.02 ! [M: nat,N: nat,A: int] :
% 12.85/7.02 ( ( bit_se2923211474154528505it_int @ M @ ( bit_se8568078237143864401it_int @ N @ A ) )
% 12.85/7.02 = ( bit_se8568078237143864401it_int @ N @ ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8763_take__bit__drop__bit,axiom,
% 12.85/7.02 ! [M: nat,N: nat,A: nat] :
% 12.85/7.02 ( ( bit_se2925701944663578781it_nat @ M @ ( bit_se8570568707652914677it_nat @ N @ A ) )
% 12.85/7.02 = ( bit_se8570568707652914677it_nat @ N @ ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N ) @ A ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8764_drop__bit__take__bit,axiom,
% 12.85/7.02 ! [M: nat,N: nat,A: int] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 12.85/7.02 = ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N @ M ) @ ( bit_se8568078237143864401it_int @ M @ A ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8765_drop__bit__take__bit,axiom,
% 12.85/7.02 ! [M: nat,N: nat,A: nat] :
% 12.85/7.02 ( ( bit_se8570568707652914677it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 12.85/7.02 = ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N @ M ) @ ( bit_se8570568707652914677it_nat @ M @ A ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8766_or__not__numerals_I1_J,axiom,
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 12.85/7.02 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8767_bit_Oxor__def2,axiom,
% 12.85/7.02 ( bit_se6526347334894502574or_int
% 12.85/7.02 = ( ^ [X3: int,Y6: int] : ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ X3 @ Y6 ) @ ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ X3 ) @ ( bit_ri7919022796975470100ot_int @ Y6 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8768_bit_Oxor__def,axiom,
% 12.85/7.02 ( bit_se6526347334894502574or_int
% 12.85/7.02 = ( ^ [X3: int,Y6: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ X3 @ ( bit_ri7919022796975470100ot_int @ Y6 ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X3 ) @ Y6 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8769_set__bit__eq__or,axiom,
% 12.85/7.02 ( bit_se2793503036327961859nteger
% 12.85/7.02 = ( ^ [N4: nat,A3: code_integer] : ( bit_se1080825931792720795nteger @ A3 @ ( bit_se7788150548672797655nteger @ N4 @ one_one_Code_integer ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8770_set__bit__eq__or,axiom,
% 12.85/7.02 ( bit_se7879613467334960850it_int
% 12.85/7.02 = ( ^ [N4: nat,A3: int] : ( bit_se1409905431419307370or_int @ A3 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8771_set__bit__eq__or,axiom,
% 12.85/7.02 ( bit_se7882103937844011126it_nat
% 12.85/7.02 = ( ^ [N4: nat,A3: nat] : ( bit_se1412395901928357646or_nat @ A3 @ ( bit_se547839408752420682it_nat @ N4 @ one_one_nat ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8772_xor__int__def,axiom,
% 12.85/7.02 ( bit_se6526347334894502574or_int
% 12.85/7.02 = ( ^ [K3: int,L3: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ L3 ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ L3 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8773_div__push__bit__of__1__eq__drop__bit,axiom,
% 12.85/7.02 ! [A: int,N: nat] :
% 12.85/7.02 ( ( divide_divide_int @ A @ ( bit_se545348938243370406it_int @ N @ one_one_int ) )
% 12.85/7.02 = ( bit_se8568078237143864401it_int @ N @ A ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8774_div__push__bit__of__1__eq__drop__bit,axiom,
% 12.85/7.02 ! [A: nat,N: nat] :
% 12.85/7.02 ( ( divide_divide_nat @ A @ ( bit_se547839408752420682it_nat @ N @ one_one_nat ) )
% 12.85/7.02 = ( bit_se8570568707652914677it_nat @ N @ A ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8775_concat__bit__def,axiom,
% 12.85/7.02 ( bit_concat_bit
% 12.85/7.02 = ( ^ [N4: nat,K3: int,L3: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N4 @ K3 ) @ ( bit_se545348938243370406it_int @ N4 @ L3 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8776_bit__iff__and__drop__bit__eq__1,axiom,
% 12.85/7.02 ( bit_se1146084159140164899it_int
% 12.85/7.02 = ( ^ [A3: int,N4: nat] :
% 12.85/7.02 ( ( bit_se725231765392027082nd_int @ ( bit_se8568078237143864401it_int @ N4 @ A3 ) @ one_one_int )
% 12.85/7.02 = one_one_int ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8777_bit__iff__and__drop__bit__eq__1,axiom,
% 12.85/7.02 ( bit_se1148574629649215175it_nat
% 12.85/7.02 = ( ^ [A3: nat,N4: nat] :
% 12.85/7.02 ( ( bit_se727722235901077358nd_nat @ ( bit_se8570568707652914677it_nat @ N4 @ A3 ) @ one_one_nat )
% 12.85/7.02 = one_one_nat ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8778_set__bit__int__def,axiom,
% 12.85/7.02 ( bit_se7879613467334960850it_int
% 12.85/7.02 = ( ^ [N4: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8779_bits__ident,axiom,
% 12.85/7.02 ! [N: nat,A: int] :
% 12.85/7.02 ( ( plus_plus_int @ ( bit_se545348938243370406it_int @ N @ ( bit_se8568078237143864401it_int @ N @ A ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 12.85/7.02 = A ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8780_bits__ident,axiom,
% 12.85/7.02 ! [N: nat,A: nat] :
% 12.85/7.02 ( ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N @ ( bit_se8570568707652914677it_nat @ N @ A ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 12.85/7.02 = A ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8781_even__or__iff,axiom,
% 12.85/7.02 ! [A: code_integer,B: code_integer] :
% 12.85/7.02 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1080825931792720795nteger @ A @ B ) )
% 12.85/7.02 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 12.85/7.02 & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8782_even__or__iff,axiom,
% 12.85/7.02 ! [A: int,B: int] :
% 12.85/7.02 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 12.85/7.02 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 12.85/7.02 & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8783_even__or__iff,axiom,
% 12.85/7.02 ! [A: nat,B: nat] :
% 12.85/7.02 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 12.85/7.02 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 12.85/7.02 & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8784_bit_Ocomplement__unique,axiom,
% 12.85/7.02 ! [A: code_integer,X: code_integer,Y4: code_integer] :
% 12.85/7.02 ( ( ( bit_se3949692690581998587nteger @ A @ X )
% 12.85/7.02 = zero_z3403309356797280102nteger )
% 12.85/7.02 => ( ( ( bit_se1080825931792720795nteger @ A @ X )
% 12.85/7.02 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.02 => ( ( ( bit_se3949692690581998587nteger @ A @ Y4 )
% 12.85/7.02 = zero_z3403309356797280102nteger )
% 12.85/7.02 => ( ( ( bit_se1080825931792720795nteger @ A @ Y4 )
% 12.85/7.02 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.02 => ( X = Y4 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8785_bit_Ocomplement__unique,axiom,
% 12.85/7.02 ! [A: int,X: int,Y4: int] :
% 12.85/7.02 ( ( ( bit_se725231765392027082nd_int @ A @ X )
% 12.85/7.02 = zero_zero_int )
% 12.85/7.02 => ( ( ( bit_se1409905431419307370or_int @ A @ X )
% 12.85/7.02 = ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.02 => ( ( ( bit_se725231765392027082nd_int @ A @ Y4 )
% 12.85/7.02 = zero_zero_int )
% 12.85/7.02 => ( ( ( bit_se1409905431419307370or_int @ A @ Y4 )
% 12.85/7.02 = ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.02 => ( X = Y4 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8786_or__not__numerals_I4_J,axiom,
% 12.85/7.02 ! [M: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 12.85/7.02 = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8787_or__not__numerals_I2_J,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.85/7.02 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8788_stable__imp__drop__bit__eq,axiom,
% 12.85/7.02 ! [A: int,N: nat] :
% 12.85/7.02 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/7.02 = A )
% 12.85/7.02 => ( ( bit_se8568078237143864401it_int @ N @ A )
% 12.85/7.02 = A ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8789_stable__imp__drop__bit__eq,axiom,
% 12.85/7.02 ! [A: nat,N: nat] :
% 12.85/7.02 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.02 = A )
% 12.85/7.02 => ( ( bit_se8570568707652914677it_nat @ N @ A )
% 12.85/7.02 = A ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8790_drop__bit__half,axiom,
% 12.85/7.02 ! [N: nat,A: int] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 12.85/7.02 = ( divide_divide_int @ ( bit_se8568078237143864401it_int @ N @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8791_drop__bit__half,axiom,
% 12.85/7.02 ! [N: nat,A: nat] :
% 12.85/7.02 ( ( bit_se8570568707652914677it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.02 = ( divide_divide_nat @ ( bit_se8570568707652914677it_nat @ N @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8792_bit_Ocompl__unique,axiom,
% 12.85/7.02 ! [X: code_integer,Y4: code_integer] :
% 12.85/7.02 ( ( ( bit_se3949692690581998587nteger @ X @ Y4 )
% 12.85/7.02 = zero_z3403309356797280102nteger )
% 12.85/7.02 => ( ( ( bit_se1080825931792720795nteger @ X @ Y4 )
% 12.85/7.02 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 12.85/7.02 => ( ( bit_ri7632146776885996613nteger @ X )
% 12.85/7.02 = Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8793_bit_Ocompl__unique,axiom,
% 12.85/7.02 ! [X: int,Y4: int] :
% 12.85/7.02 ( ( ( bit_se725231765392027082nd_int @ X @ Y4 )
% 12.85/7.02 = zero_zero_int )
% 12.85/7.02 => ( ( ( bit_se1409905431419307370or_int @ X @ Y4 )
% 12.85/7.02 = ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.02 => ( ( bit_ri7919022796975470100ot_int @ X )
% 12.85/7.02 = Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8794_or__not__numerals_I3_J,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.85/7.02 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8795_or__not__numerals_I7_J,axiom,
% 12.85/7.02 ! [M: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 12.85/7.02 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8796_drop__bit__int__def,axiom,
% 12.85/7.02 ( bit_se8568078237143864401it_int
% 12.85/7.02 = ( ^ [N4: nat,K3: int] : ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8797_signed__take__bit__eq__if__negative,axiom,
% 12.85/7.02 ! [A: int,N: nat] :
% 12.85/7.02 ( ( bit_se1146084159140164899it_int @ A @ N )
% 12.85/7.02 => ( ( bit_ri631733984087533419it_int @ N @ A )
% 12.85/7.02 = ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8798_drop__bit__Suc,axiom,
% 12.85/7.02 ! [N: nat,A: int] :
% 12.85/7.02 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ A )
% 12.85/7.02 = ( bit_se8568078237143864401it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8799_drop__bit__Suc,axiom,
% 12.85/7.02 ! [N: nat,A: nat] :
% 12.85/7.02 ( ( bit_se8570568707652914677it_nat @ ( suc @ N ) @ A )
% 12.85/7.02 = ( bit_se8570568707652914677it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8800_drop__bit__eq__div,axiom,
% 12.85/7.02 ( bit_se8568078237143864401it_int
% 12.85/7.02 = ( ^ [N4: nat,A3: int] : ( divide_divide_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8801_drop__bit__eq__div,axiom,
% 12.85/7.02 ( bit_se8570568707652914677it_nat
% 12.85/7.02 = ( ^ [N4: nat,A3: nat] : ( divide_divide_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8802_even__drop__bit__iff__not__bit,axiom,
% 12.85/7.02 ! [N: nat,A: code_integer] :
% 12.85/7.02 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3928097537394005634nteger @ N @ A ) )
% 12.85/7.02 = ( ~ ( bit_se9216721137139052372nteger @ A @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8803_even__drop__bit__iff__not__bit,axiom,
% 12.85/7.02 ! [N: nat,A: int] :
% 12.85/7.02 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se8568078237143864401it_int @ N @ A ) )
% 12.85/7.02 = ( ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8804_even__drop__bit__iff__not__bit,axiom,
% 12.85/7.02 ! [N: nat,A: nat] :
% 12.85/7.02 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se8570568707652914677it_nat @ N @ A ) )
% 12.85/7.02 = ( ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8805_bit__iff__odd__drop__bit,axiom,
% 12.85/7.02 ( bit_se9216721137139052372nteger
% 12.85/7.02 = ( ^ [A3: code_integer,N4: nat] :
% 12.85/7.02 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3928097537394005634nteger @ N4 @ A3 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8806_bit__iff__odd__drop__bit,axiom,
% 12.85/7.02 ( bit_se1146084159140164899it_int
% 12.85/7.02 = ( ^ [A3: int,N4: nat] :
% 12.85/7.02 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se8568078237143864401it_int @ N4 @ A3 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8807_bit__iff__odd__drop__bit,axiom,
% 12.85/7.02 ( bit_se1148574629649215175it_nat
% 12.85/7.02 = ( ^ [A3: nat,N4: nat] :
% 12.85/7.02 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se8570568707652914677it_nat @ N4 @ A3 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8808_mask__Suc__exp,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( bit_se2002935070580805687sk_nat @ ( suc @ N ) )
% 12.85/7.02 = ( bit_se1412395901928357646or_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8809_mask__Suc__exp,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( bit_se2000444600071755411sk_int @ ( suc @ N ) )
% 12.85/7.02 = ( bit_se1409905431419307370or_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8810_or__not__numerals_I6_J,axiom,
% 12.85/7.02 ! [M: num,N: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.85/7.02 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8811_or__one__eq,axiom,
% 12.85/7.02 ! [A: code_integer] :
% 12.85/7.02 ( ( bit_se1080825931792720795nteger @ A @ one_one_Code_integer )
% 12.85/7.02 = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8812_or__one__eq,axiom,
% 12.85/7.02 ! [A: int] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ A @ one_one_int )
% 12.85/7.02 = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8813_or__one__eq,axiom,
% 12.85/7.02 ! [A: nat] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ A @ one_one_nat )
% 12.85/7.02 = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8814_one__or__eq,axiom,
% 12.85/7.02 ! [A: code_integer] :
% 12.85/7.02 ( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ A )
% 12.85/7.02 = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8815_one__or__eq,axiom,
% 12.85/7.02 ! [A: int] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ one_one_int @ A )
% 12.85/7.02 = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8816_one__or__eq,axiom,
% 12.85/7.02 ! [A: nat] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ one_one_nat @ A )
% 12.85/7.02 = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8817_mask__Suc__double,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( bit_se2002935070580805687sk_nat @ ( suc @ N ) )
% 12.85/7.02 = ( bit_se1412395901928357646or_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8818_mask__Suc__double,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( bit_se2000444600071755411sk_int @ ( suc @ N ) )
% 12.85/7.02 = ( bit_se1409905431419307370or_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8819_OR__upper,axiom,
% 12.85/7.02 ! [X: int,N: nat,Y4: int] :
% 12.85/7.02 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/7.02 => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.02 => ( ( ord_less_int @ Y4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 12.85/7.02 => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y4 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8820_or__not__numerals_I5_J,axiom,
% 12.85/7.02 ! [M: num,N: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.85/7.02 = ( 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 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8821_slice__eq__mask,axiom,
% 12.85/7.02 ! [N: nat,M: nat,A: int] :
% 12.85/7.02 ( ( bit_se545348938243370406it_int @ N @ ( bit_se2923211474154528505it_int @ M @ ( bit_se8568078237143864401it_int @ N @ A ) ) )
% 12.85/7.02 = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ ( bit_se2000444600071755411sk_int @ ( plus_plus_nat @ M @ N ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8822_signed__take__bit__def,axiom,
% 12.85/7.02 ( bit_ri6519982836138164636nteger
% 12.85/7.02 = ( ^ [N4: nat,A3: code_integer] : ( bit_se1080825931792720795nteger @ ( bit_se1745604003318907178nteger @ N4 @ A3 ) @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( bit_se9216721137139052372nteger @ A3 @ N4 ) ) @ ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N4 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8823_signed__take__bit__def,axiom,
% 12.85/7.02 ( bit_ri631733984087533419it_int
% 12.85/7.02 = ( ^ [N4: nat,A3: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N4 @ A3 ) @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ A3 @ N4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N4 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8824_or__Suc__0__eq,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
% 12.85/7.02 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8825_Suc__0__or__eq,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.85/7.02 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8826_drop__bit__rec,axiom,
% 12.85/7.02 ( bit_se8568078237143864401it_int
% 12.85/7.02 = ( ^ [N4: nat,A3: int] : ( if_int @ ( N4 = zero_zero_nat ) @ A3 @ ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8827_drop__bit__rec,axiom,
% 12.85/7.02 ( bit_se8570568707652914677it_nat
% 12.85/7.02 = ( ^ [N4: nat,A3: nat] : ( if_nat @ ( N4 = zero_zero_nat ) @ A3 @ ( bit_se8570568707652914677it_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8828_or__not__numerals_I9_J,axiom,
% 12.85/7.02 ! [M: num,N: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.85/7.02 = ( 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 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8829_or__not__numerals_I8_J,axiom,
% 12.85/7.02 ! [M: num,N: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.85/7.02 = ( 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 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8830_or__nat__rec,axiom,
% 12.85/7.02 ( bit_se1412395901928357646or_nat
% 12.85/7.02 = ( ^ [M5: nat,N4: nat] :
% 12.85/7.02 ( plus_plus_nat
% 12.85/7.02 @ ( zero_n2687167440665602831ol_nat
% 12.85/7.02 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 )
% 12.85/7.02 | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
% 12.85/7.02 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8831_or__int__unfold,axiom,
% 12.85/7.02 ( bit_se1409905431419307370or_int
% 12.85/7.02 = ( ^ [K3: int,L3: int] :
% 12.85/7.02 ( if_int
% 12.85/7.02 @ ( ( K3
% 12.85/7.02 = ( uminus_uminus_int @ one_one_int ) )
% 12.85/7.02 | ( L3
% 12.85/7.02 = ( uminus_uminus_int @ one_one_int ) ) )
% 12.85/7.02 @ ( uminus_uminus_int @ one_one_int )
% 12.85/7.02 @ ( if_int @ ( K3 = zero_zero_int ) @ L3 @ ( if_int @ ( L3 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8832_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
% 12.85/7.02 boolea2445317508997433345nteger @ bit_se3949692690581998587nteger @ bit_se1080825931792720795nteger @ bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ bit_se3222712562003087583nteger ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8833_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
% 12.85/7.02 boolea8527374999097803216ff_int @ bit_se725231765392027082nd_int @ bit_se1409905431419307370or_int @ bit_ri7919022796975470100ot_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) @ bit_se6526347334894502574or_int ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8834_or__minus__numerals_I8_J,axiom,
% 12.85/7.02 ! [N: num,M: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 12.85/7.02 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8835_or__minus__numerals_I4_J,axiom,
% 12.85/7.02 ! [M: num,N: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.85/7.02 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8836_max_Obounded__iff,axiom,
% 12.85/7.02 ! [B: extended_enat,C: extended_enat,A: extended_enat] :
% 12.85/7.02 ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
% 12.85/7.02 = ( ( ord_le2932123472753598470d_enat @ B @ A )
% 12.85/7.02 & ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8837_max_Obounded__iff,axiom,
% 12.85/7.02 ! [B: code_integer,C: code_integer,A: code_integer] :
% 12.85/7.02 ( ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B @ C ) @ A )
% 12.85/7.02 = ( ( ord_le3102999989581377725nteger @ B @ A )
% 12.85/7.02 & ( ord_le3102999989581377725nteger @ C @ A ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8838_max_Obounded__iff,axiom,
% 12.85/7.02 ! [B: rat,C: rat,A: rat] :
% 12.85/7.02 ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 12.85/7.02 = ( ( ord_less_eq_rat @ B @ A )
% 12.85/7.02 & ( ord_less_eq_rat @ C @ A ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8839_max_Obounded__iff,axiom,
% 12.85/7.02 ! [B: num,C: num,A: num] :
% 12.85/7.02 ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 12.85/7.02 = ( ( ord_less_eq_num @ B @ A )
% 12.85/7.02 & ( ord_less_eq_num @ C @ A ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8840_max_Obounded__iff,axiom,
% 12.85/7.02 ! [B: nat,C: nat,A: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 12.85/7.02 = ( ( ord_less_eq_nat @ B @ A )
% 12.85/7.02 & ( ord_less_eq_nat @ C @ A ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8841_max_Obounded__iff,axiom,
% 12.85/7.02 ! [B: int,C: int,A: int] :
% 12.85/7.02 ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 12.85/7.02 = ( ( ord_less_eq_int @ B @ A )
% 12.85/7.02 & ( ord_less_eq_int @ C @ A ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8842_max_Obounded__iff,axiom,
% 12.85/7.02 ! [B: real,C: real,A: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A )
% 12.85/7.02 = ( ( ord_less_eq_real @ B @ A )
% 12.85/7.02 & ( ord_less_eq_real @ C @ A ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8843_max_Oabsorb2,axiom,
% 12.85/7.02 ! [A: extended_enat,B: extended_enat] :
% 12.85/7.02 ( ( ord_le2932123472753598470d_enat @ A @ B )
% 12.85/7.02 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 12.85/7.02 = B ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8844_max_Oabsorb2,axiom,
% 12.85/7.02 ! [A: code_integer,B: code_integer] :
% 12.85/7.02 ( ( ord_le3102999989581377725nteger @ A @ B )
% 12.85/7.02 => ( ( ord_max_Code_integer @ A @ B )
% 12.85/7.02 = B ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8845_max_Oabsorb2,axiom,
% 12.85/7.02 ! [A: rat,B: rat] :
% 12.85/7.02 ( ( ord_less_eq_rat @ A @ B )
% 12.85/7.02 => ( ( ord_max_rat @ A @ B )
% 12.85/7.02 = B ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8846_max_Oabsorb2,axiom,
% 12.85/7.02 ! [A: num,B: num] :
% 12.85/7.02 ( ( ord_less_eq_num @ A @ B )
% 12.85/7.02 => ( ( ord_max_num @ A @ B )
% 12.85/7.02 = B ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8847_max_Oabsorb2,axiom,
% 12.85/7.02 ! [A: nat,B: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ A @ B )
% 12.85/7.02 => ( ( ord_max_nat @ A @ B )
% 12.85/7.02 = B ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8848_max_Oabsorb2,axiom,
% 12.85/7.02 ! [A: int,B: int] :
% 12.85/7.02 ( ( ord_less_eq_int @ A @ B )
% 12.85/7.02 => ( ( ord_max_int @ A @ B )
% 12.85/7.02 = B ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8849_max_Oabsorb2,axiom,
% 12.85/7.02 ! [A: real,B: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ A @ B )
% 12.85/7.02 => ( ( ord_max_real @ A @ B )
% 12.85/7.02 = B ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8850_max_Oabsorb1,axiom,
% 12.85/7.02 ! [B: extended_enat,A: extended_enat] :
% 12.85/7.02 ( ( ord_le2932123472753598470d_enat @ B @ A )
% 12.85/7.02 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 12.85/7.02 = A ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8851_max_Oabsorb1,axiom,
% 12.85/7.02 ! [B: code_integer,A: code_integer] :
% 12.85/7.02 ( ( ord_le3102999989581377725nteger @ B @ A )
% 12.85/7.02 => ( ( ord_max_Code_integer @ A @ B )
% 12.85/7.02 = A ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8852_max_Oabsorb1,axiom,
% 12.85/7.02 ! [B: rat,A: rat] :
% 12.85/7.02 ( ( ord_less_eq_rat @ B @ A )
% 12.85/7.02 => ( ( ord_max_rat @ A @ B )
% 12.85/7.02 = A ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8853_max_Oabsorb1,axiom,
% 12.85/7.02 ! [B: num,A: num] :
% 12.85/7.02 ( ( ord_less_eq_num @ B @ A )
% 12.85/7.02 => ( ( ord_max_num @ A @ B )
% 12.85/7.02 = A ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8854_max_Oabsorb1,axiom,
% 12.85/7.02 ! [B: nat,A: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ B @ A )
% 12.85/7.02 => ( ( ord_max_nat @ A @ B )
% 12.85/7.02 = A ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8855_max_Oabsorb1,axiom,
% 12.85/7.02 ! [B: int,A: int] :
% 12.85/7.02 ( ( ord_less_eq_int @ B @ A )
% 12.85/7.02 => ( ( ord_max_int @ A @ B )
% 12.85/7.02 = A ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8856_max_Oabsorb1,axiom,
% 12.85/7.02 ! [B: real,A: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ B @ A )
% 12.85/7.02 => ( ( ord_max_real @ A @ B )
% 12.85/7.02 = A ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8857_of__bool__or__iff,axiom,
% 12.85/7.02 ! [P: $o,Q: $o] :
% 12.85/7.02 ( ( zero_n2687167440665602831ol_nat
% 12.85/7.02 @ ( P
% 12.85/7.02 | Q ) )
% 12.85/7.02 = ( ord_max_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8858_of__bool__or__iff,axiom,
% 12.85/7.02 ! [P: $o,Q: $o] :
% 12.85/7.02 ( ( zero_n2684676970156552555ol_int
% 12.85/7.02 @ ( P
% 12.85/7.02 | Q ) )
% 12.85/7.02 = ( ord_max_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8859_of__bool__or__iff,axiom,
% 12.85/7.02 ! [P: $o,Q: $o] :
% 12.85/7.02 ( ( zero_n356916108424825756nteger
% 12.85/7.02 @ ( P
% 12.85/7.02 | Q ) )
% 12.85/7.02 = ( ord_max_Code_integer @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8860_max__number__of_I1_J,axiom,
% 12.85/7.02 ! [U: num,V: num] :
% 12.85/7.02 ( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 12.85/7.02 => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 12.85/7.02 = ( numera1916890842035813515d_enat @ V ) ) )
% 12.85/7.02 & ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 12.85/7.02 => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 12.85/7.02 = ( numera1916890842035813515d_enat @ U ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8861_max__number__of_I1_J,axiom,
% 12.85/7.02 ! [U: num,V: num] :
% 12.85/7.02 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
% 12.85/7.02 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
% 12.85/7.02 = ( numera6620942414471956472nteger @ V ) ) )
% 12.85/7.02 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
% 12.85/7.02 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
% 12.85/7.02 = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8862_max__number__of_I1_J,axiom,
% 12.85/7.02 ! [U: num,V: num] :
% 12.85/7.02 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 12.85/7.02 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 12.85/7.02 = ( numeral_numeral_rat @ V ) ) )
% 12.85/7.02 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 12.85/7.02 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 12.85/7.02 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8863_max__number__of_I1_J,axiom,
% 12.85/7.02 ! [U: num,V: num] :
% 12.85/7.02 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 12.85/7.02 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 12.85/7.02 = ( numeral_numeral_nat @ V ) ) )
% 12.85/7.02 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 12.85/7.02 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 12.85/7.02 = ( numeral_numeral_nat @ U ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8864_max__number__of_I1_J,axiom,
% 12.85/7.02 ! [U: num,V: num] :
% 12.85/7.02 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 12.85/7.02 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 12.85/7.02 = ( numeral_numeral_int @ V ) ) )
% 12.85/7.02 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 12.85/7.02 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 12.85/7.02 = ( numeral_numeral_int @ U ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8865_max__number__of_I1_J,axiom,
% 12.85/7.02 ! [U: num,V: num] :
% 12.85/7.02 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 12.85/7.02 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 12.85/7.02 = ( numeral_numeral_real @ V ) ) )
% 12.85/7.02 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 12.85/7.02 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 12.85/7.02 = ( numeral_numeral_real @ U ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8866_max__0__1_I3_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X ) )
% 12.85/7.02 = ( numera1916890842035813515d_enat @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8867_max__0__1_I3_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ X ) )
% 12.85/7.02 = ( numera6620942414471956472nteger @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8868_max__0__1_I3_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
% 12.85/7.02 = ( numeral_numeral_real @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8869_max__0__1_I3_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
% 12.85/7.02 = ( numeral_numeral_rat @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8870_max__0__1_I3_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
% 12.85/7.02 = ( numeral_numeral_nat @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8871_max__0__1_I3_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
% 12.85/7.02 = ( numeral_numeral_int @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8872_max__0__1_I4_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ zero_z5237406670263579293d_enat )
% 12.85/7.02 = ( numera1916890842035813515d_enat @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8873_max__0__1_I4_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X ) @ zero_z3403309356797280102nteger )
% 12.85/7.02 = ( numera6620942414471956472nteger @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8874_max__0__1_I4_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
% 12.85/7.02 = ( numeral_numeral_real @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8875_max__0__1_I4_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
% 12.85/7.02 = ( numeral_numeral_rat @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8876_max__0__1_I4_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
% 12.85/7.02 = ( numeral_numeral_nat @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8877_max__0__1_I4_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
% 12.85/7.02 = ( numeral_numeral_int @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8878_max__0__1_I2_J,axiom,
% 12.85/7.02 ( ( ord_max_real @ one_one_real @ zero_zero_real )
% 12.85/7.02 = one_one_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8879_max__0__1_I2_J,axiom,
% 12.85/7.02 ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
% 12.85/7.02 = one_one_rat ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8880_max__0__1_I2_J,axiom,
% 12.85/7.02 ( ( ord_max_int @ one_one_int @ zero_zero_int )
% 12.85/7.02 = one_one_int ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8881_max__0__1_I2_J,axiom,
% 12.85/7.02 ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
% 12.85/7.02 = one_one_nat ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8882_max__0__1_I2_J,axiom,
% 12.85/7.02 ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
% 12.85/7.02 = one_on7984719198319812577d_enat ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8883_max__0__1_I2_J,axiom,
% 12.85/7.02 ( ( ord_max_Code_integer @ one_one_Code_integer @ zero_z3403309356797280102nteger )
% 12.85/7.02 = one_one_Code_integer ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8884_max__0__1_I1_J,axiom,
% 12.85/7.02 ( ( ord_max_real @ zero_zero_real @ one_one_real )
% 12.85/7.02 = one_one_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8885_max__0__1_I1_J,axiom,
% 12.85/7.02 ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
% 12.85/7.02 = one_one_rat ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8886_max__0__1_I1_J,axiom,
% 12.85/7.02 ( ( ord_max_int @ zero_zero_int @ one_one_int )
% 12.85/7.02 = one_one_int ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8887_max__0__1_I1_J,axiom,
% 12.85/7.02 ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
% 12.85/7.02 = one_one_nat ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8888_max__0__1_I1_J,axiom,
% 12.85/7.02 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
% 12.85/7.02 = one_on7984719198319812577d_enat ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8889_max__0__1_I1_J,axiom,
% 12.85/7.02 ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer )
% 12.85/7.02 = one_one_Code_integer ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8890_max__0__1_I5_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
% 12.85/7.02 = ( numera1916890842035813515d_enat @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8891_max__0__1_I5_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_Code_integer @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X ) )
% 12.85/7.02 = ( numera6620942414471956472nteger @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8892_max__0__1_I5_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X ) )
% 12.85/7.02 = ( numeral_numeral_real @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8893_max__0__1_I5_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
% 12.85/7.02 = ( numeral_numeral_rat @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8894_max__0__1_I5_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
% 12.85/7.02 = ( numeral_numeral_nat @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8895_max__0__1_I5_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
% 12.85/7.02 = ( numeral_numeral_int @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8896_max__0__1_I6_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat )
% 12.85/7.02 = ( numera1916890842035813515d_enat @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8897_max__0__1_I6_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer )
% 12.85/7.02 = ( numera6620942414471956472nteger @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8898_max__0__1_I6_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ one_one_real )
% 12.85/7.02 = ( numeral_numeral_real @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8899_max__0__1_I6_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
% 12.85/7.02 = ( numeral_numeral_rat @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8900_max__0__1_I6_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
% 12.85/7.02 = ( numeral_numeral_nat @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8901_max__0__1_I6_J,axiom,
% 12.85/7.02 ! [X: num] :
% 12.85/7.02 ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
% 12.85/7.02 = ( numeral_numeral_int @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8902_max__number__of_I4_J,axiom,
% 12.85/7.02 ! [U: num,V: num] :
% 12.85/7.02 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 12.85/7.02 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 12.85/7.02 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 12.85/7.02 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 12.85/7.02 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 12.85/7.02 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8903_max__number__of_I4_J,axiom,
% 12.85/7.02 ! [U: num,V: num] :
% 12.85/7.02 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 12.85/7.02 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 12.85/7.02 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 12.85/7.02 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 12.85/7.02 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 12.85/7.02 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8904_max__number__of_I4_J,axiom,
% 12.85/7.02 ! [U: num,V: num] :
% 12.85/7.02 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.85/7.02 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.85/7.02 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 12.85/7.02 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.85/7.02 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.85/7.02 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8905_max__number__of_I4_J,axiom,
% 12.85/7.02 ! [U: num,V: num] :
% 12.85/7.02 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 12.85/7.02 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 12.85/7.02 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 12.85/7.02 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 12.85/7.02 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 12.85/7.02 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8906_max__number__of_I3_J,axiom,
% 12.85/7.02 ! [U: num,V: num] :
% 12.85/7.02 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 12.85/7.02 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 12.85/7.02 = ( numera6620942414471956472nteger @ V ) ) )
% 12.85/7.02 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 12.85/7.02 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 12.85/7.02 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8907_max__number__of_I3_J,axiom,
% 12.85/7.02 ! [U: num,V: num] :
% 12.85/7.02 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 12.85/7.02 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 12.85/7.02 = ( numeral_numeral_rat @ V ) ) )
% 12.85/7.02 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 12.85/7.02 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 12.85/7.02 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8908_max__number__of_I3_J,axiom,
% 12.85/7.02 ! [U: num,V: num] :
% 12.85/7.02 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 12.85/7.02 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 12.85/7.02 = ( numeral_numeral_int @ V ) ) )
% 12.85/7.02 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 12.85/7.02 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 12.85/7.02 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8909_max__number__of_I3_J,axiom,
% 12.85/7.02 ! [U: num,V: num] :
% 12.85/7.02 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 12.85/7.02 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 12.85/7.02 = ( numeral_numeral_real @ V ) ) )
% 12.85/7.02 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 12.85/7.02 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 12.85/7.02 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8910_max__number__of_I2_J,axiom,
% 12.85/7.02 ! [U: num,V: num] :
% 12.85/7.02 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 12.85/7.02 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 12.85/7.02 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 12.85/7.02 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 12.85/7.02 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 12.85/7.02 = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8911_max__number__of_I2_J,axiom,
% 12.85/7.02 ! [U: num,V: num] :
% 12.85/7.02 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 12.85/7.02 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 12.85/7.02 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 12.85/7.02 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 12.85/7.02 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 12.85/7.02 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8912_max__number__of_I2_J,axiom,
% 12.85/7.02 ! [U: num,V: num] :
% 12.85/7.02 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.85/7.02 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.85/7.02 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 12.85/7.02 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.85/7.02 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 12.85/7.02 = ( numeral_numeral_int @ U ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8913_max__number__of_I2_J,axiom,
% 12.85/7.02 ! [U: num,V: num] :
% 12.85/7.02 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 12.85/7.02 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 12.85/7.02 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 12.85/7.02 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 12.85/7.02 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 12.85/7.02 = ( numeral_numeral_real @ U ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8914_drop__bit__of__Suc__0,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
% 12.85/7.02 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8915_of__nat__max,axiom,
% 12.85/7.02 ! [X: nat,Y4: nat] :
% 12.85/7.02 ( ( semiri4216267220026989637d_enat @ ( ord_max_nat @ X @ Y4 ) )
% 12.85/7.02 = ( ord_ma741700101516333627d_enat @ ( semiri4216267220026989637d_enat @ X ) @ ( semiri4216267220026989637d_enat @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8916_of__nat__max,axiom,
% 12.85/7.02 ! [X: nat,Y4: nat] :
% 12.85/7.02 ( ( semiri4939895301339042750nteger @ ( ord_max_nat @ X @ Y4 ) )
% 12.85/7.02 = ( ord_max_Code_integer @ ( semiri4939895301339042750nteger @ X ) @ ( semiri4939895301339042750nteger @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8917_of__nat__max,axiom,
% 12.85/7.02 ! [X: nat,Y4: nat] :
% 12.85/7.02 ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y4 ) )
% 12.85/7.02 = ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8918_of__nat__max,axiom,
% 12.85/7.02 ! [X: nat,Y4: nat] :
% 12.85/7.02 ( ( semiri681578069525770553at_rat @ ( ord_max_nat @ X @ Y4 ) )
% 12.85/7.02 = ( ord_max_rat @ ( semiri681578069525770553at_rat @ X ) @ ( semiri681578069525770553at_rat @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8919_of__nat__max,axiom,
% 12.85/7.02 ! [X: nat,Y4: nat] :
% 12.85/7.02 ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y4 ) )
% 12.85/7.02 = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8920_of__nat__max,axiom,
% 12.85/7.02 ! [X: nat,Y4: nat] :
% 12.85/7.02 ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y4 ) )
% 12.85/7.02 = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8921_max__diff__distrib__left,axiom,
% 12.85/7.02 ! [X: code_integer,Y4: code_integer,Z: code_integer] :
% 12.85/7.02 ( ( minus_8373710615458151222nteger @ ( ord_max_Code_integer @ X @ Y4 ) @ Z )
% 12.85/7.02 = ( ord_max_Code_integer @ ( minus_8373710615458151222nteger @ X @ Z ) @ ( minus_8373710615458151222nteger @ Y4 @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8922_max__diff__distrib__left,axiom,
% 12.85/7.02 ! [X: real,Y4: real,Z: real] :
% 12.85/7.02 ( ( minus_minus_real @ ( ord_max_real @ X @ Y4 ) @ Z )
% 12.85/7.02 = ( ord_max_real @ ( minus_minus_real @ X @ Z ) @ ( minus_minus_real @ Y4 @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8923_max__diff__distrib__left,axiom,
% 12.85/7.02 ! [X: rat,Y4: rat,Z: rat] :
% 12.85/7.02 ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y4 ) @ Z )
% 12.85/7.02 = ( ord_max_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y4 @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8924_max__diff__distrib__left,axiom,
% 12.85/7.02 ! [X: int,Y4: int,Z: int] :
% 12.85/7.02 ( ( minus_minus_int @ ( ord_max_int @ X @ Y4 ) @ Z )
% 12.85/7.02 = ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y4 @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8925_max__add__distrib__right,axiom,
% 12.85/7.02 ! [X: real,Y4: real,Z: real] :
% 12.85/7.02 ( ( plus_plus_real @ X @ ( ord_max_real @ Y4 @ Z ) )
% 12.85/7.02 = ( ord_max_real @ ( plus_plus_real @ X @ Y4 ) @ ( plus_plus_real @ X @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8926_max__add__distrib__right,axiom,
% 12.85/7.02 ! [X: rat,Y4: rat,Z: rat] :
% 12.85/7.02 ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y4 @ Z ) )
% 12.85/7.02 = ( ord_max_rat @ ( plus_plus_rat @ X @ Y4 ) @ ( plus_plus_rat @ X @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8927_max__add__distrib__right,axiom,
% 12.85/7.02 ! [X: int,Y4: int,Z: int] :
% 12.85/7.02 ( ( plus_plus_int @ X @ ( ord_max_int @ Y4 @ Z ) )
% 12.85/7.02 = ( ord_max_int @ ( plus_plus_int @ X @ Y4 ) @ ( plus_plus_int @ X @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8928_max__add__distrib__right,axiom,
% 12.85/7.02 ! [X: nat,Y4: nat,Z: nat] :
% 12.85/7.02 ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y4 @ Z ) )
% 12.85/7.02 = ( ord_max_nat @ ( plus_plus_nat @ X @ Y4 ) @ ( plus_plus_nat @ X @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8929_max__add__distrib__right,axiom,
% 12.85/7.02 ! [X: code_integer,Y4: code_integer,Z: code_integer] :
% 12.85/7.02 ( ( plus_p5714425477246183910nteger @ X @ ( ord_max_Code_integer @ Y4 @ Z ) )
% 12.85/7.02 = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Y4 ) @ ( plus_p5714425477246183910nteger @ X @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8930_max__add__distrib__left,axiom,
% 12.85/7.02 ! [X: real,Y4: real,Z: real] :
% 12.85/7.02 ( ( plus_plus_real @ ( ord_max_real @ X @ Y4 ) @ Z )
% 12.85/7.02 = ( ord_max_real @ ( plus_plus_real @ X @ Z ) @ ( plus_plus_real @ Y4 @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8931_max__add__distrib__left,axiom,
% 12.85/7.02 ! [X: rat,Y4: rat,Z: rat] :
% 12.85/7.02 ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y4 ) @ Z )
% 12.85/7.02 = ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y4 @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8932_max__add__distrib__left,axiom,
% 12.85/7.02 ! [X: int,Y4: int,Z: int] :
% 12.85/7.02 ( ( plus_plus_int @ ( ord_max_int @ X @ Y4 ) @ Z )
% 12.85/7.02 = ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y4 @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8933_max__add__distrib__left,axiom,
% 12.85/7.02 ! [X: nat,Y4: nat,Z: nat] :
% 12.85/7.02 ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y4 ) @ Z )
% 12.85/7.02 = ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y4 @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8934_max__add__distrib__left,axiom,
% 12.85/7.02 ! [X: code_integer,Y4: code_integer,Z: code_integer] :
% 12.85/7.02 ( ( plus_p5714425477246183910nteger @ ( ord_max_Code_integer @ X @ Y4 ) @ Z )
% 12.85/7.02 = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Z ) @ ( plus_p5714425477246183910nteger @ Y4 @ Z ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8935_max__absorb2,axiom,
% 12.85/7.02 ! [X: extended_enat,Y4: extended_enat] :
% 12.85/7.02 ( ( ord_le2932123472753598470d_enat @ X @ Y4 )
% 12.85/7.02 => ( ( ord_ma741700101516333627d_enat @ X @ Y4 )
% 12.85/7.02 = Y4 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8936_max__absorb2,axiom,
% 12.85/7.02 ! [X: code_integer,Y4: code_integer] :
% 12.85/7.02 ( ( ord_le3102999989581377725nteger @ X @ Y4 )
% 12.85/7.02 => ( ( ord_max_Code_integer @ X @ Y4 )
% 12.85/7.02 = Y4 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8937_max__absorb2,axiom,
% 12.85/7.02 ! [X: set_int,Y4: set_int] :
% 12.85/7.02 ( ( ord_less_eq_set_int @ X @ Y4 )
% 12.85/7.02 => ( ( ord_max_set_int @ X @ Y4 )
% 12.85/7.02 = Y4 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8938_max__absorb2,axiom,
% 12.85/7.02 ! [X: rat,Y4: rat] :
% 12.85/7.02 ( ( ord_less_eq_rat @ X @ Y4 )
% 12.85/7.02 => ( ( ord_max_rat @ X @ Y4 )
% 12.85/7.02 = Y4 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8939_max__absorb2,axiom,
% 12.85/7.02 ! [X: num,Y4: num] :
% 12.85/7.02 ( ( ord_less_eq_num @ X @ Y4 )
% 12.85/7.02 => ( ( ord_max_num @ X @ Y4 )
% 12.85/7.02 = Y4 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8940_max__absorb2,axiom,
% 12.85/7.02 ! [X: nat,Y4: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ X @ Y4 )
% 12.85/7.02 => ( ( ord_max_nat @ X @ Y4 )
% 12.85/7.02 = Y4 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8941_max__absorb2,axiom,
% 12.85/7.02 ! [X: int,Y4: int] :
% 12.85/7.02 ( ( ord_less_eq_int @ X @ Y4 )
% 12.85/7.02 => ( ( ord_max_int @ X @ Y4 )
% 12.85/7.02 = Y4 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8942_max__absorb2,axiom,
% 12.85/7.02 ! [X: real,Y4: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/7.02 => ( ( ord_max_real @ X @ Y4 )
% 12.85/7.02 = Y4 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8943_max__absorb1,axiom,
% 12.85/7.02 ! [Y4: extended_enat,X: extended_enat] :
% 12.85/7.02 ( ( ord_le2932123472753598470d_enat @ Y4 @ X )
% 12.85/7.02 => ( ( ord_ma741700101516333627d_enat @ X @ Y4 )
% 12.85/7.02 = X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8944_max__absorb1,axiom,
% 12.85/7.02 ! [Y4: code_integer,X: code_integer] :
% 12.85/7.02 ( ( ord_le3102999989581377725nteger @ Y4 @ X )
% 12.85/7.02 => ( ( ord_max_Code_integer @ X @ Y4 )
% 12.85/7.02 = X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8945_max__absorb1,axiom,
% 12.85/7.02 ! [Y4: set_int,X: set_int] :
% 12.85/7.02 ( ( ord_less_eq_set_int @ Y4 @ X )
% 12.85/7.02 => ( ( ord_max_set_int @ X @ Y4 )
% 12.85/7.02 = X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8946_max__absorb1,axiom,
% 12.85/7.02 ! [Y4: rat,X: rat] :
% 12.85/7.02 ( ( ord_less_eq_rat @ Y4 @ X )
% 12.85/7.02 => ( ( ord_max_rat @ X @ Y4 )
% 12.85/7.02 = X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8947_max__absorb1,axiom,
% 12.85/7.02 ! [Y4: num,X: num] :
% 12.85/7.02 ( ( ord_less_eq_num @ Y4 @ X )
% 12.85/7.02 => ( ( ord_max_num @ X @ Y4 )
% 12.85/7.02 = X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8948_max__absorb1,axiom,
% 12.85/7.02 ! [Y4: nat,X: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ Y4 @ X )
% 12.85/7.02 => ( ( ord_max_nat @ X @ Y4 )
% 12.85/7.02 = X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8949_max__absorb1,axiom,
% 12.85/7.02 ! [Y4: int,X: int] :
% 12.85/7.02 ( ( ord_less_eq_int @ Y4 @ X )
% 12.85/7.02 => ( ( ord_max_int @ X @ Y4 )
% 12.85/7.02 = X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8950_max__absorb1,axiom,
% 12.85/7.02 ! [Y4: real,X: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ Y4 @ X )
% 12.85/7.02 => ( ( ord_max_real @ X @ Y4 )
% 12.85/7.02 = X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8951_max__def,axiom,
% 12.85/7.02 ( ord_max_nat
% 12.85/7.02 = ( ^ [A3: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8952_max__def,axiom,
% 12.85/7.02 ( ord_max_int
% 12.85/7.02 = ( ^ [A3: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8953_max__def,axiom,
% 12.85/7.02 ( ord_max_real
% 12.85/7.02 = ( ^ [A3: real,B2: real] : ( if_real @ ( ord_less_eq_real @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8954_or__not__num__neg_Osimps_I1_J,axiom,
% 12.85/7.02 ( ( bit_or_not_num_neg @ one @ one )
% 12.85/7.02 = one ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8955_or__not__num__neg_Osimps_I4_J,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
% 12.85/7.02 = ( bit0 @ one ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8956_drop__bit__nat__eq,axiom,
% 12.85/7.02 ! [N: nat,K: int] :
% 12.85/7.02 ( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K ) )
% 12.85/7.02 = ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8957_or__not__num__neg_Osimps_I6_J,axiom,
% 12.85/7.02 ! [N: num,M: num] :
% 12.85/7.02 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
% 12.85/7.02 = ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8958_or__not__num__neg_Osimps_I7_J,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
% 12.85/7.02 = one ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8959_or__not__num__neg_Osimps_I3_J,axiom,
% 12.85/7.02 ! [M: num] :
% 12.85/7.02 ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
% 12.85/7.02 = ( bit1 @ M ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8960_or__not__num__neg_Osimps_I2_J,axiom,
% 12.85/7.02 ! [M: num] :
% 12.85/7.02 ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
% 12.85/7.02 = ( bit1 @ M ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8961_drop__bit__nat__def,axiom,
% 12.85/7.02 ( bit_se8570568707652914677it_nat
% 12.85/7.02 = ( ^ [N4: nat,M5: nat] : ( divide_divide_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8962_int__numeral__or__not__num__neg,axiom,
% 12.85/7.02 ! [M: num,N: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.02 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8963_int__numeral__not__or__num__neg,axiom,
% 12.85/7.02 ! [M: num,N: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.85/7.02 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8964_numeral__or__not__num__eq,axiom,
% 12.85/7.02 ! [M: num,N: num] :
% 12.85/7.02 ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
% 12.85/7.02 = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8965_or__minus__numerals_I1_J,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.85/7.02 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8966_or__minus__numerals_I5_J,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 12.85/7.02 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8967_or__nat__unfold,axiom,
% 12.85/7.02 ( bit_se1412395901928357646or_nat
% 12.85/7.02 = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M5 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8968_or__minus__numerals_I7_J,axiom,
% 12.85/7.02 ! [N: num,M: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 12.85/7.02 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8969_max__enat__simps_I2_J,axiom,
% 12.85/7.02 ! [Q2: extended_enat] :
% 12.85/7.02 ( ( ord_ma741700101516333627d_enat @ Q2 @ zero_z5237406670263579293d_enat )
% 12.85/7.02 = Q2 ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8970_max__enat__simps_I3_J,axiom,
% 12.85/7.02 ! [Q2: extended_enat] :
% 12.85/7.02 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q2 )
% 12.85/7.02 = Q2 ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8971_max__Suc__Suc,axiom,
% 12.85/7.02 ! [M: nat,N: nat] :
% 12.85/7.02 ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
% 12.85/7.02 = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8972_max__0R,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( ord_max_nat @ N @ zero_zero_nat )
% 12.85/7.02 = N ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8973_max__0L,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( ord_max_nat @ zero_zero_nat @ N )
% 12.85/7.02 = N ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8974_max__nat_Oright__neutral,axiom,
% 12.85/7.02 ! [A: nat] :
% 12.85/7.02 ( ( ord_max_nat @ A @ zero_zero_nat )
% 12.85/7.02 = A ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8975_max__nat_Oneutr__eq__iff,axiom,
% 12.85/7.02 ! [A: nat,B: nat] :
% 12.85/7.02 ( ( zero_zero_nat
% 12.85/7.02 = ( ord_max_nat @ A @ B ) )
% 12.85/7.02 = ( ( A = zero_zero_nat )
% 12.85/7.02 & ( B = zero_zero_nat ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8976_max__nat_Oleft__neutral,axiom,
% 12.85/7.02 ! [A: nat] :
% 12.85/7.02 ( ( ord_max_nat @ zero_zero_nat @ A )
% 12.85/7.02 = A ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8977_max__nat_Oeq__neutr__iff,axiom,
% 12.85/7.02 ! [A: nat,B: nat] :
% 12.85/7.02 ( ( ( ord_max_nat @ A @ B )
% 12.85/7.02 = zero_zero_nat )
% 12.85/7.02 = ( ( A = zero_zero_nat )
% 12.85/7.02 & ( B = zero_zero_nat ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8978_pred__numeral__simps_I2_J,axiom,
% 12.85/7.02 ! [K: num] :
% 12.85/7.02 ( ( pred_numeral @ ( bit0 @ K ) )
% 12.85/7.02 = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8979_max__numeral__Suc,axiom,
% 12.85/7.02 ! [K: num,N: nat] :
% 12.85/7.02 ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 12.85/7.02 = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8980_max__Suc__numeral,axiom,
% 12.85/7.02 ! [N: nat,K: num] :
% 12.85/7.02 ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 12.85/7.02 = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8981_or__minus__numerals_I3_J,axiom,
% 12.85/7.02 ! [M: num,N: num] :
% 12.85/7.02 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.85/7.02 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8982_nat__mult__max__right,axiom,
% 12.85/7.02 ! [M: nat,N: nat,Q2: nat] :
% 12.85/7.02 ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 12.85/7.02 = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8983_nat__mult__max__left,axiom,
% 12.85/7.02 ! [M: nat,N: nat,Q2: nat] :
% 12.85/7.02 ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 12.85/7.02 = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8984_nat__add__max__right,axiom,
% 12.85/7.02 ! [M: nat,N: nat,Q2: nat] :
% 12.85/7.02 ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 12.85/7.02 = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8985_nat__add__max__left,axiom,
% 12.85/7.02 ! [M: nat,N: nat,Q2: nat] :
% 12.85/7.02 ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 12.85/7.02 = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8986_semiring__norm_I26_J,axiom,
% 12.85/7.02 ( ( bitM @ one )
% 12.85/7.02 = one ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8987_nat__minus__add__max,axiom,
% 12.85/7.02 ! [N: nat,M: nat] :
% 12.85/7.02 ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 12.85/7.02 = ( ord_max_nat @ N @ M ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8988_semiring__norm_I27_J,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( bitM @ ( bit0 @ N ) )
% 12.85/7.02 = ( bit1 @ ( bitM @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8989_semiring__norm_I28_J,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( bitM @ ( bit1 @ N ) )
% 12.85/7.02 = ( bit1 @ ( bit0 @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8990_or__not__num__neg_Osimps_I5_J,axiom,
% 12.85/7.02 ! [N: num,M: num] :
% 12.85/7.02 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
% 12.85/7.02 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8991_inc__BitM__eq,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( inc @ ( bitM @ N ) )
% 12.85/7.02 = ( bit0 @ N ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8992_or__not__num__neg_Osimps_I9_J,axiom,
% 12.85/7.02 ! [N: num,M: num] :
% 12.85/7.02 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
% 12.85/7.02 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8993_BitM__inc__eq,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( bitM @ ( inc @ N ) )
% 12.85/7.02 = ( bit1 @ N ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8994_eval__nat__numeral_I2_J,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 12.85/7.02 = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8995_one__plus__BitM,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( plus_plus_num @ one @ ( bitM @ N ) )
% 12.85/7.02 = ( bit0 @ N ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8996_BitM__plus__one,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( plus_plus_num @ ( bitM @ N ) @ one )
% 12.85/7.02 = ( bit0 @ N ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8997_or__not__num__neg_Osimps_I8_J,axiom,
% 12.85/7.02 ! [N: num,M: num] :
% 12.85/7.02 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
% 12.85/7.02 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8998_or__not__num__neg_Oelims,axiom,
% 12.85/7.02 ! [X: num,Xa: num,Y4: num] :
% 12.85/7.02 ( ( ( bit_or_not_num_neg @ X @ Xa )
% 12.85/7.02 = Y4 )
% 12.85/7.02 => ( ( ( X = one )
% 12.85/7.02 => ( ( Xa = one )
% 12.85/7.02 => ( Y4 != one ) ) )
% 12.85/7.02 => ( ( ( X = one )
% 12.85/7.02 => ! [M4: num] :
% 12.85/7.02 ( ( Xa
% 12.85/7.02 = ( bit0 @ M4 ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 != ( bit1 @ M4 ) ) ) )
% 12.85/7.02 => ( ( ( X = one )
% 12.85/7.02 => ! [M4: num] :
% 12.85/7.02 ( ( Xa
% 12.85/7.02 = ( bit1 @ M4 ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 != ( bit1 @ M4 ) ) ) )
% 12.85/7.02 => ( ( ? [N2: num] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( bit0 @ N2 ) )
% 12.85/7.02 => ( ( Xa = one )
% 12.85/7.02 => ( Y4
% 12.85/7.02 != ( bit0 @ one ) ) ) )
% 12.85/7.02 => ( ! [N2: num] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( bit0 @ N2 ) )
% 12.85/7.02 => ! [M4: num] :
% 12.85/7.02 ( ( Xa
% 12.85/7.02 = ( bit0 @ M4 ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
% 12.85/7.02 => ( ! [N2: num] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( bit0 @ N2 ) )
% 12.85/7.02 => ! [M4: num] :
% 12.85/7.02 ( ( Xa
% 12.85/7.02 = ( bit1 @ M4 ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 != ( bit0 @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
% 12.85/7.02 => ( ( ? [N2: num] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( bit1 @ N2 ) )
% 12.85/7.02 => ( ( Xa = one )
% 12.85/7.02 => ( Y4 != one ) ) )
% 12.85/7.02 => ( ! [N2: num] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( bit1 @ N2 ) )
% 12.85/7.02 => ! [M4: num] :
% 12.85/7.02 ( ( Xa
% 12.85/7.02 = ( bit0 @ M4 ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
% 12.85/7.02 => ~ ! [N2: num] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( bit1 @ N2 ) )
% 12.85/7.02 => ! [M4: num] :
% 12.85/7.02 ( ( Xa
% 12.85/7.02 = ( bit1 @ M4 ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_8999_root__powr__inverse,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.02 => ( ( root @ N @ X )
% 12.85/7.02 = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9000_xor__minus__numerals_I2_J,axiom,
% 12.85/7.02 ! [K: int,N: num] :
% 12.85/7.02 ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.02 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9001_xor__minus__numerals_I1_J,axiom,
% 12.85/7.02 ! [N: num,K: int] :
% 12.85/7.02 ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
% 12.85/7.02 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9002_real__root__zero,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( root @ N @ zero_zero_real )
% 12.85/7.02 = zero_zero_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9003_real__root__Suc__0,axiom,
% 12.85/7.02 ! [X: real] :
% 12.85/7.02 ( ( root @ ( suc @ zero_zero_nat ) @ X )
% 12.85/7.02 = X ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9004_root__0,axiom,
% 12.85/7.02 ! [X: real] :
% 12.85/7.02 ( ( root @ zero_zero_nat @ X )
% 12.85/7.02 = zero_zero_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9005_real__root__eq__iff,axiom,
% 12.85/7.02 ! [N: nat,X: real,Y4: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ( root @ N @ X )
% 12.85/7.02 = ( root @ N @ Y4 ) )
% 12.85/7.02 = ( X = Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9006_real__root__eq__0__iff,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ( root @ N @ X )
% 12.85/7.02 = zero_zero_real )
% 12.85/7.02 = ( X = zero_zero_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9007_real__root__less__iff,axiom,
% 12.85/7.02 ! [N: nat,X: real,Y4: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y4 ) )
% 12.85/7.02 = ( ord_less_real @ X @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9008_real__root__le__iff,axiom,
% 12.85/7.02 ! [N: nat,X: real,Y4: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y4 ) )
% 12.85/7.02 = ( ord_less_eq_real @ X @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9009_real__root__one,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( root @ N @ one_one_real )
% 12.85/7.02 = one_one_real ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9010_real__root__eq__1__iff,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ( root @ N @ X )
% 12.85/7.02 = one_one_real )
% 12.85/7.02 = ( X = one_one_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9011_real__root__gt__0__iff,axiom,
% 12.85/7.02 ! [N: nat,Y4: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y4 ) )
% 12.85/7.02 = ( ord_less_real @ zero_zero_real @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9012_real__root__lt__0__iff,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
% 12.85/7.02 = ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9013_real__root__ge__0__iff,axiom,
% 12.85/7.02 ! [N: nat,Y4: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y4 ) )
% 12.85/7.02 = ( ord_less_eq_real @ zero_zero_real @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9014_real__root__le__0__iff,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
% 12.85/7.02 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9015_real__root__gt__1__iff,axiom,
% 12.85/7.02 ! [N: nat,Y4: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y4 ) )
% 12.85/7.02 = ( ord_less_real @ one_one_real @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9016_real__root__lt__1__iff,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
% 12.85/7.02 = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9017_real__root__ge__1__iff,axiom,
% 12.85/7.02 ! [N: nat,Y4: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y4 ) )
% 12.85/7.02 = ( ord_less_eq_real @ one_one_real @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9018_real__root__le__1__iff,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
% 12.85/7.02 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9019_real__root__pow__pos2,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.02 => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 12.85/7.02 = X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9020_real__root__inverse,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ( root @ N @ ( inverse_inverse_real @ X ) )
% 12.85/7.02 = ( inverse_inverse_real @ ( root @ N @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9021_real__root__mult__exp,axiom,
% 12.85/7.02 ! [M: nat,N: nat,X: real] :
% 12.85/7.02 ( ( root @ ( times_times_nat @ M @ N ) @ X )
% 12.85/7.02 = ( root @ M @ ( root @ N @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9022_real__root__mult,axiom,
% 12.85/7.02 ! [N: nat,X: real,Y4: real] :
% 12.85/7.02 ( ( root @ N @ ( times_times_real @ X @ Y4 ) )
% 12.85/7.02 = ( times_times_real @ ( root @ N @ X ) @ ( root @ N @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9023_real__root__minus,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ( root @ N @ ( uminus_uminus_real @ X ) )
% 12.85/7.02 = ( uminus_uminus_real @ ( root @ N @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9024_real__root__commute,axiom,
% 12.85/7.02 ! [M: nat,N: nat,X: real] :
% 12.85/7.02 ( ( root @ M @ ( root @ N @ X ) )
% 12.85/7.02 = ( root @ N @ ( root @ M @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9025_real__root__divide,axiom,
% 12.85/7.02 ! [N: nat,X: real,Y4: real] :
% 12.85/7.02 ( ( root @ N @ ( divide_divide_real @ X @ Y4 ) )
% 12.85/7.02 = ( divide_divide_real @ ( root @ N @ X ) @ ( root @ N @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9026_real__root__pos__pos__le,axiom,
% 12.85/7.02 ! [X: real,N: nat] :
% 12.85/7.02 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.02 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9027_real__root__less__mono,axiom,
% 12.85/7.02 ! [N: nat,X: real,Y4: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_real @ X @ Y4 )
% 12.85/7.02 => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9028_real__root__le__mono,axiom,
% 12.85/7.02 ! [N: nat,X: real,Y4: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_eq_real @ X @ Y4 )
% 12.85/7.02 => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9029_real__root__power,axiom,
% 12.85/7.02 ! [N: nat,X: real,K: nat] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( root @ N @ ( power_power_real @ X @ K ) )
% 12.85/7.02 = ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9030_real__root__abs,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( root @ N @ ( abs_abs_real @ X ) )
% 12.85/7.02 = ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9031_sgn__root,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( sgn_sgn_real @ ( root @ N @ X ) )
% 12.85/7.02 = ( sgn_sgn_real @ X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9032_real__root__gt__zero,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.02 => ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9033_real__root__strict__decreasing,axiom,
% 12.85/7.02 ! [N: nat,N5: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_nat @ N @ N5 )
% 12.85/7.02 => ( ( ord_less_real @ one_one_real @ X )
% 12.85/7.02 => ( ord_less_real @ ( root @ N5 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9034_sqrt__def,axiom,
% 12.85/7.02 ( sqrt
% 12.85/7.02 = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9035_root__abs__power,axiom,
% 12.85/7.02 ! [N: nat,Y4: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y4 @ N ) ) )
% 12.85/7.02 = ( abs_abs_real @ Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9036_real__root__pos__pos,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.02 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9037_real__root__strict__increasing,axiom,
% 12.85/7.02 ! [N: nat,N5: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_nat @ N @ N5 )
% 12.85/7.02 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.02 => ( ( ord_less_real @ X @ one_one_real )
% 12.85/7.02 => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9038_real__root__decreasing,axiom,
% 12.85/7.02 ! [N: nat,N5: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_eq_nat @ N @ N5 )
% 12.85/7.02 => ( ( ord_less_eq_real @ one_one_real @ X )
% 12.85/7.02 => ( ord_less_eq_real @ ( root @ N5 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9039_real__root__pow__pos,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.02 => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 12.85/7.02 = X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9040_real__root__pos__unique,axiom,
% 12.85/7.02 ! [N: nat,Y4: real,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 12.85/7.02 => ( ( ( power_power_real @ Y4 @ N )
% 12.85/7.02 = X )
% 12.85/7.02 => ( ( root @ N @ X )
% 12.85/7.02 = Y4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9041_real__root__power__cancel,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.02 => ( ( root @ N @ ( power_power_real @ X @ N ) )
% 12.85/7.02 = X ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9042_odd__real__root__pow,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/7.02 => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 12.85/7.02 = X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9043_odd__real__root__unique,axiom,
% 12.85/7.02 ! [N: nat,Y4: real,X: real] :
% 12.85/7.02 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/7.02 => ( ( ( power_power_real @ Y4 @ N )
% 12.85/7.02 = X )
% 12.85/7.02 => ( ( root @ N @ X )
% 12.85/7.02 = Y4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9044_odd__real__root__power__cancel,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/7.02 => ( ( root @ N @ ( power_power_real @ X @ N ) )
% 12.85/7.02 = X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9045_real__root__increasing,axiom,
% 12.85/7.02 ! [N: nat,N5: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_eq_nat @ N @ N5 )
% 12.85/7.02 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.02 => ( ( ord_less_eq_real @ X @ one_one_real )
% 12.85/7.02 => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9046_sub__BitM__One__eq,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
% 12.85/7.02 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9047_root__sgn__power,axiom,
% 12.85/7.02 ! [N: nat,Y4: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y4 ) @ ( power_power_real @ ( abs_abs_real @ Y4 ) @ N ) ) )
% 12.85/7.02 = Y4 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9048_sgn__power__root,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
% 12.85/7.02 = X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9049_ln__root,axiom,
% 12.85/7.02 ! [N: nat,B: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_real @ zero_zero_real @ B )
% 12.85/7.02 => ( ( ln_ln_real @ ( root @ N @ B ) )
% 12.85/7.02 = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9050_log__root,axiom,
% 12.85/7.02 ! [N: nat,A: real,B: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_real @ zero_zero_real @ A )
% 12.85/7.02 => ( ( log @ B @ ( root @ N @ A ) )
% 12.85/7.02 = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9051_log__base__root,axiom,
% 12.85/7.02 ! [N: nat,B: real,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_real @ zero_zero_real @ B )
% 12.85/7.02 => ( ( log @ ( root @ N @ B ) @ X )
% 12.85/7.02 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9052_split__root,axiom,
% 12.85/7.02 ! [P: real > $o,N: nat,X: real] :
% 12.85/7.02 ( ( P @ ( root @ N @ X ) )
% 12.85/7.02 = ( ( ( N = zero_zero_nat )
% 12.85/7.02 => ( P @ zero_zero_real ) )
% 12.85/7.02 & ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ! [Y6: real] :
% 12.85/7.02 ( ( ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N ) )
% 12.85/7.02 = X )
% 12.85/7.02 => ( P @ Y6 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9053_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
% 12.85/7.02 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 12.85/7.02 ( ( vEBT_VEBT_cnt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
% 12.85/7.02 = ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList ) @ zero_zero_real ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9054_VEBT_Oinject_I1_J,axiom,
% 12.85/7.02 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
% 12.85/7.02 ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 12.85/7.02 = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
% 12.85/7.02 = ( ( X11 = Y11 )
% 12.85/7.02 & ( X12 = Y12 )
% 12.85/7.02 & ( X13 = Y13 )
% 12.85/7.02 & ( X14 = Y14 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9055_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
% 12.85/7.02 ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
% 12.85/7.02 ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9056_both__member__options__ding,axiom,
% 12.85/7.02 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
% 12.85/7.02 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
% 12.85/7.02 => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 12.85/7.02 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.02 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9057_VEBT__internal_Ospace_Osimps_I2_J,axiom,
% 12.85/7.02 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 12.85/7.02 ( ( vEBT_VEBT_space @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
% 12.85/7.02 = ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList ) @ zero_zero_nat ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9058_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
% 12.85/7.02 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 12.85/7.02 ( ( vEBT_VEBT_space2 @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
% 12.85/7.02 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList ) @ zero_zero_nat ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9059_both__member__options__def,axiom,
% 12.85/7.02 ( vEBT_V8194947554948674370ptions
% 12.85/7.02 = ( ^ [T2: vEBT_VEBT,X3: nat] :
% 12.85/7.02 ( ( vEBT_V5719532721284313246member @ T2 @ X3 )
% 12.85/7.02 | ( vEBT_VEBT_membermima @ T2 @ X3 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9060_VEBT_Oinject_I2_J,axiom,
% 12.85/7.02 ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
% 12.85/7.02 ( ( ( vEBT_Leaf @ X21 @ X222 )
% 12.85/7.02 = ( vEBT_Leaf @ Y21 @ Y22 ) )
% 12.85/7.02 = ( ( X21 = Y21 )
% 12.85/7.02 & ( X222 = Y22 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9061_VEBT__internal_Ospace_H_Ocases,axiom,
% 12.85/7.02 ! [X: vEBT_VEBT] :
% 12.85/7.02 ( ! [A4: $o,B3: $o] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( vEBT_Leaf @ A4 @ B3 ) )
% 12.85/7.02 => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9062_VEBT_Oexhaust,axiom,
% 12.85/7.02 ! [Y4: vEBT_VEBT] :
% 12.85/7.02 ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
% 12.85/7.02 ( Y4
% 12.85/7.02 != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
% 12.85/7.02 => ~ ! [X212: $o,X223: $o] :
% 12.85/7.02 ( Y4
% 12.85/7.02 != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9063_VEBT_Odistinct_I1_J,axiom,
% 12.85/7.02 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
% 12.85/7.02 ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 12.85/7.02 != ( vEBT_Leaf @ X21 @ X222 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9064_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
% 12.85/7.02 ! [Uu: $o,Uv: $o,Uw: nat] :
% 12.85/7.02 ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9065_VEBT_Osize_I4_J,axiom,
% 12.85/7.02 ! [X21: $o,X222: $o] :
% 12.85/7.02 ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 12.85/7.02 = zero_zero_nat ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9066_vebt__buildup_Osimps_I1_J,axiom,
% 12.85/7.02 ( ( vEBT_vebt_buildup @ zero_zero_nat )
% 12.85/7.02 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9067_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 12.85/7.02 ! [Uu: $o,Uv: $o,D: nat] :
% 12.85/7.02 ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
% 12.85/7.02 = ( D = one_one_nat ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9068_VEBT__internal_Ocnt_Osimps_I1_J,axiom,
% 12.85/7.02 ! [A: $o,B: $o] :
% 12.85/7.02 ( ( vEBT_VEBT_cnt @ ( vEBT_Leaf @ A @ B ) )
% 12.85/7.02 = one_one_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9069_invar__vebt_Ointros_I1_J,axiom,
% 12.85/7.02 ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9070_vebt__buildup_Osimps_I2_J,axiom,
% 12.85/7.02 ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
% 12.85/7.02 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9071_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 12.85/7.02 ! [A: $o,B: $o,X: nat] :
% 12.85/7.02 ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
% 12.85/7.02 = ( ( ( X = zero_zero_nat )
% 12.85/7.02 => A )
% 12.85/7.02 & ( ( X != zero_zero_nat )
% 12.85/7.02 => ( ( ( X = one_one_nat )
% 12.85/7.02 => B )
% 12.85/7.02 & ( X = one_one_nat ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9072_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
% 12.85/7.02 ! [A: $o,B: $o] :
% 12.85/7.02 ( ( vEBT_VEBT_space2 @ ( vEBT_Leaf @ A @ B ) )
% 12.85/7.02 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9073_VEBT__internal_Ospace_Osimps_I1_J,axiom,
% 12.85/7.02 ! [A: $o,B: $o] :
% 12.85/7.02 ( ( vEBT_VEBT_space @ ( vEBT_Leaf @ A @ B ) )
% 12.85/7.02 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9074_VEBT__internal_Ospace_H_Oelims,axiom,
% 12.85/7.02 ! [X: vEBT_VEBT,Y4: nat] :
% 12.85/7.02 ( ( ( vEBT_VEBT_space2 @ X )
% 12.85/7.02 = Y4 )
% 12.85/7.02 => ( ( ? [A4: $o,B3: $o] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( vEBT_Leaf @ A4 @ B3 ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 12.85/7.02 => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 != ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList2 ) @ zero_zero_nat ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9075_VEBT__internal_Ospace_Oelims,axiom,
% 12.85/7.02 ! [X: vEBT_VEBT,Y4: nat] :
% 12.85/7.02 ( ( ( vEBT_VEBT_space @ X )
% 12.85/7.02 = Y4 )
% 12.85/7.02 => ( ( ? [A4: $o,B3: $o] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( vEBT_Leaf @ A4 @ B3 ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
% 12.85/7.02 => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 != ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList2 ) @ zero_zero_nat ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9076_VEBT__internal_Ocnt_Oelims,axiom,
% 12.85/7.02 ! [X: vEBT_VEBT,Y4: real] :
% 12.85/7.02 ( ( ( vEBT_VEBT_cnt @ X )
% 12.85/7.02 = Y4 )
% 12.85/7.02 => ( ( ? [A4: $o,B3: $o] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( vEBT_Leaf @ A4 @ B3 ) )
% 12.85/7.02 => ( Y4 != one_one_real ) )
% 12.85/7.02 => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 != ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary2 ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList2 ) @ zero_zero_real ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9077_in__children__def,axiom,
% 12.85/7.02 ( vEBT_V5917875025757280293ildren
% 12.85/7.02 = ( ^ [N4: nat,TreeList3: list_VEBT_VEBT,X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X3 @ N4 ) ) @ ( vEBT_VEBT_low @ X3 @ N4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9078_invar__vebt_Ointros_I3_J,axiom,
% 12.85/7.02 ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 12.85/7.02 ( ! [X4: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.85/7.02 => ( vEBT_invar_vebt @ X4 @ N ) )
% 12.85/7.02 => ( ( vEBT_invar_vebt @ Summary @ M )
% 12.85/7.02 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 12.85/7.02 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.85/7.02 => ( ( M
% 12.85/7.02 = ( suc @ N ) )
% 12.85/7.02 => ( ( Deg
% 12.85/7.02 = ( plus_plus_nat @ N @ M ) )
% 12.85/7.02 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 12.85/7.02 => ( ! [X4: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.85/7.02 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
% 12.85/7.02 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9079_VEBT__internal_Ospace_Opelims,axiom,
% 12.85/7.02 ! [X: vEBT_VEBT,Y4: nat] :
% 12.85/7.02 ( ( ( vEBT_VEBT_space @ X )
% 12.85/7.02 = Y4 )
% 12.85/7.02 => ( ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ X )
% 12.85/7.02 => ( ! [A4: $o,B3: $o] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Leaf @ A4 @ B3 ) )
% 12.85/7.02 => ( ( Y4
% 12.85/7.02 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 12.85/7.02 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
% 12.85/7.02 => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
% 12.85/7.02 => ( ( Y4
% 12.85/7.02 = ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList2 ) @ zero_zero_nat ) ) )
% 12.85/7.02 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9080_invar__vebt_Ointros_I2_J,axiom,
% 12.85/7.02 ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 12.85/7.02 ( ! [X4: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.85/7.02 => ( vEBT_invar_vebt @ X4 @ N ) )
% 12.85/7.02 => ( ( vEBT_invar_vebt @ Summary @ M )
% 12.85/7.02 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 12.85/7.02 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.85/7.02 => ( ( M = N )
% 12.85/7.02 => ( ( Deg
% 12.85/7.02 = ( plus_plus_nat @ N @ M ) )
% 12.85/7.02 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 12.85/7.02 => ( ! [X4: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.85/7.02 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
% 12.85/7.02 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9081_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
% 12.85/7.02 ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
% 12.85/7.02 ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9082_VEBT__internal_Ospace_H_Opelims,axiom,
% 12.85/7.02 ! [X: vEBT_VEBT,Y4: nat] :
% 12.85/7.02 ( ( ( vEBT_VEBT_space2 @ X )
% 12.85/7.02 = Y4 )
% 12.85/7.02 => ( ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ X )
% 12.85/7.02 => ( ! [A4: $o,B3: $o] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Leaf @ A4 @ B3 ) )
% 12.85/7.02 => ( ( Y4
% 12.85/7.02 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.85/7.02 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
% 12.85/7.02 => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
% 12.85/7.02 => ( ( Y4
% 12.85/7.02 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList2 ) @ zero_zero_nat ) ) )
% 12.85/7.02 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9083_VEBT__internal_Ocnt_Opelims,axiom,
% 12.85/7.02 ! [X: vEBT_VEBT,Y4: real] :
% 12.85/7.02 ( ( ( vEBT_VEBT_cnt @ X )
% 12.85/7.02 = Y4 )
% 12.85/7.02 => ( ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ X )
% 12.85/7.02 => ( ! [A4: $o,B3: $o] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Leaf @ A4 @ B3 ) )
% 12.85/7.02 => ( ( Y4 = one_one_real )
% 12.85/7.02 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
% 12.85/7.02 => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
% 12.85/7.02 => ( ( Y4
% 12.85/7.02 = ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary2 ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList2 ) @ zero_zero_real ) ) )
% 12.85/7.02 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9084_mi__ma__2__deg,axiom,
% 12.85/7.02 ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 12.85/7.02 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 12.85/7.02 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 12.85/7.02 & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9085_both__member__options__from__complete__tree__to__child,axiom,
% 12.85/7.02 ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 12.85/7.02 => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 12.85/7.02 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.02 | ( X = Mi )
% 12.85/7.02 | ( X = Ma ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9086_both__member__options__from__chilf__to__complete__tree,axiom,
% 12.85/7.02 ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 12.85/7.02 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 12.85/7.02 => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 12.85/7.02 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.02 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9087_prod__decode__aux_Ocases,axiom,
% 12.85/7.02 ! [X: product_prod_nat_nat] :
% 12.85/7.02 ~ ! [K2: nat,M4: nat] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( product_Pair_nat_nat @ K2 @ M4 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9088_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
% 12.85/7.02 ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
% 12.85/7.02 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va2 @ Vb ) @ X )
% 12.85/7.02 = ( ( X = Mi )
% 12.85/7.02 | ( X = Ma ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9089_invar__vebt_Ointros_I4_J,axiom,
% 12.85/7.02 ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 12.85/7.02 ( ! [X4: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.85/7.02 => ( vEBT_invar_vebt @ X4 @ N ) )
% 12.85/7.02 => ( ( vEBT_invar_vebt @ Summary @ M )
% 12.85/7.02 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 12.85/7.02 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.85/7.02 => ( ( M = N )
% 12.85/7.02 => ( ( Deg
% 12.85/7.02 = ( plus_plus_nat @ N @ M ) )
% 12.85/7.02 => ( ! [I3: nat] :
% 12.85/7.02 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.85/7.02 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
% 12.85/7.02 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 12.85/7.02 => ( ( ( Mi = Ma )
% 12.85/7.02 => ! [X4: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.85/7.02 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
% 12.85/7.02 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 12.85/7.02 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 12.85/7.02 => ( ( ( Mi != Ma )
% 12.85/7.02 => ! [I3: nat] :
% 12.85/7.02 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.85/7.02 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 12.85/7.02 = I3 )
% 12.85/7.02 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 12.85/7.02 & ! [X4: nat] :
% 12.85/7.02 ( ( ( ( vEBT_VEBT_high @ X4 @ N )
% 12.85/7.02 = I3 )
% 12.85/7.02 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
% 12.85/7.02 => ( ( ord_less_nat @ Mi @ X4 )
% 12.85/7.02 & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 12.85/7.02 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9090_invar__vebt_Ointros_I5_J,axiom,
% 12.85/7.02 ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 12.85/7.02 ( ! [X4: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.85/7.02 => ( vEBT_invar_vebt @ X4 @ N ) )
% 12.85/7.02 => ( ( vEBT_invar_vebt @ Summary @ M )
% 12.85/7.02 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 12.85/7.02 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.85/7.02 => ( ( M
% 12.85/7.02 = ( suc @ N ) )
% 12.85/7.02 => ( ( Deg
% 12.85/7.02 = ( plus_plus_nat @ N @ M ) )
% 12.85/7.02 => ( ! [I3: nat] :
% 12.85/7.02 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.85/7.02 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
% 12.85/7.02 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 12.85/7.02 => ( ( ( Mi = Ma )
% 12.85/7.02 => ! [X4: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.85/7.02 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
% 12.85/7.02 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 12.85/7.02 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 12.85/7.02 => ( ( ( Mi != Ma )
% 12.85/7.02 => ! [I3: nat] :
% 12.85/7.02 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 12.85/7.02 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 12.85/7.02 = I3 )
% 12.85/7.02 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 12.85/7.02 & ! [X4: nat] :
% 12.85/7.02 ( ( ( ( vEBT_VEBT_high @ X4 @ N )
% 12.85/7.02 = I3 )
% 12.85/7.02 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
% 12.85/7.02 => ( ( ord_less_nat @ Mi @ X4 )
% 12.85/7.02 & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 12.85/7.02 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9091_invar__vebt_Ocases,axiom,
% 12.85/7.02 ! [A1: vEBT_VEBT,A22: nat] :
% 12.85/7.02 ( ( vEBT_invar_vebt @ A1 @ A22 )
% 12.85/7.02 => ( ( ? [A4: $o,B3: $o] :
% 12.85/7.02 ( A1
% 12.85/7.02 = ( vEBT_Leaf @ A4 @ B3 ) )
% 12.85/7.02 => ( A22
% 12.85/7.02 != ( suc @ zero_zero_nat ) ) )
% 12.85/7.02 => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 12.85/7.02 ( ( A1
% 12.85/7.02 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
% 12.85/7.02 => ( ( A22 = Deg2 )
% 12.85/7.02 => ( ! [X5: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_invar_vebt @ X5 @ N2 ) )
% 12.85/7.02 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 12.85/7.02 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 12.85/7.02 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 12.85/7.02 => ( ( M4 = N2 )
% 12.85/7.02 => ( ( Deg2
% 12.85/7.02 = ( plus_plus_nat @ N2 @ M4 ) )
% 12.85/7.02 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 12.85/7.02 => ~ ! [X5: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.02 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
% 12.85/7.02 => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 12.85/7.02 ( ( A1
% 12.85/7.02 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
% 12.85/7.02 => ( ( A22 = Deg2 )
% 12.85/7.02 => ( ! [X5: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_invar_vebt @ X5 @ N2 ) )
% 12.85/7.02 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 12.85/7.02 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 12.85/7.02 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 12.85/7.02 => ( ( M4
% 12.85/7.02 = ( suc @ N2 ) )
% 12.85/7.02 => ( ( Deg2
% 12.85/7.02 = ( plus_plus_nat @ N2 @ M4 ) )
% 12.85/7.02 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 12.85/7.02 => ~ ! [X5: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.02 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
% 12.85/7.02 => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 12.85/7.02 ( ( A1
% 12.85/7.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
% 12.85/7.02 => ( ( A22 = Deg2 )
% 12.85/7.02 => ( ! [X5: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_invar_vebt @ X5 @ N2 ) )
% 12.85/7.02 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 12.85/7.02 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 12.85/7.02 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 12.85/7.02 => ( ( M4 = N2 )
% 12.85/7.02 => ( ( Deg2
% 12.85/7.02 = ( plus_plus_nat @ N2 @ M4 ) )
% 12.85/7.02 => ( ! [I4: nat] :
% 12.85/7.02 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 12.85/7.02 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
% 12.85/7.02 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 12.85/7.02 => ( ( ( Mi2 = Ma2 )
% 12.85/7.02 => ! [X5: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.02 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 12.85/7.02 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 12.85/7.02 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.85/7.02 => ~ ( ( Mi2 != Ma2 )
% 12.85/7.02 => ! [I4: nat] :
% 12.85/7.02 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 12.85/7.02 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 12.85/7.02 = I4 )
% 12.85/7.02 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 12.85/7.02 & ! [X5: nat] :
% 12.85/7.02 ( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
% 12.85/7.02 = I4 )
% 12.85/7.02 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
% 12.85/7.02 => ( ( ord_less_nat @ Mi2 @ X5 )
% 12.85/7.02 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 12.85/7.02 => ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 12.85/7.02 ( ( A1
% 12.85/7.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
% 12.85/7.02 => ( ( A22 = Deg2 )
% 12.85/7.02 => ( ! [X5: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_invar_vebt @ X5 @ N2 ) )
% 12.85/7.02 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 12.85/7.02 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 12.85/7.02 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 12.85/7.02 => ( ( M4
% 12.85/7.02 = ( suc @ N2 ) )
% 12.85/7.02 => ( ( Deg2
% 12.85/7.02 = ( plus_plus_nat @ N2 @ M4 ) )
% 12.85/7.02 => ( ! [I4: nat] :
% 12.85/7.02 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 12.85/7.02 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
% 12.85/7.02 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 12.85/7.02 => ( ( ( Mi2 = Ma2 )
% 12.85/7.02 => ! [X5: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.02 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 12.85/7.02 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 12.85/7.02 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.85/7.02 => ~ ( ( Mi2 != Ma2 )
% 12.85/7.02 => ! [I4: nat] :
% 12.85/7.02 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 12.85/7.02 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 12.85/7.02 = I4 )
% 12.85/7.02 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 12.85/7.02 & ! [X5: nat] :
% 12.85/7.02 ( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
% 12.85/7.02 = I4 )
% 12.85/7.02 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
% 12.85/7.02 => ( ( ord_less_nat @ Mi2 @ X5 )
% 12.85/7.02 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9092_invar__vebt_Osimps,axiom,
% 12.85/7.02 ( vEBT_invar_vebt
% 12.85/7.02 = ( ^ [A12: vEBT_VEBT,A23: nat] :
% 12.85/7.02 ( ( ? [A3: $o,B2: $o] :
% 12.85/7.02 ( A12
% 12.85/7.02 = ( vEBT_Leaf @ A3 @ B2 ) )
% 12.85/7.02 & ( A23
% 12.85/7.02 = ( suc @ zero_zero_nat ) ) )
% 12.85/7.02 | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
% 12.85/7.02 ( ( A12
% 12.85/7.02 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList3 @ Summary3 ) )
% 12.85/7.02 & ! [X3: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.85/7.02 => ( vEBT_invar_vebt @ X3 @ N4 ) )
% 12.85/7.02 & ( vEBT_invar_vebt @ Summary3 @ N4 )
% 12.85/7.02 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 12.85/7.02 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
% 12.85/7.02 & ( A23
% 12.85/7.02 = ( plus_plus_nat @ N4 @ N4 ) )
% 12.85/7.02 & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
% 12.85/7.02 & ! [X3: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.85/7.02 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 12.85/7.02 | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
% 12.85/7.02 ( ( A12
% 12.85/7.02 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList3 @ Summary3 ) )
% 12.85/7.02 & ! [X3: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.85/7.02 => ( vEBT_invar_vebt @ X3 @ N4 ) )
% 12.85/7.02 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N4 ) )
% 12.85/7.02 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 12.85/7.02 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
% 12.85/7.02 & ( A23
% 12.85/7.02 = ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) )
% 12.85/7.02 & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
% 12.85/7.02 & ! [X3: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.85/7.02 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 12.85/7.02 | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 12.85/7.02 ( ( A12
% 12.85/7.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
% 12.85/7.02 & ! [X3: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.85/7.02 => ( vEBT_invar_vebt @ X3 @ N4 ) )
% 12.85/7.02 & ( vEBT_invar_vebt @ Summary3 @ N4 )
% 12.85/7.02 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 12.85/7.02 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
% 12.85/7.02 & ( A23
% 12.85/7.02 = ( plus_plus_nat @ N4 @ N4 ) )
% 12.85/7.02 & ! [I2: nat] :
% 12.85/7.02 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
% 12.85/7.02 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X8 ) )
% 12.85/7.02 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
% 12.85/7.02 & ( ( Mi3 = Ma3 )
% 12.85/7.02 => ! [X3: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.85/7.02 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 12.85/7.02 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 12.85/7.02 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 12.85/7.02 & ( ( Mi3 != Ma3 )
% 12.85/7.02 => ! [I2: nat] :
% 12.85/7.02 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
% 12.85/7.02 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N4 )
% 12.85/7.02 = I2 )
% 12.85/7.02 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
% 12.85/7.02 & ! [X3: nat] :
% 12.85/7.02 ( ( ( ( vEBT_VEBT_high @ X3 @ N4 )
% 12.85/7.02 = I2 )
% 12.85/7.02 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N4 ) ) )
% 12.85/7.02 => ( ( ord_less_nat @ Mi3 @ X3 )
% 12.85/7.02 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) )
% 12.85/7.02 | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 12.85/7.02 ( ( A12
% 12.85/7.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
% 12.85/7.02 & ! [X3: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.85/7.02 => ( vEBT_invar_vebt @ X3 @ N4 ) )
% 12.85/7.02 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N4 ) )
% 12.85/7.02 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 12.85/7.02 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
% 12.85/7.02 & ( A23
% 12.85/7.02 = ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) )
% 12.85/7.02 & ! [I2: nat] :
% 12.85/7.02 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
% 12.85/7.02 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X8 ) )
% 12.85/7.02 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
% 12.85/7.02 & ( ( Mi3 = Ma3 )
% 12.85/7.02 => ! [X3: vEBT_VEBT] :
% 12.85/7.02 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 12.85/7.02 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 12.85/7.02 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 12.85/7.02 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 12.85/7.02 & ( ( Mi3 != Ma3 )
% 12.85/7.02 => ! [I2: nat] :
% 12.85/7.02 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
% 12.85/7.02 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N4 )
% 12.85/7.02 = I2 )
% 12.85/7.02 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
% 12.85/7.02 & ! [X3: nat] :
% 12.85/7.02 ( ( ( ( vEBT_VEBT_high @ X3 @ N4 )
% 12.85/7.02 = I2 )
% 12.85/7.02 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N4 ) ) )
% 12.85/7.02 => ( ( ord_less_nat @ Mi3 @ X3 )
% 12.85/7.02 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9093_VEBT__internal_Ovalid_H_Ocases,axiom,
% 12.85/7.02 ! [X: produc9072475918466114483BT_nat] :
% 12.85/7.02 ( ! [Uu2: $o,Uv2: $o,D2: nat] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D2 ) )
% 12.85/7.02 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Deg3 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9094_xor__num_Ocases,axiom,
% 12.85/7.02 ! [X: product_prod_num_num] :
% 12.85/7.02 ( ( X
% 12.85/7.02 != ( product_Pair_num_num @ one @ one ) )
% 12.85/7.02 => ( ! [N2: num] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) )
% 12.85/7.02 => ( ! [N2: num] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) )
% 12.85/7.02 => ( ! [M4: num] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) )
% 12.85/7.02 => ( ! [M4: num,N2: num] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) )
% 12.85/7.02 => ( ! [M4: num,N2: num] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) )
% 12.85/7.02 => ( ! [M4: num] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) )
% 12.85/7.02 => ( ! [M4: num,N2: num] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) )
% 12.85/7.02 => ~ ! [M4: num,N2: num] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9095_VEBT__internal_Onaive__member_Ocases,axiom,
% 12.85/7.02 ! [X: produc9072475918466114483BT_nat] :
% 12.85/7.02 ( ! [A4: $o,B3: $o,X4: nat] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ X4 ) )
% 12.85/7.02 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
% 12.85/7.02 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) @ X4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9096_VEBT__internal_Omembermima_Ocases,axiom,
% 12.85/7.02 ! [X: produc9072475918466114483BT_nat] :
% 12.85/7.02 ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
% 12.85/7.02 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
% 12.85/7.02 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X4 ) )
% 12.85/7.02 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X4: nat] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ X4 ) )
% 12.85/7.02 => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X4: nat] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ X4 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9097_neg__eucl__rel__int__mult__2,axiom,
% 12.85/7.02 ! [B: int,A: int,Q2: int,R2: int] :
% 12.85/7.02 ( ( ord_less_eq_int @ B @ zero_zero_int )
% 12.85/7.02 => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.85/7.02 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9098_Divides_Oadjust__div__eq,axiom,
% 12.85/7.02 ! [Q2: int,R2: int] :
% 12.85/7.02 ( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.85/7.02 = ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9099_numeral__div__minus__numeral,axiom,
% 12.85/7.02 ! [M: num,N: num] :
% 12.85/7.02 ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.02 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9100_minus__numeral__div__numeral,axiom,
% 12.85/7.02 ! [M: num,N: num] :
% 12.85/7.02 ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.85/7.02 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9101_one__div__minus__numeral,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.02 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9102_minus__one__div__numeral,axiom,
% 12.85/7.02 ! [N: num] :
% 12.85/7.02 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 12.85/7.02 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9103_divmod__BitM__2__eq,axiom,
% 12.85/7.02 ! [M: num] :
% 12.85/7.02 ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 12.85/7.02 = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9104_unique__quotient,axiom,
% 12.85/7.02 ! [A: int,B: int,Q2: int,R2: int,Q5: int,R4: int] :
% 12.85/7.02 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.85/7.02 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R4 ) )
% 12.85/7.02 => ( Q2 = Q5 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9105_unique__remainder,axiom,
% 12.85/7.02 ! [A: int,B: int,Q2: int,R2: int,Q5: int,R4: int] :
% 12.85/7.02 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.85/7.02 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R4 ) )
% 12.85/7.02 => ( R2 = R4 ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9106_eucl__rel__int__by0,axiom,
% 12.85/7.02 ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9107_div__int__unique,axiom,
% 12.85/7.02 ! [K: int,L: int,Q2: int,R2: int] :
% 12.85/7.02 ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.85/7.02 => ( ( divide_divide_int @ K @ L )
% 12.85/7.02 = Q2 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9108_mod__int__unique,axiom,
% 12.85/7.02 ! [K: int,L: int,Q2: int,R2: int] :
% 12.85/7.02 ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.85/7.02 => ( ( modulo_modulo_int @ K @ L )
% 12.85/7.02 = R2 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9109_eucl__rel__int__dividesI,axiom,
% 12.85/7.02 ! [L: int,K: int,Q2: int] :
% 12.85/7.02 ( ( L != zero_zero_int )
% 12.85/7.02 => ( ( K
% 12.85/7.02 = ( times_times_int @ Q2 @ L ) )
% 12.85/7.02 => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9110_eucl__rel__int,axiom,
% 12.85/7.02 ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9111_divmod__int__def,axiom,
% 12.85/7.02 ( unique5052692396658037445od_int
% 12.85/7.02 = ( ^ [M5: num,N4: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N4 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N4 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9112_divmod_H__nat__def,axiom,
% 12.85/7.02 ( unique5055182867167087721od_nat
% 12.85/7.02 = ( ^ [M5: num,N4: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N4 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N4 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9113_zminus1__lemma,axiom,
% 12.85/7.02 ! [A: int,B: int,Q2: int,R2: int] :
% 12.85/7.02 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.85/7.02 => ( ( B != zero_zero_int )
% 12.85/7.02 => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( 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 @ B @ R2 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9114_eucl__rel__int__iff,axiom,
% 12.85/7.02 ! [K: int,L: int,Q2: int,R2: int] :
% 12.85/7.02 ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.85/7.02 = ( ( K
% 12.85/7.02 = ( plus_plus_int @ ( times_times_int @ L @ Q2 ) @ R2 ) )
% 12.85/7.02 & ( ( ord_less_int @ zero_zero_int @ L )
% 12.85/7.02 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 12.85/7.02 & ( ord_less_int @ R2 @ L ) ) )
% 12.85/7.02 & ( ~ ( ord_less_int @ zero_zero_int @ L )
% 12.85/7.02 => ( ( ( ord_less_int @ L @ zero_zero_int )
% 12.85/7.02 => ( ( ord_less_int @ L @ R2 )
% 12.85/7.02 & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
% 12.85/7.02 & ( ~ ( ord_less_int @ L @ zero_zero_int )
% 12.85/7.02 => ( Q2 = zero_zero_int ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9115_eucl__rel__int__remainderI,axiom,
% 12.85/7.02 ! [R2: int,L: int,K: int,Q2: int] :
% 12.85/7.02 ( ( ( sgn_sgn_int @ R2 )
% 12.85/7.02 = ( sgn_sgn_int @ L ) )
% 12.85/7.02 => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
% 12.85/7.02 => ( ( K
% 12.85/7.02 = ( plus_plus_int @ ( times_times_int @ Q2 @ L ) @ R2 ) )
% 12.85/7.02 => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R2 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9116_eucl__rel__int_Osimps,axiom,
% 12.85/7.02 ( eucl_rel_int
% 12.85/7.02 = ( ^ [A12: int,A23: int,A32: product_prod_int_int] :
% 12.85/7.02 ( ? [K3: int] :
% 12.85/7.02 ( ( A12 = K3 )
% 12.85/7.02 & ( A23 = zero_zero_int )
% 12.85/7.02 & ( A32
% 12.85/7.02 = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
% 12.85/7.02 | ? [L3: int,K3: int,Q3: int] :
% 12.85/7.02 ( ( A12 = K3 )
% 12.85/7.02 & ( A23 = L3 )
% 12.85/7.02 & ( A32
% 12.85/7.02 = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
% 12.85/7.02 & ( L3 != zero_zero_int )
% 12.85/7.02 & ( K3
% 12.85/7.02 = ( times_times_int @ Q3 @ L3 ) ) )
% 12.85/7.02 | ? [R5: int,L3: int,K3: int,Q3: int] :
% 12.85/7.02 ( ( A12 = K3 )
% 12.85/7.02 & ( A23 = L3 )
% 12.85/7.02 & ( A32
% 12.85/7.02 = ( product_Pair_int_int @ Q3 @ R5 ) )
% 12.85/7.02 & ( ( sgn_sgn_int @ R5 )
% 12.85/7.02 = ( sgn_sgn_int @ L3 ) )
% 12.85/7.02 & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L3 ) )
% 12.85/7.02 & ( K3
% 12.85/7.02 = ( plus_plus_int @ ( times_times_int @ Q3 @ L3 ) @ R5 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9117_eucl__rel__int_Ocases,axiom,
% 12.85/7.02 ! [A1: int,A22: int,A33: product_prod_int_int] :
% 12.85/7.02 ( ( eucl_rel_int @ A1 @ A22 @ A33 )
% 12.85/7.02 => ( ( ( A22 = zero_zero_int )
% 12.85/7.02 => ( A33
% 12.85/7.02 != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
% 12.85/7.02 => ( ! [Q4: int] :
% 12.85/7.02 ( ( A33
% 12.85/7.02 = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
% 12.85/7.02 => ( ( A22 != zero_zero_int )
% 12.85/7.02 => ( A1
% 12.85/7.02 != ( times_times_int @ Q4 @ A22 ) ) ) )
% 12.85/7.02 => ~ ! [R3: int,Q4: int] :
% 12.85/7.02 ( ( A33
% 12.85/7.02 = ( product_Pair_int_int @ Q4 @ R3 ) )
% 12.85/7.02 => ( ( ( sgn_sgn_int @ R3 )
% 12.85/7.02 = ( sgn_sgn_int @ A22 ) )
% 12.85/7.02 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A22 ) )
% 12.85/7.02 => ( A1
% 12.85/7.02 != ( plus_plus_int @ ( times_times_int @ Q4 @ A22 ) @ R3 ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9118_pos__eucl__rel__int__mult__2,axiom,
% 12.85/7.02 ! [B: int,A: int,Q2: int,R2: int] :
% 12.85/7.02 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 12.85/7.02 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
% 12.85/7.02 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9119_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 12.85/7.02 ! [X: vEBT_VEBT,Xa: nat] :
% 12.85/7.02 ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
% 12.85/7.02 => ( ! [Uu2: $o,Uv2: $o] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 12.85/7.02 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 12.85/7.02 => ( ! [Mi2: nat,Ma2: nat] :
% 12.85/7.02 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 12.85/7.02 => ( ( Xa = Mi2 )
% 12.85/7.02 | ( Xa = Ma2 ) ) )
% 12.85/7.02 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT] :
% 12.85/7.02 ( ? [Vc: vEBT_VEBT] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
% 12.85/7.02 => ( ( Xa = Mi2 )
% 12.85/7.02 | ( Xa = Ma2 )
% 12.85/7.02 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
% 12.85/7.02 => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT] :
% 12.85/7.02 ( ? [Vd: vEBT_VEBT] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
% 12.85/7.02 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9120_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 12.85/7.02 ! [X: vEBT_VEBT,Xa: nat,Y4: $o] :
% 12.85/7.02 ( ( ( vEBT_VEBT_membermima @ X @ Xa )
% 12.85/7.02 = Y4 )
% 12.85/7.02 => ( ( ? [Uu2: $o,Uv2: $o] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 12.85/7.02 => Y4 )
% 12.85/7.02 => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 12.85/7.02 => Y4 )
% 12.85/7.02 => ( ! [Mi2: nat,Ma2: nat] :
% 12.85/7.02 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 = ( ~ ( ( Xa = Mi2 )
% 12.85/7.02 | ( Xa = Ma2 ) ) ) ) )
% 12.85/7.02 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT] :
% 12.85/7.02 ( ? [Vc: vEBT_VEBT] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 = ( ~ ( ( Xa = Mi2 )
% 12.85/7.02 | ( Xa = Ma2 )
% 12.85/7.02 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) )
% 12.85/7.02 => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT] :
% 12.85/7.02 ( ? [Vd: vEBT_VEBT] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9121_prod__decode__aux_Osimps,axiom,
% 12.85/7.02 ( nat_prod_decode_aux
% 12.85/7.02 = ( ^ [K3: nat,M5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M5 @ K3 ) @ ( product_Pair_nat_nat @ M5 @ ( minus_minus_nat @ K3 @ M5 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M5 @ ( suc @ K3 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9122_prod__decode__aux_Oelims,axiom,
% 12.85/7.02 ! [X: nat,Xa: nat,Y4: product_prod_nat_nat] :
% 12.85/7.02 ( ( ( nat_prod_decode_aux @ X @ Xa )
% 12.85/7.02 = Y4 )
% 12.85/7.02 => ( ( ( ord_less_eq_nat @ Xa @ X )
% 12.85/7.02 => ( Y4
% 12.85/7.02 = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
% 12.85/7.02 & ( ~ ( ord_less_eq_nat @ Xa @ X )
% 12.85/7.02 => ( Y4
% 12.85/7.02 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9123_foldr__same,axiom,
% 12.85/7.02 ! [Xs: list_real,Y4: real] :
% 12.85/7.02 ( ! [X4: real,Y: real] :
% 12.85/7.02 ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
% 12.85/7.02 => ( ( member_real @ Y @ ( set_real2 @ Xs ) )
% 12.85/7.02 => ( X4 = Y ) ) )
% 12.85/7.02 => ( ! [X4: real] :
% 12.85/7.02 ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
% 12.85/7.02 => ( X4 = Y4 ) )
% 12.85/7.02 => ( ( foldr_real_real @ plus_plus_real @ Xs @ zero_zero_real )
% 12.85/7.02 = ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_real @ Xs ) ) @ Y4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9124_foldr__mono,axiom,
% 12.85/7.02 ! [Xs: list_nat,Ys: list_nat,C: nat,D: nat] :
% 12.85/7.02 ( ( ( size_size_list_nat @ Xs )
% 12.85/7.02 = ( size_size_list_nat @ Ys ) )
% 12.85/7.02 => ( ! [I3: nat] :
% 12.85/7.02 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 12.85/7.02 => ( ord_less_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) )
% 12.85/7.02 => ( ( ord_less_eq_nat @ C @ D )
% 12.85/7.02 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( foldr_nat_nat @ plus_plus_nat @ Xs @ C ) @ ( size_size_list_nat @ Ys ) ) @ ( foldr_nat_nat @ plus_plus_nat @ Ys @ D ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9125_foldr__zero,axiom,
% 12.85/7.02 ! [Xs: list_nat,D: nat] :
% 12.85/7.02 ( ! [I3: nat] :
% 12.85/7.02 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 12.85/7.02 => ( ord_less_nat @ zero_zero_nat @ ( nth_nat @ Xs @ I3 ) ) )
% 12.85/7.02 => ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( minus_minus_nat @ ( foldr_nat_nat @ plus_plus_nat @ Xs @ D ) @ D ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9126_nat__leq__as__int,axiom,
% 12.85/7.02 ( ord_less_eq_nat
% 12.85/7.02 = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9127_nat__less__as__int,axiom,
% 12.85/7.02 ( ord_less_nat
% 12.85/7.02 = ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9128_nat__plus__as__int,axiom,
% 12.85/7.02 ( plus_plus_nat
% 12.85/7.02 = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9129_nat__times__as__int,axiom,
% 12.85/7.02 ( times_times_nat
% 12.85/7.02 = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9130_nat__minus__as__int,axiom,
% 12.85/7.02 ( minus_minus_nat
% 12.85/7.02 = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9131_nat__div__as__int,axiom,
% 12.85/7.02 ( divide_divide_nat
% 12.85/7.02 = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9132_nat__mod__as__int,axiom,
% 12.85/7.02 ( modulo_modulo_nat
% 12.85/7.02 = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9133_diff__nat__eq__if,axiom,
% 12.85/7.02 ! [Z6: int,Z: int] :
% 12.85/7.02 ( ( ( ord_less_int @ Z6 @ zero_zero_int )
% 12.85/7.02 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
% 12.85/7.02 = ( nat2 @ Z ) ) )
% 12.85/7.02 & ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
% 12.85/7.02 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
% 12.85/7.02 = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z6 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9134_set__decode__def,axiom,
% 12.85/7.02 ( nat_set_decode
% 12.85/7.02 = ( ^ [X3: nat] :
% 12.85/7.02 ( collect_nat
% 12.85/7.02 @ ^ [N4: nat] :
% 12.85/7.02 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9135_vebt__buildup_Osimps_I3_J,axiom,
% 12.85/7.02 ! [Va2: nat] :
% 12.85/7.02 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 12.85/7.02 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
% 12.85/7.02 = ( 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 ) ) ) ) ) ) )
% 12.85/7.02 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 12.85/7.02 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
% 12.85/7.02 = ( 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 ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9136_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
% 12.85/7.02 ! [Va2: nat] :
% 12.85/7.02 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 12.85/7.02 => ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va2 ) ) )
% 12.85/7.02 = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 12.85/7.02 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 12.85/7.02 => ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va2 ) ) )
% 12.85/7.02 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9137_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
% 12.85/7.02 ! [Va2: nat] :
% 12.85/7.02 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 12.85/7.02 => ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va2 ) ) )
% 12.85/7.02 = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
% 12.85/7.02 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 12.85/7.02 => ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va2 ) ) )
% 12.85/7.02 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9138_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 12.85/7.02 ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
% 12.85/7.02 ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S2 ) @ X )
% 12.85/7.02 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 12.85/7.02 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9139_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
% 12.85/7.02 ! [X: nat,Y4: nat] :
% 12.85/7.02 ( ( ( vEBT_V8646137997579335489_i_l_d @ X )
% 12.85/7.02 = Y4 )
% 12.85/7.02 => ( ( ( X = zero_zero_nat )
% 12.85/7.02 => ( Y4
% 12.85/7.02 != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 12.85/7.02 => ( ( ( X
% 12.85/7.02 = ( suc @ zero_zero_nat ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 12.85/7.02 => ~ ! [Va: nat] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( suc @ ( suc @ Va ) ) )
% 12.85/7.02 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 12.85/7.02 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9140_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
% 12.85/7.02 ! [X: nat,Y4: nat] :
% 12.85/7.02 ( ( ( vEBT_V8346862874174094_d_u_p @ X )
% 12.85/7.02 = Y4 )
% 12.85/7.02 => ( ( ( X = zero_zero_nat )
% 12.85/7.02 => ( Y4
% 12.85/7.02 != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
% 12.85/7.02 => ( ( ( X
% 12.85/7.02 = ( suc @ zero_zero_nat ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
% 12.85/7.02 => ~ ! [Va: nat] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( suc @ ( suc @ Va ) ) )
% 12.85/7.02 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
% 12.85/7.02 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9141_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 12.85/7.02 ! [V: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
% 12.85/7.02 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd2 ) @ X )
% 12.85/7.02 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 12.85/7.02 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9142_vebt__buildup_Oelims,axiom,
% 12.85/7.02 ! [X: nat,Y4: vEBT_VEBT] :
% 12.85/7.02 ( ( ( vEBT_vebt_buildup @ X )
% 12.85/7.02 = Y4 )
% 12.85/7.02 => ( ( ( X = zero_zero_nat )
% 12.85/7.02 => ( Y4
% 12.85/7.02 != ( vEBT_Leaf @ $false @ $false ) ) )
% 12.85/7.02 => ( ( ( X
% 12.85/7.02 = ( suc @ zero_zero_nat ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 != ( vEBT_Leaf @ $false @ $false ) ) )
% 12.85/7.02 => ~ ! [Va: nat] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( suc @ ( suc @ Va ) ) )
% 12.85/7.02 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 = ( 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 ) ) ) ) ) ) )
% 12.85/7.02 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9143_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 12.85/7.02 ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X: nat] :
% 12.85/7.02 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc2 ) @ X )
% 12.85/7.02 = ( ( X = Mi )
% 12.85/7.02 | ( X = Ma )
% 12.85/7.02 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 12.85/7.02 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9144_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 12.85/7.02 ! [X: vEBT_VEBT,Xa: nat,Y4: $o] :
% 12.85/7.02 ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 12.85/7.02 = Y4 )
% 12.85/7.02 => ( ! [A4: $o,B3: $o] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Leaf @ A4 @ B3 ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 = ( ~ ( ( ( Xa = zero_zero_nat )
% 12.85/7.02 => A4 )
% 12.85/7.02 & ( ( Xa != zero_zero_nat )
% 12.85/7.02 => ( ( ( Xa = one_one_nat )
% 12.85/7.02 => B3 )
% 12.85/7.02 & ( Xa = one_one_nat ) ) ) ) ) ) )
% 12.85/7.02 => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 12.85/7.02 => Y4 )
% 12.85/7.02 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT] :
% 12.85/7.02 ( ? [S3: vEBT_VEBT] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
% 12.85/7.02 => ( Y4
% 12.85/7.02 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9145_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 12.85/7.02 ! [X: vEBT_VEBT,Xa: nat] :
% 12.85/7.02 ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 12.85/7.02 => ( ! [A4: $o,B3: $o] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Leaf @ A4 @ B3 ) )
% 12.85/7.02 => ~ ( ( ( Xa = zero_zero_nat )
% 12.85/7.02 => A4 )
% 12.85/7.02 & ( ( Xa != zero_zero_nat )
% 12.85/7.02 => ( ( ( Xa = one_one_nat )
% 12.85/7.02 => B3 )
% 12.85/7.02 & ( Xa = one_one_nat ) ) ) ) )
% 12.85/7.02 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT] :
% 12.85/7.02 ( ? [S3: vEBT_VEBT] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
% 12.85/7.02 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9146_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 12.85/7.02 ! [X: vEBT_VEBT,Xa: nat] :
% 12.85/7.02 ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
% 12.85/7.02 => ( ! [A4: $o,B3: $o] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Leaf @ A4 @ B3 ) )
% 12.85/7.02 => ( ( ( Xa = zero_zero_nat )
% 12.85/7.02 => A4 )
% 12.85/7.02 & ( ( Xa != zero_zero_nat )
% 12.85/7.02 => ( ( ( Xa = one_one_nat )
% 12.85/7.02 => B3 )
% 12.85/7.02 & ( Xa = one_one_nat ) ) ) ) )
% 12.85/7.02 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 12.85/7.02 ( X
% 12.85/7.02 != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 12.85/7.02 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT] :
% 12.85/7.02 ( ? [S3: vEBT_VEBT] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
% 12.85/7.02 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9147_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 12.85/7.02 ! [X: vEBT_VEBT,Xa: nat] :
% 12.85/7.02 ( ( vEBT_VEBT_membermima @ X @ Xa )
% 12.85/7.02 => ( ! [Mi2: nat,Ma2: nat] :
% 12.85/7.02 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 12.85/7.02 => ~ ( ( Xa = Mi2 )
% 12.85/7.02 | ( Xa = Ma2 ) ) )
% 12.85/7.02 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT] :
% 12.85/7.02 ( ? [Vc: vEBT_VEBT] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
% 12.85/7.02 => ~ ( ( Xa = Mi2 )
% 12.85/7.02 | ( Xa = Ma2 )
% 12.85/7.02 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
% 12.85/7.02 => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT] :
% 12.85/7.02 ( ? [Vd: vEBT_VEBT] :
% 12.85/7.02 ( X
% 12.85/7.02 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
% 12.85/7.02 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9148_monoseq__arctan__series,axiom,
% 12.85/7.02 ! [X: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 12.85/7.02 => ( topolo6980174941875973593q_real
% 12.85/7.02 @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9149_ln__series,axiom,
% 12.85/7.02 ! [X: real] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.02 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.85/7.02 => ( ( ln_ln_real @ X )
% 12.85/7.02 = ( suminf_real
% 12.85/7.02 @ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N4 ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9150_monoseq__realpow,axiom,
% 12.85/7.02 ! [X: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.02 => ( ( ord_less_eq_real @ X @ one_one_real )
% 12.85/7.02 => ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9151_pi__series,axiom,
% 12.85/7.02 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.85/7.02 = ( suminf_real
% 12.85/7.02 @ ^ [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 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9152_arctan__series,axiom,
% 12.85/7.02 ! [X: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 12.85/7.02 => ( ( arctan @ X )
% 12.85/7.02 = ( suminf_real
% 12.85/7.02 @ ^ [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 @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9153_bij__betw__nth__root__unity,axiom,
% 12.85/7.02 ! [C: complex,N: nat] :
% 12.85/7.02 ( ( C != zero_zero_complex )
% 12.85/7.02 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 12.85/7.02 @ ( collect_complex
% 12.85/7.02 @ ^ [Z5: complex] :
% 12.85/7.02 ( ( power_power_complex @ Z5 @ N )
% 12.85/7.02 = one_one_complex ) )
% 12.85/7.02 @ ( collect_complex
% 12.85/7.02 @ ^ [Z5: complex] :
% 12.85/7.02 ( ( power_power_complex @ Z5 @ N )
% 12.85/7.02 = C ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9154_summable__arctan__series,axiom,
% 12.85/7.02 ! [X: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 12.85/7.02 => ( summable_real
% 12.85/7.02 @ ^ [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 @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9155_summable__rabs__comparison__test,axiom,
% 12.85/7.02 ! [F: nat > real,G: nat > real] :
% 12.85/7.02 ( ? [N7: nat] :
% 12.85/7.02 ! [N2: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ N7 @ N2 )
% 12.85/7.02 => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
% 12.85/7.02 => ( ( summable_real @ G )
% 12.85/7.02 => ( summable_real
% 12.85/7.02 @ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9156_summable__rabs,axiom,
% 12.85/7.02 ! [F: nat > real] :
% 12.85/7.02 ( ( summable_real
% 12.85/7.02 @ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) )
% 12.85/7.02 => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
% 12.85/7.02 @ ( suminf_real
% 12.85/7.02 @ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9157_summable__power__series,axiom,
% 12.85/7.02 ! [F: nat > real,Z: real] :
% 12.85/7.02 ( ! [I3: nat] : ( ord_less_eq_real @ ( F @ I3 ) @ one_one_real )
% 12.85/7.02 => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 12.85/7.02 => ( ( ord_less_eq_real @ zero_zero_real @ Z )
% 12.85/7.02 => ( ( ord_less_real @ Z @ one_one_real )
% 12.85/7.02 => ( summable_real
% 12.85/7.02 @ ^ [I2: nat] : ( times_times_real @ ( F @ I2 ) @ ( power_power_real @ Z @ I2 ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9158_sin__paired,axiom,
% 12.85/7.02 ! [X: real] :
% 12.85/7.02 ( sums_real
% 12.85/7.02 @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
% 12.85/7.02 @ ( sin_real @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9159_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 12.85/7.02 ! [X: vEBT_VEBT,Xa: nat] :
% 12.85/7.02 ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
% 12.85/7.02 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 12.85/7.02 => ( ! [Uu2: $o,Uv2: $o] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 12.85/7.02 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) )
% 12.85/7.02 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 12.85/7.02 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) )
% 12.85/7.02 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 12.85/7.02 => ( ( 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 ) )
% 12.85/7.02 => ( ( Xa = Mi2 )
% 12.85/7.02 | ( Xa = Ma2 ) ) ) )
% 12.85/7.02 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
% 12.85/7.02 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ Xa ) )
% 12.85/7.02 => ( ( Xa = Mi2 )
% 12.85/7.02 | ( Xa = Ma2 )
% 12.85/7.02 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
% 12.85/7.02 => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
% 12.85/7.02 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ Xa ) )
% 12.85/7.02 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9160_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 12.85/7.02 ! [X: vEBT_VEBT,Xa: nat,Y4: $o] :
% 12.85/7.02 ( ( ( vEBT_VEBT_membermima @ X @ Xa )
% 12.85/7.02 = Y4 )
% 12.85/7.02 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 12.85/7.02 => ( ! [Uu2: $o,Uv2: $o] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 12.85/7.02 => ( ~ Y4
% 12.85/7.02 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
% 12.85/7.02 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 12.85/7.02 => ( ~ Y4
% 12.85/7.02 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) ) )
% 12.85/7.02 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 12.85/7.02 => ( ( Y4
% 12.85/7.02 = ( ( Xa = Mi2 )
% 12.85/7.02 | ( Xa = Ma2 ) ) )
% 12.85/7.02 => ~ ( 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 ) ) ) )
% 12.85/7.02 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
% 12.85/7.02 => ( ( Y4
% 12.85/7.02 = ( ( Xa = Mi2 )
% 12.85/7.02 | ( Xa = Ma2 )
% 12.85/7.02 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
% 12.85/7.02 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ Xa ) ) ) )
% 12.85/7.02 => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
% 12.85/7.02 => ( ( Y4
% 12.85/7.02 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
% 12.85/7.02 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9161_power__half__series,axiom,
% 12.85/7.02 ( sums_real
% 12.85/7.02 @ ^ [N4: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N4 ) )
% 12.85/7.02 @ one_one_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9162_sums__if_H,axiom,
% 12.85/7.02 ! [G: nat > real,X: real] :
% 12.85/7.02 ( ( sums_real @ G @ X )
% 12.85/7.02 => ( sums_real
% 12.85/7.02 @ ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.02 @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9163_sums__if,axiom,
% 12.85/7.02 ! [G: nat > real,X: real,F: nat > real,Y4: real] :
% 12.85/7.02 ( ( sums_real @ G @ X )
% 12.85/7.02 => ( ( sums_real @ F @ Y4 )
% 12.85/7.02 => ( sums_real
% 12.85/7.02 @ ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( F @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.02 @ ( plus_plus_real @ X @ Y4 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9164_cos__paired,axiom,
% 12.85/7.02 ! [X: real] :
% 12.85/7.02 ( sums_real
% 12.85/7.02 @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
% 12.85/7.02 @ ( cos_real @ X ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9165_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 12.85/7.02 ! [X: vEBT_VEBT,Xa: nat] :
% 12.85/7.02 ( ( vEBT_VEBT_membermima @ X @ Xa )
% 12.85/7.02 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 12.85/7.02 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 12.85/7.02 => ( ( 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 ) )
% 12.85/7.02 => ~ ( ( Xa = Mi2 )
% 12.85/7.02 | ( Xa = Ma2 ) ) ) )
% 12.85/7.02 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
% 12.85/7.02 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ Xa ) )
% 12.85/7.02 => ~ ( ( Xa = Mi2 )
% 12.85/7.02 | ( Xa = Ma2 )
% 12.85/7.02 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
% 12.85/7.02 => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
% 12.85/7.02 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ Xa ) )
% 12.85/7.02 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9166_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 12.85/7.02 ! [X: vEBT_VEBT,Xa: nat] :
% 12.85/7.02 ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
% 12.85/7.02 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 12.85/7.02 => ( ! [A4: $o,B3: $o] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Leaf @ A4 @ B3 ) )
% 12.85/7.02 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) )
% 12.85/7.02 => ( ( ( Xa = zero_zero_nat )
% 12.85/7.02 => A4 )
% 12.85/7.02 & ( ( Xa != zero_zero_nat )
% 12.85/7.02 => ( ( ( Xa = one_one_nat )
% 12.85/7.02 => B3 )
% 12.85/7.02 & ( Xa = one_one_nat ) ) ) ) ) )
% 12.85/7.02 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 12.85/7.02 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) )
% 12.85/7.02 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
% 12.85/7.02 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) @ Xa ) )
% 12.85/7.02 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9167_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 12.85/7.02 ! [X: vEBT_VEBT,Xa: nat] :
% 12.85/7.02 ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 12.85/7.02 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 12.85/7.02 => ( ! [A4: $o,B3: $o] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Leaf @ A4 @ B3 ) )
% 12.85/7.02 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) )
% 12.85/7.02 => ~ ( ( ( Xa = zero_zero_nat )
% 12.85/7.02 => A4 )
% 12.85/7.02 & ( ( Xa != zero_zero_nat )
% 12.85/7.02 => ( ( ( Xa = one_one_nat )
% 12.85/7.02 => B3 )
% 12.85/7.02 & ( Xa = one_one_nat ) ) ) ) ) )
% 12.85/7.02 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
% 12.85/7.02 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) @ Xa ) )
% 12.85/7.02 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9168_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 12.85/7.02 ! [X: vEBT_VEBT,Xa: nat,Y4: $o] :
% 12.85/7.02 ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 12.85/7.02 = Y4 )
% 12.85/7.02 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 12.85/7.02 => ( ! [A4: $o,B3: $o] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Leaf @ A4 @ B3 ) )
% 12.85/7.02 => ( ( Y4
% 12.85/7.02 = ( ( ( Xa = zero_zero_nat )
% 12.85/7.02 => A4 )
% 12.85/7.02 & ( ( Xa != zero_zero_nat )
% 12.85/7.02 => ( ( ( Xa = one_one_nat )
% 12.85/7.02 => B3 )
% 12.85/7.02 & ( Xa = one_one_nat ) ) ) ) )
% 12.85/7.02 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
% 12.85/7.02 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 12.85/7.02 => ( ~ Y4
% 12.85/7.02 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) ) )
% 12.85/7.02 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 12.85/7.02 ( ( X
% 12.85/7.02 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
% 12.85/7.02 => ( ( Y4
% 12.85/7.02 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 12.85/7.02 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 12.85/7.02 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
% 12.85/7.02 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) @ Xa ) ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9169_Arg__def,axiom,
% 12.85/7.02 ( arg
% 12.85/7.02 = ( ^ [Z5: complex] :
% 12.85/7.02 ( if_real @ ( Z5 = zero_zero_complex ) @ zero_zero_real
% 12.85/7.02 @ ( fChoice_real
% 12.85/7.02 @ ^ [A3: real] :
% 12.85/7.02 ( ( ( sgn_sgn_complex @ Z5 )
% 12.85/7.02 = ( cis @ A3 ) )
% 12.85/7.02 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A3 )
% 12.85/7.02 & ( ord_less_eq_real @ A3 @ pi ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9170_set__vebt__def,axiom,
% 12.85/7.02 ( vEBT_set_vebt
% 12.85/7.02 = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9171_sum__choose__lower,axiom,
% 12.85/7.02 ! [R2: nat,N: nat] :
% 12.85/7.02 ( ( groups3542108847815614940at_nat
% 12.85/7.02 @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K3 ) @ K3 )
% 12.85/7.02 @ ( set_ord_atMost_nat @ N ) )
% 12.85/7.02 = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N ) ) @ N ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9172_sum__choose__upper,axiom,
% 12.85/7.02 ! [M: nat,N: nat] :
% 12.85/7.02 ( ( groups3542108847815614940at_nat
% 12.85/7.02 @ ^ [K3: nat] : ( binomial @ K3 @ M )
% 12.85/7.02 @ ( set_ord_atMost_nat @ N ) )
% 12.85/7.02 = ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9173_sum__choose__diagonal,axiom,
% 12.85/7.02 ! [M: nat,N: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ M @ N )
% 12.85/7.02 => ( ( groups3542108847815614940at_nat
% 12.85/7.02 @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
% 12.85/7.02 @ ( set_ord_atMost_nat @ M ) )
% 12.85/7.02 = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9174_vandermonde,axiom,
% 12.85/7.02 ! [M: nat,N: nat,R2: nat] :
% 12.85/7.02 ( ( groups3542108847815614940at_nat
% 12.85/7.02 @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus_nat @ R2 @ K3 ) ) )
% 12.85/7.02 @ ( set_ord_atMost_nat @ R2 ) )
% 12.85/7.02 = ( binomial @ ( plus_plus_nat @ M @ N ) @ R2 ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9175_choose__row__sum,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
% 12.85/7.02 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9176_binomial,axiom,
% 12.85/7.02 ! [A: nat,B: nat,N: nat] :
% 12.85/7.02 ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
% 12.85/7.02 = ( groups3542108847815614940at_nat
% 12.85/7.02 @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
% 12.85/7.02 @ ( set_ord_atMost_nat @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9177_polynomial__product__nat,axiom,
% 12.85/7.02 ! [M: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
% 12.85/7.02 ( ! [I3: nat] :
% 12.85/7.02 ( ( ord_less_nat @ M @ I3 )
% 12.85/7.02 => ( ( A @ I3 )
% 12.85/7.02 = zero_zero_nat ) )
% 12.85/7.02 => ( ! [J2: nat] :
% 12.85/7.02 ( ( ord_less_nat @ N @ J2 )
% 12.85/7.02 => ( ( B @ J2 )
% 12.85/7.02 = zero_zero_nat ) )
% 12.85/7.02 => ( ( times_times_nat
% 12.85/7.02 @ ( groups3542108847815614940at_nat
% 12.85/7.02 @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( power_power_nat @ X @ I2 ) )
% 12.85/7.02 @ ( set_ord_atMost_nat @ M ) )
% 12.85/7.02 @ ( groups3542108847815614940at_nat
% 12.85/7.02 @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
% 12.85/7.02 @ ( set_ord_atMost_nat @ N ) ) )
% 12.85/7.02 = ( groups3542108847815614940at_nat
% 12.85/7.02 @ ^ [R5: nat] :
% 12.85/7.02 ( times_times_nat
% 12.85/7.02 @ ( groups3542108847815614940at_nat
% 12.85/7.02 @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 12.85/7.02 @ ( set_ord_atMost_nat @ R5 ) )
% 12.85/7.02 @ ( power_power_nat @ X @ R5 ) )
% 12.85/7.02 @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9178_choose__square__sum,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( groups3542108847815614940at_nat
% 12.85/7.02 @ ^ [K3: nat] : ( power_power_nat @ ( binomial @ N @ K3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.02 @ ( set_ord_atMost_nat @ N ) )
% 12.85/7.02 = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9179_binomial__r__part__sum,axiom,
% 12.85/7.02 ! [M: nat] :
% 12.85/7.02 ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
% 12.85/7.02 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9180_choose__linear__sum,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( groups3542108847815614940at_nat
% 12.85/7.02 @ ^ [I2: nat] : ( times_times_nat @ I2 @ ( binomial @ N @ I2 ) )
% 12.85/7.02 @ ( set_ord_atMost_nat @ N ) )
% 12.85/7.02 = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9181_mask__eq__sum__exp__nat,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
% 12.85/7.02 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 12.85/7.02 @ ( collect_nat
% 12.85/7.02 @ ^ [Q3: nat] : ( ord_less_nat @ Q3 @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9182_bounded__Max__nat,axiom,
% 12.85/7.02 ! [P: nat > $o,X: nat,M10: nat] :
% 12.85/7.02 ( ( P @ X )
% 12.85/7.02 => ( ! [X4: nat] :
% 12.85/7.02 ( ( P @ X4 )
% 12.85/7.02 => ( ord_less_eq_nat @ X4 @ M10 ) )
% 12.85/7.02 => ~ ! [M4: nat] :
% 12.85/7.02 ( ( P @ M4 )
% 12.85/7.02 => ~ ! [X5: nat] :
% 12.85/7.02 ( ( P @ X5 )
% 12.85/7.02 => ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9183_sum__nth__roots,axiom,
% 12.85/7.02 ! [N: nat,C: complex] :
% 12.85/7.02 ( ( ord_less_nat @ one_one_nat @ N )
% 12.85/7.02 => ( ( groups7754918857620584856omplex
% 12.85/7.02 @ ^ [X3: complex] : X3
% 12.85/7.02 @ ( collect_complex
% 12.85/7.02 @ ^ [Z5: complex] :
% 12.85/7.02 ( ( power_power_complex @ Z5 @ N )
% 12.85/7.02 = C ) ) )
% 12.85/7.02 = zero_zero_complex ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9184_sum__roots__unity,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( ord_less_nat @ one_one_nat @ N )
% 12.85/7.02 => ( ( groups7754918857620584856omplex
% 12.85/7.02 @ ^ [X3: complex] : X3
% 12.85/7.02 @ ( collect_complex
% 12.85/7.02 @ ^ [Z5: complex] :
% 12.85/7.02 ( ( power_power_complex @ Z5 @ N )
% 12.85/7.02 = one_one_complex ) ) )
% 12.85/7.02 = zero_zero_complex ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9185_Maclaurin__minus__cos__expansion,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_real @ X @ zero_zero_real )
% 12.85/7.02 => ? [T3: real] :
% 12.85/7.02 ( ( ord_less_real @ X @ T3 )
% 12.85/7.02 & ( ord_less_real @ T3 @ zero_zero_real )
% 12.85/7.02 & ( ( cos_real @ X )
% 12.85/7.02 = ( plus_plus_real
% 12.85/7.02 @ ( groups6591440286371151544t_real
% 12.85/7.02 @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
% 12.85/7.02 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.02 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9186_Maclaurin__cos__expansion2,axiom,
% 12.85/7.02 ! [X: real,N: nat] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.02 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ? [T3: real] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ T3 )
% 12.85/7.02 & ( ord_less_real @ T3 @ X )
% 12.85/7.02 & ( ( cos_real @ X )
% 12.85/7.02 = ( plus_plus_real
% 12.85/7.02 @ ( groups6591440286371151544t_real
% 12.85/7.02 @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
% 12.85/7.02 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.02 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9187_Maclaurin__sin__expansion3,axiom,
% 12.85/7.02 ! [N: nat,X: real] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.02 => ? [T3: real] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ T3 )
% 12.85/7.02 & ( ord_less_real @ T3 @ X )
% 12.85/7.02 & ( ( sin_real @ X )
% 12.85/7.02 = ( plus_plus_real
% 12.85/7.02 @ ( groups6591440286371151544t_real
% 12.85/7.02 @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
% 12.85/7.02 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.02 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9188_Maclaurin__sin__expansion4,axiom,
% 12.85/7.02 ! [X: real,N: nat] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.02 => ? [T3: real] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ T3 )
% 12.85/7.02 & ( ord_less_eq_real @ T3 @ X )
% 12.85/7.02 & ( ( sin_real @ X )
% 12.85/7.02 = ( plus_plus_real
% 12.85/7.02 @ ( groups6591440286371151544t_real
% 12.85/7.02 @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
% 12.85/7.02 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.02 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9189_sumr__cos__zero__one,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( groups6591440286371151544t_real
% 12.85/7.02 @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ zero_zero_real @ M5 ) )
% 12.85/7.02 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 12.85/7.02 = one_one_real ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9190_lessThan__Suc__atMost,axiom,
% 12.85/7.02 ! [K: nat] :
% 12.85/7.02 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 12.85/7.02 = ( set_ord_atMost_nat @ K ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9191_Maclaurin__lemma,axiom,
% 12.85/7.02 ! [H: real,F: real > real,J: nat > real,N: nat] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ H )
% 12.85/7.02 => ? [B8: real] :
% 12.85/7.02 ( ( F @ H )
% 12.85/7.02 = ( plus_plus_real
% 12.85/7.02 @ ( groups6591440286371151544t_real
% 12.85/7.02 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H @ M5 ) )
% 12.85/7.02 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.02 @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9192_sum__split__even__odd,axiom,
% 12.85/7.02 ! [F: nat > real,G: nat > real,N: nat] :
% 12.85/7.02 ( ( groups6591440286371151544t_real
% 12.85/7.02 @ ^ [I2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ ( F @ I2 ) @ ( G @ I2 ) )
% 12.85/7.02 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.85/7.02 = ( plus_plus_real
% 12.85/7.02 @ ( groups6591440286371151544t_real
% 12.85/7.02 @ ^ [I2: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) )
% 12.85/7.02 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.02 @ ( groups6591440286371151544t_real
% 12.85/7.02 @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ one_one_nat ) )
% 12.85/7.02 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9193_Maclaurin__exp__le,axiom,
% 12.85/7.02 ! [X: real,N: nat] :
% 12.85/7.02 ? [T3: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 12.85/7.02 & ( ( exp_real @ X )
% 12.85/7.02 = ( plus_plus_real
% 12.85/7.02 @ ( groups6591440286371151544t_real
% 12.85/7.02 @ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
% 12.85/7.02 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.02 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9194_Maclaurin__sin__bound,axiom,
% 12.85/7.02 ! [X: real,N: nat] :
% 12.85/7.02 ( ord_less_eq_real
% 12.85/7.02 @ ( abs_abs_real
% 12.85/7.02 @ ( minus_minus_real @ ( sin_real @ X )
% 12.85/7.02 @ ( groups6591440286371151544t_real
% 12.85/7.02 @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
% 12.85/7.02 @ ( set_ord_lessThan_nat @ N ) ) ) )
% 12.85/7.02 @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9195_sum__pos__lt__pair,axiom,
% 12.85/7.02 ! [F: nat > real,K: nat] :
% 12.85/7.02 ( ( summable_real @ F )
% 12.85/7.02 => ( ! [D2: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D2 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D2 ) @ one_one_nat ) ) ) ) )
% 12.85/7.02 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9196_Maclaurin__exp__lt,axiom,
% 12.85/7.02 ! [X: real,N: nat] :
% 12.85/7.02 ( ( X != zero_zero_real )
% 12.85/7.02 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ? [T3: real] :
% 12.85/7.02 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
% 12.85/7.02 & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 12.85/7.02 & ( ( exp_real @ X )
% 12.85/7.02 = ( plus_plus_real
% 12.85/7.02 @ ( groups6591440286371151544t_real
% 12.85/7.02 @ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
% 12.85/7.02 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.02 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9197_Maclaurin__sin__expansion,axiom,
% 12.85/7.02 ! [X: real,N: nat] :
% 12.85/7.02 ? [T3: real] :
% 12.85/7.02 ( ( sin_real @ X )
% 12.85/7.02 = ( plus_plus_real
% 12.85/7.02 @ ( groups6591440286371151544t_real
% 12.85/7.02 @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
% 12.85/7.02 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.02 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9198_Maclaurin__sin__expansion2,axiom,
% 12.85/7.02 ! [X: real,N: nat] :
% 12.85/7.02 ? [T3: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 12.85/7.02 & ( ( sin_real @ X )
% 12.85/7.02 = ( plus_plus_real
% 12.85/7.02 @ ( groups6591440286371151544t_real
% 12.85/7.02 @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
% 12.85/7.02 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.02 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9199_Maclaurin__cos__expansion,axiom,
% 12.85/7.02 ! [X: real,N: nat] :
% 12.85/7.02 ? [T3: real] :
% 12.85/7.02 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 12.85/7.02 & ( ( cos_real @ X )
% 12.85/7.02 = ( plus_plus_real
% 12.85/7.02 @ ( groups6591440286371151544t_real
% 12.85/7.02 @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
% 12.85/7.02 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.02 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9200_bij__betw__roots__unity,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.02 => ( bij_betw_nat_complex
% 12.85/7.02 @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 12.85/7.02 @ ( set_ord_lessThan_nat @ N )
% 12.85/7.02 @ ( collect_complex
% 12.85/7.02 @ ^ [Z5: complex] :
% 12.85/7.02 ( ( power_power_complex @ Z5 @ N )
% 12.85/7.02 = one_one_complex ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9201_all__nat__less,axiom,
% 12.85/7.02 ! [N: nat,P: nat > $o] :
% 12.85/7.02 ( ( ! [M5: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ M5 @ N )
% 12.85/7.02 => ( P @ M5 ) ) )
% 12.85/7.02 = ( ! [X3: nat] :
% 12.85/7.02 ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 12.85/7.02 => ( P @ X3 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9202_ex__nat__less,axiom,
% 12.85/7.02 ! [N: nat,P: nat > $o] :
% 12.85/7.02 ( ( ? [M5: nat] :
% 12.85/7.02 ( ( ord_less_eq_nat @ M5 @ N )
% 12.85/7.02 & ( P @ M5 ) ) )
% 12.85/7.02 = ( ? [X3: nat] :
% 12.85/7.02 ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 12.85/7.02 & ( P @ X3 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9203_atMost__atLeast0,axiom,
% 12.85/7.02 ( set_ord_atMost_nat
% 12.85/7.02 = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9204_gauss__sum__nat,axiom,
% 12.85/7.02 ! [N: nat] :
% 12.85/7.02 ( ( groups3542108847815614940at_nat
% 12.85/7.02 @ ^ [X3: nat] : X3
% 12.85/7.02 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 12.85/7.02 = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9205_arith__series__nat,axiom,
% 12.85/7.02 ! [A: nat,D: nat,N: nat] :
% 12.85/7.02 ( ( groups3542108847815614940at_nat
% 12.85/7.02 @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I2 @ D ) )
% 12.85/7.02 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 12.85/7.02 = ( 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 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9206_Sum__Icc__nat,axiom,
% 12.85/7.02 ! [M: nat,N: nat] :
% 12.85/7.02 ( ( groups3542108847815614940at_nat
% 12.85/7.02 @ ^ [X3: nat] : X3
% 12.85/7.02 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 12.85/7.02 = ( 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 ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9207_bset_I1_J,axiom,
% 12.85/7.02 ! [D3: int,B4: set_int,P: int > $o,Q: int > $o] :
% 12.85/7.02 ( ! [X4: int] :
% 12.85/7.02 ( ! [Xa2: int] :
% 12.85/7.02 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.02 => ! [Xb: int] :
% 12.85/7.02 ( ( member_int @ Xb @ B4 )
% 12.85/7.02 => ( X4
% 12.85/7.02 != ( plus_plus_int @ Xb @ Xa2 ) ) ) )
% 12.85/7.02 => ( ( P @ X4 )
% 12.85/7.02 => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 12.85/7.02 => ( ! [X4: int] :
% 12.85/7.02 ( ! [Xa2: int] :
% 12.85/7.02 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.02 => ! [Xb: int] :
% 12.85/7.02 ( ( member_int @ Xb @ B4 )
% 12.85/7.02 => ( X4
% 12.85/7.02 != ( plus_plus_int @ Xb @ Xa2 ) ) ) )
% 12.85/7.02 => ( ( Q @ X4 )
% 12.85/7.02 => ( Q @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 12.85/7.02 => ! [X5: int] :
% 12.85/7.02 ( ! [Xa3: int] :
% 12.85/7.02 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.02 => ! [Xb2: int] :
% 12.85/7.02 ( ( member_int @ Xb2 @ B4 )
% 12.85/7.02 => ( X5
% 12.85/7.02 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.02 => ( ( ( P @ X5 )
% 12.85/7.02 & ( Q @ X5 ) )
% 12.85/7.02 => ( ( P @ ( minus_minus_int @ X5 @ D3 ) )
% 12.85/7.02 & ( Q @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9208_bset_I2_J,axiom,
% 12.85/7.02 ! [D3: int,B4: set_int,P: int > $o,Q: int > $o] :
% 12.85/7.02 ( ! [X4: int] :
% 12.85/7.02 ( ! [Xa2: int] :
% 12.85/7.02 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.02 => ! [Xb: int] :
% 12.85/7.02 ( ( member_int @ Xb @ B4 )
% 12.85/7.02 => ( X4
% 12.85/7.02 != ( plus_plus_int @ Xb @ Xa2 ) ) ) )
% 12.85/7.02 => ( ( P @ X4 )
% 12.85/7.02 => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 12.85/7.02 => ( ! [X4: int] :
% 12.85/7.02 ( ! [Xa2: int] :
% 12.85/7.02 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.02 => ! [Xb: int] :
% 12.85/7.02 ( ( member_int @ Xb @ B4 )
% 12.85/7.02 => ( X4
% 12.85/7.02 != ( plus_plus_int @ Xb @ Xa2 ) ) ) )
% 12.85/7.02 => ( ( Q @ X4 )
% 12.85/7.02 => ( Q @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 12.85/7.02 => ! [X5: int] :
% 12.85/7.02 ( ! [Xa3: int] :
% 12.85/7.02 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.02 => ! [Xb2: int] :
% 12.85/7.02 ( ( member_int @ Xb2 @ B4 )
% 12.85/7.02 => ( X5
% 12.85/7.02 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.02 => ( ( ( P @ X5 )
% 12.85/7.02 | ( Q @ X5 ) )
% 12.85/7.02 => ( ( P @ ( minus_minus_int @ X5 @ D3 ) )
% 12.85/7.02 | ( Q @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 12.85/7.02
% 12.85/7.02 thf(fact_9209_aset_I1_J,axiom,
% 12.85/7.02 ! [D3: int,A2: set_int,P: int > $o,Q: int > $o] :
% 12.85/7.02 ( ! [X4: int] :
% 12.85/7.02 ( ! [Xa2: int] :
% 12.85/7.02 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.02 => ! [Xb: int] :
% 12.85/7.02 ( ( member_int @ Xb @ A2 )
% 12.85/7.02 => ( X4
% 12.85/7.02 != ( minus_minus_int @ Xb @ Xa2 ) ) ) )
% 12.85/7.02 => ( ( P @ X4 )
% 12.85/7.02 => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 12.85/7.02 => ( ! [X4: int] :
% 12.85/7.02 ( ! [Xa2: int] :
% 12.85/7.02 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.02 => ! [Xb: int] :
% 12.85/7.02 ( ( member_int @ Xb @ A2 )
% 12.85/7.02 => ( X4
% 12.85/7.02 != ( minus_minus_int @ Xb @ Xa2 ) ) ) )
% 12.85/7.02 => ( ( Q @ X4 )
% 12.85/7.02 => ( Q @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 12.85/7.02 => ! [X5: int] :
% 12.85/7.02 ( ! [Xa3: int] :
% 12.85/7.02 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.02 => ! [Xb2: int] :
% 12.85/7.02 ( ( member_int @ Xb2 @ A2 )
% 12.85/7.02 => ( X5
% 12.85/7.03 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.03 => ( ( ( P @ X5 )
% 12.85/7.03 & ( Q @ X5 ) )
% 12.85/7.03 => ( ( P @ ( plus_plus_int @ X5 @ D3 ) )
% 12.85/7.03 & ( Q @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9210_aset_I2_J,axiom,
% 12.85/7.03 ! [D3: int,A2: set_int,P: int > $o,Q: int > $o] :
% 12.85/7.03 ( ! [X4: int] :
% 12.85/7.03 ( ! [Xa2: int] :
% 12.85/7.03 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb: int] :
% 12.85/7.03 ( ( member_int @ Xb @ A2 )
% 12.85/7.03 => ( X4
% 12.85/7.03 != ( minus_minus_int @ Xb @ Xa2 ) ) ) )
% 12.85/7.03 => ( ( P @ X4 )
% 12.85/7.03 => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 12.85/7.03 => ( ! [X4: int] :
% 12.85/7.03 ( ! [Xa2: int] :
% 12.85/7.03 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb: int] :
% 12.85/7.03 ( ( member_int @ Xb @ A2 )
% 12.85/7.03 => ( X4
% 12.85/7.03 != ( minus_minus_int @ Xb @ Xa2 ) ) ) )
% 12.85/7.03 => ( ( Q @ X4 )
% 12.85/7.03 => ( Q @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 12.85/7.03 => ! [X5: int] :
% 12.85/7.03 ( ! [Xa3: int] :
% 12.85/7.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb2: int] :
% 12.85/7.03 ( ( member_int @ Xb2 @ A2 )
% 12.85/7.03 => ( X5
% 12.85/7.03 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.03 => ( ( ( P @ X5 )
% 12.85/7.03 | ( Q @ X5 ) )
% 12.85/7.03 => ( ( P @ ( plus_plus_int @ X5 @ D3 ) )
% 12.85/7.03 | ( Q @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9211_bset_I9_J,axiom,
% 12.85/7.03 ! [D: int,D3: int,B4: set_int,T: int] :
% 12.85/7.03 ( ( dvd_dvd_int @ D @ D3 )
% 12.85/7.03 => ! [X5: int] :
% 12.85/7.03 ( ! [Xa3: int] :
% 12.85/7.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb2: int] :
% 12.85/7.03 ( ( member_int @ Xb2 @ B4 )
% 12.85/7.03 => ( X5
% 12.85/7.03 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.03 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 12.85/7.03 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9212_bset_I10_J,axiom,
% 12.85/7.03 ! [D: int,D3: int,B4: set_int,T: int] :
% 12.85/7.03 ( ( dvd_dvd_int @ D @ D3 )
% 12.85/7.03 => ! [X5: int] :
% 12.85/7.03 ( ! [Xa3: int] :
% 12.85/7.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb2: int] :
% 12.85/7.03 ( ( member_int @ Xb2 @ B4 )
% 12.85/7.03 => ( X5
% 12.85/7.03 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.03 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 12.85/7.03 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9213_aset_I9_J,axiom,
% 12.85/7.03 ! [D: int,D3: int,A2: set_int,T: int] :
% 12.85/7.03 ( ( dvd_dvd_int @ D @ D3 )
% 12.85/7.03 => ! [X5: int] :
% 12.85/7.03 ( ! [Xa3: int] :
% 12.85/7.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb2: int] :
% 12.85/7.03 ( ( member_int @ Xb2 @ A2 )
% 12.85/7.03 => ( X5
% 12.85/7.03 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.03 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 12.85/7.03 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9214_aset_I10_J,axiom,
% 12.85/7.03 ! [D: int,D3: int,A2: set_int,T: int] :
% 12.85/7.03 ( ( dvd_dvd_int @ D @ D3 )
% 12.85/7.03 => ! [X5: int] :
% 12.85/7.03 ( ! [Xa3: int] :
% 12.85/7.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb2: int] :
% 12.85/7.03 ( ( member_int @ Xb2 @ A2 )
% 12.85/7.03 => ( X5
% 12.85/7.03 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.03 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 12.85/7.03 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9215_periodic__finite__ex,axiom,
% 12.85/7.03 ! [D: int,P: int > $o] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ D )
% 12.85/7.03 => ( ! [X4: int,K2: int] :
% 12.85/7.03 ( ( P @ X4 )
% 12.85/7.03 = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 12.85/7.03 => ( ( ? [X8: int] : ( P @ X8 ) )
% 12.85/7.03 = ( ? [X3: int] :
% 12.85/7.03 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 12.85/7.03 & ( P @ X3 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9216_bset_I3_J,axiom,
% 12.85/7.03 ! [D3: int,T: int,B4: set_int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ D3 )
% 12.85/7.03 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B4 )
% 12.85/7.03 => ! [X5: int] :
% 12.85/7.03 ( ! [Xa3: int] :
% 12.85/7.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb2: int] :
% 12.85/7.03 ( ( member_int @ Xb2 @ B4 )
% 12.85/7.03 => ( X5
% 12.85/7.03 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.03 => ( ( X5 = T )
% 12.85/7.03 => ( ( minus_minus_int @ X5 @ D3 )
% 12.85/7.03 = T ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9217_bset_I4_J,axiom,
% 12.85/7.03 ! [D3: int,T: int,B4: set_int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ D3 )
% 12.85/7.03 => ( ( member_int @ T @ B4 )
% 12.85/7.03 => ! [X5: int] :
% 12.85/7.03 ( ! [Xa3: int] :
% 12.85/7.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb2: int] :
% 12.85/7.03 ( ( member_int @ Xb2 @ B4 )
% 12.85/7.03 => ( X5
% 12.85/7.03 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.03 => ( ( X5 != T )
% 12.85/7.03 => ( ( minus_minus_int @ X5 @ D3 )
% 12.85/7.03 != T ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9218_bset_I5_J,axiom,
% 12.85/7.03 ! [D3: int,B4: set_int,T: int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ D3 )
% 12.85/7.03 => ! [X5: int] :
% 12.85/7.03 ( ! [Xa3: int] :
% 12.85/7.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb2: int] :
% 12.85/7.03 ( ( member_int @ Xb2 @ B4 )
% 12.85/7.03 => ( X5
% 12.85/7.03 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.03 => ( ( ord_less_int @ X5 @ T )
% 12.85/7.03 => ( ord_less_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9219_bset_I7_J,axiom,
% 12.85/7.03 ! [D3: int,T: int,B4: set_int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ D3 )
% 12.85/7.03 => ( ( member_int @ T @ B4 )
% 12.85/7.03 => ! [X5: int] :
% 12.85/7.03 ( ! [Xa3: int] :
% 12.85/7.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb2: int] :
% 12.85/7.03 ( ( member_int @ Xb2 @ B4 )
% 12.85/7.03 => ( X5
% 12.85/7.03 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.03 => ( ( ord_less_int @ T @ X5 )
% 12.85/7.03 => ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9220_aset_I3_J,axiom,
% 12.85/7.03 ! [D3: int,T: int,A2: set_int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ D3 )
% 12.85/7.03 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 12.85/7.03 => ! [X5: int] :
% 12.85/7.03 ( ! [Xa3: int] :
% 12.85/7.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb2: int] :
% 12.85/7.03 ( ( member_int @ Xb2 @ A2 )
% 12.85/7.03 => ( X5
% 12.85/7.03 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.03 => ( ( X5 = T )
% 12.85/7.03 => ( ( plus_plus_int @ X5 @ D3 )
% 12.85/7.03 = T ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9221_aset_I4_J,axiom,
% 12.85/7.03 ! [D3: int,T: int,A2: set_int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ D3 )
% 12.85/7.03 => ( ( member_int @ T @ A2 )
% 12.85/7.03 => ! [X5: int] :
% 12.85/7.03 ( ! [Xa3: int] :
% 12.85/7.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb2: int] :
% 12.85/7.03 ( ( member_int @ Xb2 @ A2 )
% 12.85/7.03 => ( X5
% 12.85/7.03 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.03 => ( ( X5 != T )
% 12.85/7.03 => ( ( plus_plus_int @ X5 @ D3 )
% 12.85/7.03 != T ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9222_aset_I5_J,axiom,
% 12.85/7.03 ! [D3: int,T: int,A2: set_int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ D3 )
% 12.85/7.03 => ( ( member_int @ T @ A2 )
% 12.85/7.03 => ! [X5: int] :
% 12.85/7.03 ( ! [Xa3: int] :
% 12.85/7.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb2: int] :
% 12.85/7.03 ( ( member_int @ Xb2 @ A2 )
% 12.85/7.03 => ( X5
% 12.85/7.03 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.03 => ( ( ord_less_int @ X5 @ T )
% 12.85/7.03 => ( ord_less_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9223_aset_I7_J,axiom,
% 12.85/7.03 ! [D3: int,A2: set_int,T: int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ D3 )
% 12.85/7.03 => ! [X5: int] :
% 12.85/7.03 ( ! [Xa3: int] :
% 12.85/7.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb2: int] :
% 12.85/7.03 ( ( member_int @ Xb2 @ A2 )
% 12.85/7.03 => ( X5
% 12.85/7.03 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.03 => ( ( ord_less_int @ T @ X5 )
% 12.85/7.03 => ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9224_fact__eq__fact__times,axiom,
% 12.85/7.03 ! [N: nat,M: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ N @ M )
% 12.85/7.03 => ( ( semiri1408675320244567234ct_nat @ M )
% 12.85/7.03 = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
% 12.85/7.03 @ ( groups708209901874060359at_nat
% 12.85/7.03 @ ^ [X3: nat] : X3
% 12.85/7.03 @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9225_bset_I6_J,axiom,
% 12.85/7.03 ! [D3: int,B4: set_int,T: int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ D3 )
% 12.85/7.03 => ! [X5: int] :
% 12.85/7.03 ( ! [Xa3: int] :
% 12.85/7.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb2: int] :
% 12.85/7.03 ( ( member_int @ Xb2 @ B4 )
% 12.85/7.03 => ( X5
% 12.85/7.03 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.03 => ( ( ord_less_eq_int @ X5 @ T )
% 12.85/7.03 => ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9226_bset_I8_J,axiom,
% 12.85/7.03 ! [D3: int,T: int,B4: set_int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ D3 )
% 12.85/7.03 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B4 )
% 12.85/7.03 => ! [X5: int] :
% 12.85/7.03 ( ! [Xa3: int] :
% 12.85/7.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb2: int] :
% 12.85/7.03 ( ( member_int @ Xb2 @ B4 )
% 12.85/7.03 => ( X5
% 12.85/7.03 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.03 => ( ( ord_less_eq_int @ T @ X5 )
% 12.85/7.03 => ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9227_aset_I6_J,axiom,
% 12.85/7.03 ! [D3: int,T: int,A2: set_int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ D3 )
% 12.85/7.03 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 12.85/7.03 => ! [X5: int] :
% 12.85/7.03 ( ! [Xa3: int] :
% 12.85/7.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb2: int] :
% 12.85/7.03 ( ( member_int @ Xb2 @ A2 )
% 12.85/7.03 => ( X5
% 12.85/7.03 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.03 => ( ( ord_less_eq_int @ X5 @ T )
% 12.85/7.03 => ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9228_aset_I8_J,axiom,
% 12.85/7.03 ! [D3: int,A2: set_int,T: int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ D3 )
% 12.85/7.03 => ! [X5: int] :
% 12.85/7.03 ( ! [Xa3: int] :
% 12.85/7.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb2: int] :
% 12.85/7.03 ( ( member_int @ Xb2 @ A2 )
% 12.85/7.03 => ( X5
% 12.85/7.03 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 12.85/7.03 => ( ( ord_less_eq_int @ T @ X5 )
% 12.85/7.03 => ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9229_cppi,axiom,
% 12.85/7.03 ! [D3: int,P: int > $o,P4: int > $o,A2: set_int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ D3 )
% 12.85/7.03 => ( ? [Z3: int] :
% 12.85/7.03 ! [X4: int] :
% 12.85/7.03 ( ( ord_less_int @ Z3 @ X4 )
% 12.85/7.03 => ( ( P @ X4 )
% 12.85/7.03 = ( P4 @ X4 ) ) )
% 12.85/7.03 => ( ! [X4: int] :
% 12.85/7.03 ( ! [Xa2: int] :
% 12.85/7.03 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb: int] :
% 12.85/7.03 ( ( member_int @ Xb @ A2 )
% 12.85/7.03 => ( X4
% 12.85/7.03 != ( minus_minus_int @ Xb @ Xa2 ) ) ) )
% 12.85/7.03 => ( ( P @ X4 )
% 12.85/7.03 => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 12.85/7.03 => ( ! [X4: int,K2: int] :
% 12.85/7.03 ( ( P4 @ X4 )
% 12.85/7.03 = ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 12.85/7.03 => ( ( ? [X8: int] : ( P @ X8 ) )
% 12.85/7.03 = ( ? [X3: int] :
% 12.85/7.03 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 & ( P4 @ X3 ) )
% 12.85/7.03 | ? [X3: int] :
% 12.85/7.03 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 & ? [Y6: int] :
% 12.85/7.03 ( ( member_int @ Y6 @ A2 )
% 12.85/7.03 & ( P @ ( minus_minus_int @ Y6 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9230_cpmi,axiom,
% 12.85/7.03 ! [D3: int,P: int > $o,P4: int > $o,B4: set_int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ D3 )
% 12.85/7.03 => ( ? [Z3: int] :
% 12.85/7.03 ! [X4: int] :
% 12.85/7.03 ( ( ord_less_int @ X4 @ Z3 )
% 12.85/7.03 => ( ( P @ X4 )
% 12.85/7.03 = ( P4 @ X4 ) ) )
% 12.85/7.03 => ( ! [X4: int] :
% 12.85/7.03 ( ! [Xa2: int] :
% 12.85/7.03 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 => ! [Xb: int] :
% 12.85/7.03 ( ( member_int @ Xb @ B4 )
% 12.85/7.03 => ( X4
% 12.85/7.03 != ( plus_plus_int @ Xb @ Xa2 ) ) ) )
% 12.85/7.03 => ( ( P @ X4 )
% 12.85/7.03 => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 12.85/7.03 => ( ! [X4: int,K2: int] :
% 12.85/7.03 ( ( P4 @ X4 )
% 12.85/7.03 = ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 12.85/7.03 => ( ( ? [X8: int] : ( P @ X8 ) )
% 12.85/7.03 = ( ? [X3: int] :
% 12.85/7.03 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 & ( P4 @ X3 ) )
% 12.85/7.03 | ? [X3: int] :
% 12.85/7.03 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 12.85/7.03 & ? [Y6: int] :
% 12.85/7.03 ( ( member_int @ Y6 @ B4 )
% 12.85/7.03 & ( P @ ( plus_plus_int @ Y6 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9231_fact__div__fact,axiom,
% 12.85/7.03 ! [N: nat,M: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ N @ M )
% 12.85/7.03 => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
% 12.85/7.03 = ( groups708209901874060359at_nat
% 12.85/7.03 @ ^ [X3: nat] : X3
% 12.85/7.03 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9232_divmod__step__nat__def,axiom,
% 12.85/7.03 ( unique5026877609467782581ep_nat
% 12.85/7.03 = ( ^ [L3: num] :
% 12.85/7.03 ( produc2626176000494625587at_nat
% 12.85/7.03 @ ^ [Q3: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L3 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L3 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R5 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9233_divmod__step__int__def,axiom,
% 12.85/7.03 ( unique5024387138958732305ep_int
% 12.85/7.03 = ( ^ [L3: num] :
% 12.85/7.03 ( produc4245557441103728435nt_int
% 12.85/7.03 @ ^ [Q3: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L3 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L3 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R5 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9234_Sum__Icc__int,axiom,
% 12.85/7.03 ! [M: int,N: int] :
% 12.85/7.03 ( ( ord_less_eq_int @ M @ N )
% 12.85/7.03 => ( ( groups4538972089207619220nt_int
% 12.85/7.03 @ ^ [X3: int] : X3
% 12.85/7.03 @ ( set_or1266510415728281911st_int @ M @ N ) )
% 12.85/7.03 = ( 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 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9235_divmod__nat__if,axiom,
% 12.85/7.03 ( divmod_nat
% 12.85/7.03 = ( ^ [M5: nat,N4: nat] :
% 12.85/7.03 ( if_Pro6206227464963214023at_nat
% 12.85/7.03 @ ( ( N4 = zero_zero_nat )
% 12.85/7.03 | ( ord_less_nat @ M5 @ N4 ) )
% 12.85/7.03 @ ( product_Pair_nat_nat @ zero_zero_nat @ M5 )
% 12.85/7.03 @ ( produc2626176000494625587at_nat
% 12.85/7.03 @ ^ [Q3: nat] : ( product_Pair_nat_nat @ ( suc @ Q3 ) )
% 12.85/7.03 @ ( divmod_nat @ ( minus_minus_nat @ M5 @ N4 ) @ N4 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9236_vebt__buildup_Opelims,axiom,
% 12.85/7.03 ! [X: nat,Y4: vEBT_VEBT] :
% 12.85/7.03 ( ( ( vEBT_vebt_buildup @ X )
% 12.85/7.03 = Y4 )
% 12.85/7.03 => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
% 12.85/7.03 => ( ( ( X = zero_zero_nat )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( vEBT_Leaf @ $false @ $false ) )
% 12.85/7.03 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 12.85/7.03 => ( ( ( X
% 12.85/7.03 = ( suc @ zero_zero_nat ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( vEBT_Leaf @ $false @ $false ) )
% 12.85/7.03 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 12.85/7.03 => ~ ! [Va: nat] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( suc @ ( suc @ Va ) ) )
% 12.85/7.03 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( 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 ) ) ) ) ) ) )
% 12.85/7.03 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( 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 ) ) ) ) ) ) ) ) )
% 12.85/7.03 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9237_arctan__def,axiom,
% 12.85/7.03 ( arctan
% 12.85/7.03 = ( ^ [Y6: real] :
% 12.85/7.03 ( the_real
% 12.85/7.03 @ ^ [X3: real] :
% 12.85/7.03 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 12.85/7.03 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.03 & ( ( tan_real @ X3 )
% 12.85/7.03 = Y6 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9238_prod__int__eq,axiom,
% 12.85/7.03 ! [I: nat,J: nat] :
% 12.85/7.03 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 12.85/7.03 = ( groups1705073143266064639nt_int
% 12.85/7.03 @ ^ [X3: int] : X3
% 12.85/7.03 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9239_prod__int__plus__eq,axiom,
% 12.85/7.03 ! [I: nat,J: nat] :
% 12.85/7.03 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
% 12.85/7.03 = ( groups1705073143266064639nt_int
% 12.85/7.03 @ ^ [X3: int] : X3
% 12.85/7.03 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9240_ln__neg__is__const,axiom,
% 12.85/7.03 ! [X: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 12.85/7.03 => ( ( ln_ln_real @ X )
% 12.85/7.03 = ( the_real
% 12.85/7.03 @ ^ [X3: real] : $false ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9241_arccos__def,axiom,
% 12.85/7.03 ( arccos
% 12.85/7.03 = ( ^ [Y6: real] :
% 12.85/7.03 ( the_real
% 12.85/7.03 @ ^ [X3: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 12.85/7.03 & ( ord_less_eq_real @ X3 @ pi )
% 12.85/7.03 & ( ( cos_real @ X3 )
% 12.85/7.03 = Y6 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9242_divmod__nat__def,axiom,
% 12.85/7.03 ( divmod_nat
% 12.85/7.03 = ( ^ [M5: nat,N4: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M5 @ N4 ) @ ( modulo_modulo_nat @ M5 @ N4 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9243_pi__half,axiom,
% 12.85/7.03 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.85/7.03 = ( the_real
% 12.85/7.03 @ ^ [X3: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 12.85/7.03 & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.85/7.03 & ( ( cos_real @ X3 )
% 12.85/7.03 = zero_zero_real ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9244_pi__def,axiom,
% 12.85/7.03 ( pi
% 12.85/7.03 = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 12.85/7.03 @ ( the_real
% 12.85/7.03 @ ^ [X3: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 12.85/7.03 & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 12.85/7.03 & ( ( cos_real @ X3 )
% 12.85/7.03 = zero_zero_real ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9245_arcsin__def,axiom,
% 12.85/7.03 ( arcsin
% 12.85/7.03 = ( ^ [Y6: real] :
% 12.85/7.03 ( the_real
% 12.85/7.03 @ ^ [X3: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 12.85/7.03 & ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.03 & ( ( sin_real @ X3 )
% 12.85/7.03 = Y6 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9246_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
% 12.85/7.03 ! [X: nat,Y4: nat] :
% 12.85/7.03 ( ( ( vEBT_V8346862874174094_d_u_p @ X )
% 12.85/7.03 = Y4 )
% 12.85/7.03 => ( ( accp_nat @ vEBT_V1247956027447740395_p_rel @ X )
% 12.85/7.03 => ( ( ( X = zero_zero_nat )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 12.85/7.03 => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ zero_zero_nat ) ) )
% 12.85/7.03 => ( ( ( X
% 12.85/7.03 = ( suc @ zero_zero_nat ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 12.85/7.03 => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ zero_zero_nat ) ) ) )
% 12.85/7.03 => ~ ! [Va: nat] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( suc @ ( suc @ Va ) ) )
% 12.85/7.03 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
% 12.85/7.03 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
% 12.85/7.03 => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9247_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
% 12.85/7.03 ! [X: nat,Y4: nat] :
% 12.85/7.03 ( ( ( vEBT_V8646137997579335489_i_l_d @ X )
% 12.85/7.03 = Y4 )
% 12.85/7.03 => ( ( accp_nat @ vEBT_V5144397997797733112_d_rel @ X )
% 12.85/7.03 => ( ( ( X = zero_zero_nat )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.85/7.03 => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ zero_zero_nat ) ) )
% 12.85/7.03 => ( ( ( X
% 12.85/7.03 = ( suc @ zero_zero_nat ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 12.85/7.03 => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ zero_zero_nat ) ) ) )
% 12.85/7.03 => ~ ! [Va: nat] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( suc @ ( suc @ Va ) ) )
% 12.85/7.03 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 12.85/7.03 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 12.85/7.03 => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9248_set__encode__def,axiom,
% 12.85/7.03 ( nat_set_encode
% 12.85/7.03 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9249_set__decode__inverse,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( nat_set_encode @ ( nat_set_decode @ N ) )
% 12.85/7.03 = N ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9250_VEBT_Osize_I3_J,axiom,
% 12.85/7.03 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 12.85/7.03 ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 12.85/7.03 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9251_Sum__Ico__nat,axiom,
% 12.85/7.03 ! [M: nat,N: nat] :
% 12.85/7.03 ( ( groups3542108847815614940at_nat
% 12.85/7.03 @ ^ [X3: nat] : X3
% 12.85/7.03 @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 12.85/7.03 = ( 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 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9252_sum__power2,axiom,
% 12.85/7.03 ! [K: nat] :
% 12.85/7.03 ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 12.85/7.03 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9253_ex__nat__less__eq,axiom,
% 12.85/7.03 ! [N: nat,P: nat > $o] :
% 12.85/7.03 ( ( ? [M5: nat] :
% 12.85/7.03 ( ( ord_less_nat @ M5 @ N )
% 12.85/7.03 & ( P @ M5 ) ) )
% 12.85/7.03 = ( ? [X3: nat] :
% 12.85/7.03 ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 12.85/7.03 & ( P @ X3 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9254_all__nat__less__eq,axiom,
% 12.85/7.03 ! [N: nat,P: nat > $o] :
% 12.85/7.03 ( ( ! [M5: nat] :
% 12.85/7.03 ( ( ord_less_nat @ M5 @ N )
% 12.85/7.03 => ( P @ M5 ) ) )
% 12.85/7.03 = ( ! [X3: nat] :
% 12.85/7.03 ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 12.85/7.03 => ( P @ X3 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9255_atLeastLessThanSuc__atLeastAtMost,axiom,
% 12.85/7.03 ! [L: nat,U: nat] :
% 12.85/7.03 ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
% 12.85/7.03 = ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9256_lessThan__atLeast0,axiom,
% 12.85/7.03 ( set_ord_lessThan_nat
% 12.85/7.03 = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9257_prod__Suc__Suc__fact,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 12.85/7.03 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9258_prod__Suc__fact,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 12.85/7.03 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9259_Chebyshev__sum__upper__nat,axiom,
% 12.85/7.03 ! [N: nat,A: nat > nat,B: nat > nat] :
% 12.85/7.03 ( ! [I3: nat,J2: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ I3 @ J2 )
% 12.85/7.03 => ( ( ord_less_nat @ J2 @ N )
% 12.85/7.03 => ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
% 12.85/7.03 => ( ! [I3: nat,J2: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ I3 @ J2 )
% 12.85/7.03 => ( ( ord_less_nat @ J2 @ N )
% 12.85/7.03 => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I3 ) ) ) )
% 12.85/7.03 => ( ord_less_eq_nat
% 12.85/7.03 @ ( times_times_nat @ N
% 12.85/7.03 @ ( groups3542108847815614940at_nat
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( B @ I2 ) )
% 12.85/7.03 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 12.85/7.03 @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9260_VEBT_Osize__gen_I1_J,axiom,
% 12.85/7.03 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 12.85/7.03 ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 12.85/7.03 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9261_VEBT_Osize__gen_I2_J,axiom,
% 12.85/7.03 ! [X21: $o,X222: $o] :
% 12.85/7.03 ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 12.85/7.03 = zero_zero_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9262_int__of__nat__def,axiom,
% 12.85/7.03 code_T6385005292777649522of_nat = semiri1314217659103216013at_int ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9263_prod__decode__aux_Opelims,axiom,
% 12.85/7.03 ! [X: nat,Xa: nat,Y4: product_prod_nat_nat] :
% 12.85/7.03 ( ( ( nat_prod_decode_aux @ X @ Xa )
% 12.85/7.03 = Y4 )
% 12.85/7.03 => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 12.85/7.03 => ~ ( ( ( ( ord_less_eq_nat @ Xa @ X )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
% 12.85/7.03 & ( ~ ( ord_less_eq_nat @ Xa @ X )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) )
% 12.85/7.03 => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9264_or__not__num__neg_Opelims,axiom,
% 12.85/7.03 ! [X: num,Xa: num,Y4: num] :
% 12.85/7.03 ( ( ( bit_or_not_num_neg @ X @ Xa )
% 12.85/7.03 = Y4 )
% 12.85/7.03 => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( ( Y4 = one )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ! [M4: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M4 ) ) ) ) ) )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ! [M4: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M4 ) ) ) ) ) )
% 12.85/7.03 => ( ! [N2: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( bit0 @ one ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ one ) ) ) ) )
% 12.85/7.03 => ( ! [N2: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ! [M4: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 12.85/7.03 => ( ! [N2: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ! [M4: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( bit0 @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ ( bit1 @ M4 ) ) ) ) ) )
% 12.85/7.03 => ( ! [N2: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( ( Y4 = one )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ one ) ) ) ) )
% 12.85/7.03 => ( ! [N2: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ! [M4: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 12.85/7.03 => ~ ! [N2: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ! [M4: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ ( bit1 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9265_int__ge__less__than__def,axiom,
% 12.85/7.03 ( int_ge_less_than
% 12.85/7.03 = ( ^ [D4: int] :
% 12.85/7.03 ( collec213857154873943460nt_int
% 12.85/7.03 @ ( produc4947309494688390418_int_o
% 12.85/7.03 @ ^ [Z7: int,Z5: int] :
% 12.85/7.03 ( ( ord_less_eq_int @ D4 @ Z7 )
% 12.85/7.03 & ( ord_less_int @ Z7 @ Z5 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9266_int__ge__less__than2__def,axiom,
% 12.85/7.03 ( int_ge_less_than2
% 12.85/7.03 = ( ^ [D4: int] :
% 12.85/7.03 ( collec213857154873943460nt_int
% 12.85/7.03 @ ( produc4947309494688390418_int_o
% 12.85/7.03 @ ^ [Z7: int,Z5: int] :
% 12.85/7.03 ( ( ord_less_eq_int @ D4 @ Z5 )
% 12.85/7.03 & ( ord_less_int @ Z7 @ Z5 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9267_upto_Opinduct,axiom,
% 12.85/7.03 ! [A0: int,A1: int,P: int > int > $o] :
% 12.85/7.03 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
% 12.85/7.03 => ( ! [I3: int,J2: int] :
% 12.85/7.03 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
% 12.85/7.03 => ( ( ( ord_less_eq_int @ I3 @ J2 )
% 12.85/7.03 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
% 12.85/7.03 => ( P @ I3 @ J2 ) ) )
% 12.85/7.03 => ( P @ A0 @ A1 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9268_Code__Target__Int_Opositive__def,axiom,
% 12.85/7.03 code_Target_positive = numeral_numeral_int ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9269_divmod__step__integer__def,axiom,
% 12.85/7.03 ( unique4921790084139445826nteger
% 12.85/7.03 = ( ^ [L3: num] :
% 12.85/7.03 ( produc6916734918728496179nteger
% 12.85/7.03 @ ^ [Q3: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L3 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L3 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ R5 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9270_less__eq__integer__code_I1_J,axiom,
% 12.85/7.03 ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9271_plus__integer__code_I1_J,axiom,
% 12.85/7.03 ! [K: code_integer] :
% 12.85/7.03 ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
% 12.85/7.03 = K ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9272_plus__integer__code_I2_J,axiom,
% 12.85/7.03 ! [L: code_integer] :
% 12.85/7.03 ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L )
% 12.85/7.03 = L ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9273_times__integer__code_I2_J,axiom,
% 12.85/7.03 ! [L: code_integer] :
% 12.85/7.03 ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
% 12.85/7.03 = zero_z3403309356797280102nteger ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9274_times__integer__code_I1_J,axiom,
% 12.85/7.03 ! [K: code_integer] :
% 12.85/7.03 ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
% 12.85/7.03 = zero_z3403309356797280102nteger ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9275_minus__integer__code_I2_J,axiom,
% 12.85/7.03 ! [L: code_integer] :
% 12.85/7.03 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
% 12.85/7.03 = ( uminus1351360451143612070nteger @ L ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9276_minus__integer__code_I1_J,axiom,
% 12.85/7.03 ! [K: code_integer] :
% 12.85/7.03 ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
% 12.85/7.03 = K ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9277_sgn__integer__code,axiom,
% 12.85/7.03 ( sgn_sgn_Code_integer
% 12.85/7.03 = ( ^ [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 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9278_nat_Odisc__eq__case_I2_J,axiom,
% 12.85/7.03 ! [Nat: nat] :
% 12.85/7.03 ( ( Nat != zero_zero_nat )
% 12.85/7.03 = ( case_nat_o @ $false
% 12.85/7.03 @ ^ [Uu3: nat] : $true
% 12.85/7.03 @ Nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9279_nat_Odisc__eq__case_I1_J,axiom,
% 12.85/7.03 ! [Nat: nat] :
% 12.85/7.03 ( ( Nat = zero_zero_nat )
% 12.85/7.03 = ( case_nat_o @ $true
% 12.85/7.03 @ ^ [Uu3: nat] : $false
% 12.85/7.03 @ Nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9280_zero__natural_Orsp,axiom,
% 12.85/7.03 zero_zero_nat = zero_zero_nat ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9281_zero__integer_Orsp,axiom,
% 12.85/7.03 zero_zero_int = zero_zero_int ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9282_less__eq__nat_Osimps_I2_J,axiom,
% 12.85/7.03 ! [M: nat,N: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 12.85/7.03 = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9283_max__Suc2,axiom,
% 12.85/7.03 ! [M: nat,N: nat] :
% 12.85/7.03 ( ( ord_max_nat @ M @ ( suc @ N ) )
% 12.85/7.03 = ( case_nat_nat @ ( suc @ N )
% 12.85/7.03 @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ M6 @ N ) )
% 12.85/7.03 @ M ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9284_max__Suc1,axiom,
% 12.85/7.03 ! [N: nat,M: nat] :
% 12.85/7.03 ( ( ord_max_nat @ ( suc @ N ) @ M )
% 12.85/7.03 = ( case_nat_nat @ ( suc @ N )
% 12.85/7.03 @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ N @ M6 ) )
% 12.85/7.03 @ M ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9285_diff__Suc,axiom,
% 12.85/7.03 ! [M: nat,N: nat] :
% 12.85/7.03 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 12.85/7.03 = ( case_nat_nat @ zero_zero_nat
% 12.85/7.03 @ ^ [K3: nat] : K3
% 12.85/7.03 @ ( minus_minus_nat @ M @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9286_integer__of__int__code,axiom,
% 12.85/7.03 ( code_integer_of_int
% 12.85/7.03 = ( ^ [K3: int] :
% 12.85/7.03 ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
% 12.85/7.03 @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 12.85/7.03 @ ( if_Code_integer
% 12.85/7.03 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 12.85/7.03 = zero_zero_int )
% 12.85/7.03 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 12.85/7.03 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9287_flip__bit__integer_Oabs__eq,axiom,
% 12.85/7.03 ! [Xa: nat,X: int] :
% 12.85/7.03 ( ( bit_se1345352211410354436nteger @ Xa @ ( code_integer_of_int @ X ) )
% 12.85/7.03 = ( code_integer_of_int @ ( bit_se2159334234014336723it_int @ Xa @ X ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9288_zero__integer__def,axiom,
% 12.85/7.03 ( zero_z3403309356797280102nteger
% 12.85/7.03 = ( code_integer_of_int @ zero_zero_int ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9289_unset__bit__integer_Oabs__eq,axiom,
% 12.85/7.03 ! [Xa: nat,X: int] :
% 12.85/7.03 ( ( bit_se8260200283734997820nteger @ Xa @ ( code_integer_of_int @ X ) )
% 12.85/7.03 = ( code_integer_of_int @ ( bit_se4203085406695923979it_int @ Xa @ X ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9290_divide__integer_Oabs__eq,axiom,
% 12.85/7.03 ! [Xa: int,X: int] :
% 12.85/7.03 ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 12.85/7.03 = ( code_integer_of_int @ ( divide_divide_int @ Xa @ X ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9291_set__bit__integer_Oabs__eq,axiom,
% 12.85/7.03 ! [Xa: nat,X: int] :
% 12.85/7.03 ( ( bit_se2793503036327961859nteger @ Xa @ ( code_integer_of_int @ X ) )
% 12.85/7.03 = ( code_integer_of_int @ ( bit_se7879613467334960850it_int @ Xa @ X ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9292_uminus__integer__code_I1_J,axiom,
% 12.85/7.03 ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 12.85/7.03 = zero_z3403309356797280102nteger ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9293_less__integer__code_I1_J,axiom,
% 12.85/7.03 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9294_abs__integer__code,axiom,
% 12.85/7.03 ( abs_abs_Code_integer
% 12.85/7.03 = ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9295_times__integer_Oabs__eq,axiom,
% 12.85/7.03 ! [Xa: int,X: int] :
% 12.85/7.03 ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 12.85/7.03 = ( code_integer_of_int @ ( times_times_int @ Xa @ X ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9296_less__eq__integer_Oabs__eq,axiom,
% 12.85/7.03 ! [Xa: int,X: int] :
% 12.85/7.03 ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 12.85/7.03 = ( ord_less_eq_int @ Xa @ X ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9297_minus__integer_Oabs__eq,axiom,
% 12.85/7.03 ! [Xa: int,X: int] :
% 12.85/7.03 ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 12.85/7.03 = ( code_integer_of_int @ ( minus_minus_int @ Xa @ X ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9298_floor__real__def,axiom,
% 12.85/7.03 ( archim6058952711729229775r_real
% 12.85/7.03 = ( ^ [X3: real] :
% 12.85/7.03 ( the_int
% 12.85/7.03 @ ^ [Z5: int] :
% 12.85/7.03 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z5 ) @ X3 )
% 12.85/7.03 & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9299_floor__rat__def,axiom,
% 12.85/7.03 ( archim3151403230148437115or_rat
% 12.85/7.03 = ( ^ [X3: rat] :
% 12.85/7.03 ( the_int
% 12.85/7.03 @ ^ [Z5: int] :
% 12.85/7.03 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z5 ) @ X3 )
% 12.85/7.03 & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9300_integer__of__num_I3_J,axiom,
% 12.85/7.03 ! [N: num] :
% 12.85/7.03 ( ( code_integer_of_num @ ( bit1 @ N ) )
% 12.85/7.03 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9301_less__eq__rat__def,axiom,
% 12.85/7.03 ( ord_less_eq_rat
% 12.85/7.03 = ( ^ [X3: rat,Y6: rat] :
% 12.85/7.03 ( ( ord_less_rat @ X3 @ Y6 )
% 12.85/7.03 | ( X3 = Y6 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9302_obtain__pos__sum,axiom,
% 12.85/7.03 ! [R2: rat] :
% 12.85/7.03 ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 12.85/7.03 => ~ ! [S3: rat] :
% 12.85/7.03 ( ( ord_less_rat @ zero_zero_rat @ S3 )
% 12.85/7.03 => ! [T3: rat] :
% 12.85/7.03 ( ( ord_less_rat @ zero_zero_rat @ T3 )
% 12.85/7.03 => ( R2
% 12.85/7.03 != ( plus_plus_rat @ S3 @ T3 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9303_sgn__rat__def,axiom,
% 12.85/7.03 ( sgn_sgn_rat
% 12.85/7.03 = ( ^ [A3: rat] : ( if_rat @ ( A3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9304_abs__rat__def,axiom,
% 12.85/7.03 ( abs_abs_rat
% 12.85/7.03 = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9305_integer__of__num__triv_I1_J,axiom,
% 12.85/7.03 ( ( code_integer_of_num @ one )
% 12.85/7.03 = one_one_Code_integer ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9306_integer__of__num_I2_J,axiom,
% 12.85/7.03 ! [N: num] :
% 12.85/7.03 ( ( code_integer_of_num @ ( bit0 @ N ) )
% 12.85/7.03 = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9307_integer__of__num__triv_I2_J,axiom,
% 12.85/7.03 ( ( code_integer_of_num @ ( bit0 @ one ) )
% 12.85/7.03 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9308_pred__def,axiom,
% 12.85/7.03 ( pred
% 12.85/7.03 = ( case_nat_nat @ zero_zero_nat
% 12.85/7.03 @ ^ [X23: nat] : X23 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9309_rat__inverse__code,axiom,
% 12.85/7.03 ! [P5: rat] :
% 12.85/7.03 ( ( quotient_of @ ( inverse_inverse_rat @ P5 ) )
% 12.85/7.03 = ( produc4245557441103728435nt_int
% 12.85/7.03 @ ^ [A3: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( A3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A3 ) @ B2 ) @ ( abs_abs_int @ A3 ) ) )
% 12.85/7.03 @ ( quotient_of @ P5 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9310_normalize__negative,axiom,
% 12.85/7.03 ! [Q2: int,P5: int] :
% 12.85/7.03 ( ( ord_less_int @ Q2 @ zero_zero_int )
% 12.85/7.03 => ( ( normalize @ ( product_Pair_int_int @ P5 @ Q2 ) )
% 12.85/7.03 = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P5 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9311_bit__cut__integer__def,axiom,
% 12.85/7.03 ( code_bit_cut_integer
% 12.85/7.03 = ( ^ [K3: code_integer] :
% 12.85/7.03 ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 12.85/7.03 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9312_quotient__of__number_I3_J,axiom,
% 12.85/7.03 ! [K: num] :
% 12.85/7.03 ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
% 12.85/7.03 = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9313_normalize__denom__zero,axiom,
% 12.85/7.03 ! [P5: int] :
% 12.85/7.03 ( ( normalize @ ( product_Pair_int_int @ P5 @ zero_zero_int ) )
% 12.85/7.03 = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9314_rat__zero__code,axiom,
% 12.85/7.03 ( ( quotient_of @ zero_zero_rat )
% 12.85/7.03 = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9315_quotient__of__number_I5_J,axiom,
% 12.85/7.03 ! [K: num] :
% 12.85/7.03 ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 12.85/7.03 = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9316_diff__rat__def,axiom,
% 12.85/7.03 ( minus_minus_rat
% 12.85/7.03 = ( ^ [Q3: rat,R5: rat] : ( plus_plus_rat @ Q3 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9317_rat__times__code,axiom,
% 12.85/7.03 ! [P5: rat,Q2: rat] :
% 12.85/7.03 ( ( quotient_of @ ( times_times_rat @ P5 @ Q2 ) )
% 12.85/7.03 = ( produc4245557441103728435nt_int
% 12.85/7.03 @ ^ [A3: int,C5: int] :
% 12.85/7.03 ( produc4245557441103728435nt_int
% 12.85/7.03 @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ B2 ) @ ( times_times_int @ C5 @ D4 ) ) )
% 12.85/7.03 @ ( quotient_of @ Q2 ) )
% 12.85/7.03 @ ( quotient_of @ P5 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9318_rat__divide__code,axiom,
% 12.85/7.03 ! [P5: rat,Q2: rat] :
% 12.85/7.03 ( ( quotient_of @ ( divide_divide_rat @ P5 @ Q2 ) )
% 12.85/7.03 = ( produc4245557441103728435nt_int
% 12.85/7.03 @ ^ [A3: int,C5: int] :
% 12.85/7.03 ( produc4245557441103728435nt_int
% 12.85/7.03 @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ D4 ) @ ( times_times_int @ C5 @ B2 ) ) )
% 12.85/7.03 @ ( quotient_of @ Q2 ) )
% 12.85/7.03 @ ( quotient_of @ P5 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9319_rat__plus__code,axiom,
% 12.85/7.03 ! [P5: rat,Q2: rat] :
% 12.85/7.03 ( ( quotient_of @ ( plus_plus_rat @ P5 @ Q2 ) )
% 12.85/7.03 = ( produc4245557441103728435nt_int
% 12.85/7.03 @ ^ [A3: int,C5: int] :
% 12.85/7.03 ( produc4245557441103728435nt_int
% 12.85/7.03 @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A3 @ D4 ) @ ( times_times_int @ B2 @ C5 ) ) @ ( times_times_int @ C5 @ D4 ) ) )
% 12.85/7.03 @ ( quotient_of @ Q2 ) )
% 12.85/7.03 @ ( quotient_of @ P5 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9320_rat__minus__code,axiom,
% 12.85/7.03 ! [P5: rat,Q2: rat] :
% 12.85/7.03 ( ( quotient_of @ ( minus_minus_rat @ P5 @ Q2 ) )
% 12.85/7.03 = ( produc4245557441103728435nt_int
% 12.85/7.03 @ ^ [A3: int,C5: int] :
% 12.85/7.03 ( produc4245557441103728435nt_int
% 12.85/7.03 @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A3 @ D4 ) @ ( times_times_int @ B2 @ C5 ) ) @ ( times_times_int @ C5 @ D4 ) ) )
% 12.85/7.03 @ ( quotient_of @ Q2 ) )
% 12.85/7.03 @ ( quotient_of @ P5 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9321_quotient__of__denom__pos,axiom,
% 12.85/7.03 ! [R2: rat,P5: int,Q2: int] :
% 12.85/7.03 ( ( ( quotient_of @ R2 )
% 12.85/7.03 = ( product_Pair_int_int @ P5 @ Q2 ) )
% 12.85/7.03 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9322_normalize__denom__pos,axiom,
% 12.85/7.03 ! [R2: product_prod_int_int,P5: int,Q2: int] :
% 12.85/7.03 ( ( ( normalize @ R2 )
% 12.85/7.03 = ( product_Pair_int_int @ P5 @ Q2 ) )
% 12.85/7.03 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9323_normalize__crossproduct,axiom,
% 12.85/7.03 ! [Q2: int,S2: int,P5: int,R2: int] :
% 12.85/7.03 ( ( Q2 != zero_zero_int )
% 12.85/7.03 => ( ( S2 != zero_zero_int )
% 12.85/7.03 => ( ( ( normalize @ ( product_Pair_int_int @ P5 @ Q2 ) )
% 12.85/7.03 = ( normalize @ ( product_Pair_int_int @ R2 @ S2 ) ) )
% 12.85/7.03 => ( ( times_times_int @ P5 @ S2 )
% 12.85/7.03 = ( times_times_int @ R2 @ Q2 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9324_rat__less__code,axiom,
% 12.85/7.03 ( ord_less_rat
% 12.85/7.03 = ( ^ [P6: rat,Q3: rat] :
% 12.85/7.03 ( produc4947309494688390418_int_o
% 12.85/7.03 @ ^ [A3: int,C5: int] :
% 12.85/7.03 ( produc4947309494688390418_int_o
% 12.85/7.03 @ ^ [B2: int,D4: int] : ( ord_less_int @ ( times_times_int @ A3 @ D4 ) @ ( times_times_int @ C5 @ B2 ) )
% 12.85/7.03 @ ( quotient_of @ Q3 ) )
% 12.85/7.03 @ ( quotient_of @ P6 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9325_rat__less__eq__code,axiom,
% 12.85/7.03 ( ord_less_eq_rat
% 12.85/7.03 = ( ^ [P6: rat,Q3: rat] :
% 12.85/7.03 ( produc4947309494688390418_int_o
% 12.85/7.03 @ ^ [A3: int,C5: int] :
% 12.85/7.03 ( produc4947309494688390418_int_o
% 12.85/7.03 @ ^ [B2: int,D4: int] : ( ord_less_eq_int @ ( times_times_int @ A3 @ D4 ) @ ( times_times_int @ C5 @ B2 ) )
% 12.85/7.03 @ ( quotient_of @ Q3 ) )
% 12.85/7.03 @ ( quotient_of @ P6 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9326_bit__cut__integer__code,axiom,
% 12.85/7.03 ( code_bit_cut_integer
% 12.85/7.03 = ( ^ [K3: code_integer] :
% 12.85/7.03 ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 12.85/7.03 @ ( produc9125791028180074456eger_o
% 12.85/7.03 @ ^ [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 ) )
% 12.85/7.03 @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9327_bezw__0,axiom,
% 12.85/7.03 ! [X: nat] :
% 12.85/7.03 ( ( bezw @ X @ zero_zero_nat )
% 12.85/7.03 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9328_set__encode__inverse,axiom,
% 12.85/7.03 ! [A2: set_nat] :
% 12.85/7.03 ( ( finite_finite_nat @ A2 )
% 12.85/7.03 => ( ( nat_set_decode @ ( nat_set_encode @ A2 ) )
% 12.85/7.03 = A2 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9329_finite__set__decode,axiom,
% 12.85/7.03 ! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9330_set__encode__eq,axiom,
% 12.85/7.03 ! [A2: set_nat,B4: set_nat] :
% 12.85/7.03 ( ( finite_finite_nat @ A2 )
% 12.85/7.03 => ( ( finite_finite_nat @ B4 )
% 12.85/7.03 => ( ( ( nat_set_encode @ A2 )
% 12.85/7.03 = ( nat_set_encode @ B4 ) )
% 12.85/7.03 = ( A2 = B4 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9331_finite__nat__set__iff__bounded__le,axiom,
% 12.85/7.03 ( finite_finite_nat
% 12.85/7.03 = ( ^ [N8: set_nat] :
% 12.85/7.03 ? [M5: nat] :
% 12.85/7.03 ! [X3: nat] :
% 12.85/7.03 ( ( member_nat @ X3 @ N8 )
% 12.85/7.03 => ( ord_less_eq_nat @ X3 @ M5 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9332_finite__less__ub,axiom,
% 12.85/7.03 ! [F: nat > nat,U: nat] :
% 12.85/7.03 ( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
% 12.85/7.03 => ( finite_finite_nat
% 12.85/7.03 @ ( collect_nat
% 12.85/7.03 @ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9333_set__encode__inf,axiom,
% 12.85/7.03 ! [A2: set_nat] :
% 12.85/7.03 ( ~ ( finite_finite_nat @ A2 )
% 12.85/7.03 => ( ( nat_set_encode @ A2 )
% 12.85/7.03 = zero_zero_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9334_finite__divisors__nat,axiom,
% 12.85/7.03 ! [M: nat] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/7.03 => ( finite_finite_nat
% 12.85/7.03 @ ( collect_nat
% 12.85/7.03 @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9335_subset__eq__atLeast0__atMost__finite,axiom,
% 12.85/7.03 ! [N5: set_nat,N: nat] :
% 12.85/7.03 ( ( ord_less_eq_set_nat @ N5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 12.85/7.03 => ( finite_finite_nat @ N5 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9336_subset__eq__atLeast0__lessThan__finite,axiom,
% 12.85/7.03 ! [N5: set_nat,N: nat] :
% 12.85/7.03 ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 12.85/7.03 => ( finite_finite_nat @ N5 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9337_divmod__abs__code_I6_J,axiom,
% 12.85/7.03 ! [J: code_integer] :
% 12.85/7.03 ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
% 12.85/7.03 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9338_divmod__abs__code_I5_J,axiom,
% 12.85/7.03 ! [J: code_integer] :
% 12.85/7.03 ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
% 12.85/7.03 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9339_even__set__encode__iff,axiom,
% 12.85/7.03 ! [A2: set_nat] :
% 12.85/7.03 ( ( finite_finite_nat @ A2 )
% 12.85/7.03 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
% 12.85/7.03 = ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9340_finite__Collect__le__nat,axiom,
% 12.85/7.03 ! [K: nat] :
% 12.85/7.03 ( finite_finite_nat
% 12.85/7.03 @ ( collect_nat
% 12.85/7.03 @ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9341_finite__interval__int1,axiom,
% 12.85/7.03 ! [A: int,B: int] :
% 12.85/7.03 ( finite_finite_int
% 12.85/7.03 @ ( collect_int
% 12.85/7.03 @ ^ [I2: int] :
% 12.85/7.03 ( ( ord_less_eq_int @ A @ I2 )
% 12.85/7.03 & ( ord_less_eq_int @ I2 @ B ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9342_finite__interval__int4,axiom,
% 12.85/7.03 ! [A: int,B: int] :
% 12.85/7.03 ( finite_finite_int
% 12.85/7.03 @ ( collect_int
% 12.85/7.03 @ ^ [I2: int] :
% 12.85/7.03 ( ( ord_less_int @ A @ I2 )
% 12.85/7.03 & ( ord_less_int @ I2 @ B ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9343_finite__interval__int2,axiom,
% 12.85/7.03 ! [A: int,B: int] :
% 12.85/7.03 ( finite_finite_int
% 12.85/7.03 @ ( collect_int
% 12.85/7.03 @ ^ [I2: int] :
% 12.85/7.03 ( ( ord_less_eq_int @ A @ I2 )
% 12.85/7.03 & ( ord_less_int @ I2 @ B ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9344_finite__interval__int3,axiom,
% 12.85/7.03 ! [A: int,B: int] :
% 12.85/7.03 ( finite_finite_int
% 12.85/7.03 @ ( collect_int
% 12.85/7.03 @ ^ [I2: int] :
% 12.85/7.03 ( ( ord_less_int @ A @ I2 )
% 12.85/7.03 & ( ord_less_eq_int @ I2 @ B ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9345_finite__nth__roots,axiom,
% 12.85/7.03 ! [N: nat,C: complex] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( finite3207457112153483333omplex
% 12.85/7.03 @ ( collect_complex
% 12.85/7.03 @ ^ [Z5: complex] :
% 12.85/7.03 ( ( power_power_complex @ Z5 @ N )
% 12.85/7.03 = C ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9346_finite__atLeastZeroLessThan__int,axiom,
% 12.85/7.03 ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9347_finite__divisors__int,axiom,
% 12.85/7.03 ! [I: int] :
% 12.85/7.03 ( ( I != zero_zero_int )
% 12.85/7.03 => ( finite_finite_int
% 12.85/7.03 @ ( collect_int
% 12.85/7.03 @ ^ [D4: int] : ( dvd_dvd_int @ D4 @ I ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9348_finite__nat__iff__bounded__le,axiom,
% 12.85/7.03 ( finite_finite_nat
% 12.85/7.03 = ( ^ [S5: set_nat] :
% 12.85/7.03 ? [K3: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9349_finite__nat__bounded,axiom,
% 12.85/7.03 ! [S: set_nat] :
% 12.85/7.03 ( ( finite_finite_nat @ S )
% 12.85/7.03 => ? [K2: nat] : ( ord_less_eq_set_nat @ S @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9350_finite__nat__iff__bounded,axiom,
% 12.85/7.03 ( finite_finite_nat
% 12.85/7.03 = ( ^ [S5: set_nat] :
% 12.85/7.03 ? [K3: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9351_infinite__int__iff__unbounded__le,axiom,
% 12.85/7.03 ! [S: set_int] :
% 12.85/7.03 ( ( ~ ( finite_finite_int @ S ) )
% 12.85/7.03 = ( ! [M5: int] :
% 12.85/7.03 ? [N4: int] :
% 12.85/7.03 ( ( ord_less_eq_int @ M5 @ ( abs_abs_int @ N4 ) )
% 12.85/7.03 & ( member_int @ N4 @ S ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9352_infinite__nat__iff__unbounded__le,axiom,
% 12.85/7.03 ! [S: set_nat] :
% 12.85/7.03 ( ( ~ ( finite_finite_nat @ S ) )
% 12.85/7.03 = ( ! [M5: nat] :
% 12.85/7.03 ? [N4: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ M5 @ N4 )
% 12.85/7.03 & ( member_nat @ N4 @ S ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9353_Frct__code__post_I5_J,axiom,
% 12.85/7.03 ! [K: num] :
% 12.85/7.03 ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 12.85/7.03 = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9354_set__encode__insert,axiom,
% 12.85/7.03 ! [A2: set_nat,N: nat] :
% 12.85/7.03 ( ( finite_finite_nat @ A2 )
% 12.85/7.03 => ( ~ ( member_nat @ N @ A2 )
% 12.85/7.03 => ( ( nat_set_encode @ ( insert_nat @ N @ A2 ) )
% 12.85/7.03 = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9355_lessThan__Suc,axiom,
% 12.85/7.03 ! [K: nat] :
% 12.85/7.03 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 12.85/7.03 = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9356_atMost__Suc,axiom,
% 12.85/7.03 ! [K: nat] :
% 12.85/7.03 ( ( set_ord_atMost_nat @ ( suc @ K ) )
% 12.85/7.03 = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9357_atLeast0__atMost__Suc,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 12.85/7.03 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9358_atLeast0__lessThan__Suc,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 12.85/7.03 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9359_atLeastAtMost__insertL,axiom,
% 12.85/7.03 ! [M: nat,N: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ M @ N )
% 12.85/7.03 => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 12.85/7.03 = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9360_atLeastAtMostSuc__conv,axiom,
% 12.85/7.03 ! [M: nat,N: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 12.85/7.03 => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
% 12.85/7.03 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9361_Icc__eq__insert__lb__nat,axiom,
% 12.85/7.03 ! [M: nat,N: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ M @ N )
% 12.85/7.03 => ( ( set_or1269000886237332187st_nat @ M @ N )
% 12.85/7.03 = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9362_lessThan__nat__numeral,axiom,
% 12.85/7.03 ! [K: num] :
% 12.85/7.03 ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
% 12.85/7.03 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9363_atMost__nat__numeral,axiom,
% 12.85/7.03 ! [K: num] :
% 12.85/7.03 ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
% 12.85/7.03 = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9364_Frct__code__post_I2_J,axiom,
% 12.85/7.03 ! [A: int] :
% 12.85/7.03 ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
% 12.85/7.03 = zero_zero_rat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9365_Frct__code__post_I1_J,axiom,
% 12.85/7.03 ! [A: int] :
% 12.85/7.03 ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
% 12.85/7.03 = zero_zero_rat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9366_set__decode__plus__power__2,axiom,
% 12.85/7.03 ! [N: nat,Z: nat] :
% 12.85/7.03 ( ~ ( member_nat @ N @ ( nat_set_decode @ Z ) )
% 12.85/7.03 => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z ) )
% 12.85/7.03 = ( insert_nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9367_Frct__code__post_I4_J,axiom,
% 12.85/7.03 ! [K: num] :
% 12.85/7.03 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
% 12.85/7.03 = ( numeral_numeral_rat @ K ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9368_Frct__code__post_I6_J,axiom,
% 12.85/7.03 ! [K: num,L: num] :
% 12.85/7.03 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
% 12.85/7.03 = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9369_divmod__integer__code,axiom,
% 12.85/7.03 ( code_divmod_integer
% 12.85/7.03 = ( ^ [K3: code_integer,L3: code_integer] :
% 12.85/7.03 ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 12.85/7.03 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L3 )
% 12.85/7.03 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L3 )
% 12.85/7.03 @ ( produc6916734918728496179nteger
% 12.85/7.03 @ ^ [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 @ L3 @ S4 ) ) )
% 12.85/7.03 @ ( code_divmod_abs @ K3 @ L3 ) ) )
% 12.85/7.03 @ ( if_Pro6119634080678213985nteger @ ( L3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 12.85/7.03 @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 12.85/7.03 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L3 )
% 12.85/7.03 @ ( produc6916734918728496179nteger
% 12.85/7.03 @ ^ [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 @ L3 ) @ S4 ) ) )
% 12.85/7.03 @ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9370_card__atMost,axiom,
% 12.85/7.03 ! [U: nat] :
% 12.85/7.03 ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
% 12.85/7.03 = ( suc @ U ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9371_card__atLeastLessThan,axiom,
% 12.85/7.03 ! [L: nat,U: nat] :
% 12.85/7.03 ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
% 12.85/7.03 = ( minus_minus_nat @ U @ L ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9372_card__Collect__le__nat,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( finite_card_nat
% 12.85/7.03 @ ( collect_nat
% 12.85/7.03 @ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N ) ) )
% 12.85/7.03 = ( suc @ N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9373_card__atLeastAtMost,axiom,
% 12.85/7.03 ! [L: nat,U: nat] :
% 12.85/7.03 ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 12.85/7.03 = ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9374_card__atLeastLessThan__int,axiom,
% 12.85/7.03 ! [L: int,U: int] :
% 12.85/7.03 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L @ U ) )
% 12.85/7.03 = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9375_card__atLeastAtMost__int,axiom,
% 12.85/7.03 ! [L: int,U: int] :
% 12.85/7.03 ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
% 12.85/7.03 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9376_card__length__sum__list__rec,axiom,
% 12.85/7.03 ! [M: nat,N5: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 12.85/7.03 => ( ( finite_card_list_nat
% 12.85/7.03 @ ( collect_list_nat
% 12.85/7.03 @ ^ [L3: list_nat] :
% 12.85/7.03 ( ( ( size_size_list_nat @ L3 )
% 12.85/7.03 = M )
% 12.85/7.03 & ( ( groups4561878855575611511st_nat @ L3 )
% 12.85/7.03 = N5 ) ) ) )
% 12.85/7.03 = ( plus_plus_nat
% 12.85/7.03 @ ( finite_card_list_nat
% 12.85/7.03 @ ( collect_list_nat
% 12.85/7.03 @ ^ [L3: list_nat] :
% 12.85/7.03 ( ( ( size_size_list_nat @ L3 )
% 12.85/7.03 = ( minus_minus_nat @ M @ one_one_nat ) )
% 12.85/7.03 & ( ( groups4561878855575611511st_nat @ L3 )
% 12.85/7.03 = N5 ) ) ) )
% 12.85/7.03 @ ( finite_card_list_nat
% 12.85/7.03 @ ( collect_list_nat
% 12.85/7.03 @ ^ [L3: list_nat] :
% 12.85/7.03 ( ( ( size_size_list_nat @ L3 )
% 12.85/7.03 = M )
% 12.85/7.03 & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L3 ) @ one_one_nat )
% 12.85/7.03 = N5 ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9377_card__length__sum__list,axiom,
% 12.85/7.03 ! [M: nat,N5: nat] :
% 12.85/7.03 ( ( finite_card_list_nat
% 12.85/7.03 @ ( collect_list_nat
% 12.85/7.03 @ ^ [L3: list_nat] :
% 12.85/7.03 ( ( ( size_size_list_nat @ L3 )
% 12.85/7.03 = M )
% 12.85/7.03 & ( ( groups4561878855575611511st_nat @ L3 )
% 12.85/7.03 = N5 ) ) ) )
% 12.85/7.03 = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N5 @ M ) @ one_one_nat ) @ N5 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9378_card__less__Suc2,axiom,
% 12.85/7.03 ! [M10: set_nat,I: nat] :
% 12.85/7.03 ( ~ ( member_nat @ zero_zero_nat @ M10 )
% 12.85/7.03 => ( ( finite_card_nat
% 12.85/7.03 @ ( collect_nat
% 12.85/7.03 @ ^ [K3: nat] :
% 12.85/7.03 ( ( member_nat @ ( suc @ K3 ) @ M10 )
% 12.85/7.03 & ( ord_less_nat @ K3 @ I ) ) ) )
% 12.85/7.03 = ( finite_card_nat
% 12.85/7.03 @ ( collect_nat
% 12.85/7.03 @ ^ [K3: nat] :
% 12.85/7.03 ( ( member_nat @ K3 @ M10 )
% 12.85/7.03 & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9379_card__less__Suc,axiom,
% 12.85/7.03 ! [M10: set_nat,I: nat] :
% 12.85/7.03 ( ( member_nat @ zero_zero_nat @ M10 )
% 12.85/7.03 => ( ( suc
% 12.85/7.03 @ ( finite_card_nat
% 12.85/7.03 @ ( collect_nat
% 12.85/7.03 @ ^ [K3: nat] :
% 12.85/7.03 ( ( member_nat @ ( suc @ K3 ) @ M10 )
% 12.85/7.03 & ( ord_less_nat @ K3 @ I ) ) ) ) )
% 12.85/7.03 = ( finite_card_nat
% 12.85/7.03 @ ( collect_nat
% 12.85/7.03 @ ^ [K3: nat] :
% 12.85/7.03 ( ( member_nat @ K3 @ M10 )
% 12.85/7.03 & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9380_card__less,axiom,
% 12.85/7.03 ! [M10: set_nat,I: nat] :
% 12.85/7.03 ( ( member_nat @ zero_zero_nat @ M10 )
% 12.85/7.03 => ( ( finite_card_nat
% 12.85/7.03 @ ( collect_nat
% 12.85/7.03 @ ^ [K3: nat] :
% 12.85/7.03 ( ( member_nat @ K3 @ M10 )
% 12.85/7.03 & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
% 12.85/7.03 != zero_zero_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9381_card__atLeastZeroLessThan__int,axiom,
% 12.85/7.03 ! [U: int] :
% 12.85/7.03 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
% 12.85/7.03 = ( nat2 @ U ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9382_subset__card__intvl__is__intvl,axiom,
% 12.85/7.03 ! [A2: set_nat,K: nat] :
% 12.85/7.03 ( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
% 12.85/7.03 => ( A2
% 12.85/7.03 = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9383_atLeastAtMostPlus1__int__conv,axiom,
% 12.85/7.03 ! [M: int,N: int] :
% 12.85/7.03 ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 12.85/7.03 => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 12.85/7.03 = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9384_subset__eq__atLeast0__lessThan__card,axiom,
% 12.85/7.03 ! [N5: set_nat,N: nat] :
% 12.85/7.03 ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 12.85/7.03 => ( ord_less_eq_nat @ ( finite_card_nat @ N5 ) @ N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9385_card__sum__le__nat__sum,axiom,
% 12.85/7.03 ! [S: set_nat] :
% 12.85/7.03 ( ord_less_eq_nat
% 12.85/7.03 @ ( groups3542108847815614940at_nat
% 12.85/7.03 @ ^ [X3: nat] : X3
% 12.85/7.03 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S ) ) )
% 12.85/7.03 @ ( groups3542108847815614940at_nat
% 12.85/7.03 @ ^ [X3: nat] : X3
% 12.85/7.03 @ S ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9386_card__nth__roots,axiom,
% 12.85/7.03 ! [C: complex,N: nat] :
% 12.85/7.03 ( ( C != zero_zero_complex )
% 12.85/7.03 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( finite_card_complex
% 12.85/7.03 @ ( collect_complex
% 12.85/7.03 @ ^ [Z5: complex] :
% 12.85/7.03 ( ( power_power_complex @ Z5 @ N )
% 12.85/7.03 = C ) ) )
% 12.85/7.03 = N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9387_card__roots__unity__eq,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( finite_card_complex
% 12.85/7.03 @ ( collect_complex
% 12.85/7.03 @ ^ [Z5: complex] :
% 12.85/7.03 ( ( power_power_complex @ Z5 @ N )
% 12.85/7.03 = one_one_complex ) ) )
% 12.85/7.03 = N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9388_and__int_Osimps,axiom,
% 12.85/7.03 ( bit_se725231765392027082nd_int
% 12.85/7.03 = ( ^ [K3: int,L3: int] :
% 12.85/7.03 ( if_int
% 12.85/7.03 @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 12.85/7.03 & ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 12.85/7.03 @ ( uminus_uminus_int
% 12.85/7.03 @ ( zero_n2684676970156552555ol_int
% 12.85/7.03 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 12.85/7.03 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) ) )
% 12.85/7.03 @ ( plus_plus_int
% 12.85/7.03 @ ( zero_n2684676970156552555ol_int
% 12.85/7.03 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 12.85/7.03 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
% 12.85/7.03 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9389_and__int_Oelims,axiom,
% 12.85/7.03 ! [X: int,Xa: int,Y4: int] :
% 12.85/7.03 ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 12.85/7.03 = Y4 )
% 12.85/7.03 => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 12.85/7.03 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( uminus_uminus_int
% 12.85/7.03 @ ( zero_n2684676970156552555ol_int
% 12.85/7.03 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 12.85/7.03 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 12.85/7.03 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 12.85/7.03 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( plus_plus_int
% 12.85/7.03 @ ( zero_n2684676970156552555ol_int
% 12.85/7.03 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 12.85/7.03 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 12.85/7.03 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9390_and__int_Opsimps,axiom,
% 12.85/7.03 ! [K: int,L: int] :
% 12.85/7.03 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
% 12.85/7.03 => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 12.85/7.03 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 12.85/7.03 => ( ( bit_se725231765392027082nd_int @ K @ L )
% 12.85/7.03 = ( uminus_uminus_int
% 12.85/7.03 @ ( zero_n2684676970156552555ol_int
% 12.85/7.03 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 12.85/7.03 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
% 12.85/7.03 & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 12.85/7.03 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 12.85/7.03 => ( ( bit_se725231765392027082nd_int @ K @ L )
% 12.85/7.03 = ( plus_plus_int
% 12.85/7.03 @ ( zero_n2684676970156552555ol_int
% 12.85/7.03 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 12.85/7.03 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 12.85/7.03 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9391_and__int_Opelims,axiom,
% 12.85/7.03 ! [X: int,Xa: int,Y4: int] :
% 12.85/7.03 ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 12.85/7.03 = Y4 )
% 12.85/7.03 => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) )
% 12.85/7.03 => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 12.85/7.03 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( uminus_uminus_int
% 12.85/7.03 @ ( zero_n2684676970156552555ol_int
% 12.85/7.03 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 12.85/7.03 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 12.85/7.03 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 12.85/7.03 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( plus_plus_int
% 12.85/7.03 @ ( zero_n2684676970156552555ol_int
% 12.85/7.03 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 12.85/7.03 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 12.85/7.03 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 12.85/7.03 => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9392_and__int_Opinduct,axiom,
% 12.85/7.03 ! [A0: int,A1: int,P: int > int > $o] :
% 12.85/7.03 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
% 12.85/7.03 => ( ! [K2: int,L2: int] :
% 12.85/7.03 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L2 ) )
% 12.85/7.03 => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 12.85/7.03 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 12.85/7.03 => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 12.85/7.03 => ( P @ K2 @ L2 ) ) )
% 12.85/7.03 => ( P @ A0 @ A1 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9393_lessThan__0,axiom,
% 12.85/7.03 ( ( set_ord_lessThan_nat @ zero_zero_nat )
% 12.85/7.03 = bot_bot_set_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9394_set__decode__zero,axiom,
% 12.85/7.03 ( ( nat_set_decode @ zero_zero_nat )
% 12.85/7.03 = bot_bot_set_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9395_set__encode__empty,axiom,
% 12.85/7.03 ( ( nat_set_encode @ bot_bot_set_nat )
% 12.85/7.03 = zero_zero_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9396_atLeastLessThan__singleton,axiom,
% 12.85/7.03 ! [M: nat] :
% 12.85/7.03 ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
% 12.85/7.03 = ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9397_atMost__0,axiom,
% 12.85/7.03 ( ( set_ord_atMost_nat @ zero_zero_nat )
% 12.85/7.03 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9398_bot__enat__def,axiom,
% 12.85/7.03 bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9399_bot__nat__def,axiom,
% 12.85/7.03 bot_bot_nat = zero_zero_nat ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9400_atLeastLessThan0,axiom,
% 12.85/7.03 ! [M: nat] :
% 12.85/7.03 ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
% 12.85/7.03 = bot_bot_set_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9401_lessThan__empty__iff,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( ( set_ord_lessThan_nat @ N )
% 12.85/7.03 = bot_bot_set_nat )
% 12.85/7.03 = ( N = zero_zero_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9402_atLeastLessThanSuc,axiom,
% 12.85/7.03 ! [M: nat,N: nat] :
% 12.85/7.03 ( ( ( ord_less_eq_nat @ M @ N )
% 12.85/7.03 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 12.85/7.03 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
% 12.85/7.03 & ( ~ ( ord_less_eq_nat @ M @ N )
% 12.85/7.03 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 12.85/7.03 = bot_bot_set_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9403_atLeast1__lessThan__eq__remove0,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.85/7.03 = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9404_atLeast1__atMost__eq__remove0,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
% 12.85/7.03 = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9405_atLeastLessThan__nat__numeral,axiom,
% 12.85/7.03 ! [M: nat,K: num] :
% 12.85/7.03 ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 12.85/7.03 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 12.85/7.03 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 12.85/7.03 & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 12.85/7.03 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 12.85/7.03 = bot_bot_set_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9406_length__upt,axiom,
% 12.85/7.03 ! [I: nat,J: nat] :
% 12.85/7.03 ( ( size_size_list_nat @ ( upt @ I @ J ) )
% 12.85/7.03 = ( minus_minus_nat @ J @ I ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9407_sum__list__upt,axiom,
% 12.85/7.03 ! [M: nat,N: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ M @ N )
% 12.85/7.03 => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
% 12.85/7.03 = ( groups3542108847815614940at_nat
% 12.85/7.03 @ ^ [X3: nat] : X3
% 12.85/7.03 @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9408_map__Suc__upt,axiom,
% 12.85/7.03 ! [M: nat,N: nat] :
% 12.85/7.03 ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
% 12.85/7.03 = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9409_map__add__upt,axiom,
% 12.85/7.03 ! [N: nat,M: nat] :
% 12.85/7.03 ( ( map_nat_nat
% 12.85/7.03 @ ^ [I2: nat] : ( plus_plus_nat @ I2 @ N )
% 12.85/7.03 @ ( upt @ zero_zero_nat @ M ) )
% 12.85/7.03 = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9410_map__decr__upt,axiom,
% 12.85/7.03 ! [M: nat,N: nat] :
% 12.85/7.03 ( ( map_nat_nat
% 12.85/7.03 @ ^ [N4: nat] : ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) )
% 12.85/7.03 @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.85/7.03 = ( upt @ M @ N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9411_atLeastAtMost__upt,axiom,
% 12.85/7.03 ( set_or1269000886237332187st_nat
% 12.85/7.03 = ( ^ [N4: nat,M5: nat] : ( set_nat2 @ ( upt @ N4 @ ( suc @ M5 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9412_atLeast__upt,axiom,
% 12.85/7.03 ( set_ord_lessThan_nat
% 12.85/7.03 = ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N4 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9413_atMost__upto,axiom,
% 12.85/7.03 ( set_ord_atMost_nat
% 12.85/7.03 = ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N4 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9414_sorted__upt,axiom,
% 12.85/7.03 ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9415_sorted__wrt__less__idx,axiom,
% 12.85/7.03 ! [Ns: list_nat,I: nat] :
% 12.85/7.03 ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
% 12.85/7.03 => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
% 12.85/7.03 => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9416_binomial__def,axiom,
% 12.85/7.03 ( binomial
% 12.85/7.03 = ( ^ [N4: nat,K3: nat] :
% 12.85/7.03 ( finite_card_set_nat
% 12.85/7.03 @ ( collect_set_nat
% 12.85/7.03 @ ^ [K7: set_nat] :
% 12.85/7.03 ( ( member_set_nat @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) )
% 12.85/7.03 & ( ( finite_card_nat @ K7 )
% 12.85/7.03 = K3 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9417_Suc__0__div__numeral,axiom,
% 12.85/7.03 ! [K: num] :
% 12.85/7.03 ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 12.85/7.03 = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9418_Suc__0__mod__numeral,axiom,
% 12.85/7.03 ! [K: num] :
% 12.85/7.03 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 12.85/7.03 = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9419_fst__divmod__nat,axiom,
% 12.85/7.03 ! [M: nat,N: nat] :
% 12.85/7.03 ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
% 12.85/7.03 = ( divide_divide_nat @ M @ N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9420_snd__divmod__nat,axiom,
% 12.85/7.03 ! [M: nat,N: nat] :
% 12.85/7.03 ( ( product_snd_nat_nat @ ( divmod_nat @ M @ N ) )
% 12.85/7.03 = ( modulo_modulo_nat @ M @ N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9421_quotient__of__denom__pos_H,axiom,
% 12.85/7.03 ! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9422_bezw__non__0,axiom,
% 12.85/7.03 ! [Y4: nat,X: nat] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ Y4 )
% 12.85/7.03 => ( ( bezw @ X @ Y4 )
% 12.85/7.03 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X @ Y4 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X @ Y4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X @ Y4 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y4 ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9423_bezw_Osimps,axiom,
% 12.85/7.03 ( bezw
% 12.85/7.03 = ( ^ [X3: nat,Y6: nat] : ( if_Pro3027730157355071871nt_int @ ( Y6 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X3 @ Y6 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X3 @ Y6 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X3 @ Y6 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Y6 ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9424_bezw_Oelims,axiom,
% 12.85/7.03 ! [X: nat,Xa: nat,Y4: product_prod_int_int] :
% 12.85/7.03 ( ( ( bezw @ X @ Xa )
% 12.85/7.03 = Y4 )
% 12.85/7.03 => ( ( ( Xa = zero_zero_nat )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 12.85/7.03 & ( ( Xa != zero_zero_nat )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9425_bezw_Opelims,axiom,
% 12.85/7.03 ! [X: nat,Xa: nat,Y4: product_prod_int_int] :
% 12.85/7.03 ( ( ( bezw @ X @ Xa )
% 12.85/7.03 = Y4 )
% 12.85/7.03 => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 12.85/7.03 => ~ ( ( ( ( Xa = zero_zero_nat )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 12.85/7.03 & ( ( Xa != zero_zero_nat )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) )
% 12.85/7.03 => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9426_one__mod__minus__numeral,axiom,
% 12.85/7.03 ! [N: num] :
% 12.85/7.03 ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.03 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9427_minus__one__mod__numeral,axiom,
% 12.85/7.03 ! [N: num] :
% 12.85/7.03 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 12.85/7.03 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9428_minus__numeral__mod__numeral,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.85/7.03 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9429_numeral__mod__minus__numeral,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.03 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9430_Divides_Oadjust__mod__def,axiom,
% 12.85/7.03 ( adjust_mod
% 12.85/7.03 = ( ^ [L3: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L3 @ R5 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9431_normalize__def,axiom,
% 12.85/7.03 ( normalize
% 12.85/7.03 = ( ^ [P6: product_prod_int_int] :
% 12.85/7.03 ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P6 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) )
% 12.85/7.03 @ ( if_Pro3027730157355071871nt_int
% 12.85/7.03 @ ( ( product_snd_int_int @ P6 )
% 12.85/7.03 = zero_zero_int )
% 12.85/7.03 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 12.85/7.03 @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9432_gcd__pos__int,axiom,
% 12.85/7.03 ! [M: int,N: int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
% 12.85/7.03 = ( ( M != zero_zero_int )
% 12.85/7.03 | ( N != zero_zero_int ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9433_gcd__neg__numeral__1__int,axiom,
% 12.85/7.03 ! [N: num,X: int] :
% 12.85/7.03 ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ X )
% 12.85/7.03 = ( gcd_gcd_int @ ( numeral_numeral_int @ N ) @ X ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9434_gcd__neg__numeral__2__int,axiom,
% 12.85/7.03 ! [X: int,N: num] :
% 12.85/7.03 ( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.03 = ( gcd_gcd_int @ X @ ( numeral_numeral_int @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9435_gcd__0__int,axiom,
% 12.85/7.03 ! [X: int] :
% 12.85/7.03 ( ( gcd_gcd_int @ X @ zero_zero_int )
% 12.85/7.03 = ( abs_abs_int @ X ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9436_gcd__0__left__int,axiom,
% 12.85/7.03 ! [X: int] :
% 12.85/7.03 ( ( gcd_gcd_int @ zero_zero_int @ X )
% 12.85/7.03 = ( abs_abs_int @ X ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9437_gcd__proj2__if__dvd__int,axiom,
% 12.85/7.03 ! [Y4: int,X: int] :
% 12.85/7.03 ( ( dvd_dvd_int @ Y4 @ X )
% 12.85/7.03 => ( ( gcd_gcd_int @ X @ Y4 )
% 12.85/7.03 = ( abs_abs_int @ Y4 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9438_gcd__proj1__if__dvd__int,axiom,
% 12.85/7.03 ! [X: int,Y4: int] :
% 12.85/7.03 ( ( dvd_dvd_int @ X @ Y4 )
% 12.85/7.03 => ( ( gcd_gcd_int @ X @ Y4 )
% 12.85/7.03 = ( abs_abs_int @ X ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9439_gcd__ge__0__int,axiom,
% 12.85/7.03 ! [X: int,Y4: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y4 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9440_bezout__int,axiom,
% 12.85/7.03 ! [X: int,Y4: int] :
% 12.85/7.03 ? [U2: int,V3: int] :
% 12.85/7.03 ( ( plus_plus_int @ ( times_times_int @ U2 @ X ) @ ( times_times_int @ V3 @ Y4 ) )
% 12.85/7.03 = ( gcd_gcd_int @ X @ Y4 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9441_gcd__mult__distrib__int,axiom,
% 12.85/7.03 ! [K: int,M: int,N: int] :
% 12.85/7.03 ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
% 12.85/7.03 = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9442_gcd__le1__int,axiom,
% 12.85/7.03 ! [A: int,B: int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ A )
% 12.85/7.03 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9443_gcd__le2__int,axiom,
% 12.85/7.03 ! [B: int,A: int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/7.03 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9444_gcd__cases__int,axiom,
% 12.85/7.03 ! [X: int,Y4: int,P: int > $o] :
% 12.85/7.03 ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/7.03 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/7.03 => ( P @ ( gcd_gcd_int @ X @ Y4 ) ) ) )
% 12.85/7.03 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 12.85/7.03 => ( ( ord_less_eq_int @ Y4 @ zero_zero_int )
% 12.85/7.03 => ( P @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y4 ) ) ) ) )
% 12.85/7.03 => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 12.85/7.03 => ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
% 12.85/7.03 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y4 ) ) ) )
% 12.85/7.03 => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 12.85/7.03 => ( ( ord_less_eq_int @ Y4 @ zero_zero_int )
% 12.85/7.03 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y4 ) ) ) ) )
% 12.85/7.03 => ( P @ ( gcd_gcd_int @ X @ Y4 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9445_gcd__unique__int,axiom,
% 12.85/7.03 ! [D: int,A: int,B: int] :
% 12.85/7.03 ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 12.85/7.03 & ( dvd_dvd_int @ D @ A )
% 12.85/7.03 & ( dvd_dvd_int @ D @ B )
% 12.85/7.03 & ! [E3: int] :
% 12.85/7.03 ( ( ( dvd_dvd_int @ E3 @ A )
% 12.85/7.03 & ( dvd_dvd_int @ E3 @ B ) )
% 12.85/7.03 => ( dvd_dvd_int @ E3 @ D ) ) )
% 12.85/7.03 = ( D
% 12.85/7.03 = ( gcd_gcd_int @ A @ B ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9446_gcd__non__0__int,axiom,
% 12.85/7.03 ! [Y4: int,X: int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ Y4 )
% 12.85/7.03 => ( ( gcd_gcd_int @ X @ Y4 )
% 12.85/7.03 = ( gcd_gcd_int @ Y4 @ ( modulo_modulo_int @ X @ Y4 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9447_gcd__code__int,axiom,
% 12.85/7.03 ( gcd_gcd_int
% 12.85/7.03 = ( ^ [K3: int,L3: int] : ( abs_abs_int @ ( if_int @ ( L3 = zero_zero_int ) @ K3 @ ( gcd_gcd_int @ L3 @ ( modulo_modulo_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L3 ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9448_gcd__0__left__nat,axiom,
% 12.85/7.03 ! [X: nat] :
% 12.85/7.03 ( ( gcd_gcd_nat @ zero_zero_nat @ X )
% 12.85/7.03 = X ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9449_gcd__0__nat,axiom,
% 12.85/7.03 ! [X: nat] :
% 12.85/7.03 ( ( gcd_gcd_nat @ X @ zero_zero_nat )
% 12.85/7.03 = X ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9450_gcd__nat_Oright__neutral,axiom,
% 12.85/7.03 ! [A: nat] :
% 12.85/7.03 ( ( gcd_gcd_nat @ A @ zero_zero_nat )
% 12.85/7.03 = A ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9451_gcd__nat_Oneutr__eq__iff,axiom,
% 12.85/7.03 ! [A: nat,B: nat] :
% 12.85/7.03 ( ( zero_zero_nat
% 12.85/7.03 = ( gcd_gcd_nat @ A @ B ) )
% 12.85/7.03 = ( ( A = zero_zero_nat )
% 12.85/7.03 & ( B = zero_zero_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9452_gcd__nat_Oleft__neutral,axiom,
% 12.85/7.03 ! [A: nat] :
% 12.85/7.03 ( ( gcd_gcd_nat @ zero_zero_nat @ A )
% 12.85/7.03 = A ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9453_gcd__nat_Oeq__neutr__iff,axiom,
% 12.85/7.03 ! [A: nat,B: nat] :
% 12.85/7.03 ( ( ( gcd_gcd_nat @ A @ B )
% 12.85/7.03 = zero_zero_nat )
% 12.85/7.03 = ( ( A = zero_zero_nat )
% 12.85/7.03 & ( B = zero_zero_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9454_gcd__nat_Oabsorb1,axiom,
% 12.85/7.03 ! [A: nat,B: nat] :
% 12.85/7.03 ( ( dvd_dvd_nat @ A @ B )
% 12.85/7.03 => ( ( gcd_gcd_nat @ A @ B )
% 12.85/7.03 = A ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9455_gcd__nat_Oabsorb2,axiom,
% 12.85/7.03 ! [B: nat,A: nat] :
% 12.85/7.03 ( ( dvd_dvd_nat @ B @ A )
% 12.85/7.03 => ( ( gcd_gcd_nat @ A @ B )
% 12.85/7.03 = B ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9456_gcd__nat_Obounded__iff,axiom,
% 12.85/7.03 ! [A: nat,B: nat,C: nat] :
% 12.85/7.03 ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
% 12.85/7.03 = ( ( dvd_dvd_nat @ A @ B )
% 12.85/7.03 & ( dvd_dvd_nat @ A @ C ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9457_gcd__proj1__if__dvd__nat,axiom,
% 12.85/7.03 ! [X: nat,Y4: nat] :
% 12.85/7.03 ( ( dvd_dvd_nat @ X @ Y4 )
% 12.85/7.03 => ( ( gcd_gcd_nat @ X @ Y4 )
% 12.85/7.03 = X ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9458_gcd__proj2__if__dvd__nat,axiom,
% 12.85/7.03 ! [Y4: nat,X: nat] :
% 12.85/7.03 ( ( dvd_dvd_nat @ Y4 @ X )
% 12.85/7.03 => ( ( gcd_gcd_nat @ X @ Y4 )
% 12.85/7.03 = Y4 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9459_gcd__Suc__0,axiom,
% 12.85/7.03 ! [M: nat] :
% 12.85/7.03 ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
% 12.85/7.03 = ( suc @ zero_zero_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9460_gcd__pos__nat,axiom,
% 12.85/7.03 ! [M: nat,N: nat] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
% 12.85/7.03 = ( ( M != zero_zero_nat )
% 12.85/7.03 | ( N != zero_zero_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9461_gcd__int__int__eq,axiom,
% 12.85/7.03 ! [M: nat,N: nat] :
% 12.85/7.03 ( ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 12.85/7.03 = ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ M @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9462_gcd__nat__abs__right__eq,axiom,
% 12.85/7.03 ! [N: nat,K: int] :
% 12.85/7.03 ( ( gcd_gcd_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 12.85/7.03 = ( nat2 @ ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9463_gcd__nat__abs__left__eq,axiom,
% 12.85/7.03 ! [K: int,N: nat] :
% 12.85/7.03 ( ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 12.85/7.03 = ( nat2 @ ( gcd_gcd_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9464_gcd__mult__distrib__nat,axiom,
% 12.85/7.03 ! [K: nat,M: nat,N: nat] :
% 12.85/7.03 ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
% 12.85/7.03 = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9465_gcd__nat_Omono,axiom,
% 12.85/7.03 ! [A: nat,C: nat,B: nat,D: nat] :
% 12.85/7.03 ( ( dvd_dvd_nat @ A @ C )
% 12.85/7.03 => ( ( dvd_dvd_nat @ B @ D )
% 12.85/7.03 => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ ( gcd_gcd_nat @ C @ D ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9466_gcd__nat_OorderE,axiom,
% 12.85/7.03 ! [A: nat,B: nat] :
% 12.85/7.03 ( ( dvd_dvd_nat @ A @ B )
% 12.85/7.03 => ( A
% 12.85/7.03 = ( gcd_gcd_nat @ A @ B ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9467_gcd__nat_OorderI,axiom,
% 12.85/7.03 ! [A: nat,B: nat] :
% 12.85/7.03 ( ( A
% 12.85/7.03 = ( gcd_gcd_nat @ A @ B ) )
% 12.85/7.03 => ( dvd_dvd_nat @ A @ B ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9468_gcd__nat_Oabsorb3,axiom,
% 12.85/7.03 ! [A: nat,B: nat] :
% 12.85/7.03 ( ( ( dvd_dvd_nat @ A @ B )
% 12.85/7.03 & ( A != B ) )
% 12.85/7.03 => ( ( gcd_gcd_nat @ A @ B )
% 12.85/7.03 = A ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9469_gcd__nat_Oabsorb4,axiom,
% 12.85/7.03 ! [B: nat,A: nat] :
% 12.85/7.03 ( ( ( dvd_dvd_nat @ B @ A )
% 12.85/7.03 & ( B != A ) )
% 12.85/7.03 => ( ( gcd_gcd_nat @ A @ B )
% 12.85/7.03 = B ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9470_gcd__nat_OboundedE,axiom,
% 12.85/7.03 ! [A: nat,B: nat,C: nat] :
% 12.85/7.03 ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
% 12.85/7.03 => ~ ( ( dvd_dvd_nat @ A @ B )
% 12.85/7.03 => ~ ( dvd_dvd_nat @ A @ C ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9471_gcd__nat_OboundedI,axiom,
% 12.85/7.03 ! [A: nat,B: nat,C: nat] :
% 12.85/7.03 ( ( dvd_dvd_nat @ A @ B )
% 12.85/7.03 => ( ( dvd_dvd_nat @ A @ C )
% 12.85/7.03 => ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9472_gcd__nat_Oorder__iff,axiom,
% 12.85/7.03 ( dvd_dvd_nat
% 12.85/7.03 = ( ^ [A3: nat,B2: nat] :
% 12.85/7.03 ( A3
% 12.85/7.03 = ( gcd_gcd_nat @ A3 @ B2 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9473_gcd__nat_Ocobounded1,axiom,
% 12.85/7.03 ! [A: nat,B: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9474_gcd__nat_Ocobounded2,axiom,
% 12.85/7.03 ! [A: nat,B: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9475_gcd__nat_Oabsorb__iff1,axiom,
% 12.85/7.03 ( dvd_dvd_nat
% 12.85/7.03 = ( ^ [A3: nat,B2: nat] :
% 12.85/7.03 ( ( gcd_gcd_nat @ A3 @ B2 )
% 12.85/7.03 = A3 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9476_gcd__nat_Oabsorb__iff2,axiom,
% 12.85/7.03 ( dvd_dvd_nat
% 12.85/7.03 = ( ^ [B2: nat,A3: nat] :
% 12.85/7.03 ( ( gcd_gcd_nat @ A3 @ B2 )
% 12.85/7.03 = B2 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9477_gcd__nat_OcoboundedI1,axiom,
% 12.85/7.03 ! [A: nat,C: nat,B: nat] :
% 12.85/7.03 ( ( dvd_dvd_nat @ A @ C )
% 12.85/7.03 => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9478_gcd__nat_OcoboundedI2,axiom,
% 12.85/7.03 ! [B: nat,C: nat,A: nat] :
% 12.85/7.03 ( ( dvd_dvd_nat @ B @ C )
% 12.85/7.03 => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9479_gcd__nat_Ostrict__boundedE,axiom,
% 12.85/7.03 ! [A: nat,B: nat,C: nat] :
% 12.85/7.03 ( ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
% 12.85/7.03 & ( A
% 12.85/7.03 != ( gcd_gcd_nat @ B @ C ) ) )
% 12.85/7.03 => ~ ( ( ( dvd_dvd_nat @ A @ B )
% 12.85/7.03 & ( A != B ) )
% 12.85/7.03 => ~ ( ( dvd_dvd_nat @ A @ C )
% 12.85/7.03 & ( A != C ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9480_gcd__nat_Ostrict__order__iff,axiom,
% 12.85/7.03 ! [A: nat,B: nat] :
% 12.85/7.03 ( ( ( dvd_dvd_nat @ A @ B )
% 12.85/7.03 & ( A != B ) )
% 12.85/7.03 = ( ( A
% 12.85/7.03 = ( gcd_gcd_nat @ A @ B ) )
% 12.85/7.03 & ( A != B ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9481_gcd__nat_Ostrict__coboundedI1,axiom,
% 12.85/7.03 ! [A: nat,C: nat,B: nat] :
% 12.85/7.03 ( ( ( dvd_dvd_nat @ A @ C )
% 12.85/7.03 & ( A != C ) )
% 12.85/7.03 => ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
% 12.85/7.03 & ( ( gcd_gcd_nat @ A @ B )
% 12.85/7.03 != C ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9482_gcd__nat_Ostrict__coboundedI2,axiom,
% 12.85/7.03 ! [B: nat,C: nat,A: nat] :
% 12.85/7.03 ( ( ( dvd_dvd_nat @ B @ C )
% 12.85/7.03 & ( B != C ) )
% 12.85/7.03 => ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
% 12.85/7.03 & ( ( gcd_gcd_nat @ A @ B )
% 12.85/7.03 != C ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9483_gcd__unique__nat,axiom,
% 12.85/7.03 ! [D: nat,A: nat,B: nat] :
% 12.85/7.03 ( ( ( dvd_dvd_nat @ D @ A )
% 12.85/7.03 & ( dvd_dvd_nat @ D @ B )
% 12.85/7.03 & ! [E3: nat] :
% 12.85/7.03 ( ( ( dvd_dvd_nat @ E3 @ A )
% 12.85/7.03 & ( dvd_dvd_nat @ E3 @ B ) )
% 12.85/7.03 => ( dvd_dvd_nat @ E3 @ D ) ) )
% 12.85/7.03 = ( D
% 12.85/7.03 = ( gcd_gcd_nat @ A @ B ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9484_gcd__le2__nat,axiom,
% 12.85/7.03 ! [B: nat,A: nat] :
% 12.85/7.03 ( ( B != zero_zero_nat )
% 12.85/7.03 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9485_gcd__le1__nat,axiom,
% 12.85/7.03 ! [A: nat,B: nat] :
% 12.85/7.03 ( ( A != zero_zero_nat )
% 12.85/7.03 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9486_gcd__diff1__nat,axiom,
% 12.85/7.03 ! [N: nat,M: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ N @ M )
% 12.85/7.03 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 12.85/7.03 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9487_gcd__diff2__nat,axiom,
% 12.85/7.03 ! [M: nat,N: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ M @ N )
% 12.85/7.03 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
% 12.85/7.03 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9488_gcd__nat_Oelims,axiom,
% 12.85/7.03 ! [X: nat,Xa: nat,Y4: nat] :
% 12.85/7.03 ( ( ( gcd_gcd_nat @ X @ Xa )
% 12.85/7.03 = Y4 )
% 12.85/7.03 => ( ( ( Xa = zero_zero_nat )
% 12.85/7.03 => ( Y4 = X ) )
% 12.85/7.03 & ( ( Xa != zero_zero_nat )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9489_gcd__nat_Osimps,axiom,
% 12.85/7.03 ( gcd_gcd_nat
% 12.85/7.03 = ( ^ [X3: nat,Y6: nat] : ( if_nat @ ( Y6 = zero_zero_nat ) @ X3 @ ( gcd_gcd_nat @ Y6 @ ( modulo_modulo_nat @ X3 @ Y6 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9490_gcd__non__0__nat,axiom,
% 12.85/7.03 ! [Y4: nat,X: nat] :
% 12.85/7.03 ( ( Y4 != zero_zero_nat )
% 12.85/7.03 => ( ( gcd_gcd_nat @ X @ Y4 )
% 12.85/7.03 = ( gcd_gcd_nat @ Y4 @ ( modulo_modulo_nat @ X @ Y4 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9491_gcd__code__integer,axiom,
% 12.85/7.03 ( gcd_gcd_Code_integer
% 12.85/7.03 = ( ^ [K3: code_integer,L3: code_integer] : ( abs_abs_Code_integer @ ( if_Code_integer @ ( L3 = zero_z3403309356797280102nteger ) @ K3 @ ( gcd_gcd_Code_integer @ L3 @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L3 ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9492_bezout__nat,axiom,
% 12.85/7.03 ! [A: nat,B: nat] :
% 12.85/7.03 ( ( A != zero_zero_nat )
% 12.85/7.03 => ? [X4: nat,Y: nat] :
% 12.85/7.03 ( ( times_times_nat @ A @ X4 )
% 12.85/7.03 = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9493_bezout__gcd__nat_H,axiom,
% 12.85/7.03 ! [B: nat,A: nat] :
% 12.85/7.03 ? [X4: nat,Y: nat] :
% 12.85/7.03 ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y ) @ ( times_times_nat @ A @ X4 ) )
% 12.85/7.03 & ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y ) )
% 12.85/7.03 = ( gcd_gcd_nat @ A @ B ) ) )
% 12.85/7.03 | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y ) @ ( times_times_nat @ B @ X4 ) )
% 12.85/7.03 & ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y ) )
% 12.85/7.03 = ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9494_gcd__int__def,axiom,
% 12.85/7.03 ( gcd_gcd_int
% 12.85/7.03 = ( ^ [X3: int,Y6: int] : ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ X3 ) ) @ ( nat2 @ ( abs_abs_int @ Y6 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9495_bezw__aux,axiom,
% 12.85/7.03 ! [X: nat,Y4: nat] :
% 12.85/7.03 ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y4 ) )
% 12.85/7.03 = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y4 ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y4 ) ) @ ( semiri1314217659103216013at_int @ Y4 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9496_nat__descend__induct,axiom,
% 12.85/7.03 ! [N: nat,P: nat > $o,M: nat] :
% 12.85/7.03 ( ! [K2: nat] :
% 12.85/7.03 ( ( ord_less_nat @ N @ K2 )
% 12.85/7.03 => ( P @ K2 ) )
% 12.85/7.03 => ( ! [K2: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ K2 @ N )
% 12.85/7.03 => ( ! [I4: nat] :
% 12.85/7.03 ( ( ord_less_nat @ K2 @ I4 )
% 12.85/7.03 => ( P @ I4 ) )
% 12.85/7.03 => ( P @ K2 ) ) )
% 12.85/7.03 => ( P @ M ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9497_gcd__nat_Opelims,axiom,
% 12.85/7.03 ! [X: nat,Xa: nat,Y4: nat] :
% 12.85/7.03 ( ( ( gcd_gcd_nat @ X @ Xa )
% 12.85/7.03 = Y4 )
% 12.85/7.03 => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 12.85/7.03 => ~ ( ( ( ( Xa = zero_zero_nat )
% 12.85/7.03 => ( Y4 = X ) )
% 12.85/7.03 & ( ( Xa != zero_zero_nat )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) ) )
% 12.85/7.03 => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9498_Sup__nat__empty,axiom,
% 12.85/7.03 ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
% 12.85/7.03 = zero_zero_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9499_divmod__integer__eq__cases,axiom,
% 12.85/7.03 ( code_divmod_integer
% 12.85/7.03 = ( ^ [K3: code_integer,L3: code_integer] :
% 12.85/7.03 ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 12.85/7.03 @ ( if_Pro6119634080678213985nteger @ ( L3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 12.85/7.03 @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L3
% 12.85/7.03 @ ( if_Pro6119634080678213985nteger
% 12.85/7.03 @ ( ( sgn_sgn_Code_integer @ K3 )
% 12.85/7.03 = ( sgn_sgn_Code_integer @ L3 ) )
% 12.85/7.03 @ ( code_divmod_abs @ K3 @ L3 )
% 12.85/7.03 @ ( produc6916734918728496179nteger
% 12.85/7.03 @ ^ [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 @ L3 ) @ S4 ) ) )
% 12.85/7.03 @ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9500_card__greaterThanLessThan__int,axiom,
% 12.85/7.03 ! [L: int,U: int] :
% 12.85/7.03 ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
% 12.85/7.03 = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9501_Suc__funpow,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( compow_nat_nat @ N @ suc )
% 12.85/7.03 = ( plus_plus_nat @ N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9502_card__greaterThanLessThan,axiom,
% 12.85/7.03 ! [L: nat,U: nat] :
% 12.85/7.03 ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
% 12.85/7.03 = ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9503_card_Ocomp__fun__commute__on,axiom,
% 12.85/7.03 ( ( comp_nat_nat_nat @ suc @ suc )
% 12.85/7.03 = ( comp_nat_nat_nat @ suc @ suc ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9504_atLeastSucLessThan__greaterThanLessThan,axiom,
% 12.85/7.03 ! [L: nat,U: nat] :
% 12.85/7.03 ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
% 12.85/7.03 = ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9505_greaterThanLessThan__upt,axiom,
% 12.85/7.03 ( set_or5834768355832116004an_nat
% 12.85/7.03 = ( ^ [N4: nat,M5: nat] : ( set_nat2 @ ( upt @ ( suc @ N4 ) @ M5 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9506_Code__Target__Int_Onegative__def,axiom,
% 12.85/7.03 ( code_Target_negative
% 12.85/7.03 = ( comp_int_int_num @ uminus_uminus_int @ numeral_numeral_int ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9507_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 12.85/7.03 ! [N: nat,J: nat,I: nat] :
% 12.85/7.03 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
% 12.85/7.03 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
% 12.85/7.03 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9508_nat__of__integer__non__positive,axiom,
% 12.85/7.03 ! [K: code_integer] :
% 12.85/7.03 ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
% 12.85/7.03 => ( ( code_nat_of_integer @ K )
% 12.85/7.03 = zero_zero_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9509_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 12.85/7.03 ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 12.85/7.03 @ ^ [X3: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X3 )
% 12.85/7.03 @ ^ [X3: nat,Y6: nat] : ( ord_less_nat @ Y6 @ X3 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9510_of__nat__of__integer,axiom,
% 12.85/7.03 ! [K: code_integer] :
% 12.85/7.03 ( ( semiri4939895301339042750nteger @ ( code_nat_of_integer @ K ) )
% 12.85/7.03 = ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ K ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9511_nat__of__integer__code__post_I1_J,axiom,
% 12.85/7.03 ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
% 12.85/7.03 = zero_zero_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9512_nat__of__integer__code__post_I3_J,axiom,
% 12.85/7.03 ! [K: num] :
% 12.85/7.03 ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 12.85/7.03 = ( numeral_numeral_nat @ K ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9513_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
% 12.85/7.03 ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
% 12.85/7.03 @ ^ [M5: nat,N4: nat] :
% 12.85/7.03 ( ( dvd_dvd_nat @ M5 @ N4 )
% 12.85/7.03 & ( M5 != N4 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9514_nat__of__integer__code,axiom,
% 12.85/7.03 ( code_nat_of_integer
% 12.85/7.03 = ( ^ [K3: code_integer] :
% 12.85/7.03 ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 12.85/7.03 @ ( produc1555791787009142072er_nat
% 12.85/7.03 @ ^ [L3: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ one_one_nat ) )
% 12.85/7.03 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9515_int__of__integer__code,axiom,
% 12.85/7.03 ( code_int_of_integer
% 12.85/7.03 = ( ^ [K3: code_integer] :
% 12.85/7.03 ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
% 12.85/7.03 @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 12.85/7.03 @ ( produc1553301316500091796er_int
% 12.85/7.03 @ ^ [L3: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ one_one_int ) )
% 12.85/7.03 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9516_times__int_Oabs__eq,axiom,
% 12.85/7.03 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 12.85/7.03 ( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 12.85/7.03 = ( abs_Integ
% 12.85/7.03 @ ( produc27273713700761075at_nat
% 12.85/7.03 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.03 ( produc2626176000494625587at_nat
% 12.85/7.03 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) )
% 12.85/7.03 @ Xa
% 12.85/7.03 @ X ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9517_zero__integer_Orep__eq,axiom,
% 12.85/7.03 ( ( code_int_of_integer @ zero_z3403309356797280102nteger )
% 12.85/7.03 = zero_zero_int ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9518_int__of__integer__numeral,axiom,
% 12.85/7.03 ! [K: num] :
% 12.85/7.03 ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 12.85/7.03 = ( numeral_numeral_int @ K ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9519_times__integer_Orep__eq,axiom,
% 12.85/7.03 ! [X: code_integer,Xa: code_integer] :
% 12.85/7.03 ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X @ Xa ) )
% 12.85/7.03 = ( times_times_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9520_minus__integer_Orep__eq,axiom,
% 12.85/7.03 ! [X: code_integer,Xa: code_integer] :
% 12.85/7.03 ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X @ Xa ) )
% 12.85/7.03 = ( minus_minus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9521_divide__integer_Orep__eq,axiom,
% 12.85/7.03 ! [X: code_integer,Xa: code_integer] :
% 12.85/7.03 ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X @ Xa ) )
% 12.85/7.03 = ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9522_int__of__integer__of__nat,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( code_int_of_integer @ ( semiri4939895301339042750nteger @ N ) )
% 12.85/7.03 = ( semiri1314217659103216013at_int @ N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9523_eq__Abs__Integ,axiom,
% 12.85/7.03 ! [Z: int] :
% 12.85/7.03 ~ ! [X4: nat,Y: nat] :
% 12.85/7.03 ( Z
% 12.85/7.03 != ( abs_Integ @ ( product_Pair_nat_nat @ X4 @ Y ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9524_int_Oabs__induct,axiom,
% 12.85/7.03 ! [P: int > $o,X: int] :
% 12.85/7.03 ( ! [Y: product_prod_nat_nat] : ( P @ ( abs_Integ @ Y ) )
% 12.85/7.03 => ( P @ X ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9525_less__eq__integer_Orep__eq,axiom,
% 12.85/7.03 ( ord_le3102999989581377725nteger
% 12.85/7.03 = ( ^ [X3: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9526_integer__less__eq__iff,axiom,
% 12.85/7.03 ( ord_le3102999989581377725nteger
% 12.85/7.03 = ( ^ [K3: code_integer,L3: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9527_nat_Oabs__eq,axiom,
% 12.85/7.03 ! [X: product_prod_nat_nat] :
% 12.85/7.03 ( ( nat2 @ ( abs_Integ @ X ) )
% 12.85/7.03 = ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9528_unset__bit__integer_Orep__eq,axiom,
% 12.85/7.03 ! [X: nat,Xa: code_integer] :
% 12.85/7.03 ( ( code_int_of_integer @ ( bit_se8260200283734997820nteger @ X @ Xa ) )
% 12.85/7.03 = ( bit_se4203085406695923979it_int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9529_set__bit__integer_Orep__eq,axiom,
% 12.85/7.03 ! [X: nat,Xa: code_integer] :
% 12.85/7.03 ( ( code_int_of_integer @ ( bit_se2793503036327961859nteger @ X @ Xa ) )
% 12.85/7.03 = ( bit_se7879613467334960850it_int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9530_flip__bit__integer_Orep__eq,axiom,
% 12.85/7.03 ! [X: nat,Xa: code_integer] :
% 12.85/7.03 ( ( code_int_of_integer @ ( bit_se1345352211410354436nteger @ X @ Xa ) )
% 12.85/7.03 = ( bit_se2159334234014336723it_int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9531_zero__int__def,axiom,
% 12.85/7.03 ( zero_zero_int
% 12.85/7.03 = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9532_int__def,axiom,
% 12.85/7.03 ( semiri1314217659103216013at_int
% 12.85/7.03 = ( ^ [N4: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N4 @ zero_zero_nat ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9533_uminus__int_Oabs__eq,axiom,
% 12.85/7.03 ! [X: product_prod_nat_nat] :
% 12.85/7.03 ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
% 12.85/7.03 = ( abs_Integ
% 12.85/7.03 @ ( produc2626176000494625587at_nat
% 12.85/7.03 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 )
% 12.85/7.03 @ X ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9534_one__int__def,axiom,
% 12.85/7.03 ( one_one_int
% 12.85/7.03 = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9535_less__int_Oabs__eq,axiom,
% 12.85/7.03 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 12.85/7.03 ( ( ord_less_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 12.85/7.03 = ( produc8739625826339149834_nat_o
% 12.85/7.03 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.03 ( produc6081775807080527818_nat_o
% 12.85/7.03 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) )
% 12.85/7.03 @ Xa
% 12.85/7.03 @ X ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9536_less__eq__int_Oabs__eq,axiom,
% 12.85/7.03 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 12.85/7.03 ( ( ord_less_eq_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 12.85/7.03 = ( produc8739625826339149834_nat_o
% 12.85/7.03 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.03 ( produc6081775807080527818_nat_o
% 12.85/7.03 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) )
% 12.85/7.03 @ Xa
% 12.85/7.03 @ X ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9537_plus__int_Oabs__eq,axiom,
% 12.85/7.03 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 12.85/7.03 ( ( plus_plus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 12.85/7.03 = ( abs_Integ
% 12.85/7.03 @ ( produc27273713700761075at_nat
% 12.85/7.03 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.03 ( produc2626176000494625587at_nat
% 12.85/7.03 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) )
% 12.85/7.03 @ Xa
% 12.85/7.03 @ X ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9538_minus__int_Oabs__eq,axiom,
% 12.85/7.03 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 12.85/7.03 ( ( minus_minus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 12.85/7.03 = ( abs_Integ
% 12.85/7.03 @ ( produc27273713700761075at_nat
% 12.85/7.03 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.03 ( produc2626176000494625587at_nat
% 12.85/7.03 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) )
% 12.85/7.03 @ Xa
% 12.85/7.03 @ X ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9539_Gcd__remove0__nat,axiom,
% 12.85/7.03 ! [M10: set_nat] :
% 12.85/7.03 ( ( finite_finite_nat @ M10 )
% 12.85/7.03 => ( ( gcd_Gcd_nat @ M10 )
% 12.85/7.03 = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M10 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9540_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 12.85/7.03 ! [N: nat,J: nat,I: nat] :
% 12.85/7.03 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
% 12.85/7.03 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
% 12.85/7.03 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9541_card__greaterThanAtMost,axiom,
% 12.85/7.03 ! [L: nat,U: nat] :
% 12.85/7.03 ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
% 12.85/7.03 = ( minus_minus_nat @ U @ L ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9542_Gcd__dvd__nat,axiom,
% 12.85/7.03 ! [A: nat,A2: set_nat] :
% 12.85/7.03 ( ( member_nat @ A @ A2 )
% 12.85/7.03 => ( dvd_dvd_nat @ ( gcd_Gcd_nat @ A2 ) @ A ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9543_Gcd__greatest__nat,axiom,
% 12.85/7.03 ! [A2: set_nat,A: nat] :
% 12.85/7.03 ( ! [B3: nat] :
% 12.85/7.03 ( ( member_nat @ B3 @ A2 )
% 12.85/7.03 => ( dvd_dvd_nat @ A @ B3 ) )
% 12.85/7.03 => ( dvd_dvd_nat @ A @ ( gcd_Gcd_nat @ A2 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9544_atLeastSucAtMost__greaterThanAtMost,axiom,
% 12.85/7.03 ! [L: nat,U: nat] :
% 12.85/7.03 ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
% 12.85/7.03 = ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9545_greaterThanAtMost__upt,axiom,
% 12.85/7.03 ( set_or6659071591806873216st_nat
% 12.85/7.03 = ( ^ [N4: nat,M5: nat] : ( set_nat2 @ ( upt @ ( suc @ N4 ) @ ( suc @ M5 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9546_less__eq__int_Orep__eq,axiom,
% 12.85/7.03 ( ord_less_eq_int
% 12.85/7.03 = ( ^ [X3: int,Xa4: int] :
% 12.85/7.03 ( produc8739625826339149834_nat_o
% 12.85/7.03 @ ^ [Y6: nat,Z5: nat] :
% 12.85/7.03 ( produc6081775807080527818_nat_o
% 12.85/7.03 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y6 @ V4 ) @ ( plus_plus_nat @ U3 @ Z5 ) ) )
% 12.85/7.03 @ ( rep_Integ @ X3 )
% 12.85/7.03 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9547_less__int_Orep__eq,axiom,
% 12.85/7.03 ( ord_less_int
% 12.85/7.03 = ( ^ [X3: int,Xa4: int] :
% 12.85/7.03 ( produc8739625826339149834_nat_o
% 12.85/7.03 @ ^ [Y6: nat,Z5: nat] :
% 12.85/7.03 ( produc6081775807080527818_nat_o
% 12.85/7.03 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y6 @ V4 ) @ ( plus_plus_nat @ U3 @ Z5 ) ) )
% 12.85/7.03 @ ( rep_Integ @ X3 )
% 12.85/7.03 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9548_card__greaterThanAtMost__int,axiom,
% 12.85/7.03 ! [L: int,U: int] :
% 12.85/7.03 ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L @ U ) )
% 12.85/7.03 = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9549_Gcd__dvd__int,axiom,
% 12.85/7.03 ! [A: int,A2: set_int] :
% 12.85/7.03 ( ( member_int @ A @ A2 )
% 12.85/7.03 => ( dvd_dvd_int @ ( gcd_Gcd_int @ A2 ) @ A ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9550_Gcd__greatest__int,axiom,
% 12.85/7.03 ! [A2: set_int,A: int] :
% 12.85/7.03 ( ! [B3: int] :
% 12.85/7.03 ( ( member_int @ B3 @ A2 )
% 12.85/7.03 => ( dvd_dvd_int @ A @ B3 ) )
% 12.85/7.03 => ( dvd_dvd_int @ A @ ( gcd_Gcd_int @ A2 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9551_Gcd__int__greater__eq__0,axiom,
% 12.85/7.03 ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9552_nat_Orep__eq,axiom,
% 12.85/7.03 ( nat2
% 12.85/7.03 = ( ^ [X3: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X3 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9553_uminus__int__def,axiom,
% 12.85/7.03 ( uminus_uminus_int
% 12.85/7.03 = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
% 12.85/7.03 @ ( produc2626176000494625587at_nat
% 12.85/7.03 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9554_prod__encode__def,axiom,
% 12.85/7.03 ( nat_prod_encode
% 12.85/7.03 = ( produc6842872674320459806at_nat
% 12.85/7.03 @ ^ [M5: nat,N4: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M5 @ N4 ) ) @ M5 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9555_num__of__nat_Osimps_I2_J,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( num_of_nat @ ( suc @ N ) )
% 12.85/7.03 = ( inc @ ( num_of_nat @ N ) ) ) )
% 12.85/7.03 & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( num_of_nat @ ( suc @ N ) )
% 12.85/7.03 = one ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9556_prod__encode__eq,axiom,
% 12.85/7.03 ! [X: product_prod_nat_nat,Y4: product_prod_nat_nat] :
% 12.85/7.03 ( ( ( nat_prod_encode @ X )
% 12.85/7.03 = ( nat_prod_encode @ Y4 ) )
% 12.85/7.03 = ( X = Y4 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9557_num__of__nat__numeral__eq,axiom,
% 12.85/7.03 ! [Q2: num] :
% 12.85/7.03 ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
% 12.85/7.03 = Q2 ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9558_num__of__nat_Osimps_I1_J,axiom,
% 12.85/7.03 ( ( num_of_nat @ zero_zero_nat )
% 12.85/7.03 = one ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9559_le__prod__encode__1,axiom,
% 12.85/7.03 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9560_le__prod__encode__2,axiom,
% 12.85/7.03 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9561_numeral__num__of__nat,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
% 12.85/7.03 = N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9562_num__of__nat__One,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ N @ one_one_nat )
% 12.85/7.03 => ( ( num_of_nat @ N )
% 12.85/7.03 = one ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9563_num__of__nat__double,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
% 12.85/7.03 = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9564_num__of__nat__plus__distrib,axiom,
% 12.85/7.03 ! [M: nat,N: nat] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ M )
% 12.85/7.03 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
% 12.85/7.03 = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9565_prod__encode__prod__decode__aux,axiom,
% 12.85/7.03 ! [K: nat,M: nat] :
% 12.85/7.03 ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
% 12.85/7.03 = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9566_times__int__def,axiom,
% 12.85/7.03 ( times_times_int
% 12.85/7.03 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 12.85/7.03 @ ( produc27273713700761075at_nat
% 12.85/7.03 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.03 ( produc2626176000494625587at_nat
% 12.85/7.03 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9567_minus__int__def,axiom,
% 12.85/7.03 ( minus_minus_int
% 12.85/7.03 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 12.85/7.03 @ ( produc27273713700761075at_nat
% 12.85/7.03 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.03 ( produc2626176000494625587at_nat
% 12.85/7.03 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9568_plus__int__def,axiom,
% 12.85/7.03 ( plus_plus_int
% 12.85/7.03 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 12.85/7.03 @ ( produc27273713700761075at_nat
% 12.85/7.03 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.03 ( produc2626176000494625587at_nat
% 12.85/7.03 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9569_pred__nat__def,axiom,
% 12.85/7.03 ( pred_nat
% 12.85/7.03 = ( collec3392354462482085612at_nat
% 12.85/7.03 @ ( produc6081775807080527818_nat_o
% 12.85/7.03 @ ^ [M5: nat,N4: nat] :
% 12.85/7.03 ( N4
% 12.85/7.03 = ( suc @ M5 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9570_pow_Osimps_I3_J,axiom,
% 12.85/7.03 ! [X: num,Y4: num] :
% 12.85/7.03 ( ( pow @ X @ ( bit1 @ Y4 ) )
% 12.85/7.03 = ( times_times_num @ ( sqr @ ( pow @ X @ Y4 ) ) @ X ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9571_image__minus__const__atLeastLessThan__nat,axiom,
% 12.85/7.03 ! [C: nat,Y4: nat,X: nat] :
% 12.85/7.03 ( ( ( ord_less_nat @ C @ Y4 )
% 12.85/7.03 => ( ( image_nat_nat
% 12.85/7.03 @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
% 12.85/7.03 @ ( set_or4665077453230672383an_nat @ X @ Y4 ) )
% 12.85/7.03 = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y4 @ C ) ) ) )
% 12.85/7.03 & ( ~ ( ord_less_nat @ C @ Y4 )
% 12.85/7.03 => ( ( ( ord_less_nat @ X @ Y4 )
% 12.85/7.03 => ( ( image_nat_nat
% 12.85/7.03 @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
% 12.85/7.03 @ ( set_or4665077453230672383an_nat @ X @ Y4 ) )
% 12.85/7.03 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 12.85/7.03 & ( ~ ( ord_less_nat @ X @ Y4 )
% 12.85/7.03 => ( ( image_nat_nat
% 12.85/7.03 @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
% 12.85/7.03 @ ( set_or4665077453230672383an_nat @ X @ Y4 ) )
% 12.85/7.03 = bot_bot_set_nat ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9572_bij__betw__Suc,axiom,
% 12.85/7.03 ! [M10: set_nat,N5: set_nat] :
% 12.85/7.03 ( ( bij_betw_nat_nat @ suc @ M10 @ N5 )
% 12.85/7.03 = ( ( image_nat_nat @ suc @ M10 )
% 12.85/7.03 = N5 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9573_image__Suc__atLeastAtMost,axiom,
% 12.85/7.03 ! [I: nat,J: nat] :
% 12.85/7.03 ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 12.85/7.03 = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9574_image__Suc__atLeastLessThan,axiom,
% 12.85/7.03 ! [I: nat,J: nat] :
% 12.85/7.03 ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
% 12.85/7.03 = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9575_sqr_Osimps_I2_J,axiom,
% 12.85/7.03 ! [N: num] :
% 12.85/7.03 ( ( sqr @ ( bit0 @ N ) )
% 12.85/7.03 = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9576_sqr_Osimps_I1_J,axiom,
% 12.85/7.03 ( ( sqr @ one )
% 12.85/7.03 = one ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9577_zero__notin__Suc__image,axiom,
% 12.85/7.03 ! [A2: set_nat] :
% 12.85/7.03 ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9578_sqr__conv__mult,axiom,
% 12.85/7.03 ( sqr
% 12.85/7.03 = ( ^ [X3: num] : ( times_times_num @ X3 @ X3 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9579_image__Suc__lessThan,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.03 = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9580_image__Suc__atMost,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
% 12.85/7.03 = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9581_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 12.85/7.03 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9582_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 12.85/7.03 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9583_lessThan__Suc__eq__insert__0,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( set_ord_lessThan_nat @ ( suc @ N ) )
% 12.85/7.03 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9584_atMost__Suc__eq__insert__0,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( set_ord_atMost_nat @ ( suc @ N ) )
% 12.85/7.03 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9585_pow_Osimps_I2_J,axiom,
% 12.85/7.03 ! [X: num,Y4: num] :
% 12.85/7.03 ( ( pow @ X @ ( bit0 @ Y4 ) )
% 12.85/7.03 = ( sqr @ ( pow @ X @ Y4 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9586_sqr_Osimps_I3_J,axiom,
% 12.85/7.03 ! [N: num] :
% 12.85/7.03 ( ( sqr @ ( bit1 @ N ) )
% 12.85/7.03 = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9587_Gcd__int__eq,axiom,
% 12.85/7.03 ! [N5: set_nat] :
% 12.85/7.03 ( ( gcd_Gcd_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N5 ) )
% 12.85/7.03 = ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ N5 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9588_Inf__real__def,axiom,
% 12.85/7.03 ( comple4887499456419720421f_real
% 12.85/7.03 = ( ^ [X8: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X8 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9589_finite__int__iff__bounded,axiom,
% 12.85/7.03 ( finite_finite_int
% 12.85/7.03 = ( ^ [S5: set_int] :
% 12.85/7.03 ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S5 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9590_finite__int__iff__bounded__le,axiom,
% 12.85/7.03 ( finite_finite_int
% 12.85/7.03 = ( ^ [S5: set_int] :
% 12.85/7.03 ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S5 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9591_image__int__atLeastAtMost,axiom,
% 12.85/7.03 ! [A: nat,B: nat] :
% 12.85/7.03 ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 12.85/7.03 = ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9592_image__int__atLeastLessThan,axiom,
% 12.85/7.03 ! [A: nat,B: nat] :
% 12.85/7.03 ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B ) )
% 12.85/7.03 = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9593_image__add__int__atLeastLessThan,axiom,
% 12.85/7.03 ! [L: int,U: int] :
% 12.85/7.03 ( ( image_int_int
% 12.85/7.03 @ ^ [X3: int] : ( plus_plus_int @ X3 @ L )
% 12.85/7.03 @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
% 12.85/7.03 = ( set_or4662586982721622107an_int @ L @ U ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9594_Gcd__int__def,axiom,
% 12.85/7.03 ( gcd_Gcd_int
% 12.85/7.03 = ( ^ [K7: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K7 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9595_image__atLeastZeroLessThan__int,axiom,
% 12.85/7.03 ! [U: int] :
% 12.85/7.03 ( ( ord_less_eq_int @ zero_zero_int @ U )
% 12.85/7.03 => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
% 12.85/7.03 = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9596_range__mod,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( image_nat_nat
% 12.85/7.03 @ ^ [M5: nat] : ( modulo_modulo_nat @ M5 @ N )
% 12.85/7.03 @ top_top_set_nat )
% 12.85/7.03 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9597_suminf__eq__SUP__real,axiom,
% 12.85/7.03 ! [X9: nat > real] :
% 12.85/7.03 ( ( summable_real @ X9 )
% 12.85/7.03 => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X9 @ I3 ) )
% 12.85/7.03 => ( ( suminf_real @ X9 )
% 12.85/7.03 = ( comple1385675409528146559p_real
% 12.85/7.03 @ ( image_nat_real
% 12.85/7.03 @ ^ [I2: nat] : ( groups6591440286371151544t_real @ X9 @ ( set_ord_lessThan_nat @ I2 ) )
% 12.85/7.03 @ top_top_set_nat ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9598_UNIV__nat__eq,axiom,
% 12.85/7.03 ( top_top_set_nat
% 12.85/7.03 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9599_measure__function__int,axiom,
% 12.85/7.03 fun_is_measure_int @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9600_card__UNIV__bool,axiom,
% 12.85/7.03 ( ( finite_card_o @ top_top_set_o )
% 12.85/7.03 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9601_range__mult,axiom,
% 12.85/7.03 ! [A: real] :
% 12.85/7.03 ( ( ( A = zero_zero_real )
% 12.85/7.03 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 12.85/7.03 = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 12.85/7.03 & ( ( A != zero_zero_real )
% 12.85/7.03 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 12.85/7.03 = top_top_set_real ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9602_infinite__UNIV__int,axiom,
% 12.85/7.03 ~ ( finite_finite_int @ top_top_set_int ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9603_bij__prod__encode,axiom,
% 12.85/7.03 bij_be5333170631980326235at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat @ top_top_set_nat ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9604_surj__prod__encode,axiom,
% 12.85/7.03 ( ( image_2486076414777270412at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat )
% 12.85/7.03 = top_top_set_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9605_int__in__range__abs,axiom,
% 12.85/7.03 ! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9606_root__def,axiom,
% 12.85/7.03 ( root
% 12.85/7.03 = ( ^ [N4: nat,X3: real] :
% 12.85/7.03 ( if_real @ ( N4 = zero_zero_nat ) @ zero_zero_real
% 12.85/7.03 @ ( the_in5290026491893676941l_real @ top_top_set_real
% 12.85/7.03 @ ^ [Y6: real] : ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N4 ) )
% 12.85/7.03 @ X3 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9607_card__UNIV__char,axiom,
% 12.85/7.03 ( ( finite_card_char @ top_top_set_char )
% 12.85/7.03 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9608_UNIV__bool,axiom,
% 12.85/7.03 ( top_top_set_o
% 12.85/7.03 = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9609_UNIV__char__of__nat,axiom,
% 12.85/7.03 ( top_top_set_char
% 12.85/7.03 = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9610_char_Osize_I2_J,axiom,
% 12.85/7.03 ! [X1: $o,X2: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
% 12.85/7.03 ( ( size_size_char @ ( char2 @ X1 @ X2 @ X33 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
% 12.85/7.03 = zero_zero_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9611_nat__of__char__less__256,axiom,
% 12.85/7.03 ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9612_range__nat__of__char,axiom,
% 12.85/7.03 ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
% 12.85/7.03 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9613_integer__of__char__code,axiom,
% 12.85/7.03 ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 12.85/7.03 ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 12.85/7.03 = ( 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 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9614_char_Osize__gen,axiom,
% 12.85/7.03 ! [X1: $o,X2: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
% 12.85/7.03 ( ( size_char @ ( char2 @ X1 @ X2 @ X33 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
% 12.85/7.03 = zero_zero_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9615_String_Ochar__of__ascii__of,axiom,
% 12.85/7.03 ! [C: char] :
% 12.85/7.03 ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
% 12.85/7.03 = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9616_DERIV__real__root__generic,axiom,
% 12.85/7.03 ! [N: nat,X: real,D3: real] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( X != zero_zero_real )
% 12.85/7.03 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/7.03 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.03 => ( D3
% 12.85/7.03 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 12.85/7.03 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/7.03 => ( ( ord_less_real @ X @ zero_zero_real )
% 12.85/7.03 => ( D3
% 12.85/7.03 = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 12.85/7.03 => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/7.03 => ( D3
% 12.85/7.03 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9617_has__real__derivative__neg__dec__right,axiom,
% 12.85/7.03 ! [F: real > real,L: real,X: real,S: set_real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S ) )
% 12.85/7.03 => ( ( ord_less_real @ L @ zero_zero_real )
% 12.85/7.03 => ? [D2: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ D2 )
% 12.85/7.03 & ! [H2: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ H2 )
% 12.85/7.03 => ( ( member_real @ ( plus_plus_real @ X @ H2 ) @ S )
% 12.85/7.03 => ( ( ord_less_real @ H2 @ D2 )
% 12.85/7.03 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9618_has__real__derivative__pos__inc__right,axiom,
% 12.85/7.03 ! [F: real > real,L: real,X: real,S: set_real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S ) )
% 12.85/7.03 => ( ( ord_less_real @ zero_zero_real @ L )
% 12.85/7.03 => ? [D2: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ D2 )
% 12.85/7.03 & ! [H2: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ H2 )
% 12.85/7.03 => ( ( member_real @ ( plus_plus_real @ X @ H2 ) @ S )
% 12.85/7.03 => ( ( ord_less_real @ H2 @ D2 )
% 12.85/7.03 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H2 ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9619_has__real__derivative__pos__inc__left,axiom,
% 12.85/7.03 ! [F: real > real,L: real,X: real,S: set_real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S ) )
% 12.85/7.03 => ( ( ord_less_real @ zero_zero_real @ L )
% 12.85/7.03 => ? [D2: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ D2 )
% 12.85/7.03 & ! [H2: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ H2 )
% 12.85/7.03 => ( ( member_real @ ( minus_minus_real @ X @ H2 ) @ S )
% 12.85/7.03 => ( ( ord_less_real @ H2 @ D2 )
% 12.85/7.03 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9620_has__real__derivative__neg__dec__left,axiom,
% 12.85/7.03 ! [F: real > real,L: real,X: real,S: set_real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S ) )
% 12.85/7.03 => ( ( ord_less_real @ L @ zero_zero_real )
% 12.85/7.03 => ? [D2: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ D2 )
% 12.85/7.03 & ! [H2: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ H2 )
% 12.85/7.03 => ( ( member_real @ ( minus_minus_real @ X @ H2 ) @ S )
% 12.85/7.03 => ( ( ord_less_real @ H2 @ D2 )
% 12.85/7.03 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H2 ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9621_deriv__nonneg__imp__mono,axiom,
% 12.85/7.03 ! [A: real,B: real,G: real > real,G2: real > real] :
% 12.85/7.03 ( ! [X4: real] :
% 12.85/7.03 ( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 12.85/7.03 => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X4 ) ) )
% 12.85/7.03 => ( ( ord_less_eq_real @ A @ B )
% 12.85/7.03 => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9622_DERIV__nonneg__imp__nondecreasing,axiom,
% 12.85/7.03 ! [A: real,B: real,F: real > real] :
% 12.85/7.03 ( ( ord_less_eq_real @ A @ B )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ A @ X4 )
% 12.85/7.03 => ( ( ord_less_eq_real @ X4 @ B )
% 12.85/7.03 => ? [Y3: real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.85/7.03 & ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ) )
% 12.85/7.03 => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9623_DERIV__nonpos__imp__nonincreasing,axiom,
% 12.85/7.03 ! [A: real,B: real,F: real > real] :
% 12.85/7.03 ( ( ord_less_eq_real @ A @ B )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ A @ X4 )
% 12.85/7.03 => ( ( ord_less_eq_real @ X4 @ B )
% 12.85/7.03 => ? [Y3: real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.85/7.03 & ( ord_less_eq_real @ Y3 @ zero_zero_real ) ) ) )
% 12.85/7.03 => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9624_DERIV__pos__imp__increasing,axiom,
% 12.85/7.03 ! [A: real,B: real,F: real > real] :
% 12.85/7.03 ( ( ord_less_real @ A @ B )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ A @ X4 )
% 12.85/7.03 => ( ( ord_less_eq_real @ X4 @ B )
% 12.85/7.03 => ? [Y3: real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.85/7.03 & ( ord_less_real @ zero_zero_real @ Y3 ) ) ) )
% 12.85/7.03 => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9625_DERIV__neg__imp__decreasing,axiom,
% 12.85/7.03 ! [A: real,B: real,F: real > real] :
% 12.85/7.03 ( ( ord_less_real @ A @ B )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ A @ X4 )
% 12.85/7.03 => ( ( ord_less_eq_real @ X4 @ B )
% 12.85/7.03 => ? [Y3: real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.85/7.03 & ( ord_less_real @ Y3 @ zero_zero_real ) ) ) )
% 12.85/7.03 => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9626_DERIV__isconst__all,axiom,
% 12.85/7.03 ! [F: real > real,X: real,Y4: real] :
% 12.85/7.03 ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.85/7.03 => ( ( F @ X )
% 12.85/7.03 = ( F @ Y4 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9627_DERIV__pos__inc__right,axiom,
% 12.85/7.03 ! [F: real > real,L: real,X: real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( ( ord_less_real @ zero_zero_real @ L )
% 12.85/7.03 => ? [D2: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ D2 )
% 12.85/7.03 & ! [H2: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ H2 )
% 12.85/7.03 => ( ( ord_less_real @ H2 @ D2 )
% 12.85/7.03 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H2 ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9628_DERIV__neg__dec__right,axiom,
% 12.85/7.03 ! [F: real > real,L: real,X: real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( ( ord_less_real @ L @ zero_zero_real )
% 12.85/7.03 => ? [D2: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ D2 )
% 12.85/7.03 & ! [H2: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ H2 )
% 12.85/7.03 => ( ( ord_less_real @ H2 @ D2 )
% 12.85/7.03 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9629_DERIV__const__ratio__const2,axiom,
% 12.85/7.03 ! [A: real,B: real,F: real > real,K: real] :
% 12.85/7.03 ( ( A != B )
% 12.85/7.03 => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.85/7.03 => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
% 12.85/7.03 = K ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9630_DERIV__const__ratio__const,axiom,
% 12.85/7.03 ! [A: real,B: real,F: real > real,K: real] :
% 12.85/7.03 ( ( A != B )
% 12.85/7.03 => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.85/7.03 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 12.85/7.03 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ K ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9631_DERIV__pos__inc__left,axiom,
% 12.85/7.03 ! [F: real > real,L: real,X: real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( ( ord_less_real @ zero_zero_real @ L )
% 12.85/7.03 => ? [D2: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ D2 )
% 12.85/7.03 & ! [H2: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ H2 )
% 12.85/7.03 => ( ( ord_less_real @ H2 @ D2 )
% 12.85/7.03 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9632_DERIV__neg__dec__left,axiom,
% 12.85/7.03 ! [F: real > real,L: real,X: real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( ( ord_less_real @ L @ zero_zero_real )
% 12.85/7.03 => ? [D2: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ D2 )
% 12.85/7.03 & ! [H2: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ H2 )
% 12.85/7.03 => ( ( ord_less_real @ H2 @ D2 )
% 12.85/7.03 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H2 ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9633_DERIV__isconst3,axiom,
% 12.85/7.03 ! [A: real,B: real,X: real,Y4: real,F: real > real] :
% 12.85/7.03 ( ( ord_less_real @ A @ B )
% 12.85/7.03 => ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
% 12.85/7.03 => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 12.85/7.03 => ( ( F @ X )
% 12.85/7.03 = ( F @ Y4 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9634_MVT2,axiom,
% 12.85/7.03 ! [A: real,B: real,F: real > real,F2: real > real] :
% 12.85/7.03 ( ( ord_less_real @ A @ B )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ A @ X4 )
% 12.85/7.03 => ( ( ord_less_eq_real @ X4 @ B )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ F @ ( F2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 12.85/7.03 => ? [Z2: real] :
% 12.85/7.03 ( ( ord_less_real @ A @ Z2 )
% 12.85/7.03 & ( ord_less_real @ Z2 @ B )
% 12.85/7.03 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 12.85/7.03 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F2 @ Z2 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9635_DERIV__local__const,axiom,
% 12.85/7.03 ! [F: real > real,L: real,X: real,D: real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( ( ord_less_real @ zero_zero_real @ D )
% 12.85/7.03 => ( ! [Y: real] :
% 12.85/7.03 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) @ D )
% 12.85/7.03 => ( ( F @ X )
% 12.85/7.03 = ( F @ Y ) ) )
% 12.85/7.03 => ( L = zero_zero_real ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9636_DERIV__ln,axiom,
% 12.85/7.03 ! [X: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9637_DERIV__const__average,axiom,
% 12.85/7.03 ! [A: real,B: real,V: real > real,K: real] :
% 12.85/7.03 ( ( A != B )
% 12.85/7.03 => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.85/7.03 => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 12.85/7.03 = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9638_DERIV__local__min,axiom,
% 12.85/7.03 ! [F: real > real,L: real,X: real,D: real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( ( ord_less_real @ zero_zero_real @ D )
% 12.85/7.03 => ( ! [Y: real] :
% 12.85/7.03 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) @ D )
% 12.85/7.03 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
% 12.85/7.03 => ( L = zero_zero_real ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9639_DERIV__local__max,axiom,
% 12.85/7.03 ! [F: real > real,L: real,X: real,D: real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( ( ord_less_real @ zero_zero_real @ D )
% 12.85/7.03 => ( ! [Y: real] :
% 12.85/7.03 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) @ D )
% 12.85/7.03 => ( ord_less_eq_real @ ( F @ Y ) @ ( F @ X ) ) )
% 12.85/7.03 => ( L = zero_zero_real ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9640_DERIV__ln__divide,axiom,
% 12.85/7.03 ! [X: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9641_DERIV__pow,axiom,
% 12.85/7.03 ! [N: nat,X: real,S2: set_real] :
% 12.85/7.03 ( has_fi5821293074295781190e_real
% 12.85/7.03 @ ^ [X3: real] : ( power_power_real @ X3 @ N )
% 12.85/7.03 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ S2 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9642_DERIV__fun__pow,axiom,
% 12.85/7.03 ! [G: real > real,M: real,X: real,N: nat] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( has_fi5821293074295781190e_real
% 12.85/7.03 @ ^ [X3: real] : ( power_power_real @ ( G @ X3 ) @ N )
% 12.85/7.03 @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9643_has__real__derivative__powr,axiom,
% 12.85/7.03 ! [Z: real,R2: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ Z )
% 12.85/7.03 => ( has_fi5821293074295781190e_real
% 12.85/7.03 @ ^ [Z5: real] : ( powr_real @ Z5 @ R2 )
% 12.85/7.03 @ ( times_times_real @ R2 @ ( powr_real @ Z @ ( minus_minus_real @ R2 @ one_one_real ) ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9644_DERIV__series_H,axiom,
% 12.85/7.03 ! [F: real > nat > real,F2: real > nat > real,X0: real,A: real,B: real,L4: nat > real] :
% 12.85/7.03 ( ! [N2: nat] :
% 12.85/7.03 ( has_fi5821293074295781190e_real
% 12.85/7.03 @ ^ [X3: real] : ( F @ X3 @ N2 )
% 12.85/7.03 @ ( F2 @ X0 @ N2 )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 12.85/7.03 => ( summable_real @ ( F @ X4 ) ) )
% 12.85/7.03 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 12.85/7.03 => ( ( summable_real @ ( F2 @ X0 ) )
% 12.85/7.03 => ( ( summable_real @ L4 )
% 12.85/7.03 => ( ! [N2: nat,X4: real,Y: real] :
% 12.85/7.03 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 12.85/7.03 => ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A @ B ) )
% 12.85/7.03 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X4 @ N2 ) @ ( F @ Y @ N2 ) ) ) @ ( times_times_real @ ( L4 @ N2 ) @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y ) ) ) ) ) )
% 12.85/7.03 => ( has_fi5821293074295781190e_real
% 12.85/7.03 @ ^ [X3: real] : ( suminf_real @ ( F @ X3 ) )
% 12.85/7.03 @ ( suminf_real @ ( F2 @ X0 ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9645_DERIV__log,axiom,
% 12.85/7.03 ! [X: real,B: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9646_DERIV__fun__powr,axiom,
% 12.85/7.03 ! [G: real > real,M: real,X: real,R2: real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 12.85/7.03 => ( has_fi5821293074295781190e_real
% 12.85/7.03 @ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ R2 )
% 12.85/7.03 @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9647_DERIV__powr,axiom,
% 12.85/7.03 ! [G: real > real,M: real,X: real,F: real > real,R2: real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 12.85/7.03 => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( has_fi5821293074295781190e_real
% 12.85/7.03 @ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ ( F @ X3 ) )
% 12.85/7.03 @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9648_artanh__real__has__field__derivative,axiom,
% 12.85/7.03 ! [X: real,A2: set_real] :
% 12.85/7.03 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9649_DERIV__real__sqrt,axiom,
% 12.85/7.03 ! [X: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9650_DERIV__arctan,axiom,
% 12.85/7.03 ! [X: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9651_arsinh__real__has__field__derivative,axiom,
% 12.85/7.03 ! [X: real,A2: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9652_DERIV__real__sqrt__generic,axiom,
% 12.85/7.03 ! [X: real,D3: real] :
% 12.85/7.03 ( ( X != zero_zero_real )
% 12.85/7.03 => ( ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.03 => ( D3
% 12.85/7.03 = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 12.85/7.03 => ( ( ( ord_less_real @ X @ zero_zero_real )
% 12.85/7.03 => ( D3
% 12.85/7.03 = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ sqrt @ D3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9653_arcosh__real__has__field__derivative,axiom,
% 12.85/7.03 ! [X: real,A2: set_real] :
% 12.85/7.03 ( ( ord_less_real @ one_one_real @ X )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9654_DERIV__power__series_H,axiom,
% 12.85/7.03 ! [R: real,F: nat > real,X0: real] :
% 12.85/7.03 ( ! [X4: real] :
% 12.85/7.03 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 12.85/7.03 => ( summable_real
% 12.85/7.03 @ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( F @ N4 ) @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ ( power_power_real @ X4 @ N4 ) ) ) )
% 12.85/7.03 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 12.85/7.03 => ( ( ord_less_real @ zero_zero_real @ R )
% 12.85/7.03 => ( has_fi5821293074295781190e_real
% 12.85/7.03 @ ^ [X3: real] :
% 12.85/7.03 ( suminf_real
% 12.85/7.03 @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ X3 @ ( suc @ N4 ) ) ) )
% 12.85/7.03 @ ( suminf_real
% 12.85/7.03 @ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( F @ N4 ) @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ ( power_power_real @ X0 @ N4 ) ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9655_DERIV__real__root,axiom,
% 12.85/7.03 ! [N: nat,X: real] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( ord_less_real @ zero_zero_real @ X )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9656_DERIV__arccos,axiom,
% 12.85/7.03 ! [X: real] :
% 12.85/7.03 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 12.85/7.03 => ( ( ord_less_real @ X @ one_one_real )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9657_DERIV__arcsin,axiom,
% 12.85/7.03 ! [X: real] :
% 12.85/7.03 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 12.85/7.03 => ( ( ord_less_real @ X @ one_one_real )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9658_Maclaurin__all__le__objl,axiom,
% 12.85/7.03 ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 12.85/7.03 ( ( ( ( Diff @ zero_zero_nat )
% 12.85/7.03 = F )
% 12.85/7.03 & ! [M4: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 12.85/7.03 => ? [T3: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 12.85/7.03 & ( ( F @ X )
% 12.85/7.03 = ( plus_plus_real
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.03 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9659_Maclaurin__all__le,axiom,
% 12.85/7.03 ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 12.85/7.03 ( ( ( Diff @ zero_zero_nat )
% 12.85/7.03 = F )
% 12.85/7.03 => ( ! [M4: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.85/7.03 => ? [T3: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 12.85/7.03 & ( ( F @ X )
% 12.85/7.03 = ( plus_plus_real
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.03 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9660_DERIV__odd__real__root,axiom,
% 12.85/7.03 ! [N: nat,X: real] :
% 12.85/7.03 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/7.03 => ( ( X != zero_zero_real )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9661_Maclaurin__minus,axiom,
% 12.85/7.03 ! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
% 12.85/7.03 ( ( ord_less_real @ H @ zero_zero_real )
% 12.85/7.03 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( ( Diff @ zero_zero_nat )
% 12.85/7.03 = F )
% 12.85/7.03 => ( ! [M4: nat,T3: real] :
% 12.85/7.03 ( ( ( ord_less_nat @ M4 @ N )
% 12.85/7.03 & ( ord_less_eq_real @ H @ T3 )
% 12.85/7.03 & ( ord_less_eq_real @ T3 @ zero_zero_real ) )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 12.85/7.03 => ? [T3: real] :
% 12.85/7.03 ( ( ord_less_real @ H @ T3 )
% 12.85/7.03 & ( ord_less_real @ T3 @ zero_zero_real )
% 12.85/7.03 & ( ( F @ H )
% 12.85/7.03 = ( plus_plus_real
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H @ M5 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.03 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9662_Maclaurin2,axiom,
% 12.85/7.03 ! [H: real,Diff: nat > real > real,F: real > real,N: nat] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ H )
% 12.85/7.03 => ( ( ( Diff @ zero_zero_nat )
% 12.85/7.03 = F )
% 12.85/7.03 => ( ! [M4: nat,T3: real] :
% 12.85/7.03 ( ( ( ord_less_nat @ M4 @ N )
% 12.85/7.03 & ( ord_less_eq_real @ zero_zero_real @ T3 )
% 12.85/7.03 & ( ord_less_eq_real @ T3 @ H ) )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 12.85/7.03 => ? [T3: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ T3 )
% 12.85/7.03 & ( ord_less_eq_real @ T3 @ H )
% 12.85/7.03 & ( ( F @ H )
% 12.85/7.03 = ( plus_plus_real
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H @ M5 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.03 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9663_Maclaurin,axiom,
% 12.85/7.03 ! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ H )
% 12.85/7.03 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( ( Diff @ zero_zero_nat )
% 12.85/7.03 = F )
% 12.85/7.03 => ( ! [M4: nat,T3: real] :
% 12.85/7.03 ( ( ( ord_less_nat @ M4 @ N )
% 12.85/7.03 & ( ord_less_eq_real @ zero_zero_real @ T3 )
% 12.85/7.03 & ( ord_less_eq_real @ T3 @ H ) )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 12.85/7.03 => ? [T3: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ T3 )
% 12.85/7.03 & ( ord_less_real @ T3 @ H )
% 12.85/7.03 & ( ( F @ H )
% 12.85/7.03 = ( plus_plus_real
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H @ M5 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.03 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9664_Maclaurin__all__lt,axiom,
% 12.85/7.03 ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 12.85/7.03 ( ( ( Diff @ zero_zero_nat )
% 12.85/7.03 = F )
% 12.85/7.03 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( X != zero_zero_real )
% 12.85/7.03 => ( ! [M4: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.85/7.03 => ? [T3: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
% 12.85/7.03 & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 12.85/7.03 & ( ( F @ X )
% 12.85/7.03 = ( plus_plus_real
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.03 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9665_Maclaurin__bi__le,axiom,
% 12.85/7.03 ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 12.85/7.03 ( ( ( Diff @ zero_zero_nat )
% 12.85/7.03 = F )
% 12.85/7.03 => ( ! [M4: nat,T3: real] :
% 12.85/7.03 ( ( ( ord_less_nat @ M4 @ N )
% 12.85/7.03 & ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) ) )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 12.85/7.03 => ? [T3: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 12.85/7.03 & ( ( F @ X )
% 12.85/7.03 = ( plus_plus_real
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.03 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9666_Taylor__down,axiom,
% 12.85/7.03 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( ( Diff @ zero_zero_nat )
% 12.85/7.03 = F )
% 12.85/7.03 => ( ! [M4: nat,T3: real] :
% 12.85/7.03 ( ( ( ord_less_nat @ M4 @ N )
% 12.85/7.03 & ( ord_less_eq_real @ A @ T3 )
% 12.85/7.03 & ( ord_less_eq_real @ T3 @ B ) )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 12.85/7.03 => ( ( ord_less_real @ A @ C )
% 12.85/7.03 => ( ( ord_less_eq_real @ C @ B )
% 12.85/7.03 => ? [T3: real] :
% 12.85/7.03 ( ( ord_less_real @ A @ T3 )
% 12.85/7.03 & ( ord_less_real @ T3 @ C )
% 12.85/7.03 & ( ( F @ A )
% 12.85/7.03 = ( plus_plus_real
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M5 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.03 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9667_Taylor__up,axiom,
% 12.85/7.03 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( ( Diff @ zero_zero_nat )
% 12.85/7.03 = F )
% 12.85/7.03 => ( ! [M4: nat,T3: real] :
% 12.85/7.03 ( ( ( ord_less_nat @ M4 @ N )
% 12.85/7.03 & ( ord_less_eq_real @ A @ T3 )
% 12.85/7.03 & ( ord_less_eq_real @ T3 @ B ) )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 12.85/7.03 => ( ( ord_less_eq_real @ A @ C )
% 12.85/7.03 => ( ( ord_less_real @ C @ B )
% 12.85/7.03 => ? [T3: real] :
% 12.85/7.03 ( ( ord_less_real @ C @ T3 )
% 12.85/7.03 & ( ord_less_real @ T3 @ B )
% 12.85/7.03 & ( ( F @ B )
% 12.85/7.03 = ( plus_plus_real
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M5 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.03 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9668_Taylor,axiom,
% 12.85/7.03 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( ( Diff @ zero_zero_nat )
% 12.85/7.03 = F )
% 12.85/7.03 => ( ! [M4: nat,T3: real] :
% 12.85/7.03 ( ( ( ord_less_nat @ M4 @ N )
% 12.85/7.03 & ( ord_less_eq_real @ A @ T3 )
% 12.85/7.03 & ( ord_less_eq_real @ T3 @ B ) )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 12.85/7.03 => ( ( ord_less_eq_real @ A @ C )
% 12.85/7.03 => ( ( ord_less_eq_real @ C @ B )
% 12.85/7.03 => ( ( ord_less_eq_real @ A @ X )
% 12.85/7.03 => ( ( ord_less_eq_real @ X @ B )
% 12.85/7.03 => ( ( X != C )
% 12.85/7.03 => ? [T3: real] :
% 12.85/7.03 ( ( ( ord_less_real @ X @ C )
% 12.85/7.03 => ( ( ord_less_real @ X @ T3 )
% 12.85/7.03 & ( ord_less_real @ T3 @ C ) ) )
% 12.85/7.03 & ( ~ ( ord_less_real @ X @ C )
% 12.85/7.03 => ( ( ord_less_real @ C @ T3 )
% 12.85/7.03 & ( ord_less_real @ T3 @ X ) ) )
% 12.85/7.03 & ( ( F @ X )
% 12.85/7.03 = ( plus_plus_real
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M5 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ N ) )
% 12.85/7.03 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9669_Maclaurin__lemma2,axiom,
% 12.85/7.03 ! [N: nat,H: real,Diff: nat > real > real,K: nat,B4: real] :
% 12.85/7.03 ( ! [M4: nat,T3: real] :
% 12.85/7.03 ( ( ( ord_less_nat @ M4 @ N )
% 12.85/7.03 & ( ord_less_eq_real @ zero_zero_real @ T3 )
% 12.85/7.03 & ( ord_less_eq_real @ T3 @ H ) )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 12.85/7.03 => ( ( N
% 12.85/7.03 = ( suc @ K ) )
% 12.85/7.03 => ! [M2: nat,T4: real] :
% 12.85/7.03 ( ( ( ord_less_nat @ M2 @ N )
% 12.85/7.03 & ( ord_less_eq_real @ zero_zero_real @ T4 )
% 12.85/7.03 & ( ord_less_eq_real @ T4 @ H ) )
% 12.85/7.03 => ( has_fi5821293074295781190e_real
% 12.85/7.03 @ ^ [U3: real] :
% 12.85/7.03 ( minus_minus_real @ ( Diff @ M2 @ U3 )
% 12.85/7.03 @ ( plus_plus_real
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ U3 @ P6 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M2 ) ) )
% 12.85/7.03 @ ( times_times_real @ B4 @ ( divide_divide_real @ ( power_power_real @ U3 @ ( minus_minus_nat @ N @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) )
% 12.85/7.03 @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T4 )
% 12.85/7.03 @ ( plus_plus_real
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ T4 @ P6 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) )
% 12.85/7.03 @ ( times_times_real @ B4 @ ( divide_divide_real @ ( power_power_real @ T4 @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) ) ) ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9670_DERIV__arctan__series,axiom,
% 12.85/7.03 ! [X: real] :
% 12.85/7.03 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 12.85/7.03 => ( has_fi5821293074295781190e_real
% 12.85/7.03 @ ^ [X10: real] :
% 12.85/7.03 ( suminf_real
% 12.85/7.03 @ ^ [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 @ X10 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 12.85/7.03 @ ( suminf_real
% 12.85/7.03 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9671_DERIV__even__real__root,axiom,
% 12.85/7.03 ! [N: nat,X: real] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/7.03 => ( ( ord_less_real @ X @ zero_zero_real )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9672_isCont__Lb__Ub,axiom,
% 12.85/7.03 ! [A: real,B: real,F: real > real] :
% 12.85/7.03 ( ( ord_less_eq_real @ A @ B )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( ( ord_less_eq_real @ A @ X4 )
% 12.85/7.03 & ( ord_less_eq_real @ X4 @ B ) )
% 12.85/7.03 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
% 12.85/7.03 => ? [L5: real,M8: real] :
% 12.85/7.03 ( ! [X5: real] :
% 12.85/7.03 ( ( ( ord_less_eq_real @ A @ X5 )
% 12.85/7.03 & ( ord_less_eq_real @ X5 @ B ) )
% 12.85/7.03 => ( ( ord_less_eq_real @ L5 @ ( F @ X5 ) )
% 12.85/7.03 & ( ord_less_eq_real @ ( F @ X5 ) @ M8 ) ) )
% 12.85/7.03 & ! [Y3: real] :
% 12.85/7.03 ( ( ( ord_less_eq_real @ L5 @ Y3 )
% 12.85/7.03 & ( ord_less_eq_real @ Y3 @ M8 ) )
% 12.85/7.03 => ? [X4: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ A @ X4 )
% 12.85/7.03 & ( ord_less_eq_real @ X4 @ B )
% 12.85/7.03 & ( ( F @ X4 )
% 12.85/7.03 = Y3 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9673_LIM__fun__less__zero,axiom,
% 12.85/7.03 ! [F: real > real,L: real,C: real] :
% 12.85/7.03 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 12.85/7.03 => ( ( ord_less_real @ L @ zero_zero_real )
% 12.85/7.03 => ? [R3: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ R3 )
% 12.85/7.03 & ! [X5: real] :
% 12.85/7.03 ( ( ( X5 != C )
% 12.85/7.03 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
% 12.85/7.03 => ( ord_less_real @ ( F @ X5 ) @ zero_zero_real ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9674_LIM__fun__not__zero,axiom,
% 12.85/7.03 ! [F: real > real,L: real,C: real] :
% 12.85/7.03 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 12.85/7.03 => ( ( L != zero_zero_real )
% 12.85/7.03 => ? [R3: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ R3 )
% 12.85/7.03 & ! [X5: real] :
% 12.85/7.03 ( ( ( X5 != C )
% 12.85/7.03 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
% 12.85/7.03 => ( ( F @ X5 )
% 12.85/7.03 != zero_zero_real ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9675_LIM__fun__gt__zero,axiom,
% 12.85/7.03 ! [F: real > real,L: real,C: real] :
% 12.85/7.03 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 12.85/7.03 => ( ( ord_less_real @ zero_zero_real @ L )
% 12.85/7.03 => ? [R3: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ R3 )
% 12.85/7.03 & ! [X5: real] :
% 12.85/7.03 ( ( ( X5 != C )
% 12.85/7.03 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
% 12.85/7.03 => ( ord_less_real @ zero_zero_real @ ( F @ X5 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9676_isCont__real__sqrt,axiom,
% 12.85/7.03 ! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ sqrt ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9677_isCont__real__root,axiom,
% 12.85/7.03 ! [X: real,N: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( root @ N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9678_isCont__inverse__function2,axiom,
% 12.85/7.03 ! [A: real,X: real,B: real,G: real > real,F: real > real] :
% 12.85/7.03 ( ( ord_less_real @ A @ X )
% 12.85/7.03 => ( ( ord_less_real @ X @ B )
% 12.85/7.03 => ( ! [Z2: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ A @ Z2 )
% 12.85/7.03 => ( ( ord_less_eq_real @ Z2 @ B )
% 12.85/7.03 => ( ( G @ ( F @ Z2 ) )
% 12.85/7.03 = Z2 ) ) )
% 12.85/7.03 => ( ! [Z2: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ A @ Z2 )
% 12.85/7.03 => ( ( ord_less_eq_real @ Z2 @ B )
% 12.85/7.03 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
% 12.85/7.03 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9679_isCont__ln,axiom,
% 12.85/7.03 ! [X: real] :
% 12.85/7.03 ( ( X != zero_zero_real )
% 12.85/7.03 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ln_ln_real ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9680_LIM__cos__div__sin,axiom,
% 12.85/7.03 ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( cos_real @ X3 ) @ ( sin_real @ X3 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9681_DERIV__inverse__function,axiom,
% 12.85/7.03 ! [F: real > real,D3: real,G: real > real,X: real,A: real,B: real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ D3 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
% 12.85/7.03 => ( ( D3 != zero_zero_real )
% 12.85/7.03 => ( ( ord_less_real @ A @ X )
% 12.85/7.03 => ( ( ord_less_real @ X @ B )
% 12.85/7.03 => ( ! [Y: real] :
% 12.85/7.03 ( ( ord_less_real @ A @ Y )
% 12.85/7.03 => ( ( ord_less_real @ Y @ B )
% 12.85/7.03 => ( ( F @ ( G @ Y ) )
% 12.85/7.03 = Y ) ) )
% 12.85/7.03 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D3 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9682_LIM__less__bound,axiom,
% 12.85/7.03 ! [B: real,X: real,F: real > real] :
% 12.85/7.03 ( ( ord_less_real @ B @ X )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ B @ X ) )
% 12.85/7.03 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 12.85/7.03 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
% 12.85/7.03 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9683_isCont__inverse__function,axiom,
% 12.85/7.03 ! [D: real,X: real,G: real > real,F: real > real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ D )
% 12.85/7.03 => ( ! [Z2: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X ) ) @ D )
% 12.85/7.03 => ( ( G @ ( F @ Z2 ) )
% 12.85/7.03 = Z2 ) )
% 12.85/7.03 => ( ! [Z2: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X ) ) @ D )
% 12.85/7.03 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) )
% 12.85/7.03 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9684_GMVT_H,axiom,
% 12.85/7.03 ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F2: real > real] :
% 12.85/7.03 ( ( ord_less_real @ A @ B )
% 12.85/7.03 => ( ! [Z2: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ A @ Z2 )
% 12.85/7.03 => ( ( ord_less_eq_real @ Z2 @ B )
% 12.85/7.03 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
% 12.85/7.03 => ( ! [Z2: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ A @ Z2 )
% 12.85/7.03 => ( ( ord_less_eq_real @ Z2 @ B )
% 12.85/7.03 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ G ) ) )
% 12.85/7.03 => ( ! [Z2: real] :
% 12.85/7.03 ( ( ord_less_real @ A @ Z2 )
% 12.85/7.03 => ( ( ord_less_real @ Z2 @ B )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
% 12.85/7.03 => ( ! [Z2: real] :
% 12.85/7.03 ( ( ord_less_real @ A @ Z2 )
% 12.85/7.03 => ( ( ord_less_real @ Z2 @ B )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ F @ ( F2 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
% 12.85/7.03 => ? [C3: real] :
% 12.85/7.03 ( ( ord_less_real @ A @ C3 )
% 12.85/7.03 & ( ord_less_real @ C3 @ B )
% 12.85/7.03 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C3 ) )
% 12.85/7.03 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F2 @ C3 ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9685_summable__Leibniz_I3_J,axiom,
% 12.85/7.03 ! [A: nat > real] :
% 12.85/7.03 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.85/7.03 => ( ( topolo6980174941875973593q_real @ A )
% 12.85/7.03 => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 12.85/7.03 => ! [N6: nat] :
% 12.85/7.03 ( member_real
% 12.85/7.03 @ ( suminf_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
% 12.85/7.03 @ ( set_or1222579329274155063t_real
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) )
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9686_summable__Leibniz_I2_J,axiom,
% 12.85/7.03 ! [A: nat > real] :
% 12.85/7.03 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.85/7.03 => ( ( topolo6980174941875973593q_real @ A )
% 12.85/7.03 => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 12.85/7.03 => ! [N6: nat] :
% 12.85/7.03 ( member_real
% 12.85/7.03 @ ( suminf_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
% 12.85/7.03 @ ( set_or1222579329274155063t_real
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9687_filterlim__Suc,axiom,
% 12.85/7.03 filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9688_mult__nat__right__at__top,axiom,
% 12.85/7.03 ! [C: nat] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ C )
% 12.85/7.03 => ( filterlim_nat_nat
% 12.85/7.03 @ ^ [X3: nat] : ( times_times_nat @ X3 @ C )
% 12.85/7.03 @ at_top_nat
% 12.85/7.03 @ at_top_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9689_mult__nat__left__at__top,axiom,
% 12.85/7.03 ! [C: nat] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ C )
% 12.85/7.03 => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9690_monoseq__convergent,axiom,
% 12.85/7.03 ! [X9: nat > real,B4: real] :
% 12.85/7.03 ( ( topolo6980174941875973593q_real @ X9 )
% 12.85/7.03 => ( ! [I3: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X9 @ I3 ) ) @ B4 )
% 12.85/7.03 => ~ ! [L5: real] :
% 12.85/7.03 ~ ( filterlim_nat_real @ X9 @ ( topolo2815343760600316023s_real @ L5 ) @ at_top_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9691_LIMSEQ__root,axiom,
% 12.85/7.03 ( filterlim_nat_real
% 12.85/7.03 @ ^ [N4: nat] : ( root @ N4 @ ( semiri5074537144036343181t_real @ N4 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ one_one_real )
% 12.85/7.03 @ at_top_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9692_nested__sequence__unique,axiom,
% 12.85/7.03 ! [F: nat > real,G: nat > real] :
% 12.85/7.03 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 12.85/7.03 => ( ! [N2: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N2 ) ) @ ( G @ N2 ) )
% 12.85/7.03 => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 12.85/7.03 => ( ( filterlim_nat_real
% 12.85/7.03 @ ^ [N4: nat] : ( minus_minus_real @ ( F @ N4 ) @ ( G @ N4 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.85/7.03 @ at_top_nat )
% 12.85/7.03 => ? [L2: real] :
% 12.85/7.03 ( ! [N6: nat] : ( ord_less_eq_real @ ( F @ N6 ) @ L2 )
% 12.85/7.03 & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat )
% 12.85/7.03 & ! [N6: nat] : ( ord_less_eq_real @ L2 @ ( G @ N6 ) )
% 12.85/7.03 & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9693_LIMSEQ__inverse__zero,axiom,
% 12.85/7.03 ! [X9: nat > real] :
% 12.85/7.03 ( ! [R3: real] :
% 12.85/7.03 ? [N7: nat] :
% 12.85/7.03 ! [N2: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ N7 @ N2 )
% 12.85/7.03 => ( ord_less_real @ R3 @ ( X9 @ N2 ) ) )
% 12.85/7.03 => ( filterlim_nat_real
% 12.85/7.03 @ ^ [N4: nat] : ( inverse_inverse_real @ ( X9 @ N4 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.85/7.03 @ at_top_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9694_lim__inverse__n_H,axiom,
% 12.85/7.03 ( filterlim_nat_real
% 12.85/7.03 @ ^ [N4: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N4 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.85/7.03 @ at_top_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9695_LIMSEQ__root__const,axiom,
% 12.85/7.03 ! [C: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ C )
% 12.85/7.03 => ( filterlim_nat_real
% 12.85/7.03 @ ^ [N4: nat] : ( root @ N4 @ C )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ one_one_real )
% 12.85/7.03 @ at_top_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9696_LIMSEQ__inverse__real__of__nat,axiom,
% 12.85/7.03 ( filterlim_nat_real
% 12.85/7.03 @ ^ [N4: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.85/7.03 @ at_top_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9697_LIMSEQ__inverse__real__of__nat__add,axiom,
% 12.85/7.03 ! [R2: real] :
% 12.85/7.03 ( filterlim_nat_real
% 12.85/7.03 @ ^ [N4: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ R2 )
% 12.85/7.03 @ at_top_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9698_increasing__LIMSEQ,axiom,
% 12.85/7.03 ! [F: nat > real,L: real] :
% 12.85/7.03 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 12.85/7.03 => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ L )
% 12.85/7.03 => ( ! [E: real] :
% 12.85/7.03 ( ( ord_less_real @ zero_zero_real @ E )
% 12.85/7.03 => ? [N6: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N6 ) @ E ) ) )
% 12.85/7.03 => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9699_LIMSEQ__realpow__zero,axiom,
% 12.85/7.03 ! [X: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.03 => ( ( ord_less_real @ X @ one_one_real )
% 12.85/7.03 => ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9700_LIMSEQ__divide__realpow__zero,axiom,
% 12.85/7.03 ! [X: real,A: real] :
% 12.85/7.03 ( ( ord_less_real @ one_one_real @ X )
% 12.85/7.03 => ( filterlim_nat_real
% 12.85/7.03 @ ^ [N4: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N4 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.85/7.03 @ at_top_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9701_LIMSEQ__abs__realpow__zero,axiom,
% 12.85/7.03 ! [C: real] :
% 12.85/7.03 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 12.85/7.03 => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9702_LIMSEQ__abs__realpow__zero2,axiom,
% 12.85/7.03 ! [C: real] :
% 12.85/7.03 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 12.85/7.03 => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9703_LIMSEQ__inverse__realpow__zero,axiom,
% 12.85/7.03 ! [X: real] :
% 12.85/7.03 ( ( ord_less_real @ one_one_real @ X )
% 12.85/7.03 => ( filterlim_nat_real
% 12.85/7.03 @ ^ [N4: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N4 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.85/7.03 @ at_top_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9704_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
% 12.85/7.03 ! [R2: real] :
% 12.85/7.03 ( filterlim_nat_real
% 12.85/7.03 @ ^ [N4: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ R2 )
% 12.85/7.03 @ at_top_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9705_tendsto__exp__limit__sequentially,axiom,
% 12.85/7.03 ! [X: real] :
% 12.85/7.03 ( filterlim_nat_real
% 12.85/7.03 @ ^ [N4: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ) @ N4 )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 12.85/7.03 @ at_top_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9706_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
% 12.85/7.03 ! [R2: real] :
% 12.85/7.03 ( filterlim_nat_real
% 12.85/7.03 @ ^ [N4: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) ) ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ R2 )
% 12.85/7.03 @ at_top_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9707_summable__Leibniz_I1_J,axiom,
% 12.85/7.03 ! [A: nat > real] :
% 12.85/7.03 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.85/7.03 => ( ( topolo6980174941875973593q_real @ A )
% 12.85/7.03 => ( summable_real
% 12.85/7.03 @ ^ [N4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( A @ N4 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9708_summable,axiom,
% 12.85/7.03 ! [A: nat > real] :
% 12.85/7.03 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.85/7.03 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 12.85/7.03 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 12.85/7.03 => ( summable_real
% 12.85/7.03 @ ^ [N4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( A @ N4 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9709_cos__diff__limit__1,axiom,
% 12.85/7.03 ! [Theta: nat > real,Theta2: real] :
% 12.85/7.03 ( ( filterlim_nat_real
% 12.85/7.03 @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ one_one_real )
% 12.85/7.03 @ at_top_nat )
% 12.85/7.03 => ~ ! [K2: nat > int] :
% 12.85/7.03 ~ ( filterlim_nat_real
% 12.85/7.03 @ ^ [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 ) ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ Theta2 )
% 12.85/7.03 @ at_top_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9710_cos__limit__1,axiom,
% 12.85/7.03 ! [Theta: nat > real] :
% 12.85/7.03 ( ( filterlim_nat_real
% 12.85/7.03 @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ one_one_real )
% 12.85/7.03 @ at_top_nat )
% 12.85/7.03 => ? [K2: nat > int] :
% 12.85/7.03 ( filterlim_nat_real
% 12.85/7.03 @ ^ [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 ) ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.85/7.03 @ at_top_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9711_summable__Leibniz_I4_J,axiom,
% 12.85/7.03 ! [A: nat > real] :
% 12.85/7.03 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.85/7.03 => ( ( topolo6980174941875973593q_real @ A )
% 12.85/7.03 => ( filterlim_nat_real
% 12.85/7.03 @ ^ [N4: nat] :
% 12.85/7.03 ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real
% 12.85/7.03 @ ( suminf_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 12.85/7.03 @ at_top_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9712_zeroseq__arctan__series,axiom,
% 12.85/7.03 ! [X: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 12.85/7.03 => ( filterlim_nat_real
% 12.85/7.03 @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.85/7.03 @ at_top_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9713_summable__Leibniz_H_I2_J,axiom,
% 12.85/7.03 ! [A: nat > real,N: nat] :
% 12.85/7.03 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.85/7.03 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 12.85/7.03 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 12.85/7.03 => ( ord_less_eq_real
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 12.85/7.03 @ ( suminf_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9714_summable__Leibniz_H_I3_J,axiom,
% 12.85/7.03 ! [A: nat > real] :
% 12.85/7.03 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.85/7.03 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 12.85/7.03 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 12.85/7.03 => ( filterlim_nat_real
% 12.85/7.03 @ ^ [N4: nat] :
% 12.85/7.03 ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real
% 12.85/7.03 @ ( suminf_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 12.85/7.03 @ at_top_nat ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9715_sums__alternating__upper__lower,axiom,
% 12.85/7.03 ! [A: nat > real] :
% 12.85/7.03 ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 12.85/7.03 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 12.85/7.03 => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.85/7.03 => ? [L2: real] :
% 12.85/7.03 ( ! [N6: nat] :
% 12.85/7.03 ( ord_less_eq_real
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
% 12.85/7.03 @ L2 )
% 12.85/7.03 & ( filterlim_nat_real
% 12.85/7.03 @ ^ [N4: nat] :
% 12.85/7.03 ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ L2 )
% 12.85/7.03 @ at_top_nat )
% 12.85/7.03 & ! [N6: nat] :
% 12.85/7.03 ( ord_less_eq_real @ L2
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) )
% 12.85/7.03 & ( filterlim_nat_real
% 12.85/7.03 @ ^ [N4: nat] :
% 12.85/7.03 ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ L2 )
% 12.85/7.03 @ at_top_nat ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9716_summable__Leibniz_I5_J,axiom,
% 12.85/7.03 ! [A: nat > real] :
% 12.85/7.03 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.85/7.03 => ( ( topolo6980174941875973593q_real @ A )
% 12.85/7.03 => ( filterlim_nat_real
% 12.85/7.03 @ ^ [N4: nat] :
% 12.85/7.03 ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real
% 12.85/7.03 @ ( suminf_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 12.85/7.03 @ at_top_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9717_summable__Leibniz_H_I5_J,axiom,
% 12.85/7.03 ! [A: nat > real] :
% 12.85/7.03 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.85/7.03 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 12.85/7.03 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 12.85/7.03 => ( filterlim_nat_real
% 12.85/7.03 @ ^ [N4: nat] :
% 12.85/7.03 ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real
% 12.85/7.03 @ ( suminf_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 12.85/7.03 @ at_top_nat ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9718_summable__Leibniz_H_I4_J,axiom,
% 12.85/7.03 ! [A: nat > real,N: nat] :
% 12.85/7.03 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 12.85/7.03 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 12.85/7.03 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 12.85/7.03 => ( ord_less_eq_real
% 12.85/7.03 @ ( suminf_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
% 12.85/7.03 @ ( groups6591440286371151544t_real
% 12.85/7.03 @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 12.85/7.03 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9719_real__bounded__linear,axiom,
% 12.85/7.03 ( real_V5970128139526366754l_real
% 12.85/7.03 = ( ^ [F3: real > real] :
% 12.85/7.03 ? [C5: real] :
% 12.85/7.03 ( F3
% 12.85/7.03 = ( ^ [X3: real] : ( times_times_real @ X3 @ C5 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9720_tendsto__arctan__at__bot,axiom,
% 12.85/7.03 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9721_dist__real__def,axiom,
% 12.85/7.03 ( real_V975177566351809787t_real
% 12.85/7.03 = ( ^ [X3: real,Y6: real] : ( abs_abs_real @ ( minus_minus_real @ X3 @ Y6 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9722_dist__complex__def,axiom,
% 12.85/7.03 ( real_V3694042436643373181omplex
% 12.85/7.03 = ( ^ [X3: complex,Y6: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y6 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9723_exp__at__bot,axiom,
% 12.85/7.03 filterlim_real_real @ exp_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_bot_real ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9724_filterlim__inverse__at__bot__neg,axiom,
% 12.85/7.03 filterlim_real_real @ inverse_inverse_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5984915006950818249n_real @ zero_zero_real ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9725_DERIV__pos__imp__increasing__at__bot,axiom,
% 12.85/7.03 ! [B: real,F: real > real,Flim: real] :
% 12.85/7.03 ( ! [X4: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ X4 @ B )
% 12.85/7.03 => ? [Y3: real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.85/7.03 & ( ord_less_real @ zero_zero_real @ Y3 ) ) )
% 12.85/7.03 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 12.85/7.03 => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9726_filterlim__pow__at__bot__odd,axiom,
% 12.85/7.03 ! [N: nat,F: real > real,F4: filter_real] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
% 12.85/7.03 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/7.03 => ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N )
% 12.85/7.03 @ at_bot_real
% 12.85/7.03 @ F4 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9727_tendsto__exp__limit__at__right,axiom,
% 12.85/7.03 ! [X: real] :
% 12.85/7.03 ( filterlim_real_real
% 12.85/7.03 @ ^ [Y6: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y6 ) ) @ ( divide_divide_real @ one_one_real @ Y6 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9728_ln__at__0,axiom,
% 12.85/7.03 filterlim_real_real @ ln_ln_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9729_tendsto__arcosh__at__left__1,axiom,
% 12.85/7.03 filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9730_filterlim__tan__at__right,axiom,
% 12.85/7.03 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 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9731_atLeast__0,axiom,
% 12.85/7.03 ( ( set_ord_atLeast_nat @ zero_zero_nat )
% 12.85/7.03 = top_top_set_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9732_atLeast__Suc__greaterThan,axiom,
% 12.85/7.03 ! [K: nat] :
% 12.85/7.03 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 12.85/7.03 = ( set_or1210151606488870762an_nat @ K ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9733_greaterThan__0,axiom,
% 12.85/7.03 ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
% 12.85/7.03 = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9734_greaterThan__Suc,axiom,
% 12.85/7.03 ! [K: nat] :
% 12.85/7.03 ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 12.85/7.03 = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9735_atLeast__Suc,axiom,
% 12.85/7.03 ! [K: nat] :
% 12.85/7.03 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 12.85/7.03 = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9736_filterlim__pow__at__bot__even,axiom,
% 12.85/7.03 ! [N: nat,F: real > real,F4: filter_real] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
% 12.85/7.03 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 12.85/7.03 => ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N )
% 12.85/7.03 @ at_top_real
% 12.85/7.03 @ F4 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9737_filterlim__tan__at__left,axiom,
% 12.85/7.03 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 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9738_sqrt__at__top,axiom,
% 12.85/7.03 filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9739_filterlim__real__sequentially,axiom,
% 12.85/7.03 filterlim_nat_real @ semiri5074537144036343181t_real @ at_top_real @ at_top_nat ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9740_ln__x__over__x__tendsto__0,axiom,
% 12.85/7.03 ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( ln_ln_real @ X3 ) @ X3 )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.85/7.03 @ at_top_real ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9741_filterlim__inverse__at__right__top,axiom,
% 12.85/7.03 filterlim_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) @ at_top_real ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9742_filterlim__inverse__at__top__right,axiom,
% 12.85/7.03 filterlim_real_real @ inverse_inverse_real @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9743_tendsto__power__div__exp__0,axiom,
% 12.85/7.03 ! [K: nat] :
% 12.85/7.03 ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( power_power_real @ X3 @ K ) @ ( exp_real @ X3 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 12.85/7.03 @ at_top_real ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9744_tendsto__exp__limit__at__top,axiom,
% 12.85/7.03 ! [X: real] :
% 12.85/7.03 ( filterlim_real_real
% 12.85/7.03 @ ^ [Y6: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y6 ) ) @ Y6 )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 12.85/7.03 @ at_top_real ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9745_DERIV__neg__imp__decreasing__at__top,axiom,
% 12.85/7.03 ! [B: real,F: real > real,Flim: real] :
% 12.85/7.03 ( ! [X4: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ B @ X4 )
% 12.85/7.03 => ? [Y3: real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.85/7.03 & ( ord_less_real @ Y3 @ zero_zero_real ) ) )
% 12.85/7.03 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 12.85/7.03 => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9746_tendsto__arctan__at__top,axiom,
% 12.85/7.03 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9747_lhopital__left__at__top,axiom,
% 12.85/7.03 ! [G: real > real,X: real,G2: real > real,F: real > real,F2: real > real,Y4: real] :
% 12.85/7.03 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] :
% 12.85/7.03 ( ( G2 @ X3 )
% 12.85/7.03 != zero_zero_real )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 12.85/7.03 => ( ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ Y4 )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 12.85/7.03 => ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ Y4 )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9748_at__top__le__at__infinity,axiom,
% 12.85/7.03 ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9749_at__bot__le__at__infinity,axiom,
% 12.85/7.03 ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9750_filterlim__real__at__infinity__sequentially,axiom,
% 12.85/7.03 filterlim_nat_real @ semiri5074537144036343181t_real @ at_infinity_real @ at_top_nat ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9751_eventually__at__right__to__0,axiom,
% 12.85/7.03 ! [P: real > $o,A: real] :
% 12.85/7.03 ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.85/7.03 = ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( P @ ( plus_plus_real @ X3 @ A ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9752_eventually__at__top__to__right,axiom,
% 12.85/7.03 ! [P: real > $o] :
% 12.85/7.03 ( ( eventually_real @ P @ at_top_real )
% 12.85/7.03 = ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( P @ ( inverse_inverse_real @ X3 ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9753_eventually__at__right__to__top,axiom,
% 12.85/7.03 ! [P: real > $o] :
% 12.85/7.03 ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.85/7.03 = ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( P @ ( inverse_inverse_real @ X3 ) )
% 12.85/7.03 @ at_top_real ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9754_lhopital__at__top__at__top,axiom,
% 12.85/7.03 ! [F: real > real,A: real,G: real > real,F2: real > real,G2: real > real] :
% 12.85/7.03 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.85/7.03 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.85/7.03 => ( ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 12.85/7.03 @ at_top_real
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.85/7.03 => ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 12.85/7.03 @ at_top_real
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9755_lhopital,axiom,
% 12.85/7.03 ! [F: real > real,X: real,G: real > real,G2: real > real,F2: real > real,F4: filter_real] :
% 12.85/7.03 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] :
% 12.85/7.03 ( ( G @ X3 )
% 12.85/7.03 != zero_zero_real )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] :
% 12.85/7.03 ( ( G2 @ X3 )
% 12.85/7.03 != zero_zero_real )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 12.85/7.03 @ F4
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 12.85/7.03 @ F4
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9756_lhopital__right__at__top__at__top,axiom,
% 12.85/7.03 ! [F: real > real,A: real,G: real > real,F2: real > real,G2: real > real] :
% 12.85/7.03 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.85/7.03 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.85/7.03 => ( ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 12.85/7.03 @ at_top_real
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.85/7.03 => ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 12.85/7.03 @ at_top_real
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9757_lhopital__at__top__at__bot,axiom,
% 12.85/7.03 ! [F: real > real,A: real,G: real > real,F2: real > real,G2: real > real] :
% 12.85/7.03 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.85/7.03 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.85/7.03 => ( ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 12.85/7.03 @ at_bot_real
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 12.85/7.03 => ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 12.85/7.03 @ at_bot_real
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9758_lhopital__left__at__top__at__top,axiom,
% 12.85/7.03 ! [F: real > real,A: real,G: real > real,F2: real > real,G2: real > real] :
% 12.85/7.03 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.85/7.03 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.85/7.03 => ( ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 12.85/7.03 @ at_top_real
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.85/7.03 => ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 12.85/7.03 @ at_top_real
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9759_lhospital__at__top__at__top,axiom,
% 12.85/7.03 ! [G: real > real,G2: real > real,F: real > real,F2: real > real,X: real] :
% 12.85/7.03 ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] :
% 12.85/7.03 ( ( G2 @ X3 )
% 12.85/7.03 != zero_zero_real )
% 12.85/7.03 @ at_top_real )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ at_top_real )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ at_top_real )
% 12.85/7.03 => ( ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ X )
% 12.85/7.03 @ at_top_real )
% 12.85/7.03 => ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ X )
% 12.85/7.03 @ at_top_real ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9760_lhopital__at__top,axiom,
% 12.85/7.03 ! [G: real > real,X: real,G2: real > real,F: real > real,F2: real > real,Y4: real] :
% 12.85/7.03 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] :
% 12.85/7.03 ( ( G2 @ X3 )
% 12.85/7.03 != zero_zero_real )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ Y4 )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 12.85/7.03 => ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ Y4 )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9761_lhopital__right__0,axiom,
% 12.85/7.03 ! [F0: real > real,G0: real > real,G2: real > real,F2: real > real,F4: filter_real] :
% 12.85/7.03 ( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.85/7.03 => ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] :
% 12.85/7.03 ( ( G0 @ X3 )
% 12.85/7.03 != zero_zero_real )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] :
% 12.85/7.03 ( ( G2 @ X3 )
% 12.85/7.03 != zero_zero_real )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.85/7.03 => ( ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 12.85/7.03 @ F4
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.85/7.03 => ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F0 @ X3 ) @ ( G0 @ X3 ) )
% 12.85/7.03 @ F4
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9762_lhopital__right,axiom,
% 12.85/7.03 ! [F: real > real,X: real,G: real > real,G2: real > real,F2: real > real,F4: filter_real] :
% 12.85/7.03 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 12.85/7.03 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] :
% 12.85/7.03 ( ( G @ X3 )
% 12.85/7.03 != zero_zero_real )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] :
% 12.85/7.03 ( ( G2 @ X3 )
% 12.85/7.03 != zero_zero_real )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 12.85/7.03 => ( ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 12.85/7.03 @ F4
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 12.85/7.03 => ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 12.85/7.03 @ F4
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9763_lhopital__left,axiom,
% 12.85/7.03 ! [F: real > real,X: real,G: real > real,G2: real > real,F2: real > real,F4: filter_real] :
% 12.85/7.03 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 12.85/7.03 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] :
% 12.85/7.03 ( ( G @ X3 )
% 12.85/7.03 != zero_zero_real )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] :
% 12.85/7.03 ( ( G2 @ X3 )
% 12.85/7.03 != zero_zero_real )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 12.85/7.03 => ( ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 12.85/7.03 @ F4
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 12.85/7.03 => ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 12.85/7.03 @ F4
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9764_lhopital__right__at__top__at__bot,axiom,
% 12.85/7.03 ! [F: real > real,A: real,G: real > real,F2: real > real,G2: real > real] :
% 12.85/7.03 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.85/7.03 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.85/7.03 => ( ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 12.85/7.03 @ at_bot_real
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.85/7.03 => ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 12.85/7.03 @ at_bot_real
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9765_lhopital__left__at__top__at__bot,axiom,
% 12.85/7.03 ! [F: real > real,A: real,G: real > real,F2: real > real,G2: real > real] :
% 12.85/7.03 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.85/7.03 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.85/7.03 => ( ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 12.85/7.03 @ at_bot_real
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 12.85/7.03 => ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 12.85/7.03 @ at_bot_real
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9766_lhopital__right__at__top,axiom,
% 12.85/7.03 ! [G: real > real,X: real,G2: real > real,F: real > real,F2: real > real,Y4: real] :
% 12.85/7.03 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] :
% 12.85/7.03 ( ( G2 @ X3 )
% 12.85/7.03 != zero_zero_real )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 12.85/7.03 => ( ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ Y4 )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 12.85/7.03 => ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ Y4 )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9767_lhopital__right__0__at__top,axiom,
% 12.85/7.03 ! [G: real > real,G2: real > real,F: real > real,F2: real > real,X: real] :
% 12.85/7.03 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] :
% 12.85/7.03 ( ( G2 @ X3 )
% 12.85/7.03 != zero_zero_real )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.85/7.03 => ( ( eventually_real
% 12.85/7.03 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.85/7.03 => ( ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ X )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.85/7.03 => ( filterlim_real_real
% 12.85/7.03 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 12.85/7.03 @ ( topolo2815343760600316023s_real @ X )
% 12.85/7.03 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9768_eventually__sequentially__Suc,axiom,
% 12.85/7.03 ! [P: nat > $o] :
% 12.85/7.03 ( ( eventually_nat
% 12.85/7.03 @ ^ [I2: nat] : ( P @ ( suc @ I2 ) )
% 12.85/7.03 @ at_top_nat )
% 12.85/7.03 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9769_le__sequentially,axiom,
% 12.85/7.03 ! [F4: filter_nat] :
% 12.85/7.03 ( ( ord_le2510731241096832064er_nat @ F4 @ at_top_nat )
% 12.85/7.03 = ( ! [N8: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N8 ) @ F4 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9770_eventually__sequentially,axiom,
% 12.85/7.03 ! [P: nat > $o] :
% 12.85/7.03 ( ( eventually_nat @ P @ at_top_nat )
% 12.85/7.03 = ( ? [N8: nat] :
% 12.85/7.03 ! [N4: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ N8 @ N4 )
% 12.85/7.03 => ( P @ N4 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9771_eventually__sequentiallyI,axiom,
% 12.85/7.03 ! [C: nat,P: nat > $o] :
% 12.85/7.03 ( ! [X4: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ C @ X4 )
% 12.85/7.03 => ( P @ X4 ) )
% 12.85/7.03 => ( eventually_nat @ P @ at_top_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9772_filterlim__int__sequentially,axiom,
% 12.85/7.03 filterlim_nat_int @ semiri1314217659103216013at_int @ at_top_int @ at_top_nat ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9773_Bseq__eq__bounded,axiom,
% 12.85/7.03 ! [F: nat > real,A: real,B: real] :
% 12.85/7.03 ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A @ B ) )
% 12.85/7.03 => ( bfun_nat_real @ F @ at_top_nat ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9774_Bseq__realpow,axiom,
% 12.85/7.03 ! [X: real] :
% 12.85/7.03 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.03 => ( ( ord_less_eq_real @ X @ one_one_real )
% 12.85/7.03 => ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9775_GreatestI__nat,axiom,
% 12.85/7.03 ! [P: nat > $o,K: nat,B: nat] :
% 12.85/7.03 ( ( P @ K )
% 12.85/7.03 => ( ! [Y: nat] :
% 12.85/7.03 ( ( P @ Y )
% 12.85/7.03 => ( ord_less_eq_nat @ Y @ B ) )
% 12.85/7.03 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9776_Greatest__le__nat,axiom,
% 12.85/7.03 ! [P: nat > $o,K: nat,B: nat] :
% 12.85/7.03 ( ( P @ K )
% 12.85/7.03 => ( ! [Y: nat] :
% 12.85/7.03 ( ( P @ Y )
% 12.85/7.03 => ( ord_less_eq_nat @ Y @ B ) )
% 12.85/7.03 => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9777_GreatestI__ex__nat,axiom,
% 12.85/7.03 ! [P: nat > $o,B: nat] :
% 12.85/7.03 ( ? [X_12: nat] : ( P @ X_12 )
% 12.85/7.03 => ( ! [Y: nat] :
% 12.85/7.03 ( ( P @ Y )
% 12.85/7.03 => ( ord_less_eq_nat @ Y @ B ) )
% 12.85/7.03 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9778_GMVT,axiom,
% 12.85/7.03 ! [A: real,B: real,F: real > real,G: real > real] :
% 12.85/7.03 ( ( ord_less_real @ A @ B )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( ( ord_less_eq_real @ A @ X4 )
% 12.85/7.03 & ( ord_less_eq_real @ X4 @ B ) )
% 12.85/7.03 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( ( ord_less_real @ A @ X4 )
% 12.85/7.03 & ( ord_less_real @ X4 @ B ) )
% 12.85/7.03 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( ( ord_less_eq_real @ A @ X4 )
% 12.85/7.03 & ( ord_less_eq_real @ X4 @ B ) )
% 12.85/7.03 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G ) )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( ( ord_less_real @ A @ X4 )
% 12.85/7.03 & ( ord_less_real @ X4 @ B ) )
% 12.85/7.03 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 12.85/7.03 => ? [G_c: real,F_c: real,C3: real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 12.85/7.03 & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 12.85/7.03 & ( ord_less_real @ A @ C3 )
% 12.85/7.03 & ( ord_less_real @ C3 @ B )
% 12.85/7.03 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
% 12.85/7.03 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9779_MVT,axiom,
% 12.85/7.03 ! [A: real,B: real,F: real > real] :
% 12.85/7.03 ( ( ord_less_real @ A @ B )
% 12.85/7.03 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( ord_less_real @ A @ X4 )
% 12.85/7.03 => ( ( ord_less_real @ X4 @ B )
% 12.85/7.03 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 12.85/7.03 => ? [L2: real,Z2: real] :
% 12.85/7.03 ( ( ord_less_real @ A @ Z2 )
% 12.85/7.03 & ( ord_less_real @ Z2 @ B )
% 12.85/7.03 & ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) )
% 12.85/7.03 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 12.85/7.03 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L2 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9780_continuous__on__arcosh_H,axiom,
% 12.85/7.03 ! [A2: set_real,F: real > real] :
% 12.85/7.03 ( ( topolo5044208981011980120l_real @ A2 @ F )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( member_real @ X4 @ A2 )
% 12.85/7.03 => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 12.85/7.03 => ( topolo5044208981011980120l_real @ A2
% 12.85/7.03 @ ^ [X3: real] : ( arcosh_real @ ( F @ X3 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9781_continuous__image__closed__interval,axiom,
% 12.85/7.03 ! [A: real,B: real,F: real > real] :
% 12.85/7.03 ( ( ord_less_eq_real @ A @ B )
% 12.85/7.03 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 12.85/7.03 => ? [C3: real,D2: real] :
% 12.85/7.03 ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
% 12.85/7.03 = ( set_or1222579329274155063t_real @ C3 @ D2 ) )
% 12.85/7.03 & ( ord_less_eq_real @ C3 @ D2 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9782_continuous__on__arcosh,axiom,
% 12.85/7.03 ! [A2: set_real] :
% 12.85/7.03 ( ( ord_less_eq_set_real @ A2 @ ( set_ord_atLeast_real @ one_one_real ) )
% 12.85/7.03 => ( topolo5044208981011980120l_real @ A2 @ arcosh_real ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9783_Rolle__deriv,axiom,
% 12.85/7.03 ! [A: real,B: real,F: real > real,F2: real > real > real] :
% 12.85/7.03 ( ( ord_less_real @ A @ B )
% 12.85/7.03 => ( ( ( F @ A )
% 12.85/7.03 = ( F @ B ) )
% 12.85/7.03 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( ord_less_real @ A @ X4 )
% 12.85/7.03 => ( ( ord_less_real @ X4 @ B )
% 12.85/7.03 => ( has_de1759254742604945161l_real @ F @ ( F2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 12.85/7.03 => ? [Z2: real] :
% 12.85/7.03 ( ( ord_less_real @ A @ Z2 )
% 12.85/7.03 & ( ord_less_real @ Z2 @ B )
% 12.85/7.03 & ( ( F2 @ Z2 )
% 12.85/7.03 = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9784_mvt,axiom,
% 12.85/7.03 ! [A: real,B: real,F: real > real,F2: real > real > real] :
% 12.85/7.03 ( ( ord_less_real @ A @ B )
% 12.85/7.03 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( ord_less_real @ A @ X4 )
% 12.85/7.03 => ( ( ord_less_real @ X4 @ B )
% 12.85/7.03 => ( has_de1759254742604945161l_real @ F @ ( F2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 12.85/7.03 => ~ ! [Xi: real] :
% 12.85/7.03 ( ( ord_less_real @ A @ Xi )
% 12.85/7.03 => ( ( ord_less_real @ Xi @ B )
% 12.85/7.03 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 12.85/7.03 != ( F2 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9785_DERIV__pos__imp__increasing__open,axiom,
% 12.85/7.03 ! [A: real,B: real,F: real > real] :
% 12.85/7.03 ( ( ord_less_real @ A @ B )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( ord_less_real @ A @ X4 )
% 12.85/7.03 => ( ( ord_less_real @ X4 @ B )
% 12.85/7.03 => ? [Y3: real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.85/7.03 & ( ord_less_real @ zero_zero_real @ Y3 ) ) ) )
% 12.85/7.03 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 12.85/7.03 => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9786_DERIV__neg__imp__decreasing__open,axiom,
% 12.85/7.03 ! [A: real,B: real,F: real > real] :
% 12.85/7.03 ( ( ord_less_real @ A @ B )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( ord_less_real @ A @ X4 )
% 12.85/7.03 => ( ( ord_less_real @ X4 @ B )
% 12.85/7.03 => ? [Y3: real] :
% 12.85/7.03 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.85/7.03 & ( ord_less_real @ Y3 @ zero_zero_real ) ) ) )
% 12.85/7.03 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 12.85/7.03 => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9787_DERIV__isconst__end,axiom,
% 12.85/7.03 ! [A: real,B: real,F: real > real] :
% 12.85/7.03 ( ( ord_less_real @ A @ B )
% 12.85/7.03 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( ord_less_real @ A @ X4 )
% 12.85/7.03 => ( ( ord_less_real @ X4 @ B )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 12.85/7.03 => ( ( F @ B )
% 12.85/7.03 = ( F @ A ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9788_continuous__on__artanh,axiom,
% 12.85/7.03 ! [A2: set_real] :
% 12.85/7.03 ( ( ord_less_eq_set_real @ A2 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
% 12.85/7.03 => ( topolo5044208981011980120l_real @ A2 @ artanh_real ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9789_DERIV__isconst2,axiom,
% 12.85/7.03 ! [A: real,B: real,F: real > real,X: real] :
% 12.85/7.03 ( ( ord_less_real @ A @ B )
% 12.85/7.03 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( ord_less_real @ A @ X4 )
% 12.85/7.03 => ( ( ord_less_real @ X4 @ B )
% 12.85/7.03 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 12.85/7.03 => ( ( ord_less_eq_real @ A @ X )
% 12.85/7.03 => ( ( ord_less_eq_real @ X @ B )
% 12.85/7.03 => ( ( F @ X )
% 12.85/7.03 = ( F @ A ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9790_Rolle,axiom,
% 12.85/7.03 ! [A: real,B: real,F: real > real] :
% 12.85/7.03 ( ( ord_less_real @ A @ B )
% 12.85/7.03 => ( ( ( F @ A )
% 12.85/7.03 = ( F @ B ) )
% 12.85/7.03 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 12.85/7.03 => ( ! [X4: real] :
% 12.85/7.03 ( ( ord_less_real @ A @ X4 )
% 12.85/7.03 => ( ( ord_less_real @ X4 @ B )
% 12.85/7.03 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 12.85/7.03 => ? [Z2: real] :
% 12.85/7.03 ( ( ord_less_real @ A @ Z2 )
% 12.85/7.03 & ( ord_less_real @ Z2 @ B )
% 12.85/7.03 & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9791_surj__int__encode,axiom,
% 12.85/7.03 ( ( image_int_nat @ nat_int_encode @ top_top_set_int )
% 12.85/7.03 = top_top_set_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9792_int__encode__eq,axiom,
% 12.85/7.03 ! [X: int,Y4: int] :
% 12.85/7.03 ( ( ( nat_int_encode @ X )
% 12.85/7.03 = ( nat_int_encode @ Y4 ) )
% 12.85/7.03 = ( X = Y4 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9793_bij__int__encode,axiom,
% 12.85/7.03 bij_betw_int_nat @ nat_int_encode @ top_top_set_int @ top_top_set_nat ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9794_uniformity__real__def,axiom,
% 12.85/7.03 ( topolo1511823702728130853y_real
% 12.85/7.03 = ( comple2936214249959783750l_real
% 12.85/7.03 @ ( image_2178119161166701260l_real
% 12.85/7.03 @ ^ [E3: real] :
% 12.85/7.03 ( princi6114159922880469582l_real
% 12.85/7.03 @ ( collec3799799289383736868l_real
% 12.85/7.03 @ ( produc5414030515140494994real_o
% 12.85/7.03 @ ^ [X3: real,Y6: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X3 @ Y6 ) @ E3 ) ) ) )
% 12.85/7.03 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9795_uniformity__complex__def,axiom,
% 12.85/7.03 ( topolo896644834953643431omplex
% 12.85/7.03 = ( comple8358262395181532106omplex
% 12.85/7.03 @ ( image_5971271580939081552omplex
% 12.85/7.03 @ ^ [E3: real] :
% 12.85/7.03 ( princi3496590319149328850omplex
% 12.85/7.03 @ ( collec8663557070575231912omplex
% 12.85/7.03 @ ( produc6771430404735790350plex_o
% 12.85/7.03 @ ^ [X3: complex,Y6: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X3 @ Y6 ) @ E3 ) ) ) )
% 12.85/7.03 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9796_range__abs__Nats,axiom,
% 12.85/7.03 ( ( image_int_int @ abs_abs_int @ top_top_set_int )
% 12.85/7.03 = semiring_1_Nats_int ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9797_inj__sgn__power,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( inj_on_real_real
% 12.85/7.03 @ ^ [Y6: real] : ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N ) )
% 12.85/7.03 @ top_top_set_real ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9798_surj__int__decode,axiom,
% 12.85/7.03 ( ( image_nat_int @ nat_int_decode @ top_top_set_nat )
% 12.85/7.03 = top_top_set_int ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9799_int__decode__inverse,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( nat_int_encode @ ( nat_int_decode @ N ) )
% 12.85/7.03 = N ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9800_int__encode__inverse,axiom,
% 12.85/7.03 ! [X: int] :
% 12.85/7.03 ( ( nat_int_decode @ ( nat_int_encode @ X ) )
% 12.85/7.03 = X ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9801_int__decode__eq,axiom,
% 12.85/7.03 ! [X: nat,Y4: nat] :
% 12.85/7.03 ( ( ( nat_int_decode @ X )
% 12.85/7.03 = ( nat_int_decode @ Y4 ) )
% 12.85/7.03 = ( X = Y4 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9802_bij__int__decode,axiom,
% 12.85/7.03 bij_betw_nat_int @ nat_int_decode @ top_top_set_nat @ top_top_set_int ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9803_log__inj,axiom,
% 12.85/7.03 ! [B: real] :
% 12.85/7.03 ( ( ord_less_real @ one_one_real @ B )
% 12.85/7.03 => ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9804_complex__is__Nat__iff,axiom,
% 12.85/7.03 ! [Z: complex] :
% 12.85/7.03 ( ( member_complex @ Z @ semiri3842193898606819883omplex )
% 12.85/7.03 = ( ( ( im @ Z )
% 12.85/7.03 = zero_zero_real )
% 12.85/7.03 & ? [I2: nat] :
% 12.85/7.03 ( ( re @ Z )
% 12.85/7.03 = ( semiri5074537144036343181t_real @ I2 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9805_inj__int__encode,axiom,
% 12.85/7.03 ! [A2: set_int] : ( inj_on_int_nat @ nat_int_encode @ A2 ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9806_inj__on__set__encode,axiom,
% 12.85/7.03 inj_on_set_nat_nat @ nat_set_encode @ ( collect_set_nat @ finite_finite_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9807_inj__prod__encode,axiom,
% 12.85/7.03 ! [A2: set_Pr1261947904930325089at_nat] : ( inj_on2178005380612969504at_nat @ nat_prod_encode @ A2 ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9808_inj__Suc,axiom,
% 12.85/7.03 ! [N5: set_nat] : ( inj_on_nat_nat @ suc @ N5 ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9809_inj__int__decode,axiom,
% 12.85/7.03 ! [A2: set_nat] : ( inj_on_nat_int @ nat_int_decode @ A2 ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9810_inj__on__diff__nat,axiom,
% 12.85/7.03 ! [N5: set_nat,K: nat] :
% 12.85/7.03 ( ! [N2: nat] :
% 12.85/7.03 ( ( member_nat @ N2 @ N5 )
% 12.85/7.03 => ( ord_less_eq_nat @ K @ N2 ) )
% 12.85/7.03 => ( inj_on_nat_nat
% 12.85/7.03 @ ^ [N4: nat] : ( minus_minus_nat @ N4 @ K )
% 12.85/7.03 @ N5 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9811_summable__reindex,axiom,
% 12.85/7.03 ! [F: nat > real,G: nat > nat] :
% 12.85/7.03 ( ( summable_real @ F )
% 12.85/7.03 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 12.85/7.03 => ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 12.85/7.03 => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9812_suminf__reindex__mono,axiom,
% 12.85/7.03 ! [F: nat > real,G: nat > nat] :
% 12.85/7.03 ( ( summable_real @ F )
% 12.85/7.03 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 12.85/7.03 => ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 12.85/7.03 => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9813_inj__on__char__of__nat,axiom,
% 12.85/7.03 inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9814_suminf__reindex,axiom,
% 12.85/7.03 ! [F: nat > real,G: nat > nat] :
% 12.85/7.03 ( ( summable_real @ F )
% 12.85/7.03 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 12.85/7.03 => ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 12.85/7.03 => ( ! [X4: nat] :
% 12.85/7.03 ( ~ ( member_nat @ X4 @ ( image_nat_nat @ G @ top_top_set_nat ) )
% 12.85/7.03 => ( ( F @ X4 )
% 12.85/7.03 = zero_zero_real ) )
% 12.85/7.03 => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
% 12.85/7.03 = ( suminf_real @ F ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9815_powr__real__of__int_H,axiom,
% 12.85/7.03 ! [X: real,N: int] :
% 12.85/7.03 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.03 => ( ( ( X != zero_zero_real )
% 12.85/7.03 | ( ord_less_int @ zero_zero_int @ N ) )
% 12.85/7.03 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 12.85/7.03 = ( power_int_real @ X @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9816_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 12.85/7.03 ! [X: vEBT_VEBT,Xa: nat] :
% 12.85/7.03 ( ~ ( vEBT_VEBT_valid @ X @ Xa )
% 12.85/7.03 => ( ( ? [Uu2: $o,Uv2: $o] :
% 12.85/7.03 ( X
% 12.85/7.03 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 12.85/7.03 => ( Xa = one_one_nat ) )
% 12.85/7.03 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 12.85/7.03 => ( ( Deg2 = Xa )
% 12.85/7.03 & ! [X4: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.03 => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.03 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.03 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 12.85/7.03 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.85/7.03 & ( case_o184042715313410164at_nat
% 12.85/7.03 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 12.85/7.03 & ! [X3: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.03 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 12.85/7.03 @ ( produc6081775807080527818_nat_o
% 12.85/7.03 @ ^ [Mi3: nat,Ma3: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 12.85/7.03 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.85/7.03 & ! [I2: nat] :
% 12.85/7.03 ( ( 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 ) ) ) ) ) )
% 12.85/7.03 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 12.85/7.03 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 12.85/7.03 & ( ( Mi3 = Ma3 )
% 12.85/7.03 => ! [X3: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.03 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 12.85/7.03 & ( ( Mi3 != Ma3 )
% 12.85/7.03 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 12.85/7.03 & ! [X3: nat] :
% 12.85/7.03 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.85/7.03 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
% 12.85/7.03 => ( ( ord_less_nat @ Mi3 @ X3 )
% 12.85/7.03 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 12.85/7.03 @ Mima ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9817_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 12.85/7.03 ! [X: vEBT_VEBT,Xa: nat] :
% 12.85/7.03 ( ( vEBT_VEBT_valid @ X @ Xa )
% 12.85/7.03 => ( ( ? [Uu2: $o,Uv2: $o] :
% 12.85/7.03 ( X
% 12.85/7.03 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 12.85/7.03 => ( Xa != one_one_nat ) )
% 12.85/7.03 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 12.85/7.03 => ~ ( ( Deg2 = Xa )
% 12.85/7.03 & ! [X5: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.03 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.03 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.03 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 12.85/7.03 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.85/7.03 & ( case_o184042715313410164at_nat
% 12.85/7.03 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 12.85/7.03 & ! [X3: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.03 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 12.85/7.03 @ ( produc6081775807080527818_nat_o
% 12.85/7.03 @ ^ [Mi3: nat,Ma3: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 12.85/7.03 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.85/7.03 & ! [I2: nat] :
% 12.85/7.03 ( ( 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 ) ) ) ) ) )
% 12.85/7.03 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 12.85/7.03 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 12.85/7.03 & ( ( Mi3 = Ma3 )
% 12.85/7.03 => ! [X3: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.03 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 12.85/7.03 & ( ( Mi3 != Ma3 )
% 12.85/7.03 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 12.85/7.03 & ! [X3: nat] :
% 12.85/7.03 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.85/7.03 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
% 12.85/7.03 => ( ( ord_less_nat @ Mi3 @ X3 )
% 12.85/7.03 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 12.85/7.03 @ Mima ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9818_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 12.85/7.03 ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
% 12.85/7.03 ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg4 )
% 12.85/7.03 = ( ( Deg = Deg4 )
% 12.85/7.03 & ! [X3: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.85/7.03 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.03 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.03 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 12.85/7.03 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.85/7.03 & ( case_o184042715313410164at_nat
% 12.85/7.03 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
% 12.85/7.03 & ! [X3: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.85/7.03 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 12.85/7.03 @ ( produc6081775807080527818_nat_o
% 12.85/7.03 @ ^ [Mi3: nat,Ma3: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 12.85/7.03 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 12.85/7.03 & ! [I2: nat] :
% 12.85/7.03 ( ( 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 ) ) ) ) ) )
% 12.85/7.03 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X8 ) )
% 12.85/7.03 = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 12.85/7.03 & ( ( Mi3 = Ma3 )
% 12.85/7.03 => ! [X3: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 12.85/7.03 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 12.85/7.03 & ( ( Mi3 != Ma3 )
% 12.85/7.03 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
% 12.85/7.03 & ! [X3: nat] :
% 12.85/7.03 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 12.85/7.03 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X3 )
% 12.85/7.03 => ( ( ord_less_nat @ Mi3 @ X3 )
% 12.85/7.03 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 12.85/7.03 @ Mima2 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9819_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 12.85/7.03 ! [X: vEBT_VEBT,Xa: nat,Y4: $o] :
% 12.85/7.03 ( ( ( vEBT_VEBT_valid @ X @ Xa )
% 12.85/7.03 = Y4 )
% 12.85/7.03 => ( ( ? [Uu2: $o,Uv2: $o] :
% 12.85/7.03 ( X
% 12.85/7.03 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( Xa != one_one_nat ) ) )
% 12.85/7.03 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 = ( ~ ( ( Deg2 = Xa )
% 12.85/7.03 & ! [X3: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.03 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.03 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.03 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 12.85/7.03 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.85/7.03 & ( case_o184042715313410164at_nat
% 12.85/7.03 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 12.85/7.03 & ! [X3: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.03 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 12.85/7.03 @ ( produc6081775807080527818_nat_o
% 12.85/7.03 @ ^ [Mi3: nat,Ma3: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 12.85/7.03 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.85/7.03 & ! [I2: nat] :
% 12.85/7.03 ( ( 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 ) ) ) ) ) )
% 12.85/7.03 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 12.85/7.03 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 12.85/7.03 & ( ( Mi3 = Ma3 )
% 12.85/7.03 => ! [X3: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.03 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 12.85/7.03 & ( ( Mi3 != Ma3 )
% 12.85/7.03 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 12.85/7.03 & ! [X3: nat] :
% 12.85/7.03 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.85/7.03 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
% 12.85/7.03 => ( ( ord_less_nat @ Mi3 @ X3 )
% 12.85/7.03 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 12.85/7.03 @ Mima ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9820_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 12.85/7.03 ! [X: vEBT_VEBT,Xa: nat] :
% 12.85/7.03 ( ~ ( vEBT_VEBT_valid @ X @ Xa )
% 12.85/7.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 12.85/7.03 => ( ! [Uu2: $o,Uv2: $o] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 12.85/7.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
% 12.85/7.03 => ( Xa = one_one_nat ) ) )
% 12.85/7.03 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 12.85/7.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) )
% 12.85/7.03 => ( ( Deg2 = Xa )
% 12.85/7.03 & ! [X4: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.03 => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.03 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.03 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 12.85/7.03 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.85/7.03 & ( case_o184042715313410164at_nat
% 12.85/7.03 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 12.85/7.03 & ! [X3: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.03 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 12.85/7.03 @ ( produc6081775807080527818_nat_o
% 12.85/7.03 @ ^ [Mi3: nat,Ma3: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 12.85/7.03 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.85/7.03 & ! [I2: nat] :
% 12.85/7.03 ( ( 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 ) ) ) ) ) )
% 12.85/7.03 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 12.85/7.03 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 12.85/7.03 & ( ( Mi3 = Ma3 )
% 12.85/7.03 => ! [X3: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.03 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 12.85/7.03 & ( ( Mi3 != Ma3 )
% 12.85/7.03 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 12.85/7.03 & ! [X3: nat] :
% 12.85/7.03 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.85/7.03 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
% 12.85/7.03 => ( ( ord_less_nat @ Mi3 @ X3 )
% 12.85/7.03 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 12.85/7.03 @ Mima ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9821_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 12.85/7.03 ! [X: vEBT_VEBT,Xa: nat] :
% 12.85/7.03 ( ( vEBT_VEBT_valid @ X @ Xa )
% 12.85/7.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 12.85/7.03 => ( ! [Uu2: $o,Uv2: $o] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 12.85/7.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
% 12.85/7.03 => ( Xa != one_one_nat ) ) )
% 12.85/7.03 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 12.85/7.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) )
% 12.85/7.03 => ~ ( ( Deg2 = Xa )
% 12.85/7.03 & ! [X5: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.03 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.03 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.03 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 12.85/7.03 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.85/7.03 & ( case_o184042715313410164at_nat
% 12.85/7.03 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 12.85/7.03 & ! [X3: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.03 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 12.85/7.03 @ ( produc6081775807080527818_nat_o
% 12.85/7.03 @ ^ [Mi3: nat,Ma3: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 12.85/7.03 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.85/7.03 & ! [I2: nat] :
% 12.85/7.03 ( ( 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 ) ) ) ) ) )
% 12.85/7.03 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 12.85/7.03 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 12.85/7.03 & ( ( Mi3 = Ma3 )
% 12.85/7.03 => ! [X3: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.03 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 12.85/7.03 & ( ( Mi3 != Ma3 )
% 12.85/7.03 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 12.85/7.03 & ! [X3: nat] :
% 12.85/7.03 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.85/7.03 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
% 12.85/7.03 => ( ( ord_less_nat @ Mi3 @ X3 )
% 12.85/7.03 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 12.85/7.03 @ Mima ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9822_Sup__int__def,axiom,
% 12.85/7.03 ( complete_Sup_Sup_int
% 12.85/7.03 = ( ^ [X8: set_int] :
% 12.85/7.03 ( the_int
% 12.85/7.03 @ ^ [X3: int] :
% 12.85/7.03 ( ( member_int @ X3 @ X8 )
% 12.85/7.03 & ! [Y6: int] :
% 12.85/7.03 ( ( member_int @ Y6 @ X8 )
% 12.85/7.03 => ( ord_less_eq_int @ Y6 @ X3 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9823_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 12.85/7.03 ! [X: vEBT_VEBT,Xa: nat,Y4: $o] :
% 12.85/7.03 ( ( ( vEBT_VEBT_valid @ X @ Xa )
% 12.85/7.03 = Y4 )
% 12.85/7.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 12.85/7.03 => ( ! [Uu2: $o,Uv2: $o] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( Xa = one_one_nat ) )
% 12.85/7.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
% 12.85/7.03 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( ( Deg2 = Xa )
% 12.85/7.03 & ! [X3: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.03 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.03 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 12.85/7.03 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 12.85/7.03 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 12.85/7.03 & ( case_o184042715313410164at_nat
% 12.85/7.03 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 12.85/7.03 & ! [X3: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.03 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 12.85/7.03 @ ( produc6081775807080527818_nat_o
% 12.85/7.03 @ ^ [Mi3: nat,Ma3: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 12.85/7.03 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.85/7.03 & ! [I2: nat] :
% 12.85/7.03 ( ( 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 ) ) ) ) ) )
% 12.85/7.03 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 12.85/7.03 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 12.85/7.03 & ( ( Mi3 = Ma3 )
% 12.85/7.03 => ! [X3: vEBT_VEBT] :
% 12.85/7.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 12.85/7.03 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 12.85/7.03 & ( ( Mi3 != Ma3 )
% 12.85/7.03 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 12.85/7.03 & ! [X3: nat] :
% 12.85/7.03 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 12.85/7.03 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
% 12.85/7.03 => ( ( ord_less_nat @ Mi3 @ X3 )
% 12.85/7.03 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 12.85/7.03 @ Mima ) ) )
% 12.85/7.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9824_Rats__eq__int__div__nat,axiom,
% 12.85/7.03 ( field_5140801741446780682s_real
% 12.85/7.03 = ( collect_real
% 12.85/7.03 @ ^ [Uu3: real] :
% 12.85/7.03 ? [I2: int,N4: nat] :
% 12.85/7.03 ( ( Uu3
% 12.85/7.03 = ( divide_divide_real @ ( ring_1_of_int_real @ I2 ) @ ( semiri5074537144036343181t_real @ N4 ) ) )
% 12.85/7.03 & ( N4 != zero_zero_nat ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9825_Rats__abs__iff,axiom,
% 12.85/7.03 ! [X: real] :
% 12.85/7.03 ( ( member_real @ ( abs_abs_real @ X ) @ field_5140801741446780682s_real )
% 12.85/7.03 = ( member_real @ X @ field_5140801741446780682s_real ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9826_Rats__dense__in__real,axiom,
% 12.85/7.03 ! [X: real,Y4: real] :
% 12.85/7.03 ( ( ord_less_real @ X @ Y4 )
% 12.85/7.03 => ? [X4: real] :
% 12.85/7.03 ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 12.85/7.03 & ( ord_less_real @ X @ X4 )
% 12.85/7.03 & ( ord_less_real @ X4 @ Y4 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9827_Rats__no__bot__less,axiom,
% 12.85/7.03 ! [X: real] :
% 12.85/7.03 ? [X4: real] :
% 12.85/7.03 ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 12.85/7.03 & ( ord_less_real @ X4 @ X ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9828_Rats__no__top__le,axiom,
% 12.85/7.03 ! [X: real] :
% 12.85/7.03 ? [X4: real] :
% 12.85/7.03 ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 12.85/7.03 & ( ord_less_eq_real @ X @ X4 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9829_Rats__eq__int__div__int,axiom,
% 12.85/7.03 ( field_5140801741446780682s_real
% 12.85/7.03 = ( collect_real
% 12.85/7.03 @ ^ [Uu3: real] :
% 12.85/7.03 ? [I2: int,J3: int] :
% 12.85/7.03 ( ( Uu3
% 12.85/7.03 = ( divide_divide_real @ ( ring_1_of_int_real @ I2 ) @ ( ring_1_of_int_real @ J3 ) ) )
% 12.85/7.03 & ( J3 != zero_zero_int ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9830_rat__floor__lemma,axiom,
% 12.85/7.03 ! [A: int,B: int] :
% 12.85/7.03 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
% 12.85/7.03 & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9831_mult__rat,axiom,
% 12.85/7.03 ! [A: int,B: int,C: int,D: int] :
% 12.85/7.03 ( ( times_times_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 12.85/7.03 = ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9832_divide__rat,axiom,
% 12.85/7.03 ! [A: int,B: int,C: int,D: int] :
% 12.85/7.03 ( ( divide_divide_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 12.85/7.03 = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9833_floor__Fract,axiom,
% 12.85/7.03 ! [A: int,B: int] :
% 12.85/7.03 ( ( archim3151403230148437115or_rat @ ( fract @ A @ B ) )
% 12.85/7.03 = ( divide_divide_int @ A @ B ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9834_less__rat,axiom,
% 12.85/7.03 ! [B: int,D: int,A: int,C: int] :
% 12.85/7.03 ( ( B != zero_zero_int )
% 12.85/7.03 => ( ( D != zero_zero_int )
% 12.85/7.03 => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 12.85/7.03 = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9835_add__rat,axiom,
% 12.85/7.03 ! [B: int,D: int,A: int,C: int] :
% 12.85/7.03 ( ( B != zero_zero_int )
% 12.85/7.03 => ( ( D != zero_zero_int )
% 12.85/7.03 => ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 12.85/7.03 = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9836_le__rat,axiom,
% 12.85/7.03 ! [B: int,D: int,A: int,C: int] :
% 12.85/7.03 ( ( B != zero_zero_int )
% 12.85/7.03 => ( ( D != zero_zero_int )
% 12.85/7.03 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 12.85/7.03 = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9837_diff__rat,axiom,
% 12.85/7.03 ! [B: int,D: int,A: int,C: int] :
% 12.85/7.03 ( ( B != zero_zero_int )
% 12.85/7.03 => ( ( D != zero_zero_int )
% 12.85/7.03 => ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 12.85/7.03 = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9838_sgn__rat,axiom,
% 12.85/7.03 ! [A: int,B: int] :
% 12.85/7.03 ( ( sgn_sgn_rat @ ( fract @ A @ B ) )
% 12.85/7.03 = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9839_Rat__induct__pos,axiom,
% 12.85/7.03 ! [P: rat > $o,Q2: rat] :
% 12.85/7.03 ( ! [A4: int,B3: int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ B3 )
% 12.85/7.03 => ( P @ ( fract @ A4 @ B3 ) ) )
% 12.85/7.03 => ( P @ Q2 ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9840_Fract__coprime,axiom,
% 12.85/7.03 ! [A: int,B: int] :
% 12.85/7.03 ( ( fract @ ( divide_divide_int @ A @ ( gcd_gcd_int @ A @ B ) ) @ ( divide_divide_int @ B @ ( gcd_gcd_int @ A @ B ) ) )
% 12.85/7.03 = ( fract @ A @ B ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9841_rat__number__collapse_I1_J,axiom,
% 12.85/7.03 ! [K: int] :
% 12.85/7.03 ( ( fract @ zero_zero_int @ K )
% 12.85/7.03 = zero_zero_rat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9842_rat__number__collapse_I6_J,axiom,
% 12.85/7.03 ! [K: int] :
% 12.85/7.03 ( ( fract @ K @ zero_zero_int )
% 12.85/7.03 = zero_zero_rat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9843_eq__rat_I2_J,axiom,
% 12.85/7.03 ! [A: int] :
% 12.85/7.03 ( ( fract @ A @ zero_zero_int )
% 12.85/7.03 = ( fract @ zero_zero_int @ one_one_int ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9844_mult__rat__cancel,axiom,
% 12.85/7.03 ! [C: int,A: int,B: int] :
% 12.85/7.03 ( ( C != zero_zero_int )
% 12.85/7.03 => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 12.85/7.03 = ( fract @ A @ B ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9845_eq__rat_I1_J,axiom,
% 12.85/7.03 ! [B: int,D: int,A: int,C: int] :
% 12.85/7.03 ( ( B != zero_zero_int )
% 12.85/7.03 => ( ( D != zero_zero_int )
% 12.85/7.03 => ( ( ( fract @ A @ B )
% 12.85/7.03 = ( fract @ C @ D ) )
% 12.85/7.03 = ( ( times_times_int @ A @ D )
% 12.85/7.03 = ( times_times_int @ C @ B ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9846_eq__rat_I3_J,axiom,
% 12.85/7.03 ! [A: int,C: int] :
% 12.85/7.03 ( ( fract @ zero_zero_int @ A )
% 12.85/7.03 = ( fract @ zero_zero_int @ C ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9847_Fract__of__nat__eq,axiom,
% 12.85/7.03 ! [K: nat] :
% 12.85/7.03 ( ( fract @ ( semiri1314217659103216013at_int @ K ) @ one_one_int )
% 12.85/7.03 = ( semiri681578069525770553at_rat @ K ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9848_Zero__rat__def,axiom,
% 12.85/7.03 ( zero_zero_rat
% 12.85/7.03 = ( fract @ zero_zero_int @ one_one_int ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9849_atLeastLessThan__add__Un,axiom,
% 12.85/7.03 ! [I: nat,J: nat,K: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ I @ J )
% 12.85/7.03 => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
% 12.85/7.03 = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9850_rat__number__collapse_I3_J,axiom,
% 12.85/7.03 ! [W: num] :
% 12.85/7.03 ( ( fract @ ( numeral_numeral_int @ W ) @ one_one_int )
% 12.85/7.03 = ( numeral_numeral_rat @ W ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9851_rat__number__expand_I3_J,axiom,
% 12.85/7.03 ( numeral_numeral_rat
% 12.85/7.03 = ( ^ [K3: num] : ( fract @ ( numeral_numeral_int @ K3 ) @ one_one_int ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9852_zero__less__Fract__iff,axiom,
% 12.85/7.03 ! [B: int,A: int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/7.03 => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 12.85/7.03 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9853_Fract__less__zero__iff,axiom,
% 12.85/7.03 ! [B: int,A: int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/7.03 => ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 12.85/7.03 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9854_Fract__less__one__iff,axiom,
% 12.85/7.03 ! [B: int,A: int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/7.03 => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
% 12.85/7.03 = ( ord_less_int @ A @ B ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9855_one__less__Fract__iff,axiom,
% 12.85/7.03 ! [B: int,A: int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/7.03 => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
% 12.85/7.03 = ( ord_less_int @ B @ A ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9856_Fract__add__one,axiom,
% 12.85/7.03 ! [N: int,M: int] :
% 12.85/7.03 ( ( N != zero_zero_int )
% 12.85/7.03 => ( ( fract @ ( plus_plus_int @ M @ N ) @ N )
% 12.85/7.03 = ( plus_plus_rat @ ( fract @ M @ N ) @ one_one_rat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9857_Fract__le__zero__iff,axiom,
% 12.85/7.03 ! [B: int,A: int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/7.03 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 12.85/7.03 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9858_zero__le__Fract__iff,axiom,
% 12.85/7.03 ! [B: int,A: int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/7.03 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 12.85/7.03 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9859_one__le__Fract__iff,axiom,
% 12.85/7.03 ! [B: int,A: int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/7.03 => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
% 12.85/7.03 = ( ord_less_eq_int @ B @ A ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9860_Fract__le__one__iff,axiom,
% 12.85/7.03 ! [B: int,A: int] :
% 12.85/7.03 ( ( ord_less_int @ zero_zero_int @ B )
% 12.85/7.03 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
% 12.85/7.03 = ( ord_less_eq_int @ A @ B ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9861_rat__number__expand_I5_J,axiom,
% 12.85/7.03 ! [K: num] :
% 12.85/7.03 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) )
% 12.85/7.03 = ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9862_rat__number__collapse_I4_J,axiom,
% 12.85/7.03 ! [W: num] :
% 12.85/7.03 ( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ one_one_int )
% 12.85/7.03 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9863_sup__int__def,axiom,
% 12.85/7.03 sup_sup_int = ord_max_int ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9864_sup__enat__def,axiom,
% 12.85/7.03 sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9865_sup__nat__def,axiom,
% 12.85/7.03 sup_sup_nat = ord_max_nat ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9866_pos__deriv__imp__strict__mono,axiom,
% 12.85/7.03 ! [F: real > real,F2: real > real] :
% 12.85/7.03 ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 12.85/7.03 => ( ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( F2 @ X4 ) )
% 12.85/7.03 => ( order_7092887310737990675l_real @ F ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9867_take__bit__numeral__minus__numeral__int,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.03 = ( case_option_int_num @ zero_zero_int
% 12.85/7.03 @ ^ [Q3: 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 @ Q3 ) ) )
% 12.85/7.03 @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9868_of__nat__eq__id,axiom,
% 12.85/7.03 semiri1316708129612266289at_nat = id_nat ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9869_take__bit__num__simps_I1_J,axiom,
% 12.85/7.03 ! [M: num] :
% 12.85/7.03 ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 12.85/7.03 = none_num ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9870_take__bit__num__simps_I2_J,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( bit_take_bit_num @ ( suc @ N ) @ one )
% 12.85/7.03 = ( some_num @ one ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9871_take__bit__num__simps_I5_J,axiom,
% 12.85/7.03 ! [R2: num] :
% 12.85/7.03 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ one )
% 12.85/7.03 = ( some_num @ one ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9872_strict__mono__imp__increasing,axiom,
% 12.85/7.03 ! [F: nat > nat,N: nat] :
% 12.85/7.03 ( ( order_5726023648592871131at_nat @ F )
% 12.85/7.03 => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9873_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( bit_take_bit_num @ N @ one )
% 12.85/7.03 = ( case_nat_option_num @ none_num
% 12.85/7.03 @ ^ [N4: nat] : ( some_num @ one )
% 12.85/7.03 @ N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9874_less__int__def,axiom,
% 12.85/7.03 ( ord_less_int
% 12.85/7.03 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 12.85/7.03 @ ( produc8739625826339149834_nat_o
% 12.85/7.03 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.03 ( produc6081775807080527818_nat_o
% 12.85/7.03 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9875_nat__def,axiom,
% 12.85/7.03 ( nat2
% 12.85/7.03 = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9876_less__eq__int__def,axiom,
% 12.85/7.03 ( ord_less_eq_int
% 12.85/7.03 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 12.85/7.03 @ ( produc8739625826339149834_nat_o
% 12.85/7.03 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.03 ( produc6081775807080527818_nat_o
% 12.85/7.03 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9877_take__bit__num__def,axiom,
% 12.85/7.03 ( bit_take_bit_num
% 12.85/7.03 = ( ^ [N4: nat,M5: num] :
% 12.85/7.03 ( if_option_num
% 12.85/7.03 @ ( ( bit_se2925701944663578781it_nat @ N4 @ ( numeral_numeral_nat @ M5 ) )
% 12.85/7.03 = zero_zero_nat )
% 12.85/7.03 @ none_num
% 12.85/7.03 @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N4 @ ( numeral_numeral_nat @ M5 ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9878_and__minus__numerals_I3_J,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 12.85/7.03 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9879_and__minus__numerals_I7_J,axiom,
% 12.85/7.03 ! [N: num,M: num] :
% 12.85/7.03 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 12.85/7.03 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9880_take__bit__num__simps_I4_J,axiom,
% 12.85/7.03 ! [N: nat,M: num] :
% 12.85/7.03 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
% 12.85/7.03 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9881_take__bit__num__simps_I3_J,axiom,
% 12.85/7.03 ! [N: nat,M: num] :
% 12.85/7.03 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
% 12.85/7.03 = ( case_o6005452278849405969um_num @ none_num
% 12.85/7.03 @ ^ [Q3: num] : ( some_num @ ( bit0 @ Q3 ) )
% 12.85/7.03 @ ( bit_take_bit_num @ N @ M ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9882_take__bit__num__simps_I7_J,axiom,
% 12.85/7.03 ! [R2: num,M: num] :
% 12.85/7.03 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M ) )
% 12.85/7.03 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9883_take__bit__num__simps_I6_J,axiom,
% 12.85/7.03 ! [R2: num,M: num] :
% 12.85/7.03 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
% 12.85/7.03 = ( case_o6005452278849405969um_num @ none_num
% 12.85/7.03 @ ^ [Q3: num] : ( some_num @ ( bit0 @ Q3 ) )
% 12.85/7.03 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9884_and__minus__numerals_I8_J,axiom,
% 12.85/7.03 ! [N: num,M: num] :
% 12.85/7.03 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 12.85/7.03 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9885_and__minus__numerals_I4_J,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 12.85/7.03 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9886_and__not__num_Osimps_I8_J,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.85/7.03 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 12.85/7.03 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 12.85/7.03 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9887_and__not__num_Osimps_I1_J,axiom,
% 12.85/7.03 ( ( bit_and_not_num @ one @ one )
% 12.85/7.03 = none_num ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9888_and__not__num_Osimps_I3_J,axiom,
% 12.85/7.03 ! [N: num] :
% 12.85/7.03 ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
% 12.85/7.03 = none_num ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9889_and__not__num_Osimps_I4_J,axiom,
% 12.85/7.03 ! [M: num] :
% 12.85/7.03 ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
% 12.85/7.03 = ( some_num @ ( bit0 @ M ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9890_and__not__num_Osimps_I2_J,axiom,
% 12.85/7.03 ! [N: num] :
% 12.85/7.03 ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
% 12.85/7.03 = ( some_num @ one ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9891_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 12.85/7.03 ! [N: nat,M: num] :
% 12.85/7.03 ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
% 12.85/7.03 = ( case_nat_option_num @ none_num
% 12.85/7.03 @ ^ [N4: nat] :
% 12.85/7.03 ( case_o6005452278849405969um_num @ none_num
% 12.85/7.03 @ ^ [Q3: num] : ( some_num @ ( bit0 @ Q3 ) )
% 12.85/7.03 @ ( bit_take_bit_num @ N4 @ M ) )
% 12.85/7.03 @ N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9892_and__not__num_Osimps_I7_J,axiom,
% 12.85/7.03 ! [M: num] :
% 12.85/7.03 ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
% 12.85/7.03 = ( some_num @ ( bit0 @ M ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9893_and__not__num__eq__Some__iff,axiom,
% 12.85/7.03 ! [M: num,N: num,Q2: num] :
% 12.85/7.03 ( ( ( bit_and_not_num @ M @ N )
% 12.85/7.03 = ( some_num @ Q2 ) )
% 12.85/7.03 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.03 = ( numeral_numeral_int @ Q2 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9894_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 12.85/7.03 ! [N: nat,M: num] :
% 12.85/7.03 ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
% 12.85/7.03 = ( case_nat_option_num @ none_num
% 12.85/7.03 @ ^ [N4: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N4 @ M ) ) )
% 12.85/7.03 @ N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9895_and__not__num__eq__None__iff,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( ( bit_and_not_num @ M @ N )
% 12.85/7.03 = none_num )
% 12.85/7.03 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.03 = zero_zero_int ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9896_int__numeral__and__not__num,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.03 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9897_int__numeral__not__and__num,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.85/7.03 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9898_positive__rat,axiom,
% 12.85/7.03 ! [A: int,B: int] :
% 12.85/7.03 ( ( positive @ ( fract @ A @ B ) )
% 12.85/7.03 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9899_Rat_Opositive__zero,axiom,
% 12.85/7.03 ~ ( positive @ zero_zero_rat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9900_Rat_Opositive__minus,axiom,
% 12.85/7.03 ! [X: rat] :
% 12.85/7.03 ( ~ ( positive @ X )
% 12.85/7.03 => ( ( X != zero_zero_rat )
% 12.85/7.03 => ( positive @ ( uminus_uminus_rat @ X ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9901_less__rat__def,axiom,
% 12.85/7.03 ( ord_less_rat
% 12.85/7.03 = ( ^ [X3: rat,Y6: rat] : ( positive @ ( minus_minus_rat @ Y6 @ X3 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9902_Rat_Opositive_Orep__eq,axiom,
% 12.85/7.03 ( positive
% 12.85/7.03 = ( ^ [X3: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X3 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X3 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9903_and__not__num_Oelims,axiom,
% 12.85/7.03 ! [X: num,Xa: num,Y4: option_num] :
% 12.85/7.03 ( ( ( bit_and_not_num @ X @ Xa )
% 12.85/7.03 = Y4 )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( Y4 != none_num ) ) )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ( ? [N2: num] :
% 12.85/7.03 ( Xa
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( some_num @ one ) ) ) )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ( ? [N2: num] :
% 12.85/7.03 ( Xa
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ( Y4 != none_num ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 12.85/7.03 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 12.85/7.03 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) )
% 12.85/7.03 => ~ ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9904_and__not__num_Osimps_I5_J,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.85/7.03 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9905_and__not__num_Osimps_I9_J,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.85/7.03 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9906_and__not__num_Osimps_I6_J,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.85/7.03 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9907_and__not__num_Opelims,axiom,
% 12.85/7.03 ! [X: num,Xa: num,Y4: option_num] :
% 12.85/7.03 ( ( ( bit_and_not_num @ X @ Xa )
% 12.85/7.03 = Y4 )
% 12.85/7.03 => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( ( Y4 = none_num )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( some_num @ one ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ( ( Y4 = none_num )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( some_num @ ( bit0 @ M4 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( some_num @ ( bit0 @ M4 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 12.85/7.03 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 12.85/7.03 @ ( bit_and_not_num @ M4 @ N2 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 12.85/7.03 => ~ ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9908_Rat_Opositive__def,axiom,
% 12.85/7.03 ( positive
% 12.85/7.03 = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
% 12.85/7.03 @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9909_and__num_Oelims,axiom,
% 12.85/7.03 ! [X: num,Xa: num,Y4: option_num] :
% 12.85/7.03 ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 12.85/7.03 = Y4 )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( some_num @ one ) ) ) )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ( ? [N2: num] :
% 12.85/7.03 ( Xa
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ( Y4 != none_num ) ) )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ( ? [N2: num] :
% 12.85/7.03 ( Xa
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( some_num @ one ) ) ) )
% 12.85/7.03 => ( ( ? [M4: num] :
% 12.85/7.03 ( X
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( Y4 != none_num ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) )
% 12.85/7.03 => ( ( ? [M4: num] :
% 12.85/7.03 ( X
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( some_num @ one ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) )
% 12.85/7.03 => ~ ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 12.85/7.03 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 12.85/7.03 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9910_xor__num_Oelims,axiom,
% 12.85/7.03 ! [X: num,Xa: num,Y4: option_num] :
% 12.85/7.03 ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 12.85/7.03 = Y4 )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( Y4 != none_num ) ) )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( some_num @ ( bit1 @ N2 ) ) ) ) )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( some_num @ ( bit0 @ N2 ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( some_num @ ( bit1 @ M4 ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) ) )
% 12.85/7.03 => ~ ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ( Y4
% 12.85/7.03 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9911_and__num_Osimps_I1_J,axiom,
% 12.85/7.03 ( ( bit_un7362597486090784418nd_num @ one @ one )
% 12.85/7.03 = ( some_num @ one ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9912_xor__num_Osimps_I1_J,axiom,
% 12.85/7.03 ( ( bit_un2480387367778600638or_num @ one @ one )
% 12.85/7.03 = none_num ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9913_xor__num_Osimps_I5_J,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.85/7.03 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9914_and__num_Osimps_I5_J,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.85/7.03 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9915_and__num_Osimps_I7_J,axiom,
% 12.85/7.03 ! [M: num] :
% 12.85/7.03 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
% 12.85/7.03 = ( some_num @ one ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9916_and__num_Osimps_I3_J,axiom,
% 12.85/7.03 ! [N: num] :
% 12.85/7.03 ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
% 12.85/7.03 = ( some_num @ one ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9917_and__num_Osimps_I4_J,axiom,
% 12.85/7.03 ! [M: num] :
% 12.85/7.03 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 12.85/7.03 = none_num ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9918_and__num_Osimps_I2_J,axiom,
% 12.85/7.03 ! [N: num] :
% 12.85/7.03 ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
% 12.85/7.03 = none_num ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9919_and__num_Osimps_I8_J,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.85/7.03 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9920_and__num_Osimps_I6_J,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.85/7.03 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9921_xor__num_Osimps_I9_J,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.85/7.03 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9922_xor__num_Osimps_I2_J,axiom,
% 12.85/7.03 ! [N: num] :
% 12.85/7.03 ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
% 12.85/7.03 = ( some_num @ ( bit1 @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9923_xor__num_Osimps_I3_J,axiom,
% 12.85/7.03 ! [N: num] :
% 12.85/7.03 ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
% 12.85/7.03 = ( some_num @ ( bit0 @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9924_xor__num_Osimps_I4_J,axiom,
% 12.85/7.03 ! [M: num] :
% 12.85/7.03 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
% 12.85/7.03 = ( some_num @ ( bit1 @ M ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9925_xor__num_Osimps_I7_J,axiom,
% 12.85/7.03 ! [M: num] :
% 12.85/7.03 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
% 12.85/7.03 = ( some_num @ ( bit0 @ M ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9926_and__num_Osimps_I9_J,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.85/7.03 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 12.85/7.03 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 12.85/7.03 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9927_xor__num_Osimps_I8_J,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.85/7.03 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9928_xor__num_Osimps_I6_J,axiom,
% 12.85/7.03 ! [M: num,N: num] :
% 12.85/7.03 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.85/7.03 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9929_and__num_Opelims,axiom,
% 12.85/7.03 ! [X: num,Xa: num,Y4: option_num] :
% 12.85/7.03 ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 12.85/7.03 = Y4 )
% 12.85/7.03 => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( some_num @ one ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ( ( Y4 = none_num )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( some_num @ one ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( ( Y4 = none_num )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( some_num @ one ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 12.85/7.03 => ~ ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 12.85/7.03 @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
% 12.85/7.03 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9930_xor__num_Opelims,axiom,
% 12.85/7.03 ! [X: num,Xa: num,Y4: option_num] :
% 12.85/7.03 ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 12.85/7.03 = Y4 )
% 12.85/7.03 => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( ( Y4 = none_num )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( some_num @ ( bit1 @ N2 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 12.85/7.03 => ( ( ( X = one )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( some_num @ ( bit0 @ N2 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( some_num @ ( bit1 @ M4 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit0 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ( ( Xa = one )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( some_num @ ( bit0 @ M4 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 12.85/7.03 => ( ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit0 @ N2 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 12.85/7.03 => ~ ! [M4: num] :
% 12.85/7.03 ( ( X
% 12.85/7.03 = ( bit1 @ M4 ) )
% 12.85/7.03 => ! [N2: num] :
% 12.85/7.03 ( ( Xa
% 12.85/7.03 = ( bit1 @ N2 ) )
% 12.85/7.03 => ( ( Y4
% 12.85/7.03 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) )
% 12.85/7.03 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9931_and__num__rel__dict,axiom,
% 12.85/7.03 bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9932_xor__num__rel__dict,axiom,
% 12.85/7.03 bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9933_xor__num__dict,axiom,
% 12.85/7.03 bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9934_and__num__dict,axiom,
% 12.85/7.03 bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9935_Bit__Operations_Otake__bit__num__code,axiom,
% 12.85/7.03 ( bit_take_bit_num
% 12.85/7.03 = ( ^ [N4: nat,M5: num] :
% 12.85/7.03 ( produc478579273971653890on_num
% 12.85/7.03 @ ^ [A3: nat,X3: num] :
% 12.85/7.03 ( case_nat_option_num @ none_num
% 12.85/7.03 @ ^ [O: nat] :
% 12.85/7.03 ( case_num_option_num @ ( some_num @ one )
% 12.85/7.03 @ ^ [P6: num] :
% 12.85/7.03 ( case_o6005452278849405969um_num @ none_num
% 12.85/7.03 @ ^ [Q3: num] : ( some_num @ ( bit0 @ Q3 ) )
% 12.85/7.03 @ ( bit_take_bit_num @ O @ P6 ) )
% 12.85/7.03 @ ^ [P6: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) )
% 12.85/7.03 @ X3 )
% 12.85/7.03 @ A3 )
% 12.85/7.03 @ ( product_Pair_nat_num @ N4 @ M5 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9936_of__rat__dense,axiom,
% 12.85/7.03 ! [X: real,Y4: real] :
% 12.85/7.03 ( ( ord_less_real @ X @ Y4 )
% 12.85/7.03 => ? [Q4: rat] :
% 12.85/7.03 ( ( ord_less_real @ X @ ( field_7254667332652039916t_real @ Q4 ) )
% 12.85/7.03 & ( ord_less_real @ ( field_7254667332652039916t_real @ Q4 ) @ Y4 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9937_plus__rat__def,axiom,
% 12.85/7.03 ( plus_plus_rat
% 12.85/7.03 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 12.85/7.03 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9938_inverse__rat__def,axiom,
% 12.85/7.03 ( inverse_inverse_rat
% 12.85/7.03 = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
% 12.85/7.03 @ ^ [X3: product_prod_int_int] :
% 12.85/7.03 ( if_Pro3027730157355071871nt_int
% 12.85/7.03 @ ( ( product_fst_int_int @ X3 )
% 12.85/7.03 = zero_zero_int )
% 12.85/7.03 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 12.85/7.03 @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9939_Fract_Oabs__eq,axiom,
% 12.85/7.03 ( fract
% 12.85/7.03 = ( ^ [Xa4: int,X3: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X3 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9940_zero__rat__def,axiom,
% 12.85/7.03 ( zero_zero_rat
% 12.85/7.03 = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9941_times__rat__def,axiom,
% 12.85/7.03 ( times_times_rat
% 12.85/7.03 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 12.85/7.03 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9942_plus__rat_Oabs__eq,axiom,
% 12.85/7.03 ! [Xa: product_prod_int_int,X: product_prod_int_int] :
% 12.85/7.03 ( ( ratrel @ Xa @ Xa )
% 12.85/7.03 => ( ( ratrel @ X @ X )
% 12.85/7.03 => ( ( plus_plus_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 12.85/7.03 = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9943_inverse__rat_Oabs__eq,axiom,
% 12.85/7.03 ! [X: product_prod_int_int] :
% 12.85/7.03 ( ( ratrel @ X @ X )
% 12.85/7.03 => ( ( inverse_inverse_rat @ ( abs_Rat @ X ) )
% 12.85/7.03 = ( abs_Rat
% 12.85/7.03 @ ( if_Pro3027730157355071871nt_int
% 12.85/7.03 @ ( ( product_fst_int_int @ X )
% 12.85/7.03 = zero_zero_int )
% 12.85/7.03 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 12.85/7.03 @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9944_ratrel__iff,axiom,
% 12.85/7.03 ( ratrel
% 12.85/7.03 = ( ^ [X3: product_prod_int_int,Y6: product_prod_int_int] :
% 12.85/7.03 ( ( ( product_snd_int_int @ X3 )
% 12.85/7.03 != zero_zero_int )
% 12.85/7.03 & ( ( product_snd_int_int @ Y6 )
% 12.85/7.03 != zero_zero_int )
% 12.85/7.03 & ( ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) )
% 12.85/7.03 = ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9945_zero__rat_Orsp,axiom,
% 12.85/7.03 ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9946_ratrel__def,axiom,
% 12.85/7.03 ( ratrel
% 12.85/7.03 = ( ^ [X3: product_prod_int_int,Y6: product_prod_int_int] :
% 12.85/7.03 ( ( ( product_snd_int_int @ X3 )
% 12.85/7.03 != zero_zero_int )
% 12.85/7.03 & ( ( product_snd_int_int @ Y6 )
% 12.85/7.03 != zero_zero_int )
% 12.85/7.03 & ( ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) )
% 12.85/7.03 = ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9947_times__rat_Oabs__eq,axiom,
% 12.85/7.03 ! [Xa: product_prod_int_int,X: product_prod_int_int] :
% 12.85/7.03 ( ( ratrel @ Xa @ Xa )
% 12.85/7.03 => ( ( ratrel @ X @ X )
% 12.85/7.03 => ( ( times_times_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 12.85/7.03 = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_fst_int_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9948_Rat_Opositive_Oabs__eq,axiom,
% 12.85/7.03 ! [X: product_prod_int_int] :
% 12.85/7.03 ( ( ratrel @ X @ X )
% 12.85/7.03 => ( ( positive @ ( abs_Rat @ X ) )
% 12.85/7.03 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9949_num__of__integer__code,axiom,
% 12.85/7.03 ( code_num_of_integer
% 12.85/7.03 = ( ^ [K3: code_integer] :
% 12.85/7.03 ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
% 12.85/7.03 @ ( produc7336495610019696514er_num
% 12.85/7.03 @ ^ [L3: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ one ) )
% 12.85/7.03 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9950_take__upt,axiom,
% 12.85/7.03 ! [I: nat,M: nat,N: nat] :
% 12.85/7.03 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
% 12.85/7.03 => ( ( take_nat @ M @ ( upt @ I @ N ) )
% 12.85/7.03 = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9951_min__Suc__Suc,axiom,
% 12.85/7.03 ! [M: nat,N: nat] :
% 12.85/7.03 ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
% 12.85/7.03 = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9952_min__0R,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( ord_min_nat @ N @ zero_zero_nat )
% 12.85/7.03 = zero_zero_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9953_min__0L,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( ord_min_nat @ zero_zero_nat @ N )
% 12.85/7.03 = zero_zero_nat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9954_min__Suc__numeral,axiom,
% 12.85/7.03 ! [N: nat,K: num] :
% 12.85/7.03 ( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 12.85/7.03 = ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9955_min__numeral__Suc,axiom,
% 12.85/7.03 ! [K: num,N: nat] :
% 12.85/7.03 ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 12.85/7.03 = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9956_inf__nat__def,axiom,
% 12.85/7.03 inf_inf_nat = ord_min_nat ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9957_concat__bit__assoc__sym,axiom,
% 12.85/7.03 ! [M: nat,N: nat,K: int,L: int,R2: int] :
% 12.85/7.03 ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L ) @ R2 )
% 12.85/7.03 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N ) @ L @ R2 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9958_min__diff,axiom,
% 12.85/7.03 ! [M: nat,I: nat,N: nat] :
% 12.85/7.03 ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
% 12.85/7.03 = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9959_nat__mult__min__right,axiom,
% 12.85/7.03 ! [M: nat,N: nat,Q2: nat] :
% 12.85/7.03 ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q2 ) )
% 12.85/7.03 = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9960_nat__mult__min__left,axiom,
% 12.85/7.03 ! [M: nat,N: nat,Q2: nat] :
% 12.85/7.03 ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q2 )
% 12.85/7.03 = ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9961_take__bit__concat__bit__eq,axiom,
% 12.85/7.03 ! [M: nat,N: nat,K: int,L: int] :
% 12.85/7.03 ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N @ K @ L ) )
% 12.85/7.03 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ L ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9962_min__Suc2,axiom,
% 12.85/7.03 ! [M: nat,N: nat] :
% 12.85/7.03 ( ( ord_min_nat @ M @ ( suc @ N ) )
% 12.85/7.03 = ( case_nat_nat @ zero_zero_nat
% 12.85/7.03 @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ M6 @ N ) )
% 12.85/7.03 @ M ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9963_min__Suc1,axiom,
% 12.85/7.03 ! [N: nat,M: nat] :
% 12.85/7.03 ( ( ord_min_nat @ ( suc @ N ) @ M )
% 12.85/7.03 = ( case_nat_nat @ zero_zero_nat
% 12.85/7.03 @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ N @ M6 ) )
% 12.85/7.03 @ M ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9964_min__enat__simps_I2_J,axiom,
% 12.85/7.03 ! [Q2: extended_enat] :
% 12.85/7.03 ( ( ord_mi8085742599997312461d_enat @ Q2 @ zero_z5237406670263579293d_enat )
% 12.85/7.03 = zero_z5237406670263579293d_enat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9965_min__enat__simps_I3_J,axiom,
% 12.85/7.03 ! [Q2: extended_enat] :
% 12.85/7.03 ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q2 )
% 12.85/7.03 = zero_z5237406670263579293d_enat ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9966_inf__int__def,axiom,
% 12.85/7.03 inf_inf_int = ord_min_int ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9967_Max__divisors__self__nat,axiom,
% 12.85/7.03 ! [N: nat] :
% 12.85/7.03 ( ( N != zero_zero_nat )
% 12.85/7.03 => ( ( lattic8265883725875713057ax_nat
% 12.85/7.03 @ ( collect_nat
% 12.85/7.03 @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ N ) ) )
% 12.85/7.03 = N ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9968_inf__enat__def,axiom,
% 12.85/7.03 inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9969_card__le__Suc__Max,axiom,
% 12.85/7.03 ! [S: set_nat] :
% 12.85/7.03 ( ( finite_finite_nat @ S )
% 12.85/7.03 => ( ord_less_eq_nat @ ( finite_card_nat @ S ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9970_Sup__nat__def,axiom,
% 12.85/7.03 ( complete_Sup_Sup_nat
% 12.85/7.03 = ( ^ [X8: set_nat] : ( if_nat @ ( X8 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X8 ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9971_divide__nat__def,axiom,
% 12.85/7.03 ( divide_divide_nat
% 12.85/7.03 = ( ^ [M5: nat,N4: nat] :
% 12.85/7.03 ( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat
% 12.85/7.03 @ ( lattic8265883725875713057ax_nat
% 12.85/7.03 @ ( collect_nat
% 12.85/7.03 @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N4 ) @ M5 ) ) ) ) ) ) ).
% 12.85/7.03
% 12.85/7.03 thf(fact_9972_gcd__is__Max__divisors__nat,axiom,
% 12.85/7.03 ! [N: nat,M: nat] :
% 12.85/7.03 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.03 => ( ( gcd_gcd_nat @ M @ N )
% 12.85/7.03 = ( lattic8265883725875713057ax_nat
% 12.85/7.03 @ ( collect_nat
% 12.85/7.03 @ ^ [D4: nat] :
% 12.85/7.04 ( ( dvd_dvd_nat @ D4 @ M )
% 12.85/7.04 & ( dvd_dvd_nat @ D4 @ N ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9973_Gcd__eq__Max,axiom,
% 12.85/7.04 ! [M10: set_nat] :
% 12.85/7.04 ( ( finite_finite_nat @ M10 )
% 12.85/7.04 => ( ( M10 != bot_bot_set_nat )
% 12.85/7.04 => ( ~ ( member_nat @ zero_zero_nat @ M10 )
% 12.85/7.04 => ( ( gcd_Gcd_nat @ M10 )
% 12.85/7.04 = ( lattic8265883725875713057ax_nat
% 12.85/7.04 @ ( comple7806235888213564991et_nat
% 12.85/7.04 @ ( image_nat_set_nat
% 12.85/7.04 @ ^ [M5: nat] :
% 12.85/7.04 ( collect_nat
% 12.85/7.04 @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M5 ) )
% 12.85/7.04 @ M10 ) ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9974_Max__divisors__self__int,axiom,
% 12.85/7.04 ! [N: int] :
% 12.85/7.04 ( ( N != zero_zero_int )
% 12.85/7.04 => ( ( lattic8263393255366662781ax_int
% 12.85/7.04 @ ( collect_int
% 12.85/7.04 @ ^ [D4: int] : ( dvd_dvd_int @ D4 @ N ) ) )
% 12.85/7.04 = ( abs_abs_int @ N ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9975_upt__conv__Cons__Cons,axiom,
% 12.85/7.04 ! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
% 12.85/7.04 ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
% 12.85/7.04 = ( upt @ M @ Q2 ) )
% 12.85/7.04 = ( ( cons_nat @ N @ Ns )
% 12.85/7.04 = ( upt @ ( suc @ M ) @ Q2 ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9976_upt__conv__Cons,axiom,
% 12.85/7.04 ! [I: nat,J: nat] :
% 12.85/7.04 ( ( ord_less_nat @ I @ J )
% 12.85/7.04 => ( ( upt @ I @ J )
% 12.85/7.04 = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9977_gcd__is__Max__divisors__int,axiom,
% 12.85/7.04 ! [N: int,M: int] :
% 12.85/7.04 ( ( N != zero_zero_int )
% 12.85/7.04 => ( ( gcd_gcd_int @ M @ N )
% 12.85/7.04 = ( lattic8263393255366662781ax_int
% 12.85/7.04 @ ( collect_int
% 12.85/7.04 @ ^ [D4: int] :
% 12.85/7.04 ( ( dvd_dvd_int @ D4 @ M )
% 12.85/7.04 & ( dvd_dvd_int @ D4 @ N ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9978_sorted__list__of__set__greaterThanAtMost,axiom,
% 12.85/7.04 ! [I: nat,J: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
% 12.85/7.04 => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
% 12.85/7.04 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9979_sorted__list__of__set__greaterThanLessThan,axiom,
% 12.85/7.04 ! [I: nat,J: nat] :
% 12.85/7.04 ( ( ord_less_nat @ ( suc @ I ) @ J )
% 12.85/7.04 => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
% 12.85/7.04 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9980_upto__aux__rec,axiom,
% 12.85/7.04 ( upto_aux
% 12.85/7.04 = ( ^ [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 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9981_list__encode_Osimps_I2_J,axiom,
% 12.85/7.04 ! [X: nat,Xs: list_nat] :
% 12.85/7.04 ( ( nat_list_encode @ ( cons_nat @ X @ Xs ) )
% 12.85/7.04 = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9982_upt__conv__Nil,axiom,
% 12.85/7.04 ! [J: nat,I: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ J @ I )
% 12.85/7.04 => ( ( upt @ I @ J )
% 12.85/7.04 = nil_nat ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9983_upt__eq__Nil__conv,axiom,
% 12.85/7.04 ! [I: nat,J: nat] :
% 12.85/7.04 ( ( ( upt @ I @ J )
% 12.85/7.04 = nil_nat )
% 12.85/7.04 = ( ( J = zero_zero_nat )
% 12.85/7.04 | ( ord_less_eq_nat @ J @ I ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9984_sorted__list__of__set__lessThan__Suc,axiom,
% 12.85/7.04 ! [K: nat] :
% 12.85/7.04 ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
% 12.85/7.04 = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9985_sorted__list__of__set__atMost__Suc,axiom,
% 12.85/7.04 ! [K: nat] :
% 12.85/7.04 ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
% 12.85/7.04 = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9986_upt__rec__numeral,axiom,
% 12.85/7.04 ! [M: num,N: num] :
% 12.85/7.04 ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 12.85/7.04 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 12.85/7.04 = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
% 12.85/7.04 & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 12.85/7.04 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 12.85/7.04 = nil_nat ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9987_upt__Suc,axiom,
% 12.85/7.04 ! [I: nat,J: nat] :
% 12.85/7.04 ( ( ( ord_less_eq_nat @ I @ J )
% 12.85/7.04 => ( ( upt @ I @ ( suc @ J ) )
% 12.85/7.04 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
% 12.85/7.04 & ( ~ ( ord_less_eq_nat @ I @ J )
% 12.85/7.04 => ( ( upt @ I @ ( suc @ J ) )
% 12.85/7.04 = nil_nat ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9988_upt__Suc__append,axiom,
% 12.85/7.04 ! [I: nat,J: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ I @ J )
% 12.85/7.04 => ( ( upt @ I @ ( suc @ J ) )
% 12.85/7.04 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9989_list__encode_Ocases,axiom,
% 12.85/7.04 ! [X: list_nat] :
% 12.85/7.04 ( ( X != nil_nat )
% 12.85/7.04 => ~ ! [X4: nat,Xs2: list_nat] :
% 12.85/7.04 ( X
% 12.85/7.04 != ( cons_nat @ X4 @ Xs2 ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9990_list__encode_Osimps_I1_J,axiom,
% 12.85/7.04 ( ( nat_list_encode @ nil_nat )
% 12.85/7.04 = zero_zero_nat ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9991_upt__0,axiom,
% 12.85/7.04 ! [I: nat] :
% 12.85/7.04 ( ( upt @ I @ zero_zero_nat )
% 12.85/7.04 = nil_nat ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9992_inj__list__encode,axiom,
% 12.85/7.04 ! [A2: set_list_nat] : ( inj_on_list_nat_nat @ nat_list_encode @ A2 ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9993_list__encode__eq,axiom,
% 12.85/7.04 ! [X: list_nat,Y4: list_nat] :
% 12.85/7.04 ( ( ( nat_list_encode @ X )
% 12.85/7.04 = ( nat_list_encode @ Y4 ) )
% 12.85/7.04 = ( X = Y4 ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9994_bij__list__encode,axiom,
% 12.85/7.04 bij_be8532844293280997160at_nat @ nat_list_encode @ top_top_set_list_nat @ top_top_set_nat ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9995_surj__list__encode,axiom,
% 12.85/7.04 ( ( image_list_nat_nat @ nat_list_encode @ top_top_set_list_nat )
% 12.85/7.04 = top_top_set_nat ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9996_upt__add__eq__append,axiom,
% 12.85/7.04 ! [I: nat,J: nat,K: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ I @ J )
% 12.85/7.04 => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
% 12.85/7.04 = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9997_list__encode_Oelims,axiom,
% 12.85/7.04 ! [X: list_nat,Y4: nat] :
% 12.85/7.04 ( ( ( nat_list_encode @ X )
% 12.85/7.04 = Y4 )
% 12.85/7.04 => ( ( ( X = nil_nat )
% 12.85/7.04 => ( Y4 != zero_zero_nat ) )
% 12.85/7.04 => ~ ! [X4: nat,Xs2: list_nat] :
% 12.85/7.04 ( ( X
% 12.85/7.04 = ( cons_nat @ X4 @ Xs2 ) )
% 12.85/7.04 => ( Y4
% 12.85/7.04 != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9998_upt__rec,axiom,
% 12.85/7.04 ( upt
% 12.85/7.04 = ( ^ [I2: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J3 ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J3 ) ) @ nil_nat ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_9999_list__encode_Opelims,axiom,
% 12.85/7.04 ! [X: list_nat,Y4: nat] :
% 12.85/7.04 ( ( ( nat_list_encode @ X )
% 12.85/7.04 = Y4 )
% 12.85/7.04 => ( ( accp_list_nat @ nat_list_encode_rel @ X )
% 12.85/7.04 => ( ( ( X = nil_nat )
% 12.85/7.04 => ( ( Y4 = zero_zero_nat )
% 12.85/7.04 => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
% 12.85/7.04 => ~ ! [X4: nat,Xs2: list_nat] :
% 12.85/7.04 ( ( X
% 12.85/7.04 = ( cons_nat @ X4 @ Xs2 ) )
% 12.85/7.04 => ( ( Y4
% 12.85/7.04 = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) )
% 12.85/7.04 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X4 @ Xs2 ) ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10000_upto_Opelims,axiom,
% 12.85/7.04 ! [X: int,Xa: int,Y4: list_int] :
% 12.85/7.04 ( ( ( upto @ X @ Xa )
% 12.85/7.04 = Y4 )
% 12.85/7.04 => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) )
% 12.85/7.04 => ~ ( ( ( ( ord_less_eq_int @ X @ Xa )
% 12.85/7.04 => ( Y4
% 12.85/7.04 = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
% 12.85/7.04 & ( ~ ( ord_less_eq_int @ X @ Xa )
% 12.85/7.04 => ( Y4 = nil_int ) ) )
% 12.85/7.04 => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10001_nth__upto,axiom,
% 12.85/7.04 ! [I: int,K: nat,J: int] :
% 12.85/7.04 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 12.85/7.04 => ( ( nth_int @ ( upto @ I @ J ) @ K )
% 12.85/7.04 = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10002_length__upto,axiom,
% 12.85/7.04 ! [I: int,J: int] :
% 12.85/7.04 ( ( size_size_list_int @ ( upto @ I @ J ) )
% 12.85/7.04 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10003_upto__rec__numeral_I1_J,axiom,
% 12.85/7.04 ! [M: num,N: num] :
% 12.85/7.04 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 12.85/7.04 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 12.85/7.04 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 12.85/7.04 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 12.85/7.04 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 12.85/7.04 = nil_int ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10004_upto__rec__numeral_I4_J,axiom,
% 12.85/7.04 ! [M: num,N: num] :
% 12.85/7.04 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.04 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.04 = ( 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 ) ) ) ) ) )
% 12.85/7.04 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.04 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.04 = nil_int ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10005_upto__rec__numeral_I3_J,axiom,
% 12.85/7.04 ! [M: num,N: num] :
% 12.85/7.04 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.85/7.04 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.85/7.04 = ( 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 ) ) ) ) )
% 12.85/7.04 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.85/7.04 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 12.85/7.04 = nil_int ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10006_upto__rec__numeral_I2_J,axiom,
% 12.85/7.04 ! [M: num,N: num] :
% 12.85/7.04 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.04 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.04 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 12.85/7.04 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.04 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 12.85/7.04 = nil_int ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10007_sorted__upto,axiom,
% 12.85/7.04 ! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10008_upto__split2,axiom,
% 12.85/7.04 ! [I: int,J: int,K: int] :
% 12.85/7.04 ( ( ord_less_eq_int @ I @ J )
% 12.85/7.04 => ( ( ord_less_eq_int @ J @ K )
% 12.85/7.04 => ( ( upto @ I @ K )
% 12.85/7.04 = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10009_upto__split1,axiom,
% 12.85/7.04 ! [I: int,J: int,K: int] :
% 12.85/7.04 ( ( ord_less_eq_int @ I @ J )
% 12.85/7.04 => ( ( ord_less_eq_int @ J @ K )
% 12.85/7.04 => ( ( upto @ I @ K )
% 12.85/7.04 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10010_atLeastLessThan__upto,axiom,
% 12.85/7.04 ( set_or4662586982721622107an_int
% 12.85/7.04 = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10011_upto_Osimps,axiom,
% 12.85/7.04 ( upto
% 12.85/7.04 = ( ^ [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 ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10012_upto_Oelims,axiom,
% 12.85/7.04 ! [X: int,Xa: int,Y4: list_int] :
% 12.85/7.04 ( ( ( upto @ X @ Xa )
% 12.85/7.04 = Y4 )
% 12.85/7.04 => ( ( ( ord_less_eq_int @ X @ Xa )
% 12.85/7.04 => ( Y4
% 12.85/7.04 = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
% 12.85/7.04 & ( ~ ( ord_less_eq_int @ X @ Xa )
% 12.85/7.04 => ( Y4 = nil_int ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10013_upto__rec1,axiom,
% 12.85/7.04 ! [I: int,J: int] :
% 12.85/7.04 ( ( ord_less_eq_int @ I @ J )
% 12.85/7.04 => ( ( upto @ I @ J )
% 12.85/7.04 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10014_upto__rec2,axiom,
% 12.85/7.04 ! [I: int,J: int] :
% 12.85/7.04 ( ( ord_less_eq_int @ I @ J )
% 12.85/7.04 => ( ( upto @ I @ J )
% 12.85/7.04 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10015_greaterThanLessThan__upto,axiom,
% 12.85/7.04 ( set_or5832277885323065728an_int
% 12.85/7.04 = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10016_upto__split3,axiom,
% 12.85/7.04 ! [I: int,J: int,K: int] :
% 12.85/7.04 ( ( ord_less_eq_int @ I @ J )
% 12.85/7.04 => ( ( ord_less_eq_int @ J @ K )
% 12.85/7.04 => ( ( upto @ I @ K )
% 12.85/7.04 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10017_upto_Opsimps,axiom,
% 12.85/7.04 ! [I: int,J: int] :
% 12.85/7.04 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
% 12.85/7.04 => ( ( ( ord_less_eq_int @ I @ J )
% 12.85/7.04 => ( ( upto @ I @ J )
% 12.85/7.04 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
% 12.85/7.04 & ( ~ ( ord_less_eq_int @ I @ J )
% 12.85/7.04 => ( ( upto @ I @ J )
% 12.85/7.04 = nil_int ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10018_tl__upt,axiom,
% 12.85/7.04 ! [M: nat,N: nat] :
% 12.85/7.04 ( ( tl_nat @ ( upt @ M @ N ) )
% 12.85/7.04 = ( upt @ ( suc @ M ) @ N ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10019_quotient__of__def,axiom,
% 12.85/7.04 ( quotient_of
% 12.85/7.04 = ( ^ [X3: rat] :
% 12.85/7.04 ( the_Pr4378521158711661632nt_int
% 12.85/7.04 @ ^ [Pair: product_prod_int_int] :
% 12.85/7.04 ( ( X3
% 12.85/7.04 = ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
% 12.85/7.04 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
% 12.85/7.04 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10020_coprime__abs__left__iff,axiom,
% 12.85/7.04 ! [K: int,L: int] :
% 12.85/7.04 ( ( algebr932160517623751201me_int @ ( abs_abs_int @ K ) @ L )
% 12.85/7.04 = ( algebr932160517623751201me_int @ K @ L ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10021_coprime__abs__right__iff,axiom,
% 12.85/7.04 ! [K: int,L: int] :
% 12.85/7.04 ( ( algebr932160517623751201me_int @ K @ ( abs_abs_int @ L ) )
% 12.85/7.04 = ( algebr932160517623751201me_int @ K @ L ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10022_normalize__stable,axiom,
% 12.85/7.04 ! [Q2: int,P5: int] :
% 12.85/7.04 ( ( ord_less_int @ zero_zero_int @ Q2 )
% 12.85/7.04 => ( ( algebr932160517623751201me_int @ P5 @ Q2 )
% 12.85/7.04 => ( ( normalize @ ( product_Pair_int_int @ P5 @ Q2 ) )
% 12.85/7.04 = ( product_Pair_int_int @ P5 @ Q2 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10023_mono__Suc,axiom,
% 12.85/7.04 order_mono_nat_nat @ suc ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10024_coprime__crossproduct__int,axiom,
% 12.85/7.04 ! [A: int,D: int,B: int,C: int] :
% 12.85/7.04 ( ( algebr932160517623751201me_int @ A @ D )
% 12.85/7.04 => ( ( algebr932160517623751201me_int @ B @ C )
% 12.85/7.04 => ( ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ C ) )
% 12.85/7.04 = ( times_times_int @ ( abs_abs_int @ B ) @ ( abs_abs_int @ D ) ) )
% 12.85/7.04 = ( ( ( abs_abs_int @ A )
% 12.85/7.04 = ( abs_abs_int @ B ) )
% 12.85/7.04 & ( ( abs_abs_int @ C )
% 12.85/7.04 = ( abs_abs_int @ D ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10025_mono__times__nat,axiom,
% 12.85/7.04 ! [N: nat] :
% 12.85/7.04 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.04 => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10026_coprime__common__divisor__int,axiom,
% 12.85/7.04 ! [A: int,B: int,X: int] :
% 12.85/7.04 ( ( algebr932160517623751201me_int @ A @ B )
% 12.85/7.04 => ( ( dvd_dvd_int @ X @ A )
% 12.85/7.04 => ( ( dvd_dvd_int @ X @ B )
% 12.85/7.04 => ( ( abs_abs_int @ X )
% 12.85/7.04 = one_one_int ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10027_Rat__cases,axiom,
% 12.85/7.04 ! [Q2: rat] :
% 12.85/7.04 ~ ! [A4: int,B3: int] :
% 12.85/7.04 ( ( Q2
% 12.85/7.04 = ( fract @ A4 @ B3 ) )
% 12.85/7.04 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 12.85/7.04 => ~ ( algebr932160517623751201me_int @ A4 @ B3 ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10028_Rat__induct,axiom,
% 12.85/7.04 ! [P: rat > $o,Q2: rat] :
% 12.85/7.04 ( ! [A4: int,B3: int] :
% 12.85/7.04 ( ( ord_less_int @ zero_zero_int @ B3 )
% 12.85/7.04 => ( ( algebr932160517623751201me_int @ A4 @ B3 )
% 12.85/7.04 => ( P @ ( fract @ A4 @ B3 ) ) ) )
% 12.85/7.04 => ( P @ Q2 ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10029_Rat__cases__nonzero,axiom,
% 12.85/7.04 ! [Q2: rat] :
% 12.85/7.04 ( ! [A4: int,B3: int] :
% 12.85/7.04 ( ( Q2
% 12.85/7.04 = ( fract @ A4 @ B3 ) )
% 12.85/7.04 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 12.85/7.04 => ( ( A4 != zero_zero_int )
% 12.85/7.04 => ~ ( algebr932160517623751201me_int @ A4 @ B3 ) ) ) )
% 12.85/7.04 => ( Q2 = zero_zero_rat ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10030_quotient__of__unique,axiom,
% 12.85/7.04 ! [R2: rat] :
% 12.85/7.04 ? [X4: product_prod_int_int] :
% 12.85/7.04 ( ( R2
% 12.85/7.04 = ( fract @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) )
% 12.85/7.04 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X4 ) )
% 12.85/7.04 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) )
% 12.85/7.04 & ! [Y3: product_prod_int_int] :
% 12.85/7.04 ( ( ( R2
% 12.85/7.04 = ( fract @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ Y3 ) ) )
% 12.85/7.04 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y3 ) )
% 12.85/7.04 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ Y3 ) ) )
% 12.85/7.04 => ( Y3 = X4 ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10031_mono__ge2__power__minus__self,axiom,
% 12.85/7.04 ! [K: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 12.85/7.04 => ( order_mono_nat_nat
% 12.85/7.04 @ ^ [M5: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M5 ) @ M5 ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10032_tendsto__at__topI__sequentially__real,axiom,
% 12.85/7.04 ! [F: real > real,Y4: real] :
% 12.85/7.04 ( ( order_mono_real_real @ F )
% 12.85/7.04 => ( ( filterlim_nat_real
% 12.85/7.04 @ ^ [N4: nat] : ( F @ ( semiri5074537144036343181t_real @ N4 ) )
% 12.85/7.04 @ ( topolo2815343760600316023s_real @ Y4 )
% 12.85/7.04 @ at_top_nat )
% 12.85/7.04 => ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Y4 ) @ at_top_real ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10033_coprime__int__iff,axiom,
% 12.85/7.04 ! [M: nat,N: nat] :
% 12.85/7.04 ( ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 12.85/7.04 = ( algebr934650988132801477me_nat @ M @ N ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10034_coprime__nat__abs__left__iff,axiom,
% 12.85/7.04 ! [K: int,N: nat] :
% 12.85/7.04 ( ( algebr934650988132801477me_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 12.85/7.04 = ( algebr932160517623751201me_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10035_coprime__nat__abs__right__iff,axiom,
% 12.85/7.04 ! [N: nat,K: int] :
% 12.85/7.04 ( ( algebr934650988132801477me_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 12.85/7.04 = ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10036_coprime__Suc__0__right,axiom,
% 12.85/7.04 ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10037_coprime__Suc__0__left,axiom,
% 12.85/7.04 ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10038_coprime__crossproduct__nat,axiom,
% 12.85/7.04 ! [A: nat,D: nat,B: nat,C: nat] :
% 12.85/7.04 ( ( algebr934650988132801477me_nat @ A @ D )
% 12.85/7.04 => ( ( algebr934650988132801477me_nat @ B @ C )
% 12.85/7.04 => ( ( ( times_times_nat @ A @ C )
% 12.85/7.04 = ( times_times_nat @ B @ D ) )
% 12.85/7.04 = ( ( A = B )
% 12.85/7.04 & ( C = D ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10039_coprime__Suc__left__nat,axiom,
% 12.85/7.04 ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ N ) @ N ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10040_coprime__Suc__right__nat,axiom,
% 12.85/7.04 ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ N ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10041_coprime__common__divisor__nat,axiom,
% 12.85/7.04 ! [A: nat,B: nat,X: nat] :
% 12.85/7.04 ( ( algebr934650988132801477me_nat @ A @ B )
% 12.85/7.04 => ( ( dvd_dvd_nat @ X @ A )
% 12.85/7.04 => ( ( dvd_dvd_nat @ X @ B )
% 12.85/7.04 => ( X = one_one_nat ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10042_incseq__bounded,axiom,
% 12.85/7.04 ! [X9: nat > real,B4: real] :
% 12.85/7.04 ( ( order_mono_nat_real @ X9 )
% 12.85/7.04 => ( ! [I3: nat] : ( ord_less_eq_real @ ( X9 @ I3 ) @ B4 )
% 12.85/7.04 => ( bfun_nat_real @ X9 @ at_top_nat ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10043_coprime__diff__one__right__nat,axiom,
% 12.85/7.04 ! [N: nat] :
% 12.85/7.04 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.04 => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10044_coprime__diff__one__left__nat,axiom,
% 12.85/7.04 ! [N: nat] :
% 12.85/7.04 ( ( ord_less_nat @ zero_zero_nat @ N )
% 12.85/7.04 => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10045_incseq__convergent,axiom,
% 12.85/7.04 ! [X9: nat > real,B4: real] :
% 12.85/7.04 ( ( order_mono_nat_real @ X9 )
% 12.85/7.04 => ( ! [I3: nat] : ( ord_less_eq_real @ ( X9 @ I3 ) @ B4 )
% 12.85/7.04 => ~ ! [L5: real] :
% 12.85/7.04 ( ( filterlim_nat_real @ X9 @ ( topolo2815343760600316023s_real @ L5 ) @ at_top_nat )
% 12.85/7.04 => ~ ! [I4: nat] : ( ord_less_eq_real @ ( X9 @ I4 ) @ L5 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10046_Rats__abs__nat__div__natE,axiom,
% 12.85/7.04 ! [X: real] :
% 12.85/7.04 ( ( member_real @ X @ field_5140801741446780682s_real )
% 12.85/7.04 => ~ ! [M4: nat,N2: nat] :
% 12.85/7.04 ( ( N2 != zero_zero_nat )
% 12.85/7.04 => ( ( ( abs_abs_real @ X )
% 12.85/7.04 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M4 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 12.85/7.04 => ~ ( algebr934650988132801477me_nat @ M4 @ N2 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10047_nonneg__incseq__Bseq__subseq__iff,axiom,
% 12.85/7.04 ! [F: nat > real,G: nat > nat] :
% 12.85/7.04 ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 12.85/7.04 => ( ( order_mono_nat_real @ F )
% 12.85/7.04 => ( ( order_5726023648592871131at_nat @ G )
% 12.85/7.04 => ( ( bfun_nat_real
% 12.85/7.04 @ ^ [X3: nat] : ( F @ ( G @ X3 ) )
% 12.85/7.04 @ at_top_nat )
% 12.85/7.04 = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10048_Rat_Opositive_Orsp,axiom,
% 12.85/7.04 ( bNF_re8699439704749558557nt_o_o @ ratrel
% 12.85/7.04 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
% 12.85/7.04 @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10049_vanishes__mult__bounded,axiom,
% 12.85/7.04 ! [X9: nat > rat,Y7: nat > rat] :
% 12.85/7.04 ( ? [A7: rat] :
% 12.85/7.04 ( ( ord_less_rat @ zero_zero_rat @ A7 )
% 12.85/7.04 & ! [N2: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N2 ) ) @ A7 ) )
% 12.85/7.04 => ( ( vanishes @ Y7 )
% 12.85/7.04 => ( vanishes
% 12.85/7.04 @ ^ [N4: nat] : ( times_times_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10050_vanishes__const,axiom,
% 12.85/7.04 ! [C: rat] :
% 12.85/7.04 ( ( vanishes
% 12.85/7.04 @ ^ [N4: nat] : C )
% 12.85/7.04 = ( C = zero_zero_rat ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10051_vanishes__diff,axiom,
% 12.85/7.04 ! [X9: nat > rat,Y7: nat > rat] :
% 12.85/7.04 ( ( vanishes @ X9 )
% 12.85/7.04 => ( ( vanishes @ Y7 )
% 12.85/7.04 => ( vanishes
% 12.85/7.04 @ ^ [N4: nat] : ( minus_minus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10052_vanishes__add,axiom,
% 12.85/7.04 ! [X9: nat > rat,Y7: nat > rat] :
% 12.85/7.04 ( ( vanishes @ X9 )
% 12.85/7.04 => ( ( vanishes @ Y7 )
% 12.85/7.04 => ( vanishes
% 12.85/7.04 @ ^ [N4: nat] : ( plus_plus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10053_vanishes__minus,axiom,
% 12.85/7.04 ! [X9: nat > rat] :
% 12.85/7.04 ( ( vanishes @ X9 )
% 12.85/7.04 => ( vanishes
% 12.85/7.04 @ ^ [N4: nat] : ( uminus_uminus_rat @ ( X9 @ N4 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10054_vanishes__def,axiom,
% 12.85/7.04 ( vanishes
% 12.85/7.04 = ( ^ [X8: nat > rat] :
% 12.85/7.04 ! [R5: rat] :
% 12.85/7.04 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.85/7.04 => ? [K3: nat] :
% 12.85/7.04 ! [N4: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K3 @ N4 )
% 12.85/7.04 => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N4 ) ) @ R5 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10055_vanishesI,axiom,
% 12.85/7.04 ! [X9: nat > rat] :
% 12.85/7.04 ( ! [R3: rat] :
% 12.85/7.04 ( ( ord_less_rat @ zero_zero_rat @ R3 )
% 12.85/7.04 => ? [K4: nat] :
% 12.85/7.04 ! [N2: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K4 @ N2 )
% 12.85/7.04 => ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N2 ) ) @ R3 ) ) )
% 12.85/7.04 => ( vanishes @ X9 ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10056_vanishesD,axiom,
% 12.85/7.04 ! [X9: nat > rat,R2: rat] :
% 12.85/7.04 ( ( vanishes @ X9 )
% 12.85/7.04 => ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 12.85/7.04 => ? [K2: nat] :
% 12.85/7.04 ! [N6: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K2 @ N6 )
% 12.85/7.04 => ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N6 ) ) @ R2 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10057_Fract_Orsp,axiom,
% 12.85/7.04 ( bNF_re157797125943740599nt_int
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 12.85/7.04 @ ( bNF_re6250860962936578807nt_int
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 12.85/7.04 @ ratrel )
% 12.85/7.04 @ ^ [A3: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B2 ) )
% 12.85/7.04 @ ^ [A3: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B2 ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10058_integer__of__natural_Orsp,axiom,
% 12.85/7.04 ( bNF_re6650684261131312217nt_int
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 12.85/7.04 @ semiri1314217659103216013at_int
% 12.85/7.04 @ semiri1314217659103216013at_int ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10059_less__eq__integer_Orsp,axiom,
% 12.85/7.04 ( bNF_re3403563459893282935_int_o
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 12.85/7.04 @ ( bNF_re5089333283451836215nt_o_o
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 12.85/7.04 @ ord_less_eq_int
% 12.85/7.04 @ ord_less_eq_int ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10060_less__eq__natural_Orsp,axiom,
% 12.85/7.04 ( bNF_re578469030762574527_nat_o
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ( bNF_re4705727531993890431at_o_o
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 12.85/7.04 @ ord_less_eq_nat
% 12.85/7.04 @ ord_less_eq_nat ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10061_times__natural_Orsp,axiom,
% 12.85/7.04 ( bNF_re1345281282404953727at_nat
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ( bNF_re5653821019739307937at_nat
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 12.85/7.04 @ times_times_nat
% 12.85/7.04 @ times_times_nat ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10062_times__integer_Orsp,axiom,
% 12.85/7.04 ( bNF_re711492959462206631nt_int
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 12.85/7.04 @ ( bNF_re4712519889275205905nt_int
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 12.85/7.04 @ times_times_int
% 12.85/7.04 @ times_times_int ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10063_divide__integer_Orsp,axiom,
% 12.85/7.04 ( bNF_re711492959462206631nt_int
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 12.85/7.04 @ ( bNF_re4712519889275205905nt_int
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 12.85/7.04 @ divide_divide_int
% 12.85/7.04 @ divide_divide_int ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10064_divide__natural_Orsp,axiom,
% 12.85/7.04 ( bNF_re1345281282404953727at_nat
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ( bNF_re5653821019739307937at_nat
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 12.85/7.04 @ divide_divide_nat
% 12.85/7.04 @ divide_divide_nat ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10065_set__bit__natural_Orsp,axiom,
% 12.85/7.04 ( bNF_re1345281282404953727at_nat
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ( bNF_re5653821019739307937at_nat
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 12.85/7.04 @ bit_se7882103937844011126it_nat
% 12.85/7.04 @ bit_se7882103937844011126it_nat ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10066_set__bit__integer_Orsp,axiom,
% 12.85/7.04 ( bNF_re4785983289428654063nt_int
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ( bNF_re4712519889275205905nt_int
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 12.85/7.04 @ bit_se7879613467334960850it_int
% 12.85/7.04 @ bit_se7879613467334960850it_int ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10067_unset__bit__natural_Orsp,axiom,
% 12.85/7.04 ( bNF_re1345281282404953727at_nat
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ( bNF_re5653821019739307937at_nat
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 12.85/7.04 @ bit_se4205575877204974255it_nat
% 12.85/7.04 @ bit_se4205575877204974255it_nat ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10068_unset__bit__integer_Orsp,axiom,
% 12.85/7.04 ( bNF_re4785983289428654063nt_int
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ( bNF_re4712519889275205905nt_int
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 12.85/7.04 @ bit_se4203085406695923979it_int
% 12.85/7.04 @ bit_se4203085406695923979it_int ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10069_flip__bit__natural_Orsp,axiom,
% 12.85/7.04 ( bNF_re1345281282404953727at_nat
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ( bNF_re5653821019739307937at_nat
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 12.85/7.04 @ bit_se2161824704523386999it_nat
% 12.85/7.04 @ bit_se2161824704523386999it_nat ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10070_flip__bit__integer_Orsp,axiom,
% 12.85/7.04 ( bNF_re4785983289428654063nt_int
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ( bNF_re4712519889275205905nt_int
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 12.85/7.04 @ bit_se2159334234014336723it_int
% 12.85/7.04 @ bit_se2159334234014336723it_int ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10071_Suc_Orsp,axiom,
% 12.85/7.04 ( bNF_re5653821019739307937at_nat
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ suc
% 12.85/7.04 @ suc ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10072_minus__integer_Orsp,axiom,
% 12.85/7.04 ( bNF_re711492959462206631nt_int
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 12.85/7.04 @ ( bNF_re4712519889275205905nt_int
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 12.85/7.04 @ minus_minus_int
% 12.85/7.04 @ minus_minus_int ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10073_minus__natural_Orsp,axiom,
% 12.85/7.04 ( bNF_re1345281282404953727at_nat
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ( bNF_re5653821019739307937at_nat
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 12.85/7.04 @ minus_minus_nat
% 12.85/7.04 @ minus_minus_nat ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10074_sub_Orsp,axiom,
% 12.85/7.04 ( bNF_re8402795839162346335um_int
% 12.85/7.04 @ ^ [Y5: num,Z4: num] : ( Y5 = Z4 )
% 12.85/7.04 @ ( bNF_re1822329894187522285nt_int
% 12.85/7.04 @ ^ [Y5: num,Z4: num] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 12.85/7.04 @ ^ [M5: num,N4: num] : ( minus_minus_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N4 ) )
% 12.85/7.04 @ ^ [M5: num,N4: num] : ( minus_minus_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N4 ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10075_times__rat_Orsp,axiom,
% 12.85/7.04 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 12.85/7.04 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) )
% 12.85/7.04 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10076_plus__rat_Orsp,axiom,
% 12.85/7.04 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 12.85/7.04 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) )
% 12.85/7.04 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10077_inverse__rat_Orsp,axiom,
% 12.85/7.04 ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
% 12.85/7.04 @ ^ [X3: product_prod_int_int] :
% 12.85/7.04 ( if_Pro3027730157355071871nt_int
% 12.85/7.04 @ ( ( product_fst_int_int @ X3 )
% 12.85/7.04 = zero_zero_int )
% 12.85/7.04 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 12.85/7.04 @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
% 12.85/7.04 @ ^ [X3: product_prod_int_int] :
% 12.85/7.04 ( if_Pro3027730157355071871nt_int
% 12.85/7.04 @ ( ( product_fst_int_int @ X3 )
% 12.85/7.04 = zero_zero_int )
% 12.85/7.04 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 12.85/7.04 @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10078_plus__rat_Otransfer,axiom,
% 12.85/7.04 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 12.85/7.04 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) )
% 12.85/7.04 @ plus_plus_rat ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10079_inverse__rat_Otransfer,axiom,
% 12.85/7.04 ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
% 12.85/7.04 @ ^ [X3: product_prod_int_int] :
% 12.85/7.04 ( if_Pro3027730157355071871nt_int
% 12.85/7.04 @ ( ( product_fst_int_int @ X3 )
% 12.85/7.04 = zero_zero_int )
% 12.85/7.04 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 12.85/7.04 @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
% 12.85/7.04 @ inverse_inverse_rat ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10080_zero__rat_Otransfer,axiom,
% 12.85/7.04 pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10081_Fract_Otransfer,axiom,
% 12.85/7.04 ( bNF_re3461391660133120880nt_rat
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 12.85/7.04 @ ( bNF_re2214769303045360666nt_rat
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 12.85/7.04 @ pcr_rat )
% 12.85/7.04 @ ^ [A3: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B2 ) )
% 12.85/7.04 @ fract ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10082_times__rat_Otransfer,axiom,
% 12.85/7.04 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 12.85/7.04 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) )
% 12.85/7.04 @ times_times_rat ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10083_Rat_Opositive_Otransfer,axiom,
% 12.85/7.04 ( bNF_re1494630372529172596at_o_o @ pcr_rat
% 12.85/7.04 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
% 12.85/7.04 @ positive ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10084_times__int_Otransfer,axiom,
% 12.85/7.04 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 12.85/7.04 @ ( produc27273713700761075at_nat
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.04 ( produc2626176000494625587at_nat
% 12.85/7.04 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) )
% 12.85/7.04 @ times_times_int ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10085_zero__int_Otransfer,axiom,
% 12.85/7.04 pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10086_int__transfer,axiom,
% 12.85/7.04 ( bNF_re6830278522597306478at_int
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ pcr_int
% 12.85/7.04 @ ^ [N4: nat] : ( product_Pair_nat_nat @ N4 @ zero_zero_nat )
% 12.85/7.04 @ semiri1314217659103216013at_int ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10087_uminus__int_Otransfer,axiom,
% 12.85/7.04 ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
% 12.85/7.04 @ ( produc2626176000494625587at_nat
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 ) )
% 12.85/7.04 @ uminus_uminus_int ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10088_nat_Otransfer,axiom,
% 12.85/7.04 ( bNF_re4555766996558763186at_nat @ pcr_int
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 12.85/7.04 @ nat2 ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10089_one__int_Otransfer,axiom,
% 12.85/7.04 pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10090_less__int_Otransfer,axiom,
% 12.85/7.04 ( bNF_re717283939379294677_int_o @ pcr_int
% 12.85/7.04 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 12.85/7.04 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 12.85/7.04 @ ( produc8739625826339149834_nat_o
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.04 ( produc6081775807080527818_nat_o
% 12.85/7.04 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
% 12.85/7.04 @ ord_less_int ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10091_less__eq__int_Otransfer,axiom,
% 12.85/7.04 ( bNF_re717283939379294677_int_o @ pcr_int
% 12.85/7.04 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 12.85/7.04 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 12.85/7.04 @ ( produc8739625826339149834_nat_o
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.04 ( produc6081775807080527818_nat_o
% 12.85/7.04 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
% 12.85/7.04 @ ord_less_eq_int ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10092_plus__int_Otransfer,axiom,
% 12.85/7.04 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 12.85/7.04 @ ( produc27273713700761075at_nat
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.04 ( produc2626176000494625587at_nat
% 12.85/7.04 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) )
% 12.85/7.04 @ plus_plus_int ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10093_minus__int_Otransfer,axiom,
% 12.85/7.04 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 12.85/7.04 @ ( produc27273713700761075at_nat
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.04 ( produc2626176000494625587at_nat
% 12.85/7.04 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) )
% 12.85/7.04 @ minus_minus_int ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10094_product__atMost__eq__Un,axiom,
% 12.85/7.04 ! [A2: set_nat,M: nat] :
% 12.85/7.04 ( ( produc457027306803732586at_nat @ A2
% 12.85/7.04 @ ^ [Uu3: nat] : ( set_ord_atMost_nat @ M ) )
% 12.85/7.04 = ( sup_su6327502436637775413at_nat
% 12.85/7.04 @ ( produc457027306803732586at_nat @ A2
% 12.85/7.04 @ ^ [I2: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ I2 ) ) )
% 12.85/7.04 @ ( produc457027306803732586at_nat @ A2
% 12.85/7.04 @ ^ [I2: nat] : ( set_or6659071591806873216st_nat @ ( minus_minus_nat @ M @ I2 ) @ M ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10095_pairs__le__eq__Sigma,axiom,
% 12.85/7.04 ! [M: nat] :
% 12.85/7.04 ( ( collec3392354462482085612at_nat
% 12.85/7.04 @ ( produc6081775807080527818_nat_o
% 12.85/7.04 @ ^ [I2: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ J3 ) @ M ) ) )
% 12.85/7.04 = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
% 12.85/7.04 @ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R5 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10096_inverse__real_Otransfer,axiom,
% 12.85/7.04 ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
% 12.85/7.04 @ ^ [X8: nat > rat] :
% 12.85/7.04 ( if_nat_rat @ ( vanishes @ X8 )
% 12.85/7.04 @ ^ [N4: nat] : zero_zero_rat
% 12.85/7.04 @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X8 @ N4 ) ) )
% 12.85/7.04 @ inverse_inverse_real ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10097_zero__real_Otransfer,axiom,
% 12.85/7.04 ( pcr_real
% 12.85/7.04 @ ^ [N4: nat] : zero_zero_rat
% 12.85/7.04 @ zero_zero_real ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10098_one__real_Otransfer,axiom,
% 12.85/7.04 ( pcr_real
% 12.85/7.04 @ ^ [N4: nat] : one_one_rat
% 12.85/7.04 @ one_one_real ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10099_uminus__real_Otransfer,axiom,
% 12.85/7.04 ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
% 12.85/7.04 @ ^ [X8: nat > rat,N4: nat] : ( uminus_uminus_rat @ ( X8 @ N4 ) )
% 12.85/7.04 @ uminus_uminus_real ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10100_plus__real_Otransfer,axiom,
% 12.85/7.04 ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
% 12.85/7.04 @ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( plus_plus_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) )
% 12.85/7.04 @ plus_plus_real ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10101_times__real_Otransfer,axiom,
% 12.85/7.04 ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
% 12.85/7.04 @ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) )
% 12.85/7.04 @ times_times_real ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10102_rat__floor__code,axiom,
% 12.85/7.04 ( archim3151403230148437115or_rat
% 12.85/7.04 = ( ^ [P6: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P6 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10103_Divides_Oadjust__div__def,axiom,
% 12.85/7.04 ( adjust_div
% 12.85/7.04 = ( produc8211389475949308722nt_int
% 12.85/7.04 @ ^ [Q3: int,R5: int] : ( plus_plus_int @ Q3 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10104_Real_Opositive_Otransfer,axiom,
% 12.85/7.04 ( bNF_re4297313714947099218al_o_o @ pcr_real
% 12.85/7.04 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [X8: nat > rat] :
% 12.85/7.04 ? [R5: rat] :
% 12.85/7.04 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.85/7.04 & ? [K3: nat] :
% 12.85/7.04 ! [N4: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K3 @ N4 )
% 12.85/7.04 => ( ord_less_rat @ R5 @ ( X8 @ N4 ) ) ) )
% 12.85/7.04 @ positive2 ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10105_times__int_Orsp,axiom,
% 12.85/7.04 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 12.85/7.04 @ ( produc27273713700761075at_nat
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.04 ( produc2626176000494625587at_nat
% 12.85/7.04 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) )
% 12.85/7.04 @ ( produc27273713700761075at_nat
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.04 ( produc2626176000494625587at_nat
% 12.85/7.04 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10106_intrel__iff,axiom,
% 12.85/7.04 ! [X: nat,Y4: nat,U: nat,V: nat] :
% 12.85/7.04 ( ( intrel @ ( product_Pair_nat_nat @ X @ Y4 ) @ ( product_Pair_nat_nat @ U @ V ) )
% 12.85/7.04 = ( ( plus_plus_nat @ X @ V )
% 12.85/7.04 = ( plus_plus_nat @ U @ Y4 ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10107_uminus__int_Orsp,axiom,
% 12.85/7.04 ( bNF_re2241393799969408733at_nat @ intrel @ intrel
% 12.85/7.04 @ ( produc2626176000494625587at_nat
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 ) )
% 12.85/7.04 @ ( produc2626176000494625587at_nat
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10108_zero__int_Orsp,axiom,
% 12.85/7.04 intrel @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10109_int_Oabs__eq__iff,axiom,
% 12.85/7.04 ! [X: product_prod_nat_nat,Y4: product_prod_nat_nat] :
% 12.85/7.04 ( ( ( abs_Integ @ X )
% 12.85/7.04 = ( abs_Integ @ Y4 ) )
% 12.85/7.04 = ( intrel @ X @ Y4 ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10110_Real_Opositive__mult,axiom,
% 12.85/7.04 ! [X: real,Y4: real] :
% 12.85/7.04 ( ( positive2 @ X )
% 12.85/7.04 => ( ( positive2 @ Y4 )
% 12.85/7.04 => ( positive2 @ ( times_times_real @ X @ Y4 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10111_Real_Opositive__zero,axiom,
% 12.85/7.04 ~ ( positive2 @ zero_zero_real ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10112_Real_Opositive__add,axiom,
% 12.85/7.04 ! [X: real,Y4: real] :
% 12.85/7.04 ( ( positive2 @ X )
% 12.85/7.04 => ( ( positive2 @ Y4 )
% 12.85/7.04 => ( positive2 @ ( plus_plus_real @ X @ Y4 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10113_Real_Opositive__minus,axiom,
% 12.85/7.04 ! [X: real] :
% 12.85/7.04 ( ~ ( positive2 @ X )
% 12.85/7.04 => ( ( X != zero_zero_real )
% 12.85/7.04 => ( positive2 @ ( uminus_uminus_real @ X ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10114_nat_Orsp,axiom,
% 12.85/7.04 ( bNF_re8246922863344978751at_nat @ intrel
% 12.85/7.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 12.85/7.04 @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 12.85/7.04 @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10115_less__real__def,axiom,
% 12.85/7.04 ( ord_less_real
% 12.85/7.04 = ( ^ [X3: real,Y6: real] : ( positive2 @ ( minus_minus_real @ Y6 @ X3 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10116_one__int_Orsp,axiom,
% 12.85/7.04 intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10117_intrel__def,axiom,
% 12.85/7.04 ( intrel
% 12.85/7.04 = ( produc8739625826339149834_nat_o
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.04 ( produc6081775807080527818_nat_o
% 12.85/7.04 @ ^ [U3: nat,V4: nat] :
% 12.85/7.04 ( ( plus_plus_nat @ X3 @ V4 )
% 12.85/7.04 = ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10118_less__int_Orsp,axiom,
% 12.85/7.04 ( bNF_re4202695980764964119_nat_o @ intrel
% 12.85/7.04 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 12.85/7.04 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 12.85/7.04 @ ( produc8739625826339149834_nat_o
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.04 ( produc6081775807080527818_nat_o
% 12.85/7.04 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
% 12.85/7.04 @ ( produc8739625826339149834_nat_o
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.04 ( produc6081775807080527818_nat_o
% 12.85/7.04 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10119_less__eq__int_Orsp,axiom,
% 12.85/7.04 ( bNF_re4202695980764964119_nat_o @ intrel
% 12.85/7.04 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 12.85/7.04 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 12.85/7.04 @ ( produc8739625826339149834_nat_o
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.04 ( produc6081775807080527818_nat_o
% 12.85/7.04 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
% 12.85/7.04 @ ( produc8739625826339149834_nat_o
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.04 ( produc6081775807080527818_nat_o
% 12.85/7.04 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10120_int_Orel__eq__transfer,axiom,
% 12.85/7.04 ( bNF_re717283939379294677_int_o @ pcr_int
% 12.85/7.04 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 12.85/7.04 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 12.85/7.04 @ intrel
% 12.85/7.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10121_plus__int_Orsp,axiom,
% 12.85/7.04 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 12.85/7.04 @ ( produc27273713700761075at_nat
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.04 ( produc2626176000494625587at_nat
% 12.85/7.04 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) )
% 12.85/7.04 @ ( produc27273713700761075at_nat
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.04 ( produc2626176000494625587at_nat
% 12.85/7.04 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10122_minus__int_Orsp,axiom,
% 12.85/7.04 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 12.85/7.04 @ ( produc27273713700761075at_nat
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.04 ( produc2626176000494625587at_nat
% 12.85/7.04 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) )
% 12.85/7.04 @ ( produc27273713700761075at_nat
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] :
% 12.85/7.04 ( produc2626176000494625587at_nat
% 12.85/7.04 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10123_Real_Opositive_Orep__eq,axiom,
% 12.85/7.04 ( positive2
% 12.85/7.04 = ( ^ [X3: real] :
% 12.85/7.04 ? [R5: rat] :
% 12.85/7.04 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.85/7.04 & ? [K3: nat] :
% 12.85/7.04 ! [N4: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K3 @ N4 )
% 12.85/7.04 => ( ord_less_rat @ R5 @ ( rep_real @ X3 @ N4 ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10124_Real_Opositive_Orsp,axiom,
% 12.85/7.04 ( bNF_re728719798268516973at_o_o @ realrel
% 12.85/7.04 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 )
% 12.85/7.04 @ ^ [X8: nat > rat] :
% 12.85/7.04 ? [R5: rat] :
% 12.85/7.04 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.85/7.04 & ? [K3: nat] :
% 12.85/7.04 ! [N4: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K3 @ N4 )
% 12.85/7.04 => ( ord_less_rat @ R5 @ ( X8 @ N4 ) ) ) )
% 12.85/7.04 @ ^ [X8: nat > rat] :
% 12.85/7.04 ? [R5: rat] :
% 12.85/7.04 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.85/7.04 & ? [K3: nat] :
% 12.85/7.04 ! [N4: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K3 @ N4 )
% 12.85/7.04 => ( ord_less_rat @ R5 @ ( X8 @ N4 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10125_times__real_Orsp,axiom,
% 12.85/7.04 ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
% 12.85/7.04 @ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) )
% 12.85/7.04 @ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10126_plus__real_Orsp,axiom,
% 12.85/7.04 ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
% 12.85/7.04 @ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( plus_plus_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) )
% 12.85/7.04 @ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( plus_plus_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10127_uminus__real_Orsp,axiom,
% 12.85/7.04 ( bNF_re895249473297799549at_rat @ realrel @ realrel
% 12.85/7.04 @ ^ [X8: nat > rat,N4: nat] : ( uminus_uminus_rat @ ( X8 @ N4 ) )
% 12.85/7.04 @ ^ [X8: nat > rat,N4: nat] : ( uminus_uminus_rat @ ( X8 @ N4 ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10128_zero__real_Orsp,axiom,
% 12.85/7.04 ( realrel
% 12.85/7.04 @ ^ [N4: nat] : zero_zero_rat
% 12.85/7.04 @ ^ [N4: nat] : zero_zero_rat ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10129_one__real_Orsp,axiom,
% 12.85/7.04 ( realrel
% 12.85/7.04 @ ^ [N4: nat] : one_one_rat
% 12.85/7.04 @ ^ [N4: nat] : one_one_rat ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10130_real_Orel__eq__transfer,axiom,
% 12.85/7.04 ( bNF_re4521903465945308077real_o @ pcr_real
% 12.85/7.04 @ ( bNF_re4297313714947099218al_o_o @ pcr_real
% 12.85/7.04 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 12.85/7.04 @ realrel
% 12.85/7.04 @ ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10131_inverse__real_Orsp,axiom,
% 12.85/7.04 ( bNF_re895249473297799549at_rat @ realrel @ realrel
% 12.85/7.04 @ ^ [X8: nat > rat] :
% 12.85/7.04 ( if_nat_rat @ ( vanishes @ X8 )
% 12.85/7.04 @ ^ [N4: nat] : zero_zero_rat
% 12.85/7.04 @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X8 @ N4 ) ) )
% 12.85/7.04 @ ^ [X8: nat > rat] :
% 12.85/7.04 ( if_nat_rat @ ( vanishes @ X8 )
% 12.85/7.04 @ ^ [N4: nat] : zero_zero_rat
% 12.85/7.04 @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X8 @ N4 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10132_Real_Opositive__def,axiom,
% 12.85/7.04 ( positive2
% 12.85/7.04 = ( map_fu1856342031159181835at_o_o @ rep_real @ id_o
% 12.85/7.04 @ ^ [X8: nat > rat] :
% 12.85/7.04 ? [R5: rat] :
% 12.85/7.04 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.85/7.04 & ? [K3: nat] :
% 12.85/7.04 ! [N4: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K3 @ N4 )
% 12.85/7.04 => ( ord_less_rat @ R5 @ ( X8 @ N4 ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10133_inverse__real_Oabs__eq,axiom,
% 12.85/7.04 ! [X: nat > rat] :
% 12.85/7.04 ( ( realrel @ X @ X )
% 12.85/7.04 => ( ( inverse_inverse_real @ ( real2 @ X ) )
% 12.85/7.04 = ( real2
% 12.85/7.04 @ ( if_nat_rat @ ( vanishes @ X )
% 12.85/7.04 @ ^ [N4: nat] : zero_zero_rat
% 12.85/7.04 @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X @ N4 ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10134_real_Oabs__induct,axiom,
% 12.85/7.04 ! [P: real > $o,X: real] :
% 12.85/7.04 ( ! [Y: nat > rat] :
% 12.85/7.04 ( ( realrel @ Y @ Y )
% 12.85/7.04 => ( P @ ( real2 @ Y ) ) )
% 12.85/7.04 => ( P @ X ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10135_of__int__Real,axiom,
% 12.85/7.04 ( ring_1_of_int_real
% 12.85/7.04 = ( ^ [X3: int] :
% 12.85/7.04 ( real2
% 12.85/7.04 @ ^ [N4: nat] : ( ring_1_of_int_rat @ X3 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10136_of__rat__Real,axiom,
% 12.85/7.04 ( field_7254667332652039916t_real
% 12.85/7.04 = ( ^ [X3: rat] :
% 12.85/7.04 ( real2
% 12.85/7.04 @ ^ [N4: nat] : X3 ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10137_one__real__def,axiom,
% 12.85/7.04 ( one_one_real
% 12.85/7.04 = ( real2
% 12.85/7.04 @ ^ [N4: nat] : one_one_rat ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10138_zero__real__def,axiom,
% 12.85/7.04 ( zero_zero_real
% 12.85/7.04 = ( real2
% 12.85/7.04 @ ^ [N4: nat] : zero_zero_rat ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10139_of__nat__Real,axiom,
% 12.85/7.04 ( semiri5074537144036343181t_real
% 12.85/7.04 = ( ^ [X3: nat] :
% 12.85/7.04 ( real2
% 12.85/7.04 @ ^ [N4: nat] : ( semiri681578069525770553at_rat @ X3 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10140_uminus__real_Oabs__eq,axiom,
% 12.85/7.04 ! [X: nat > rat] :
% 12.85/7.04 ( ( realrel @ X @ X )
% 12.85/7.04 => ( ( uminus_uminus_real @ ( real2 @ X ) )
% 12.85/7.04 = ( real2
% 12.85/7.04 @ ^ [N4: nat] : ( uminus_uminus_rat @ ( X @ N4 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10141_plus__real_Oabs__eq,axiom,
% 12.85/7.04 ! [Xa: nat > rat,X: nat > rat] :
% 12.85/7.04 ( ( realrel @ Xa @ Xa )
% 12.85/7.04 => ( ( realrel @ X @ X )
% 12.85/7.04 => ( ( plus_plus_real @ ( real2 @ Xa ) @ ( real2 @ X ) )
% 12.85/7.04 = ( real2
% 12.85/7.04 @ ^ [N4: nat] : ( plus_plus_rat @ ( Xa @ N4 ) @ ( X @ N4 ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10142_times__real_Oabs__eq,axiom,
% 12.85/7.04 ! [Xa: nat > rat,X: nat > rat] :
% 12.85/7.04 ( ( realrel @ Xa @ Xa )
% 12.85/7.04 => ( ( realrel @ X @ X )
% 12.85/7.04 => ( ( times_times_real @ ( real2 @ Xa ) @ ( real2 @ X ) )
% 12.85/7.04 = ( real2
% 12.85/7.04 @ ^ [N4: nat] : ( times_times_rat @ ( Xa @ N4 ) @ ( X @ N4 ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10143_Real_Opositive_Oabs__eq,axiom,
% 12.85/7.04 ! [X: nat > rat] :
% 12.85/7.04 ( ( realrel @ X @ X )
% 12.85/7.04 => ( ( positive2 @ ( real2 @ X ) )
% 12.85/7.04 = ( ? [R5: rat] :
% 12.85/7.04 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.85/7.04 & ? [K3: nat] :
% 12.85/7.04 ! [N4: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K3 @ N4 )
% 12.85/7.04 => ( ord_less_rat @ R5 @ ( X @ N4 ) ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10144_inverse__real__def,axiom,
% 12.85/7.04 ( inverse_inverse_real
% 12.85/7.04 = ( map_fu7146612038024189824t_real @ rep_real @ real2
% 12.85/7.04 @ ^ [X8: nat > rat] :
% 12.85/7.04 ( if_nat_rat @ ( vanishes @ X8 )
% 12.85/7.04 @ ^ [N4: nat] : zero_zero_rat
% 12.85/7.04 @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X8 @ N4 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10145_le__Real,axiom,
% 12.85/7.04 ! [X9: nat > rat,Y7: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( ( cauchy @ Y7 )
% 12.85/7.04 => ( ( ord_less_eq_real @ ( real2 @ X9 ) @ ( real2 @ Y7 ) )
% 12.85/7.04 = ( ! [R5: rat] :
% 12.85/7.04 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.85/7.04 => ? [K3: nat] :
% 12.85/7.04 ! [N4: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K3 @ N4 )
% 12.85/7.04 => ( ord_less_eq_rat @ ( X9 @ N4 ) @ ( plus_plus_rat @ ( Y7 @ N4 ) @ R5 ) ) ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10146_cr__real__eq,axiom,
% 12.85/7.04 ( pcr_real
% 12.85/7.04 = ( ^ [X3: nat > rat,Y6: real] :
% 12.85/7.04 ( ( cauchy @ X3 )
% 12.85/7.04 & ( ( real2 @ X3 )
% 12.85/7.04 = Y6 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10147_Real__induct,axiom,
% 12.85/7.04 ! [P: real > $o,X: real] :
% 12.85/7.04 ( ! [X15: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X15 )
% 12.85/7.04 => ( P @ ( real2 @ X15 ) ) )
% 12.85/7.04 => ( P @ X ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10148_cauchy__inverse,axiom,
% 12.85/7.04 ! [X9: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( ~ ( vanishes @ X9 )
% 12.85/7.04 => ( cauchy
% 12.85/7.04 @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X9 @ N4 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10149_cauchy__add,axiom,
% 12.85/7.04 ! [X9: nat > rat,Y7: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( ( cauchy @ Y7 )
% 12.85/7.04 => ( cauchy
% 12.85/7.04 @ ^ [N4: nat] : ( plus_plus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10150_cauchy__minus,axiom,
% 12.85/7.04 ! [X9: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( cauchy
% 12.85/7.04 @ ^ [N4: nat] : ( uminus_uminus_rat @ ( X9 @ N4 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10151_cauchy__const,axiom,
% 12.85/7.04 ! [X: rat] :
% 12.85/7.04 ( cauchy
% 12.85/7.04 @ ^ [N4: nat] : X ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10152_cauchy__mult,axiom,
% 12.85/7.04 ! [X9: nat > rat,Y7: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( ( cauchy @ Y7 )
% 12.85/7.04 => ( cauchy
% 12.85/7.04 @ ^ [N4: nat] : ( times_times_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10153_cauchy__diff,axiom,
% 12.85/7.04 ! [X9: nat > rat,Y7: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( ( cauchy @ Y7 )
% 12.85/7.04 => ( cauchy
% 12.85/7.04 @ ^ [N4: nat] : ( minus_minus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10154_realrel__refl,axiom,
% 12.85/7.04 ! [X9: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( realrel @ X9 @ X9 ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10155_cauchy__imp__bounded,axiom,
% 12.85/7.04 ! [X9: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ? [B3: rat] :
% 12.85/7.04 ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 12.85/7.04 & ! [N6: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N6 ) ) @ B3 ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10156_less__RealD,axiom,
% 12.85/7.04 ! [Y7: nat > rat,X: real] :
% 12.85/7.04 ( ( cauchy @ Y7 )
% 12.85/7.04 => ( ( ord_less_real @ X @ ( real2 @ Y7 ) )
% 12.85/7.04 => ? [N2: nat] : ( ord_less_real @ X @ ( field_7254667332652039916t_real @ ( Y7 @ N2 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10157_Real__leI,axiom,
% 12.85/7.04 ! [X9: nat > rat,Y4: real] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( ! [N2: nat] : ( ord_less_eq_real @ ( field_7254667332652039916t_real @ ( X9 @ N2 ) ) @ Y4 )
% 12.85/7.04 => ( ord_less_eq_real @ ( real2 @ X9 ) @ Y4 ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10158_le__RealI,axiom,
% 12.85/7.04 ! [Y7: nat > rat,X: real] :
% 12.85/7.04 ( ( cauchy @ Y7 )
% 12.85/7.04 => ( ! [N2: nat] : ( ord_less_eq_real @ X @ ( field_7254667332652039916t_real @ ( Y7 @ N2 ) ) )
% 12.85/7.04 => ( ord_less_eq_real @ X @ ( real2 @ Y7 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10159_minus__Real,axiom,
% 12.85/7.04 ! [X9: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( ( uminus_uminus_real @ ( real2 @ X9 ) )
% 12.85/7.04 = ( real2
% 12.85/7.04 @ ^ [N4: nat] : ( uminus_uminus_rat @ ( X9 @ N4 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10160_add__Real,axiom,
% 12.85/7.04 ! [X9: nat > rat,Y7: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( ( cauchy @ Y7 )
% 12.85/7.04 => ( ( plus_plus_real @ ( real2 @ X9 ) @ ( real2 @ Y7 ) )
% 12.85/7.04 = ( real2
% 12.85/7.04 @ ^ [N4: nat] : ( plus_plus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10161_mult__Real,axiom,
% 12.85/7.04 ! [X9: nat > rat,Y7: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( ( cauchy @ Y7 )
% 12.85/7.04 => ( ( times_times_real @ ( real2 @ X9 ) @ ( real2 @ Y7 ) )
% 12.85/7.04 = ( real2
% 12.85/7.04 @ ^ [N4: nat] : ( times_times_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10162_diff__Real,axiom,
% 12.85/7.04 ! [X9: nat > rat,Y7: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( ( cauchy @ Y7 )
% 12.85/7.04 => ( ( minus_minus_real @ ( real2 @ X9 ) @ ( real2 @ Y7 ) )
% 12.85/7.04 = ( real2
% 12.85/7.04 @ ^ [N4: nat] : ( minus_minus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10163_realrelI,axiom,
% 12.85/7.04 ! [X9: nat > rat,Y7: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( ( cauchy @ Y7 )
% 12.85/7.04 => ( ( vanishes
% 12.85/7.04 @ ^ [N4: nat] : ( minus_minus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) )
% 12.85/7.04 => ( realrel @ X9 @ Y7 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10164_eq__Real,axiom,
% 12.85/7.04 ! [X9: nat > rat,Y7: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( ( cauchy @ Y7 )
% 12.85/7.04 => ( ( ( real2 @ X9 )
% 12.85/7.04 = ( real2 @ Y7 ) )
% 12.85/7.04 = ( vanishes
% 12.85/7.04 @ ^ [N4: nat] : ( minus_minus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10165_vanishes__diff__inverse,axiom,
% 12.85/7.04 ! [X9: nat > rat,Y7: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( ~ ( vanishes @ X9 )
% 12.85/7.04 => ( ( cauchy @ Y7 )
% 12.85/7.04 => ( ~ ( vanishes @ Y7 )
% 12.85/7.04 => ( ( vanishes
% 12.85/7.04 @ ^ [N4: nat] : ( minus_minus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) )
% 12.85/7.04 => ( vanishes
% 12.85/7.04 @ ^ [N4: nat] : ( minus_minus_rat @ ( inverse_inverse_rat @ ( X9 @ N4 ) ) @ ( inverse_inverse_rat @ ( Y7 @ N4 ) ) ) ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10166_realrel__def,axiom,
% 12.85/7.04 ( realrel
% 12.85/7.04 = ( ^ [X8: nat > rat,Y8: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X8 )
% 12.85/7.04 & ( cauchy @ Y8 )
% 12.85/7.04 & ( vanishes
% 12.85/7.04 @ ^ [N4: nat] : ( minus_minus_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10167_cauchy__not__vanishes__cases,axiom,
% 12.85/7.04 ! [X9: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( ~ ( vanishes @ X9 )
% 12.85/7.04 => ? [B3: rat] :
% 12.85/7.04 ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 12.85/7.04 & ? [K2: nat] :
% 12.85/7.04 ( ! [N6: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K2 @ N6 )
% 12.85/7.04 => ( ord_less_rat @ B3 @ ( uminus_uminus_rat @ ( X9 @ N6 ) ) ) )
% 12.85/7.04 | ! [N6: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K2 @ N6 )
% 12.85/7.04 => ( ord_less_rat @ B3 @ ( X9 @ N6 ) ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10168_positive__Real,axiom,
% 12.85/7.04 ! [X9: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( ( positive2 @ ( real2 @ X9 ) )
% 12.85/7.04 = ( ? [R5: rat] :
% 12.85/7.04 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.85/7.04 & ? [K3: nat] :
% 12.85/7.04 ! [N4: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K3 @ N4 )
% 12.85/7.04 => ( ord_less_rat @ R5 @ ( X9 @ N4 ) ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10169_uminus__real__def,axiom,
% 12.85/7.04 ( uminus_uminus_real
% 12.85/7.04 = ( map_fu7146612038024189824t_real @ rep_real @ real2
% 12.85/7.04 @ ^ [X8: nat > rat,N4: nat] : ( uminus_uminus_rat @ ( X8 @ N4 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10170_cauchy__not__vanishes,axiom,
% 12.85/7.04 ! [X9: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( ~ ( vanishes @ X9 )
% 12.85/7.04 => ? [B3: rat] :
% 12.85/7.04 ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 12.85/7.04 & ? [K2: nat] :
% 12.85/7.04 ! [N6: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K2 @ N6 )
% 12.85/7.04 => ( ord_less_rat @ B3 @ ( abs_abs_rat @ ( X9 @ N6 ) ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10171_cauchyD,axiom,
% 12.85/7.04 ! [X9: nat > rat,R2: rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 12.85/7.04 => ? [K2: nat] :
% 12.85/7.04 ! [M2: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K2 @ M2 )
% 12.85/7.04 => ! [N6: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K2 @ N6 )
% 12.85/7.04 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X9 @ M2 ) @ ( X9 @ N6 ) ) ) @ R2 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10172_cauchyI,axiom,
% 12.85/7.04 ! [X9: nat > rat] :
% 12.85/7.04 ( ! [R3: rat] :
% 12.85/7.04 ( ( ord_less_rat @ zero_zero_rat @ R3 )
% 12.85/7.04 => ? [K4: nat] :
% 12.85/7.04 ! [M4: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K4 @ M4 )
% 12.85/7.04 => ! [N2: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K4 @ N2 )
% 12.85/7.04 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X9 @ M4 ) @ ( X9 @ N2 ) ) ) @ R3 ) ) ) )
% 12.85/7.04 => ( cauchy @ X9 ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10173_cauchy__def,axiom,
% 12.85/7.04 ( cauchy
% 12.85/7.04 = ( ^ [X8: nat > rat] :
% 12.85/7.04 ! [R5: rat] :
% 12.85/7.04 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.85/7.04 => ? [K3: nat] :
% 12.85/7.04 ! [M5: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K3 @ M5 )
% 12.85/7.04 => ! [N4: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K3 @ N4 )
% 12.85/7.04 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M5 ) @ ( X8 @ N4 ) ) ) @ R5 ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10174_inverse__Real,axiom,
% 12.85/7.04 ! [X9: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( ( ( vanishes @ X9 )
% 12.85/7.04 => ( ( inverse_inverse_real @ ( real2 @ X9 ) )
% 12.85/7.04 = zero_zero_real ) )
% 12.85/7.04 & ( ~ ( vanishes @ X9 )
% 12.85/7.04 => ( ( inverse_inverse_real @ ( real2 @ X9 ) )
% 12.85/7.04 = ( real2
% 12.85/7.04 @ ^ [N4: nat] : ( inverse_inverse_rat @ ( X9 @ N4 ) ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10175_not__positive__Real,axiom,
% 12.85/7.04 ! [X9: nat > rat] :
% 12.85/7.04 ( ( cauchy @ X9 )
% 12.85/7.04 => ( ( ~ ( positive2 @ ( real2 @ X9 ) ) )
% 12.85/7.04 = ( ! [R5: rat] :
% 12.85/7.04 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 12.85/7.04 => ? [K3: nat] :
% 12.85/7.04 ! [N4: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ K3 @ N4 )
% 12.85/7.04 => ( ord_less_eq_rat @ ( X9 @ N4 ) @ R5 ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10176_times__real__def,axiom,
% 12.85/7.04 ( times_times_real
% 12.85/7.04 = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
% 12.85/7.04 @ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10177_plus__real__def,axiom,
% 12.85/7.04 ( plus_plus_real
% 12.85/7.04 = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
% 12.85/7.04 @ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( plus_plus_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10178_cr__real__def,axiom,
% 12.85/7.04 ( cr_real
% 12.85/7.04 = ( ^ [X3: nat > rat,Y6: real] :
% 12.85/7.04 ( ( realrel @ X3 @ X3 )
% 12.85/7.04 & ( ( real2 @ X3 )
% 12.85/7.04 = Y6 ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10179_int_Obi__total,axiom,
% 12.85/7.04 bi_tot896582865486249351at_int @ pcr_int ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10180_real_Opcr__cr__eq,axiom,
% 12.85/7.04 pcr_real = cr_real ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10181_rcis__inverse,axiom,
% 12.85/7.04 ! [R2: real,A: real] :
% 12.85/7.04 ( ( invers8013647133539491842omplex @ ( rcis @ R2 @ A ) )
% 12.85/7.04 = ( rcis @ ( divide_divide_real @ one_one_real @ R2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10182_rcis__zero__arg,axiom,
% 12.85/7.04 ! [R2: real] :
% 12.85/7.04 ( ( rcis @ R2 @ zero_zero_real )
% 12.85/7.04 = ( real_V4546457046886955230omplex @ R2 ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10183_rcis__eq__zero__iff,axiom,
% 12.85/7.04 ! [R2: real,A: real] :
% 12.85/7.04 ( ( ( rcis @ R2 @ A )
% 12.85/7.04 = zero_zero_complex )
% 12.85/7.04 = ( R2 = zero_zero_real ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10184_rcis__zero__mod,axiom,
% 12.85/7.04 ! [A: real] :
% 12.85/7.04 ( ( rcis @ zero_zero_real @ A )
% 12.85/7.04 = zero_zero_complex ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10185_Re__rcis,axiom,
% 12.85/7.04 ! [R2: real,A: real] :
% 12.85/7.04 ( ( re @ ( rcis @ R2 @ A ) )
% 12.85/7.04 = ( times_times_real @ R2 @ ( cos_real @ A ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10186_Im__rcis,axiom,
% 12.85/7.04 ! [R2: real,A: real] :
% 12.85/7.04 ( ( im @ ( rcis @ R2 @ A ) )
% 12.85/7.04 = ( times_times_real @ R2 @ ( sin_real @ A ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10187_rcis__mult,axiom,
% 12.85/7.04 ! [R1: real,A: real,R22: real,B: real] :
% 12.85/7.04 ( ( times_times_complex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B ) )
% 12.85/7.04 = ( rcis @ ( times_times_real @ R1 @ R22 ) @ ( plus_plus_real @ A @ B ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10188_rcis__divide,axiom,
% 12.85/7.04 ! [R1: real,A: real,R22: real,B: real] :
% 12.85/7.04 ( ( divide1717551699836669952omplex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B ) )
% 12.85/7.04 = ( rcis @ ( divide_divide_real @ R1 @ R22 ) @ ( minus_minus_real @ A @ B ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10189_DeMoivre2,axiom,
% 12.85/7.04 ! [R2: real,A: real,N: nat] :
% 12.85/7.04 ( ( power_power_complex @ ( rcis @ R2 @ A ) @ N )
% 12.85/7.04 = ( rcis @ ( power_power_real @ R2 @ N ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10190_pred__nat__trancl__eq__le,axiom,
% 12.85/7.04 ! [M: nat,N: nat] :
% 12.85/7.04 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
% 12.85/7.04 = ( ord_less_eq_nat @ M @ N ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10191_le__enumerate,axiom,
% 12.85/7.04 ! [S: set_nat,N: nat] :
% 12.85/7.04 ( ~ ( finite_finite_nat @ S )
% 12.85/7.04 => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10192_finite__le__enumerate,axiom,
% 12.85/7.04 ! [S: set_nat,N: nat] :
% 12.85/7.04 ( ( finite_finite_nat @ S )
% 12.85/7.04 => ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
% 12.85/7.04 => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10193_Least__eq__0,axiom,
% 12.85/7.04 ! [P: nat > $o] :
% 12.85/7.04 ( ( P @ zero_zero_nat )
% 12.85/7.04 => ( ( ord_Least_nat @ P )
% 12.85/7.04 = zero_zero_nat ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10194_Least__Suc2,axiom,
% 12.85/7.04 ! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
% 12.85/7.04 ( ( P @ N )
% 12.85/7.04 => ( ( Q @ M )
% 12.85/7.04 => ( ~ ( P @ zero_zero_nat )
% 12.85/7.04 => ( ! [K2: nat] :
% 12.85/7.04 ( ( P @ ( suc @ K2 ) )
% 12.85/7.04 = ( Q @ K2 ) )
% 12.85/7.04 => ( ( ord_Least_nat @ P )
% 12.85/7.04 = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10195_Least__Suc,axiom,
% 12.85/7.04 ! [P: nat > $o,N: nat] :
% 12.85/7.04 ( ( P @ N )
% 12.85/7.04 => ( ~ ( P @ zero_zero_nat )
% 12.85/7.04 => ( ( ord_Least_nat @ P )
% 12.85/7.04 = ( suc
% 12.85/7.04 @ ( ord_Least_nat
% 12.85/7.04 @ ^ [M5: nat] : ( P @ ( suc @ M5 ) ) ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10196_Sup__real__def,axiom,
% 12.85/7.04 ( comple1385675409528146559p_real
% 12.85/7.04 = ( ^ [X8: set_real] :
% 12.85/7.04 ( ord_Least_real
% 12.85/7.04 @ ^ [Z5: real] :
% 12.85/7.04 ! [X3: real] :
% 12.85/7.04 ( ( member_real @ X3 @ X8 )
% 12.85/7.04 => ( ord_less_eq_real @ X3 @ Z5 ) ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10197_last__upt,axiom,
% 12.85/7.04 ! [I: nat,J: nat] :
% 12.85/7.04 ( ( ord_less_nat @ I @ J )
% 12.85/7.04 => ( ( last_nat @ ( upt @ I @ J ) )
% 12.85/7.04 = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10198_vimage__Suc__insert__0,axiom,
% 12.85/7.04 ! [A2: set_nat] :
% 12.85/7.04 ( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A2 ) )
% 12.85/7.04 = ( vimage_nat_nat @ suc @ A2 ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10199_vimage__Suc__insert__Suc,axiom,
% 12.85/7.04 ! [N: nat,A2: set_nat] :
% 12.85/7.04 ( ( vimage_nat_nat @ suc @ ( insert_nat @ ( suc @ N ) @ A2 ) )
% 12.85/7.04 = ( insert_nat @ N @ ( vimage_nat_nat @ suc @ A2 ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10200_finite__vimage__Suc__iff,axiom,
% 12.85/7.04 ! [F4: set_nat] :
% 12.85/7.04 ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F4 ) )
% 12.85/7.04 = ( finite_finite_nat @ F4 ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10201_set__decode__div__2,axiom,
% 12.85/7.04 ! [X: nat] :
% 12.85/7.04 ( ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 12.85/7.04 = ( vimage_nat_nat @ suc @ ( nat_set_decode @ X ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10202_set__encode__vimage__Suc,axiom,
% 12.85/7.04 ! [A2: set_nat] :
% 12.85/7.04 ( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A2 ) )
% 12.85/7.04 = ( divide_divide_nat @ ( nat_set_encode @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10203_Bseq__monoseq__convergent_H__dec,axiom,
% 12.85/7.04 ! [F: nat > real,M10: nat] :
% 12.85/7.04 ( ( bfun_nat_real
% 12.85/7.04 @ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ M10 ) )
% 12.85/7.04 @ at_top_nat )
% 12.85/7.04 => ( ! [M4: nat,N2: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ M10 @ M4 )
% 12.85/7.04 => ( ( ord_less_eq_nat @ M4 @ N2 )
% 12.85/7.04 => ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ M4 ) ) ) )
% 12.85/7.04 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10204_Bseq__mono__convergent,axiom,
% 12.85/7.04 ! [X9: nat > real] :
% 12.85/7.04 ( ( bfun_nat_real @ X9 @ at_top_nat )
% 12.85/7.04 => ( ! [M4: nat,N2: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ M4 @ N2 )
% 12.85/7.04 => ( ord_less_eq_real @ ( X9 @ M4 ) @ ( X9 @ N2 ) ) )
% 12.85/7.04 => ( topolo7531315842566124627t_real @ X9 ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10205_convergent__realpow,axiom,
% 12.85/7.04 ! [X: real] :
% 12.85/7.04 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 12.85/7.04 => ( ( ord_less_eq_real @ X @ one_one_real )
% 12.85/7.04 => ( topolo7531315842566124627t_real @ ( power_power_real @ X ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10206_Bseq__monoseq__convergent_H__inc,axiom,
% 12.85/7.04 ! [F: nat > real,M10: nat] :
% 12.85/7.04 ( ( bfun_nat_real
% 12.85/7.04 @ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ M10 ) )
% 12.85/7.04 @ at_top_nat )
% 12.85/7.04 => ( ! [M4: nat,N2: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ M10 @ M4 )
% 12.85/7.04 => ( ( ord_less_eq_nat @ M4 @ N2 )
% 12.85/7.04 => ( ord_less_eq_real @ ( F @ M4 ) @ ( F @ N2 ) ) ) )
% 12.85/7.04 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10207_pair__lessI2,axiom,
% 12.85/7.04 ! [A: nat,B: nat,S2: nat,T: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ A @ B )
% 12.85/7.04 => ( ( ord_less_nat @ S2 @ T )
% 12.85/7.04 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10208_pair__leqI2,axiom,
% 12.85/7.04 ! [A: nat,B: nat,S2: nat,T: nat] :
% 12.85/7.04 ( ( ord_less_eq_nat @ A @ B )
% 12.85/7.04 => ( ( ord_less_eq_nat @ S2 @ T )
% 12.85/7.04 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10209_filtermap__at__right__shift,axiom,
% 12.85/7.04 ! [D: real,A: real] :
% 12.85/7.04 ( ( filtermap_real_real
% 12.85/7.04 @ ^ [X3: real] : ( minus_minus_real @ X3 @ D )
% 12.85/7.04 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 12.85/7.04 = ( topolo2177554685111907308n_real @ ( minus_minus_real @ A @ D ) @ ( set_or5849166863359141190n_real @ ( minus_minus_real @ A @ D ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10210_at__right__to__0,axiom,
% 12.85/7.04 ! [A: real] :
% 12.85/7.04 ( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
% 12.85/7.04 = ( filtermap_real_real
% 12.85/7.04 @ ^ [X3: real] : ( plus_plus_real @ X3 @ A )
% 12.85/7.04 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10211_at__right__to__top,axiom,
% 12.85/7.04 ( ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) )
% 12.85/7.04 = ( filtermap_real_real @ inverse_inverse_real @ at_top_real ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10212_at__top__to__right,axiom,
% 12.85/7.04 ( at_top_real
% 12.85/7.04 = ( filtermap_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10213_filtermap__ln__at__right,axiom,
% 12.85/7.04 ( ( filtermap_real_real @ ln_ln_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 12.85/7.04 = at_bot_real ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10214_gcd__nat_Oordering__top__axioms,axiom,
% 12.85/7.04 ( ordering_top_nat @ dvd_dvd_nat
% 12.85/7.04 @ ^ [M5: nat,N4: nat] :
% 12.85/7.04 ( ( dvd_dvd_nat @ M5 @ N4 )
% 12.85/7.04 & ( M5 != N4 ) )
% 12.85/7.04 @ zero_zero_nat ) ).
% 12.85/7.04
% 12.85/7.04 thf(fact_10215_bot__nat__0_Oordering__top__axioms,axiom,
% 12.85/7.04 ( ordering_top_nat
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X3 )
% 12.85/7.04 @ ^ [X3: nat,Y6: nat] : ( ord_less_nat @ Y6 @ X3 )
% 12.85/7.04 @ zero_zero_nat ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 12.85/7.04 ! [X: int,Y4: int] :
% 12.85/7.04 ( ( if_int @ $false @ X @ Y4 )
% 12.85/7.04 = Y4 ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 12.85/7.04 ! [X: int,Y4: int] :
% 12.85/7.04 ( ( if_int @ $true @ X @ Y4 )
% 12.85/7.04 = X ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 12.85/7.04 ! [X: nat,Y4: nat] :
% 12.85/7.04 ( ( if_nat @ $false @ X @ Y4 )
% 12.85/7.04 = Y4 ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 12.85/7.04 ! [X: nat,Y4: nat] :
% 12.85/7.04 ( ( if_nat @ $true @ X @ Y4 )
% 12.85/7.04 = X ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
% 12.85/7.04 ! [X: num,Y4: num] :
% 12.85/7.04 ( ( if_num @ $false @ X @ Y4 )
% 12.85/7.04 = Y4 ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
% 12.85/7.04 ! [X: num,Y4: num] :
% 12.85/7.04 ( ( if_num @ $true @ X @ Y4 )
% 12.85/7.04 = X ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 12.85/7.04 ! [X: rat,Y4: rat] :
% 12.85/7.04 ( ( if_rat @ $false @ X @ Y4 )
% 12.85/7.04 = Y4 ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 12.85/7.04 ! [X: rat,Y4: rat] :
% 12.85/7.04 ( ( if_rat @ $true @ X @ Y4 )
% 12.85/7.04 = X ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 12.85/7.04 ! [X: real,Y4: real] :
% 12.85/7.04 ( ( if_real @ $false @ X @ Y4 )
% 12.85/7.04 = Y4 ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 12.85/7.04 ! [X: real,Y4: real] :
% 12.85/7.04 ( ( if_real @ $true @ X @ Y4 )
% 12.85/7.04 = X ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 12.85/7.04 ! [P: real > $o] :
% 12.85/7.04 ( ( P @ ( fChoice_real @ P ) )
% 12.85/7.04 = ( ? [X8: real] : ( P @ X8 ) ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 12.85/7.04 ! [X: complex,Y4: complex] :
% 12.85/7.04 ( ( if_complex @ $false @ X @ Y4 )
% 12.85/7.04 = Y4 ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 12.85/7.04 ! [X: complex,Y4: complex] :
% 12.85/7.04 ( ( if_complex @ $true @ X @ Y4 )
% 12.85/7.04 = X ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 12.85/7.04 ! [X: code_integer,Y4: code_integer] :
% 12.85/7.04 ( ( if_Code_integer @ $false @ X @ Y4 )
% 12.85/7.04 = Y4 ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 12.85/7.04 ! [X: code_integer,Y4: code_integer] :
% 12.85/7.04 ( ( if_Code_integer @ $true @ X @ Y4 )
% 12.85/7.04 = X ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 12.85/7.04 ! [X: list_int,Y4: list_int] :
% 12.85/7.04 ( ( if_list_int @ $false @ X @ Y4 )
% 12.85/7.04 = Y4 ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 12.85/7.04 ! [X: list_int,Y4: list_int] :
% 12.85/7.04 ( ( if_list_int @ $true @ X @ Y4 )
% 12.85/7.04 = X ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 12.85/7.04 ! [X: list_nat,Y4: list_nat] :
% 12.85/7.04 ( ( if_list_nat @ $false @ X @ Y4 )
% 12.85/7.04 = Y4 ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 12.85/7.04 ! [X: list_nat,Y4: list_nat] :
% 12.85/7.04 ( ( if_list_nat @ $true @ X @ Y4 )
% 12.85/7.04 = X ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 12.85/7.04 ! [X: int > int,Y4: int > int] :
% 12.85/7.04 ( ( if_int_int @ $false @ X @ Y4 )
% 12.85/7.04 = Y4 ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 12.85/7.04 ! [X: int > int,Y4: int > int] :
% 12.85/7.04 ( ( if_int_int @ $true @ X @ Y4 )
% 12.85/7.04 = X ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_2_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
% 12.85/7.04 ! [X: nat > rat,Y4: nat > rat] :
% 12.85/7.04 ( ( if_nat_rat @ $false @ X @ Y4 )
% 12.85/7.04 = Y4 ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_1_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
% 12.85/7.04 ! [X: nat > rat,Y4: nat > rat] :
% 12.85/7.04 ( ( if_nat_rat @ $true @ X @ Y4 )
% 12.85/7.04 = X ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 12.85/7.04 ! [X: option_num,Y4: option_num] :
% 12.85/7.04 ( ( if_option_num @ $false @ X @ Y4 )
% 12.85/7.04 = Y4 ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 12.85/7.04 ! [X: option_num,Y4: option_num] :
% 12.85/7.04 ( ( if_option_num @ $true @ X @ Y4 )
% 12.85/7.04 = X ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 12.85/7.04 ! [X: product_prod_int_int,Y4: product_prod_int_int] :
% 12.85/7.04 ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y4 )
% 12.85/7.04 = Y4 ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 12.85/7.04 ! [X: product_prod_int_int,Y4: product_prod_int_int] :
% 12.85/7.04 ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y4 )
% 12.85/7.04 = X ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 12.85/7.04 ! [X: product_prod_nat_nat,Y4: product_prod_nat_nat] :
% 12.85/7.04 ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y4 )
% 12.85/7.04 = Y4 ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 12.85/7.04 ! [X: product_prod_nat_nat,Y4: product_prod_nat_nat] :
% 12.85/7.04 ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y4 )
% 12.85/7.04 = X ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
% 12.85/7.04 ! [X: nat > int > int,Y4: nat > int > int] :
% 12.85/7.04 ( ( if_nat_int_int @ $false @ X @ Y4 )
% 12.85/7.04 = Y4 ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
% 12.85/7.04 ! [X: nat > int > int,Y4: nat > int > int] :
% 12.85/7.04 ( ( if_nat_int_int @ $true @ X @ Y4 )
% 12.85/7.04 = X ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
% 12.85/7.04 ! [X: nat > nat > nat,Y4: nat > nat > nat] :
% 12.85/7.04 ( ( if_nat_nat_nat @ $false @ X @ Y4 )
% 12.85/7.04 = Y4 ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
% 12.85/7.04 ! [X: nat > nat > nat,Y4: nat > nat > nat] :
% 12.85/7.04 ( ( if_nat_nat_nat @ $true @ X @ Y4 )
% 12.85/7.04 = X ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 12.85/7.04 ! [X: produc6271795597528267376eger_o,Y4: produc6271795597528267376eger_o] :
% 12.85/7.04 ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y4 )
% 12.85/7.04 = Y4 ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 12.85/7.04 ! [X: produc6271795597528267376eger_o,Y4: produc6271795597528267376eger_o] :
% 12.85/7.04 ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y4 )
% 12.85/7.04 = X ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 12.85/7.04 ! [X: produc8923325533196201883nteger,Y4: produc8923325533196201883nteger] :
% 12.85/7.04 ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y4 )
% 12.85/7.04 = Y4 ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 12.85/7.04 ! [X: produc8923325533196201883nteger,Y4: produc8923325533196201883nteger] :
% 12.85/7.04 ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y4 )
% 12.85/7.04 = X ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_3_1_If_001_062_It__Nat__Onat_M_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_T,axiom,
% 12.85/7.04 ! [P: $o] :
% 12.85/7.04 ( ( P = $true )
% 12.85/7.04 | ( P = $false ) ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_T,axiom,
% 12.85/7.04 ! [X: nat > code_integer > code_integer,Y4: nat > code_integer > code_integer] :
% 12.85/7.04 ( ( if_nat5617392847756311170nteger @ $false @ X @ Y4 )
% 12.85/7.04 = Y4 ) ).
% 12.85/7.04
% 12.85/7.04 thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_T,axiom,
% 12.85/7.04 ! [X: nat > code_integer > code_integer,Y4: nat > code_integer > code_integer] :
% 12.85/7.04 ( ( if_nat5617392847756311170nteger @ $true @ X @ Y4 )
% 12.85/7.04 = X ) ).
% 12.85/7.04
% 12.85/7.04 thf(conj_0,conjecture,
% 12.85/7.04 ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ ( suc @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 12.85/7.04 ------------------------
% 12.85/7.36 % (24498)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_24167 for (2997ds/183Mi)
% 12.85/7.36 % (24499)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_24167 for (2997ds/4Mi)
% 12.85/7.36 % (24500)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_24167 for (2997ds/27Mi)
% 12.85/7.36 % (24502)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_24167 for (2997ds/2Mi)
% 12.85/7.36 % (24501)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_24167 for (2997ds/2Mi)
% 12.85/7.36 % (24503)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_24167 for (2997ds/275Mi)
% 12.85/7.36 % (24504)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_24167 for (2997ds/18Mi)
% 12.85/7.36 % (24501)Instruction limit reached!
% 12.85/7.36 % (24501)------------------------------
% 12.85/7.36 % (24501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.36 % (24501)Termination reason: Unknown
% 12.85/7.36 % (24501)Termination phase: shuffling
% 12.85/7.36
% 12.85/7.36 % (24501)Memory used [KB]: 10618
% 12.85/7.36 % (24501)Time elapsed: 0.003 s
% 12.85/7.36 % (24501)Instructions burned: 2 (million)
% 12.85/7.36 % (24501)------------------------------
% 12.85/7.36 % (24501)------------------------------
% 12.85/7.36 % (24502)Instruction limit reached!
% 12.85/7.36 % (24502)------------------------------
% 12.85/7.36 % (24502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.36 % (24502)Termination reason: Unknown
% 12.85/7.36 % (24502)Termination phase: shuffling
% 12.85/7.36
% 12.85/7.36 % (24502)Memory used [KB]: 10618
% 12.85/7.36 % (24502)Time elapsed: 0.003 s
% 12.85/7.36 % (24502)Instructions burned: 2 (million)
% 12.85/7.36 % (24502)------------------------------
% 12.85/7.36 % (24502)------------------------------
% 12.85/7.36 % (24499)Instruction limit reached!
% 12.85/7.36 % (24499)------------------------------
% 12.85/7.36 % (24499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.36 % (24499)Termination reason: Unknown
% 12.85/7.36 % (24499)Termination phase: shuffling
% 12.85/7.36
% 12.85/7.36 % (24499)Memory used [KB]: 10746
% 12.85/7.36 % (24499)Time elapsed: 0.007 s
% 12.85/7.36 % (24499)Instructions burned: 4 (million)
% 12.85/7.36 % (24499)------------------------------
% 12.85/7.36 % (24499)------------------------------
% 12.85/7.38 % (24511)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_24167 for (2996ds/3Mi)
% 12.85/7.38 % (24512)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_24167 for (2996ds/37Mi)
% 12.85/7.38 % (24504)Instruction limit reached!
% 12.85/7.38 % (24504)------------------------------
% 12.85/7.38 % (24504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.38 % (24504)Termination reason: Unknown
% 12.85/7.38 % (24504)Termination phase: shuffling
% 12.85/7.38
% 12.85/7.38 % (24504)Memory used [KB]: 11385
% 12.85/7.38 % (24504)Time elapsed: 0.022 s
% 12.85/7.38 % (24504)Instructions burned: 18 (million)
% 12.85/7.38 % (24504)------------------------------
% 12.85/7.38 % (24504)------------------------------
% 12.85/7.38 % (24511)Instruction limit reached!
% 12.85/7.38 % (24511)------------------------------
% 12.85/7.38 % (24511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.38 % (24511)Termination reason: Unknown
% 12.85/7.38 % (24511)Termination phase: shuffling
% 12.85/7.38
% 12.85/7.38 % (24511)Memory used [KB]: 10746
% 12.85/7.38 % (24511)Time elapsed: 0.005 s
% 12.85/7.38 % (24511)Instructions burned: 3 (million)
% 12.85/7.38 % (24511)------------------------------
% 12.85/7.38 % (24511)------------------------------
% 12.85/7.38 % (24514)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_24167 for (2996ds/15Mi)
% 12.85/7.38 % (24500)Instruction limit reached!
% 12.85/7.38 % (24500)------------------------------
% 12.85/7.38 % (24500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.38 % (24500)Termination reason: Unknown
% 12.85/7.38 % (24500)Termination phase: shuffling
% 12.85/7.38
% 12.85/7.38 % (24500)Memory used [KB]: 12537
% 12.85/7.38 % (24500)Time elapsed: 0.029 s
% 12.85/7.38 % (24500)Instructions burned: 27 (million)
% 12.85/7.38 % (24500)------------------------------
% 12.85/7.38 % (24500)------------------------------
% 12.85/7.40 % (24515)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_24167 for (2996ds/3Mi)
% 12.85/7.40 % (24514)Instruction limit reached!
% 12.85/7.40 % (24514)------------------------------
% 12.85/7.40 % (24514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.40 % (24514)Termination reason: Unknown
% 12.85/7.40 % (24514)Termination phase: shuffling
% 12.85/7.40
% 12.85/7.40 % (24514)Memory used [KB]: 11385
% 12.85/7.40 % (24514)Time elapsed: 0.020 s
% 12.85/7.40 % (24514)Instructions burned: 15 (million)
% 12.85/7.40 % (24514)------------------------------
% 12.85/7.40 % (24514)------------------------------
% 12.85/7.40 % (24516)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_24167 for (2996ds/1041Mi)
% 12.85/7.40 % (24515)Instruction limit reached!
% 12.85/7.40 % (24515)------------------------------
% 12.85/7.40 % (24515)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.40 % (24515)Termination reason: Unknown
% 12.85/7.40 % (24515)Termination phase: shuffling
% 12.85/7.40
% 12.85/7.40 % (24515)Memory used [KB]: 10746
% 12.85/7.40 % (24515)Time elapsed: 0.005 s
% 12.85/7.40 % (24515)Instructions burned: 3 (million)
% 12.85/7.40 % (24515)------------------------------
% 12.85/7.40 % (24515)------------------------------
% 12.85/7.40 % (24517)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_24167 for (2996ds/7Mi)
% 12.85/7.41 % (24512)Instruction limit reached!
% 12.85/7.41 % (24512)------------------------------
% 12.85/7.41 % (24512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.41 % (24512)Termination reason: Unknown
% 12.85/7.41 % (24512)Termination phase: shuffling
% 12.85/7.41
% 12.85/7.41 % (24512)Memory used [KB]: 12792
% 12.85/7.41 % (24512)Time elapsed: 0.033 s
% 12.85/7.41 % (24512)Instructions burned: 38 (million)
% 12.85/7.41 % (24512)------------------------------
% 12.85/7.41 % (24512)------------------------------
% 12.85/7.41 % (24517)Instruction limit reached!
% 12.85/7.41 % (24517)------------------------------
% 12.85/7.41 % (24517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.41 % (24517)Termination reason: Unknown
% 12.85/7.41 % (24517)Termination phase: shuffling
% 12.85/7.41
% 12.85/7.41 % (24517)Memory used [KB]: 11001
% 12.85/7.41 % (24517)Time elapsed: 0.011 s
% 12.85/7.41 % (24517)Instructions burned: 7 (million)
% 12.85/7.41 % (24517)------------------------------
% 12.85/7.41 % (24517)------------------------------
% 12.85/7.42 % (24518)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_24167 for (2996ds/16Mi)
% 12.85/7.42 % (24519)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_24167 for (2996ds/3Mi)
% 12.85/7.42 % (24519)Instruction limit reached!
% 12.85/7.42 % (24519)------------------------------
% 12.85/7.42 % (24519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.42 % (24519)Termination reason: Unknown
% 12.85/7.42 % (24519)Termination phase: shuffling
% 12.85/7.42
% 12.85/7.42 % (24519)Memory used [KB]: 10618
% 12.85/7.42 % (24519)Time elapsed: 0.005 s
% 12.85/7.42 % (24519)Instructions burned: 3 (million)
% 12.85/7.42 % (24519)------------------------------
% 12.85/7.42 % (24519)------------------------------
% 12.85/7.43 % (24520)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_24167 for (2996ds/3Mi)
% 12.85/7.43 % (24520)Instruction limit reached!
% 12.85/7.43 % (24520)------------------------------
% 12.85/7.43 % (24520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.43 % (24520)Termination reason: Unknown
% 12.85/7.43 % (24520)Termination phase: shuffling
% 12.85/7.43
% 12.85/7.43 % (24520)Memory used [KB]: 10746
% 12.85/7.43 % (24520)Time elapsed: 0.005 s
% 12.85/7.43 % (24520)Instructions burned: 3 (million)
% 12.85/7.43 % (24520)------------------------------
% 12.85/7.43 % (24520)------------------------------
% 12.85/7.43 % (24521)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_24167 for (2996ds/7Mi)
% 12.85/7.44 % (24518)Instruction limit reached!
% 12.85/7.44 % (24518)------------------------------
% 12.85/7.44 % (24518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.44 % (24518)Termination reason: Unknown
% 12.85/7.44 % (24518)Termination phase: shuffling
% 12.85/7.44
% 12.85/7.44 % (24518)Memory used [KB]: 11385
% 12.85/7.44 % (24518)Time elapsed: 0.021 s
% 12.85/7.44 % (24518)Instructions burned: 17 (million)
% 12.85/7.44 % (24518)------------------------------
% 12.85/7.44 % (24518)------------------------------
% 12.85/7.44 % (24522)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_24167 for (2996ds/3Mi)
% 12.85/7.44 % (24521)Instruction limit reached!
% 12.85/7.44 % (24521)------------------------------
% 12.85/7.44 % (24521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.44 % (24521)Termination reason: Unknown
% 12.85/7.44 % (24521)Termination phase: shuffling
% 12.85/7.44
% 12.85/7.44 % (24521)Memory used [KB]: 11001
% 12.85/7.44 % (24521)Time elapsed: 0.011 s
% 12.85/7.44 % (24521)Instructions burned: 7 (million)
% 12.85/7.44 % (24521)------------------------------
% 12.85/7.44 % (24521)------------------------------
% 12.85/7.44 % (24522)Instruction limit reached!
% 12.85/7.44 % (24522)------------------------------
% 12.85/7.44 % (24522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.44 % (24522)Termination reason: Unknown
% 12.85/7.44 % (24522)Termination phase: shuffling
% 12.85/7.44
% 12.85/7.44 % (24522)Memory used [KB]: 10618
% 12.85/7.44 % (24522)Time elapsed: 0.005 s
% 12.85/7.44 % (24522)Instructions burned: 3 (million)
% 12.85/7.44 % (24522)------------------------------
% 12.85/7.44 % (24522)------------------------------
% 12.85/7.45 % (24523)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_24167 for (2996ds/4Mi)
% 12.85/7.45 % (24523)Instruction limit reached!
% 12.85/7.45 % (24523)------------------------------
% 12.85/7.45 % (24523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.45 % (24523)Termination reason: Unknown
% 12.85/7.45 % (24523)Termination phase: shuffling
% 12.85/7.45
% 12.85/7.45 % (24523)Memory used [KB]: 10746
% 12.85/7.45 % (24523)Time elapsed: 0.007 s
% 12.85/7.45 % (24523)Instructions burned: 4 (million)
% 12.85/7.45 % (24523)------------------------------
% 12.85/7.45 % (24523)------------------------------
% 12.85/7.46 % (24524)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_24167 for (2996ds/18Mi)
% 12.85/7.46 % (24525)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_24167 for (2996ds/710Mi)
% 12.85/7.46 % (24526)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_24167 for (2996ds/6Mi)
% 12.85/7.46 % (24498)Instruction limit reached!
% 12.85/7.46 % (24498)------------------------------
% 12.85/7.46 % (24498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.46 % (24498)Termination reason: Unknown
% 12.85/7.46 % (24498)Termination phase: shuffling
% 12.85/7.46
% 12.85/7.46 % (24498)Memory used [KB]: 14711
% 12.85/7.46 % (24498)Time elapsed: 0.111 s
% 12.85/7.46 % (24498)Instructions burned: 185 (million)
% 12.85/7.46 % (24498)------------------------------
% 12.85/7.46 % (24498)------------------------------
% 12.85/7.47 % (24526)Instruction limit reached!
% 12.85/7.47 % (24526)------------------------------
% 12.85/7.47 % (24526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.47 % (24526)Termination reason: Unknown
% 12.85/7.47 % (24526)Termination phase: shuffling
% 12.85/7.47
% 12.85/7.47 % (24526)Memory used [KB]: 10874
% 12.85/7.47 % (24526)Time elapsed: 0.010 s
% 12.85/7.47 % (24526)Instructions burned: 6 (million)
% 12.85/7.47 % (24526)------------------------------
% 12.85/7.47 % (24526)------------------------------
% 12.85/7.47 % (24527)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_24167 for (2995ds/902Mi)
% 12.85/7.47 % (24524)Instruction limit reached!
% 12.85/7.47 % (24524)------------------------------
% 12.85/7.47 % (24524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.47 % (24524)Termination reason: Unknown
% 12.85/7.47 % (24524)Termination phase: shuffling
% 12.85/7.47
% 12.85/7.47 % (24524)Memory used [KB]: 11385
% 12.85/7.47 % (24524)Time elapsed: 0.020 s
% 12.85/7.47 % (24524)Instructions burned: 19 (million)
% 12.85/7.47 % (24524)------------------------------
% 12.85/7.47 % (24524)------------------------------
% 12.85/7.48 % (24528)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_24167 for (2995ds/21Mi)
% 12.85/7.48 % (24530)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_24167 for (2995ds/6Mi)
% 12.85/7.48 % (24529)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_24167 for (2995ds/5Mi)
% 12.85/7.49 % (24529)Instruction limit reached!
% 12.85/7.49 % (24529)------------------------------
% 12.85/7.49 % (24529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.49 % (24529)Termination reason: Unknown
% 12.85/7.49 % (24529)Termination phase: shuffling
% 12.85/7.49
% 12.85/7.49 % (24529)Memory used [KB]: 10874
% 12.85/7.49 % (24529)Time elapsed: 0.006 s
% 12.85/7.49 % (24529)Instructions burned: 5 (million)
% 12.85/7.49 % (24529)------------------------------
% 12.85/7.49 % (24529)------------------------------
% 12.85/7.49 % (24528)Instruction limit reached!
% 12.85/7.49 % (24528)------------------------------
% 12.85/7.49 % (24528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.49 % (24528)Termination reason: Unknown
% 12.85/7.49 % (24528)Termination phase: shuffling
% 12.85/7.49
% 12.85/7.49 % (24528)Memory used [KB]: 11385
% 12.85/7.49 % (24528)Time elapsed: 0.014 s
% 12.85/7.49 % (24528)Instructions burned: 21 (million)
% 12.85/7.49 % (24528)------------------------------
% 12.85/7.49 % (24528)------------------------------
% 12.85/7.49 % (24530)Instruction limit reached!
% 12.85/7.49 % (24530)------------------------------
% 12.85/7.49 % (24530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.49 % (24530)Termination reason: Unknown
% 12.85/7.49 % (24530)Termination phase: shuffling
% 12.85/7.49
% 12.85/7.49 % (24530)Memory used [KB]: 11001
% 12.85/7.49 % (24530)Time elapsed: 0.007 s
% 12.85/7.49 % (24530)Instructions burned: 6 (million)
% 12.85/7.49 % (24530)------------------------------
% 12.85/7.49 % (24530)------------------------------
% 12.85/7.50 % (24532)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_24167 for (2995ds/779Mi)
% 12.85/7.50 % (24531)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_24167 for (2995ds/377Mi)
% 12.85/7.50 % (24533)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_24167 for (2995ds/19Mi)
% 12.85/7.51 % (24533)Instruction limit reached!
% 12.85/7.51 % (24533)------------------------------
% 12.85/7.51 % (24533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.51 % (24533)Termination reason: Unknown
% 12.85/7.51 % (24533)Termination phase: shuffling
% 12.85/7.51
% 12.85/7.51 % (24533)Memory used [KB]: 12409
% 12.85/7.51 % (24533)Time elapsed: 0.015 s
% 12.85/7.51 % (24533)Instructions burned: 21 (million)
% 12.85/7.51 % (24533)------------------------------
% 12.85/7.51 % (24533)------------------------------
% 12.85/7.51 % (24503)Instruction limit reached!
% 12.85/7.51 % (24503)------------------------------
% 12.85/7.51 % (24503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.51 % (24503)Termination reason: Unknown
% 12.85/7.51 % (24503)Termination phase: shuffling
% 12.85/7.51
% 12.85/7.51 % (24503)Memory used [KB]: 15991
% 12.85/7.51 % (24503)Time elapsed: 0.160 s
% 12.85/7.51 % (24503)Instructions burned: 276 (million)
% 12.85/7.51 % (24503)------------------------------
% 12.85/7.51 % (24503)------------------------------
% 12.85/7.53 % (24534)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_24167 for (2995ds/879Mi)
% 12.85/7.53 % (24535)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_24167 for (2995ds/17Mi)
% 12.85/7.54 % (24535)Instruction limit reached!
% 12.85/7.54 % (24535)------------------------------
% 12.85/7.54 % (24535)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.54 % (24535)Termination reason: Unknown
% 12.85/7.54 % (24535)Termination phase: shuffling
% 12.85/7.54
% 12.85/7.54 % (24535)Memory used [KB]: 11385
% 12.85/7.54 % (24535)Time elapsed: 0.014 s
% 12.85/7.54 % (24535)Instructions burned: 19 (million)
% 12.85/7.54 % (24535)------------------------------
% 12.85/7.54 % (24535)------------------------------
% 12.85/7.55 % (24536)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_24167 for (2994ds/3Mi)
% 12.85/7.55 % (24536)Instruction limit reached!
% 12.85/7.55 % (24536)------------------------------
% 12.85/7.55 % (24536)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.55 % (24536)Termination reason: Unknown
% 12.85/7.55 % (24536)Termination phase: shuffling
% 12.85/7.55
% 12.85/7.55 % (24536)Memory used [KB]: 10746
% 12.85/7.55 % (24536)Time elapsed: 0.003 s
% 12.85/7.55 % (24536)Instructions burned: 3 (million)
% 12.85/7.55 % (24536)------------------------------
% 12.85/7.55 % (24536)------------------------------
% 12.85/7.56 % (24537)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_24167 for (2994ds/30Mi)
% 12.85/7.58 % (24537)Instruction limit reached!
% 12.85/7.58 % (24537)------------------------------
% 12.85/7.58 % (24537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.58 % (24537)Termination reason: Unknown
% 12.85/7.58 % (24537)Termination phase: shuffling
% 12.85/7.58
% 12.85/7.58 % (24537)Memory used [KB]: 12537
% 12.85/7.58 % (24537)Time elapsed: 0.019 s
% 12.85/7.58 % (24537)Instructions burned: 30 (million)
% 12.85/7.58 % (24537)------------------------------
% 12.85/7.58 % (24537)------------------------------
% 12.85/7.59 % (24538)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_24167 for (2994ds/127Mi)
% 12.85/7.63 % (24531)Instruction limit reached!
% 12.85/7.63 % (24531)------------------------------
% 12.85/7.63 % (24531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.63 % (24531)Termination reason: Unknown
% 12.85/7.63 % (24531)Termination phase: shuffling
% 12.85/7.63
% 12.85/7.63 % (24531)Memory used [KB]: 17270
% 12.85/7.63 % (24531)Time elapsed: 0.136 s
% 12.85/7.63 % (24531)Instructions burned: 379 (million)
% 12.85/7.63 % (24531)------------------------------
% 12.85/7.63 % (24531)------------------------------
% 12.85/7.64 % (24538)Instruction limit reached!
% 12.85/7.64 % (24538)------------------------------
% 12.85/7.64 % (24538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.64 % (24538)Termination reason: Unknown
% 12.85/7.64 % (24538)Termination phase: shuffling
% 12.85/7.64
% 12.85/7.64 % (24538)Memory used [KB]: 13944
% 12.85/7.64 % (24538)Time elapsed: 0.050 s
% 12.85/7.64 % (24538)Instructions burned: 128 (million)
% 12.85/7.64 % (24538)------------------------------
% 12.85/7.64 % (24538)------------------------------
% 12.85/7.65 % (24539)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_24167 for (2994ds/100Mi)
% 12.85/7.65 % (24540)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_24167 for (2993ds/3Mi)
% 12.85/7.65 % (24540)Instruction limit reached!
% 12.85/7.65 % (24540)------------------------------
% 12.85/7.65 % (24540)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.65 % (24540)Termination reason: Unknown
% 12.85/7.65 % (24540)Termination phase: shuffling
% 12.85/7.65
% 12.85/7.65 % (24540)Memory used [KB]: 10746
% 12.85/7.65 % (24540)Time elapsed: 0.004 s
% 12.85/7.65 % (24540)Instructions burned: 4 (million)
% 12.85/7.65 % (24540)------------------------------
% 12.85/7.65 % (24540)------------------------------
% 12.85/7.66 % (24541)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_24167 for (2993ds/20Mi)
% 12.85/7.68 % (24541)Instruction limit reached!
% 12.85/7.68 % (24541)------------------------------
% 12.85/7.68 % (24541)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.68 % (24541)Termination reason: Unknown
% 12.85/7.68 % (24541)Termination phase: shuffling
% 12.85/7.68
% 12.85/7.68 % (24541)Memory used [KB]: 11513
% 12.85/7.68 % (24541)Time elapsed: 0.015 s
% 12.85/7.68 % (24541)Instructions burned: 22 (million)
% 12.85/7.68 % (24541)------------------------------
% 12.85/7.68 % (24541)------------------------------
% 12.85/7.68 % (24539)Instruction limit reached!
% 12.85/7.68 % (24539)------------------------------
% 12.85/7.68 % (24539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.68 % (24539)Termination reason: Unknown
% 12.85/7.68 % (24539)Termination phase: shuffling
% 12.85/7.68
% 12.85/7.68 % (24539)Memory used [KB]: 13688
% 12.85/7.68 % (24539)Time elapsed: 0.037 s
% 12.85/7.68 % (24539)Instructions burned: 101 (million)
% 12.85/7.68 % (24539)------------------------------
% 12.85/7.68 % (24539)------------------------------
% 12.85/7.69 % (24542)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_24167 for (2993ds/86Mi)
% 12.85/7.69 % (24543)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_24167 for (2993ds/107Mi)
% 12.85/7.72 % (24525)Instruction limit reached!
% 12.85/7.72 % (24525)------------------------------
% 12.85/7.72 % (24525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.72 % (24525)Termination reason: Unknown
% 12.85/7.72 % (24525)Termination phase: Property scanning
% 12.85/7.72
% 12.85/7.72 % (24525)Memory used [KB]: 20852
% 12.85/7.72 % (24525)Time elapsed: 0.260 s
% 12.85/7.72 % (24525)Instructions burned: 712 (million)
% 12.85/7.72 % (24525)------------------------------
% 12.85/7.72 % (24525)------------------------------
% 12.85/7.73 % (24542)Instruction limit reached!
% 12.85/7.73 % (24542)------------------------------
% 12.85/7.73 % (24542)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.73 % (24542)Termination reason: Unknown
% 12.85/7.73 % (24542)Termination phase: shuffling
% 12.85/7.73
% 12.85/7.73 % (24542)Memory used [KB]: 13432
% 12.85/7.73 % (24542)Time elapsed: 0.038 s
% 12.85/7.73 % (24542)Instructions burned: 88 (million)
% 12.85/7.73 % (24542)------------------------------
% 12.85/7.73 % (24542)------------------------------
% 12.85/7.73 % (24544)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_24167 for (2993ds/39Mi)
% 12.85/7.73 % (24532)Instruction limit reached!
% 12.85/7.73 % (24532)------------------------------
% 12.85/7.73 % (24532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.73 % (24532)Termination reason: Unknown
% 12.85/7.73 % (24532)Termination phase: shuffling
% 12.85/7.73
% 12.85/7.73 % (24532)Memory used [KB]: 25074
% 12.85/7.73 % (24532)Time elapsed: 0.235 s
% 12.85/7.73 % (24532)Instructions burned: 781 (million)
% 12.85/7.73 % (24532)------------------------------
% 12.85/7.73 % (24532)------------------------------
% 12.85/7.74 % (24543)Instruction limit reached!
% 12.85/7.74 % (24543)------------------------------
% 12.85/7.74 % (24543)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.74 % (24543)Termination reason: Unknown
% 12.85/7.74 % (24543)Termination phase: shuffling
% 12.85/7.74
% 12.85/7.74 % (24543)Memory used [KB]: 13688
% 12.85/7.74 % (24543)Time elapsed: 0.045 s
% 12.85/7.74 % (24543)Instructions burned: 107 (million)
% 12.85/7.74 % (24543)------------------------------
% 12.85/7.74 % (24543)------------------------------
% 12.85/7.74 % (24545)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_24167 for (2993ds/448Mi)
% 12.85/7.74 % (24516)Instruction limit reached!
% 12.85/7.74 % (24516)------------------------------
% 12.85/7.74 % (24516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.74 % (24516)Termination reason: Unknown
% 12.85/7.74 % (24516)Termination phase: shuffling
% 12.85/7.74
% 12.85/7.74 % (24516)Memory used [KB]: 32622
% 12.85/7.74 % (24516)Time elapsed: 0.349 s
% 12.85/7.74 % (24516)Instructions burned: 1041 (million)
% 12.85/7.74 % (24516)------------------------------
% 12.85/7.74 % (24516)------------------------------
% 12.85/7.75 % (24546)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_24167 for (2993ds/46Mi)
% 12.85/7.75 % (24547)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_24167 for (2993ds/98Mi)
% 12.85/7.75 % (24544)Instruction limit reached!
% 12.85/7.75 % (24544)------------------------------
% 12.85/7.75 % (24544)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.75 % (24544)Termination reason: Unknown
% 12.85/7.75 % (24544)Termination phase: shuffling
% 12.85/7.75
% 12.85/7.75 % (24544)Memory used [KB]: 12792
% 12.85/7.75 % (24544)Time elapsed: 0.020 s
% 12.85/7.75 % (24544)Instructions burned: 41 (million)
% 12.85/7.75 % (24544)------------------------------
% 12.85/7.75 % (24544)------------------------------
% 12.85/7.76 % (24548)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_24167 for (2993ds/507Mi)
% 12.85/7.76 % (24549)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_24167 for (2992ds/149Mi)
% 12.85/7.77 % (24546)Instruction limit reached!
% 12.85/7.77 % (24546)------------------------------
% 12.85/7.77 % (24546)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.77 % (24546)Termination reason: Unknown
% 12.85/7.77 % (24546)Termination phase: shuffling
% 12.85/7.77
% 12.85/7.77 % (24546)Memory used [KB]: 12792
% 12.85/7.77 % (24546)Time elapsed: 0.025 s
% 12.85/7.77 % (24546)Instructions burned: 47 (million)
% 12.85/7.77 % (24546)------------------------------
% 12.85/7.77 % (24546)------------------------------
% 12.85/7.78 % (24527)Instruction limit reached!
% 12.85/7.78 % (24527)------------------------------
% 12.85/7.78 % (24527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.78 % (24527)Termination reason: Unknown
% 12.85/7.78 % (24527)Termination phase: Property scanning
% 12.85/7.78
% 12.85/7.78 % (24527)Memory used [KB]: 20852
% 12.85/7.78 % (24527)Time elapsed: 0.308 s
% 12.85/7.78 % (24527)Instructions burned: 902 (million)
% 12.85/7.78 % (24527)------------------------------
% 12.85/7.78 % (24527)------------------------------
% 12.85/7.78 % (24550)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_24167 for (2992ds/39Mi)
% 12.85/7.78 % (24547)Instruction limit reached!
% 12.85/7.78 % (24547)------------------------------
% 12.85/7.78 % (24547)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.78 % (24547)Termination reason: Unknown
% 12.85/7.78 % (24547)Termination phase: shuffling
% 12.85/7.78
% 12.85/7.78 % (24547)Memory used [KB]: 13688
% 12.85/7.78 % (24547)Time elapsed: 0.037 s
% 12.85/7.78 % (24547)Instructions burned: 100 (million)
% 12.85/7.78 % (24547)------------------------------
% 12.85/7.78 % (24547)------------------------------
% 12.85/7.79 % (24551)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_24167 for (2992ds/6Mi)
% 12.85/7.79 % (24552)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_24167 for (2992ds/1087Mi)
% 12.85/7.80 % (24551)Instruction limit reached!
% 12.85/7.80 % (24551)------------------------------
% 12.85/7.80 % (24551)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.80 % (24551)Termination reason: Unknown
% 12.85/7.80 % (24551)Termination phase: shuffling
% 12.85/7.80
% 12.85/7.80 % (24551)Memory used [KB]: 11001
% 12.85/7.80 % (24551)Time elapsed: 0.007 s
% 12.85/7.80 % (24551)Instructions burned: 6 (million)
% 12.85/7.80 % (24551)------------------------------
% 12.85/7.80 % (24551)------------------------------
% 12.85/7.80 % (24550)Instruction limit reached!
% 12.85/7.80 % (24550)------------------------------
% 12.85/7.80 % (24550)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.80 % (24550)Termination reason: Unknown
% 12.85/7.80 % (24550)Termination phase: shuffling
% 12.85/7.80
% 12.85/7.80 % (24550)Memory used [KB]: 12665
% 12.85/7.80 % (24550)Time elapsed: 0.022 s
% 12.85/7.80 % (24550)Instructions burned: 39 (million)
% 12.85/7.80 % (24550)------------------------------
% 12.85/7.80 % (24550)------------------------------
% 12.85/7.81 % (24553)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_24167 for (2992ds/47Mi)
% 12.85/7.81 % (24554)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_24167 for (2992ds/44Mi)
% 12.85/7.82 % (24534)Instruction limit reached!
% 12.85/7.82 % (24534)------------------------------
% 12.85/7.82 % (24534)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.82 % (24534)Termination reason: Unknown
% 12.85/7.82 % (24534)Termination phase: Property scanning
% 12.85/7.82
% 12.85/7.82 % (24534)Memory used [KB]: 20724
% 12.85/7.82 % (24534)Time elapsed: 0.294 s
% 12.85/7.82 % (24534)Instructions burned: 882 (million)
% 12.85/7.82 % (24534)------------------------------
% 12.85/7.82 % (24534)------------------------------
% 12.85/7.82 % (24549)Instruction limit reached!
% 12.85/7.82 % (24549)------------------------------
% 12.85/7.82 % (24549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.82 % (24549)Termination reason: Unknown
% 12.85/7.82 % (24549)Termination phase: shuffling
% 12.85/7.82
% 12.85/7.82 % (24549)Memory used [KB]: 14200
% 12.85/7.82 % (24549)Time elapsed: 0.059 s
% 12.85/7.82 % (24549)Instructions burned: 149 (million)
% 12.85/7.82 % (24549)------------------------------
% 12.85/7.82 % (24549)------------------------------
% 12.85/7.83 % (24555)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_24167 for (2992ds/432Mi)
% 12.85/7.83 % (24556)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_24167 for (2992ds/62Mi)
% 12.85/7.83 % (24553)Instruction limit reached!
% 12.85/7.83 % (24553)------------------------------
% 12.85/7.83 % (24553)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.83 % (24553)Termination reason: Unknown
% 12.85/7.83 % (24553)Termination phase: shuffling
% 12.85/7.83
% 12.85/7.83 % (24553)Memory used [KB]: 12920
% 12.85/7.83 % (24553)Time elapsed: 0.025 s
% 12.85/7.83 % (24553)Instructions burned: 48 (million)
% 12.85/7.83 % (24553)------------------------------
% 12.85/7.83 % (24553)------------------------------
% 12.85/7.83 % (24554)Instruction limit reached!
% 12.85/7.83 % (24554)------------------------------
% 12.85/7.83 % (24554)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.83 % (24554)Termination reason: Unknown
% 12.85/7.83 % (24554)Termination phase: shuffling
% 12.85/7.83
% 12.85/7.83 % (24554)Memory used [KB]: 12792
% 12.85/7.83 % (24554)Time elapsed: 0.021 s
% 12.85/7.83 % (24554)Instructions burned: 44 (million)
% 12.85/7.83 % (24554)------------------------------
% 12.85/7.83 % (24554)------------------------------
% 12.85/7.84 % (24557)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_24167 for (2992ds/31Mi)
% 12.85/7.84 % (24558)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_24167 for (2992ds/96Mi)
% 12.85/7.86 % (24556)Instruction limit reached!
% 12.85/7.86 % (24556)------------------------------
% 12.85/7.86 % (24556)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.86 % (24556)Termination reason: Unknown
% 12.85/7.86 % (24556)Termination phase: shuffling
% 12.85/7.86
% 12.85/7.86 % (24556)Memory used [KB]: 13176
% 12.85/7.86 % (24556)Time elapsed: 0.027 s
% 12.85/7.86 % (24556)Instructions burned: 64 (million)
% 12.85/7.86 % (24556)------------------------------
% 12.85/7.86 % (24556)------------------------------
% 12.85/7.86 % (24557)Instruction limit reached!
% 12.85/7.86 % (24557)------------------------------
% 12.85/7.86 % (24557)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.86 % (24557)Termination reason: Unknown
% 12.85/7.86 % (24557)Termination phase: shuffling
% 12.85/7.86
% 12.85/7.86 % (24557)Memory used [KB]: 12665
% 12.85/7.86 % (24557)Time elapsed: 0.020 s
% 12.85/7.86 % (24557)Instructions burned: 33 (million)
% 12.85/7.86 % (24557)------------------------------
% 12.85/7.86 % (24557)------------------------------
% 12.85/7.87 % (24559)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_24167 for (2991ds/453Mi)
% 12.85/7.87 % (24560)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_24167 for (2991ds/71Mi)
% 12.85/7.88 % (24545)Instruction limit reached!
% 12.85/7.88 % (24545)------------------------------
% 12.85/7.88 % (24545)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.88 % (24545)Termination reason: Unknown
% 12.85/7.88 % (24545)Termination phase: shuffling
% 12.85/7.88
% 12.85/7.88 % (24545)Memory used [KB]: 20596
% 12.85/7.88 % (24545)Time elapsed: 0.140 s
% 12.85/7.88 % (24545)Instructions burned: 451 (million)
% 12.85/7.88 % (24545)------------------------------
% 12.85/7.88 % (24545)------------------------------
% 12.85/7.88 % (24558)Instruction limit reached!
% 12.85/7.88 % (24558)------------------------------
% 12.85/7.88 % (24558)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.88 % (24558)Termination reason: Unknown
% 12.85/7.88 % (24558)Termination phase: shuffling
% 12.85/7.88
% 12.85/7.88 % (24558)Memory used [KB]: 13560
% 12.85/7.88 % (24558)Time elapsed: 0.040 s
% 12.85/7.88 % (24558)Instructions burned: 97 (million)
% 12.85/7.88 % (24558)------------------------------
% 12.85/7.88 % (24558)------------------------------
% 12.85/7.89 % (24561)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_24167 for (2991ds/83Mi)
% 12.85/7.89 % (24562)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_24167 for (2991ds/11Mi)
% 12.85/7.90 % (24560)Instruction limit reached!
% 12.85/7.90 % (24560)------------------------------
% 12.85/7.90 % (24560)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.90 % (24560)Termination reason: Unknown
% 12.85/7.90 % (24560)Termination phase: shuffling
% 12.85/7.90
% 12.85/7.90 % (24560)Memory used [KB]: 13176
% 12.85/7.90 % (24560)Time elapsed: 0.033 s
% 12.85/7.90 % (24560)Instructions burned: 73 (million)
% 12.85/7.90 % (24560)------------------------------
% 12.85/7.90 % (24560)------------------------------
% 12.85/7.90 % (24562)Instruction limit reached!
% 12.85/7.90 % (24562)------------------------------
% 12.85/7.90 % (24562)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.90 % (24562)Termination reason: Unknown
% 12.85/7.90 % (24562)Termination phase: shuffling
% 12.85/7.90
% 12.85/7.90 % (24562)Memory used [KB]: 11385
% 12.85/7.90 % (24562)Time elapsed: 0.011 s
% 12.85/7.90 % (24562)Instructions burned: 11 (million)
% 12.85/7.90 % (24562)------------------------------
% 12.85/7.90 % (24562)------------------------------
% 12.85/7.91 % (24563)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_24167 for (2991ds/41Mi)
% 12.85/7.91 % (24564)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_24167 for (2991ds/6Mi)
% 12.85/7.92 % (24564)Instruction limit reached!
% 12.85/7.92 % (24564)------------------------------
% 12.85/7.92 % (24564)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.92 % (24564)Termination reason: Unknown
% 12.85/7.92 % (24564)Termination phase: shuffling
% 12.85/7.92
% 12.85/7.92 % (24564)Memory used [KB]: 11001
% 12.85/7.92 % (24564)Time elapsed: 0.006 s
% 12.85/7.92 % (24564)Instructions burned: 6 (million)
% 12.85/7.92 % (24564)------------------------------
% 12.85/7.92 % (24564)------------------------------
% 12.85/7.92 % (24561)Instruction limit reached!
% 12.85/7.92 % (24561)------------------------------
% 12.85/7.92 % (24561)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.92 % (24561)Termination reason: Unknown
% 12.85/7.92 % (24561)Termination phase: shuffling
% 12.85/7.92
% 12.85/7.92 % (24561)Memory used [KB]: 13304
% 12.85/7.92 % (24561)Time elapsed: 0.036 s
% 12.85/7.92 % (24561)Instructions burned: 83 (million)
% 12.85/7.92 % (24561)------------------------------
% 12.85/7.92 % (24561)------------------------------
% 12.85/7.93 % (24565)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_24167 for (2991ds/150Mi)
% 12.85/7.93 % (24563)Instruction limit reached!
% 12.85/7.93 % (24563)------------------------------
% 12.85/7.93 % (24563)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.93 % (24563)Termination reason: Unknown
% 12.85/7.93 % (24563)Termination phase: shuffling
% 12.85/7.93
% 12.85/7.93 % (24563)Memory used [KB]: 12792
% 12.85/7.93 % (24563)Time elapsed: 0.021 s
% 12.85/7.93 % (24563)Instructions burned: 43 (million)
% 12.85/7.93 % (24563)------------------------------
% 12.85/7.93 % (24563)------------------------------
% 12.85/7.93 % (24566)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_24167 for (2991ds/343Mi)
% 12.85/7.94 % (24548)Instruction limit reached!
% 12.85/7.94 % (24548)------------------------------
% 12.85/7.94 % (24548)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.94 % (24548)Termination reason: Unknown
% 12.85/7.94 % (24548)Termination phase: shuffling
% 12.85/7.94
% 12.85/7.94 % (24548)Memory used [KB]: 18677
% 12.85/7.94 % (24548)Time elapsed: 0.180 s
% 12.85/7.94 % (24548)Instructions burned: 507 (million)
% 12.85/7.94 % (24548)------------------------------
% 12.85/7.94 % (24548)------------------------------
% 12.85/7.94 % (24567)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_24167 for (2991ds/109Mi)
% 12.85/7.95 % (24568)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_24167 for (2991ds/253Mi)
% 12.85/7.96 % (24555)Instruction limit reached!
% 12.85/7.96 % (24555)------------------------------
% 12.85/7.96 % (24555)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.96 % (24555)Termination reason: Unknown
% 12.85/7.96 % (24555)Termination phase: shuffling
% 12.85/7.96
% 12.85/7.96 % (24555)Memory used [KB]: 20340
% 12.85/7.96 % (24555)Time elapsed: 0.134 s
% 12.85/7.96 % (24555)Instructions burned: 434 (million)
% 12.85/7.96 % (24555)------------------------------
% 12.85/7.96 % (24555)------------------------------
% 12.85/7.97 % (24569)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_24167 for (2990ds/140Mi)
% 12.85/7.99 % (24565)Instruction limit reached!
% 12.85/7.99 % (24565)------------------------------
% 12.85/7.99 % (24565)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.99 % (24565)Termination reason: Unknown
% 12.85/7.99 % (24565)Termination phase: shuffling
% 12.85/7.99
% 12.85/7.99 % (24565)Memory used [KB]: 14200
% 12.85/7.99 % (24565)Time elapsed: 0.059 s
% 12.85/7.99 % (24565)Instructions burned: 151 (million)
% 12.85/7.99 % (24565)------------------------------
% 12.85/7.99 % (24565)------------------------------
% 12.85/7.99 % (24567)Instruction limit reached!
% 12.85/7.99 % (24567)------------------------------
% 12.85/7.99 % (24567)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/7.99 % (24567)Termination reason: Unknown
% 12.85/7.99 % (24567)Termination phase: shuffling
% 12.85/7.99
% 12.85/7.99 % (24567)Memory used [KB]: 13688
% 12.85/7.99 % (24567)Time elapsed: 0.045 s
% 12.85/7.99 % (24567)Instructions burned: 110 (million)
% 12.85/7.99 % (24567)------------------------------
% 12.85/7.99 % (24567)------------------------------
% 12.85/8.00 % (24570)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_24167 for (2990ds/614Mi)
% 12.85/8.00 % (24571)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_24167 for (2990ds/202Mi)
% 12.85/8.00 % (24559)Instruction limit reached!
% 12.85/8.00 % (24559)------------------------------
% 12.85/8.00 % (24559)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/8.01 % (24559)Termination reason: Unknown
% 12.85/8.01 % (24559)Termination phase: shuffling
% 12.85/8.01
% 12.85/8.01 % (24559)Memory used [KB]: 20596
% 12.85/8.01 % (24559)Time elapsed: 0.140 s
% 12.85/8.01 % (24559)Instructions burned: 453 (million)
% 12.85/8.01 % (24559)------------------------------
% 12.85/8.01 % (24559)------------------------------
% 17.81/8.02 % (24572)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_24167 for (2990ds/339Mi)
% 17.81/8.04 % (24568)Instruction limit reached!
% 17.81/8.04 % (24568)------------------------------
% 17.81/8.04 % (24568)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.81/8.04 % (24568)Termination reason: Unknown
% 17.81/8.04 % (24568)Termination phase: shuffling
% 17.81/8.04
% 17.81/8.04 % (24568)Memory used [KB]: 15735
% 17.81/8.04 % (24568)Time elapsed: 0.095 s
% 17.81/8.04 % (24568)Instructions burned: 255 (million)
% 17.81/8.04 % (24568)------------------------------
% 17.81/8.04 % (24568)------------------------------
% 17.81/8.05 % (24573)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_24167 for (2990ds/283Mi)
% 17.81/8.06 % (24566)Instruction limit reached!
% 17.81/8.06 % (24566)------------------------------
% 17.81/8.06 % (24566)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.81/8.06 % (24566)Termination reason: Unknown
% 17.81/8.06 % (24566)Termination phase: shuffling
% 17.81/8.06
% 17.81/8.06 % (24566)Memory used [KB]: 16758
% 17.81/8.06 % (24566)Time elapsed: 0.124 s
% 17.81/8.06 % (24566)Instructions burned: 345 (million)
% 17.81/8.06 % (24566)------------------------------
% 17.81/8.06 % (24566)------------------------------
% 17.81/8.06 % (24569)Instruction limit reached!
% 17.81/8.06 % (24569)------------------------------
% 17.81/8.06 % (24569)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.81/8.06 % (24569)Termination reason: Unknown
% 17.81/8.06 % (24569)Termination phase: shuffling
% 17.81/8.06
% 17.81/8.06 % (24569)Memory used [KB]: 14072
% 17.81/8.06 % (24569)Time elapsed: 0.084 s
% 17.81/8.06 % (24569)Instructions burned: 141 (million)
% 17.81/8.06 % (24569)------------------------------
% 17.81/8.06 % (24569)------------------------------
% 17.81/8.07 % (24574)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_24167 for (2989ds/134Mi)
% 17.81/8.07 % (24571)Instruction limit reached!
% 17.81/8.07 % (24571)------------------------------
% 17.81/8.07 % (24571)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.81/8.07 % (24571)Termination reason: Unknown
% 17.81/8.07 % (24571)Termination phase: shuffling
% 17.81/8.07
% 17.81/8.07 % (24571)Memory used [KB]: 14967
% 17.81/8.07 % (24571)Time elapsed: 0.077 s
% 17.81/8.07 % (24571)Instructions burned: 202 (million)
% 17.81/8.07 % (24571)------------------------------
% 17.81/8.07 % (24571)------------------------------
% 17.81/8.08 % (24575)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_24167 for (2989ds/298Mi)
% 17.81/8.09 % (24576)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_24167 for (2989ds/192Mi)
% 18.23/8.12 % (24574)Instruction limit reached!
% 18.23/8.12 % (24574)------------------------------
% 18.23/8.12 % (24574)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.23/8.12 % (24574)Termination reason: Unknown
% 18.23/8.12 % (24574)Termination phase: shuffling
% 18.23/8.12
% 18.23/8.12 % (24574)Memory used [KB]: 14072
% 18.23/8.12 % (24574)Time elapsed: 0.055 s
% 18.23/8.12 % (24574)Instructions burned: 135 (million)
% 18.23/8.12 % (24574)------------------------------
% 18.23/8.12 % (24574)------------------------------
% 18.23/8.13 % (24577)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_24167 for (2989ds/258Mi)
% 18.23/8.14 % (24573)Instruction limit reached!
% 18.23/8.14 % (24573)------------------------------
% 18.23/8.14 % (24573)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.23/8.14 % (24573)Termination reason: Unknown
% 18.23/8.14 % (24573)Termination phase: shuffling
% 18.23/8.14
% 18.23/8.14 % (24573)Memory used [KB]: 18293
% 18.23/8.14 % (24573)Time elapsed: 0.090 s
% 18.23/8.14 % (24573)Instructions burned: 284 (million)
% 18.23/8.14 % (24573)------------------------------
% 18.23/8.14 % (24573)------------------------------
% 18.23/8.14 % (24572)Instruction limit reached!
% 18.23/8.14 % (24572)------------------------------
% 18.23/8.14 % (24572)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.23/8.14 % (24572)Termination reason: Unknown
% 18.23/8.14 % (24572)Termination phase: shuffling
% 18.23/8.14
% 18.23/8.14 % (24572)Memory used [KB]: 16758
% 18.23/8.14 % (24572)Time elapsed: 0.125 s
% 18.23/8.14 % (24572)Instructions burned: 341 (million)
% 18.23/8.14 % (24572)------------------------------
% 18.23/8.14 % (24572)------------------------------
% 18.23/8.15 % (24552)Instruction limit reached!
% 18.23/8.15 % (24552)------------------------------
% 18.23/8.15 % (24552)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.23/8.15 % (24552)Termination reason: Unknown
% 18.23/8.15 % (24552)Termination phase: Property scanning
% 18.23/8.15
% 18.23/8.15 % (24552)Memory used [KB]: 20852
% 18.23/8.15 % (24552)Time elapsed: 0.353 s
% 18.23/8.15 % (24552)Instructions burned: 1090 (million)
% 18.23/8.15 % (24552)------------------------------
% 18.23/8.15 % (24552)------------------------------
% 18.23/8.15 % (24579)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_24167 for (2989ds/231Mi)
% 18.23/8.15 % (24578)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_24167 for (2989ds/90Mi)
% 18.23/8.16 % (24580)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_24167 for (2989ds/458Mi)
% 18.23/8.16 % (24576)Instruction limit reached!
% 18.23/8.16 % (24576)------------------------------
% 18.23/8.16 % (24576)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.23/8.16 % (24576)Termination reason: Unknown
% 18.23/8.16 % (24576)Termination phase: shuffling
% 18.23/8.16
% 18.23/8.16 % (24576)Memory used [KB]: 14839
% 18.23/8.16 % (24576)Time elapsed: 0.073 s
% 18.23/8.16 % (24576)Instructions burned: 192 (million)
% 18.23/8.16 % (24576)------------------------------
% 18.23/8.16 % (24576)------------------------------
% 18.23/8.17 % (24581)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_24167 for (2988ds/31Mi)
% 18.23/8.19 % (24575)Instruction limit reached!
% 18.23/8.19 % (24575)------------------------------
% 18.23/8.19 % (24575)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.23/8.19 % (24575)Termination reason: Unknown
% 18.23/8.19 % (24575)Termination phase: shuffling
% 18.23/8.19
% 18.23/8.19 % (24575)Memory used [KB]: 18549
% 18.23/8.19 % (24575)Time elapsed: 0.110 s
% 18.23/8.19 % (24575)Instructions burned: 300 (million)
% 18.23/8.19 % (24575)------------------------------
% 18.23/8.19 % (24575)------------------------------
% 18.23/8.19 % (24581)Instruction limit reached!
% 18.23/8.19 % (24581)------------------------------
% 18.23/8.19 % (24581)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.23/8.19 % (24581)Termination reason: Unknown
% 18.23/8.19 % (24581)Termination phase: shuffling
% 18.23/8.19
% 18.23/8.19 % (24581)Memory used [KB]: 12665
% 18.23/8.19 % (24581)Time elapsed: 0.020 s
% 18.23/8.19 % (24581)Instructions burned: 34 (million)
% 18.23/8.19 % (24581)------------------------------
% 18.23/8.19 % (24581)------------------------------
% 18.23/8.19 % (24578)Instruction limit reached!
% 18.23/8.19 % (24578)------------------------------
% 18.23/8.19 % (24578)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.23/8.19 % (24578)Termination reason: Unknown
% 18.23/8.19 % (24578)Termination phase: shuffling
% 18.23/8.19
% 18.23/8.19 % (24578)Memory used [KB]: 13432
% 18.23/8.19 % (24578)Time elapsed: 0.038 s
% 18.23/8.19 % (24578)Instructions burned: 90 (million)
% 18.23/8.19 % (24578)------------------------------
% 18.23/8.19 % (24578)------------------------------
% 18.94/8.20 % (24582)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_24167 for (2988ds/297Mi)
% 18.94/8.20 % (24583)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_24167 for (2988ds/987Mi)
% 18.94/8.20 % (24584)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_24167 for (2988ds/182Mi)
% 18.94/8.22 % (24570)Instruction limit reached!
% 18.94/8.22 % (24570)------------------------------
% 18.94/8.22 % (24570)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.94/8.22 % (24570)Termination reason: Unknown
% 18.94/8.22 % (24570)Termination phase: shuffling
% 18.94/8.22
% 18.94/8.22 % (24570)Memory used [KB]: 20212
% 18.94/8.22 % (24570)Time elapsed: 0.219 s
% 18.94/8.22 % (24570)Instructions burned: 615 (million)
% 18.94/8.22 % (24570)------------------------------
% 18.94/8.22 % (24570)------------------------------
% 18.94/8.23 % (24579)Instruction limit reached!
% 18.94/8.23 % (24579)------------------------------
% 18.94/8.23 % (24579)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.94/8.23 % (24579)Termination reason: Unknown
% 18.94/8.23 % (24579)Termination phase: shuffling
% 18.94/8.23
% 18.94/8.23 % (24579)Memory used [KB]: 15607
% 18.94/8.23 % (24579)Time elapsed: 0.076 s
% 18.94/8.23 % (24579)Instructions burned: 233 (million)
% 18.94/8.23 % (24579)------------------------------
% 18.94/8.23 % (24579)------------------------------
% 18.94/8.23 % (24577)Instruction limit reached!
% 18.94/8.23 % (24577)------------------------------
% 18.94/8.23 % (24577)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.94/8.23 % (24577)Termination reason: Unknown
% 18.94/8.23 % (24577)Termination phase: shuffling
% 18.94/8.23
% 18.94/8.23 % (24577)Memory used [KB]: 15735
% 18.94/8.23 % (24577)Time elapsed: 0.098 s
% 18.94/8.23 % (24577)Instructions burned: 259 (million)
% 18.94/8.23 % (24577)------------------------------
% 18.94/8.23 % (24577)------------------------------
% 18.94/8.23 % (24585)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_24167 for (2988ds/182Mi)
% 18.94/8.24 % (24586)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_24167 for (2988ds/154Mi)
% 18.94/8.24 % (24587)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_24167 for (2988ds/79Mi)
% 18.94/8.27 % (24584)Instruction limit reached!
% 18.94/8.27 % (24584)------------------------------
% 18.94/8.27 % (24584)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.94/8.27 % (24584)Termination reason: Unknown
% 18.94/8.27 % (24584)Termination phase: shuffling
% 18.94/8.27
% 18.94/8.27 % (24584)Memory used [KB]: 14711
% 18.94/8.27 % (24584)Time elapsed: 0.072 s
% 18.94/8.27 % (24584)Instructions burned: 184 (million)
% 18.94/8.27 % (24584)------------------------------
% 18.94/8.27 % (24584)------------------------------
% 18.94/8.27 % (24587)Instruction limit reached!
% 18.94/8.27 % (24587)------------------------------
% 18.94/8.27 % (24587)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.94/8.27 % (24587)Termination reason: Unknown
% 18.94/8.27 % (24587)Termination phase: shuffling
% 18.94/8.27
% 18.94/8.27 % (24587)Memory used [KB]: 13304
% 18.94/8.27 % (24587)Time elapsed: 0.036 s
% 18.94/8.27 % (24587)Instructions burned: 80 (million)
% 18.94/8.27 % (24587)------------------------------
% 18.94/8.27 % (24587)------------------------------
% 18.94/8.28 % (24588)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_24167 for (2987ds/216Mi)
% 18.94/8.29 % (24589)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_24167 for (2987ds/114Mi)
% 18.94/8.29 % (24586)Instruction limit reached!
% 18.94/8.29 % (24586)------------------------------
% 18.94/8.29 % (24586)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.94/8.29 % (24586)Termination reason: Unknown
% 18.94/8.29 % (24586)Termination phase: shuffling
% 18.94/8.29
% 18.94/8.29 % (24586)Memory used [KB]: 14456
% 18.94/8.29 % (24586)Time elapsed: 0.054 s
% 18.94/8.29 % (24586)Instructions burned: 157 (million)
% 18.94/8.29 % (24586)------------------------------
% 18.94/8.29 % (24586)------------------------------
% 18.94/8.29 % (24582)Instruction limit reached!
% 18.94/8.29 % (24582)------------------------------
% 18.94/8.29 % (24582)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.94/8.29 % (24582)Termination reason: Unknown
% 18.94/8.29 % (24582)Termination phase: shuffling
% 18.94/8.29
% 18.94/8.29 % (24582)Memory used [KB]: 18549
% 18.94/8.29 % (24582)Time elapsed: 0.097 s
% 18.94/8.29 % (24582)Instructions burned: 299 (million)
% 18.94/8.29 % (24582)------------------------------
% 18.94/8.29 % (24582)------------------------------
% 19.37/8.30 % (24585)Instruction limit reached!
% 19.37/8.30 % (24585)------------------------------
% 19.37/8.30 % (24585)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.37/8.30 % (24585)Termination reason: Unknown
% 19.37/8.30 % (24585)Termination phase: shuffling
% 19.37/8.30
% 19.37/8.30 % (24585)Memory used [KB]: 14711
% 19.37/8.30 % (24585)Time elapsed: 0.072 s
% 19.37/8.30 % (24585)Instructions burned: 184 (million)
% 19.37/8.30 % (24585)------------------------------
% 19.37/8.30 % (24585)------------------------------
% 19.37/8.30 % (24590)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_24167 for (2987ds/922Mi)
% 19.37/8.31 % (24592)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_24167 for (2987ds/164Mi)
% 19.37/8.31 % (24591)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_24167 for (2987ds/182Mi)
% 19.37/8.32 % (24580)Instruction limit reached!
% 19.37/8.32 % (24580)------------------------------
% 19.37/8.32 % (24580)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.37/8.32 % (24580)Termination reason: Unknown
% 19.37/8.32 % (24580)Termination phase: shuffling
% 19.37/8.32
% 19.37/8.32 % (24580)Memory used [KB]: 18293
% 19.37/8.32 % (24580)Time elapsed: 0.167 s
% 19.37/8.32 % (24580)Instructions burned: 460 (million)
% 19.37/8.32 % (24580)------------------------------
% 19.37/8.32 % (24580)------------------------------
% 19.37/8.33 % (24589)Instruction limit reached!
% 19.37/8.33 % (24589)------------------------------
% 19.37/8.33 % (24589)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.37/8.33 % (24589)Termination reason: Unknown
% 19.37/8.33 % (24589)Termination phase: shuffling
% 19.37/8.33
% 19.37/8.33 % (24589)Memory used [KB]: 13816
% 19.37/8.33 % (24589)Time elapsed: 0.047 s
% 19.37/8.33 % (24589)Instructions burned: 115 (million)
% 19.37/8.33 % (24589)------------------------------
% 19.37/8.33 % (24589)------------------------------
% 19.37/8.33 % (24593)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_24167 for (2987ds/50Mi)
% 19.37/8.34 % (24594)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_24167 for (2987ds/368Mi)
% 19.37/8.35 % (24588)Instruction limit reached!
% 19.37/8.35 % (24588)------------------------------
% 19.37/8.35 % (24588)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.37/8.35 % (24588)Termination reason: Unknown
% 19.37/8.35 % (24588)Termination phase: shuffling
% 19.37/8.35
% 19.37/8.35 % (24588)Memory used [KB]: 15351
% 19.37/8.35 % (24588)Time elapsed: 0.070 s
% 19.37/8.35 % (24588)Instructions burned: 217 (million)
% 19.37/8.35 % (24588)------------------------------
% 19.37/8.35 % (24588)------------------------------
% 19.37/8.36 % (24593)Instruction limit reached!
% 19.37/8.36 % (24593)------------------------------
% 19.37/8.36 % (24593)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.37/8.36 % (24593)Termination reason: Unknown
% 19.37/8.36 % (24593)Termination phase: shuffling
% 19.37/8.36
% 19.37/8.36 % (24593)Memory used [KB]: 12920
% 19.37/8.36 % (24593)Time elapsed: 0.023 s
% 19.37/8.36 % (24593)Instructions burned: 50 (million)
% 19.37/8.36 % (24593)------------------------------
% 19.37/8.36 % (24593)------------------------------
% 19.37/8.36 % (24595)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_24167 for (2986ds/1021Mi)
% 19.37/8.37 % (24592)Instruction limit reached!
% 19.37/8.37 % (24592)------------------------------
% 19.37/8.37 % (24592)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.37/8.37 % (24592)Termination reason: Unknown
% 19.37/8.37 % (24592)Termination phase: shuffling
% 19.37/8.37
% 19.37/8.37 % (24592)Memory used [KB]: 14583
% 19.37/8.37 % (24592)Time elapsed: 0.056 s
% 19.37/8.37 % (24592)Instructions burned: 166 (million)
% 19.37/8.37 % (24592)------------------------------
% 19.37/8.37 % (24592)------------------------------
% 19.37/8.37 % (24596)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_24167 for (2986ds/267Mi)
% 19.37/8.38 % (24597)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_24167 for (2986ds/204Mi)
% 19.97/8.40 % (24591)Instruction limit reached!
% 19.97/8.40 % (24591)------------------------------
% 19.97/8.40 % (24591)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.97/8.40 % (24591)Termination reason: Unknown
% 19.97/8.40 % (24591)Termination phase: shuffling
% 19.97/8.40
% 19.97/8.40 % (24591)Memory used [KB]: 14839
% 19.97/8.40 % (24591)Time elapsed: 0.086 s
% 19.97/8.40 % (24591)Instructions burned: 182 (million)
% 19.97/8.40 % (24591)------------------------------
% 19.97/8.40 % (24591)------------------------------
% 19.97/8.41 % (24598)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_24167 for (2986ds/26Mi)
% 19.97/8.43 % (24598)Instruction limit reached!
% 19.97/8.43 % (24598)------------------------------
% 19.97/8.43 % (24598)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.97/8.43 % (24598)Termination reason: Unknown
% 19.97/8.43 % (24598)Termination phase: shuffling
% 19.97/8.43
% 19.97/8.43 % (24598)Memory used [KB]: 12537
% 19.97/8.43 % (24598)Time elapsed: 0.017 s
% 19.97/8.43 % (24598)Instructions burned: 28 (million)
% 19.97/8.43 % (24598)------------------------------
% 19.97/8.43 % (24598)------------------------------
% 19.97/8.44 % (24599)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_24167 for (2986ds/53Mi)
% 19.97/8.45 % (24596)Instruction limit reached!
% 19.97/8.45 % (24596)------------------------------
% 19.97/8.45 % (24596)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.97/8.45 % (24596)Termination reason: Unknown
% 19.97/8.45 % (24596)Termination phase: shuffling
% 19.97/8.45
% 19.97/8.45 % (24596)Memory used [KB]: 16119
% 19.97/8.45 % (24596)Time elapsed: 0.085 s
% 19.97/8.45 % (24596)Instructions burned: 270 (million)
% 19.97/8.45 % (24596)------------------------------
% 19.97/8.45 % (24596)------------------------------
% 19.97/8.45 % (24597)Instruction limit reached!
% 19.97/8.45 % (24597)------------------------------
% 19.97/8.45 % (24597)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.97/8.45 % (24597)Termination reason: Unknown
% 19.97/8.45 % (24597)Termination phase: shuffling
% 19.97/8.45
% 19.97/8.45 % (24597)Memory used [KB]: 14967
% 19.97/8.45 % (24597)Time elapsed: 0.077 s
% 19.97/8.45 % (24597)Instructions burned: 205 (million)
% 19.97/8.45 % (24597)------------------------------
% 19.97/8.45 % (24597)------------------------------
% 19.97/8.46 % (24599)Instruction limit reached!
% 19.97/8.46 % (24599)------------------------------
% 19.97/8.46 % (24599)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.97/8.46 % (24599)Termination reason: Unknown
% 19.97/8.46 % (24599)Termination phase: shuffling
% 19.97/8.46
% 19.97/8.46 % (24599)Memory used [KB]: 12920
% 19.97/8.46 % (24599)Time elapsed: 0.024 s
% 19.97/8.46 % (24599)Instructions burned: 53 (million)
% 19.97/8.46 % (24599)------------------------------
% 19.97/8.46 % (24599)------------------------------
% 19.97/8.46 % (24600)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_24167 for (2985ds/4436Mi)
% 19.97/8.46 % (24601)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_24167 for (2985ds/1539Mi)
% 19.97/8.47 % (24602)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_24167 for (2985ds/408Mi)
% 19.97/8.47 % (24594)Instruction limit reached!
% 19.97/8.47 % (24594)------------------------------
% 19.97/8.47 % (24594)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.97/8.47 % (24594)Termination reason: Unknown
% 19.97/8.47 % (24594)Termination phase: shuffling
% 19.97/8.47
% 19.97/8.47 % (24594)Memory used [KB]: 17142
% 19.97/8.47 % (24594)Time elapsed: 0.134 s
% 19.97/8.47 % (24594)Instructions burned: 370 (million)
% 19.97/8.47 % (24594)------------------------------
% 19.97/8.47 % (24594)------------------------------
% 19.97/8.49 % (24603)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_24167 for (2985ds/5205Mi)
% 20.45/8.50 % (24583)Instruction limit reached!
% 20.45/8.50 % (24583)------------------------------
% 20.45/8.50 % (24583)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.45/8.50 % (24583)Termination reason: Unknown
% 20.45/8.50 % (24583)Termination phase: shuffling
% 20.45/8.50
% 20.45/8.50 % (24583)Memory used [KB]: 31854
% 20.45/8.50 % (24583)Time elapsed: 0.299 s
% 20.45/8.50 % (24583)Instructions burned: 987 (million)
% 20.45/8.50 % (24583)------------------------------
% 20.45/8.50 % (24583)------------------------------
% 20.45/8.51 % (24604)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_24167 for (2985ds/1913Mi)
% 20.77/8.60 % (24590)Instruction limit reached!
% 20.77/8.60 % (24590)------------------------------
% 20.77/8.60 % (24590)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.77/8.60 % (24590)Termination reason: Unknown
% 20.77/8.60 % (24590)Termination phase: Property scanning
% 20.77/8.61
% 20.77/8.61 % (24590)Memory used [KB]: 20852
% 20.77/8.61 % (24590)Time elapsed: 0.304 s
% 20.77/8.61 % (24590)Instructions burned: 922 (million)
% 20.77/8.61 % (24590)------------------------------
% 20.77/8.61 % (24590)------------------------------
% 20.77/8.62 % (24602)Instruction limit reached!
% 20.77/8.62 % (24602)------------------------------
% 20.77/8.62 % (24602)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.77/8.62 % (24602)Termination reason: Unknown
% 20.77/8.62 % (24605)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_24167 for (2984ds/4350Mi)
% 20.77/8.62 % (24602)Termination phase: shuffling
% 20.77/8.62
% 20.77/8.62 % (24602)Memory used [KB]: 17782
% 20.77/8.62 % (24602)Time elapsed: 0.149 s
% 20.77/8.62 % (24602)Instructions burned: 410 (million)
% 20.77/8.62 % (24602)------------------------------
% 20.77/8.62 % (24602)------------------------------
% 20.77/8.63 % (24606)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_24167 for (2984ds/68Mi)
% 20.77/8.66 % (24606)Instruction limit reached!
% 20.77/8.66 % (24606)------------------------------
% 20.77/8.66 % (24606)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.77/8.66 % (24606)Termination reason: Unknown
% 20.77/8.66 % (24606)Termination phase: shuffling
% 20.77/8.66
% 20.77/8.66 % (24606)Memory used [KB]: 13176
% 20.77/8.66 % (24606)Time elapsed: 0.031 s
% 20.77/8.66 % (24606)Instructions burned: 70 (million)
% 20.77/8.66 % (24606)------------------------------
% 20.77/8.66 % (24606)------------------------------
% 20.77/8.67 % (24607)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_24167 for (2983ds/358Mi)
% 20.77/8.69 % (24595)Instruction limit reached!
% 20.77/8.69 % (24595)------------------------------
% 20.77/8.69 % (24595)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.77/8.69 % (24595)Termination reason: Unknown
% 20.77/8.69 % (24595)Termination phase: Property scanning
% 20.77/8.69
% 20.77/8.69 % (24595)Memory used [KB]: 20852
% 20.77/8.69 % (24595)Time elapsed: 0.331 s
% 20.77/8.69 % (24595)Instructions burned: 1021 (million)
% 20.77/8.69 % (24595)------------------------------
% 20.77/8.69 % (24595)------------------------------
% 21.65/8.71 % (24608)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_24167 for (2983ds/31Mi)
% 21.65/8.72 % (24608)Instruction limit reached!
% 21.65/8.72 % (24608)------------------------------
% 21.65/8.72 % (24608)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.65/8.72 % (24608)Termination reason: Unknown
% 21.65/8.72 % (24608)Termination phase: shuffling
% 21.65/8.72
% 21.65/8.72 % (24608)Memory used [KB]: 12665
% 21.65/8.72 % (24608)Time elapsed: 0.019 s
% 21.65/8.72 % (24608)Instructions burned: 32 (million)
% 21.65/8.72 % (24608)------------------------------
% 21.65/8.72 % (24608)------------------------------
% 21.65/8.73 % (24609)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_24167 for (2983ds/279Mi)
% 21.67/8.80 % (24607)Instruction limit reached!
% 21.67/8.80 % (24607)------------------------------
% 21.67/8.80 % (24607)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.67/8.80 % (24607)Termination reason: Unknown
% 21.67/8.80 % (24607)Termination phase: shuffling
% 21.67/8.80
% 21.67/8.80 % (24607)Memory used [KB]: 17014
% 21.67/8.80 % (24607)Time elapsed: 0.131 s
% 21.67/8.80 % (24607)Instructions burned: 359 (million)
% 21.67/8.80 % (24607)------------------------------
% 21.67/8.80 % (24607)------------------------------
% 21.67/8.81 % (24610)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_24167 for (2982ds/258Mi)
% 21.67/8.82 % (24609)Instruction limit reached!
% 21.67/8.82 % (24609)------------------------------
% 21.67/8.82 % (24609)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.67/8.82 % (24609)Termination reason: Unknown
% 21.67/8.82 % (24609)Termination phase: shuffling
% 21.67/8.82
% 21.67/8.82 % (24609)Memory used [KB]: 16247
% 21.67/8.82 % (24609)Time elapsed: 0.089 s
% 21.67/8.82 % (24609)Instructions burned: 282 (million)
% 21.67/8.82 % (24609)------------------------------
% 21.67/8.82 % (24609)------------------------------
% 21.67/8.83 % (24611)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_24167 for (2982ds/26Mi)
% 21.67/8.85 % (24611)Instruction limit reached!
% 21.67/8.85 % (24611)------------------------------
% 21.67/8.85 % (24611)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.67/8.85 % (24611)Termination reason: Unknown
% 21.67/8.85 % (24611)Termination phase: shuffling
% 21.67/8.85
% 21.67/8.85 % (24611)Memory used [KB]: 12537
% 21.67/8.85 % (24611)Time elapsed: 0.017 s
% 21.67/8.85 % (24611)Instructions burned: 27 (million)
% 21.67/8.85 % (24611)------------------------------
% 21.67/8.85 % (24611)------------------------------
% 21.67/8.86 % (24612)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_24167 for (2981ds/995Mi)
% 21.93/8.91 % (24610)Instruction limit reached!
% 21.93/8.91 % (24610)------------------------------
% 21.93/8.91 % (24610)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.93/8.91 % (24610)Termination reason: Unknown
% 21.93/8.91 % (24610)Termination phase: shuffling
% 21.93/8.91
% 21.93/8.91 % (24610)Memory used [KB]: 15735
% 21.93/8.91 % (24610)Time elapsed: 0.096 s
% 21.93/8.91 % (24610)Instructions burned: 260 (million)
% 21.93/8.91 % (24610)------------------------------
% 21.93/8.91 % (24610)------------------------------
% 21.93/8.92 % (24613)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_24167 for (2981ds/373Mi)
% 21.93/8.96 % (24601)Instruction limit reached!
% 21.93/8.96 % (24601)------------------------------
% 21.93/8.96 % (24601)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.93/8.96 % (24601)Termination reason: Unknown
% 21.93/8.96 % (24601)Termination phase: Preprocessing 2
% 21.93/8.96
% 21.93/8.96 % (24601)Memory used [KB]: 26481
% 21.93/8.96 % (24601)Time elapsed: 0.495 s
% 21.93/8.96 % (24601)Instructions burned: 1540 (million)
% 21.93/8.96 % (24601)------------------------------
% 21.93/8.96 % (24601)------------------------------
% 21.93/8.97 % (24614)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_24167 for (2980ds/253Mi)
% 24.79/9.05 % (24614)Instruction limit reached!
% 24.79/9.05 % (24614)------------------------------
% 24.79/9.05 % (24614)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.79/9.05 % (24614)Termination reason: Unknown
% 24.79/9.05 % (24614)Termination phase: shuffling
% 24.79/9.05
% 24.79/9.05 % (24614)Memory used [KB]: 15863
% 24.79/9.05 % (24614)Time elapsed: 0.081 s
% 24.79/9.05 % (24614)Instructions burned: 253 (million)
% 24.79/9.05 % (24614)------------------------------
% 24.79/9.05 % (24614)------------------------------
% 24.79/9.06 % (24615)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_24167 for (2979ds/832Mi)
% 24.79/9.10 % (24613)Instruction limit reached!
% 24.79/9.10 % (24613)------------------------------
% 24.79/9.10 % (24613)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.79/9.10 % (24613)Termination reason: Unknown
% 24.79/9.10 % (24613)Termination phase: shuffling
% 24.79/9.10
% 24.79/9.10 % (24613)Memory used [KB]: 17270
% 24.79/9.10 % (24613)Time elapsed: 0.183 s
% 24.79/9.10 % (24613)Instructions burned: 373 (million)
% 24.79/9.10 % (24613)------------------------------
% 24.79/9.10 % (24613)------------------------------
% 25.45/9.11 % (24616)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_24167 for (2979ds/668Mi)
% 25.45/9.12 % (24604)Instruction limit reached!
% 25.45/9.12 % (24604)------------------------------
% 25.45/9.12 % (24604)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.45/9.12 % (24604)Termination reason: Unknown
% 25.45/9.12 % (24604)Termination phase: Naming
% 25.45/9.12
% 25.45/9.12 % (24604)Memory used [KB]: 26481
% 25.45/9.12 % (24604)Time elapsed: 0.613 s
% 25.45/9.12 % (24604)Instructions burned: 1913 (million)
% 25.45/9.12 % (24604)------------------------------
% 25.45/9.12 % (24604)------------------------------
% 25.45/9.13 % (24617)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_24167 for (2979ds/40Mi)
% 25.45/9.15 % (24617)Instruction limit reached!
% 25.45/9.15 % (24617)------------------------------
% 25.45/9.15 % (24617)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.45/9.15 % (24617)Termination reason: Unknown
% 25.45/9.15 % (24617)Termination phase: shuffling
% 25.45/9.15
% 25.45/9.15 % (24617)Memory used [KB]: 12792
% 25.45/9.15 % (24617)Time elapsed: 0.020 s
% 25.45/9.15 % (24617)Instructions burned: 41 (million)
% 25.45/9.15 % (24617)------------------------------
% 25.45/9.15 % (24617)------------------------------
% 25.45/9.16 % (24612)Instruction limit reached!
% 25.45/9.16 % (24612)------------------------------
% 25.45/9.16 % (24612)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.45/9.16 % (24612)Termination reason: Unknown
% 25.45/9.16 % (24612)Termination phase: shuffling
% 25.45/9.16
% 25.45/9.16 % (24612)Memory used [KB]: 32110
% 25.45/9.16 % (24612)Time elapsed: 0.300 s
% 25.45/9.16 % (24612)Instructions burned: 996 (million)
% 25.45/9.16 % (24612)------------------------------
% 25.45/9.16 % (24612)------------------------------
% 25.45/9.16 % (24618)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_24167 for (2978ds/294Mi)
% 25.45/9.17 % (24619)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_24167 for (2978ds/187Mi)
% 25.83/9.24 % (24619)Instruction limit reached!
% 25.83/9.24 % (24619)------------------------------
% 25.83/9.24 % (24619)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.83/9.24 % (24619)Termination reason: Unknown
% 25.83/9.24 % (24619)Termination phase: shuffling
% 25.83/9.24
% 25.83/9.24 % (24619)Memory used [KB]: 14839
% 25.83/9.24 % (24619)Time elapsed: 0.070 s
% 25.83/9.24 % (24619)Instructions burned: 189 (million)
% 25.83/9.24 % (24619)------------------------------
% 25.83/9.24 % (24619)------------------------------
% 25.83/9.25 % (24620)lrs+1010_2:3_au=on:fsr=off:sd=4:ss=axioms:st=1.5:i=371:si=on:rtra=on_0 on DTF2THF_24167 for (2978ds/371Mi)
% 25.83/9.25 % (24618)Instruction limit reached!
% 25.83/9.25 % (24618)------------------------------
% 25.83/9.25 % (24618)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.83/9.25 % (24618)Termination reason: Unknown
% 25.83/9.25 % (24618)Termination phase: shuffling
% 25.83/9.25
% 25.83/9.25 % (24618)Memory used [KB]: 18421
% 25.83/9.25 % (24618)Time elapsed: 0.091 s
% 25.83/9.25 % (24618)Instructions burned: 296 (million)
% 25.83/9.25 % (24618)------------------------------
% 25.83/9.25 % (24618)------------------------------
% 25.83/9.26 % (24621)dis+1003_1:1_au=on:bd=preordered:fde=none:ixr=off:nwc=10.0:i=1827:si=on:rtra=on_0 on DTF2THF_24167 for (2977ds/1827Mi)
% 26.31/9.31 % (24616)Instruction limit reached!
% 26.31/9.31 % (24616)------------------------------
% 26.31/9.31 % (24616)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.31/9.31 % (24616)Termination reason: Unknown
% 26.31/9.31 % (24616)Termination phase: shuffling
% 26.31/9.31
% 26.31/9.31 % (24616)Memory used [KB]: 23539
% 26.31/9.31 % (24616)Time elapsed: 0.197 s
% 26.31/9.31 % (24616)Instructions burned: 669 (million)
% 26.31/9.31 % (24616)------------------------------
% 26.31/9.31 % (24616)------------------------------
% 26.31/9.32 % (24622)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_24167 for (2977ds/70Mi)
% 26.31/9.34 % (24615)Instruction limit reached!
% 26.31/9.34 % (24615)------------------------------
% 26.31/9.34 % (24615)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.31/9.34 % (24615)Termination reason: Unknown
% 26.31/9.34 % (24615)Termination phase: Property scanning
% 26.31/9.34
% 26.31/9.34 % (24615)Memory used [KB]: 20852
% 26.31/9.34 % (24615)Time elapsed: 0.279 s
% 26.31/9.34 % (24615)Instructions burned: 834 (million)
% 26.31/9.34 % (24615)------------------------------
% 26.31/9.34 % (24615)------------------------------
% 26.31/9.35 % (24622)Instruction limit reached!
% 26.31/9.35 % (24622)------------------------------
% 26.31/9.35 % (24622)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.31/9.35 % (24622)Termination reason: Unknown
% 26.31/9.35 % (24622)Termination phase: shuffling
% 26.31/9.35
% 26.31/9.35 % (24622)Memory used [KB]: 13176
% 26.31/9.35 % (24622)Time elapsed: 0.028 s
% 26.31/9.35 % (24622)Instructions burned: 70 (million)
% 26.31/9.35 % (24622)------------------------------
% 26.31/9.35 % (24622)------------------------------
% 26.31/9.35 % (24623)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_24167 for (2977ds/316Mi)
% 26.31/9.36 % (24624)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_24167 for (2977ds/932Mi)
% 26.31/9.38 % (24620)Instruction limit reached!
% 26.31/9.38 % (24620)------------------------------
% 26.31/9.38 % (24620)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.31/9.38 % (24620)Termination reason: Unknown
% 26.31/9.38 % (24620)Termination phase: shuffling
% 26.31/9.38
% 26.31/9.38 % (24620)Memory used [KB]: 17142
% 26.31/9.38 % (24620)Time elapsed: 0.132 s
% 26.31/9.38 % (24620)Instructions burned: 371 (million)
% 26.31/9.38 % (24620)------------------------------
% 26.31/9.38 % (24620)------------------------------
% 26.31/9.40 % (24625)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_24167 for (2976ds/4714Mi)
% 27.30/9.45 % (24623)Instruction limit reached!
% 27.30/9.45 % (24623)------------------------------
% 27.30/9.45 % (24623)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.30/9.45 % (24623)Termination reason: Unknown
% 27.30/9.45 % (24623)Termination phase: shuffling
% 27.30/9.45
% 27.30/9.45 % (24623)Memory used [KB]: 18805
% 27.30/9.45 % (24623)Time elapsed: 0.098 s
% 27.30/9.45 % (24623)Instructions burned: 318 (million)
% 27.30/9.45 % (24623)------------------------------
% 27.30/9.45 % (24623)------------------------------
% 27.30/9.46 % (24626)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_24167 for (2975ds/8250Mi)
% 28.00/9.64 % (24624)Instruction limit reached!
% 28.00/9.64 % (24624)------------------------------
% 28.00/9.64 % (24624)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.00/9.64 % (24624)Termination reason: Unknown
% 28.00/9.64 % (24624)Termination phase: shuffling
% 28.00/9.64
% 28.00/9.64 % (24624)Memory used [KB]: 31214
% 28.00/9.64 % (24624)Time elapsed: 0.281 s
% 28.00/9.64 % (24624)Instructions burned: 932 (million)
% 28.00/9.64 % (24624)------------------------------
% 28.00/9.64 % (24624)------------------------------
% 28.00/9.65 % (24627)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_24167 for (2973ds/770Mi)
% 28.90/9.83 % (24621)Instruction limit reached!
% 28.90/9.83 % (24621)------------------------------
% 28.90/9.83 % (24621)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.90/9.83 % (24621)Termination reason: Unknown
% 28.90/9.83 % (24621)Termination phase: Property scanning
% 28.90/9.83
% 28.90/9.83 % (24621)Memory used [KB]: 30958
% 28.90/9.83 % (24621)Time elapsed: 0.569 s
% 28.90/9.83 % (24621)Instructions burned: 1828 (million)
% 28.90/9.83 % (24621)------------------------------
% 28.90/9.83 % (24621)------------------------------
% 28.90/9.83 % (24603)Instruction limit reached!
% 28.90/9.83 % (24603)------------------------------
% 28.90/9.83 % (24603)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.90/9.83 % (24603)Termination reason: Unknown
% 28.90/9.83 % (24603)Termination phase: Function definition elimination
% 28.90/9.83
% 28.90/9.83 % (24603)Memory used [KB]: 45415
% 28.90/9.83 % (24603)Time elapsed: 1.350 s
% 28.90/9.83 % (24603)Instructions burned: 5205 (million)
% 28.90/9.83 % (24603)------------------------------
% 28.90/9.83 % (24603)------------------------------
% 28.90/9.85 % (24629)dis+1002_1:1_au=on:avsq=on:avsqr=1,16:cond=on:sp=frequency:i=464:si=on:rtra=on_0 on DTF2THF_24167 for (2972ds/464Mi)
% 28.90/9.85 % (24628)ott+1010_5:1_av=off:fe=off:ins=1:kws=precedence:sp=reverse_arity:i=794:si=on:rtra=on_0 on DTF2THF_24167 for (2972ds/794Mi)
% 29.74/9.91 % (24600)Instruction limit reached!
% 29.74/9.91 % (24600)------------------------------
% 29.74/9.91 % (24600)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.74/9.91 % (24600)Termination reason: Unknown
% 29.74/9.91 % (24600)Termination phase: Saturation
% 29.74/9.91
% 29.74/9.91 % (24600)Memory used [KB]: 33133
% 29.74/9.91 % (24600)Time elapsed: 1.451 s
% 29.74/9.91 % (24600)Instructions burned: 4437 (million)
% 29.74/9.91 % (24600)------------------------------
% 29.74/9.91 % (24600)------------------------------
% 29.74/9.91 % (24627)Instruction limit reached!
% 29.74/9.91 % (24627)------------------------------
% 29.74/9.91 % (24627)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.74/9.91 % (24627)Termination reason: Unknown
% 29.74/9.91 % (24627)Termination phase: shuffling
% 29.74/9.91
% 29.74/9.91 % (24627)Memory used [KB]: 23922
% 29.74/9.91 % (24627)Time elapsed: 0.263 s
% 29.74/9.91 % (24627)Instructions burned: 771 (million)
% 29.74/9.91 % (24627)------------------------------
% 29.74/9.91 % (24627)------------------------------
% 29.74/9.93 % (24630)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_24167 for (2971ds/1376Mi)
% 29.74/9.93 % (24631)dis+10_1:1_etr=on:plsq=on:plsqc=1:plsqr=32,1:i=906:si=on:rtra=on_0 on DTF2THF_24167 for (2971ds/906Mi)
% 29.74/9.97 % (24605)Instruction limit reached!
% 29.74/9.97 % (24605)------------------------------
% 29.74/9.97 % (24605)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.74/9.97 % (24605)Termination reason: Unknown
% 29.74/9.97 % (24605)Termination phase: Saturation
% 29.74/9.97
% 29.74/9.97 % (24605)Memory used [KB]: 30831
% 29.74/9.97 % (24605)Time elapsed: 1.349 s
% 29.74/9.97 % (24605)Instructions burned: 4352 (million)
% 29.74/9.97 % (24605)------------------------------
% 29.74/9.97 % (24605)------------------------------
% 29.74/9.98 % (24632)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_24167 for (2970ds/434Mi)
% 31.81/10.01 % (24629)Instruction limit reached!
% 31.81/10.01 % (24629)------------------------------
% 31.81/10.01 % (24629)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.81/10.01 % (24629)Termination reason: Unknown
% 31.81/10.01 % (24629)Termination phase: shuffling
% 31.81/10.01
% 31.81/10.01 % (24629)Memory used [KB]: 18421
% 31.81/10.01 % (24629)Time elapsed: 0.167 s
% 31.81/10.01 % (24629)Instructions burned: 466 (million)
% 31.81/10.01 % (24629)------------------------------
% 31.81/10.01 % (24629)------------------------------
% 31.81/10.03 % (24633)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_24167 for (2970ds/865Mi)
% 32.45/10.13 % (24632)Instruction limit reached!
% 32.45/10.13 % (24632)------------------------------
% 32.45/10.13 % (24632)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.45/10.13 % (24632)Termination reason: Unknown
% 32.45/10.13 % (24632)Termination phase: shuffling
% 32.45/10.13
% 32.45/10.13 % (24632)Memory used [KB]: 18166
% 32.45/10.13 % (24632)Time elapsed: 0.155 s
% 32.45/10.13 % (24632)Instructions burned: 434 (million)
% 32.45/10.13 % (24632)------------------------------
% 32.45/10.13 % (24632)------------------------------
% 32.45/10.15 % (24634)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_24167 for (2969ds/593Mi)
% 32.45/10.16 % (24628)Instruction limit reached!
% 32.45/10.16 % (24628)------------------------------
% 32.45/10.16 % (24628)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.45/10.16 % (24628)Termination reason: Unknown
% 32.45/10.16 % (24628)Termination phase: Property scanning
% 32.45/10.16
% 32.45/10.16 % (24628)Memory used [KB]: 20724
% 32.45/10.16 % (24628)Time elapsed: 0.310 s
% 32.45/10.16 % (24628)Instructions burned: 797 (million)
% 32.45/10.16 % (24628)------------------------------
% 32.45/10.16 % (24628)------------------------------
% 32.45/10.17 % (24635)dis+10_1:1_sd=2:sgt=20:slsq=on:ss=axioms:st=6.0:i=443:si=on:rtra=on_0 on DTF2THF_24167 for (2968ds/443Mi)
% 32.89/10.22 % (24631)Instruction limit reached!
% 32.89/10.22 % (24631)------------------------------
% 32.89/10.22 % (24631)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.89/10.22 % (24631)Termination reason: Unknown
% 32.89/10.22 % (24631)Termination phase: Property scanning
% 32.89/10.22
% 32.89/10.22 % (24631)Memory used [KB]: 20724
% 32.89/10.22 % (24631)Time elapsed: 0.297 s
% 32.89/10.22 % (24631)Instructions burned: 908 (million)
% 32.89/10.22 % (24631)------------------------------
% 32.89/10.22 % (24631)------------------------------
% 32.89/10.23 % (24636)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_24167 for (2968ds/594Mi)
% 32.89/10.28 % (24633)Instruction limit reached!
% 32.89/10.28 % (24633)------------------------------
% 32.89/10.28 % (24633)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.89/10.28 % (24633)Termination reason: Unknown
% 32.89/10.28 % (24633)Termination phase: shuffling
% 32.89/10.28
% 32.89/10.28 % (24633)Memory used [KB]: 26225
% 32.89/10.28 % (24633)Time elapsed: 0.255 s
% 32.89/10.28 % (24633)Instructions burned: 866 (million)
% 32.89/10.28 % (24633)------------------------------
% 32.89/10.28 % (24633)------------------------------
% 32.89/10.29 % (24637)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_24167 for (2967ds/327Mi)
% 33.48/10.32 % (24634)Instruction limit reached!
% 33.48/10.32 % (24634)------------------------------
% 33.48/10.32 % (24634)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.48/10.32 % (24634)Termination reason: Unknown
% 33.48/10.32 % (24634)Termination phase: shuffling
% 33.48/10.32
% 33.48/10.32 % (24634)Memory used [KB]: 22643
% 33.48/10.32 % (24634)Time elapsed: 0.179 s
% 33.48/10.32 % (24634)Instructions burned: 594 (million)
% 33.48/10.32 % (24634)------------------------------
% 33.48/10.32 % (24634)------------------------------
% 33.48/10.33 % (24630)Instruction limit reached!
% 33.48/10.33 % (24630)------------------------------
% 33.48/10.33 % (24630)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.48/10.33 % (24630)Termination reason: Unknown
% 33.48/10.33 % (24630)Termination phase: shuffling
% 33.48/10.33
% 33.48/10.33 % (24630)Memory used [KB]: 37099
% 33.48/10.33 % (24630)Time elapsed: 0.405 s
% 33.48/10.33 % (24630)Instructions burned: 1376 (million)
% 33.48/10.33 % (24630)------------------------------
% 33.48/10.33 % (24630)------------------------------
% 33.48/10.33 % (24635)Instruction limit reached!
% 33.48/10.33 % (24635)------------------------------
% 33.48/10.33 % (24635)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.48/10.33 % (24635)Termination reason: Unknown
% 33.48/10.33 % (24635)Termination phase: shuffling
% 33.48/10.33
% 33.48/10.33 % (24635)Memory used [KB]: 18038
% 33.48/10.33 % (24635)Time elapsed: 0.158 s
% 33.48/10.33 % (24635)Instructions burned: 445 (million)
% 33.48/10.33 % (24635)------------------------------
% 33.48/10.33 % (24635)------------------------------
% 33.48/10.34 % (24638)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_24167 for (2967ds/5411Mi)
% 33.48/10.34 % (24639)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_24167 for (2967ds/1762Mi)
% 33.48/10.34 % (24640)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_24167 for (2967ds/96Mi)
% 33.48/10.38 % (24640)Instruction limit reached!
% 33.48/10.38 % (24640)------------------------------
% 33.48/10.38 % (24640)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.48/10.38 % (24640)Termination reason: Unknown
% 33.48/10.38 % (24640)Termination phase: shuffling
% 33.48/10.38
% 33.48/10.38 % (24640)Memory used [KB]: 13560
% 33.48/10.38 % (24640)Time elapsed: 0.035 s
% 33.48/10.38 % (24640)Instructions burned: 96 (million)
% 33.48/10.38 % (24640)------------------------------
% 33.48/10.38 % (24640)------------------------------
% 33.48/10.39 % (24641)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_24167 for (2966ds/503Mi)
% 34.14/10.41 % (24637)Instruction limit reached!
% 34.14/10.41 % (24637)------------------------------
% 34.14/10.41 % (24637)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.14/10.41 % (24637)Termination reason: Unknown
% 34.14/10.41 % (24637)Termination phase: shuffling
% 34.14/10.41
% 34.14/10.41 % (24637)Memory used [KB]: 16758
% 34.14/10.41 % (24637)Time elapsed: 0.120 s
% 34.14/10.41 % (24637)Instructions burned: 329 (million)
% 34.14/10.41 % (24637)------------------------------
% 34.14/10.41 % (24637)------------------------------
% 34.14/10.42 % (24642)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_24167 for (2966ds/90Mi)
% 34.14/10.44 % (24636)Instruction limit reached!
% 34.14/10.44 % (24636)------------------------------
% 34.14/10.44 % (24636)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.14/10.44 % (24636)Termination reason: Unknown
% 34.14/10.44 % (24636)Termination phase: shuffling
% 34.14/10.44
% 34.14/10.44 % (24636)Memory used [KB]: 19957
% 34.14/10.44 % (24636)Time elapsed: 0.208 s
% 34.14/10.44 % (24636)Instructions burned: 596 (million)
% 34.14/10.44 % (24636)------------------------------
% 34.14/10.44 % (24636)------------------------------
% 34.14/10.45 % (24643)lrs+10_1:64_hud=5:plsq=on:plsqr=2011171,524288:i=593:si=on:rtra=on_0 on DTF2THF_24167 for (2966ds/593Mi)
% 34.14/10.46 % (24642)Instruction limit reached!
% 34.14/10.46 % (24642)------------------------------
% 34.14/10.46 % (24642)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.14/10.46 % (24642)Termination reason: Unknown
% 34.14/10.46 % (24642)Termination phase: shuffling
% 34.14/10.46
% 34.14/10.46 % (24642)Memory used [KB]: 13432
% 34.14/10.46 % (24642)Time elapsed: 0.039 s
% 34.14/10.46 % (24642)Instructions burned: 91 (million)
% 34.14/10.46 % (24642)------------------------------
% 34.14/10.46 % (24642)------------------------------
% 34.14/10.47 % (24644)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_24167 for (2965ds/1512Mi)
% 34.74/10.56 % (24641)Instruction limit reached!
% 34.74/10.56 % (24641)------------------------------
% 34.74/10.56 % (24641)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.74/10.56 % (24641)Termination reason: Unknown
% 34.74/10.56 % (24641)Termination phase: shuffling
% 34.74/10.56
% 34.74/10.56 % (24641)Memory used [KB]: 18805
% 34.74/10.56 % (24641)Time elapsed: 0.176 s
% 34.74/10.56 % (24641)Instructions burned: 503 (million)
% 34.74/10.56 % (24641)------------------------------
% 34.74/10.56 % (24641)------------------------------
% 34.74/10.58 % (24645)lrs+1002_1:128_cbe=off:cnfonf=lazy_not_gen_be_off:ins=1:i=569:si=on:rtra=on_0 on DTF2THF_24167 for (2964ds/569Mi)
% 35.12/10.60 % (24625)Instruction limit reached!
% 35.12/10.60 % (24625)------------------------------
% 35.12/10.60 % (24625)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.12/10.60 % (24625)Termination reason: Unknown
% 35.12/10.60 % (24625)Termination phase: Function definition elimination
% 35.12/10.60
% 35.12/10.60 % (24625)Memory used [KB]: 40553
% 35.12/10.60 % (24625)Time elapsed: 1.209 s
% 35.12/10.60 % (24625)Instructions burned: 4714 (million)
% 35.12/10.60 % (24625)------------------------------
% 35.12/10.60 % (24625)------------------------------
% 35.12/10.62 % (24646)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_24167 for (2964ds/501Mi)
% 35.12/10.66 % (24643)Instruction limit reached!
% 35.12/10.66 % (24643)------------------------------
% 35.12/10.66 % (24643)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.12/10.66 % (24643)Termination reason: Unknown
% 35.12/10.66 % (24643)Termination phase: shuffling
% 35.12/10.66
% 35.12/10.66 % (24643)Memory used [KB]: 19957
% 35.12/10.66 % (24643)Time elapsed: 0.207 s
% 35.12/10.66 % (24643)Instructions burned: 594 (million)
% 35.12/10.66 % (24643)------------------------------
% 35.12/10.66 % (24643)------------------------------
% 35.12/10.67 % (24647)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_24167 for (2963ds/562Mi)
% 35.45/10.74 % (24645)Instruction limit reached!
% 35.45/10.74 % (24645)------------------------------
% 35.45/10.74 % (24645)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.45/10.74 % (24645)Termination reason: Unknown
% 35.45/10.74 % (24645)Termination phase: shuffling
% 35.45/10.74
% 35.45/10.74 % (24645)Memory used [KB]: 22259
% 35.45/10.74 % (24645)Time elapsed: 0.169 s
% 35.45/10.74 % (24645)Instructions burned: 572 (million)
% 35.45/10.74 % (24645)------------------------------
% 35.45/10.74 % (24645)------------------------------
% 35.45/10.76 % (24648)lrs+10_1:1_au=on:cnfonf=off:fde=unused:kws=precedence:tgt=full:i=406:si=on:rtra=on_0 on DTF2THF_24167 for (2963ds/406Mi)
% 35.45/10.79 % (24646)Instruction limit reached!
% 35.45/10.79 % (24646)------------------------------
% 35.45/10.79 % (24646)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.45/10.79 % (24646)Termination reason: Unknown
% 35.45/10.79 % (24646)Termination phase: shuffling
% 35.45/10.79
% 35.45/10.79 % (24646)Memory used [KB]: 18677
% 35.45/10.79 % (24646)Time elapsed: 0.177 s
% 35.45/10.79 % (24646)Instructions burned: 501 (million)
% 35.45/10.79 % (24646)------------------------------
% 35.45/10.79 % (24646)------------------------------
% 35.93/10.80 % (24649)dis+1004_1:128_au=on:cbe=off:e2e=on:sac=on:i=582:si=on:rtra=on_0 on DTF2THF_24167 for (2962ds/582Mi)
% 35.93/10.84 % (24647)Instruction limit reached!
% 35.93/10.84 % (24647)------------------------------
% 35.93/10.84 % (24647)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.93/10.84 % (24647)Termination reason: Unknown
% 35.93/10.84 % (24647)Termination phase: shuffling
% 35.93/10.84
% 35.93/10.84 % (24647)Memory used [KB]: 22131
% 35.93/10.84 % (24647)Time elapsed: 0.171 s
% 35.93/10.84 % (24647)Instructions burned: 564 (million)
% 35.93/10.84 % (24647)------------------------------
% 35.93/10.84 % (24647)------------------------------
% 35.93/10.84 % (24639)Instruction limit reached!
% 35.93/10.84 % (24639)------------------------------
% 35.93/10.84 % (24639)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.93/10.84 % (24639)Termination reason: Unknown
% 35.93/10.84 % (24639)Termination phase: Property scanning
% 35.93/10.84
% 35.93/10.84 % (24639)Memory used [KB]: 37483
% 35.93/10.84 % (24639)Time elapsed: 0.504 s
% 35.93/10.84 % (24639)Instructions burned: 1765 (million)
% 35.93/10.84 % (24639)------------------------------
% 35.93/10.84 % (24639)------------------------------
% 35.93/10.85 % (24650)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_24167 for (2962ds/456Mi)
% 35.93/10.86 % (24651)lrs+1010_1:1_au=on:cbe=off:cnfonf=conj_eager:tgt=ground:i=393:si=on:rtra=on_0 on DTF2THF_24167 for (2962ds/393Mi)
% 36.74/10.90 % (24648)Instruction limit reached!
% 36.74/10.90 % (24648)------------------------------
% 36.74/10.90 % (24648)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 36.74/10.90 % (24648)Termination reason: Unknown
% 36.74/10.90 % (24648)Termination phase: shuffling
% 36.74/10.90
% 36.74/10.90 % (24648)Memory used [KB]: 17398
% 36.74/10.90 % (24648)Time elapsed: 0.145 s
% 36.74/10.90 % (24648)Instructions burned: 407 (million)
% 36.74/10.90 % (24648)------------------------------
% 36.74/10.90 % (24648)------------------------------
% 36.74/10.91 % (24652)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_24167 for (2961ds/1335Mi)
% 36.74/10.95 % (24644)Instruction limit reached!
% 36.74/10.95 % (24644)------------------------------
% 36.74/10.95 % (24644)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 36.74/10.95 % (24644)Termination reason: Unknown
% 36.74/10.95 % (24644)Termination phase: SInE selection
% 36.74/10.95
% 36.74/10.95 % (24644)Memory used [KB]: 25841
% 36.74/10.95 % (24644)Time elapsed: 0.481 s
% 36.74/10.95 % (24644)Instructions burned: 1512 (million)
% 36.74/10.95 % (24644)------------------------------
% 36.74/10.95 % (24644)------------------------------
% 36.74/10.96 % (24653)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_24167 for (2960ds/3373Mi)
% 36.74/10.98 % (24651)Instruction limit reached!
% 36.74/10.98 % (24651)------------------------------
% 36.74/10.98 % (24651)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 36.74/10.98 % (24651)Termination reason: Unknown
% 36.74/10.98 % (24651)Termination phase: shuffling
% 36.74/10.98
% 36.74/10.98 % (24651)Memory used [KB]: 19829
% 36.74/10.98 % (24651)Time elapsed: 0.121 s
% 36.74/10.98 % (24651)Instructions burned: 396 (million)
% 36.74/10.98 % (24651)------------------------------
% 36.74/10.98 % (24651)------------------------------
% 36.74/10.99 % (24654)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_24167 for (2960ds/474Mi)
% 36.74/11.01 % (24649)Instruction limit reached!
% 36.74/11.01 % (24649)------------------------------
% 36.74/11.01 % (24649)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 36.74/11.01 % (24649)Termination reason: Unknown
% 36.74/11.01 % (24649)Termination phase: shuffling
% 36.74/11.01
% 36.74/11.01 % (24649)Memory used [KB]: 20084
% 36.74/11.01 % (24649)Time elapsed: 0.210 s
% 36.74/11.01 % (24649)Instructions burned: 583 (million)
% 36.74/11.01 % (24650)Instruction limit reached!
% 36.74/11.01 % (24650)------------------------------
% 36.74/11.01 % (24650)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 36.74/11.01 % (24649)------------------------------
% 36.74/11.01 % (24649)------------------------------
% 36.74/11.01 % (24650)Termination reason: Unknown
% 36.74/11.01 % (24650)Termination phase: shuffling
% 36.74/11.01
% 36.74/11.01 % (24650)Memory used [KB]: 18293
% 36.74/11.01 % (24650)Time elapsed: 0.162 s
% 36.74/11.01 % (24650)Instructions burned: 458 (million)
% 36.74/11.01 % (24650)------------------------------
% 36.74/11.01 % (24650)------------------------------
% 38.92/11.03 % (24655)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_24167 for (2960ds/157Mi)
% 38.92/11.03 % (24656)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_24167 for (2960ds/590Mi)
% 38.92/11.08 % (24655)Instruction limit reached!
% 38.92/11.08 % (24655)------------------------------
% 38.92/11.08 % (24655)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 38.92/11.08 % (24655)Termination reason: Unknown
% 38.92/11.08 % (24655)Termination phase: shuffling
% 38.92/11.08
% 38.92/11.08 % (24655)Memory used [KB]: 14456
% 38.92/11.08 % (24655)Time elapsed: 0.052 s
% 38.92/11.08 % (24655)Instructions burned: 157 (million)
% 38.92/11.08 % (24655)------------------------------
% 38.92/11.08 % (24655)------------------------------
% 38.92/11.09 % (24657)dis+10_1:1024_kws=precedence:prag=on:rawr=on:sos=all:sp=unary_first:i=104:si=on:rtra=on_0 on DTF2THF_24167 for (2959ds/104Mi)
% 39.52/11.13 % (24657)Instruction limit reached!
% 39.52/11.13 % (24657)------------------------------
% 39.52/11.13 % (24657)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.52/11.13 % (24657)Termination reason: Unknown
% 39.52/11.13 % (24657)Termination phase: shuffling
% 39.52/11.13
% 39.52/11.13 % (24657)Memory used [KB]: 13688
% 39.52/11.13 % (24657)Time elapsed: 0.042 s
% 39.52/11.13 % (24657)Instructions burned: 104 (million)
% 39.52/11.13 % (24657)------------------------------
% 39.52/11.13 % (24657)------------------------------
% 39.52/11.13 % (24654)Instruction limit reached!
% 39.52/11.13 % (24654)------------------------------
% 39.52/11.13 % (24654)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.52/11.13 % (24654)Termination reason: Unknown
% 39.52/11.13 % (24654)Termination phase: shuffling
% 39.52/11.13
% 39.52/11.13 % (24654)Memory used [KB]: 20980
% 39.52/11.13 % (24654)Time elapsed: 0.141 s
% 39.52/11.13 % (24654)Instructions burned: 474 (million)
% 39.52/11.13 % (24654)------------------------------
% 39.52/11.13 % (24654)------------------------------
% 39.52/11.14 % (24658)lrs+1010_1:10_au=on:bd=off:cnfonf=lazy_pi_sigma_gen:sp=occurrence:i=414:si=on:rtra=on_0 on DTF2THF_24167 for (2959ds/414Mi)
% 39.52/11.14 % (24659)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_24167 for (2959ds/715Mi)
% 39.86/11.23 % (24656)Instruction limit reached!
% 39.86/11.23 % (24656)------------------------------
% 39.86/11.23 % (24656)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.86/11.23 % (24656)Termination reason: Unknown
% 39.86/11.23 % (24656)Termination phase: shuffling
% 39.86/11.23
% 39.86/11.23 % (24656)Memory used [KB]: 19957
% 39.86/11.23 % (24656)Time elapsed: 0.206 s
% 39.86/11.23 % (24656)Instructions burned: 592 (million)
% 39.86/11.23 % (24656)------------------------------
% 39.86/11.23 % (24656)------------------------------
% 39.86/11.24 % (24660)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_24167 for (2958ds/961Mi)
% 39.86/11.26 % (24658)Instruction limit reached!
% 39.86/11.26 % (24658)------------------------------
% 39.86/11.26 % (24658)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.86/11.26 % (24658)Termination reason: Unknown
% 39.86/11.26 % (24658)Termination phase: shuffling
% 39.86/11.26
% 39.86/11.26 % (24658)Memory used [KB]: 20084
% 39.86/11.26 % (24658)Time elapsed: 0.127 s
% 39.86/11.26 % (24658)Instructions burned: 416 (million)
% 39.86/11.26 % (24658)------------------------------
% 39.86/11.26 % (24658)------------------------------
% 39.86/11.28 % (24661)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_24167 for (2957ds/1213Mi)
% 40.62/11.30 % (24652)Instruction limit reached!
% 40.62/11.30 % (24652)------------------------------
% 40.62/11.30 % (24652)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.62/11.30 % (24652)Termination reason: Unknown
% 40.62/11.30 % (24652)Termination phase: shuffling
% 40.62/11.30
% 40.62/11.30 % (24652)Memory used [KB]: 36587
% 40.62/11.30 % (24652)Time elapsed: 0.393 s
% 40.62/11.30 % (24652)Instructions burned: 1336 (million)
% 40.62/11.30 % (24652)------------------------------
% 40.62/11.30 % (24652)------------------------------
% 40.62/11.32 % (24662)lrs+1002_1:12_bd=off:hud=1:ntd=on:piset=equals:prag=on:i=1979:si=on:rtra=on_0 on DTF2THF_24167 for (2957ds/1979Mi)
% 40.62/11.39 % (24659)Instruction limit reached!
% 40.62/11.39 % (24659)------------------------------
% 40.62/11.39 % (24659)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.62/11.39 % (24659)Termination reason: Unknown
% 40.62/11.39 % (24659)Termination phase: Property scanning
% 40.62/11.39
% 40.62/11.39 % (24659)Memory used [KB]: 20724
% 40.62/11.39 % (24659)Time elapsed: 0.246 s
% 40.62/11.39 % (24659)Instructions burned: 717 (million)
% 40.62/11.39 % (24659)------------------------------
% 40.62/11.39 % (24659)------------------------------
% 41.20/11.40 % (24663)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_24167 for (2956ds/481Mi)
% 41.50/11.54 % (24663)Instruction limit reached!
% 41.50/11.54 % (24663)------------------------------
% 41.50/11.54 % (24663)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.50/11.54 % (24663)Termination reason: Unknown
% 41.50/11.54 % (24663)Termination phase: shuffling
% 41.50/11.54
% 41.50/11.54 % (24663)Memory used [KB]: 20980
% 41.50/11.54 % (24663)Time elapsed: 0.143 s
% 41.50/11.54 % (24663)Instructions burned: 483 (million)
% 41.50/11.54 % (24663)------------------------------
% 41.50/11.54 % (24663)------------------------------
% 41.50/11.55 % (24664)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_24167 for (2955ds/783Mi)
% 41.50/11.55 % (24660)Instruction limit reached!
% 41.50/11.55 % (24660)------------------------------
% 41.50/11.55 % (24660)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.50/11.55 % (24660)Termination reason: Unknown
% 41.50/11.55 % (24660)Termination phase: Property scanning
% 41.50/11.55
% 41.50/11.55 % (24660)Memory used [KB]: 20724
% 41.50/11.55 % (24660)Time elapsed: 0.311 s
% 41.50/11.55 % (24660)Instructions burned: 963 (million)
% 41.50/11.55 % (24660)------------------------------
% 41.50/11.55 % (24660)------------------------------
% 41.50/11.56 % (24665)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_24167 for (2954ds/432Mi)
% 42.28/11.66 % (24661)Instruction limit reached!
% 42.28/11.66 % (24661)------------------------------
% 42.28/11.66 % (24661)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.28/11.66 % (24661)Termination reason: Unknown
% 42.28/11.66 % (24661)Termination phase: Preprocessing 1
% 42.28/11.66
% 42.28/11.66 % (24661)Memory used [KB]: 21875
% 42.28/11.66 % (24661)Time elapsed: 0.383 s
% 42.28/11.66 % (24661)Instructions burned: 1214 (million)
% 42.28/11.66 % (24661)------------------------------
% 42.28/11.66 % (24661)------------------------------
% 42.28/11.67 % (24666)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_24167 for (2953ds/1670Mi)
% 42.97/11.72 % (24665)Instruction limit reached!
% 42.97/11.72 % (24665)------------------------------
% 42.97/11.72 % (24665)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.97/11.72 % (24665)Termination reason: Unknown
% 42.97/11.72 % (24665)Termination phase: shuffling
% 42.97/11.72
% 42.97/11.72 % (24665)Memory used [KB]: 17910
% 42.97/11.72 % (24665)Time elapsed: 0.153 s
% 42.97/11.72 % (24665)Instructions burned: 434 (million)
% 42.97/11.72 % (24665)------------------------------
% 42.97/11.72 % (24665)------------------------------
% 42.97/11.73 % (24667)dis+2_1:1_au=on:hfaw=0:hfsq=on:sp=weighted_frequency:i=90:si=on:rtra=on_0 on DTF2THF_24167 for (2953ds/90Mi)
% 42.97/11.76 % (24667)Instruction limit reached!
% 42.97/11.76 % (24667)------------------------------
% 42.97/11.76 % (24667)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.97/11.76 % (24667)Termination reason: Unknown
% 42.97/11.76 % (24667)Termination phase: shuffling
% 42.97/11.76
% 42.97/11.76 % (24667)Memory used [KB]: 13432
% 42.97/11.76 % (24667)Time elapsed: 0.038 s
% 42.97/11.76 % (24667)Instructions burned: 90 (million)
% 42.97/11.76 % (24667)------------------------------
% 42.97/11.76 % (24667)------------------------------
% 42.97/11.78 % (24668)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_24167 for (2952ds/346Mi)
% 43.48/11.82 % (24664)Instruction limit reached!
% 43.48/11.82 % (24664)------------------------------
% 43.48/11.82 % (24664)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.48/11.82 % (24664)Termination reason: Unknown
% 43.48/11.82 % (24664)Termination phase: shuffling
% 43.48/11.82
% 43.48/11.82 % (24664)Memory used [KB]: 22131
% 43.48/11.82 % (24664)Time elapsed: 0.270 s
% 43.48/11.82 % (24664)Instructions burned: 783 (million)
% 43.48/11.82 % (24664)------------------------------
% 43.48/11.82 % (24664)------------------------------
% 43.48/11.83 % (24669)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_24167 for (2952ds/401Mi)
% 43.48/11.87 % (24653)Instruction limit reached!
% 43.48/11.87 % (24653)------------------------------
% 43.48/11.87 % (24653)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.48/11.87 % (24653)Termination reason: Unknown
% 43.48/11.87 % (24653)Termination phase: Function definition elimination
% 43.48/11.87
% 43.48/11.87 % (24653)Memory used [KB]: 32366
% 43.48/11.87 % (24653)Time elapsed: 0.907 s
% 43.48/11.87 % (24653)Instructions burned: 3374 (million)
% 43.48/11.87 % (24653)------------------------------
% 43.48/11.87 % (24653)------------------------------
% 43.48/11.88 % (24668)Instruction limit reached!
% 43.48/11.88 % (24668)------------------------------
% 43.48/11.88 % (24668)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.48/11.88 % (24668)Termination reason: Unknown
% 43.48/11.88 % (24668)Termination phase: shuffling
% 43.48/11.88
% 43.48/11.88 % (24668)Memory used [KB]: 19189
% 43.48/11.88 % (24668)Time elapsed: 0.106 s
% 43.48/11.88 % (24668)Instructions burned: 349 (million)
% 43.48/11.88 % (24668)------------------------------
% 43.48/11.88 % (24668)------------------------------
% 43.48/11.89 % (24670)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=6454:si=on:rtra=on_0 on DTF2THF_24167 for (2951ds/6454Mi)
% 43.48/11.89 % (24671)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_24167 for (2951ds/283Mi)
% 44.17/11.93 % (24662)Instruction limit reached!
% 44.17/11.93 % (24662)------------------------------
% 44.17/11.93 % (24662)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 44.17/11.93 % (24662)Termination reason: Unknown
% 44.17/11.93 % (24662)Termination phase: Property scanning
% 44.17/11.93
% 44.17/11.93 % (24662)Memory used [KB]: 30958
% 44.17/11.93 % (24662)Time elapsed: 0.610 s
% 44.17/11.93 % (24662)Instructions burned: 1980 (million)
% 44.17/11.93 % (24662)------------------------------
% 44.17/11.93 % (24662)------------------------------
% 44.17/11.94 % (24672)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_24167 for (2951ds/865Mi)
% 44.17/11.95 % (24669)Instruction limit reached!
% 44.17/11.95 % (24669)------------------------------
% 44.17/11.95 % (24669)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 44.17/11.95 % (24669)Termination reason: Unknown
% 44.17/11.95 % (24669)Termination phase: shuffling
% 44.17/11.95
% 44.17/11.95 % (24669)Memory used [KB]: 19829
% 44.17/11.95 % (24669)Time elapsed: 0.122 s
% 44.17/11.95 % (24669)Instructions burned: 402 (million)
% 44.17/11.95 % (24669)------------------------------
% 44.17/11.95 % (24669)------------------------------
% 44.17/11.96 % (24673)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_24167 for (2950ds/4528Mi)
% 44.17/11.99 % (24671)Instruction limit reached!
% 44.17/11.99 % (24671)------------------------------
% 44.17/11.99 % (24671)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 44.17/11.99 % (24671)Termination reason: Unknown
% 44.17/11.99 % (24671)Termination phase: shuffling
% 44.17/11.99
% 44.17/11.99 % (24671)Memory used [KB]: 15991
% 44.17/11.99 % (24671)Time elapsed: 0.103 s
% 44.17/11.99 % (24671)Instructions burned: 285 (million)
% 44.17/11.99 % (24671)------------------------------
% 44.17/11.99 % (24671)------------------------------
% 44.17/12.01 % (24638)Instruction limit reached!
% 44.17/12.01 % (24638)------------------------------
% 44.17/12.01 % (24638)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 44.17/12.01 % (24638)Termination reason: Unknown
% 44.17/12.01 % (24638)Termination phase: Saturation
% 44.17/12.01
% 44.17/12.01 % (24638)Memory used [KB]: 36204
% 44.17/12.01 % (24638)Time elapsed: 1.671 s
% 44.17/12.01 % (24638)Instructions burned: 5412 (million)
% 44.17/12.01 % (24638)------------------------------
% 44.17/12.01 % (24638)------------------------------
% 44.17/12.01 % (24674)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_24167 for (2950ds/1001Mi)
% 44.17/12.02 % (24675)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=102:si=on:rtra=on_0 on DTF2THF_24167 for (2950ds/102Mi)
% 45.90/12.06 % (24675)Instruction limit reached!
% 45.90/12.06 % (24675)------------------------------
% 45.90/12.06 % (24675)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 45.90/12.06 % (24675)Termination reason: Unknown
% 45.90/12.06 % (24675)Termination phase: shuffling
% 45.90/12.06
% 45.90/12.06 % (24675)Memory used [KB]: 13688
% 45.90/12.06 % (24675)Time elapsed: 0.039 s
% 45.90/12.06 % (24675)Instructions burned: 103 (million)
% 45.90/12.06 % (24675)------------------------------
% 45.90/12.06 % (24675)------------------------------
% 45.90/12.07 % (24676)ott+1010_3:1_nm=20:s2a=on:s2at=3.0:sp=const_frequency:i=952:si=on:rtra=on_0 on DTF2THF_24167 for (2949ds/952Mi)
% 45.90/12.10 % (24626)Instruction limit reached!
% 45.90/12.10 % (24626)------------------------------
% 45.90/12.10 % (24626)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 45.90/12.10 % (24626)Termination reason: Unknown
% 45.90/12.10 % (24626)Termination phase: Saturation
% 45.90/12.10
% 45.90/12.10 % (24626)Memory used [KB]: 51427
% 45.90/12.10 % (24626)Time elapsed: 2.637 s
% 45.90/12.10 % (24626)Instructions burned: 8250 (million)
% 45.90/12.10 % (24626)------------------------------
% 45.90/12.10 % (24626)------------------------------
% 45.90/12.11 % (24677)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_24167 for (2949ds/578Mi)
% 46.54/12.15 % (24666)Instruction limit reached!
% 46.54/12.15 % (24666)------------------------------
% 46.54/12.15 % (24666)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.54/12.15 % (24666)Termination reason: Unknown
% 46.54/12.15 % (24666)Termination phase: Property scanning
% 46.54/12.15
% 46.54/12.15 % (24666)Memory used [KB]: 37483
% 46.54/12.15 % (24666)Time elapsed: 0.479 s
% 46.54/12.15 % (24666)Instructions burned: 1671 (million)
% 46.54/12.15 % (24666)------------------------------
% 46.54/12.15 % (24666)------------------------------
% 46.54/12.16 % (24678)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_24167 for (2949ds/643Mi)
% 46.91/12.24 % (24672)Instruction limit reached!
% 46.91/12.24 % (24672)------------------------------
% 46.91/12.24 % (24672)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.91/12.24 % (24672)Termination reason: Unknown
% 46.91/12.24 % (24672)Termination phase: shuffling
% 46.91/12.24
% 46.91/12.24 % (24672)Memory used [KB]: 24946
% 46.91/12.24 % (24672)Time elapsed: 0.301 s
% 46.91/12.24 % (24672)Instructions burned: 866 (million)
% 46.91/12.24 % (24672)------------------------------
% 46.91/12.24 % (24672)------------------------------
% 46.91/12.25 % (24679)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_24167 for (2948ds/865Mi)
% 46.91/12.29 % (24677)Instruction limit reached!
% 46.91/12.29 % (24677)------------------------------
% 46.91/12.29 % (24677)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.91/12.29 % (24677)Termination reason: Unknown
% 46.91/12.29 % (24677)Termination phase: shuffling
% 46.91/12.29
% 46.91/12.29 % (24677)Memory used [KB]: 22387
% 46.91/12.29 % (24677)Time elapsed: 0.177 s
% 46.91/12.29 % (24677)Instructions burned: 580 (million)
% 46.91/12.29 % (24677)------------------------------
% 46.91/12.29 % (24677)------------------------------
% 47.46/12.30 % (24680)lrs+10_1:1_au=on:erd=off:s2a=on:s2at=3.0:sos=all:i=1174:si=on:rtra=on_0 on DTF2THF_24167 for (2947ds/1174Mi)
% 47.46/12.33 % (24674)Instruction limit reached!
% 47.46/12.33 % (24674)------------------------------
% 47.46/12.33 % (24674)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 47.46/12.33 % (24674)Termination reason: Unknown
% 47.46/12.33 % (24674)Termination phase: Property scanning
% 47.46/12.33
% 47.46/12.33 % (24674)Memory used [KB]: 20724
% 47.46/12.33 % (24674)Time elapsed: 0.328 s
% 47.46/12.33 % (24674)Instructions burned: 1001 (million)
% 47.46/12.33 % (24674)------------------------------
% 47.46/12.33 % (24674)------------------------------
% 47.46/12.34 % (24681)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_24167 for (2947ds/644Mi)
% 47.46/12.36 % (24678)Instruction limit reached!
% 47.46/12.36 % (24678)------------------------------
% 47.46/12.36 % (24678)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 47.46/12.36 % (24678)Termination reason: Unknown
% 47.46/12.36 % (24678)Termination phase: shuffling
% 47.46/12.36
% 47.46/12.36 % (24678)Memory used [KB]: 23283
% 47.46/12.36 % (24678)Time elapsed: 0.198 s
% 47.46/12.36 % (24678)Instructions burned: 644 (million)
% 47.46/12.36 % (24678)------------------------------
% 47.46/12.36 % (24678)------------------------------
% 47.46/12.37 % (24682)lrs+10_1:1_alpa=true:au=on:cs=on:tgt=ground:i=1201:si=on:rtra=on_0 on DTF2THF_24167 for (2946ds/1201Mi)
% 47.46/12.38 % (24676)Instruction limit reached!
% 47.46/12.38 % (24676)------------------------------
% 47.46/12.38 % (24676)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 47.46/12.38 % (24676)Termination reason: Unknown
% 47.46/12.38 % (24676)Termination phase: Property scanning
% 47.46/12.38
% 47.46/12.38 % (24676)Memory used [KB]: 20724
% 47.46/12.38 % (24676)Time elapsed: 0.313 s
% 47.46/12.38 % (24676)Instructions burned: 952 (million)
% 47.46/12.38 % (24676)------------------------------
% 47.46/12.38 % (24676)------------------------------
% 47.46/12.39 % (24683)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_24167 for (2946ds/1046Mi)
% 48.44/12.56 % (24679)Instruction limit reached!
% 48.44/12.56 % (24679)------------------------------
% 48.44/12.56 % (24679)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.44/12.56 % (24679)Termination reason: Unknown
% 48.44/12.56 % (24679)Termination phase: shuffling
% 48.44/12.56
% 48.44/12.56 % (24679)Memory used [KB]: 24946
% 48.44/12.56 % (24679)Time elapsed: 0.306 s
% 48.44/12.56 % (24679)Instructions burned: 865 (million)
% 48.44/12.56 % (24679)------------------------------
% 48.44/12.56 % (24679)------------------------------
% 48.44/12.57 % (24684)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_24167 for (2944ds/222Mi)
% 48.44/12.57 % (24681)Instruction limit reached!
% 48.44/12.57 % (24681)------------------------------
% 48.44/12.57 % (24681)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.44/12.57 % (24681)Termination reason: Unknown
% 48.44/12.57 % (24681)Termination phase: shuffling
% 48.44/12.57
% 48.44/12.57 % (24681)Memory used [KB]: 20340
% 48.44/12.57 % (24681)Time elapsed: 0.229 s
% 48.44/12.57 % (24681)Instructions burned: 644 (million)
% 48.44/12.57 % (24681)------------------------------
% 48.44/12.57 % (24681)------------------------------
% 48.44/12.58 % (24685)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_24167 for (2944ds/68Mi)
% 49.21/12.61 % (24685)Instruction limit reached!
% 49.21/12.61 % (24685)------------------------------
% 49.21/12.61 % (24685)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.21/12.61 % (24685)Termination reason: Unknown
% 49.21/12.61 % (24685)Termination phase: shuffling
% 49.21/12.61
% 49.21/12.61 % (24685)Memory used [KB]: 13176
% 49.21/12.61 % (24685)Time elapsed: 0.031 s
% 49.21/12.61 % (24685)Instructions burned: 69 (million)
% 49.21/12.61 % (24685)------------------------------
% 49.21/12.61 % (24685)------------------------------
% 49.21/12.63 % (24686)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_24167 for (2944ds/454Mi)
% 49.21/12.64 % (24684)Instruction limit reached!
% 49.21/12.64 % (24684)------------------------------
% 49.21/12.64 % (24684)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.21/12.64 % (24684)Termination reason: Unknown
% 49.21/12.64 % (24684)Termination phase: shuffling
% 49.21/12.64
% 49.21/12.64 % (24684)Memory used [KB]: 15351
% 49.21/12.64 % (24684)Time elapsed: 0.071 s
% 49.21/12.64 % (24684)Instructions burned: 222 (million)
% 49.21/12.64 % (24684)------------------------------
% 49.21/12.64 % (24684)------------------------------
% 49.21/12.65 % (24687)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_24167 for (2943ds/360Mi)
% 49.21/12.67 % (24680)Instruction limit reached!
% 49.21/12.67 % (24680)------------------------------
% 49.21/12.67 % (24680)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.21/12.67 % (24680)Termination reason: Unknown
% 49.21/12.67 % (24680)Termination phase: Property scanning
% 49.21/12.67
% 49.21/12.67 % (24680)Memory used [KB]: 20724
% 49.21/12.67 % (24680)Time elapsed: 0.372 s
% 49.21/12.67 % (24680)Instructions burned: 1176 (million)
% 49.21/12.67 % (24680)------------------------------
% 49.21/12.67 % (24680)------------------------------
% 49.21/12.68 % (24688)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_24167 for (2943ds/157Mi)
% 49.91/12.70 % (24683)Instruction limit reached!
% 49.91/12.70 % (24683)------------------------------
% 49.91/12.70 % (24683)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.91/12.70 % (24683)Termination reason: Unknown
% 49.91/12.70 % (24683)Termination phase: shuffling
% 49.91/12.70
% 49.91/12.70 % (24683)Memory used [KB]: 32750
% 49.91/12.70 % (24683)Time elapsed: 0.313 s
% 49.91/12.70 % (24683)Instructions burned: 1047 (million)
% 49.91/12.70 % (24683)------------------------------
% 49.91/12.70 % (24683)------------------------------
% 49.91/12.72 % (24689)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_24167 for (2943ds/5061Mi)
% 49.91/12.73 % (24688)Instruction limit reached!
% 49.91/12.73 % (24688)------------------------------
% 49.91/12.73 % (24688)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.91/12.73 % (24688)Termination reason: Unknown
% 49.91/12.73 % (24688)Termination phase: shuffling
% 49.91/12.73
% 49.91/12.73 % (24688)Memory used [KB]: 14456
% 49.91/12.73 % (24688)Time elapsed: 0.052 s
% 49.91/12.73 % (24688)Instructions burned: 157 (million)
% 49.91/12.73 % (24688)------------------------------
% 49.91/12.73 % (24688)------------------------------
% 49.91/12.75 % (24690)lrs+1002_1:8_av=off:fd=off:hud=5:tnu=1:i=2401:si=on:rtra=on_0 on DTF2THF_24167 for (2943ds/2401Mi)
% 49.91/12.75 % (24682)Instruction limit reached!
% 49.91/12.75 % (24682)------------------------------
% 49.91/12.75 % (24682)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.91/12.75 % (24682)Termination reason: Unknown
% 49.91/12.75 % (24682)Termination phase: Property scanning
% 49.91/12.75
% 49.91/12.75 % (24682)Memory used [KB]: 20724
% 49.91/12.75 % (24682)Time elapsed: 0.376 s
% 49.91/12.75 % (24682)Instructions burned: 1201 (million)
% 49.91/12.75 % (24682)------------------------------
% 49.91/12.75 % (24682)------------------------------
% 49.91/12.76 % (24691)dis+10_1:1_av=off:e2e=on:nm=4:s2a=on:i=5798:si=on:rtra=on_0 on DTF2THF_24167 for (2942ds/5798Mi)
% 49.91/12.76 % (24687)Instruction limit reached!
% 49.91/12.76 % (24687)------------------------------
% 49.91/12.76 % (24687)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.91/12.76 % (24687)Termination reason: Unknown
% 49.91/12.76 % (24687)Termination phase: shuffling
% 49.91/12.76
% 49.91/12.76 % (24687)Memory used [KB]: 19317
% 49.91/12.76 % (24687)Time elapsed: 0.110 s
% 49.91/12.76 % (24687)Instructions burned: 362 (million)
% 49.91/12.76 % (24687)------------------------------
% 49.91/12.76 % (24687)------------------------------
% 49.91/12.76 % (24686)Instruction limit reached!
% 49.91/12.76 % (24686)------------------------------
% 49.91/12.76 % (24686)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.91/12.76 % (24686)Termination reason: Unknown
% 49.91/12.76 % (24686)Termination phase: shuffling
% 49.91/12.76
% 49.91/12.76 % (24686)Memory used [KB]: 20724
% 49.91/12.76 % (24686)Time elapsed: 0.138 s
% 49.91/12.76 % (24686)Instructions burned: 457 (million)
% 49.91/12.76 % (24686)------------------------------
% 49.91/12.76 % (24686)------------------------------
% 49.91/12.77 % (24692)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_24167 for (2942ds/346Mi)
% 49.91/12.77 % (24693)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_24167 for (2942ds/1321Mi)
% 51.15/12.88 % (24692)Instruction limit reached!
% 51.15/12.88 % (24692)------------------------------
% 51.15/12.88 % (24692)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 51.15/12.88 % (24692)Termination reason: Unknown
% 51.15/12.88 % (24692)Termination phase: shuffling
% 51.15/12.88
% 51.15/12.88 % (24692)Memory used [KB]: 19189
% 51.15/12.88 % (24692)Time elapsed: 0.105 s
% 51.15/12.88 % (24692)Instructions burned: 346 (million)
% 51.15/12.88 % (24692)------------------------------
% 51.15/12.88 % (24692)------------------------------
% 51.15/12.89 % (24694)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:i=1312:si=on:rtra=on_0 on DTF2THF_24167 for (2941ds/1312Mi)
% 51.46/13.02 % (24670)First to succeed.
% 52.96/13.03 % (24670)Refutation found. Thanks to Tanya!
% 52.96/13.03 % SZS status Theorem for DTF2THF_24167
% 52.96/13.03 % SZS output start Proof for DTF2THF_24167
% 52.96/13.03 thf(type_def_5, type, rat: $tType).
% 52.96/13.03 thf(type_def_6, type, int: $tType).
% 52.96/13.03 thf(type_def_8, type, real: $tType).
% 52.96/13.03 thf(type_def_9, type, nat: $tType).
% 52.96/13.03 thf(type_def_10, type, product_prod_int_int: $tType).
% 52.96/13.03 thf(type_def_11, type, product_prod_nat_nat: $tType).
% 52.96/13.03 thf(type_def_12, type, num: $tType).
% 52.96/13.03 thf(type_def_13, type, complex: $tType).
% 52.96/13.03 thf(type_def_14, type, option_num: $tType).
% 52.96/13.03 thf(type_def_15, type, product_prod_num_num: $tType).
% 52.96/13.03 thf(type_def_16, type, code_integer: $tType).
% 52.96/13.03 thf(type_def_17, type, produc6271795597528267376eger_o: $tType).
% 52.96/13.03 thf(type_def_18, type, produc8923325533196201883nteger: $tType).
% 52.96/13.03 thf(type_def_19, type, set_fi4554929511873752355omplex: $tType).
% 52.96/13.03 thf(type_def_20, type, filter6041513312241820739omplex: $tType).
% 52.96/13.03 thf(type_def_21, type, set_fi7789364187291644575l_real: $tType).
% 52.96/13.03 thf(type_def_22, type, filter2146258269922977983l_real: $tType).
% 52.96/13.03 thf(type_def_23, type, set_real: $tType).
% 52.96/13.03 thf(type_def_24, type, set_set_nat: $tType).
% 52.96/13.03 thf(type_def_25, type, set_nat: $tType).
% 52.96/13.03 thf(type_def_26, type, set_int: $tType).
% 52.96/13.03 thf(type_def_27, type, filter_real: $tType).
% 52.96/13.03 thf(type_def_28, type, filter_int: $tType).
% 52.96/13.03 thf(type_def_29, type, filter_nat: $tType).
% 52.96/13.03 thf(type_def_30, type, set_Pr5085853215250843933omplex: $tType).
% 52.96/13.03 thf(type_def_31, type, set_Pr6218003697084177305l_real: $tType).
% 52.96/13.03 thf(type_def_32, type, set_o: $tType).
% 52.96/13.03 thf(type_def_33, type, set_complex: $tType).
% 52.96/13.03 thf(type_def_34, type, set_list_nat: $tType).
% 52.96/13.03 thf(type_def_35, type, set_char: $tType).
% 52.96/13.03 thf(type_def_36, type, list_nat: $tType).
% 52.96/13.03 thf(type_def_37, type, set_Pr1261947904930325089at_nat: $tType).
% 52.96/13.03 thf(type_def_38, type, char: $tType).
% 52.96/13.03 thf(type_def_39, type, set_Pr8693737435421807431at_nat: $tType).
% 52.96/13.03 thf(type_def_40, type, extended_enat: $tType).
% 52.96/13.03 thf(type_def_41, type, list_o: $tType).
% 52.96/13.03 thf(type_def_42, type, list_int: $tType).
% 52.96/13.03 thf(type_def_43, type, set_Pr958786334691620121nt_int: $tType).
% 52.96/13.03 thf(type_def_44, type, set_Code_integer: $tType).
% 52.96/13.03 thf(type_def_45, type, set_rat: $tType).
% 52.96/13.03 thf(type_def_46, type, list_list_o: $tType).
% 52.96/13.03 thf(type_def_47, type, list_list_int: $tType).
% 52.96/13.03 thf(type_def_48, type, list_list_nat: $tType).
% 52.96/13.03 thf(type_def_49, type, list_list_real: $tType).
% 52.96/13.03 thf(type_def_50, type, list_real: $tType).
% 52.96/13.03 thf(type_def_51, type, list_complex: $tType).
% 52.96/13.03 thf(type_def_52, type, list_set_nat: $tType).
% 52.96/13.03 thf(type_def_53, type, vEBT_VEBT: $tType).
% 52.96/13.03 thf(type_def_54, type, list_VEBT_VEBT: $tType).
% 52.96/13.03 thf(type_def_55, type, set_list_o: $tType).
% 52.96/13.03 thf(type_def_56, type, set_list_int: $tType).
% 52.96/13.03 thf(type_def_57, type, set_list_real: $tType).
% 52.96/13.03 thf(type_def_58, type, set_VEBT_VEBT: $tType).
% 52.96/13.03 thf(type_def_59, type, option4927543243414619207at_nat: $tType).
% 52.96/13.03 thf(type_def_60, type, product_prod_nat_num: $tType).
% 52.96/13.03 thf(type_def_61, type, produc859450856879609959at_nat: $tType).
% 52.96/13.03 thf(type_def_62, type, produc9072475918466114483BT_nat: $tType).
% 52.96/13.03 thf(type_def_63, type, produc4411394909380815293omplex: $tType).
% 52.96/13.03 thf(type_def_64, type, produc2422161461964618553l_real: $tType).
% 52.96/13.03 thf(func_def_0, type, set_Pr8693737435421807431at_nat: $tType).
% 52.96/13.03 thf(func_def_1, type, produc859450856879609959at_nat: $tType).
% 52.96/13.03 thf(func_def_2, type, set_fi4554929511873752355omplex: $tType).
% 52.96/13.03 thf(func_def_3, type, set_fi7789364187291644575l_real: $tType).
% 52.96/13.03 thf(func_def_4, type, filter6041513312241820739omplex: $tType).
% 52.96/13.03 thf(func_def_5, type, set_Pr5085853215250843933omplex: $tType).
% 52.96/13.03 thf(func_def_6, type, produc8923325533196201883nteger: $tType).
% 52.96/13.03 thf(func_def_7, type, filter2146258269922977983l_real: $tType).
% 52.96/13.03 thf(func_def_8, type, option4927543243414619207at_nat: $tType).
% 52.96/13.03 thf(func_def_9, type, set_Pr6218003697084177305l_real: $tType).
% 52.96/13.03 thf(func_def_10, type, produc9072475918466114483BT_nat: $tType).
% 52.96/13.03 thf(func_def_11, type, set_Pr1261947904930325089at_nat: $tType).
% 52.96/13.03 thf(func_def_12, type, set_Pr958786334691620121nt_int: $tType).
% 52.96/13.03 thf(func_def_13, type, produc4411394909380815293omplex: $tType).
% 52.96/13.03 thf(func_def_14, type, produc6271795597528267376eger_o: $tType).
% 52.96/13.03 thf(func_def_15, type, produc2422161461964618553l_real: $tType).
% 52.96/13.03 thf(func_def_16, type, product_prod_num_num: $tType).
% 52.96/13.03 thf(func_def_17, type, product_prod_nat_num: $tType).
% 52.96/13.03 thf(func_def_18, type, product_prod_nat_nat: $tType).
% 52.96/13.03 thf(func_def_19, type, product_prod_int_int: $tType).
% 52.96/13.03 thf(func_def_20, type, list_list_real: $tType).
% 52.96/13.03 thf(func_def_21, type, set_list_real: $tType).
% 52.96/13.03 thf(func_def_22, type, list_list_nat: $tType).
% 52.96/13.03 thf(func_def_23, type, list_list_int: $tType).
% 52.96/13.03 thf(func_def_24, type, list_VEBT_VEBT: $tType).
% 52.96/13.03 thf(func_def_25, type, set_list_nat: $tType).
% 52.96/13.03 thf(func_def_26, type, set_list_int: $tType).
% 52.96/13.03 thf(func_def_27, type, list_set_nat: $tType).
% 52.96/13.03 thf(func_def_28, type, set_VEBT_VEBT: $tType).
% 52.96/13.03 thf(func_def_29, type, set_set_nat: $tType).
% 52.96/13.03 thf(func_def_30, type, set_Code_integer: $tType).
% 52.96/13.03 thf(func_def_31, type, list_list_o: $tType).
% 52.96/13.03 thf(func_def_32, type, list_complex: $tType).
% 52.96/13.03 thf(func_def_33, type, set_list_o: $tType).
% 52.96/13.03 thf(func_def_34, type, set_complex: $tType).
% 52.96/13.03 thf(func_def_35, type, filter_real: $tType).
% 52.96/13.03 thf(func_def_36, type, option_num: $tType).
% 52.96/13.03 thf(func_def_37, type, filter_nat: $tType).
% 52.96/13.03 thf(func_def_38, type, filter_int: $tType).
% 52.96/13.03 thf(func_def_39, type, set_char: $tType).
% 52.96/13.03 thf(func_def_40, type, list_real: $tType).
% 52.96/13.03 thf(func_def_41, type, set_real: $tType).
% 52.96/13.03 thf(func_def_42, type, list_nat: $tType).
% 52.96/13.03 thf(func_def_43, type, list_int: $tType).
% 52.96/13.03 thf(func_def_44, type, vEBT_VEBT: $tType).
% 52.96/13.03 thf(func_def_45, type, set_rat: $tType).
% 52.96/13.03 thf(func_def_46, type, set_nat: $tType).
% 52.96/13.03 thf(func_def_47, type, set_int: $tType).
% 52.96/13.03 thf(func_def_48, type, code_integer: $tType).
% 52.96/13.03 thf(func_def_49, type, extended_enat: $tType).
% 52.96/13.03 thf(func_def_50, type, list_o: $tType).
% 52.96/13.03 thf(func_def_51, type, complex: $tType).
% 52.96/13.03 thf(func_def_52, type, set_o: $tType).
% 52.96/13.03 thf(func_def_53, type, char: $tType).
% 52.96/13.03 thf(func_def_54, type, real: $tType).
% 52.96/13.03 thf(func_def_55, type, rat: $tType).
% 52.96/13.03 thf(func_def_56, type, num: $tType).
% 52.96/13.03 thf(func_def_57, type, nat: $tType).
% 52.96/13.03 thf(func_def_58, type, int: $tType).
% 52.96/13.03 thf(func_def_59, type, archim2889992004027027881ng_rat: rat > int).
% 52.96/13.03 thf(func_def_60, type, archim7802044766580827645g_real: real > int).
% 52.96/13.03 thf(func_def_61, type, archim3151403230148437115or_rat: rat > int).
% 52.96/13.03 thf(func_def_62, type, archim6058952711729229775r_real: real > int).
% 52.96/13.03 thf(func_def_63, type, archimedean_frac_rat: rat > rat).
% 52.96/13.03 thf(func_def_64, type, archim2898591450579166408c_real: real > real).
% 52.96/13.03 thf(func_def_65, type, archim7778729529865785530nd_rat: rat > int).
% 52.96/13.03 thf(func_def_66, type, archim8280529875227126926d_real: real > int).
% 52.96/13.03 thf(func_def_67, 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).
% 52.96/13.03 thf(func_def_68, type, bNF_re895249473297799549at_rat: ((nat > rat) > (nat > rat) > $o) > ((nat > rat) > (nat > rat) > $o) > ((nat > rat) > nat > rat) > ((nat > rat) > nat > rat) > $o).
% 52.96/13.03 thf(func_def_69, type, bNF_re728719798268516973at_o_o: ((nat > rat) > (nat > rat) > $o) > ($o > $o > $o) > ((nat > rat) > $o) > ((nat > rat) > $o) > $o).
% 52.96/13.03 thf(func_def_70, 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).
% 52.96/13.03 thf(func_def_71, type, bNF_re4521903465945308077real_o: ((nat > rat) > real > $o) > (((nat > rat) > $o) > (real > $o) > $o) > ((nat > rat) > (nat > rat) > $o) > (real > real > $o) > $o).
% 52.96/13.03 thf(func_def_72, type, bNF_re3023117138289059399t_real: ((nat > rat) > real > $o) > ((nat > rat) > real > $o) > ((nat > rat) > nat > rat) > (real > real) > $o).
% 52.96/13.03 thf(func_def_73, type, bNF_re4297313714947099218al_o_o: ((nat > rat) > real > $o) > ($o > $o > $o) > ((nat > rat) > $o) > (real > $o) > $o).
% 52.96/13.03 thf(func_def_74, type, bNF_re3403563459893282935_int_o: (int > int > $o) > ((int > $o) > (int > $o) > $o) > (int > int > $o) > (int > int > $o) > $o).
% 52.96/13.03 thf(func_def_75, type, bNF_re711492959462206631nt_int: (int > int > $o) > ((int > int) > (int > int) > $o) > (int > int > int) > (int > int > int) > $o).
% 52.96/13.03 thf(func_def_76, type, bNF_re157797125943740599nt_int: (int > int > $o) > ((int > product_prod_int_int) > (int > product_prod_int_int) > $o) > (int > int > product_prod_int_int) > (int > int > product_prod_int_int) > $o).
% 52.96/13.03 thf(func_def_77, type, bNF_re3461391660133120880nt_rat: (int > int > $o) > ((int > product_prod_int_int) > (int > rat) > $o) > (int > int > product_prod_int_int) > (int > int > rat) > $o).
% 52.96/13.03 thf(func_def_78, type, bNF_re5089333283451836215nt_o_o: (int > int > $o) > ($o > $o > $o) > (int > $o) > (int > $o) > $o).
% 52.96/13.03 thf(func_def_79, type, bNF_re4712519889275205905nt_int: (int > int > $o) > (int > int > $o) > (int > int) > (int > int) > $o).
% 52.96/13.03 thf(func_def_80, type, bNF_re6250860962936578807nt_int: (int > int > $o) > (product_prod_int_int > product_prod_int_int > $o) > (int > product_prod_int_int) > (int > product_prod_int_int) > $o).
% 52.96/13.03 thf(func_def_81, type, bNF_re2214769303045360666nt_rat: (int > int > $o) > (product_prod_int_int > rat > $o) > (int > product_prod_int_int) > (int > rat) > $o).
% 52.96/13.03 thf(func_def_82, type, bNF_re4785983289428654063nt_int: (nat > nat > $o) > ((int > int) > (int > int) > $o) > (nat > int > int) > (nat > int > int) > $o).
% 52.96/13.03 thf(func_def_83, type, bNF_re578469030762574527_nat_o: (nat > nat > $o) > ((nat > $o) > (nat > $o) > $o) > (nat > nat > $o) > (nat > nat > $o) > $o).
% 52.96/13.03 thf(func_def_84, type, bNF_re1345281282404953727at_nat: (nat > nat > $o) > ((nat > nat) > (nat > nat) > $o) > (nat > nat > nat) > (nat > nat > nat) > $o).
% 52.96/13.03 thf(func_def_85, type, bNF_re4705727531993890431at_o_o: (nat > nat > $o) > ($o > $o > $o) > (nat > $o) > (nat > $o) > $o).
% 52.96/13.03 thf(func_def_86, type, bNF_re6650684261131312217nt_int: (nat > nat > $o) > (int > int > $o) > (nat > int) > (nat > int) > $o).
% 52.96/13.03 thf(func_def_87, type, bNF_re5653821019739307937at_nat: (nat > nat > $o) > (nat > nat > $o) > (nat > nat) > (nat > nat) > $o).
% 52.96/13.03 thf(func_def_88, type, bNF_re6830278522597306478at_int: (nat > nat > $o) > (product_prod_nat_nat > int > $o) > (nat > product_prod_nat_nat) > (nat > int) > $o).
% 52.96/13.03 thf(func_def_89, type, bNF_re8402795839162346335um_int: (num > num > $o) > ((num > int) > (num > int) > $o) > (num > num > int) > (num > num > int) > $o).
% 52.96/13.03 thf(func_def_90, type, bNF_re1822329894187522285nt_int: (num > num > $o) > (int > int > $o) > (num > int) > (num > int) > $o).
% 52.96/13.03 thf(func_def_91, 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).
% 52.96/13.03 thf(func_def_92, 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).
% 52.96/13.03 thf(func_def_93, 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).
% 52.96/13.03 thf(func_def_94, 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).
% 52.96/13.03 thf(func_def_95, type, bNF_re1494630372529172596at_o_o: (product_prod_int_int > rat > $o) > ($o > $o > $o) > (product_prod_int_int > $o) > (rat > $o) > $o).
% 52.96/13.03 thf(func_def_96, 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).
% 52.96/13.03 thf(func_def_97, 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).
% 52.96/13.03 thf(func_def_98, 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).
% 52.96/13.03 thf(func_def_99, type, bNF_re6644619430987730960nt_o_o: (product_prod_nat_nat > int > $o) > ($o > $o > $o) > (product_prod_nat_nat > $o) > (int > $o) > $o).
% 52.96/13.03 thf(func_def_100, type, bNF_re4555766996558763186at_nat: (product_prod_nat_nat > int > $o) > (nat > nat > $o) > (product_prod_nat_nat > nat) > (int > nat) > $o).
% 52.96/13.03 thf(func_def_101, 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).
% 52.96/13.03 thf(func_def_102, 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).
% 52.96/13.03 thf(func_def_103, 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).
% 52.96/13.03 thf(func_def_104, 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).
% 52.96/13.03 thf(func_def_105, 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).
% 52.96/13.03 thf(func_def_106, 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).
% 52.96/13.03 thf(func_def_107, type, binomial: nat > nat > nat).
% 52.96/13.03 thf(func_def_108, type, gbinomial_complex: complex > nat > complex).
% 52.96/13.03 thf(func_def_109, type, gbinomial_int: int > nat > int).
% 52.96/13.03 thf(func_def_110, type, gbinomial_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_111, type, gbinomial_rat: rat > nat > rat).
% 52.96/13.03 thf(func_def_112, type, gbinomial_real: real > nat > real).
% 52.96/13.03 thf(func_def_113, type, bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o).
% 52.96/13.03 thf(func_def_114, type, bit_and_not_num: num > num > option_num).
% 52.96/13.03 thf(func_def_115, type, bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o).
% 52.96/13.03 thf(func_def_116, type, bit_concat_bit: nat > int > int > int).
% 52.96/13.03 thf(func_def_117, type, bit_or_not_num_neg: num > num > num).
% 52.96/13.03 thf(func_def_118, type, bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o).
% 52.96/13.03 thf(func_def_119, type, bit_ri7632146776885996613nteger: code_integer > code_integer).
% 52.96/13.03 thf(func_def_120, type, bit_ri7919022796975470100ot_int: int > int).
% 52.96/13.03 thf(func_def_121, type, bit_ri6519982836138164636nteger: nat > code_integer > code_integer).
% 52.96/13.03 thf(func_def_122, type, bit_ri631733984087533419it_int: nat > int > int).
% 52.96/13.03 thf(func_def_123, type, bit_se3949692690581998587nteger: code_integer > code_integer > code_integer).
% 52.96/13.03 thf(func_def_124, type, bit_se725231765392027082nd_int: int > int > int).
% 52.96/13.03 thf(func_def_125, type, bit_se727722235901077358nd_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_126, type, bit_se3928097537394005634nteger: nat > code_integer > code_integer).
% 52.96/13.03 thf(func_def_127, type, bit_se8568078237143864401it_int: nat > int > int).
% 52.96/13.03 thf(func_def_128, type, bit_se8570568707652914677it_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_129, type, bit_se1345352211410354436nteger: nat > code_integer > code_integer).
% 52.96/13.03 thf(func_def_130, type, bit_se2159334234014336723it_int: nat > int > int).
% 52.96/13.03 thf(func_def_131, type, bit_se2161824704523386999it_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_132, type, bit_se2119862282449309892nteger: nat > code_integer).
% 52.96/13.03 thf(func_def_133, type, bit_se2000444600071755411sk_int: nat > int).
% 52.96/13.03 thf(func_def_134, type, bit_se2002935070580805687sk_nat: nat > nat).
% 52.96/13.03 thf(func_def_135, type, bit_se1080825931792720795nteger: code_integer > code_integer > code_integer).
% 52.96/13.03 thf(func_def_136, type, bit_se1409905431419307370or_int: int > int > int).
% 52.96/13.03 thf(func_def_137, type, bit_se1412395901928357646or_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_138, type, bit_se7788150548672797655nteger: nat > code_integer > code_integer).
% 52.96/13.03 thf(func_def_139, type, bit_se545348938243370406it_int: nat > int > int).
% 52.96/13.03 thf(func_def_140, type, bit_se547839408752420682it_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_141, type, bit_se2793503036327961859nteger: nat > code_integer > code_integer).
% 52.96/13.03 thf(func_def_142, type, bit_se7879613467334960850it_int: nat > int > int).
% 52.96/13.03 thf(func_def_143, type, bit_se7882103937844011126it_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_144, type, bit_se1745604003318907178nteger: nat > code_integer > code_integer).
% 52.96/13.03 thf(func_def_145, type, bit_se2923211474154528505it_int: nat > int > int).
% 52.96/13.03 thf(func_def_146, type, bit_se2925701944663578781it_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_147, type, bit_se8260200283734997820nteger: nat > code_integer > code_integer).
% 52.96/13.03 thf(func_def_148, type, bit_se4203085406695923979it_int: nat > int > int).
% 52.96/13.03 thf(func_def_149, type, bit_se4205575877204974255it_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_150, type, bit_se3222712562003087583nteger: code_integer > code_integer > code_integer).
% 52.96/13.03 thf(func_def_151, type, bit_se6526347334894502574or_int: int > int > int).
% 52.96/13.03 thf(func_def_152, type, bit_se6528837805403552850or_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_153, type, bit_se9216721137139052372nteger: code_integer > nat > $o).
% 52.96/13.03 thf(func_def_154, type, bit_se1146084159140164899it_int: int > nat > $o).
% 52.96/13.03 thf(func_def_155, type, bit_se1148574629649215175it_nat: nat > nat > $o).
% 52.96/13.03 thf(func_def_156, type, bit_take_bit_num: nat > num > option_num).
% 52.96/13.03 thf(func_def_157, type, bit_un1837492267222099188nd_num: num > num > option_num).
% 52.96/13.03 thf(func_def_158, type, bit_un5425074673868309765um_rel: product_prod_num_num > product_prod_num_num > $o).
% 52.96/13.03 thf(func_def_159, type, bit_un6178654185764691216or_num: num > num > option_num).
% 52.96/13.03 thf(func_def_160, type, bit_un3595099601533988841um_rel: product_prod_num_num > product_prod_num_num > $o).
% 52.96/13.03 thf(func_def_161, type, bit_un7362597486090784418nd_num: num > num > option_num).
% 52.96/13.03 thf(func_def_162, type, bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o).
% 52.96/13.03 thf(func_def_163, type, bit_un2480387367778600638or_num: num > num > option_num).
% 52.96/13.03 thf(func_def_164, type, bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o).
% 52.96/13.03 thf(func_def_165, type, boolea2445317508997433345nteger: (code_integer > code_integer > code_integer) > (code_integer > code_integer > code_integer) > (code_integer > code_integer) > code_integer > code_integer > (code_integer > code_integer > code_integer) > $o).
% 52.96/13.03 thf(func_def_166, type, boolea8527374999097803216ff_int: (int > int > int) > (int > int > int) > (int > int) > int > int > (int > int > int) > $o).
% 52.96/13.03 thf(func_def_167, type, code_bit_cut_integer: code_integer > produc6271795597528267376eger_o).
% 52.96/13.03 thf(func_def_168, type, code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger).
% 52.96/13.03 thf(func_def_169, type, code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger).
% 52.96/13.03 thf(func_def_170, type, code_int_of_integer: code_integer > int).
% 52.96/13.03 thf(func_def_171, type, code_integer_of_int: int > code_integer).
% 52.96/13.03 thf(func_def_172, type, code_integer_of_num: num > code_integer).
% 52.96/13.03 thf(func_def_173, type, code_nat_of_integer: code_integer > nat).
% 52.96/13.03 thf(func_def_174, type, code_num_of_integer: code_integer > num).
% 52.96/13.03 thf(func_def_175, type, code_Target_negative: num > int).
% 52.96/13.03 thf(func_def_176, type, code_Target_positive: num > int).
% 52.96/13.03 thf(func_def_177, type, code_T6385005292777649522of_nat: nat > int).
% 52.96/13.03 thf(func_def_178, type, comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex).
% 52.96/13.03 thf(func_def_179, type, comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real).
% 52.96/13.03 thf(func_def_180, type, comple4887499456419720421f_real: set_real > real).
% 52.96/13.03 thf(func_def_181, type, comple7806235888213564991et_nat: set_set_nat > set_nat).
% 52.96/13.03 thf(func_def_182, type, complete_Sup_Sup_int: set_int > int).
% 52.96/13.03 thf(func_def_183, type, complete_Sup_Sup_nat: set_nat > nat).
% 52.96/13.03 thf(func_def_184, type, comple1385675409528146559p_real: set_real > real).
% 52.96/13.03 thf(func_def_185, type, arg: complex > real).
% 52.96/13.03 thf(func_def_186, type, cis: real > complex).
% 52.96/13.03 thf(func_def_187, type, cnj: complex > complex).
% 52.96/13.03 thf(func_def_188, type, complex2: real > real > complex).
% 52.96/13.03 thf(func_def_189, type, im: complex > real).
% 52.96/13.03 thf(func_def_190, type, re: complex > real).
% 52.96/13.03 thf(func_def_191, type, csqrt: complex > complex).
% 52.96/13.03 thf(func_def_192, type, imaginary_unit: complex).
% 52.96/13.03 thf(func_def_193, type, rcis: real > real > complex).
% 52.96/13.03 thf(func_def_194, type, differ6690327859849518006l_real: (real > real) > filter_real > $o).
% 52.96/13.03 thf(func_def_195, type, has_de1759254742604945161l_real: (real > real) > (real > real) > filter_real > $o).
% 52.96/13.03 thf(func_def_196, type, has_fi5821293074295781190e_real: (real > real) > real > filter_real > $o).
% 52.96/13.03 thf(func_def_197, type, adjust_div: product_prod_int_int > int).
% 52.96/13.03 thf(func_def_198, type, adjust_mod: int > int > int).
% 52.96/13.03 thf(func_def_199, type, divmod_nat: nat > nat > product_prod_nat_nat).
% 52.96/13.03 thf(func_def_200, type, eucl_rel_int: int > int > product_prod_int_int > $o).
% 52.96/13.03 thf(func_def_201, type, unique5052692396658037445od_int: num > num > product_prod_int_int).
% 52.96/13.03 thf(func_def_202, type, unique5055182867167087721od_nat: num > num > product_prod_nat_nat).
% 52.96/13.03 thf(func_def_203, type, unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 52.96/13.03 thf(func_def_204, type, unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int).
% 52.96/13.03 thf(func_def_205, type, unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat).
% 52.96/13.03 thf(func_def_206, type, comm_s8582702949713902594nteger: code_integer > nat > code_integer).
% 52.96/13.03 thf(func_def_207, type, comm_s2602460028002588243omplex: complex > nat > complex).
% 52.96/13.03 thf(func_def_208, type, comm_s4660882817536571857er_int: int > nat > int).
% 52.96/13.03 thf(func_def_209, type, comm_s4663373288045622133er_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_210, type, comm_s4028243227959126397er_rat: rat > nat > rat).
% 52.96/13.03 thf(func_def_211, type, comm_s7457072308508201937r_real: real > nat > real).
% 52.96/13.03 thf(func_def_212, type, semiri3624122377584611663nteger: nat > code_integer).
% 52.96/13.03 thf(func_def_213, type, semiri5044797733671781792omplex: nat > complex).
% 52.96/13.03 thf(func_def_214, type, semiri1406184849735516958ct_int: nat > int).
% 52.96/13.03 thf(func_def_215, type, semiri1408675320244567234ct_nat: nat > nat).
% 52.96/13.03 thf(func_def_216, type, semiri773545260158071498ct_rat: nat > rat).
% 52.96/13.03 thf(func_def_217, type, semiri2265585572941072030t_real: nat > real).
% 52.96/13.03 thf(func_def_218, type, invers8013647133539491842omplex: complex > complex).
% 52.96/13.03 thf(func_def_219, type, inverse_inverse_rat: rat > rat).
% 52.96/13.03 thf(func_def_220, type, inverse_inverse_real: real > real).
% 52.96/13.03 thf(func_def_221, type, at_bot_real: filter_real).
% 52.96/13.03 thf(func_def_222, type, at_top_int: filter_int).
% 52.96/13.03 thf(func_def_223, type, at_top_nat: filter_nat).
% 52.96/13.03 thf(func_def_224, type, at_top_real: filter_real).
% 52.96/13.03 thf(func_def_225, type, eventually_nat: (nat > $o) > filter_nat > $o).
% 52.96/13.03 thf(func_def_226, type, eventually_real: (real > $o) > filter_real > $o).
% 52.96/13.03 thf(func_def_227, type, filterlim_nat_int: (nat > int) > filter_int > filter_nat > $o).
% 52.96/13.03 thf(func_def_228, type, filterlim_nat_nat: (nat > nat) > filter_nat > filter_nat > $o).
% 52.96/13.03 thf(func_def_229, type, filterlim_nat_real: (nat > real) > filter_real > filter_nat > $o).
% 52.96/13.03 thf(func_def_230, type, filterlim_real_real: (real > real) > filter_real > filter_real > $o).
% 52.96/13.03 thf(func_def_231, type, filtermap_real_real: (real > real) > filter_real > filter_real).
% 52.96/13.03 thf(func_def_232, type, princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex).
% 52.96/13.03 thf(func_def_233, type, princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real).
% 52.96/13.03 thf(func_def_234, type, finite_card_o: set_o > nat).
% 52.96/13.03 thf(func_def_235, type, finite_card_complex: set_complex > nat).
% 52.96/13.03 thf(func_def_236, type, finite_card_int: set_int > nat).
% 52.96/13.03 thf(func_def_237, type, finite_card_list_nat: set_list_nat > nat).
% 52.96/13.03 thf(func_def_238, type, finite_card_nat: set_nat > nat).
% 52.96/13.03 thf(func_def_239, type, finite_card_set_nat: set_set_nat > nat).
% 52.96/13.03 thf(func_def_240, type, finite_card_char: set_char > nat).
% 52.96/13.03 thf(func_def_241, type, finite3207457112153483333omplex: set_complex > $o).
% 52.96/13.03 thf(func_def_242, type, finite_finite_int: set_int > $o).
% 52.96/13.03 thf(func_def_243, type, finite_finite_nat: set_nat > $o).
% 52.96/13.03 thf(func_def_244, type, bij_be1856998921033663316omplex: (complex > complex) > set_complex > set_complex > $o).
% 52.96/13.03 thf(func_def_245, type, bij_betw_int_nat: (int > nat) > set_int > set_nat > $o).
% 52.96/13.03 thf(func_def_246, type, bij_be8532844293280997160at_nat: (list_nat > nat) > set_list_nat > set_nat > $o).
% 52.96/13.03 thf(func_def_247, type, bij_betw_nat_complex: (nat > complex) > set_nat > set_complex > $o).
% 52.96/13.03 thf(func_def_248, type, bij_betw_nat_int: (nat > int) > set_nat > set_int > $o).
% 52.96/13.03 thf(func_def_249, type, bij_betw_nat_nat: (nat > nat) > set_nat > set_nat > $o).
% 52.96/13.03 thf(func_def_250, type, bij_be5333170631980326235at_nat: (product_prod_nat_nat > nat) > set_Pr1261947904930325089at_nat > set_nat > $o).
% 52.96/13.03 thf(func_def_251, type, comp_C8797469213163452608nteger: ((code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 52.96/13.03 thf(func_def_252, type, comp_C1593894019821074884nteger: (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 52.96/13.03 thf(func_def_253, type, comp_int_int_num: (int > int) > (num > int) > num > int).
% 52.96/13.03 thf(func_def_254, type, comp_int_nat_int: (int > nat) > (int > int) > int > nat).
% 52.96/13.03 thf(func_def_255, type, comp_nat_nat_nat: (nat > nat) > (nat > nat) > nat > nat).
% 52.96/13.03 thf(func_def_256, type, comp_nat_real_nat: (nat > real) > (nat > nat) > nat > real).
% 52.96/13.03 thf(func_def_257, type, id_o: $o > $o).
% 52.96/13.03 thf(func_def_258, type, id_nat: nat > nat).
% 52.96/13.03 thf(func_def_259, type, inj_on_int_nat: (int > nat) > set_int > $o).
% 52.96/13.03 thf(func_def_260, type, inj_on_list_nat_nat: (list_nat > nat) > set_list_nat > $o).
% 52.96/13.03 thf(func_def_261, type, inj_on_nat_int: (nat > int) > set_nat > $o).
% 52.96/13.03 thf(func_def_262, type, inj_on_nat_nat: (nat > nat) > set_nat > $o).
% 52.96/13.03 thf(func_def_263, type, inj_on_nat_char: (nat > char) > set_nat > $o).
% 52.96/13.03 thf(func_def_264, type, inj_on2178005380612969504at_nat: (product_prod_nat_nat > nat) > set_Pr1261947904930325089at_nat > $o).
% 52.96/13.03 thf(func_def_265, type, inj_on_real_real: (real > real) > set_real > $o).
% 52.96/13.03 thf(func_def_266, type, inj_on_set_nat_nat: (set_nat > nat) > set_set_nat > $o).
% 52.96/13.03 thf(func_def_267, 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 > $o).
% 52.96/13.03 thf(func_def_268, type, map_fu4960017516451851995nt_int: (int > product_prod_nat_nat) > ((product_prod_nat_nat > product_prod_nat_nat) > int > int) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > int > int > int).
% 52.96/13.03 thf(func_def_269, type, map_fu4826362097070443709at_o_o: (int > product_prod_nat_nat) > ($o > $o) > (product_prod_nat_nat > $o) > int > $o).
% 52.96/13.03 thf(func_def_270, type, map_fu2345160673673942751at_nat: (int > product_prod_nat_nat) > (nat > nat) > (product_prod_nat_nat > nat) > int > nat).
% 52.96/13.03 thf(func_def_271, type, map_fu3667384564859982768at_int: (int > product_prod_nat_nat) > (product_prod_nat_nat > int) > (product_prod_nat_nat > product_prod_nat_nat) > int > int).
% 52.96/13.03 thf(func_def_272, type, map_fu4333342158222067775at_rat: (rat > product_prod_int_int) > ((product_prod_int_int > product_prod_int_int) > rat > rat) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > rat > rat > rat).
% 52.96/13.03 thf(func_def_273, type, map_fu898904425404107465nt_o_o: (rat > product_prod_int_int) > ($o > $o) > (product_prod_int_int > $o) > rat > $o).
% 52.96/13.03 thf(func_def_274, type, map_fu5673905371560938248nt_rat: (rat > product_prod_int_int) > (product_prod_int_int > rat) > (product_prod_int_int > product_prod_int_int) > rat > rat).
% 52.96/13.03 thf(func_def_275, type, map_fu1532550112467129777l_real: (real > nat > rat) > (((nat > rat) > nat > rat) > real > real) > ((nat > rat) > (nat > rat) > nat > rat) > real > real > real).
% 52.96/13.03 thf(func_def_276, type, map_fu7146612038024189824t_real: (real > nat > rat) > ((nat > rat) > real) > ((nat > rat) > nat > rat) > real > real).
% 52.96/13.03 thf(func_def_277, type, map_fu1856342031159181835at_o_o: (real > nat > rat) > ($o > $o) > ((nat > rat) > $o) > real > $o).
% 52.96/13.03 thf(func_def_278, type, the_in5290026491893676941l_real: set_real > (real > real) > real > real).
% 52.96/13.03 thf(func_def_279, type, fun_is_measure_int: (int > nat) > $o).
% 52.96/13.03 thf(func_def_280, type, fun_pair_leq: set_Pr8693737435421807431at_nat).
% 52.96/13.03 thf(func_def_281, type, fun_pair_less: set_Pr8693737435421807431at_nat).
% 52.96/13.03 thf(func_def_282, type, gcd_Gcd_int: set_int > int).
% 52.96/13.03 thf(func_def_283, type, gcd_Gcd_nat: set_nat > nat).
% 52.96/13.03 thf(func_def_284, type, bezw: nat > nat > product_prod_int_int).
% 52.96/13.03 thf(func_def_285, type, bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 52.96/13.03 thf(func_def_286, type, gcd_gcd_Code_integer: code_integer > code_integer > code_integer).
% 52.96/13.03 thf(func_def_287, type, gcd_gcd_int: int > int > int).
% 52.96/13.03 thf(func_def_288, type, gcd_gcd_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_289, type, gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 52.96/13.03 thf(func_def_290, type, abs_abs_Code_integer: code_integer > code_integer).
% 52.96/13.03 thf(func_def_291, type, abs_abs_complex: complex > complex).
% 52.96/13.03 thf(func_def_292, type, abs_abs_int: int > int).
% 52.96/13.03 thf(func_def_293, type, abs_abs_rat: rat > rat).
% 52.96/13.03 thf(func_def_294, type, abs_abs_real: real > real).
% 52.96/13.03 thf(func_def_295, type, minus_8373710615458151222nteger: code_integer > code_integer > code_integer).
% 52.96/13.03 thf(func_def_296, type, minus_minus_complex: complex > complex > complex).
% 52.96/13.03 thf(func_def_297, type, minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat).
% 52.96/13.03 thf(func_def_298, type, minus_minus_int: int > int > int).
% 52.96/13.03 thf(func_def_299, type, minus_minus_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_300, type, minus_minus_rat: rat > rat > rat).
% 52.96/13.03 thf(func_def_301, type, minus_minus_real: real > real > real).
% 52.96/13.03 thf(func_def_302, type, minus_811609699411566653omplex: set_complex > set_complex > set_complex).
% 52.96/13.03 thf(func_def_303, type, minus_minus_set_int: set_int > set_int > set_int).
% 52.96/13.03 thf(func_def_304, type, minus_minus_set_nat: set_nat > set_nat > set_nat).
% 52.96/13.03 thf(func_def_305, type, minus_minus_set_real: set_real > set_real > set_real).
% 52.96/13.03 thf(func_def_306, type, minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat).
% 52.96/13.03 thf(func_def_307, type, one_one_Code_integer: code_integer).
% 52.96/13.03 thf(func_def_308, type, one_one_complex: complex).
% 52.96/13.03 thf(func_def_309, type, one_on7984719198319812577d_enat: extended_enat).
% 52.96/13.03 thf(func_def_310, type, one_one_int: int).
% 52.96/13.03 thf(func_def_311, type, one_one_nat: nat).
% 52.96/13.03 thf(func_def_312, type, one_one_rat: rat).
% 52.96/13.03 thf(func_def_313, type, one_one_real: real).
% 52.96/13.03 thf(func_def_314, type, plus_p5714425477246183910nteger: code_integer > code_integer > code_integer).
% 52.96/13.03 thf(func_def_315, type, plus_plus_complex: complex > complex > complex).
% 52.96/13.03 thf(func_def_316, type, plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat).
% 52.96/13.03 thf(func_def_317, type, plus_plus_int: int > int > int).
% 52.96/13.03 thf(func_def_318, type, plus_plus_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_319, type, plus_plus_num: num > num > num).
% 52.96/13.03 thf(func_def_320, type, plus_plus_rat: rat > rat > rat).
% 52.96/13.03 thf(func_def_321, type, plus_plus_real: real > real > real).
% 52.96/13.03 thf(func_def_322, type, sgn_sgn_Code_integer: code_integer > code_integer).
% 52.96/13.03 thf(func_def_323, type, sgn_sgn_complex: complex > complex).
% 52.96/13.03 thf(func_def_324, type, sgn_sgn_int: int > int).
% 52.96/13.03 thf(func_def_325, type, sgn_sgn_rat: rat > rat).
% 52.96/13.03 thf(func_def_326, type, sgn_sgn_real: real > real).
% 52.96/13.03 thf(func_def_327, type, times_3573771949741848930nteger: code_integer > code_integer > code_integer).
% 52.96/13.03 thf(func_def_328, type, times_times_complex: complex > complex > complex).
% 52.96/13.03 thf(func_def_329, type, times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat).
% 52.96/13.03 thf(func_def_330, type, times_times_int: int > int > int).
% 52.96/13.03 thf(func_def_331, type, times_times_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_332, type, times_times_num: num > num > num).
% 52.96/13.03 thf(func_def_333, type, times_times_rat: rat > rat > rat).
% 52.96/13.03 thf(func_def_334, type, times_times_real: real > real > real).
% 52.96/13.03 thf(func_def_335, type, uminus1351360451143612070nteger: code_integer > code_integer).
% 52.96/13.03 thf(func_def_336, type, uminus1482373934393186551omplex: complex > complex).
% 52.96/13.03 thf(func_def_337, type, uminus_uminus_int: int > int).
% 52.96/13.03 thf(func_def_338, type, uminus_uminus_rat: rat > rat).
% 52.96/13.03 thf(func_def_339, type, uminus_uminus_real: real > real).
% 52.96/13.03 thf(func_def_340, type, uminus8566677241136511917omplex: set_complex > set_complex).
% 52.96/13.03 thf(func_def_341, type, uminus1532241313380277803et_int: set_int > set_int).
% 52.96/13.03 thf(func_def_342, type, uminus5710092332889474511et_nat: set_nat > set_nat).
% 52.96/13.03 thf(func_def_343, type, uminus612125837232591019t_real: set_real > set_real).
% 52.96/13.03 thf(func_def_344, type, uminus613421341184616069et_nat: set_set_nat > set_set_nat).
% 52.96/13.03 thf(func_def_345, type, zero_z3403309356797280102nteger: code_integer).
% 52.96/13.03 thf(func_def_346, type, zero_zero_complex: complex).
% 52.96/13.03 thf(func_def_347, type, zero_z5237406670263579293d_enat: extended_enat).
% 52.96/13.03 thf(func_def_348, type, zero_zero_int: int).
% 52.96/13.03 thf(func_def_349, type, zero_zero_nat: nat).
% 52.96/13.03 thf(func_def_350, type, zero_zero_rat: rat).
% 52.96/13.03 thf(func_def_351, type, zero_zero_real: real).
% 52.96/13.03 thf(func_def_352, type, groups7754918857620584856omplex: (complex > complex) > set_complex > complex).
% 52.96/13.03 thf(func_def_353, type, groups4538972089207619220nt_int: (int > int) > set_int > int).
% 52.96/13.03 thf(func_def_354, type, groups3542108847815614940at_nat: (nat > nat) > set_nat > nat).
% 52.96/13.03 thf(func_def_355, type, groups6591440286371151544t_real: (nat > real) > set_nat > real).
% 52.96/13.03 thf(func_def_356, type, groups1705073143266064639nt_int: (int > int) > set_int > int).
% 52.96/13.03 thf(func_def_357, type, groups705719431365010083at_int: (nat > int) > set_nat > int).
% 52.96/13.03 thf(func_def_358, type, groups708209901874060359at_nat: (nat > nat) > set_nat > nat).
% 52.96/13.03 thf(func_def_359, type, groups3417619833198082522nteger: ($o > code_integer) > code_integer > list_o > code_integer).
% 52.96/13.03 thf(func_def_360, type, groups9116527308978886569_o_int: ($o > int) > int > list_o > int).
% 52.96/13.03 thf(func_def_361, type, groups9119017779487936845_o_nat: ($o > nat) > nat > list_o > nat).
% 52.96/13.03 thf(func_def_362, type, groups4561878855575611511st_nat: list_nat > nat).
% 52.96/13.03 thf(func_def_363, type, the_int: (int > $o) > int).
% 52.96/13.03 thf(func_def_364, type, the_Pr4378521158711661632nt_int: (product_prod_int_int > $o) > product_prod_int_int).
% 52.96/13.03 thf(func_def_365, type, the_real: (real > $o) > real).
% 52.96/13.03 thf(func_def_366, type, if_int_int: $o > (int > int) > (int > int) > int > int).
% 52.96/13.03 thf(func_def_367, type, if_nat5617392847756311170nteger: $o > (nat > code_integer > code_integer) > (nat > code_integer > code_integer) > nat > code_integer > code_integer).
% 52.96/13.03 thf(func_def_368, type, if_nat_int_int: $o > (nat > int > int) > (nat > int > int) > nat > int > int).
% 52.96/13.03 thf(func_def_369, type, if_nat_nat_nat: $o > (nat > nat > nat) > (nat > nat > nat) > nat > nat > nat).
% 52.96/13.03 thf(func_def_370, type, if_nat_rat: $o > (nat > rat) > (nat > rat) > nat > rat).
% 52.96/13.03 thf(func_def_371, type, if_Code_integer: $o > code_integer > code_integer > code_integer).
% 52.96/13.03 thf(func_def_372, type, if_complex: $o > complex > complex > complex).
% 52.96/13.03 thf(func_def_373, type, if_int: $o > int > int > int).
% 52.96/13.03 thf(func_def_374, type, if_list_int: $o > list_int > list_int > list_int).
% 52.96/13.03 thf(func_def_375, type, if_list_nat: $o > list_nat > list_nat > list_nat).
% 52.96/13.03 thf(func_def_376, type, if_nat: $o > nat > nat > nat).
% 52.96/13.03 thf(func_def_377, type, if_num: $o > num > num > num).
% 52.96/13.03 thf(func_def_378, type, if_option_num: $o > option_num > option_num > option_num).
% 52.96/13.03 thf(func_def_379, type, if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o).
% 52.96/13.03 thf(func_def_380, type, if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 52.96/13.03 thf(func_def_381, type, if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int).
% 52.96/13.03 thf(func_def_382, type, if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat).
% 52.96/13.03 thf(func_def_383, type, if_rat: $o > rat > rat > rat).
% 52.96/13.03 thf(func_def_384, type, if_real: $o > real > real > real).
% 52.96/13.03 thf(func_def_385, type, infini8530281810654367211te_nat: set_nat > nat > nat).
% 52.96/13.03 thf(func_def_386, type, abs_Integ: product_prod_nat_nat > int).
% 52.96/13.03 thf(func_def_387, type, rep_Integ: int > product_prod_nat_nat).
% 52.96/13.03 thf(func_def_388, type, int_ge_less_than: int > set_Pr958786334691620121nt_int).
% 52.96/13.03 thf(func_def_389, type, int_ge_less_than2: int > set_Pr958786334691620121nt_int).
% 52.96/13.03 thf(func_def_390, type, intrel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 52.96/13.03 thf(func_def_391, type, nat2: int > nat).
% 52.96/13.03 thf(func_def_392, type, pcr_int: product_prod_nat_nat > int > $o).
% 52.96/13.03 thf(func_def_393, type, power_int_real: real > int > real).
% 52.96/13.03 thf(func_def_394, type, ring_11222124179247155820nteger: set_Code_integer).
% 52.96/13.03 thf(func_def_395, type, ring_1_Ints_complex: set_complex).
% 52.96/13.03 thf(func_def_396, type, ring_1_Ints_int: set_int).
% 52.96/13.03 thf(func_def_397, type, ring_1_Ints_rat: set_rat).
% 52.96/13.03 thf(func_def_398, type, ring_1_Ints_real: set_real).
% 52.96/13.03 thf(func_def_399, type, ring_18347121197199848620nteger: int > code_integer).
% 52.96/13.03 thf(func_def_400, type, ring_17405671764205052669omplex: int > complex).
% 52.96/13.03 thf(func_def_401, type, ring_1_of_int_int: int > int).
% 52.96/13.03 thf(func_def_402, type, ring_1_of_int_rat: int > rat).
% 52.96/13.03 thf(func_def_403, type, ring_1_of_int_real: int > real).
% 52.96/13.03 thf(func_def_404, type, inf_in1870772243966228564d_enat: extended_enat > extended_enat > extended_enat).
% 52.96/13.03 thf(func_def_405, type, inf_inf_int: int > int > int).
% 52.96/13.03 thf(func_def_406, type, inf_inf_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_407, type, semila1623282765462674594er_nat: (nat > nat > nat) > nat > (nat > nat > $o) > (nat > nat > $o) > $o).
% 52.96/13.03 thf(func_def_408, type, sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat).
% 52.96/13.03 thf(func_def_409, type, sup_sup_int: int > int > int).
% 52.96/13.03 thf(func_def_410, type, sup_sup_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_411, type, sup_sup_set_nat: set_nat > set_nat > set_nat).
% 52.96/13.03 thf(func_def_412, type, sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 52.96/13.03 thf(func_def_413, type, lattic8263393255366662781ax_int: set_int > int).
% 52.96/13.03 thf(func_def_414, type, lattic8265883725875713057ax_nat: set_nat > nat).
% 52.96/13.03 thf(func_def_415, type, bfun_nat_real: (nat > real) > filter_nat > $o).
% 52.96/13.03 thf(func_def_416, type, at_infinity_real: filter_real).
% 52.96/13.03 thf(func_def_417, type, append_int: list_int > list_int > list_int).
% 52.96/13.03 thf(func_def_418, type, append_nat: list_nat > list_nat > list_nat).
% 52.96/13.03 thf(func_def_419, type, concat_o: list_list_o > list_o).
% 52.96/13.03 thf(func_def_420, type, concat_int: list_list_int > list_int).
% 52.96/13.03 thf(func_def_421, type, concat_nat: list_list_nat > list_nat).
% 52.96/13.03 thf(func_def_422, type, concat_real: list_list_real > list_real).
% 52.96/13.03 thf(func_def_423, type, foldr_nat_nat: (nat > nat > nat) > list_nat > nat > nat).
% 52.96/13.03 thf(func_def_424, type, foldr_real_real: (real > real > real) > list_real > real > real).
% 52.96/13.03 thf(func_def_425, type, last_nat: list_nat > nat).
% 52.96/13.03 thf(func_def_426, type, linord2614967742042102400et_nat: set_nat > list_nat).
% 52.96/13.03 thf(func_def_427, type, cons_int: int > list_int > list_int).
% 52.96/13.03 thf(func_def_428, type, cons_nat: nat > list_nat > list_nat).
% 52.96/13.03 thf(func_def_429, type, nil_int: list_int).
% 52.96/13.03 thf(func_def_430, type, nil_nat: list_nat).
% 52.96/13.03 thf(func_def_431, type, map_complex_real: (complex > real) > list_complex > list_real).
% 52.96/13.03 thf(func_def_432, type, map_int_real: (int > real) > list_int > list_real).
% 52.96/13.03 thf(func_def_433, type, map_nat_nat: (nat > nat) > list_nat > list_nat).
% 52.96/13.03 thf(func_def_434, type, map_nat_real: (nat > real) > list_nat > list_real).
% 52.96/13.03 thf(func_def_435, type, map_real_real: (real > real) > list_real > list_real).
% 52.96/13.03 thf(func_def_436, type, map_set_nat_real: (set_nat > real) > list_set_nat > list_real).
% 52.96/13.03 thf(func_def_437, type, map_VEBT_VEBT_nat: (vEBT_VEBT > nat) > list_VEBT_VEBT > list_nat).
% 52.96/13.03 thf(func_def_438, type, map_VEBT_VEBT_real: (vEBT_VEBT > real) > list_VEBT_VEBT > list_real).
% 52.96/13.03 thf(func_def_439, type, set_o2: list_o > set_o).
% 52.96/13.03 thf(func_def_440, type, set_complex2: list_complex > set_complex).
% 52.96/13.03 thf(func_def_441, type, set_int2: list_int > set_int).
% 52.96/13.03 thf(func_def_442, type, set_list_o2: list_list_o > set_list_o).
% 52.96/13.03 thf(func_def_443, type, set_list_int2: list_list_int > set_list_int).
% 52.96/13.03 thf(func_def_444, type, set_list_nat2: list_list_nat > set_list_nat).
% 52.96/13.03 thf(func_def_445, type, set_list_real2: list_list_real > set_list_real).
% 52.96/13.03 thf(func_def_446, type, set_nat2: list_nat > set_nat).
% 52.96/13.03 thf(func_def_447, type, set_real2: list_real > set_real).
% 52.96/13.03 thf(func_def_448, type, set_set_nat2: list_set_nat > set_set_nat).
% 52.96/13.03 thf(func_def_449, type, set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT).
% 52.96/13.03 thf(func_def_450, type, size_list_VEBT_VEBT: (vEBT_VEBT > nat) > list_VEBT_VEBT > nat).
% 52.96/13.03 thf(func_def_451, type, tl_nat: list_nat > list_nat).
% 52.96/13.03 thf(func_def_452, type, nth_o: list_o > nat > $o).
% 52.96/13.03 thf(func_def_453, type, nth_complex: list_complex > nat > complex).
% 52.96/13.03 thf(func_def_454, type, nth_int: list_int > nat > int).
% 52.96/13.03 thf(func_def_455, type, nth_nat: list_nat > nat > nat).
% 52.96/13.03 thf(func_def_456, type, nth_real: list_real > nat > real).
% 52.96/13.03 thf(func_def_457, type, nth_set_nat: list_set_nat > nat > set_nat).
% 52.96/13.03 thf(func_def_458, type, nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT).
% 52.96/13.03 thf(func_def_459, type, replicate_complex: nat > complex > list_complex).
% 52.96/13.03 thf(func_def_460, type, replicate_int: nat > int > list_int).
% 52.96/13.03 thf(func_def_461, type, replicate_nat: nat > nat > list_nat).
% 52.96/13.03 thf(func_def_462, type, replicate_real: nat > real > list_real).
% 52.96/13.03 thf(func_def_463, type, replicate_set_nat: nat > set_nat > list_set_nat).
% 52.96/13.03 thf(func_def_464, type, replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT).
% 52.96/13.03 thf(func_def_465, type, sorted_wrt_int: (int > int > $o) > list_int > $o).
% 52.96/13.03 thf(func_def_466, type, sorted_wrt_nat: (nat > nat > $o) > list_nat > $o).
% 52.96/13.03 thf(func_def_467, type, subseqs_o: list_o > list_list_o).
% 52.96/13.03 thf(func_def_468, type, subseqs_int: list_int > list_list_int).
% 52.96/13.03 thf(func_def_469, type, subseqs_nat: list_nat > list_list_nat).
% 52.96/13.03 thf(func_def_470, type, subseqs_real: list_real > list_list_real).
% 52.96/13.03 thf(func_def_471, type, take_nat: nat > list_nat > list_nat).
% 52.96/13.03 thf(func_def_472, type, upt: nat > nat > list_nat).
% 52.96/13.03 thf(func_def_473, type, upto: int > int > list_int).
% 52.96/13.03 thf(func_def_474, type, upto_aux: int > int > list_int > list_int).
% 52.96/13.03 thf(func_def_475, type, upto_rel: product_prod_int_int > product_prod_int_int > $o).
% 52.96/13.03 thf(func_def_476, type, suc: nat > nat).
% 52.96/13.03 thf(func_def_477, type, compow_nat_nat: nat > (nat > nat) > nat > nat).
% 52.96/13.03 thf(func_def_478, type, case_nat_o: $o > (nat > $o) > nat > $o).
% 52.96/13.03 thf(func_def_479, type, case_nat_nat: nat > (nat > nat) > nat > nat).
% 52.96/13.03 thf(func_def_480, type, case_nat_option_num: option_num > (nat > option_num) > nat > option_num).
% 52.96/13.03 thf(func_def_481, type, pred: nat > nat).
% 52.96/13.03 thf(func_def_482, type, semiri3842193898606819883omplex: set_complex).
% 52.96/13.03 thf(func_def_483, type, semiring_1_Nats_int: set_int).
% 52.96/13.03 thf(func_def_484, type, semiri4939895301339042750nteger: nat > code_integer).
% 52.96/13.03 thf(func_def_485, type, semiri8010041392384452111omplex: nat > complex).
% 52.96/13.03 thf(func_def_486, type, semiri4216267220026989637d_enat: nat > extended_enat).
% 52.96/13.03 thf(func_def_487, type, semiri1314217659103216013at_int: nat > int).
% 52.96/13.03 thf(func_def_488, type, semiri1316708129612266289at_nat: nat > nat).
% 52.96/13.03 thf(func_def_489, type, semiri681578069525770553at_rat: nat > rat).
% 52.96/13.03 thf(func_def_490, type, semiri5074537144036343181t_real: nat > real).
% 52.96/13.03 thf(func_def_491, type, size_size_list_o: list_o > nat).
% 52.96/13.03 thf(func_def_492, type, size_s3451745648224563538omplex: list_complex > nat).
% 52.96/13.03 thf(func_def_493, type, size_size_list_int: list_int > nat).
% 52.96/13.03 thf(func_def_494, type, size_s2710708370519433104list_o: list_list_o > nat).
% 52.96/13.03 thf(func_def_495, type, size_s533118279054570080st_int: list_list_int > nat).
% 52.96/13.03 thf(func_def_496, type, size_s3023201423986296836st_nat: list_list_nat > nat).
% 52.96/13.03 thf(func_def_497, type, size_s6660260683639930848t_real: list_list_real > nat).
% 52.96/13.03 thf(func_def_498, type, size_size_list_nat: list_nat > nat).
% 52.96/13.03 thf(func_def_499, type, size_size_list_real: list_real > nat).
% 52.96/13.03 thf(func_def_500, type, size_s3254054031482475050et_nat: list_set_nat > nat).
% 52.96/13.03 thf(func_def_501, type, size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat).
% 52.96/13.03 thf(func_def_502, type, size_size_num: num > nat).
% 52.96/13.03 thf(func_def_503, type, size_size_char: char > nat).
% 52.96/13.03 thf(func_def_504, type, size_size_VEBT_VEBT: vEBT_VEBT > nat).
% 52.96/13.03 thf(func_def_505, type, nat_int_decode: nat > int).
% 52.96/13.03 thf(func_def_506, type, nat_int_encode: int > nat).
% 52.96/13.03 thf(func_def_507, type, nat_list_encode: list_nat > nat).
% 52.96/13.03 thf(func_def_508, type, nat_list_encode_rel: list_nat > list_nat > $o).
% 52.96/13.03 thf(func_def_509, type, nat_prod_decode_aux: nat > nat > product_prod_nat_nat).
% 52.96/13.03 thf(func_def_510, type, nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 52.96/13.03 thf(func_def_511, type, nat_prod_encode: product_prod_nat_nat > nat).
% 52.96/13.03 thf(func_def_512, type, nat_set_decode: nat > set_nat).
% 52.96/13.03 thf(func_def_513, type, nat_set_encode: set_nat > nat).
% 52.96/13.03 thf(func_def_514, type, nat_triangle: nat > nat).
% 52.96/13.03 thf(func_def_515, type, root: nat > real > real).
% 52.96/13.03 thf(func_def_516, type, sqrt: real > real).
% 52.96/13.03 thf(func_def_517, type, bitM: num > num).
% 52.96/13.03 thf(func_def_518, type, inc: num > num).
% 52.96/13.03 thf(func_def_519, type, neg_nu8804712462038260780nteger: code_integer > code_integer).
% 52.96/13.03 thf(func_def_520, type, neg_nu7009210354673126013omplex: complex > complex).
% 52.96/13.03 thf(func_def_521, type, neg_numeral_dbl_int: int > int).
% 52.96/13.03 thf(func_def_522, type, neg_numeral_dbl_rat: rat > rat).
% 52.96/13.03 thf(func_def_523, type, neg_numeral_dbl_real: real > real).
% 52.96/13.03 thf(func_def_524, type, neg_nu7757733837767384882nteger: code_integer > code_integer).
% 52.96/13.03 thf(func_def_525, type, neg_nu6511756317524482435omplex: complex > complex).
% 52.96/13.03 thf(func_def_526, type, neg_nu3811975205180677377ec_int: int > int).
% 52.96/13.03 thf(func_def_527, type, neg_nu3179335615603231917ec_rat: rat > rat).
% 52.96/13.03 thf(func_def_528, type, neg_nu6075765906172075777c_real: real > real).
% 52.96/13.03 thf(func_def_529, type, neg_nu5831290666863070958nteger: code_integer > code_integer).
% 52.96/13.03 thf(func_def_530, type, neg_nu8557863876264182079omplex: complex > complex).
% 52.96/13.03 thf(func_def_531, type, neg_nu5851722552734809277nc_int: int > int).
% 52.96/13.03 thf(func_def_532, type, neg_nu5219082963157363817nc_rat: rat > rat).
% 52.96/13.03 thf(func_def_533, type, neg_nu8295874005876285629c_real: real > real).
% 52.96/13.03 thf(func_def_534, type, neg_numeral_sub_int: num > num > int).
% 52.96/13.03 thf(func_def_535, type, bit0: num > num).
% 52.96/13.03 thf(func_def_536, type, bit1: num > num).
% 52.96/13.03 thf(func_def_537, type, one: num).
% 52.96/13.03 thf(func_def_538, type, case_num_option_num: option_num > (num > option_num) > (num > option_num) > num > option_num).
% 52.96/13.03 thf(func_def_539, type, size_num: num > nat).
% 52.96/13.03 thf(func_def_540, type, num_of_nat: nat > num).
% 52.96/13.03 thf(func_def_541, type, numera6620942414471956472nteger: num > code_integer).
% 52.96/13.03 thf(func_def_542, type, numera6690914467698888265omplex: num > complex).
% 52.96/13.03 thf(func_def_543, type, numera1916890842035813515d_enat: num > extended_enat).
% 52.96/13.03 thf(func_def_544, type, numeral_numeral_int: num > int).
% 52.96/13.03 thf(func_def_545, type, numeral_numeral_nat: num > nat).
% 52.96/13.03 thf(func_def_546, type, numeral_numeral_rat: num > rat).
% 52.96/13.03 thf(func_def_547, type, numeral_numeral_real: num > real).
% 52.96/13.03 thf(func_def_548, type, pow: num > num > num).
% 52.96/13.03 thf(func_def_549, type, pred_numeral: num > nat).
% 52.96/13.03 thf(func_def_550, type, sqr: num > num).
% 52.96/13.03 thf(func_def_551, type, none_num: option_num).
% 52.96/13.03 thf(func_def_552, type, none_P5556105721700978146at_nat: option4927543243414619207at_nat).
% 52.96/13.03 thf(func_def_553, type, some_num: num > option_num).
% 52.96/13.03 thf(func_def_554, type, some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat).
% 52.96/13.03 thf(func_def_555, type, case_o184042715313410164at_nat: $o > (product_prod_nat_nat > $o) > option4927543243414619207at_nat > $o).
% 52.96/13.03 thf(func_def_556, type, case_option_int_num: int > (num > int) > option_num > int).
% 52.96/13.03 thf(func_def_557, type, case_option_num_num: num > (num > num) > option_num > num).
% 52.96/13.03 thf(func_def_558, type, case_o6005452278849405969um_num: option_num > (num > option_num) > option_num > option_num).
% 52.96/13.03 thf(func_def_559, type, map_option_num_num: (num > num) > option_num > option_num).
% 52.96/13.03 thf(func_def_560, type, bot_bo4199563552545308370d_enat: extended_enat).
% 52.96/13.03 thf(func_def_561, type, bot_bot_nat: nat).
% 52.96/13.03 thf(func_def_562, type, bot_bot_set_o: set_o).
% 52.96/13.03 thf(func_def_563, type, bot_bot_set_int: set_int).
% 52.96/13.03 thf(func_def_564, type, bot_bot_set_nat: set_nat).
% 52.96/13.03 thf(func_def_565, type, bot_bot_set_real: set_real).
% 52.96/13.03 thf(func_def_566, type, ord_Least_nat: (nat > $o) > nat).
% 52.96/13.03 thf(func_def_567, type, ord_Least_real: (real > $o) > real).
% 52.96/13.03 thf(func_def_568, type, ord_le6747313008572928689nteger: code_integer > code_integer > $o).
% 52.96/13.03 thf(func_def_569, type, ord_le72135733267957522d_enat: extended_enat > extended_enat > $o).
% 52.96/13.03 thf(func_def_570, type, ord_less_int: int > int > $o).
% 52.96/13.03 thf(func_def_571, type, ord_less_nat: nat > nat > $o).
% 52.96/13.03 thf(func_def_572, type, ord_less_num: num > num > $o).
% 52.96/13.03 thf(func_def_573, type, ord_less_rat: rat > rat > $o).
% 52.96/13.03 thf(func_def_574, type, ord_less_real: real > real > $o).
% 52.96/13.03 thf(func_def_575, type, ord_less_set_complex: set_complex > set_complex > $o).
% 52.96/13.03 thf(func_def_576, type, ord_less_set_int: set_int > set_int > $o).
% 52.96/13.03 thf(func_def_577, type, ord_less_set_nat: set_nat > set_nat > $o).
% 52.96/13.03 thf(func_def_578, type, ord_less_set_real: set_real > set_real > $o).
% 52.96/13.03 thf(func_def_579, type, ord_less_set_set_nat: set_set_nat > set_set_nat > $o).
% 52.96/13.03 thf(func_def_580, type, ord_le3102999989581377725nteger: code_integer > code_integer > $o).
% 52.96/13.03 thf(func_def_581, type, ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o).
% 52.96/13.03 thf(func_def_582, type, ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o).
% 52.96/13.03 thf(func_def_583, type, ord_le4104064031414453916r_real: filter_real > filter_real > $o).
% 52.96/13.03 thf(func_def_584, type, ord_less_eq_int: int > int > $o).
% 52.96/13.03 thf(func_def_585, type, ord_less_eq_nat: nat > nat > $o).
% 52.96/13.03 thf(func_def_586, type, ord_less_eq_num: num > num > $o).
% 52.96/13.03 thf(func_def_587, type, ord_less_eq_rat: rat > rat > $o).
% 52.96/13.03 thf(func_def_588, type, ord_less_eq_real: real > real > $o).
% 52.96/13.03 thf(func_def_589, type, ord_le211207098394363844omplex: set_complex > set_complex > $o).
% 52.96/13.03 thf(func_def_590, type, ord_less_eq_set_int: set_int > set_int > $o).
% 52.96/13.03 thf(func_def_591, type, ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o).
% 52.96/13.03 thf(func_def_592, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 52.96/13.03 thf(func_def_593, type, ord_less_eq_set_real: set_real > set_real > $o).
% 52.96/13.03 thf(func_def_594, type, ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o).
% 52.96/13.03 thf(func_def_595, type, ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o).
% 52.96/13.03 thf(func_def_596, type, ord_max_Code_integer: code_integer > code_integer > code_integer).
% 52.96/13.03 thf(func_def_597, type, ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat).
% 52.96/13.03 thf(func_def_598, type, ord_max_int: int > int > int).
% 52.96/13.03 thf(func_def_599, type, ord_max_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_600, type, ord_max_num: num > num > num).
% 52.96/13.03 thf(func_def_601, type, ord_max_rat: rat > rat > rat).
% 52.96/13.03 thf(func_def_602, type, ord_max_real: real > real > real).
% 52.96/13.03 thf(func_def_603, type, ord_max_set_int: set_int > set_int > set_int).
% 52.96/13.03 thf(func_def_604, type, ord_mi8085742599997312461d_enat: extended_enat > extended_enat > extended_enat).
% 52.96/13.03 thf(func_def_605, type, ord_min_int: int > int > int).
% 52.96/13.03 thf(func_def_606, type, ord_min_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_607, type, order_Greatest_nat: (nat > $o) > nat).
% 52.96/13.03 thf(func_def_608, type, order_mono_nat_nat: (nat > nat) > $o).
% 52.96/13.03 thf(func_def_609, type, order_mono_nat_real: (nat > real) > $o).
% 52.96/13.03 thf(func_def_610, type, order_mono_real_real: (real > real) > $o).
% 52.96/13.03 thf(func_def_611, type, order_5726023648592871131at_nat: (nat > nat) > $o).
% 52.96/13.03 thf(func_def_612, type, order_7092887310737990675l_real: (real > real) > $o).
% 52.96/13.03 thf(func_def_613, type, ordering_top_nat: (nat > nat > $o) > (nat > nat > $o) > nat > $o).
% 52.96/13.03 thf(func_def_614, type, top_top_set_o: set_o).
% 52.96/13.03 thf(func_def_615, type, top_top_set_int: set_int).
% 52.96/13.03 thf(func_def_616, type, top_top_set_list_nat: set_list_nat).
% 52.96/13.03 thf(func_def_617, type, top_top_set_nat: set_nat).
% 52.96/13.03 thf(func_def_618, type, top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat).
% 52.96/13.03 thf(func_def_619, type, top_top_set_real: set_real).
% 52.96/13.03 thf(func_def_620, type, top_top_set_char: set_char).
% 52.96/13.03 thf(func_def_621, type, power_8256067586552552935nteger: code_integer > nat > code_integer).
% 52.96/13.03 thf(func_def_622, type, power_power_complex: complex > nat > complex).
% 52.96/13.03 thf(func_def_623, type, power_power_int: int > nat > int).
% 52.96/13.03 thf(func_def_624, type, power_power_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_625, type, power_power_rat: rat > nat > rat).
% 52.96/13.03 thf(func_def_626, type, power_power_real: real > nat > real).
% 52.96/13.03 thf(func_def_627, type, produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o).
% 52.96/13.03 thf(func_def_628, type, produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger).
% 52.96/13.03 thf(func_def_629, type, product_Pair_int_int: int > int > product_prod_int_int).
% 52.96/13.03 thf(func_def_630, type, product_Pair_nat_nat: nat > nat > product_prod_nat_nat).
% 52.96/13.03 thf(func_def_631, type, product_Pair_nat_num: nat > num > product_prod_nat_num).
% 52.96/13.03 thf(func_def_632, type, product_Pair_num_num: num > num > product_prod_num_num).
% 52.96/13.03 thf(func_def_633, type, produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat).
% 52.96/13.03 thf(func_def_634, type, produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat).
% 52.96/13.03 thf(func_def_635, type, produc457027306803732586at_nat: set_nat > (nat > set_nat) > set_Pr1261947904930325089at_nat).
% 52.96/13.03 thf(func_def_636, type, produc6499014454317279255nteger: (code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 52.96/13.03 thf(func_def_637, type, produc1553301316500091796er_int: (code_integer > code_integer > int) > produc8923325533196201883nteger > int).
% 52.96/13.03 thf(func_def_638, type, produc1555791787009142072er_nat: (code_integer > code_integer > nat) > produc8923325533196201883nteger > nat).
% 52.96/13.03 thf(func_def_639, type, produc7336495610019696514er_num: (code_integer > code_integer > num) > produc8923325533196201883nteger > num).
% 52.96/13.03 thf(func_def_640, type, produc9125791028180074456eger_o: (code_integer > code_integer > produc6271795597528267376eger_o) > produc8923325533196201883nteger > produc6271795597528267376eger_o).
% 52.96/13.03 thf(func_def_641, type, produc6916734918728496179nteger: (code_integer > code_integer > produc8923325533196201883nteger) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 52.96/13.03 thf(func_def_642, type, produc6771430404735790350plex_o: (complex > complex > $o) > produc4411394909380815293omplex > $o).
% 52.96/13.03 thf(func_def_643, type, produc4947309494688390418_int_o: (int > int > $o) > product_prod_int_int > $o).
% 52.96/13.03 thf(func_def_644, type, produc8211389475949308722nt_int: (int > int > int) > product_prod_int_int > int).
% 52.96/13.03 thf(func_def_645, type, produc4245557441103728435nt_int: (int > int > product_prod_int_int) > product_prod_int_int > product_prod_int_int).
% 52.96/13.03 thf(func_def_646, type, produc8739625826339149834_nat_o: (nat > nat > product_prod_nat_nat > $o) > product_prod_nat_nat > product_prod_nat_nat > $o).
% 52.96/13.03 thf(func_def_647, 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).
% 52.96/13.03 thf(func_def_648, type, produc6081775807080527818_nat_o: (nat > nat > $o) > product_prod_nat_nat > $o).
% 52.96/13.03 thf(func_def_649, type, produc6842872674320459806at_nat: (nat > nat > nat) > product_prod_nat_nat > nat).
% 52.96/13.03 thf(func_def_650, type, produc2626176000494625587at_nat: (nat > nat > product_prod_nat_nat) > product_prod_nat_nat > product_prod_nat_nat).
% 52.96/13.03 thf(func_def_651, type, produc478579273971653890on_num: (nat > num > option_num) > product_prod_nat_num > option_num).
% 52.96/13.03 thf(func_def_652, type, produc5414030515140494994real_o: (real > real > $o) > produc2422161461964618553l_real > $o).
% 52.96/13.03 thf(func_def_653, type, product_fst_int_int: product_prod_int_int > int).
% 52.96/13.03 thf(func_def_654, type, product_fst_nat_nat: product_prod_nat_nat > nat).
% 52.96/13.03 thf(func_def_655, type, product_snd_int_int: product_prod_int_int > int).
% 52.96/13.03 thf(func_def_656, type, product_snd_nat_nat: product_prod_nat_nat > nat).
% 52.96/13.03 thf(func_def_657, type, abs_Rat: product_prod_int_int > rat).
% 52.96/13.03 thf(func_def_658, type, fract: int > int > rat).
% 52.96/13.03 thf(func_def_659, type, frct: product_prod_int_int > rat).
% 52.96/13.03 thf(func_def_660, type, rep_Rat: rat > product_prod_int_int).
% 52.96/13.03 thf(func_def_661, type, field_5140801741446780682s_real: set_real).
% 52.96/13.03 thf(func_def_662, type, field_7254667332652039916t_real: rat > real).
% 52.96/13.03 thf(func_def_663, type, normalize: product_prod_int_int > product_prod_int_int).
% 52.96/13.03 thf(func_def_664, type, pcr_rat: product_prod_int_int > rat > $o).
% 52.96/13.03 thf(func_def_665, type, positive: rat > $o).
% 52.96/13.03 thf(func_def_666, type, quotient_of: rat > product_prod_int_int).
% 52.96/13.03 thf(func_def_667, type, ratrel: product_prod_int_int > product_prod_int_int > $o).
% 52.96/13.03 thf(func_def_668, type, real2: (nat > rat) > real).
% 52.96/13.03 thf(func_def_669, type, cauchy: (nat > rat) > $o).
% 52.96/13.03 thf(func_def_670, type, cr_real: (nat > rat) > real > $o).
% 52.96/13.03 thf(func_def_671, type, pcr_real: (nat > rat) > real > $o).
% 52.96/13.03 thf(func_def_672, type, positive2: real > $o).
% 52.96/13.03 thf(func_def_673, type, realrel: (nat > rat) > (nat > rat) > $o).
% 52.96/13.03 thf(func_def_674, type, rep_real: real > nat > rat).
% 52.96/13.03 thf(func_def_675, type, vanishes: (nat > rat) > $o).
% 52.96/13.03 thf(func_def_676, type, real_V2521375963428798218omplex: set_complex).
% 52.96/13.03 thf(func_def_677, type, real_V470468836141973256s_real: set_real).
% 52.96/13.03 thf(func_def_678, type, real_V5970128139526366754l_real: (real > real) > $o).
% 52.96/13.03 thf(func_def_679, type, real_V3694042436643373181omplex: complex > complex > real).
% 52.96/13.03 thf(func_def_680, type, real_V975177566351809787t_real: real > real > real).
% 52.96/13.03 thf(func_def_681, type, real_V1022390504157884413omplex: complex > real).
% 52.96/13.03 thf(func_def_682, type, real_V7735802525324610683m_real: real > real).
% 52.96/13.03 thf(func_def_683, type, real_V4546457046886955230omplex: real > complex).
% 52.96/13.03 thf(func_def_684, type, real_V1803761363581548252l_real: real > real).
% 52.96/13.03 thf(func_def_685, type, real_V2046097035970521341omplex: real > complex > complex).
% 52.96/13.03 thf(func_def_686, type, real_V1485227260804924795R_real: real > real > real).
% 52.96/13.03 thf(func_def_687, type, algebr932160517623751201me_int: int > int > $o).
% 52.96/13.03 thf(func_def_688, type, algebr934650988132801477me_nat: nat > nat > $o).
% 52.96/13.03 thf(func_def_689, type, divide6298287555418463151nteger: code_integer > code_integer > code_integer).
% 52.96/13.03 thf(func_def_690, type, divide1717551699836669952omplex: complex > complex > complex).
% 52.96/13.03 thf(func_def_691, type, divide_divide_int: int > int > int).
% 52.96/13.03 thf(func_def_692, type, divide_divide_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_693, type, divide_divide_rat: rat > rat > rat).
% 52.96/13.03 thf(func_def_694, type, divide_divide_real: real > real > real).
% 52.96/13.03 thf(func_def_695, type, dvd_dvd_Code_integer: code_integer > code_integer > $o).
% 52.96/13.03 thf(func_def_696, type, dvd_dvd_complex: complex > complex > $o).
% 52.96/13.03 thf(func_def_697, type, dvd_dvd_int: int > int > $o).
% 52.96/13.03 thf(func_def_698, type, dvd_dvd_nat: nat > nat > $o).
% 52.96/13.03 thf(func_def_699, type, dvd_dvd_rat: rat > rat > $o).
% 52.96/13.03 thf(func_def_700, type, dvd_dvd_real: real > real > $o).
% 52.96/13.03 thf(func_def_701, type, modulo364778990260209775nteger: code_integer > code_integer > code_integer).
% 52.96/13.03 thf(func_def_702, type, modulo_modulo_int: int > int > int).
% 52.96/13.03 thf(func_def_703, type, modulo_modulo_nat: nat > nat > nat).
% 52.96/13.03 thf(func_def_704, type, zero_n356916108424825756nteger: $o > code_integer).
% 52.96/13.03 thf(func_def_705, type, zero_n1201886186963655149omplex: $o > complex).
% 52.96/13.03 thf(func_def_706, type, zero_n2684676970156552555ol_int: $o > int).
% 52.96/13.03 thf(func_def_707, type, zero_n2687167440665602831ol_nat: $o > nat).
% 52.96/13.03 thf(func_def_708, type, zero_n2052037380579107095ol_rat: $o > rat).
% 52.96/13.03 thf(func_def_709, type, zero_n3304061248610475627l_real: $o > real).
% 52.96/13.03 thf(func_def_710, type, suminf_real: (nat > real) > real).
% 52.96/13.03 thf(func_def_711, type, summable_real: (nat > real) > $o).
% 52.96/13.03 thf(func_def_712, type, sums_real: (nat > real) > real > $o).
% 52.96/13.03 thf(func_def_713, type, collect_complex: (complex > $o) > set_complex).
% 52.96/13.03 thf(func_def_714, type, collect_int: (int > $o) > set_int).
% 52.96/13.03 thf(func_def_715, type, collect_list_nat: (list_nat > $o) > set_list_nat).
% 52.96/13.03 thf(func_def_716, type, collect_nat: (nat > $o) > set_nat).
% 52.96/13.03 thf(func_def_717, type, collec8663557070575231912omplex: (produc4411394909380815293omplex > $o) > set_Pr5085853215250843933omplex).
% 52.96/13.03 thf(func_def_718, type, collec213857154873943460nt_int: (product_prod_int_int > $o) > set_Pr958786334691620121nt_int).
% 52.96/13.03 thf(func_def_719, type, collec3392354462482085612at_nat: (product_prod_nat_nat > $o) > set_Pr1261947904930325089at_nat).
% 52.96/13.03 thf(func_def_720, type, collec3799799289383736868l_real: (produc2422161461964618553l_real > $o) > set_Pr6218003697084177305l_real).
% 52.96/13.03 thf(func_def_721, type, collect_real: (real > $o) > set_real).
% 52.96/13.03 thf(func_def_722, type, collect_set_nat: (set_nat > $o) > set_set_nat).
% 52.96/13.03 thf(func_def_723, type, pow_nat: set_nat > set_set_nat).
% 52.96/13.03 thf(func_def_724, type, image_int_int: (int > int) > set_int > set_int).
% 52.96/13.03 thf(func_def_725, type, image_int_nat: (int > nat) > set_int > set_nat).
% 52.96/13.03 thf(func_def_726, type, image_list_nat_nat: (list_nat > nat) > set_list_nat > set_nat).
% 52.96/13.03 thf(func_def_727, type, image_nat_int: (nat > int) > set_nat > set_int).
% 52.96/13.03 thf(func_def_728, type, image_nat_nat: (nat > nat) > set_nat > set_nat).
% 52.96/13.03 thf(func_def_729, type, image_nat_real: (nat > real) > set_nat > set_real).
% 52.96/13.03 thf(func_def_730, type, image_nat_set_nat: (nat > set_nat) > set_nat > set_set_nat).
% 52.96/13.03 thf(func_def_731, type, image_nat_char: (nat > char) > set_nat > set_char).
% 52.96/13.03 thf(func_def_732, type, image_2486076414777270412at_nat: (product_prod_nat_nat > nat) > set_Pr1261947904930325089at_nat > set_nat).
% 52.96/13.03 thf(func_def_733, type, image_5971271580939081552omplex: (real > filter6041513312241820739omplex) > set_real > set_fi4554929511873752355omplex).
% 52.96/13.03 thf(func_def_734, type, image_2178119161166701260l_real: (real > filter2146258269922977983l_real) > set_real > set_fi7789364187291644575l_real).
% 52.96/13.03 thf(func_def_735, type, image_real_real: (real > real) > set_real > set_real).
% 52.96/13.03 thf(func_def_736, type, image_char_nat: (char > nat) > set_char > set_nat).
% 52.96/13.03 thf(func_def_737, type, insert_o: $o > set_o > set_o).
% 52.96/13.03 thf(func_def_738, type, insert_int: int > set_int > set_int).
% 52.96/13.03 thf(func_def_739, type, insert_nat: nat > set_nat > set_nat).
% 52.96/13.03 thf(func_def_740, type, insert_real: real > set_real > set_real).
% 52.96/13.03 thf(func_def_741, type, vimage_nat_nat: (nat > nat) > set_nat > set_nat).
% 52.96/13.03 thf(func_def_742, type, set_fo2584398358068434914at_nat: (nat > nat > nat) > nat > nat > nat > nat).
% 52.96/13.03 thf(func_def_743, type, set_or1266510415728281911st_int: int > int > set_int).
% 52.96/13.03 thf(func_def_744, type, set_or1269000886237332187st_nat: nat > nat > set_nat).
% 52.96/13.03 thf(func_def_745, type, set_or1222579329274155063t_real: real > real > set_real).
% 52.96/13.03 thf(func_def_746, type, set_or4662586982721622107an_int: int > int > set_int).
% 52.96/13.03 thf(func_def_747, type, set_or4665077453230672383an_nat: nat > nat > set_nat).
% 52.96/13.03 thf(func_def_748, type, set_ord_atLeast_nat: nat > set_nat).
% 52.96/13.03 thf(func_def_749, type, set_ord_atLeast_real: real > set_real).
% 52.96/13.03 thf(func_def_750, type, set_ord_atMost_int: int > set_int).
% 52.96/13.03 thf(func_def_751, type, set_ord_atMost_nat: nat > set_nat).
% 52.96/13.03 thf(func_def_752, type, set_or6656581121297822940st_int: int > int > set_int).
% 52.96/13.03 thf(func_def_753, type, set_or6659071591806873216st_nat: nat > nat > set_nat).
% 52.96/13.03 thf(func_def_754, type, set_or5832277885323065728an_int: int > int > set_int).
% 52.96/13.03 thf(func_def_755, type, set_or5834768355832116004an_nat: nat > nat > set_nat).
% 52.96/13.03 thf(func_def_756, type, set_or1633881224788618240n_real: real > real > set_real).
% 52.96/13.03 thf(func_def_757, type, set_or1210151606488870762an_nat: nat > set_nat).
% 52.96/13.03 thf(func_def_758, type, set_or5849166863359141190n_real: real > set_real).
% 52.96/13.03 thf(func_def_759, type, set_ord_lessThan_int: int > set_int).
% 52.96/13.03 thf(func_def_760, type, set_ord_lessThan_nat: nat > set_nat).
% 52.96/13.03 thf(func_def_761, type, set_or5984915006950818249n_real: real > set_real).
% 52.96/13.03 thf(func_def_762, type, ascii_of: char > char).
% 52.96/13.03 thf(func_def_763, type, char2: $o > $o > $o > $o > $o > $o > $o > $o > char).
% 52.96/13.03 thf(func_def_764, type, size_char: char > nat).
% 52.96/13.03 thf(func_def_765, type, comm_s629917340098488124ar_nat: char > nat).
% 52.96/13.03 thf(func_def_766, type, integer_of_char: char > code_integer).
% 52.96/13.03 thf(func_def_767, type, unique3096191561947761185of_nat: nat > char).
% 52.96/13.03 thf(func_def_768, type, topolo4422821103128117721l_real: filter_real > (real > real) > $o).
% 52.96/13.03 thf(func_def_769, type, topolo5044208981011980120l_real: set_real > (real > real) > $o).
% 52.96/13.03 thf(func_def_770, type, topolo6980174941875973593q_real: (nat > real) > $o).
% 52.96/13.03 thf(func_def_771, type, topolo2177554685111907308n_real: real > set_real > filter_real).
% 52.96/13.03 thf(func_def_772, type, topolo7531315842566124627t_real: (nat > real) > $o).
% 52.96/13.03 thf(func_def_773, type, topolo2815343760600316023s_real: real > filter_real).
% 52.96/13.03 thf(func_def_774, type, topolo6517432010174082258omplex: (nat > complex) > $o).
% 52.96/13.03 thf(func_def_775, type, topolo4055970368930404560y_real: (nat > real) > $o).
% 52.96/13.03 thf(func_def_776, type, topolo896644834953643431omplex: filter6041513312241820739omplex).
% 52.96/13.03 thf(func_def_777, type, topolo1511823702728130853y_real: filter2146258269922977983l_real).
% 52.96/13.03 thf(func_def_778, type, arccos: real > real).
% 52.96/13.03 thf(func_def_779, type, arcosh_real: real > real).
% 52.96/13.03 thf(func_def_780, type, arcsin: real > real).
% 52.96/13.03 thf(func_def_781, type, arctan: real > real).
% 52.96/13.03 thf(func_def_782, type, arsinh_real: real > real).
% 52.96/13.03 thf(func_def_783, type, artanh_real: real > real).
% 52.96/13.03 thf(func_def_784, type, cos_complex: complex > complex).
% 52.96/13.03 thf(func_def_785, type, cos_real: real > real).
% 52.96/13.03 thf(func_def_786, type, cos_coeff: nat > real).
% 52.96/13.03 thf(func_def_787, type, cosh_complex: complex > complex).
% 52.96/13.03 thf(func_def_788, type, cosh_real: real > real).
% 52.96/13.03 thf(func_def_789, type, cot_complex: complex > complex).
% 52.96/13.03 thf(func_def_790, type, cot_real: real > real).
% 52.96/13.03 thf(func_def_791, type, exp_complex: complex > complex).
% 52.96/13.03 thf(func_def_792, type, exp_real: real > real).
% 52.96/13.03 thf(func_def_793, type, ln_ln_real: real > real).
% 52.96/13.03 thf(func_def_794, type, log: real > real > real).
% 52.96/13.03 thf(func_def_795, type, pi: real).
% 52.96/13.03 thf(func_def_796, type, powr_real: real > real > real).
% 52.96/13.03 thf(func_def_797, type, sin_complex: complex > complex).
% 52.96/13.03 thf(func_def_798, type, sin_real: real > real).
% 52.96/13.03 thf(func_def_799, type, sin_coeff: nat > real).
% 52.96/13.03 thf(func_def_800, type, sinh_complex: complex > complex).
% 52.96/13.03 thf(func_def_801, type, sinh_real: real > real).
% 52.96/13.03 thf(func_def_802, type, tan_complex: complex > complex).
% 52.96/13.03 thf(func_def_803, type, tan_real: real > real).
% 52.96/13.03 thf(func_def_804, type, tanh_complex: complex > complex).
% 52.96/13.03 thf(func_def_805, type, tanh_real: real > real).
% 52.96/13.03 thf(func_def_806, type, bi_tot896582865486249351at_int: (product_prod_nat_nat > int > $o) > $o).
% 52.96/13.03 thf(func_def_807, type, transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 52.96/13.03 thf(func_def_808, type, vEBT_Leaf: $o > $o > vEBT_VEBT).
% 52.96/13.03 thf(func_def_809, type, vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 52.96/13.03 thf(func_def_810, type, vEBT_size_VEBT: vEBT_VEBT > nat).
% 52.96/13.03 thf(func_def_811, type, vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o).
% 52.96/13.03 thf(func_def_812, type, vEBT_VEBT_high: nat > nat > nat).
% 52.96/13.03 thf(func_def_813, type, vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o).
% 52.96/13.03 thf(func_def_814, type, vEBT_VEBT_low: nat > nat > nat).
% 52.96/13.03 thf(func_def_815, type, vEBT_VEBT_membermima: vEBT_VEBT > nat > $o).
% 52.96/13.03 thf(func_def_816, type, vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 52.96/13.03 thf(func_def_817, type, vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o).
% 52.96/13.03 thf(func_def_818, type, vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 52.96/13.03 thf(func_def_819, type, vEBT_VEBT_valid: vEBT_VEBT > nat > $o).
% 52.96/13.03 thf(func_def_820, type, vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 52.96/13.03 thf(func_def_821, type, vEBT_invar_vebt: vEBT_VEBT > nat > $o).
% 52.96/13.03 thf(func_def_822, type, vEBT_set_vebt: vEBT_VEBT > set_nat).
% 52.96/13.03 thf(func_def_823, type, vEBT_vebt_buildup: nat > vEBT_VEBT).
% 52.96/13.03 thf(func_def_824, type, vEBT_v4011308405150292612up_rel: nat > nat > $o).
% 52.96/13.03 thf(func_def_825, type, vEBT_V8646137997579335489_i_l_d: nat > nat).
% 52.96/13.03 thf(func_def_826, type, vEBT_V8346862874174094_d_u_p: nat > nat).
% 52.96/13.03 thf(func_def_827, type, vEBT_V1247956027447740395_p_rel: nat > nat > $o).
% 52.96/13.03 thf(func_def_828, type, vEBT_V5144397997797733112_d_rel: nat > nat > $o).
% 52.96/13.03 thf(func_def_829, type, vEBT_VEBT_cnt: vEBT_VEBT > real).
% 52.96/13.03 thf(func_def_830, type, vEBT_VEBT_cnt_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 52.96/13.03 thf(func_def_831, type, vEBT_VEBT_space: vEBT_VEBT > nat).
% 52.96/13.03 thf(func_def_832, type, vEBT_VEBT_space2: vEBT_VEBT > nat).
% 52.96/13.03 thf(func_def_833, type, vEBT_VEBT_space_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 52.96/13.03 thf(func_def_834, type, vEBT_VEBT_space_rel2: vEBT_VEBT > vEBT_VEBT > $o).
% 52.96/13.03 thf(func_def_835, type, accp_list_nat: (list_nat > list_nat > $o) > list_nat > $o).
% 52.96/13.03 thf(func_def_836, type, accp_nat: (nat > nat > $o) > nat > $o).
% 52.96/13.03 thf(func_def_837, type, accp_P1096762738010456898nt_int: (product_prod_int_int > product_prod_int_int > $o) > product_prod_int_int > $o).
% 52.96/13.03 thf(func_def_838, type, accp_P4275260045618599050at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > product_prod_nat_nat > $o).
% 52.96/13.03 thf(func_def_839, type, accp_P3113834385874906142um_num: (product_prod_num_num > product_prod_num_num > $o) > product_prod_num_num > $o).
% 52.96/13.03 thf(func_def_840, type, accp_P2887432264394892906BT_nat: (produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o) > produc9072475918466114483BT_nat > $o).
% 52.96/13.03 thf(func_def_841, type, accp_VEBT_VEBT: (vEBT_VEBT > vEBT_VEBT > $o) > vEBT_VEBT > $o).
% 52.96/13.03 thf(func_def_842, type, pred_nat: set_Pr1261947904930325089at_nat).
% 52.96/13.03 thf(func_def_843, type, fChoice_real: (real > $o) > real).
% 52.96/13.03 thf(func_def_844, type, member_o: $o > set_o > $o).
% 52.96/13.03 thf(func_def_845, type, member_Code_integer: code_integer > set_Code_integer > $o).
% 52.96/13.03 thf(func_def_846, type, member_complex: complex > set_complex > $o).
% 52.96/13.03 thf(func_def_847, type, member_int: int > set_int > $o).
% 52.96/13.03 thf(func_def_848, type, member_list_o: list_o > set_list_o > $o).
% 52.96/13.03 thf(func_def_849, type, member_list_int: list_int > set_list_int > $o).
% 52.96/13.03 thf(func_def_850, type, member_list_nat: list_nat > set_list_nat > $o).
% 52.96/13.03 thf(func_def_851, type, member_list_real: list_real > set_list_real > $o).
% 52.96/13.03 thf(func_def_852, type, member_nat: nat > set_nat > $o).
% 52.96/13.03 thf(func_def_853, type, member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o).
% 52.96/13.03 thf(func_def_854, type, member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o).
% 52.96/13.03 thf(func_def_855, type, member_rat: rat > set_rat > $o).
% 52.96/13.03 thf(func_def_856, type, member_real: real > set_real > $o).
% 52.96/13.03 thf(func_def_857, type, member_set_nat: set_nat > set_set_nat > $o).
% 52.96/13.03 thf(func_def_858, type, member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o).
% 52.96/13.03 thf(func_def_859, type, va: nat).
% 52.96/13.03 thf(func_def_878, type, sP0: nat > vEBT_VEBT > $o).
% 52.96/13.03 thf(func_def_879, type, sP1: real > real > real > $o).
% 52.96/13.03 thf(func_def_880, type, sP2: real > real > real > $o).
% 52.96/13.03 thf(func_def_881, type, sP3: real > real > real > $o).
% 52.96/13.03 thf(func_def_882, type, sP4: nat > nat > $o).
% 52.96/13.03 thf(func_def_883, type, sP5: nat > nat > $o).
% 52.96/13.03 thf(func_def_884, type, sP6: nat > nat > $o).
% 52.96/13.03 thf(func_def_885, type, sP7: nat > vEBT_VEBT > $o).
% 52.96/13.03 thf(func_def_886, type, sP8: nat > vEBT_VEBT > $o).
% 52.96/13.03 thf(func_def_887, type, sP9: vEBT_VEBT > real > $o).
% 52.96/13.03 thf(func_def_888, type, sK10: (nat > nat > $o) > nat).
% 52.96/13.03 thf(func_def_889, type, sK11: (nat > nat > $o) > nat).
% 52.96/13.03 thf(func_def_890, type, sK12: (nat > nat > $o) > nat).
% 52.96/13.03 thf(func_def_891, type, sK13: (nat > nat > $o) > nat).
% 52.96/13.03 thf(func_def_892, type, sK14: real > real > nat).
% 52.96/13.03 thf(func_def_893, type, sK15: real > nat).
% 52.96/13.03 thf(func_def_894, type, sK16: (nat > real) > nat).
% 52.96/13.03 thf(func_def_895, type, sK17: real > real > nat).
% 52.96/13.03 thf(func_def_896, type, sK18: num > num).
% 52.96/13.03 thf(func_def_897, type, sK19: num > num).
% 52.96/13.03 thf(func_def_898, type, sK20: nat > nat).
% 52.96/13.03 thf(func_def_899, type, sK21: nat > nat).
% 52.96/13.03 thf(func_def_900, type, sK22: (nat > $o) > nat).
% 52.96/13.03 thf(func_def_901, type, sK23: nat > nat).
% 52.96/13.03 thf(func_def_902, type, sK24: nat > nat > nat).
% 52.96/13.03 thf(func_def_903, type, sK25: nat > vEBT_VEBT > nat).
% 52.96/13.03 thf(func_def_904, type, sK26: (nat > real) > nat).
% 52.96/13.03 thf(func_def_905, type, sK27: nat > nat).
% 52.96/13.03 thf(func_def_906, type, sK28: real > real > nat).
% 52.96/13.03 thf(func_def_907, type, sK29: nat > nat > nat).
% 52.96/13.03 thf(func_def_908, type, sK30: nat > nat > (nat > $o) > nat).
% 52.96/13.03 thf(func_def_909, type, sK31: nat > nat).
% 52.96/13.03 thf(func_def_910, type, sK32: nat > nat > (nat > $o) > nat).
% 52.96/13.03 thf(func_def_911, type, sK33: (nat > real) > nat).
% 52.96/13.03 thf(func_def_912, type, sK34: (nat > nat) > nat).
% 52.96/13.03 thf(func_def_913, type, sK35: real > vEBT_VEBT > vEBT_VEBT).
% 52.96/13.03 thf(func_def_914, type, sK36: real > vEBT_VEBT > nat).
% 52.96/13.03 thf(func_def_915, type, sK37: real > vEBT_VEBT > option4927543243414619207at_nat).
% 52.96/13.03 thf(func_def_916, type, sK38: real > vEBT_VEBT > list_VEBT_VEBT).
% 52.96/13.03 thf(func_def_917, type, sK39: vEBT_VEBT > $o).
% 52.96/13.03 thf(func_def_918, type, sK40: vEBT_VEBT > $o).
% 52.96/13.03 thf(func_def_919, type, sK41: (nat > nat > $o) > nat).
% 52.96/13.03 thf(func_def_920, type, sK42: (nat > nat > $o) > nat).
% 52.96/13.03 thf(func_def_921, type, sK43: (nat > nat > $o) > nat).
% 52.96/13.03 thf(func_def_922, type, sK44: (nat > nat > $o) > nat).
% 52.96/13.03 thf(func_def_923, type, sK45: (nat > $o) > nat > nat).
% 52.96/13.03 thf(func_def_924, type, sK46: nat > nat > nat).
% 52.96/13.03 thf(func_def_925, type, sK47: (nat > $o) > nat > nat).
% 52.96/13.03 thf(func_def_926, type, sK48: (nat > $o) > nat > nat).
% 52.96/13.03 thf(func_def_927, type, sK49: nat > nat).
% 52.96/13.03 thf(func_def_928, type, sK50: (nat > nat) > nat).
% 52.96/13.03 thf(func_def_929, type, sK51: (nat > $o) > nat).
% 52.96/13.03 thf(func_def_930, type, sK52: (nat > nat) > nat).
% 52.96/13.03 thf(func_def_931, type, sK53: (nat > $o) > nat > nat).
% 52.96/13.03 thf(func_def_932, type, sK54: (nat > nat) > nat).
% 52.96/13.03 thf(func_def_933, type, sK55: (nat > $o) > nat).
% 52.96/13.03 thf(func_def_934, type, sK56: (nat > nat > $o) > nat).
% 52.96/13.03 thf(func_def_935, type, sK57: (nat > nat > $o) > nat).
% 52.96/13.03 thf(func_def_936, type, sK58: (nat > nat > $o) > nat).
% 52.96/13.03 thf(func_def_937, type, sK59: (nat > nat > $o) > nat).
% 52.96/13.03 thf(func_def_938, type, sK60: (nat > nat > $o) > nat).
% 52.96/13.03 thf(func_def_939, type, sK61: nat > nat > nat).
% 52.96/13.03 thf(func_def_940, type, sK62: nat > nat > nat).
% 52.96/13.03 thf(func_def_941, type, sK63: nat > vEBT_VEBT > nat).
% 52.96/13.03 thf(func_def_942, type, sK64: real > nat).
% 52.96/13.03 thf(func_def_943, type, sK65: real > nat).
% 52.96/13.03 thf(func_def_944, type, sK66: (nat > $o) > nat).
% 52.96/13.03 thf(func_def_945, type, sK67: vEBT_VEBT > real > vEBT_VEBT).
% 52.96/13.03 thf(func_def_946, type, sK68: vEBT_VEBT > real > nat).
% 52.96/13.03 thf(func_def_947, type, sK69: vEBT_VEBT > real > option4927543243414619207at_nat).
% 52.96/13.03 thf(func_def_948, type, sK70: vEBT_VEBT > real > list_VEBT_VEBT).
% 52.96/13.03 thf(func_def_949, type, sK71: real > vEBT_VEBT > $o).
% 52.96/13.03 thf(func_def_950, type, sK72: real > vEBT_VEBT > $o).
% 52.96/13.03 thf(func_def_951, type, sK73: (real > real > $o) > real).
% 52.96/13.03 thf(func_def_952, type, sK74: (real > real > $o) > real).
% 52.96/13.03 thf(func_def_953, type, sK75: (real > real > $o) > real).
% 52.96/13.03 thf(func_def_954, type, sK76: (real > real > $o) > real).
% 52.96/13.03 thf(func_def_955, type, sK77: (nat > real) > nat).
% 52.96/13.03 thf(func_def_957, type, ph79: !>[X0: $tType]:(X0)).
% 52.96/13.03 thf(f26624,plain,(
% 52.96/13.03 $false),
% 52.96/13.03 inference(subsumption_resolution,[status(thm)],[f26623,f24630])).
% 52.96/13.03 thf(f24630,plain,(
% 52.96/13.03 ($true != (ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ (suc @ (divide_divide_nat @ va @ (suc @ (numeral_numeral_nat @ one))))))) @ (times_times_real @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one)))) @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ (suc @ (divide_divide_nat @ va @ (suc @ (numeral_numeral_nat @ one))))))))))),
% 52.96/13.03 inference(forward_demodulation,[status(thm)],[f23258,f24379])).
% 52.96/13.03 thf(f24379,plain,(
% 52.96/13.03 ((numeral_numeral_nat @ (bit0 @ one)) = (suc @ (numeral_numeral_nat @ one)))),
% 52.96/13.03 inference(forward_demodulation,[status(thm)],[f23785,f23657])).
% 52.96/13.03 thf(f23657,plain,(
% 52.96/13.03 ((numeral_numeral_nat @ one) = (suc @ (divide_divide_nat @ one_one_nat @ (numeral_numeral_nat @ (bit0 @ one)))))),
% 52.96/13.03 inference(definition_unfolding,[status(thm)],[f23179,f23338])).
% 52.96/13.03 thf(f23338,plain,(
% 52.96/13.03 (zero_zero_nat = (divide_divide_nat @ one_one_nat @ (numeral_numeral_nat @ (bit0 @ one))))),
% 52.96/13.03 inference(cnf_transformation,[status(thm)],[f1305])).
% 52.96/13.03 thf(f1305,axiom,(
% 52.96/13.03 (zero_zero_nat = (divide_divide_nat @ one_one_nat @ (numeral_numeral_nat @ (bit0 @ one))))),
% 52.96/13.03 file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_1304_bits__1__div__2)).
% 52.96/13.03 thf(f23179,plain,(
% 52.96/13.03 ((numeral_numeral_nat @ one) = (suc @ zero_zero_nat))),
% 52.96/13.03 inference(cnf_transformation,[status(thm)],[f866])).
% 52.96/13.03 thf(f866,axiom,(
% 52.96/13.03 ((numeral_numeral_nat @ one) = (suc @ zero_zero_nat))),
% 52.96/13.03 file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_865_numeral__1__eq__Suc__0)).
% 52.96/13.03 thf(f23785,plain,(
% 52.96/13.03 ((numeral_numeral_nat @ (bit0 @ one)) = (suc @ (suc @ (divide_divide_nat @ one_one_nat @ (numeral_numeral_nat @ (bit0 @ one))))))),
% 52.96/13.03 inference(definition_unfolding,[status(thm)],[f23400,f23338])).
% 52.96/13.03 thf(f23400,plain,(
% 52.96/13.03 ((numeral_numeral_nat @ (bit0 @ one)) = (suc @ (suc @ zero_zero_nat)))),
% 52.96/13.03 inference(cnf_transformation,[status(thm)],[f872])).
% 52.96/13.03 thf(f872,axiom,(
% 52.96/13.03 ((numeral_numeral_nat @ (bit0 @ one)) = (suc @ (suc @ zero_zero_nat)))),
% 52.96/13.03 file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_871_numeral__2__eq__2)).
% 52.96/13.03 thf(f23258,plain,(
% 52.96/13.03 ((ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ (suc @ (divide_divide_nat @ va @ (numeral_numeral_nat @ (bit0 @ one))))))) @ (times_times_real @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one)))) @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ (suc @ (divide_divide_nat @ va @ (numeral_numeral_nat @ (bit0 @ one))))))))) != $true)),
% 52.96/13.03 inference(cnf_transformation,[status(thm)],[f22547])).
% 52.96/13.03 thf(f22547,plain,(
% 52.96/13.03 ((ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ (suc @ (divide_divide_nat @ va @ (numeral_numeral_nat @ (bit0 @ one))))))) @ (times_times_real @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one)))) @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ (suc @ (divide_divide_nat @ va @ (numeral_numeral_nat @ (bit0 @ one))))))))) != $true)),
% 52.96/13.03 inference(flattening,[status(thm)],[f11353])).
% 52.96/13.03 thf(f11353,plain,(
% 52.96/13.03 ~((ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ (suc @ (divide_divide_nat @ va @ (numeral_numeral_nat @ (bit0 @ one))))))) @ (times_times_real @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one)))) @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ (suc @ (divide_divide_nat @ va @ (numeral_numeral_nat @ (bit0 @ one))))))))) = $true)),
% 52.96/13.03 inference(fool_elimination,[status(thm)],[f11352])).
% 52.96/13.03 thf(f11352,plain,(
% 52.96/13.03 ~(ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ (suc @ (divide_divide_nat @ va @ (numeral_numeral_nat @ (bit0 @ one))))))) @ (times_times_real @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one)))) @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ (suc @ (divide_divide_nat @ va @ (numeral_numeral_nat @ (bit0 @ one)))))))))),
% 52.96/13.03 inference(rectify,[status(thm)],[f10258])).
% 52.96/13.03 thf(f10258,negated_conjecture,(
% 52.96/13.03 ~(ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ (suc @ (divide_divide_nat @ va @ (numeral_numeral_nat @ (bit0 @ one))))))) @ (times_times_real @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one)))) @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ (suc @ (divide_divide_nat @ va @ (numeral_numeral_nat @ (bit0 @ one)))))))))),
% 52.96/13.03 inference(negated_conjecture,[status(cth)],[f10257])).
% 52.96/13.03 thf(f10257,conjecture,(
% 52.96/13.03 (ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ (suc @ (divide_divide_nat @ va @ (numeral_numeral_nat @ (bit0 @ one))))))) @ (times_times_real @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one)))) @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ (suc @ (divide_divide_nat @ va @ (numeral_numeral_nat @ (bit0 @ one)))))))))),
% 52.96/13.03 file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',conj_0)).
% 52.96/13.03 thf(f26623,plain,(
% 52.96/13.03 ($true = (ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ (suc @ (divide_divide_nat @ va @ (suc @ (numeral_numeral_nat @ one))))))) @ (times_times_real @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one)))) @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ (suc @ (divide_divide_nat @ va @ (suc @ (numeral_numeral_nat @ one))))))))))),
% 52.96/13.03 inference(backward_demodulation,[status(thm)],[f25106,f26621])).
% 52.96/13.03 thf(f26621,plain,(
% 52.96/13.03 ( ! [X0 : real,X1 : real] : (((times_times_real @ X0 @ X1) = (times_times_real @ X1 @ X0))) )),
% 52.96/13.03 inference(forward_demodulation,[status(thm)],[f26597,f24728])).
% 52.96/13.03 thf(f24728,plain,(
% 52.96/13.03 ( ! [X0 : real] : (((times_times_real @ X0 @ (semiri5074537144036343181t_real @ one_one_nat)) = X0)) )),
% 52.96/13.03 inference(backward_demodulation,[status(thm)],[f23381,f23753])).
% 52.96/13.03 thf(f23753,plain,(
% 52.96/13.03 ((numeral_numeral_real @ one) = (semiri5074537144036343181t_real @ one_one_nat))),
% 52.96/13.03 inference(definition_unfolding,[status(thm)],[f23350,f23173])).
% 52.96/13.03 thf(f23173,plain,(
% 52.96/13.03 ((numeral_numeral_real @ one) = one_one_real)),
% 52.96/13.03 inference(cnf_transformation,[status(thm)],[f2101])).
% 52.96/13.03 thf(f2101,axiom,(
% 52.96/13.03 ((numeral_numeral_real @ one) = one_one_real)),
% 52.96/13.03 file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_2100_numeral__One)).
% 52.96/13.03 thf(f23350,plain,(
% 52.96/13.03 (one_one_real = (semiri5074537144036343181t_real @ one_one_nat))),
% 52.96/13.03 inference(cnf_transformation,[status(thm)],[f1235])).
% 52.96/13.03 thf(f1235,axiom,(
% 52.96/13.03 (one_one_real = (semiri5074537144036343181t_real @ one_one_nat))),
% 52.96/13.03 file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_1234_of__nat__1)).
% 52.96/13.03 thf(f23381,plain,(
% 52.96/13.03 ( ! [X0 : real] : (((times_times_real @ X0 @ (numeral_numeral_real @ one)) = X0)) )),
% 52.96/13.03 inference(cnf_transformation,[status(thm)],[f22630])).
% 52.96/13.03 thf(f22630,plain,(
% 52.96/13.03 ! [X0 : real] : ((times_times_real @ X0 @ (numeral_numeral_real @ one)) = X0)),
% 52.96/13.03 inference(rectify,[status(thm)],[f151])).
% 52.96/13.03 thf(f151,axiom,(
% 52.96/13.03 ! [X5 : real] : ((times_times_real @ X5 @ (numeral_numeral_real @ one)) = X5)),
% 52.96/13.03 file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_150_mult__numeral__1__right)).
% 52.96/13.03 thf(f26597,plain,(
% 52.96/13.03 ( ! [X0 : real,X1 : real] : (((times_times_real @ X1 @ X0) = (times_times_real @ X0 @ (times_times_real @ X1 @ (semiri5074537144036343181t_real @ one_one_nat))))) )),
% 52.96/13.03 inference(superposition,[status(thm)],[f23425,f24728])).
% 52.96/13.03 thf(f23425,plain,(
% 52.96/13.03 ( ! [X2 : real,X0 : real,X1 : real] : (((times_times_real @ X1 @ (times_times_real @ X0 @ X2)) = (times_times_real @ X0 @ (times_times_real @ X1 @ X2)))) )),
% 52.96/13.03 inference(cnf_transformation,[status(thm)],[f22555])).
% 52.96/13.03 thf(f22555,plain,(
% 52.96/13.03 ! [X0 : real,X1 : real,X2 : real] : ((times_times_real @ X1 @ (times_times_real @ X0 @ X2)) = (times_times_real @ X0 @ (times_times_real @ X1 @ X2)))),
% 52.96/13.03 inference(rectify,[status(thm)],[f1042])).
% 52.96/13.03 thf(f1042,axiom,(
% 52.96/13.03 ! [X5 : real,X3 : real,X36 : real] : ((times_times_real @ X3 @ (times_times_real @ X5 @ X36)) = (times_times_real @ X5 @ (times_times_real @ X3 @ X36)))),
% 52.96/13.03 file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_1041_mult_Oleft__commute)).
% 52.96/13.03 thf(f25106,plain,(
% 52.96/13.03 ((ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ (suc @ (divide_divide_nat @ va @ (suc @ (numeral_numeral_nat @ one))))))) @ (times_times_real @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ (suc @ (divide_divide_nat @ va @ (suc @ (numeral_numeral_nat @ one))))))) @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one)))))) = $true)),
% 52.96/13.03 inference(forward_demodulation,[status(thm)],[f25105,f24453])).
% 52.96/13.03 thf(f24453,plain,(
% 52.96/13.03 ( ! [X0 : nat] : (((divide_divide_nat @ (suc @ (suc @ X0)) @ (suc @ (numeral_numeral_nat @ one))) = (suc @ (divide_divide_nat @ X0 @ (suc @ (numeral_numeral_nat @ one)))))) )),
% 52.96/13.03 inference(forward_demodulation,[status(thm)],[f23593,f24379])).
% 52.96/13.03 thf(f23593,plain,(
% 52.96/13.03 ( ! [X0 : nat] : (((divide_divide_nat @ (suc @ (suc @ X0)) @ (numeral_numeral_nat @ (bit0 @ one))) = (suc @ (divide_divide_nat @ X0 @ (numeral_numeral_nat @ (bit0 @ one)))))) )),
% 52.96/13.03 inference(cnf_transformation,[status(thm)],[f22662])).
% 52.96/13.03 thf(f22662,plain,(
% 52.96/13.03 ! [X0 : nat] : ((divide_divide_nat @ (suc @ (suc @ X0)) @ (numeral_numeral_nat @ (bit0 @ one))) = (suc @ (divide_divide_nat @ X0 @ (numeral_numeral_nat @ (bit0 @ one)))))),
% 52.96/13.03 inference(rectify,[status(thm)],[f6])).
% 52.96/13.03 thf(f6,axiom,(
% 52.96/13.03 ! [X2 : nat] : ((divide_divide_nat @ (suc @ (suc @ X2)) @ (numeral_numeral_nat @ (bit0 @ one))) = (suc @ (divide_divide_nat @ X2 @ (numeral_numeral_nat @ (bit0 @ one)))))),
% 52.96/13.03 file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_5_div2__Suc__Suc)).
% 52.96/13.03 thf(f25105,plain,(
% 52.96/13.03 ($true = (ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ (divide_divide_nat @ (suc @ (suc @ va)) @ (suc @ (numeral_numeral_nat @ one)))))) @ (times_times_real @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ (divide_divide_nat @ (suc @ (suc @ va)) @ (suc @ (numeral_numeral_nat @ one)))))) @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one)))))))),
% 52.96/13.03 inference(forward_demodulation,[status(thm)],[f25104,f24379])).
% 52.96/13.03 thf(f25104,plain,(
% 52.96/13.03 ((ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ (divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one)))))) @ (times_times_real @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ (divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one)))))) @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one)))))) = $true)),
% 52.96/13.03 inference(subsumption_resolution,[status(thm)],[f25103,f24488])).
% 52.96/13.03 thf(f24488,plain,(
% 52.96/13.03 ((dvd_dvd_nat @ (suc @ (numeral_numeral_nat @ one)) @ (suc @ (suc @ va))) != $true)),
% 52.96/13.03 inference(forward_demodulation,[status(thm)],[f23614,f24379])).
% 52.96/13.03 thf(f23614,plain,(
% 52.96/13.03 ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) != $true)),
% 52.96/13.03 inference(cnf_transformation,[status(thm)],[f22696])).
% 52.96/13.03 thf(f22696,plain,(
% 52.96/13.03 ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) != $true)),
% 52.96/13.03 inference(flattening,[status(thm)],[f16640])).
% 52.96/13.03 thf(f16640,plain,(
% 52.96/13.03 ~((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) = $true)),
% 52.96/13.03 inference(fool_elimination,[status(thm)],[f16639])).
% 52.96/13.03 thf(f16639,plain,(
% 52.96/13.03 ~(dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va)))),
% 52.96/13.03 inference(rectify,[status(thm)],[f1])).
% 52.96/13.03 thf(f1,axiom,(
% 52.96/13.03 ~(dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va)))),
% 52.96/13.03 file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_0_False)).
% 52.96/13.03 thf(f25103,plain,(
% 52.96/13.03 ((ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ (divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one)))))) @ (times_times_real @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ (divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one)))))) @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one)))))) = $true) | ((dvd_dvd_nat @ (suc @ (numeral_numeral_nat @ one)) @ (suc @ (suc @ va))) = $true)),
% 52.96/13.03 inference(forward_demodulation,[status(thm)],[f23941,f24379])).
% 52.96/13.03 thf(f23941,plain,(
% 52.96/13.03 ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) = $true) | ((ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ (divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one)))))) @ (times_times_real @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ (divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one)))))) @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one)))))) = $true)),
% 52.96/13.03 inference(equality_resolution,[status(thm)],[f23349])).
% 52.96/13.03 thf(f23349,plain,(
% 52.96/13.03 ( ! [X0 : nat] : (((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) != X0) | ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) = $true) | ((ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ X0))) @ (times_times_real @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ X0))) @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one)))))) = $true)) )),
% 52.96/13.03 inference(cnf_transformation,[status(thm)],[f22949])).
% 52.96/13.03 thf(f22949,plain,(
% 52.96/13.03 ! [X0 : nat] : (((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) != X0) | ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) = $true) | ((ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ X0))) @ (times_times_real @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ X0))) @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one)))))) = $true))),
% 52.96/13.03 inference(flattening,[status(thm)],[f22948])).
% 52.96/13.03 thf(f22948,plain,(
% 52.96/13.03 ! [X0 : nat] : ((((ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ X0))) @ (times_times_real @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ X0))) @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one)))))) = $true) | ((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) != X0)) | ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) = $true))),
% 52.96/13.03 inference(ennf_transformation,[status(thm)],[f22688])).
% 52.96/13.03 thf(f22688,plain,(
% 52.96/13.03 ! [X0 : nat] : (((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) != $true) => (((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) = X0) => ((ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ X0))) @ (times_times_real @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ X0))) @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one)))))) = $true)))),
% 52.96/13.03 inference(flattening,[status(thm)],[f13406])).
% 52.96/13.03 thf(f13406,plain,(
% 52.96/13.03 ! [X0 : nat] : (~((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) = $true) => (((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) = X0) => ((ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ X0))) @ (times_times_real @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ X0))) @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one)))))) = $true)))),
% 52.96/13.03 inference(fool_elimination,[status(thm)],[f13405])).
% 52.96/13.03 thf(f13405,plain,(
% 52.96/13.03 ! [X0 : nat] : (~(dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) => (((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) = X0) => (ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ X0))) @ (times_times_real @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ X0))) @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one))))))))),
% 52.96/13.03 inference(rectify,[status(thm)],[f2])).
% 52.96/13.03 thf(f2,axiom,(
% 52.96/13.03 ! [X0 : nat] : (~(dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) => (((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) = X0) => (ord_less_eq_real @ (semiri5074537144036343181t_real @ (vEBT_V8646137997579335489_i_l_d @ (suc @ X0))) @ (times_times_real @ (vEBT_VEBT_cnt @ (vEBT_vebt_buildup @ (suc @ X0))) @ (numeral_numeral_real @ (bit1 @ (bit0 @ (bit1 @ one))))))))),
% 52.96/13.03 file('/export/starexec/sandbox2/tmp/tmp.nY5YnvUMSA/DTF2THF_24167.p',fact_1__C3_OIH_C_I3_J)).
% 52.96/13.03 % SZS output end Proof for DTF2THF_24167
% 52.96/13.03 % (24670)------------------------------
% 52.96/13.03 % (24670)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 52.96/13.03 % (24670)Termination reason: Refutation
% 52.96/13.03
% 52.96/13.03 % (24670)Memory used [KB]: 29679
% 52.96/13.03 % (24670)Time elapsed: 1.147 s
% 52.96/13.03 % (24670)Instructions burned: 3290 (million)
% 52.96/13.03 % (24670)------------------------------
% 52.96/13.03 % (24670)------------------------------
% 52.96/13.03 % (24372)Success in time 5.976 s
%------------------------------------------------------------------------------